Month: February 2025

It is important to remember that these quantum states are just as real as Bloch states, and apart from the short list of differences discussed above, they can be analyzed and understood using many of the same tools. For example, in a metallic system, the Fermi level can be raised by exposing a crystal to a large population of free electrons and using an electrostatic gate to draw electrons into the crystal. This process populates previously empty Bloch states with electrons. These Bloch states have a fixed set of allowed momenta associated with their energies, and experiments that probe the momenta of electrons in a crystal will subsequently detect the presence of electrons in newly populated momentum eigenstates. Similarly, attaching a Chern magnet to a reservoir of electronsand using an electrostatic gate to draw electrons into the magnet will populate additional chiral edge states. Properties that depend on the number of electrons occupying these special quantum states will change accordingly. In all of these systems, conductivity strongly depends on the number of quantum states available at the Fermi level. For metallic systems, the number of Bloch states available at any particular energy depends on details of the band structure. The total conductance between any two points within the crystal depends on the relative positions of the two points and the geometry of the crystal. Thus conductivity is an intrinsic property of a metal, hydroponic nft system but conductance is an extrinsic property of a metal, and both are challenging to compute precisely from first principles.

When the Fermi level is in the gap of a Chern magnet, there exists a number of quantum states at the Fermi level exactly equal to the Chern number. The conductance through a small number of delocalized quantum states is quite generally equal to e 2 h per quantum channel, and so the conductance between two electronic reservoirs in contact with the same edge of a Chern magnet is equal to C e 2 h . These facts together make the conductance and not just the conductivity an intrinsic property of a Chern magnet. This is a remarkable fact; indeed, as long as a researcher has access to a tool for measuring electrical resistance and a Chern magnet, they can directly measure a combination of the fundamental constants e and h through a simple electronic transport experiment- they don’t need to know anything about the geometry or band structure of the Chern magnet. This phenomenon is known as the quantized anomalous Hall effect. Chern magnets often support magnetic hysteresis, just like trivial magnetic insulators. Because the chirality of the edge state is determined by the sign of the Chern number, and the sign of the Chern number is determined by the bulk magnetization, quantized anomalous Hall effects usually exhibit magnetic hysteresis. The quantized anomalous Hall effect is so unique to Chern mangets that that it is sometimes used as to define the entire class of systems; i.e., researchers historically have described these magnets as ‘QAH materials,’ or ‘QAH magnets.’ At finite temperature, electrons occupying Bloch states in metals can dissipate energy by scattering off of phonons, other electrons, or defects into different nearby Bloch states.

This is possible because at every position in real space and momentum space there is a near-continuum of available quantum states available for an electron to scatter into with arbitrarily similar momentum and en-ergy. This is not the case for electrons in chiral edge states of Chern magnets, which do not have available quantum states in the bulk. As a result, electrons that enter chiral edge state wave functions do not dissipate energy. There is a dissipative cost for getting electrons into these wave functionsthis was discussed in the previous paragraph- but this energetic cost is independent of all details of the shape and environment of the chiral edge state, even at finite temperature. This is why the Hall resistance Rxy in a Chern magnet is so precisely quantized; it must take on a value of C 1 e h 2 , and processes that would modify the resistance in other materials are strictly forbidden in Chern magnets. All bands have finite degeneracy- that is, they can only accommodate a certain number of electrons per unit area or volume of crystal. If electrons are forced into a crystal after a particular band is full, they will end up in a different band, generally the band that is next lowest in energy. This degeneracy depends only on the properties of the crystal. Chern bands have electronic degeneracies that change in response to an applied magnetic field; that is to say, when Chern magnets are exposed to an external magnetic field, their electronic bands will change to accommodate more electrons.Simple theoretical models that produce quantized anomalous Hall effects have been known for decades. The challenge, then, lay in realizing real materials with all of the ingredients necessary to produce a Chern magnet. These are, in short: high Berry curvature, a two-dimensional or nearly two-dimensional crystal, and an interaction-driven gap coupled to magnetic order.

It turns out that a variety of material systems with high Berry curvature are known in three dimensions; three dimensional topological insulators satisfy the first criterion, and are relatively straightforward to produce and deposit in thin film form using molecular beam epitaxy, satisfying the second. These systems do not, however, have magnetic order. Researchers attempted to induce magnetic order in these materials with the addition of magnetic dopants. It was hoped that by peppering the lattice with ions with large magnetic moments and strong exchange interactions that magnetic order could be induced in the band structure of the material, as illustrated in Fig. 3.11. This approach ultimately succeeded in producing the first material ever shown to support a quantized anomalous Hall effect. An image of a film of this material and associated electronic transport data are shown in Fig. 3.12. We now have all of the tools we need to begin discussing real examples of Chern magnets on moir´e superlattices. Our discussion will begin with twisted bilayer graphene. We have already discussed the notion that moir´e superlattices can support electronic bands, and that we can expect these bands to accommodate far fewer electrons per unit area than bands in atomic lattices . This was pointed out in 2011 by Rafi Bistritzer and Allan MacDonald, but they also made another interesting observation: the band structure of the moir´e superlattice is highly sensitive to the relative twist angle of the two lattices, and the bandwidth of the resulting moir´e bands can be finely tuned using the relative twist angle as a variational parameter. It turns out twisted bilayer graphene moir´e superlattice bands can be made to have vanishingly small bandwidth by tuning the twist angle to the so-called ‘magic angle.’ The magic angle is around 1.10- 1.15◦ and a schematic of magic angle twisted bilayer graphene is shown in Fig. 4.2A. The computed band structure of twisted bilayer graphene is illustrated in Fig. 4.2B for a few different twist angles, nft channel including the magic angle. The other bands are grayed out at the magic angle to illustrate the low bandwidth of the moir´e superlattice bands. The low bandwidth of the moir´e superlattice bands combined with the low electronic density required to fill them makes them especially appealing targets for electrostatic gating experiments. The system isrelatively easy to prepare; twisted bilayer graphene devices are produced by ripping a monolayer of graphene in half, rotating one crystal relative to the other using a mechanical goniometer, and then overlaying it on the other. The ‘flatness’ of the band also makes this system especially likely to support interaction-driven electronic phases like magnetism or superconductivity .It is also worth mentioning that it is extremely easy to identify situations in which interactions produce gaps in these systems. Because gaps appear when the moir´e superlattice bands are completely filled with electrons or with holes, and we already know that the moir´e superlattice bands are fourfold degenerate, we can expect any interaction-driven insulating phase to appear as an insulating phase at precisely 1/4, 1/2, or 3/4 of the electron density required to reach full filling of the moir´e superlattice band. These are sometimes called ‘filling factors’ of 1, 2, and 3, respectively, referencing the number of electrons per moir´e unit cell. This argument is presented in schematic form in Fig. 4.2 in the context of experimental data.

Interaction-driven gaps were first discovered in 2018, and this discovery was quickly followed by the dramatic discovery of superconductivity in twisted bilayer graphene. Other researchers predicted that breaking inversion symmetry in graphene would open a gap nearcharge neutrality with strong Berry curvature at the band edges. The graphene heterostructures we make in this field are almost always encapsulated in the two dimensional crystal hBN, which has a lattice constant quite close to that of graphene. The presence of this two dimensional crystal technically always does break inversion symmetry for graphene crystals, but this effect is averaged out over many graphene unit cells whenever the lattices of hBN and graphene are not aligned with each other. Therefore the simplest way to break inversion symmetry in graphene systems is to align the graphene lattice with the lattice of one of its encapsulating hBN crystals. Experiments on such a device indeed realized a large valley hall effect, an analogue for the valley degree of freedom of the spin Hall effect discussed in the previous chapter, a tantalizing clue that the researchers had indeed produced high Berry curvature bands in graphene. Twisted bilayer graphene aligned to hBN thus has all of the ingredients necessary for realizing an intrinsic Chern magnet: it has flat bands for realizing a magnetic insulator, it has strong Berry curvature, and it is highly gate tunable so that we can easily reach the Fermi level at which an interaction-driven gap is realized. Magnetism with a strong anomalous Hall effect was first realized in hBN-aligned twisted bilayer graphene in 2019 . Some basic properties of this phase are illustrated in Fig. 4.3. This system was clearly a magnet with strong Berry curvature; it was not gapped and thus did not realize a quantized anomalous Hall effect, but it was unknown whether this was because of disorder or because the system did not have strong enough interactions or small enough bandwidth to realize a gap. The stage was set for the discovery of a quantized anomalous Hall effect in an intrinsic Chern magnet in hBN-aligned twisted bilayer graphene.An optical microscope image of the tBLG device discussed here is shown in Fig. 4.4A. The device is made using the “tear-and-stack” technique, in which one half of a graphene monolayer is torn off, rotated by a precise relative alignment angle , and then placed on top of the other half of the monolayer. The tBLG layer is sandwiched between two hBN flakes with thickness 40 and 70 nm, as shown in Fig. 4.4B. A few-layer-thick graphite flake is used as the bottom gate of the device, which has been shown to produce devices with low charge disorder.The stack rests on a Si/SiO2 wafer, which is also used to gate the contact regions of the device. The stack was assembled at 60◦ C using a dry-transfer technique with a poly film on top of a polydimethylsiloxane stamp. In an exfoliated heterostructure, the orientation of the crystal lattice relative to the edges of the flake can often be determined by investigating the natural cleavage planes of the flake. Graphene and hBN, being hexagonal lattices, have two easy cleavage planes – zigzag and armchair, each with six-fold symmetry, that together produce cleavage planes for every 30◦ relative rotation of the lattice. We tentatively identify crystallographic directions by finding edges of the flakes with relative anglesof 30◦ . From the optical image we find that the cleavage planes of the tBLG layer and the top hBN are aligned. Four-terminal resistance measurements were carried out in a liquid helium cryostat with a 1 K pot and a base temperature of 1.6 K. The measurement was done using AC current excitations of 0.1 – 20 nA at 0.5 – 5.55 Hz using a DL 1211 current preamplifier, SR560 voltage preamplifier, and SR830 and SR860 lock-in amplifiers.

At the time I started my PhD, experiments in condensed matter physics had already begun a rapid expansion in their ability to conduct in situ modifications of the electronic structure of crystals, and in particular the electron density within crystals. The number of electrons per unit volume is an extremely important property of a crystal; it determines which quantum states within the crystal are filled and which are empty, and thus whether the system is metallic, insulating, or even something more exotic. Almost all of the properties of a crystal are impacted by electron density. Historically the number of electrons in a crystalline system has primarily been modified by adding dopants, i.e., additional atoms with more or fewer conduction electrons than the rest of the crystal .One can achieve dramatic changes in charge density using this technique, but that comes with a heavy cost- the crystal is no longer uniform, as every dopant contributes to disorder, and at high doping levels the band structure itself can be modified by the dopant atoms. More important than all of this for the purposes of experimental physics, however, is that under most circumstances the dopant concentration within a crystal can only be modified through laborious chemical treatments of a particular sample. Materials scientists working under these constraints who wish to explore electron density as an independent variable must either find ingenious material-specific techniques for modifying the dopant concentration in situ , or else they must make a separate sample for each data point they would like to present in their experiment. This is an incredibly labor-intensive process, hydroponic indoor growing system and it also comes with another significant downside: comparing the properties of two different samples with different doping densities exposes results to systematic differences in sample geometry and imperfections in protocol repeatability, and it is difficult to deconvolute these from the effects of changes in electron density.

For these reasons electron density has generally been an awkward and labor intensive independent variable to manipulate. I have found that there are a few ideas that occur naturally to newcomers and outsiders to the field that insiders know enough to immediately discount, and I’d like to discuss one of those ideas here. Chemical doping to manipulate electron density is an ingenious and important technique, but suppose we tried something much sillier- suppose we simply forcefully deposit electrons onto a crystal using some mechanical or electrical process. Would this not achieve our goal? In fact this does indeed work, we have machines that can do this- van de Graff generators can deposit charge onto a piece of metal mechanically, and a variety of other machines can mimic this behavior electronically. So why aren’t condensed matter physicists going around gluing interesting crystals to van de Graff generators so that we can controllably charge them up and measure their responses to changes in charge density? There are a few reasons, but the most important one is that there is a fundamental issue with manipulating charge density this way in three dimensions: this process does not produce a uniform distribution of electron density within the crystal we’d like to study. In three dimensional systems subjected to this treatment, as illustrated in Fig. 1.1B, excess charge accumulates on the surfaces of the crystal, and although we can force additional electrons into acrystal this way we do not ultimately get a system with a modified but still uniform electron density for us to study. This is not the case for two dimensional systems.

