Indications for Topological Physics in High-$T_{c}$ Materials

Many experiments on cuprates have reported that the Fermi surface is in fact four Fermi arcs: four curves which each terminate at two end points, with the end point location changing with temperature and doping. Norman et al. (1998); Vishik (2018); Yazdani et al. (2016); Kordyuk (2015) Within a weakly interacting approach it is impossible to account for Fermi arcs that are truly incomplete, at least in the bulk in the material. The difficulty is that the Fermi surface defines a boundary between the part of the Brillouin zone where all states are occupied and the part of the Brillouin zone where all states are unoccupied. If the Fermi surface is incomplete then it cannot be interpreted as the boundary between occupied and unoccupied sectors. Therefore it is widely understood that Fermi arcs, if they do exist in the cuprates, are a manifestation of strongly correlated physics. The details of how such physics would result in Fermi arcs are not well understood or agreed on.

There is, however, a well-understood way to explain Fermi arcs within the theory of weakly interacting materials and band structure. This explanation is a topological explanation, and it puts the Fermi arcs on the material’s surface rather than in its bulk. The theory of how such Fermi arcs arise has a solid foundation, and has been verified by experimental observation of Fermi arcs in several topological materials. Armitage et al. (2018) Since there is a clear theory and experimental realization of Fermi arcs in topological materials, and the theories of strongly interacting Fermi arcs are on a weaker footing, some attention should be be given to the possibility that the Fermi arcs seen in the cuprates are caused by topology. This possibility apparently has not been discussed, so here we breach the question, putting it baldly. ¹¹1We did many detailed and lengthy searches of citations of the original article on Fermi arcs in cuprates Norman et al. (1998), and found no discussion of the possibility that these might be associated with Weyl fermions. A hypothesis can not be proven wrong until it has been considered.

The cuprates’ Fermi arcs are seen at a range of temperatures above the superconducting temperature $T_{c}$ and below another temperature $T^{*}$ which decreases with doping. At temperatures above $T=T^{*}$ there is a single Fermi surface, a circle around the $\left[\pi,\pi\right]$ point in the Brillioun zone. This point is equivalent also to three other points $\left[\pm\pi,\pm\pi\right]$. At $T=T^{*}$ the circle separates into four arcs, with pieces missing near the boundaries of the Brillouin zone. The arc length decreases with temperature until at $T=T_{c}$ there remain only four Fermi points. The arc length also depends on doping.

There are several difficulties with analysis of the Fermi arcs: (1) They are observed using surface not bulk spectroscopy: ARPES or STM. These experimental tools do not see into the bulk. (2) It is very hard to get clean stable surfaces in La_2-xSr_xCuO₄ (LSCO) or YBa₂Cu₃O_6+δ (YBCO), so almost all experimental observations of Fermi arcs have been in Bi₂Sr₂Ca_n-1Cu_nO_2n+4+x (BSCCO). (3) ARPES and STM report results that do not entirely agree with each other. (4) Doping is hard to control so it is hard to measure the doping dependence of the Fermi arcs. (5) Most of the spectral weight seen in ARPES is spread smoothly in energy, as is typical in strongly correlated materials. The Fermi surface signal strength, i.e. the height in the small peak interpreted as a Fermi surface, is a small fraction of the smooth background. Therefore a lot depends on how one fits the signal. (6) Some claim that the disappearance of portions of the Fermi surface is not real - it is caused by broadening [associated with temperature and interactions] which washes out the peak in the spectral density; if the broadening did not occur then the Fermi surface would remain intact. Others claim that between the two endpoints of the Fermi arc there is a second connecting arc which for various reasons is less visible to ARPES and STM, so that there are four complete Fermi surfaces, and no real Fermi arcs. Vishik (2018); Yazdani et al. (2016); Kordyuk (2015)

We leave aside these concerns and take a naive approach to the overall story of Fermi arcs, taking at face value the claim that the Fermi surface in BSCCO really is incomplete.

The only way to explain Fermi arcs, within a weakly interacting approach, is if they reside not in the material’s bulk but instead on its surface. Certain topological materials host in their bulk Weyl fermions, i.e. fermions with a well-defined chirality [either right-handed or left-handed] and obeying a Dirac cone linear energy dispersion. Weyl fermions always come in pairs, with the pair separation [within momentum space] determined by the strength of time-reversal symmetry breaking via a magnetization or magnetic field. Kim et al. (2016) [Inversion symmetry breaking is also an alternative to time reversal symmetry breaking.] Any number of Weyl fermion pairs can be realized in a material. On the surface of a Weyl material Fermi arcs connect each pair of Weyl fermions. Armitage et al. (2018)

In BSCCO there are four Fermi arcs, each with two end points. If the Fermi arcs seen in BSCCO are the surface manifestation of bulk Weyl physics, then BSCCO must host eight Weyl cones in its bulk.