Those readers with any exposure to introductory physics have likely encountered parallel plate capacitors; these are highly idealized systems composed of a pair of infinitely thin conducting sheets separated by a small insulating space of consistent thickness. When a voltage is applied to one of these sheets with the other connected to a reservoir of mobile electrons, a uniform charge density per unit area appears on both sheets . Of course, in real metallic capacitors the charge density per unit volume is often still not microscopically uniform because the sheets are not actually infinitely thin, so electrons can redistribute themselves in the out-of-plane direction. To achieve true uniformity one of the plates of the capacitor must be atomically thin, so that electrons simply cannot redistribute themselves in the out-of-plane direction in response to the local electric field. An efficient technique for preparing atomically thin pieces of crystalline graphite was discovered in 2004 by Dr. Andre Geim and Dr. Konstantin Novoselov, an achievement for which they shared the Nobel prize in physics in 2010. The technique involves encapsulating a crystal within a piece of scotch tape and repeatedly ripping the tape apart; it works because the out-of-plane bonds in graphite are much weaker than the in-plane bonds. Graphite represents something of an extreme example of this condition, but it is satisfied to varying extents by a large class of other materials, and as a result the technique was rapidly generalized to produce a variety of other two-dimensional crystals. By constructing a capacitor with one gate replaced with one of these two dimensional crystals, as shown in Fig. 1.1D, researchers can easily access electron density as an independent variable in a condensed matter system.

These systems also facilitate an additional degree of control, with no real analogue in three dimensional systems. By placing capacitor plates on both sides of the two dimensional crystal and applying opposite voltages to the opposing gates, researchers can apply out-of-plane electric fields to these systems . A semiclassical model- in which electrons within the system redistribute themselves in the out-of-plane direction to screen this electric field- does not apply; instead, the wave functions hosted by the two dimensional crystal are themselves deformed in response to the applied electric field . This changes the electronic band structure of the crystal directly, without affecting the electron density. So to summarize, when a two dimensional crystal is encapsulated with gates to produce a three-layer capacitor, researchers can tune both the electron density and the band structure of the crystal at their pleasure. In the first case, this represents a degree of control that would require the creation of many separate samples to replicate in a three dimensional system. The second effect cannot be replicated in three dimensional systems with any known technique.There is a temptation to focus on the exotic phenomena that these techniques for manipulating the electronic structure of two dimensional crystals have allowed us to discover, and there will be plenty of time for that. I’d first like to take a moment to impress upon the reader the remarkable degree of control and extent of theoretical understanding these technologies have allowed us to achieve over those condensed matter systems that are known not to host any new physics. I’ve included several figures from a publication produced by Andrea’s lab with which I was completely uninvolved. It contains precise calculations of the compressibility of a particular allotrope of trilayer graphene as a function of electron density and out-of-plane electric field based on the band structure of the system . It also contains a measurement of compressibility as a function of electron density and out-of-plane electric field, performed using the techniques discussed above . The details of the physics discussed in that publication aren’t important for my point here; the observation I’d like to focus on is the fact that, for this particular condensed matter system, quantitatively accurate agreement between the predictions of our models and the real behavior of the system has been achieved. I remember sitting in a group meeting early in my time working with Andrea’s lab, long before I understood much about Chern magnets or any of the other ideas that would come to define my graduate research work, and marvelling at that fact. Experimental condensed matter physics necessarily involves the study of systems with an enormous number of degrees of freedom and countless opportunities for disorder and complexity to contaminate results. Too often work in this field feels uncomfortably close to gluing wires to rocks and then arguing about how to interpret the results, grow hydroponic with no real hope of achieving full understanding, or closure, or even agreement about the conclusions we can extract from our experiments.

Within the field of exfoliated heterostructures, it is now clear that we really can hope to pursue true quantitative accuracy in calculations of the properties of condensed matter systems. Rich datasets like these, with a variety of impactful independent variables, produce extremely strong limits on theories. They allow us to be precise in our comparisons of theory to experiment, and as a result they have allowed us to bring models based on band structure theory to new heights of predictive power. But most importantly, under these conditions we can easily identify deviations from our expectations with interesting new phenomena- in particular, situations in which electronic interactions produce even subtle deviations from the predictions of single particle band structure theory.This is more or less how I would explain the explosion of interest in the physics of two dimensional crystalline systems within experimental condensed matter physics over the past decade. If you ask a theorist if two dimensional physical systems have any special properties, they will tell you that they do. They might say that the magnetic phase transitions in a Heisenberg model on a two dimensional lattice differ dramatically from those on a three dimensional one. They might say thathigh Tc superconductivity is apparently a two dimensional phenomenon. They might note that two dimensional electronic systems can support quantum Hall effects and even be Chern magnets , while three dimensional systems cannot. But it is easy to miss the forest for the trees here, and I would argue that interest in these particular physical phenomena is not behind the recent explosion in the popularity of the study of exfoliated two dimensional crystals in condensed matter physics. Instead, much more basic technical considerations are largely responsible- it is simply much easier for us to use charge density and band structure as independent variables in two dimensional crystals than in three dimensional crystals, and that capability has facilitated rapid progress in our understanding of these systems. The techniques described above still have some limitations, and chief among them is a limited range of electronic densities that they can reach. Of course, the gold standard of electron density modulation is the ability to completely fill or deplete an electronic band, which requires about one electron per unit cell in the lattice. Chemical doping can achieve enormous offsets in charge density, sometimes as high as one electron per unit cell. This limitation isn’t fundamental and there are some ideas in the community for ways to improve it, but for now it remains true that electrostatic gates can modify electron densities only slightly relative to the total electron densities of real two dimensional crystals. As it stands, electrostatic gating can only substantially modify the properties of a crystal if the crystal happens to have large variations in the number and nature of available quantum states near charge neutrality. For many crystals this is not the case; thankfully it is for graphene, and for a wide variety of synthetic crystals we will discuss shortly. Electrostatic gating of two dimensional crystals was rapidly becoming a mature technology by the time I started my PhD. So where does nanoSQUID magnetometry fit into all of this? A variety of other techniques exist for microscopic imaging of magnetic fields; the most capable of these other technologies recently developed the sensitivity and spatial resolution necessary to image stray magnetic fields from a fully polarized two dimensional magnet, with a magnetization of about one electron spin per crystalline unit cell, and this was widely viewed within the community as a remarkable achievement. We will shortly be discussing several ferromagnets composed entirely of electrons we have added to a two dimensional crystal using electrostatic gates. Because of the a fore-mentioned limitations of electrostatic gating as a technology, this necessarily means that these will be extremely low density magnets with vanishingly small magnetizations, at least 100 times smaller than those produced by a fully polarized two dimensional magnet like the one in the reference above. It is difficult to summarize performance metrics for magnetometers, especially those used for microscopy.

The basic dynamics are illustrated in cartoon form in figure 8b. To the extent there exists evidence to support the ideas of chaos and basin or boundary collision as represented in figure 8, we display some data in figure 9. The expectation from a chaotic approach to a basin boundary collision is complicated and requires a far more dense data set than we currently have, but what is available certainly supports the idea that, as the system approaches the refuge, it undergoes complicated and relatively unpredictable behavior within a hysteretic zone. A complicating issue is the fact that the scale insect population density is the main determinant of the probability of infection, and that density increases as the refuge is approached. As the refuge is approached, we expect an increase in total scale population and an increase in the fungal infection rate. With the increase in total scale population we expect an increase in susceptible scales as the refuge is approached. Within the refuge, we expect the signal of flickering , which, in this case, would result in a bimodal distribution of bushes, many with no infection and many with high infection with few intermediates. That is precisely the pattern suggested by the data presented in figure 9. At the periphery of the refuge , we generally have very few scales and low levels of disease incidence. Moving further into the refuge area, we see a gradual buildup of disease incidence , culminating in bimodality when we arrive at a distance of 0–1 meters. What is not immediately obvious from this analysis is that there are really three qualitatively distinct outcomes at a local level, which is to say on a particular coffee bush. If the bush is very close to the Azteca nest, large round pot the ants forage vigorously and, therefore, the scale insects are highly protected from the beetle predator. Consequently, they build up very high local populations, and are subjected, eventually, to the white halo fungus disease, effectively eliminating the entire population .

At the other extreme, when the coffee bush is far removed from the ant nest, the scale insect is constantly attacked by the adult beetles and never is able to build up a substantial population. Therefore, it would appear that either very close to an Azteca nest or very far away from an Azteca nest thegreen coffee scale is kept under control. However, in the real world, there is no such thing as only far away or very near; many coffee bushes are neither. The result is a complex system in which the Azteca ant forms a reaction or diffusion Turing-like pattern-forming complex that acts as a pilot structure, driving the spatially dependent direct control system. The ant exerts a behavioral restriction on the beetle , but the phorid exerts a behavioral restriction on the ant , causing what has been referred to as a trait-mediated cascade of effects and imposing a hypergraph-like structure on the system . The concentration of ants creates refuges within which the adult beetle predator is restricted from active predation but within which the pest, the scale insect, builds up very dense local populations, the consequence of which is a high attack rate of the white halo fungus disease. Regulation of this herbivore is therefore effected through a complex system involving a Turing process, nonlinear indirect interactions, critical transitions, hysteresis, chaos, basin or boundary collisions, and a hypergraph, all elements of the burgeoning field of complex systems. The elements of the system are illustrated in figure 10. Note the central role of the Azteca ants. It is worth emphasizing that these ants are obligate tree nesters, which suggests that the trend to eliminate the shade trees in the system, thought to be a modernizing effort, completely breaks down this complex system. An interesting complication emerges as we understand the importance of these two obvious natural enemies of the scale insect . We began by proposing a predator–prey driven Turing mechanism to generate the clustered distribution of ant nests, a clear application of what has previously been noted; predator–prey systems distributed in space can generate Turing-like patterns.

The predator was the phorid parasitoid, and the prey was the Azteca ant. However, the spirit of the Turing mechanism involves only generalized reaction and diffusion terms, wherein the reaction is thought to be a coupled positive or negative effect. And in the present example, there are two clear negative effects on the ant through the attack on its food. Both the predatory beetle A. orbigera and the white halo fungus are enemies of the scale insect, the main food of the Azteca, and therefore both constitute a negative effect. It has been independently suggested that either the beetle or the fungus could be the repression agent that generates the Turing-like pattern. If the beetle is the cause of the pattern formation, it is an especially interesting situation in that the beetle population itself is dependent on the existence of the spatial pattern for its own survival but is the cause of the formation of that pattern in the first place .Perhaps the most directly obvious of the three pests is the infamous coffee berry borer, because of its habit of drilling directly into the seed, which is the basic commodity that goes to market. It emerged as a major pest in the 1980s and is regarded as far more important, on most farms, than the green coffee scale. The literature on the coffee berry borer is now enormous because of its sometimes devastating effect . A variety of natural enemies have been reported, including the fungus Beauveria bassiana , anole lizards , birds , possibly bats , and parasitic Hymenoptera . But, by far, the most obvious natural enemies are ants. There is now a substantial literature documenting the general category of ants as major predators on this seed-eating herbivore . As in the case of the predacious beetle on the scale insect, some rather casual observations can easilyconvince one that, in particular, the Azteca ants are major predators to the coffee berry borer, and a variety of detailed studies support that conviction . However, further examination reveals another major ant predator, Pheidole synanthropica, a rather large-body species that nests in the ground but forages vigorously both on the ground and in the coffee bushes. Detailed observations established that this species is a major predator of the berry borer. It takes the berry borer approximately 1–2 hours to completely burrow into the fruit , which means it is unprotected and unable to escape the predacious activity of the ants for that period of time. Both Azteca and P. synanthropica, if they encounter a berry borer trying to burrow into a seed, grab the borer by its posterior end and pull it out of the fruit. Azteca tends to simply throw the borer off the tree , whereas P. synanthropica almost inevitably takes the borer back to its nest.