There is no conceptual difficulty with producing a Hamiltonian which has the same four Fermi arcs and eight Weyl cones seen in BSCCO. A simple model with the copper oxide layer’s $C_{4}$ symmetry and four Fermi arcs arranged around $\left[\pm\pi,\pm\pi\right]$ has already been reported - see Fig. 3-d1 in Ref. Li et al., 2017. Exact reproduction of the experimental data on cuprate Fermi arcs and their dependence on doping and temperature is simply a matter of choosing a convenient Hamiltonian. The Hamiltonian’s parameters must be tuned to depend on temperature and doping, in order to reproduce the fragmentation of the Fermi surface into Fermi arcs at the pseudogap temperature, and also to reproduce the reduction of the arcs to nodes at $T_{c}$.

In summary, if the experimental evidence for Fermi arcs is taken naively, then it implies that BSSCO at least, and possibly all the cuprates, are Weyl materials in the bulk, as long as the temperature lies above $T_{c}$ and below the pseudogap temperature $T^{*}$. In this naive interpretation, ARPES and STM measurements on BSCCO are reporting surface not bulk Fermi arcs.

Section IX has some brief comments about how Weyl fermions might arise at long distance scales as the result of renormalization group flow from a non-Weyl Hamiltonian.

This author recently performed a comprehensive survey of all experimental reports of characteristic temperatures well above the superconducting transition $T_{c}$ in strontium doped lanthanum cuprate (LSCO) and oxygen doped YBCO. Sacksteder (2020) We report only characteristic temperatures at which clearly identifiable signals occur, for example opening of a pseudogap, peaks or kinks in the temperature dependence of various observables, or extinction of a diffraction peak at a particular temperature. These characteristic temperatures have been found using a wide variety of measurement techniques over a period of about three decades. The reported temperatures have been realized experimentally, rather than being derived by extrapolation from lower temperatures. We used only characteristic temperatures that were reported numerically in the original reports, rather than extracting temperatures from data ourselves. [There was one exception to this rule.] We omitted the experimental literature that focuses on ionic behavior, but were very thorough about including all other experimental data sets. The experimental corpus on characteristic temperatures from 1990 until 2018 contains twenty separate data sets from thirteen experimental groups on LSCO, and twenty-six distinct data sets on YBCO.

When the experimental corpus for LSCO is plotted all on one graph, with temperature as the $y$ axis and hole doping as the $x$ axis, a remarkable pattern emerges. The reported characteristic temperatures lie on a sequence of distinct well-separated straight lines which radiate from a common intersection near the high-doping end of the superconducting dome. The highest line is associated with a peak in the magnetic susceptibility and transport signatures, the second line has symmetry breaking from tetragonal to orthorhombic crystal structure, the third line is the pseudogap transition seen in ARPES, and the fourth line is seen in transport signatures and NMR. There is some evidence suggesting lower lines as well. We call this family of lines the pseudogap family.

This family of pseudogap lines in LSCO obeys a quantization rule: starting from the highest characteristic temperature, the second temperature’s slope and value are one half those of the highest temperature, the third line’s slope and value are one third, and the fourth line’s slope and value are one fourth.

A similar pattern is visible in YBCO. Here three well-separated lines are visible, again with a common intersection at high doping. The highest line is the pseudogap transition accompanied by symmetry breaking, the next shows nematicity in transport and also time reversal symmetry breaking, and the next is seen in transport signals. The slopes of the three YBCO lines have the the same fractional relation to each other as is seen in the second, third, and fourth lines in LSCO. This suggests the existence of another higher line at twice the temperature of the highest observed line. There is some data suggestive of the higher line, but it is sparse. Possibly this is because YBCO is rarely measured above $300\,K$, unlike LSCO which has been measured at temperatures as high as $T=700\,K$. In any case the family of pseudogap lines observed in YBCO show the same common intersection and quantized slopes that are seen in LSCO’s pseudogap family.

A pattern of lines with quantized $1/n$ slope and common intersection is known in another context: the energies of Landau levels of a fermion moving in a plane, with a Berry phase like that caused by a Dirac point. Mikitik and Sharlai (1999); Wang et al. (2010) The analogy between the pseudogap lines and the Landau levels would require that hole doping map to the Dirac fermion’s Fermi energy, and that temperature map to the Dirac fermion’s magnetic field $B$. While this mapping is unmotivated and obscure, the observed pattern of quantized lines suggests a $U(1)$ phase and a winding number associated with that phase, i.e. the winding number is the number of times that the phase goes through $2\pi$. For Landau levels the $U(1)$ phase is of course tied to magnetic flux. Winding numbers are one of the most elementary realizations of topology seen in physics. If the hierarchy of pseudogap lines seen in LSCO and YBCO really is caused by winding number physics, this would explain why the observed family of pseudogap temperatures is so robust against interactions.

In addition to the possible winding number discussed above, it is worth dwelling a little more on the fact that the slopes of the pseudogap lines seen in LSCO and YBCO follow a $1/n$ pattern of $1/1,\,1/2,1/3,\,1/4$. 2-D Landau levels reproduce this pattern only if the electronic motion is affected by either a Zeeman term or a Berry phase like that seen when there is a topologically protected Dirac point. Mikitik and Sharlai (1999) Otherwise 2-D Landau levels follow a $1/(n+1/2)$ pattern: $(1/2)^{-1},\,(3/2)^{-1},\,(5/2)^{-1},\,(7/2)^{-1}$. The experimental data do not support this pattern and support the $1/n$ pattern instead. The existence of a $1/n$ not $1/(n+1/2)$ pattern in the pseudogap lines may be a hint of topological physics in the cuprates.

One hallmark of topology is reduced dimensionality. In a 2-D material topological states appear at the edges and corners, while in a 3-D material topological states appear at the surface, edges, and corners. When the key physics happens at a dimensionality that is smaller than the apparent dimensionality of a system, this can indicate that topology is at work.

The Copper Oxide Plane. In this respect it is interesting that the essential physics of the cuprates lies in the copper oxide CuO₂ plane, even though the actual material is 3-D. Some variation in $T_{c}$ occurs as the number of CuO₂ planes is varied, but the variation in $T_{c}$ is relatively weak, suggesting that it is a second order effect. In LSCO all the physics required for its high-$T_{c}$ superconductivity lies in the plane, since monolayer LSCO reaches the same $T_{c}$ as bulk LSCO. Božović et al. (2016a) The focus on 2-D physics is also reminiscent of the IQHE and FQHE systems, which also work on topological grounds.

Linear Resistance in Bad Metals May be a Sign of Quasi-1-D Conducting Channels. At temperatures above $T_{c}$ the cuprate and pnictide superconductors have a resistance that increases linearly with temperature, and this linear growth continues to large temperatures without saturating. Linear-in-temperature resistance has become known as a hallmark of strong correlations, including in materials that do not superconduct. Materials with this signature are called bad metals. Hussey et al. (2004)

In recent years experiments have found that several materials with linear-in-temperature resistance also exhibit linear-in-magnetic field resistance. This was verified in the high-$T_{c}$ cuprate superconductor La_2-xSr_xCuO₄ (LSCO), the high-$T_{c}$ pnictide superconductor BaFe₂(As_1-xP_x)₂, and also the heavy fermion material Yb_1-xLa_xRh₂Si₂. Hayes et al. (2016); Giraldo-Gallo et al. (2017); Kumar et al. (2018) The resistance is linear in both temperature and field. It is possible that generally linear-in-field and linear-in-temperature resistance go together in strongly correlated materials. Studies of magnetoresistance in bad metals have not been common, and the data is not yet available to decide whether linear-in-field resistance is the norm in these materials.

In any case the discovery of linear-in-field resistance in the cuprates and pnictides unites the mystery of their linear resistance with the ongoing mystery of linear-in-field resistance in a wide variety of experimental realizations, many of which have Dirac cones. Zhang et al. (2012a, 2011); Wang et al. (2012); Feng et al. (2015); Kisslinger et al. (2015); Zhang et al. (2015); Wang et al. (2014); Xu et al. (2015); Sun et al. (2016); Tanabe et al. (2011); Bhoi et al. (2011). Several explanations of linear-in-field resistance have been given which focused on specific mechanisms which could be responsible for linear magnetoresistance in specific materials. These include the cases of ballistic conduction when the Fermi surface has a cusp, of a 3-D Dirac cone in the presence of a strong magnetic field, of a density gradient across the sample, and of classical transport with strong sample inhomogeneities. Fenton and Schofield (2005); Koshelev (2013); Abrikosov (1998); Khouri et al. (2016); Parish and Littlewood (2003)

In a recent work we gave a more generic explanation of linear resistance [linear both in field and in temperature]. Sacksteder (2018) Our explanation is in terms of quantum interference; it is not specific to a particular microscopic mechanism such as details of the Fermi surface. It does, however, hint that the charge carriers responsible for linear resistance may be confined to move along locally one dimensional pathways.

We argued that linear resistance is caused by the same combination of scattering and quantum interference which produces weak (anti)localization in ordinary materials. Weak (anti)localization expresses the quantum interference which occurs when an electron after several scatterings returns to its starting position, forming a loop. In this circumstance where an electron’s sequence of scatterings forms a loop, a special quantum interference will occur if a hole follows the same sequence of scatterings but in the opposite order. The net effect is that disorder causes electrons and holes to form pairs called Cooperons. It is Cooperon loops [electrons following loops in one direction and holes following in the opposite direction] that mediate weak (anti)localization and alter electronic conduction.