Therefore, we might say that Azteca is mainly a general antagonist to the berry borer, whereas P. synanthropica is definitely a predator . The consequences of this difference are quite important. Although Azteca provides some protection to the coffee from the ravages of the berry borer, when the borer is simply thrown to the ground it can easily climb back up and try again to bore into the seed. Indeed, there is some reason to suspect that the borer actually prefers to burrow in seeds that are protected by ants, presumably taking advantage of the ant’s mutualistic behavior toward the scale insects and protecting it from other predators that may attempt to enter the seed in which it is eating endosperm and creating the new generation of berry borers. However, this strategy is compromised by other species of predators that can directly prey on the borers when they are thrown to the ground, one of which is P. synanthropica. And this species is highly aggressive, big round plant pot limiting the activity of the smaller species . A number of these smaller species of ants are also known predators of the coffee berryborer . These species offer considerable regulatory potential because they are capable of entering the coffee seed through the hole that the borer makes . One group is the twig-nesting complex, including the genus Pseudomyrmex , and Procryptocerus scabriusculus, all adept at entering hollow arboreal structures because they normally nest in hollow twigs . Other small arboreal ants capable of entering the hole made by the berry borer include the arboreally nesting Solenopsis picea, which nests in superficial structures, such as moss, surrounding the branches of the coffee bushes . On the ground, a variety of ground foraging ants, including Pheidole protensa , and a variety of other species in that same genus are small enough to enter the borer’s hole. Of particular interest is the well-known Wasmannia auropunctata , which nests and forages on both the ground and arboreally . There are many other potential ant predators in the system, but these are the ones we have studied in particular. Azteca clearly dominates over P. synanthropica, and both of them dominate over the smaller species in the system, reducing their nest density significantly . In summary, there are at least six species of ants that are predators on the coffee berry borer, suggesting that ants represent an excellent natural enemy to regulate the coffee berry borer. However, the foregoing natural history suggests that the system is not so simple. Although several of the smaller arboreal species could be effective predators on adults, larvae and pupae of the berry borer within the fruit on the bush, they are effectively unable to engage in such predation if Azteca or P. synanthropica ants are around. Fruits that are not harvested tend to dry out and fall to the ground, providing a refuge for the beetles during the dry season but also being exposed to the potential predation from the smaller ants . However, those smaller ants have dramatically reduced populations if they are forced to compete with P. synanthropica, which, because of its larger size, is unable to penetrate the borer hole in the fruit. In other words, the whole system seems to be operating in a complicated fashion with potential predators interfering with one another but perhaps acting in an emergent fashion to at least partially regulate this key herbivore, the coffee berry borer.Adding to this complication is the phorid fly. As was noted earlier, this fly has an important trait-mediated effect on the Azteca ants, the foundation of the Turing patterns we seem to see at a large scale and effectively contributes to the maintenance of the major predator of the green coffee scale . However, because, as all evidence suggests, the Azteca ants so dominate the coffee bushes where they forage that the smaller ants are unable to persist there, we might expect the same sort of trait-mediated cascade we saw with the control of the green coffee scale. Indeed, in controlled laboratory settings, the coffee berry borer has its success rate of penetrating coffee fruits reduced in the presence of the phorid flies. The importance of this effect is in the addition of what has been referred to as vertical biodiversity to the system , the smaller ants who had been effecting control over the berry borer were restricted from doing so by the action of the Azteca, but when the phorids were introduced, the smaller ants again became effective predators. Although the Azteca ants reduced the effectiveness of berry borer predation from these smaller ants , the phorids reduced the effectiveness of the Azteca in reducing the effectiveness of the smaller ants in their effectiveness in controlling the berry borers, a similar trait-mediated cascade that we saw for the green scale control. All of this is summarized in figure 12. It is tempting to conclude something like “the more ants the better.” However, cascading indirect effects sometimes can have unexpected consequences, meaning that such a conclusion ought to be tempered with more careful analysis. Although we acknowledge this system as complex, it makes some sense to try and simplify it a little to perhaps gain some deeper insight into its operation.

Previous works show berry sunlight exposure to alter the composition of anthocyanins, such as the proportion of acetylated and coumarylated forms . Modulation of acylated, methylated, and hydroxylated forms of anthocyanins result from the synergistic effect of solar radiation exposure and the coupled increases in berry temperature . Generally, high berry temperatures resulting from increased solar exposure results in increased acylated anthocyanins in the grape berry, particularly coumarylated forms . Also, high temperatures result in accumulation of highly methylated anthocyanins such as malvidin derivatives, as these compounds are less likely to degrade than their counterparts . In 2020, D1 and D5 wines demonstrated highest concentrations of acetylates, coumarylates, and methylated anthocyanins compared to C0 wines. While D1 and D5 treatments demonstrated cluster temperatures less than those from C0 treatments , the concomitant thermal degradation of total anthocyanins in C0 treatments proved to negate any modulation towards acylated or methylated forms in resultant wines. Similarly in 2021, C0, D1 and D5 wines exhibited reduced acylation compared to D4 wines. Again, while D4 consistently exhibited less intense cluster temperatures, square plastic plant pots the thermal degradation in more exposed treatments eclipsed any identifiable acylation modulation from hot growing conditions. Acylated anthocyanins are more stable compounds and provide color stability and increase blueness in wine . However, an increase in methylated anthocyanins will lead to redder hues in wine .

Therefore, the improvement in acylated and methylated anthocyanin content due to partial solar radiation exclusion may enhance color perception in young red wines through color stabilization and alteration of wine hue. Likewise, anthocyanin hydroxylation is also directly influenced by temperature and solar radiation exposure. Previous studies on berry exposure utilizing UV selective shade nets as well as leaf removal, demonstrated anthocyanin tri-hydroxylation increases with increasing berry temperature . Increases in trihydroxylation are driven by accumulation of malvidin derivatives and the temperature sensitivity of F3’H, the catalyzing enzyme for 3’- hydroxylated anthocyanin biosynthesis . The highest ratio of tri- to di-hydroxylated anthocyanins in 2020 C0 wines were driven by higher concentrations of 3-pcoumaroyl-glucoside derivatives of delphinidin, petunidin and malvidin, despite the ratio of tri- to di-hydroxylated anthocyanins being unaffected at harvest in the grape berry in 2020 . Among shade film treatments in 2020, the reduction of UV light exposure, was the determining factor in anthocyanin hydroxylation patterns rather than berry temperature. Previous shade net studies at the experimental site showed a reduction in UV radiation with black-40% and blue-40% shade nets led to higher anthocyanin tri-hydroxylation in the grape berry compared to control vines at harvest . With the reduction of UVB and UVC radiation in D4 and D5 vines, anthocyanin trihydroxylation was reduced, regardless of temperature. Ultimately, the upregulation of F3’H from sun exposure could be negated by the reduced catalytic activity of this enzyme under high temperatures experienced in 2020.

In the cooler 2021 vintage, the ratio of tri- to dihydroxylated anthocyanins was unaffected, due to non-significant effect of shade films on acetylated anthocyanins. Ultimately, increased tri-hydroxylation in young red wines will also impact wine hue, resulting in more purple wines . Flavonols in the grape berry skin act as a photoprotectant and are strongly induced by ultraviolet radiation . Flavonol composition in the grape berry can be used to determine overexposure, specifically by quantifying the molar abundance of kaempferol. C0 berries in this study were shown to be overexposed by surpassing the previously described threshold of approximately 7% molar abundance of kaempferol . In both years of the study, flavonol composition in grape berries was maximized in C0 fruit, but D4 and D5 fruit contained the most flavonols across the shade films with minimal thermal degradation of the compounds on the vine . Likewise in both wine vintages, flavonol concentration was modulated by UV radiation exposure, proportional to the amount of UV radiation transmitted to the grapevine. Of the wines produced from shade films treatments, D4 allowed for the most UV transmission while subsequently reducing near infrared transmission by approximately 15%. These light conditions ultimately optimized flavonol content in D4 wines compared to the other shade treatments from both wine vintages. As such, this demonstrated the transmissibility of berry composition under shade treatments to directly improve wine flavonoid profiles. For hot viticulture regions, photoselective solar radiation exclusion provides a strategy to improve not only flavonoid profile but also wine color intensity through copigmentation with anthocyanins.These compounds are derived from microbial mediated cleavage of the C-C double bonds in linoleic and linolenic acids, by lipoxygenase and alcohol dehydrogenate enzymes in yeast . Compounds such as 1-hexanol and -2-hexen-1-ol are associated with aromas such as cut grass, green, fat and herbaceous aromas and their OAV thresholds are 8000 and 400 ug/ L, respectively . The effect of shade films on C6-alcohols was evident in both years; however, there was a yearly effect on which alcohol was altered by the treatment. In 2020, -2- hexen-1-ol was the lowest in D4. In 2021, -2-hexen-1-ol was unaffected by shade films, while 1-hexanol was highest in C0, D4 and D5. Although there was a statistical difference in C6 alcohols, the differences were not large enough between C0 and treatments to crossthe OAV thresholds for these compounds. Increases of C6-alcohols in C0, D4, and D5 wines may be explained by solar radiation overexposure in the treated clusters. L. He et al. reported higher linoleic and linolenic acid biosynthesis with leaf removal at veraison. Subsequently, fruit exposed to increased solar radiation had elevated precursors for C6- alcohol production during yeast metabolism. Additionally, L. He et al. showed higher initial concentration of C6-alcohols in grape berries from leaf removal treatments due to modulation of the volatile compound metabolome and transcriptome in grape berries exposed to sunlight under dry-hot conditions. Therefore, in our experiment which has similar climatic conditions to L. He et al. , fruit from shade films with higher percentages of UV radiation may have both an increase in linoleic and linolenic acids to act as C6-alcohols aromas precursors and increased C6-alcohols in the exposed grape berries. Ultimately, overexposure of the grape berry led to more green and grassy aromas in wine, which may lead to an unripe perception of these wines. Higher alcohols are also produced during fermentation from yeast metabolism of amino acids. These compounds are generally pleasant aromas including mushroom, roses, honey, candy and fruity notes. Of these compounds, shade treatments increased isoamyl alcohol concentration in 2020 and benzyl alcohol concentration in wines from both vintages. Isoamyl alcohol is associated with solvent and cheese aromas and, while benzyl alcohol is characterized as being citrusy and sweet . The odor active thresholds for these compounds are 30000 mg/L and 10000 mg/L, respectively . In 2020, C0 had the lowest concentration of isoamyl alcohol in wines. The effect of shading on the concentration of isoamyl alcohol in wines varies in literature . In hot growing regions, 75% of total solar radiation exclusion with black polyethylene canopy side shade nets resulted in wines with reduced isoamyl alcohol compared to the uncovered control vines .

However, this experimental site was in a region that received approximately 704.5° C less growing degree days than the present experimental site in the hotter 2020 season, and 514.1°C growing degree days less than the cooler 2021 season. In the study by Lu et al., 2021, reduced solar radiation exposure in a cooler growing region may have resulted in reduced isoamyl alcohol in shaded fruit. When cluster temperatures exceed 42°C in exposed vines, there is a reduction in isoamyl alcohol in resultant wines compared to wines produced from fruit under red and black shade nets . With cluster temperatures of C0 fruit exceeding 42°C, excessive cluster temperatures may be prompting the reduction in isoamyl alcohol and overall wine fruitiness from those produced from overexposed clusters. However, square pot plastic while there was a statistical difference in isoamyl alcohol concentrations between C0 and treatment wines, the effect was not large enough to exceed the OAV threshold for this compound . Shade films affected the ester composition predominantly in 2020 wines. Pleasant esters in red wines include ethyl acetate which has a OAV threshold of 12264 µg•L-1 and is described as fruity and balsamic , as well as isoamyl acetate, described as banana aroma with a OAV threshold of 30 µg•L-1 . In 2020, ethyl acetate was reduced in C0 and D5 wines, shading and reduced cluster temperatures preserved isoamyl acetate aromas in D1, D3 and D5 wines. When compared to wines from 2021, cooler vintage conditions did not result in ester compositional changes in exposed and shaded wines. Similarly, fatty acid esters were preserved in shaded wines, while 2020 C0 wines consistently had the lowest concentration of all measured fatty acid ethyl esters and various esters, all of which are associated with fruity and candy-like aromas . Concentrations of ethyl octanoate and ethyl decanoate remained beneath the reported perception threshold, thus observed shifts in composition with shading may be undetectable in Cabernet Sauvignon wines. However, ethyl hexanoate and ethyl isovalerate have remarkably low OAV thresholds of 5 µg•L-1 and 1 µg•L-1, respectively . In the present study, all wines were above these thresholds, indicating that reductions in fruitiness may be perceived. This overall decrease in fruity aromas with cluster exposure and excess temperatures may negatively impact the marketability of Cabernet Sauvignon wines from hot viticulture regions with increasingly more frequent heat wave events associated with climate change. Unpleasant and rancid aromas include isobutyric acid which imparts a cheese aroma and benzaldehyde which is associated with almond aroma in red wines . In this study, isobutyric acid concentrations were only affected in 2020, with D4 having the highest isobutyric acid concentration. The detection threshold for this aroma compound is 2300µg•L-1 . Concentrations detected in the experimental wines were substantially below this threshold, indicating that this slight increase in rancid aromas in D4 wines may not negatively impact overall wine perception. Given that D4 wines also exhibited enhanced fruitiness in with improved ester composition, the trade-off of slight increases in rancid aromas may be offset by the net benefit from increased fruity aromas in the wine aroma profile. While terpenes are often critical in white wines, these compounds when present in red wines have a large effect on wine aromas as their OAV thresholds are relatively low . The OAV threshold for a-terpinene, cis-rose-oxide and linalool are 250 µg•L-1 , 0.2 µg•L-1 and 25.2 µg•L-1 , respectively . The In 2020, aterpinene, cis-rose-oxide and linalool were all reduced in C0 wines compared D4 and D5 wines, however concentrations of these compounds did not exceed the OAV threshold. These compounds produce odors such as peach, citrus, rose, and floral aromas in red wines . Previous work indicated an increase in terpenoids, particularly linalool in wines produced from fruit under black and red shade nets . It was demonstrated that heat treatment will down-regulate genes encoding key enzymes in terpenoid metabolism in Cabernet Sauvignon grapevines . Thus, increases in terpenoid content in shade film wines in 2020 may be due to reduced cluster temperature in a growing season with frequent heat wave events. In 2021, C0 wines exhibited the highest concentration aterpinene, while cis-rose-oxide concentrations remained low in C0, and linalool was unaffected. In 2021, a cooler growing season with fewer days above 38°C may have resulted in less variation in terpenoid composition and net accumulation of terpenoids in exposed fruit . Ultimately, climatic shifts towards more frequent heat wave events will reduce floral and citrus aromas in wines produced from overexposed clusters. However, the year-to-year weather variation will enhance the unpredictability of the development of these compounds, leading to challenges for wine producers looking to produce a consistent product.As carotenoid breakdown products, C-13 norisoprenoids like bdamascenone often described by sweet and floral aromas and has an OAV threshold of 0.05 µg•L-1 . C-13 norisoprenoids have been shown to have a positive linear relationship with sunlight exposure to the grape cluster . Under extreme light intensity and temperature conditions, there are decreases in carotenoid concentration in the berry, thus reducing C-13 norisoprenoid precursors. In the present study, b-damascenone was highest in C0, D4 and D5 wines in 2020, while b-damascenone was highest in C0 and D1 wines in 2021, contrary to previous findings in hot viticultural areas. Lee et al. reported that grape clusters without leaf removal and inner canopy clusters contained more b-damascenone than south-facing clusters exposed to solar radiation by leaf removal. Likewise, black cloth and red shade net enhanced b-damascenone concentration compared to uncovered control .