In ordinary 2-D and 3-D materials scattering causes quantum decoherence, so that when a Cooperon completes a loop, it is unable to repeat the same loop again. This limit on Cooperon loops controls the resistance signature of weak localization - for instance in two dimensions the resistance varies logarithmically. We argued that when linear resistance is measured, one can deduce from the linear resistance that Cooperons are repeating their loops, and are not experiencing the quantum decoherence that prohibits loop repetitions in ordinary 2-D and 3-D materials.

The arguments of the above article were based entirely on analysis of the linear profile of the resistance, which does not in itself provide material for discerning which mechanism allows linear resistance materials to maintain quantum coherence even in the presence of scattering. However one way of realizing robust quantum coherence, i.e. of allowing Cooperons to repeat their loops, is if conduction is confined to move along tracks that are locally one-dimensional. There are several known ways to confine electrons to quasi-1-D channels. One such way is thin nanowires: if they are made long enough quantum interference takes over resulting in either Anderson localization or a perfectly conducting channel. Ishii (1973); Simon and Spencer (1989); Beenakker (1997); Ando and Suzuura (2002); Takane (2004) Two other ways of confining electrons to quasi-1-D channels are the edges of topological insulators, and the snake states that are generated when graphene or TIs are subjected to various control mechanisms. Oroszlány et al. (2008); Ghosh et al. (2008); Williams and Marcus (2011) In each of these 1-D realizations quantum interference is very robust, no matter how strong the scattering may be.

In this respect we are reminded of very striking STM images of electron density on the surfaces of Ca_1.88Na_0.12CuO₂Cl₂, Bi₂Sr₂Dy_0.2Ca_0.8Cu₂O_8+δ, Kohsaka et al. (2007) and Bi₂Sr₂(Ca,Dy)Cu₂O_8+x. Parker et al. (2010) These experiments show electron density focused on line segments oriented along the bonds in the CuO₂ plane, which have two directions that are perpendicular to each other. The line segments can be quite long, 15 bonds or more. The STM image reveals that the plane is covered with these line segments, with apparently random length, angular orientation, and placement. Perhaps charge conduction is guided along these quasi-1-D segments, producing the quantum coherence necessary for linear resistance.

If the linear resistance seen in strongly correlated materials, including high $T_{c}$ materials above $T_{c}$, is caused by quasi 1-D conduction, then this is evidence that topology may be at work in these materials. Topology is a very robust way of reducing dimensionality.

There are several qualitative similarities between high-$T_{c}$ superconductors and topological insulators.

Resilience Against Strong Scattering. The first qualitative similarity is that in both classes of materials electronic conduction is resilient against very strong scattering. Even very strong disorder is unable to stop topologically protected conduction on the surface/edge of a TI, as long as the bulk of the TI does not conduct. Chen et al. (2012); Xu et al. (2012a); Sacksteder et al. (2015) Unlike some TIs, high $T_{c}$ materials may not have strong static disorder. However high $T_{c}$ materials, and strongly correlated materials, do have very strong scattering. This is seen very clearly in ARPES studies of strongly correlated materials, which show that most of the spectral weight is not at the Fermi energy, but instead washed out into a smooth background. The smooth background is evidence of very strong scattering. It is remarkable, then, that the cuprates are very good superconductors even though they also show such very strong scattering. The combination of very strong scattering with good conduction suggests that high-$T_{c}$ superconductors may have something in common with TIs.

Linear Resistance. A second qualitative similarity between high-$T_{c}$ materials and topological insulators is that both classes of materials are able to manifest linear resistance. In fact, of the materials where linear magnetoresistance has been reported, most either host Dirac fermions, or it has been suggested that they host Dirac fermions. Some theoretical explanation for this comes from Abrikosov’s quantum theory of linear magnetoresistance, which builds on the foundational assumption that the linear resistance material hosts 3-D Dirac electrons in its bulk. Therefore when linear magnetoresistance is reported in strongly correlated materials, it is natural to look for links between those materials and topological physics.

This sort of reasoning, from linear resistance to topology, has had some success in the case of certain pnictide superconductors where linear magnetoresistance is seen, and where both experimental evidence and theoretical work indicate the presence of Dirac cones and topological phases. Morinari et al. (2010); Richard et al. (2010); Bhoi et al. (2011); Tanabe et al. (2011); Lau and Timm (2013); Pallecchi et al. (2013); Lau and Timm (2014); Sun et al. (2016); Tan et al. (2016); Kanayama et al. (2017); Chen et al. (2017); Terashima et al. (2018); Zhang et al. (2019); Hao and Hu (2019) Much of this work belongs in the category of work that we here call band topology, since the theoretical work is largely based in band theory, and many of the experiments use techniques such as quantum oscillations and ARPES which are essentially probes of band structure. Richard et al. (2010); Tan et al. (2016); Kanayama et al. (2017); Terashima et al. (2018); Zhang et al. (2019). We are suggesting here that the link from linear magnetoresistance to topology may have a broader relevance going beyond band-oriented analysis. The pnictides are encouraging in this respect because they are strongly correlated, and their physics goes well beyond standard band structure, as witnessed by the ongoing discrepancies between ab initio predictions and experimental measurements of the bands. Watson et al. (2016); Coldea and Watson (2018); Rhodes et al. (2020)