We note that different grape varieties were used as recipient test vines in the field and laboratory studies, which limits the direct comparison of the two studies. In addition, plants used in the laboratory study were only tested for GLRaV-3 ; it is possible that interactions among virus species could have influenced vector transmission and pathogen establishment. The physiological status of mature field vines compared to greenhouse cuttings that were several weeks old at the time of inoculation may also influence virus transmission. Despite these relevant caveats, the overall finding is suggestive that laboratory studies may overestimate vector transmission efficiency. For example, Hooks et al. also found a higher transmission efficiency of Banana bunchy top virus by aphids based on laboratory experiments compared to commercial agricultural conditions. Another study that included transmission of Cucumber mosaic virus and Zucchini yellow mosaic virus by multiple aphid species found that the relative transmission rates between field and laboratory conditions depended both on aphid and virus species . We propose that this question should be addressed in more detail in future studies due to its epidemiological relevance. While our findings are informative, similar studies should be performed with other mealybug species, genetically distinct variants of GLRaV-3, grape cultivars, and in different climatic conditions to evaluate the general applicability of the results . For example, black plant pots plastic berry quality of white wine varieties is affected by GLRaV-3 infection, yet resulting disease symptoms are not readily identifiable .

Another open question with respect to disease progression is the amount of time that passes after graft-mediated infections before symptoms can be identified or diagnostic tools can detect new GLRaV-3 infections in the recipient vine. Even though spread of grapevine leafroll disease was documented 25 years ago , many parameters that contribute to spread and progression remain poorly understood. In Napa Valley, Ps. maritimus produces two synchronous generations per year that do not overlap, and our inoculation study coincided with the emergence of the second generation in July . Differing climatic conditions during the first generation, which usually emerges in March, could potentially lead to different transmission efficiency and different timing of disease development. Varied climatic conditions could lead to variation in host traits and resulting host susceptibility, variation in the virus life cycle in response to varying environmental conditions within host and vector, and variation in mealybug activity that could lead to higher or lower transmission efficiencies. There is a need for understanding how the interactions between host, vector, and pathogen are affected by climate and seasonality. Other mealybug species in California produce higher numbers of overlapping generations; therefore vines may be vulnerable to new infections throughout the year . For example, Pl. ficus produces four to seven overlapping generations per year, and is not yet present in most areas of Napa Valley, CA, where our field study was conducted . If the invasive Pl. ficus establishes more widespread populations in Napa Valley and other areas of Northern California at the higher densities typically associated with this species, spread of GLRaV-3 could dramatically increase. In summary, the information provided by our study regarding key biological traits of GLRaV-3 can inform sound management practices. For example, when certified virus-free material has been used for planting, newly symptomatic vines can be used to infer that the infection resulted from insect-borne inoculations made during the previous growing season, and that the newly symptomatic vines can be an efficient source for further disease spread.

Decline in crop quality can be expected during the same growing season in which symptoms first appear, which may influence roguing strategies based on economic models .There is natural spatial variability present in vineyards due to the variations in soil characteristics and topography . Soil characteristics are too complex to be thoroughly surveyed effortlessly. With traditional destructive methods, it is difficult to obtain enough comprehensive information from the soil pits at the field scale. These soil characteristics may directly affect the water availability for grapevines, which eventually determine the physiological performance of the plants . However, there is no variable management practices currently available to accommodate the natural spatial variability. Thus, the spatial variability derived from vineyard soils will inevitably be expressed in the whole plant physiology at the cost of homogeneity of vineyard productivity and quality. We previously reported the spatial variation of midday stem water potential affecting grapevine carbon assimilation and stomatal conductance of grapevine . The resultant variations in whole plant physiology were associated to flavonoid composition and concentration at the farm gate. However, there is a lack of information about the effects on the chemical composition in the final wine, which would ultimately determine wine quality as perceived by consumers. Georeferenced proximal sensing tools can capture the spatial and temporal variability in vineyards, making it possible to supervise and manage variations at the field scale . Previous studies showed that soil bulk electrical conductivity may be used to evaluate many soil attributes, including soil moisture content, salinity, and texture . Soil electromagnetic induction sensing has been used in precision agriculture to acquire soil bulk EC at the field scale due to its non-invasive and prompt attributes .

Although research had been conducted on the relationships between soil electrical properties with plant water status, they were mostly point measurements and the results were rarely interpolated to whole fields. There were only a few studies that investigated the EMI sensing and soil-plant water relationships over a vineyard . Previous research suggested that the connection between soil water content and soil bulk EC could have relied on specific soil profiles, and needed to include soil physical and chemical properties to complete this connection . Nevertheless, there is evidence that soil bulk EC may still be useful not only to identify the variability in soil, but also in the plant response affected by vineyard soils such as yield, plant physiology, and grape berry chemistry . Plant available water is a determinant factor on grapevine physiology, together with nitrogen availability in semi-arid regions . Wine grapes are usually grown under a moderate degree of water deficits as yields were optimized at 80% of crop evapotranspiration demand with sustained deficit irrigation . Water deficits would limit leaf stomatal conductance and carbon assimilation rate that sustain grapevines’ vegetative and reproductive growth and development . When grapevines are under water deficits, carbohydrates repartitioned into the smaller berries would enhance berry soluble solids content . Sucrose and fructose, which are the major components of total soluble solids in grape berry, can act as a signaling factor to stimulate anthocyanin accumulation . The effects on grapevine physiology and berry composition also depend on the phenological stages they occur and how severe and prolonged the water deficits are . Flavonoids are the most critical compounds dictating many qualitative traits in both grape berries and wine . The variations in environmental factors could alter the concentration and biosynthesis of flavonoids and can be extrapolated spatially within the same vineyard, including water deficits , solar radiation , and air temperature . Among flavonoid compounds, anthocyanins are responsible for the color of berry skin as well as wine . Moderate water deficits during growing season can increase anthocyanin concentration in berry skin and wine . However, water deficits can impair plant temperature regulation through evaporative cooling . They may also inhibit berry growth by limiting berry size and altering berry skin weight . Thus, in some cases it may be uncertain if water deficit promotes anthocyanins biosynthesis or reduces berry growth, or contributes to anthocyanin degradation . Applying water deficit on grapevines can contribute to greater proportion in tri-hydroxylated over dihydroxylated anthocyanins due to the up-regulation of F30 5 0 H. Another major class in flavonoids, proanthocyanidins, are polymers of flavan-3-ol monomers and they contributes mainly toward astringency or bitterness in wine . Compared to anthocyanins, water deficits showed mild effects on proanthocyanidins . However, water deficits with great severity can still alter the concentration and composition of proanthocyanidins in both berries and wine . Selective harvest is one of the targeted management strategies to minimize the spatial variation in berry chemistry in vineyards . By differentially harvesting or segregating the fruits into batches prior to vinification, large plastic pots for plants the berry composition can be artificially set at a more uniform stage with minimal variations . In our previous work, we reported the use of plant water status to determine the spatial variation of grape berry flavonoids . The goal of this study was to deduce if the spatial variability of soil bulk EC and differences in soil texture can be related to plant physiology and grape and wine composition.

The specific objective of the study was to determine if the spatial variability of proximally sensed vineyard soil bulk EC would affect plantwater status, and if this relation would affect leaf gas exchange, components of yield, berry composition, and flavonoids in both berries and wine.Soil bulk EC was assessed with EM38 in 2016 when the vineyard soil was at field capacity condition. Both vertical dipole mode and horizontal dipole mode were used to assess EC at two depths, including deep soil and shallow soil . The instrument was calibrated according to manufacturer instructions. The device was placed on a PVC sled and driven through the vineyard with an allterrain vehicle along the inter-rows. A distance of approximately 0.5 m from the vehicle to the device was maintained to avoid interference with the vehicle. A stratified grid was used to collect soil samples corresponding to the two depths at which we measured soil bulk EC. Soil texture was assessed according to the soil analysis method: hydrometer analysis in the North American Proficiency Testing program.Geostatistical analysis was performed in the R language by using package “gstat” 1.1-6 . The bulk EC data were filtered by Tukey’s rule to remove outliers either below the first quartile by 1.5 inter-quartile range or above the third quartile by 1.5 inter-quartile range. To further remove the outliers, the data were filtered by the speed that the vehicle was driving, which was between 3.2 km per hour to 8.0 km per hour. Variograms were assessed by “automap” package 1.0-14 , and fitted to perform kriging. The soil bulk EC values were extracted from the location of each experimental unit, these values were further used to perform regression analysis. Kriging and k-means clustering on plant physiology variables were performed with the R packages “gstat” and “NbClust,” v3.0 . Universal kriging was utilized on plant water status because of the existing trend in longitude and latitude. Variograms were assessed by “automap” package 1.0-14 , and fitted to perform universal kriging. The vineyard was delineated into two clusters by k-means clustering, including Zone 1 with higher water deficit and Zone 2 with lower water deficits. The separation described 78.1% in 2017 of the variability in the plant water status according to the result of between sum of squares/total sum of squares. The resulting maps were organized and displayed by using QGIS software . Cluster comparison was analyzed by “raster” package reported as Pearson’s Correlation between two cluster maps . Data were tested for normality by using Shapiro-Wilk’s test, and subjected to mean separation by using two-way ANOVA with the package “stats” in RStudio . Significant statistical differences were determined when p values acquired from ANOVA were <0.05, and the zones were classified according to Tukey’s honestly significant difference test. Regression analysis was performed by SigmaPlot 13.0 . Correlation coefficient between variables were calculated in by Pearson’s correlation analysis, and p-values were acquired to present the significances of the linear fittings.Berry skin anthocyanins were different between the two zones in 2016 . Total delphinidins, petunidins, malvidins, and the sum of them as trihydroxylated anthocyanins were all higher in Zone 2 than Zone 1 . Total cyanidins, peonidins, and the sum of them as di-hydroxylated anthocyanins were greater in Zone 2 on 23 August, 15 September, and at harvest . Total skin anthocyanins were 2.2 mg per g of berry fresh weight in Zone 2 which was higher than the 1.85 mg measured in Zone 1 . In 2017, there were no differences between the two zones in delphinidin, cyanidin, petunidin, or peonidin at harvest . Zone 1 had higher malvidins from 24 August until harvest, and tri-hydroxylated anthocyanins, total anthocyanins from 7 September until harvest . Conversely, total malvidins, tri-hydroxylated anthocyanins, and total anthocyanins were higher in Zone 1 at harvest .

We excluded hospitalizations that included liver transplantation , and used the cohort of those who did not receive liver transplantation as the primary cohort. The primary outcome was ODS at any point during hospitalization . Patient age, sex, race, etiology of liver disease, general medical comorbidities, complications of cirrhosis, and hospital outcomes including discharge disposition, length of stay, and inflation-adjusted total cost were also extracted as described in depth elsewhere. In brief, we identified specific complications of cirrhosis using discharge diagnosis codes ; we identified specific patient comorbidities such as congestive heart failure using the Clinical Classification Software; and identified paracenteses and thoracenteses with procedure codes. Charlson Comorbidity Index was used as a marker of general illness severity/degree of comorbidities, stratified into three groups: mild , moderate ,severe . Baveno IV consensus criteria was used as a marker of cirrhosis illness severity, where those with Stages 3 , and 4 represented decompensated cirrhosis. For descriptive statistics we presented categorical variables as percentages and continuous variables as medians with respective interquartile ranges . To compare characteristics between patients with vs. without ODS, we used Pearson chi-square test for dichotomous variables; non-parametric Kruskal-Wallis to compare categorical variables; and Wilcoxon rank-sum for continuous variables. We used univariable logistic regression to assess unadjusted odds ratios associated with ODS, square plant pot and used stepwise backward selection to determine the final multi-variable logistic model. Statistical analysis were performed using Stata .

In this investigation of the National Inpatient Sample, 2009-2013, we found that the prevalence of ODS in hospitalized patients with cirrhosis was extremely rare, and much lower than the prevalence reported in patients undergoing liver transplantation.2-6,8 Alcohol related cirrhosis, younger age, and female gender were associated with an inpatient diagnosis of ODS. Notably, markers for cirrhotic decompensation and severity of comorbid illness were not found to be associated with ODS. This included no evidence for an association with ascites, which ran counter to our hypothesis that those with portal hypertension might be at higher risk for ODS due to labile serum sodium levels during hospitalization. Our findings that those with vs. without ODS experienced longer hospital stays, higher hospitalizations costs, and increased chance of receiving a discharge disposition to somewhere other than home help to quantify the burden of ODS on the health care system. Furthermore, the increased health care burden and poorer outcomes we found in those with vs. without ODS provide additional evidence that we appropriately identified pronounced cases of ODS with our selection methods. We acknowledge our study’s limitations. As with all large database investigations, our results are susceptible to case ascertainment and measurement biases. While ICD-9 codes have been well-validated for the selected measures of cirrhosis and overall disease,7 ICD-9 codes for ODS have not been systematically validated. In particular, we cannot know whether subtler cases of ODS might have gone unrecognized in the hospital setting only to be diagnosed at outpatient follow-up upon review of MRI imaging. Our low prevalence estimate likely reflects this, underestimating the total prevalence of ODS by failing to detect these subtler cases. That being said, we aimed to capture clinically-apparent cases of ODS, for which ICD-9 coding would be most specific. Finally, our study was limited by a paucity of sodium level data.

Unfortunately, the NIS does not contain laboratory values, so we couldn’t associate serum sodium changes with ODS. Additionally, hyponatremia as detected by ICD9 coding has been demonstrated to be variable and often lacking, representing perhaps only one third of inpatients experiencing hyponatremia.9 Because of this, and because hyponatremia is already known as a major precipitant of ODS, we elected to focus our research questions on other risk factors beyond it.1 In conclusion, our investigation of a large nationwide database demonstrates that ODS is extremely rare, occurring in 0.02% of hospitalized patients with cirrhosis. ODS is associated with alcohol-related cirrhosis, younger age, and female gender. ODS is not associated with specific cirrhosis complications including ascites, nor with overall liver disease severity or general comorbid disease severity. These data may help inform management of hyponatremia in patients with cirrhosis by reassuring providers of the rarity of ODS, while reinforcing the consideration of a broad range of differential diagnoses in cirrhosis patients exhibiting altered mental status after hyponatremia correction.Cobalt monosilicide crystalizes in a chiral structure in the P213 space group inset. It has been intensely studied as a potential thermoelectric material due to its large power factor at room temperature. Recent theoretical works have found that CoSi and its isostructural siblings possess a chiral double sixfold-degenerate spin-1 Weyl fermion at R point and a fourfold-degenerate chiral fermion at Γ point in their Brillouin zone. These “new fermions” with large topological charges are connected by long, robust Fermi arcs on the surface which have been later confirmed by angle-resolved photoemission spectroscopy experiments. However, no transport properties of CoSi directly related to its topological nature have been reported until now. As far as we are aware, previous reported CoSi single crystals manifest relatively low carrier mobilities and no quantum oscillations have been observed in their electrical properties under magnetic field. It is difficult to bridge the transport properties with its topological band structure due to low sample quality.

Here we report our crystal growth and a survey of the electrical transport properties of single-crystalline CoSi. Tellurium was found to be an appropriate metal flux for the growth of CoSi which yields high-quality single crystals with large magneto-resistance and carrier mobilities. Although there is plenty of research work on the thermopower of CoSi, few of them have paid attention to its magneto-Seebeck and Nernst effect. Combining the high quality of our Te-flux grown samples and the sensitivity of thermoelectricity measurement, we are able to observe, for the first time, QOs in the thermoelectric signals of CoSi. By analyzing the QOs in magneto-Seebeck and Nernst signals at different temperatures and magnetic field directions, we reveal two spherical Fermi surfaces around the BZ corner R point, which is consistent with our density functional theory calculations. The extracted Berry phases of electron orbit equal zero, agreeing well with the scenario of −2 chiral charge at R point. We also found that the spin-orbit coupling induced band-splitting is lessthan 2 meV near the Fermi level and this result is one order smaller than our DFT calculations. We also report a large Nernst effect in CoSi due to the combination of high mobility and phonon-drag contribution at intermediate temperatures. As a consequence, a relatively large Nernst-Ettinshausen figure of merit of around 0.03 is achieved at 42 K in 14 T.Grape berries undergo a series of complex physiological and biochemical changes during their development that determine their characteristics at harvest . Genome-wide expression studies using microarray and, more recently, RNA sequencing revealed that berry development involves the expression and modulation of approximately 23,000 genes and that the ripening transition is associated with a major transcriptome shift . Transcriptomic studies characterized the ripening program across grapevine cultivars , identifying key ripening-related genes and determining the impact of stress and viticultural practices on ripening . This knowledge increases the possibility of exerting control over the ripening process, improving fruit composition under suboptimal or adverse conditions, and enhancing desirable traits in a crop with outstanding cultural and commercial significance . These genome-wide expression analyses were possible because a highly contiguous assembly for the species was produced ;this first effort used a grape line created by several rounds of back crossing to reduce heterozygosity, facilitating genome assembly . Though poor by current standards, this pioneering, square pot chromosome-resolved assembly served as the basis for numerous publications. However, the structural diversity of grape genomes makes using a single one-size-fits-all reference genome inappropriate . There is substantial unshared gene content between cultivars, with 8–10% of the genes missing when two cultivars are compared . Although many of these genes are not essential for plant survival, they can account for 80% of the expression within their respective families and expand key gene families possibly associated with cultivar-specific traits . Assembling genome references for all interesting cultivars is impractical, in part because its cost remains prohibitive and because of genomic features that impede the development of high-quality genome assemblies for any grape cultivar. Although the V. vinifera genome is relatively small and as repetitive as other plant genomes of similar size , it is highly heterozygous . Most domesticated grape cultivars are crosses between distantly related parents; this and clonal propagation cause the high heterozygosity observed in the species . Earlier attempts using short reads struggled to resolve complex, highly heterozygous genomes . A limited ability to call consensus polymorphic regions yields highly fragmented assemblies where structural ambiguity occurs and alternative alleles at heterozygous sites are excluded altogether . Single Molecule Real Time DNA sequencing has emerged as the leading technology for reconstructing highly contiguous, diploid assemblies of long, repetitive genomes that include phased information about heterozygous sites . Recently, we used Vitis vinifera cv. Cabernet Sauvignon to test the ability of SMRT reads and the FALCON-Unzip assembly pipeline to resolve both alleles at heterozygous sites in the genome .

The assembly produced was significantly more contiguous than the original PN40024 assembly and provided the first phased sequences of the diploid V. vinifera genome . Despite recent advances in genome reconstruction methodologies, assembling a complex plant genome is still costly. Transcriptome reconstruction is the only alternative strategy to depict known and unknown gene content information . De novo assembly of RNA-seq reads is widely used for this purpose . SMRT technology was recently deployed to investigate expressed gene isoforms in a variety of organisms, including a handful of plant species . Long reads delivered by this methodology report full-length transcripts sequenced from their 59-ends to polyadenylated tails , making Iso-Seq an ideal technology for reconstructing a transcriptome without a reference genome sequence and without assembling fragments to resolve the complete isoform sequence . Moreover, alternative transcripts that contribute to the gene space complexity and vary with cell type , developmental stage , and stress cannot be definitively characterized without full-length transcript information. The objective of this study was to test whether full-length cDNA sequencing with Iso-Seq technology is a suitable alternative to traditional genome sequencing, assembly, and annotation for reconstructing a grape transcriptome reference for transcriptional profiling. We compared how Cabernet Sauvignon’s Iso-Seq transcriptome fares as a reference for RNA-seq analysis vs. its annotated genome. We sequenced the full-length transcripts of ripening berries with Iso-Seq and Illumina RNA-seq reads. The high-coverage short-read data were used to profile gene expression and to error-correct low-expression isoforms that would have been otherwise lost by the standard Iso-Seq pipeline. The transcriptome reference built with Iso-Seq data represented most of the expressed genes in the grape berries and included cultivarspecific or “private” genes. When used as the reference for RNAseq, Iso-Seq generated transcriptome profiles quantitatively similar to those obtained by mapping on a complete genome reference. These results support using Iso-Seq to capture the gene space of a plant and build a comprehensive reference for transcriptional pro- filing without a pre-defined reference genome.Grape berries from Cabernet Sauvignon FPS clone 08 were collected in Summer 2016 from vines grown in the Foundation Plant Services Classic Foundation Vineyard . Between 10 and 15 berries were sampled at pre-véraison, véraison, post-véraison, and at commercial maturity. Table S1 provides weather information for the sampling days. The ripening stages were visually assessed based on color development and confirmed by measurements of soluble solids . On the day of sampling, berries were deseeded, frozen in liquid nitrogen, and ground to powder . Total RNA was isolated using a Cetyltrimethyl Ammonium Bromide -based extraction protocol as described in Blanco-Ulate et al. . RNA purity was evaluated with a Nanodrop 2000 spectrophotometer . RNA was quantified with a Qubit 2.0 Fluorometer using the RNA broad range kit . RNA integrity was assessed using electrophoresis and an Agilent 2100 Bioanalyzer . Only RNA with integrity number greater than 8.0 was used for SMRTbell library preparation. One mg of the pooled RNA was used for cDNA synthesis and for SMRTbell library construction using the SMARTer PCR cDNA synthesis kit . First-strand cDNA synthesis was performed using the SMRTScribe Reverse Transcriptase . Each developmental stage was individually barcoded . To minimize artifacts during largescale amplification, a cycle optimization step was performed by collecting five 5 ml aliquots at 10, 12, 14, 16, and 18 PCR cycles.

Given the later fruit development of grapevine and the grape chemistry requirements for red wine making , the length of the growing season is often a limitation for achieving the desired ripening and vintage quality in cool climates . Thus, yield is often sacrificed to balance source-to-sink ratio in favor of accelerated fruit ripening or to mitigate the effects of early fall frosts . Although the initial control of crop level comes during pruning , the number of dormant buds retained at pruning time is maintained constant through the years in warm climate regions. Cluster thinning is a management practice fine-tuned each year to achieve vine balance . This effect could be exacerbated with high nitrogen amounts inhibiting anthocyanin biosynthesis , the absence of water stress, or changes of cluster microclimate due to mutual shading , and thus, not by the under cropping itself. Therefore, grapevine canopy development is managed through the control of inputs, vine spacing, irrigation, rootstocks, pruning, leaf removal, hedging, or cover crops, among others. A great part of the carbon assimilated through the growing season is incorporated into cellulose or lignin in roots, trunks, and shoots . However, blueberry grow bag resumption of a new season’s growth depends on the carbon stored as nonstructural carbohydrates, majorly in the form of starch, but also soluble carbohydrates such as sucrose, glucose, and fructose .