Weak Antilocalization and Fast Spin Relaxation. The third and last qualitative similarity between high-$T_{c}$ materials and topological insulators is that the linear resistance always goes up, not down, with magnetic field and with temperature. This shared behavior supplies information about spin physics in both materials. As outlined above, in Ref. Sacksteder, 2018 we have argued that the linear resistance is a manifestation of cooperon physics, the same physics which produces weak localization in ordinary materials. Since the cooperon is a pairing of an electron with a hole, and since both the electron and the hole are spin $1/2$ particles with two states, the cooperon is a $2\times 2$ matrix. The four numbers in the Cooperon matrix encode both electronic charge [ a singlet ] and spin polarization [ a triplet ].

When both the charge singlet and spin triplet components of the Cooperon have similar decay times, then the Cooperon causes the resistance to decrease with magnetic field - this is called weak localization. Hikami et al. (1980); Bergman (1982) In contrast, when the relaxation time of the Cooperon’s spin triplet is short compared to the singlet’s decay time, then the Cooperon causes the resistance to increase with magnetic field - this is called weak antilocalization. When weak antilocalization is observed instead of weak localization, i.e. when the resistance increases not decreases with field, this is an unambiguous signal that the relaxation time of the Cooperon’s spin triplet is quite short. It must be so short that the Cooperon’s spin polarization relaxes to zero before a Cooperon is able to return to its starting point and form a loop.

The reason why in TIs the Cooperon’s spin triplet has a very short relaxation time is that TIs have very strong spin-orbit couplings. Sacksteder et al. (2012) In other words, the spin state of an electron has a very strong impact on the direction of movement that is energetically preferred by the electron. Whenever an electron scatters and therefore changes direction, it also changes its spin state. In TIs the Cooperon’s spin relaxation time is very close to the scattering time, i.e. much smaller than the time required for a Cooperon to return to its origin. We turn briefly to the diffuson, the classical partner to the cooperon. Both the diffuson and the cooperon have the same spin triplet relaxation time, so if the cooperon’s spin relaxation time is short then so is the spin relaxation time of the diffuson. In summary, when weak antilocalization is observed, the role of spin polarization in long distance conduction [either via the cooperon or via the diffuson] is negligible. Chen et al. (2011)

If the linear resistance seen in high-$T_{c}$ materials is really caused by the Cooperon, then it is a kind of weak antilocalization, not weak localization. This in turn implies that the relaxation time of the Cooperon’s spin triplet is quite short - so short that spin polarization plays no part in long distance conduction. Moreover some very fast-acting physics must be responsible for Cooperon spin triplet relaxation in the high-$T_{c}$ materials. These are very interesting qualitative similarities between the high-$T_{c}$ materials and TIs, concerning both spin physics and conduction.

We continue our discussion of Cooperons, which we claim are responsible for the linear resistance [linear both in field and in temperature] seen in high $T_{c}$ materials above the cuperconducting transition. Cooperons are electron-hole pairs whose pairing is induced by scattering centers. Because of this pairing, they are able to conduct charge over distances and times that are much longer than the scattering length and time. This is in contrast to single electrons or holes, whose phase is disrupted at each scattering event and therefore are unable to conduct at scales significantly exceeding the scattering scale. The characteristic scales [both distance and time] of Cooperons are far longer than the scattering scale or the crystalline unit cell.

Because of this difference in scales, the details of the individual scattering events suffered by a Cooperon, and the kinetics of a Cooperon’s movement between scattering centers, do not determine its contribution to weak (anti)localization. These details belong to length and time scales that are much shorter than the Cooperon’s scale, and therefore their only influence on the Cooperon are to determine the diffusion constant, relaxation times, and similar constants describing mixing of charge and spin. Even the shape traced in real space by a Cooperon loop does not matter, because a description of this shape would necessarily refer to individual scattering events. These facts can be verified by computing the perturbative bubble diagram that describes the Cooperon.