Roots are the greatest sink of non-structural carbohydrates and root-derived carbohydrates constitute the principal reserve source for annual resumption of growth in the spring. The grapevine’s capacity for replenishment of these carbohydrate reserves increases at mid-ripening, when canopies are at their maximum and fruit demand slows down sugar accumulation in perennial parts . Therefore, the loss of photosynthetically active leaf area or excessive number of clusters may impair the reconstitution of reserves . In addition, high crop levels may delay fruit maturation and shorten the post-harvest period and subsequently reduce the time needed to accumulate reserve carbohydrates. Grape growing systems based on high yields are typically in warm to hot regions, relying on early harvest to replenish these reserves. However, it is common that excessive yields lead to a reduction in yields the following season . Loss of photosynthetically active leaf area or excessive number of clusters may deplete these reserves. High crop levels may reduce the reserve carbohydrate accumulation and delayed fruit maturation and may shorten the postharvest period. Therefore, the grapevine may not have sufficient time to accumulate carbohydrates for the following season in cool climates. Conversely, there is not consensus in literature regarding the effect of high cropping levels on storage reserves . This was explained by sink limitation as the grapevine was able to maintain equilibrium by adjusting physiological processes . In addition to the modulation of berry ripening and storage reserves, other compensatory mechanisms have been described in response to over and under cropping. Components of yield, which include clusters per vine, berries per cluster, and berry mass, are susceptible to change together with berry ripening in compensation of each other . Although grapevine pruning, canopy, and crop load management are the most frequently reported case of study for source-to-sink ratios, most studies may not offer direct observations , enough combinations, duration of the study, or range of source-to-sink ratios to respond to some fundamental questions.

The aim of this study was to determine the in-season and carryover effects of carbon source and sink imbalances in grapevine. Specifically, we investigated the combined effects of defoliation and fruit removal on components of yield, canopy area, and seasonal integrals of leaf gas exchange, shifts in phenology, carbohydrate, and soluble sugar concentration in the roots.The experimental design was a factorial arrangement of treatments. There were three levels of manual defoliation by three levels of manual fruit removal applied . The treatments were applied as follows. Leaves were removed on every shoot in an alternating pattern. For instance, 66% of leaf treatments retained leaves in positions 1st, 2nd, 4th, 5th, 7th, 8th etc. while 33% of leaf treatments kept leaves in positions 1st, 4th, 7th, etc. in every shoot . The fruit removal treatments retained a percentage of clusters after standardizing the cluster numbers in each year. Each treatment combination was replicated four times and each treatment-replicate consisted of three experimental units. In 2017, all vines were standardized at fruit set to 20 shoots and 30 clusters per vine, and laterals were removed prior to defoliation and fruit removal treatments. In 2018, all vines were standardized to 24 shoots and 45 clusters and laterals were removed prior to treatment application. Treatments were applied at pepper-corn size . In 2019, after two seasons of growth under the nine combinations of treatments, the carryover effects were studied by leaving all vines untreated . For each experimental unit one vine was shoot thinned to 24 shoots, and others were left unmanaged . All clusters at pepper-corn size in all treatment-replicates were dipped in a 5.5% kaolin solution to provide protection from the afternoon sun due to the row orientation of the vineyard in every year of the experiment.Percentage of bud break , flowering , veraison , and leaf senescence per plant were recorded at time intervals of either 1, 2, or 3 times a week depending on weather and phenology events . A leaf was considered senescent when 50% of its area was yellow.

Measurements started soon after the application of treatments in 2017 until leaf senescence 2019. In 2019, only grapevines thinned to 24 shoots per vine were followed.Leaf gas exchange was measured bi-weekly in all years of the experiment with an infra-red gas analyzer . Three sun-exposed leaves were selected from the main shoot axis in each experimental unit, and three readings were taken from each leaf. Gas exchange measurements were taken when the sunlight conditions were close to saturating levels in all instances. The relative humidity was set at 40%, the reference CO2 concentration was set at 400 µmol CO2 mol−1 as the standard environmental condition setting in CIRAS-3. Net carbon assimilation rate and stomatal conductance were obtained. To express the season-long response of AN, and gs , their integrals were calculated by using natural cubic splines for plant water status and gas exchange measurements to assess the cumulative values for these parameters over the whole experiment period during the growing season. Then, these cumulative values were normalized as divided by the number of days elapsed between the first measurement date and the last measurement date to make the data comparable to each individual measurement.After harvest, leaves from one vine per replicate were collected, weighted, and dried in a forced-air oven at 80◦C for 3 days. Dry leaf weights were converted into area by measuring the area of a subsample of 50 random leaves with a leaf area meter as reported previously . On 12 December 2018, after the second season of treatments, one vine per experimental unit of the most extreme treatments were pruned, coppiced, blueberry grow bag size and the root systems were removed with a back-hoe. The sectioned grapevine portions were weighed on a top loading scale, and dried in a forced-air oven at 60◦C until no weight change of tissue was detected. At harvest , clusters were removed, counted, and weighed for each plant in the experiment. Total soluble solids were measured from 55 berries collected randomly at harvest point. The berries were crushed by hand and filtered to obtain must. A digital refractometer was then used to measure total soluble solids of must.Soon after the harvest of 2017 was completed, root tissues were sampled every 2 months. The top layer of soil was removed until the roots were visible. Each grapevine root zone was divided into four quadrants and on each date and one single quadrant was sampled, leaving the other 11 quadrants undisturbed. Roots were gently cleaned with water, freeze-dried, and ground to a fine powder with a tissue lyser . Thirty milligrams of the resultant powder were extracted in 80:20 ethanol solution. A 1.5 mL aliquot of the extract was then placed in a 90◦C water bath for 10 min, then centrifuged at 10,000 rpm for 1 min. The supernatant was collected for total soluble sugars determination. The same procedure was repeated for starch determination, in which the pellet was collected for its determination.

Total soluble sugars in the roots were determined as reported elsewhere by Torres et al. . Briefly, the 1.5 mL sample was filtered by PTFE membrane filters and transferred into high performance liquid chromatography vials. Equipment consisted of a reversed-phase HPLC system Agilent 1100 coupled to a diode array detector and an Agilent Infinity Refractive Index Detector . The reversed-phase column was Luna Omega Sugar with a guard column of 5 mm. The temperature of the column compartment was maintained at 40◦C and the RID flow cell was kept at 35◦C. The mobile phase system consisted in an isocratic elution with acetonitrile:water at a flow rate of 1.0 mL•min−1 with a run time of 22 min. Standard solutions of 10 mg/L of D-glucose, D-fructose, D-sucrose, and D-raffinose were injected to obtain the retention time for each compound, and detection was conducted by RID. Sugar standards were purchased from VWR . Sugar concentration of each sample was determined by comparison of the peak area and retention time with standard sample curves. Starch content of roots was measured using the Starch Assay Kit SA-20 following the manufacturer’s instructions. Briefly, pellets of root tissues were dissolved in 1 mL DMSO, and incubated for 5 min in a water bath at 100◦C. Starch digestion was started by adding 10 µL α- amylase and incubated in boiling water for another 5 min. then, the ddH2O was added to a total volume of 5 mL. Then, 500 µL of the above sample and 500 µL of starch assay reagent were mixed and incubated for 15 min at 60◦C. Negative controls with the starch assay reagent blank, sample blank, and glucose assay reagent blank and positive controls with starch from wheat and corn were performed. Reaction started with the incubation of 500 µL of each sample and 1 ml of glucose assay reagent at 37◦C and was stopped with the addition of 1 mL of 6 M Sulfuric acid after 30 min. Reaction was followed with analytical measurements with a Cary 100 Series UV-Vis Spectrophotometer and starch content expressed as mg of starch per tissue dried weight.In our experiment the results indicated that there was an interaction of year and defoliation on cluster weight, berries per cluster and yield per vine . When we analyzed the data by year, the effect of defoliation was clearer. In both experimental years , there was a strong linear trend of defoliationon all components of yield except for cluster number; which was only affected by the fruit removal treatments. In 2017 defoliating 66% of the leaves resulted in an 8% decrease in berry weight. The differences were exacerbated in cluster weight , berries/cluster , and yield when 66% of the leaf area was removed. In 2018, the effect of defoliation was evident with a 12% decrease in berry weight. As in the previous year, we saw a diminution in cluster weight, berries/cluster, and yield. However, the decline in yield in 2018 was 56% when 66% of the leaves were defoliated. Fruit removal was effective in modulating the cluster number and thus the yield in both experimental years as expected . Furthermore, we measured a strong linear decrease in cluster weight in 2017. However, the same response was not evident in 2018. Removing 66% of the cluster resulted in a 55 and 60% decrease in yield of Cabernet Sauvignon in 2017 and 2018, respectively. Surprisingly, we did not measure an interaction of defoliation and fruit removal on components of year in either of the experimental years . The carry-over effects of source-sink adjustments on components of yield in 2019 were strongly evident ; even though no defoliation or fruit removal treatments were applied. Berry weight, cluster number per vine, cluster weight, and yield per vine were all affected by the carry over effects of defoliation from the previous 2 years. They all declined linearly with the 33% defoliation treatment. Conversely, we did not measure a carryover effect of fruit removal in 2019 in the majority of components of yield monitored. There was an interaction of defoliation and fruit removal in 2019 on the number of berries per cluster.

Although such effects with regard to child care type have been rather mixed, there is some reason to suspect that heightened exposure to peers—such as often found in center-based child care—may be a salient stressor for young children . For instance, at abehavioral level, the replicated link between greater exposure to center-based child care and subsequently heightened levels of aggression in childhood has been shown to be at least partially explained by the typically higher levels of peer exposure faced by these children . That said, the potential effects of peer exposure on HPA axis functioning in infancy and toddlerhood are largely unknown. Indeed, some work has suggested that heightened peer exposure may be associated with lower levels of HPA activity. For instance, in their generative study of infants and toddlers, Watamura and colleagues found that toddlers who spent more time playing with their peers tended to show lower cortisol levels than their less social classmates. Of course, the direction of this relation remains unclear; for example, perhaps less physiologically stressed children choose to play more with peers . However, it nonetheless raises the possibility that peer exposure may provide valuable opportunities for young children to learn to negotiate such complex social contexts.With rare exception, the extant literature concerning child care and children’s early adrenocortical functioning has been based on rather small, homogeneous samples of children from middle- to upper income families. Informed by findings from studies of children’s behavioral outcomes, plastic grow bag there is increasing evidence to suggest that child care effects for children growing up in poverty may be quite different those for children from more affluent home families .

Accumulating findings indicate that the beneficial effects of high-quality child care may be comparatively more pronounced for children from high-risk contexts. For instance, some work has shown that greater exposure to high-quality child care , or even simply attending regular non-maternal care , may mitigate the detrimental effects of economic adversity or low levels of maternal education on children’s subsequent academic achievement. Similar buffering effects have been noted with respect to children’s language development, such that exposure to high-quality language environments in child care may mitigate the negative effects of low-quality language environments experienced at home . Related findings also extend to children’s social development. For example, contrary to the replicated finding that more extensive hours in child care are predictive of heightened levels of aggression , increasing evidence suggests that the opposite may be true for children from high-risk home environments. Greater hours in high-quality child care have been linked with lower levels of internalizing and externalizing behavior in samples of low-income children . Similarly, using data from a large Canadian sample, Côté and colleagues found that the prototypically positive relation between non-maternal care and aggression in childhood was evident only for children from middle- to upper class families. Indeed, although it failed to reach statistical significance, there was a descriptive indication that non-maternal care was associated with better socialoutcomes for low-income children. We and our colleagues have recently shown similar findings with respect to child care exposure and several outcomes in pre-kindergarten .

Specifically, we found that for children experiencing high levels of household chaos across early childhood, greater weekly hours in child care were predictive of comparatively fewer behavior problems. In particular, consistent with the idea of a buffering effect, greater child care exposure ameliorated the detrimental relation between household chaos and children’s social problems. We have also recently shown similar interactive relations with respect to children’s HPA axis functioning at 48 months of age ; specifically, using the same sample as in the present study, we found that the direction of the relation between child care exposure and children’s cortisol levels at 48 months varied as a function children’s broader environmental risk. For children from low-risk households, greater weekly hours of child care were predictive of higher cortisol levels. In contrast, for children facing substantial cumulative risks at home, greater hours of child care exposure were predictive of lower cortisol levels. Of note, and contrary to our hypotheses, is that other aspects of children’s experiences, such as caregiver responsivity and child care type, were unrelated to children’s cortisol level, irrespective of children’s home contexts. As such, the potential mechanisms underlying these relations remain unclear. Furthermore, this work was concerned with HPA axis functioning just prior to children’s transition to school in relation to their average child care experiences across early childhood. Although this represents an important development span, we know little about the extent to which similar conditional relations are evident much earlier in development. Also, unlike the elegant within-person designs adopted by prior work that has considered links between child care and children’s diurnal rhythms on child care versus non–child care days, our findings comprised only between-child analyses. Within person designs afford important methodological advantages.

Most notably, they strengthen the internal validity of one’s inferences by essentially treating each individual as his or her own control group. In so doing, this holds all possible time-invariant confounds constant.Informed by our prior work, we hypothesized that increases in child care exposure would be associated with contemporaneous increases in children’s cortisol levels for those facing low levels of environmental risk. In contrast, we hypothesized that this within-child relation of child care exposure and cortisol levels would be comparatively more negative for those experiencing higher levels of risk—possibly to the extent to which the conditional relation reverses direction, such that increases in child care are predictive of contemporaneous decreases in children’s cortisol levels. Given that our prior work showed evidence of a similar interaction effect at approximately 48 months of age , we expected this conditional relation to extend downwardly to children in toddlerhood. However, consistent with cross-sectional findings by others , we allowed for the possibility that the relation might be comparatively weaker in early infancy. In our prior work we have found little evidence of effects of child care type or caregiver responsivity with respect to 48-month cortisol. Yet, informed by prior studies by others , we nonetheless conjectured that the high-quality and center-based care might play a particularly important role in children’s cortisol levels for those from high-risk home environments. Finally, informed by that idea that peer relations may support optimal behavioral and physiological regulation for children in child care , we hypothesized that within-child increases in positive peer exposure would be associated with cotemporaneous decreases in children’s cortisol levels—perhaps, particularly so for children experiencing heightened levels environmental risk. Our hypotheses with respect to between-child differences in children’s child care experiences were substantively identical to the within-person representations above.The Family Life Project was designed to study young children and their families in two of the four major geographical areas of the United States with high poverty rates —eastern North Carolina and central Pennsylvania; specifically, 1,292 children whose families resided in one of the six counties at the time of the child’s birth were sampled. Lowincome families in both states and African American families in NC were oversampled. A comprehensive description of the sampling plan was provided by Vernon-Feagans and colleagues . The present analytic sample comprises the 1,155 children with child care data for at least one of the three points at which child care information was collected between the time children were 7 and 24 months of age. Those excluded from the analytic sample did not differ from the present sample with respect to race or family income or primary caregiver education level at 7 months.In the present study we focused on data collected across infancy and toddlerhood. Children’s mothers were interviewed regarding demographic and personal information when their children were approximately 2 months of age. The demographic data used to calculate economic adversity, pe grow bag as well as information about child care usage, were collected from mothers during interviews when their children were 7, 15, and 24 months of age. At the ages of 7, 15, and 24 months, saliva samples were collected from children during home visits to assess resting or “non- stimulated” cortisol levels. Saliva samples were collected after the data collectors had been in the home for at least 1 hour interviewing the primary caregiver and prior to conducting a number of assessments with children.

The modal time of collection was approximately 10:00 a.m. at each wave of collection; however, there was some variability. Within-child differences in saliva sampling times across the three home observation points, as well as between-child differences in the saliva sampling times were adjusted statistically in all models. We use the term resting given that no active, purposeful stimulation was present, and children were given ample time to return to baseline after the arrival of the research assistants . We cannot rule out, however, that our “resting” cortisol levels partly capture HPA axis stimulation due to the visit. Unstimulated whole saliva was collected using either cotton or hydrocellulose absorbent material and expressing the sample into 2-ml cryogenic storage vials using a needleless syringe or by centrifugation . Prior studies have indicated no differences in cortisol concentrations associated with the two collection techniques . Children’s saliva samples were immediately frozen at −20° C and subsequently stored at −80° C. Intra- and interassay coefficients of variability for cortisol were, on average, less than 10% and 15%, respectively. When children were in non-parental care for at least 10 hours per week, interactions between the child care caregiver and the child were rated for caregiver responsivity by independent raters.Salivary cortisol—All samples from the 7-, 15-, and 24-month assessments were assayed for salivary cortisol using a highly sensitive enzyme immunoassay US FDA 510k cleared for use as an in vitro diagnostic measure of adrenal function . The samples were assayed in duplicate, with the average of the duplicates was used in all analyses. We examined child temperature and use of medications as influences on child cortisol); no relations were evident, after adjusting for the time of day when the saliva sample was collected. Saliva collection times were included as both time varying and time-invariant control covariates. Child care quantity—Children’s primary caregivers provided the average hours per week that the child attended non-parental child care when the child was 7, 15, and 24 months old. non-parental caregiver responsivity—Traditional measures of child care quality, such as the Early Childhood Environment Rating Scale or the Observational Record of the Caregiving Environment were unavailable in these data. We thus used independent ratings of the non-maternal caregivers’ behavior toward the child, as scored with Home Observation Measure of the Environment scale , as a proxy for process quality in child care. Indeed, although the HOME Responsivity scale is not a measure of child care quality, per se—like the process quality measure used in the NICHD Study of Early Child Care and Youth Development —it taps the sensitivity and responsivity of the non-parental caregiver to the child’s needs in ways that are comparable across different types of child care. HOME scale observations were made when the child was 7, 15, and 24 months old. Informed by prior psychometric work , we fitted a series of confirmatory factor analytic models, based on 10 items tapping caregiver responsiveness and affection toward the child. We hypothesized that a single latent factor explained observed caregiver responsivity across the items within a given time point. Because items were scored dichotomously these models were fitted using a robust weighted least-squares estimator . Descriptions of the CFA models are provided in the online supporting materials on the Family Relations website. In short, the results discussed below are based on factor scores estimated from a longitudinal CFA, constrained for partial measurement invariance. Higher values reflect higher levels of caregiver responsivity. Child care type—Child care type was reported by independent observers during child care visits conducted when the target child was 7, 15, and 24 months of age . Child care peer exposure—The numbers of peers in the classroom were based on counts conducted by independent observers during child care visits at 7, 15, and 24 months of age. Cumulative risk—Informed by extensive prior work with these data , we created a cumulative-risk composite comprising seven variables— family income, maternal education, constant spouse/partner living in the home, hours of employment, occupational prestige, household density, and neighborhood noise and safety—that were measured across infancy and toddlerhood. As described in detail in the online supporting materials, we created a continuous cumulative-risk index by reverse scoring the positively framed variables, standardizing each risk measure, and averaging across the risks.

Overall, insectivorous birds are the most abundant on coffee farms and hold great potential as bio-control of many insect pests . Details on bird densities on Costa Rica coffee farms used in the model are expanded on below .To our knowledge, there is little information about population densities of CBB in coffee plantations at the start of the growing season. We first initialized the coffee berry borer population model with 100 dispersing females . The start of CBB reproduction commenced 120 days after coffee flowering and continued until 305 days after flowering, yielding a 185-day CBB breeding season. We confirmed CBB reproduction was possible within this period for Central Valley Costa Rica using degree day calculations from Jaramillo et al. based on CBB thermal tolerance. We then calculated how much the dispersing adult survival rate would have to be reduced to cause a 50% reduction in adult female borer population size on day 185. To determine how many CBB would need to be consumed by birds to achieve this goal, we found the difference between daily borer population sizes of unsuppressed and suppressed populations and summed the differences across the CBB reproductive season. We used sensitivity analysis to estimate the degree to which changes in each vital rate affects population growth rate . All models were implemented using the popbio package in R . R code for all analyses is provided in the Supporting Information . We also wanted our model to project CBB population growth that represented “low” and “high” infestations observed in the field. To start, we estimated probable CBB densities using data on the number of dispersing females collected in alcohol-lure traps. At peak dispersal, CBB numbers have been recorded as high as 1000–6120 CBB/trap/week to as low as 50–105 CBB/trap/week .

Using these trap counts, plastic square flower bucket we calculated potential CBB densities per hectare via reported trap densities and converted weekly capture estimates to the number of daily dispersers to complement our daily population model. We used a density independent model, a standard first step in many population models. However, note that we would need to divide CBB numbers by plant density to evaluate the impacts of CBB population growth on yield. We also would need empirical data on how the demography of CBB populations change with coffee-plant density to implement a revised model, and we are unaware of published data on this. Consequently, this analysis is beyond the scope of this paper . Using data from Aristizabal et al. , we selected a high peak dispersal count from farms with large infestations and a low peak dispersal count from farms with small infestations to represent peak dispersal on Day 185 in our model. We used our calculated values of 269 and 5 as our “high” and “low” initial population sizes of gravid females at the start of the coffee season and used 100 CBB to represent “medium” initial population size.Parameters for our Leslie matrix for coffee berry borers are broadly consistent with expectations and general knowledge . For example, our conversion of fecundity to a daily value, F1 = 1.341, is consistent with published literature stating that 1–2 eggs are laid per day by CBB . Model projections showed that across a 185-day CBB breeding period starting at the point of first ovipositing, an initial population size of 100 female dispersers would produce 1.3 million offspring, resulting in a new adult population of 70,245 females . Assuming
99% of colonizing females successfully bore and oviposit in a coffee cherry on Day 0, the first generation of new dispersing females does not appear until day 37.

At Day 38, the adult population begins to increase, and continues to do so exponentially.The daily growth rate of this population converged on 1.042. Sensitivity analysis revealed that survival of adult females had the largest impact on overall population growth , followed by daily survival of pupa , juveniles , eggs and larvae and dispersing females . In addition to modeling growth with 100 initial colonists , we projected the population growth of low and high starting populations calculated from observed weekly alcohol-lure trap catches during peak dispersal . Comparing the three population projections, peak number of dispersers at Day 185 varied considerably, with 162, 3259, and 8768 daily dispersers for low, medium, and high colonizing populations, respectively. In the high population projection, the adult population toward the end of the growing season reached over 18,800 individuals. Note that because these are density-independent models, the number of CBB does not depend on plant density. However, the impacts of the CBB population on yield would depend on coffee plant density. To reduce the final adult population by 50%, the daily survival rate of dispersing females would have to be reduced from 0.99602 to 0.83202. This change represents a 16.4% reduction in daily survival when dispersing. The number of CBB that birds need to eat to reduce the adult population at this rate was driven by the initial population size as a straight line, y = 79.23 N0 . At medium starting population , birds need to consume 7628 CBB during the borer breeding season, while at high starting population , about 20,500 dispersing CBB must be consumed by birds. Daily consumption rates by birds would have to increase over time as the CBB population grows and could vary from 15 to 750 CBB being consumed a day, depending on starting population size . Overall, we calculated that for every female CBB in the initial colonization, birds need to consume 79 CBB to reduce the end of season population by half.We estimated that the caloric content of a 195 μg adult CBB to be 1.09 calories per gram dry weight, or 0.00109 kcal. At 5%–10% of a bird’s daily diet based on number of prey items, birds would consume <7 CBB per day. This represents 0.03%–0.05% of daily caloric requirements of our average insectivorous bird. At these feeding rates, our models suggest that by the time of peak dispersal, 4, 88, and 236 birds are required at low, medium, and high starting population sizes, respectively, to reduce CBB populations by 50% on day 185 .Our model suggests that avian predation is likely to be effective at reducing CBB populations by 50% only during small infestations , or during the early stages of larger infestations . Birds appear unable to successfully suppress medium and large infestations because the number of CBB that need to be eaten in a season requires higher bird densities than are reported in the literature. Karp et al. estimated 4–12 birds/ha of species that are confirmed or suspected CBB predators. Flocks of migratory birds on coffee farms are estimated at 19/ha and 24/ha , but these values are also short of our estimates of necessary densities for suppressing larger CBB outbreaks. One caveat to our conclusions is that our calculations were based on CBB accounting for 5%–10% of a bird’s daily diet . This assumption meant birds would only eat a set maximum of 7 CBB per day. Sherry et al. reported up to 116 CBB in the stomach contents of a single warbler, suggesting under certain circumstances in the field, birds eat more CBB. Generalist insectivores, particularly Neotropical migrants, have flexible foraging preferences and would likely feed opportunistically on CBB in response to dramatic dispersal peaks. Therefore, birds might be expected to increase feeding rates as CBB disperser abundances increase, though it may depend on the relative abundances of other prey. Better data on CBB consumption rates by birds under different circumstances would improve our estimates of the circumstances under which birds can control CBB populations.