There are only three quantities that that affect the Cooperon’s contribution to weak (anti)localization: the area of the loop which it traces, the statistics of which loop areas are more probable, and the relaxation times of its charge singlet and the spin triplet components. In materials showing weak antilocalization, such as linear resistance materials including high $T_{c}$ superconductors, the spin triplet relaxes very quickly leaving only the charge singlet, so there remain only two quantities which determine weak antilocalization: the loop area and the statistics of which loop areas are more probable. This information can be encoded in a probability density $P(A)$ giving the probability that a Cooperon loop has area $A$. Sacksteder (2018)

In materials displaying weak antilocalization the area distribution $P(A)$ contains a complete description of the WAL signature seen in electrical conduction. Magnetic field $B$ couples to a Cooperon loop via a phase factor $\exp(\imath 2\pi AB/\Phi_{0})$, where $\Phi_{0}=h/e$ is the magnetic flux quantum. Therefore, up to a multiplicative constant, the magnetoconductance signal $G(B)$ caused by WAL is equal to the sum of the phase factors of each Cooperon loop, i.e. $G(B)\propto\sum_{loops}\exp(\imath 2\pi A_{loop}B/\Phi_{0})$. This can be rewritten in terms of the area distribution:

This focus on counting loops and their areas is a very peculiar aspect of weak antilocalization, and is very far from the feeling of most other condensed matter physics. Loops are geometrical and topological objects, and the area of a loop is also a geometrical quantity. The shape of the loops is not important, and neither are the details of the Hamiltonian that determines electronic motion. This focus on geometry rather than ordinary physics is suggestive of topology, and in fact topological insulators are one the most significant realizations of weak antilocalization.

Of the two quantities determining weak antilocalization, i.e. Cooperon loop area and the number of Cooperon loops with a particular area, only the former quantity area is dimensionful. The inverse or conjugate of area is two dimensional sheet density. Because of sheet density’s status as the conjugate of area, the special emphasis that weak antilocalization gives to area also applies to sheet density.

One example of the focus on sheet density is equation 1, which is a Fourier transform that starts with a probability distribution $P(A)$ that is a function of area, and ends with magnetoconductance $G(B)$ which is a function of magnetic field. The reason why this equation is mathematically correct is that $B/\Phi_{0}$ has units of sheet density. When loops are the focus of attention, it is natural to expect not only their area but also sheet density to have a key role in determing experimental phenomena.

In this connection it is striking that the cuprates display several linear relations that connect sheet density to other quantities.

1. 1.

   Resistance’s Dependence on Temperature, Magnetic Field, and Hole Sheet Density.

   • •

     Resistance is linearly related to magnetic field in La_2-xSr_xCuO₄ (LSCO), BaFe₂(As_1-xP_x)₂, and Yb_1-xLa_xRh₂Si₂ at temperatures above $T_{c}$. Hayes et al. (2016); Giraldo-Gallo et al. (2017); Kumar et al. (2018) Equation 1 shows that, in the context of weak antilocalization, magnetic field is equivalent to sheet density, so the linear relation between resistance and field can be understood as a linear relation between resistance and sheet density.

   • •

     Resistance is linearly related to temperature in many bad metals, including the ones mentioned above where linearity in field has been reported.

   • •

     Ref. Barišić et al., 2013 found that the coefficient of linear-in-temperature resistance is inversely related to hole doping in four cuprates: La_2-xSr_xCuO₄ (LSCO), YBa₂Cu₃O_6+δ (YBCO), Tl₂Ba₂CuO_6+δ (Tl2201), and HgBa₂CuO_4+δ (Hg1201). Since all four cuprates have the same copper oxide plane with the same cross-sectional area, the linear relation to hole doping is also a linear relation to the sheet density of holes. Performing this conversion from hole doping $p$ to hole sheet density $\rho_{2D}$, Ref. Barišić et al., 2013’s result is $R_{xx}(T)\propto T/\rho_{2D}$.

   • •

     The coefficient of the resistance’s dependence on temperature has been found to be proportional to the coefficient of the resistance’s dependence on magnetic field. Therefore Ref. Barišić et al., 2013’s result implies that $R_{xx}(B)\propto B/\rho_{2D}$, where $\rho_{2D}$ is hole sheet density.

2. 2.

   The Pseudogap Family of Temperatures that Decrease with Hole Sheet Density. The characteristic temperatures found well above $T_{c}$ in the cuprates, including the pseudogap temperature, are linearly related to the sheet density of holes. As discussed earlier, the experimental data sets organize in four straight lines [four lines for LSCO, three or four for YBCO] that are highest at small doping and decrease with doping towards a meeting point near the the high-doping edge of the superconducting dome. This suggests the presence of a sheet density that is high at low hole sheet density and decreases as hole sheet density increases.

3. 3.

   Temperatures that Increase with Hole Sheet Density. Ref. Sacksteder, 2020’s appendix A reports six data sets with characteristic temperatures that increase as the hole sheet density increases. Five of these, measured with ARPES, the thermoelectric power, and the magnetic susceptibility, seem to be direct manifestations of the density of mobile holes because they intercept $T=0$ near the low doping edge of the superconducting dome. Hashimoto et al. (2009); Chatterjee et al. (2011); Kim et al. (2004); Ohsugi et al. (1991) The sixth data set, a pairing temperature measured using magnetic hysteresis, extrapolates to $T=0$ at a lower doping $p=0.023$, presumably because of a sensitivity to pinned holes. Panagopoulos et al. (2006, 2004)

4. 4.