A second caveat is that bird densities used in the model may not represent the potential for CBB control because bird densities depend on the structure of the agricultural landscape, which the current model does not consider. On coffee farms, plastic plant pot birds are more abundant when native tree cover is highest and natural forests are close by . Across tropical and temperate regions, the propensity for birds to forage on farms, and thus exert pressure on agricultural pests, is correlated with the physical complexity and diversity of the agroecosystem . For example, birds make more frequent foraging trips to apple orchards with high native tree coverage . In alfalfa fields, edge habitat complexity supports greater avian richness leading to lower pest abundances . Under some circumstances, the density of birds foraging in certain areas may be higher than average densities would imply, leading to greater control potential than our models suggest. More generally, our CBB population model is density independent and assumes environmental conditions and sufficient resources to allow CBB populations to increase without restriction. As a result, our model is limited, as it does not consider localized effects of weather and temperature fluctuations on CBB developmental time , nor characteristics of coffee farms that influence both CBB infestation and bird density. We assumed maximal capacity for CBB population growth and used estimates of bird densities from the literature that only included birds known to consume CBB, perhaps underestimating the potential for avian control. Models are an important tool for estimating population dynamics, but as with any species, the growth potential for CBB and availability of its predators, is context dependent. Our study echoes Kendall et al.’s conclusion that, even though errors in model construction are common, these seldom change qualitative conclusions. From our population matrix, CBB daily growth rate converged on λdaily = 1.042 around day 124, with an observed rate of population change across the entire coffee-growing season of 705 . Our λdaily is higher than Mariño et al.’s reported lambda of 1.32 over 50– 56 days, which corresponds to λdaily ≈ 1.006 . Part of this discrepancy may come from the fact that Marino et al. combined vital rates across life stages with different time steps. Nonetheless, both models are consistent in predicting rapidly growing populations. Observed CBB population growth rates are similar to ours: Baker, Barrera, & Rivas, calculated a 1.067 growth rate in wild populations and RuizC ardenas and Baker reported 1.047 in CBB reared in laboratory settings. In their sensitivity analysis, Mariño et al. reported that adult female survival, and transitions from larva to pupa and pupa to juvenile had high sensitivity in contributing to population growth rate, with adult survival the highest . We found a similar peak sensitivity value for female adult survival in our matrix , supporting the idea that CBB population growth is most sensitive to adult survival rate. Interestingly, dispersal survival from our matrix was estimated to have low impact on population growth , even though this life stage is when CBB are vulnerable to bird predation. Thus, our analysis superficially suggests that population control once CBB are established should focus on reducing adult survival rather than on trapping dispersing females , if the same impact on numbers could be achieved. However, dispersing females are much more accessible to control methods like spraying fungal bio-insecticide than are adult females, which are inside the coffee cherries, so despite the tremendous difference in sensitivity values, management of an established population is likely to be more cost effective by continuing to focus on dispersing females . Population models specific to CBB have been criticized for not being representative of wild populations, since more generations are estimated through modeling than are observed in field studies . We analyzed CBB population growth using a deterministic model, with an even distribution of dispersal and a fixed predation pressure. While CBB dispersal is continuous, there can be dramatic intraseasonal peaks in numbers that were not captured by our model . In addition, reported longevity of female CBB varies widely from 55 to 380 days, though some studies looked at CBB reared on artificial diet . Refinements of survival in natural settings would, therefore, improve models of CBB population growth, and the potential for control by birds. If field data on CBB vital rate stochasticity become available, and bird densities opportunistically increase during CBB peak numbers, it could affect our conclusions about the capacity of birds to control larger CBB outbreaks. Based on our analyses, there is a population density of CBB above which their capacity to produce more adults exceeds the ability of birds to control their numbers, at least to limit the population size by 50%.

As pointed out in Ref., the Berry phase has a profound geometrical origin because an adiabatic and cyclic process of a quantum state is mathematically equivalent to parallel transporting it along a loop, which connects to the concept of holonomy in geometry. Hence, the Berry phase bridges physics and geometry, making it extremely important in the understanding of topological phenomena, such as integer quantum Hall effect, topological insulators and superconductors, and others. The description of the Berry phase relies on the properties of a pure state of a quantum systems at zero temperature. Meanwhile, mixed quantum states, including thermal state at finite temperatures, are more common. Therefore, mixed-state generalizations of the Berry phase have been an important task. Uhlmann made a breakthrough by constructing the Uhlmann connection for exploring the topology of finite-temperature systems. As the Berry holonomy arises from parallel transport of a state-vector along a closed path, the Uhlmann holonomy is generated by parallel-transporting the amplitude of a density matrix. defined by W = √ρU. Here the amplitude W is the mixed-state counterpart of the wavefunction, and U is a phase factor. A geometrical phase is deduced from the initial and final amplitudes. However, Uhlmann’s definition of parallel transport is rather abstract and may involve non-unitary processes, complicating a direct and clear physical interpretation. Moreover, the fiber bundle built upon Uhlmann’s formalism is trivial, which severely restricts its applications in physical systems.Purification of a mixed state leads to purified state, round flower buckets a state-vector equivalent to the amplitude of a density matrix. The lack of a one-to-one correspondence between the density matrix and its purified states gives rise to a phase factor, similar to the phase of a wave function.

In a branch of quantum field theory called thermal field theory, there is a similar structure for describing the thermal-equilibrium state of a system by constructing the corresponding thermal vacuum by duplicating the system state as an ancilla and forming a composite state. It plays a crucial role in the formalism of traversable wormholes induced by the holographic correspondence between a quantum field theory and a gravitational theory of one higher dimensions. Importantly, purified states of a two level system has been demonstrated on the IBM quantum computer while the thermal vacuum of a transverse field Ising model in its approximate form has been realized on a trapped-ion quantum computer. Despite the superficial similarity, a major difference between a thermal vacuum and a purified state is a partial transposition of the ancilla to ensure the Hilbert-Schmidt product is well defined. In quantum information theory, a partial transposition is closely related to entanglement of mixed states. Importantly, partial transpositions of composite systems have been approximately realized in experiments by utilizing structural physical approximations in suitable quantum computing platforms. Although ordinary observables cannot discern the partial transposition between the purified state and thermal vacuum, here we will show that at least two types of generalizations of the Berry phase to mixed states are capable of differentiating the two representations of finite temperature systems. Among many attempts to generalize the Berry phase or related geometric concepts to mixed states , a frequently mentioned approach was proposed in Ref.. Instead of decomposing the density matrix to obtain a matrix-valued phase factor, a geometrical phase is di-rectly assigned to a mixed state after parallel transport by an analogue of the optical process of the MachZehnder interferometer. Hence, the geometrical phase generated in this way is often referred to as the interferometric phase. The interferometric phase has been generalized to nonunitary processes, but the transformations are still on the system only. Moreover, it is essentially different from Uhlmann’s theory since the conceptual structure of holonomy is not incorporated.

We will first derive a mixed-state generalization of the parallel-transport condition for generalizing the Berry phase without invoking holonomy. This approach unifies the necessary condition for both the interferometric phase and Uhlmann phase . Two ways to implement the parallel-transport condition based on how the system of interest undergoes adiabatic evolution will be introduced, and they lead to different generalizations of the Berry phase. We will name one thermal Berry phase and the other generalized Berry phase. Importantly, the partial transposition of the ancilla between a purified state and thermal vacuum will be shown to produces observable geometrical effects in both thermal Berry phase and generalized Berry phase. Through explicit examples, the two generalized phases are shown to differentiate the two finite-temperature representations, a task beyond the capability of the conventional interferometric phase or Uhlmann phase. The rest of the paper is organized as follows. Sec. II summarizes the Berry phase in a geometrical framework with an introduction of the parallel-transport condition for pure quantum states. In Sec. III, we review the representations of mixed states via purified states and thermal vacua and then explain the difference of the partial transposition of the ancilla. In Sec. IV, we introduce the thermal Berry phase via generalized adiabatic processes. While the thermal Berry phase can differentiate a purified state from a thermal vacuum, it may contain non-geometrical contributions. In Sec. V, we generalize the parallel-transport condition to involve the system and ancilla and derive the general Berry phase according to the generalized condition. While the generalized Berry phase only carries geometrical information, its ability of differentiating a purified state from a thermal vacuum depends on the setup and protocol. We present examples of the thermal and generalized Berry phases. Sec. VI concludes our study. Some details and derivations are given in the Appendix.While purified states of a two-level system incorporating environmental effects have been simulated on the IBM quantum platform, thermal vacua of the transverse Ising model has been experimentally realized on an ion-trap quantum computer by the quantum approximate optimization algorithm. Moreover, partial transposition of a composite system has been approximately realized on quantum computers with various numbers of qubits. Therefore, a comparison of the geometric effects reflected by the generalizations of the Berry phase of purified states or thermal vacua is expected to be achievable in future experiments on quantum computers or quantum simulators. For example, one may consider two identical composite quantum systems of Example V.1 of the generalized Berry phase and then apply a partial transposition to one of the composite systems. As a consequence, the composite system with a partial transposition corresponds to a purified state while the one without partial transposition may be viewed as a thermal vacuum. By applying parallel transport that involves the ancilla to both composite systems and extract their generalized Berry phase after a cycle, a π-phase difference is expected between the two composite systems. Given the large phase difference between them after a cycle, the result is robust against small perturbations or noise from the hardware and offers another demonstration of geometrical protection of information. We have presented two generalizations of the Berry phase, the thermal Berry phase and generalized Berry phase, for distinguishing the two state-vector representations of mixed states via the purified state and thermal vacuum. From the geometrical and physical points of view, the generalized Berry phase has more desirable properties since the thermal Berry phase is generated by a temperature-dependent thermal Hamiltonian and may carry non-geometrical information. We caution that while the transformations can be on the system, ancilla, or both in the construction of the generalized Berry phase, an operation on the ancilla is necessary if we want to differentiate the purified state and thermal vacuum.The earliest classification of the forms of matter we see around us, typically presented to us in our early school days, consists of solids, liquids and gases.

High school physics textbooks and experience later teach us that solids can be further classified based on their electronic properties as conductors and insulators. Solid state physics courses in college add semiconductors, semimetals and superconductors to that list, plastic flower buckets wholesale and explain the basic physics that governs the electronic properties of these phases. More precisely, as long as the electrons in a solid are non-interacting, solids with partially filled bands are shown to be metals or conductors while those with no partially filled bands and a gap between the valence and the conduction bands are insulators or semiconductors. If the gap is extremely small or vanishing or if there is a very small overlap between the valence and the conduction bands, the material is semimetallic. Superconductors are argued to be the fate of a metal cooled to extremely low temperatures. However, vast quantitative as well as qualitative differences between the properties of materials within a single category are often observed. For instance, some insulators have conducting surfaces while others do not. Moreover, the surface conduction is stable against perturbations as well as deformations in the band structure as long as the underlying symmetries of the system are preserved and the system remains insulating after the deformations. Similarly, interactions and certain perturbations can gap out some semimetals and turn them insulating but not others . These observations call for further refinement in the classification of solids, especially one that explains why the properties of some phases are robust to certain deformations, interactions and perturbations. In other words, an understanding for why some phases are topological while other are not, is required. The study of the topology of the bands provides a powerful unifying framework for accomplishing this task. Topology shows up in two distinct ways in the band structures of non-interacting Hamiltonians depending on whether the spectrum is gapped or gapless. In systems with gapped band structures, which include insulators and most superconductors, the wavefunctions wind non-trivially across the Brillouin zone in the topological phases, as described in Sec. 1.1. What constitutes a non-trivial winding depends strongly on the symmetries and the dimensionality of the system under consideration, thus, revealing a rich substrucutre within the insulating and superconducting phases. On the other hand, gapless band structures contain topological objects in momentum space which can be characterized by the winding of theGreen’s function around the object, as exemplified in Sec. 1.2. A key feature of topological media is that they typically have unconventional surface states which, in many cases, cannot exist independently of the bulk phase in one lower dimension. For instance, the two dimensional surface of a three dimensional topological insulator hosts pseudorelativistic electrons and cannot exist as an independent two dimensional system. Because of this intimate surface-bulk connection and because the surface is usually more accessible than the bulk, experiments usually probe the surface states in order to identify the bulk topological phase. The surface states are also extremely valuable from a practical point of view, since the unconventional properties bestowed upon them by the non-trivial bulk topology may be exploited to design novel electronic devices.The role topology plays in the band structures of gapped non-interacting Hamiltonians is analogous to its manifestation in an early example of topology in physics – Gauss’s law in electrostatics. Gauss’s law states that the total electric flux piercing a closed surface is determined only by the charge enclosed by it and is independent of its shape or the precise charge distribution. Thus, the total flux through each surface in Figure 1.1.1 is four units. For gapped noninteracting lattice Hamiltonians, the analog of the Gaussian surface is the Brillouin zone, and different Gaussian surfaces correspond to different Brillouin zones in the extended zone scheme or to different bands in the reduced zone scheme. The electric charge, then, maps to an appropriate topological invariant whose exact form depends on the symmetries of the system under consideration. This topological invariant can be written as an integral over the Brillouin zone of an appropriate field derived from the Bloch functions of the occupied bands, analogous to how the electric charge enclosed by a Gaussian surface is equal to an integral of the electric field over it. The analogy is generalizable to continuum Hamiltonians, defined over all of momentum space, as well, provided the points at infinity are identified. This identification compactifies momentum space to the topological equivalent of a sphere, which, being a closed surface, permits the application of Gauss’s law. Next, assuming every charge that exists in the universe is inside a Gaussian surface, the only way to change the amount of charge enclosed by a single surface is by fusing it with another Gaussian surface, moving charges across the junction and pinching the junction ofto get the original surfaces back, as illustrated in Figure 1.1.2. Similarly, for band structures, topological invariants can only be defined for Brillouin zones wave functions in the extended zone scheme or equivalently, for bands in the more common reduced zone scheme.