   $T_{c}$’s Dependence on Superfluid Phase Stiffness. In Ref. Božović et al. (2016b), Bozovic et al measured the 2-D superfluid phase stiffness $\rho_{s}$ in LSCO thin films at many different dopings on the overdoped side of the phase diagram. Phase stiffness is given in units of temperature, but this is artificial - the definition of phase stiffness contains a factor of $k_{B}$. Bozovic et al’s phase stiffness experiments were probes not of temperature but instead of a 2-D [sheet] density, which was obtained by measuring the magnetic inductance between two coils. The inductance is sensitive to the length scales of the coils themselves and of the superconducting order parameter in the superconducting film interposed between the coils.

   Bozovic et al reported that $T_{c}$ is linearly related to the $T=0$ value of $\rho_{s}$. Bozovic et al’s data also shows that in most of the superconducting sector of the phase diagram $\rho_{s}$ is linearly related to temperature $T$.

In summary, in the bad metal regime above $T_{c}$ there are linear relations between hole sheet density, magnetic field, temperature as a thermodynamic quantity, and the several characteristic temperatures found above $T_{c}$ including the pseudogap temperature. The linear resistance is controlled by ratios of these quantities. Below $T_{c}$ there is a linear relation between superfluid phase stiffness $\rho_{s}$ and temperature, and $\rho_{s}$’s value at $T=0$ is linearly related to $T_{c}$.

These experimental results show some kind of equivalence between hole sheet density, superfluid phase stiffness which is a sheet density, magnetic field which is a sheet density after division by $\Phi_{0}$, temperature as a thermodynamic quantity, and several characteristic temperatures including both the pseudogap temperature and also $T_{c}$. The linearity of these relations suggests that the equivalence between these quantities may be true in a sense that is mathematically and physically precise.

These results can also be read as saying that hole sheet density controls the phase diagram of the cuprates. As we discussed earlier, sheet density is conjugate to area, and both are geometric quantities. It is very striking to see a geometric quantity such as sheet density or area control the phase diagram. This complete ascendance of geometry over all other kinds of physics, especially at temperatures above $T_{c}$, strongly suggests that topology is a key determinant of the cuprate phase diagram.

The linear relations between sheet density and other quantities bear closer examination, because their units and the numerical values of their slopes indicate that in the cuprates the system of atomic units $m_{e}=e=\hbar=1$, supplemented with $k_{B}=1$, is somehow preferred over other systems of units.

It should be mentioned that even conventional superconductivity has features that seem reminiscent of topology: a contrast between bulk physics and surface physics, and guaranteed conduction on the surface. Where a TI bulk band gap expels charge carriers from the bulk, superconductors expel magnetic field from the bulk. In a TI charge conduction on the surface is guaranteed by topology and the bulk gap, while in a superconductor Meissner currents on the surface are required in order to expel magnetic field from the the bulk.

While a comparison between topology and conventional superconductors is in some respects attractive, this comparison has two fundamental difficulties. The first is that superconductors can not be directly compared to TIs, since one does not allow charge conduction in the bulk and the other does. Perhaps the solution to this difficulty is think of Weyl materials instead of TIs. The second and more profound difficulty is that this sort of reasoning can be used to suggest that ordinary conductors have topological features. Ordinary conductors expel electric fields from their bulk, and use surface charge accumulation to achieve that expulsion.

Band topology is not a terribly promising tool for understanding high $T_{c}$ materials, because these materials are so strongly interacting that the concepts of bands and Fermi surfaces are not reliable. Nonetheless the evidences reviewed in this article suggest that topology may be important. Here we briefly outline a few possible directions for doing topology without band structure. This is not an exhaustive list.

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  Focus on Many-Particle Aspects. One approach is to attack strong interactions head-on: first write an interacting hamiltonian or a many-particle ground state, and then within this context work on developing a topological perspective. One example of this approach is development of many-body topological invariants, where the idea is to develop mathematical machinery for analyzing a multiparticle ground state and determining whether it implements topological order. For examples of this work see Refs. Senthil, 2015; Chiu et al., 2016; Shapourian et al., 2017.

  Another example is the Kitaev model, whose solutions are spin liquids, and are intrinsically topological. The Kitaev model can be solved exactly when it is on a honeycomb lattice. Depending on the lattice, the Kitaev model implements nodal lines, Dirac cones, Majorana Fermi surfaces, or Weyl nodes.Hermanns et al. (2018)

  While this approach is perhaps the most obvious, it might not always be the most productive. The challenges and technology of topology [either in the mathematical sense or the physics sense] are relatively well understood, in comparison to those of strong correlations and interactions. Therefore attacking the combination of topology with strong interactions by concentrating on strong interactions first may deliver practical results more slowly than starting with the aspects of this combination that are relatively tractable.

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  Focus on Broken Translational Invariance and the Absence of Band Structure. Another approach is to focus on how to do topology when translational invariance is broken by scattering and therefore there is no band structure. The source of scattering can be static, i.e. disorder, or it can be dynamic, i.e. interactions. If the characteristic time scale of interactions and their fluctuations is longer than that of electron motion, then the effects of the two types of scattering may be similar.

  Topological properties can be extraordinarily resilient against scattering. This can be observed by directly calculating topological invariants using formalisms which admit formulations that are independent of band structure and can be calculated in disordered materials. Xu et al. (2012b); Yi-Fu et al. (2013); Prodan et al. (2010)

  Resilience against scattering can also be observed by measuring conduction and the energy-resolved density of states, neither of which is tied to translational invariance. The next three points discuss this in more detail:

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    Conduction. Topology can guarantee conduction on the surface of a material even when scattering completely erases the band structure and band gap both on the surface and in the bulk; the only requirement is that the bulk must not conduct, which can be arranged by making it sufficiently disordered. Chen et al. (2012); Xu et al. (2012a); Zhang et al. (2012b)

    Topology can also guarantee conduction in the bulk of a material, if the Fermi energy is tuned to a topological phase transition. Because topological numbers cannot change without conduction through the bulk, at a topological phase transition the bulk is guaranteed to conduct. Chen et al. (2012); Xu et al. (2012a)

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    Density of States as a Function of Energy. The characteristic signature of a Dirac cone [constant density of states for 1-D edge states, linear in energy for 2-D surface states, quadratic in energy for 3-D bulk states] can be maintained by topology even when scattering is so strong that there is no dispersion relation between momentum and energy. In this case the only effect of scattering on the density of states is to multiply by a constant, corresponding to multiplying the Fermi velocity by a constant. Wu et al. (2013); Sacksteder et al. (2015) This result applies within a phase, while at boundaries between two phases non-trivial scaling relations may control the DOS. Kobayashi et al. (2014)

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    Effective Models. In topological insulators the effects of scattering on the density of states and conduction can be summarized by renormalizing the parameters of a disorder-free Dirac hamiltonian. In other words, even when translational invariance is completely absent and there is no dispersion correlating momentum with energy, topology’s invisible hand can ensure that the material acts as if a scattering-free Dirac particle were present. Wu et al. (2013); Sacksteder et al. (2015)

    Weyl materials take this a step further. In 3-D Dirac materials disorder is an irrelevant operator, so its effect on the Weyl cone [except at the Weyl point itself] can be made arbitrarily small by going to large enough distance scales and small enough momentum scales. Shindou and Murakami (2009); Goswami and Chakravarty (2011) For some observables such as conductance disorder may have no effect at all. Kobayashi et al. (2013)

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  Renormalization Group Flow to Weyl Semimetals. A material’s properties at long length scales may be much different than its properties at short length scales. For example, recent work has suggested that materials with spin-orbit coupling, magnetic impurities, and disorder may naturally flow into a Weyl semimetal phase. Kim et al. (2018) Topological insulators with magnetic impurities can also flow into a Weyl semimetal phase as the system’s length scale is increased. Kim et al. (2015) This sort of scenario could produce a Weyl phase and Fermi arcs in the cuprates even if at short distances there are no Dirac or Weyl fermions.

  Moreover when stacks of 2-D TIs are coupled together into a 3-D material, they can form Weyl semimetals or topological insulators. Yoshimura et al. (2016); Kobayashi et al. (2015) If there were something topological at work in individual cuprate copper oxide layers, for instance a Chern-Simons term or some kind of vortices, then possibly when the layers are stacked they could produce a Weyl phase and Fermi arcs. Morinari (2001, 2003); Kou et al. (2005)

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  Electron/hole loops. As briefly outlined earlier in this text, weak (anti)localization is caused by Cooperons tracing loops. We also briefly discussed the fact that electrons and holes form loops in space-time, which gives a perspective on electronic transport that is founded on charge conservation and therefore is very solid. These loops are an elementary topological object. Studies of how the space-time loops of electrons and holes can be braided or knotted or woven together may be able to offer new insight into strongly correlated electron transport.

  We used an analysis of Cooperon loops and their areas to understand linear resistance. A key step in the analysis was that weak antilocalization not weak localization is seen experimentally, which implies that spin relaxes very quickly, long before a loop is completed. This obtained a simplification: in the case of weak antilocalization, the loops are spinless. Further work is needed on generalizing this loop analysis to systems where spin polarization persists longer and its effects on transport can not be neglected.