Probing topological superconductors with emergent gravity

Probing topological superconductors with emergent gravity

Omri Golan    Ady Stern Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
July 14, 2019
Abstract

We study the interplay of three phenomena that occur in 2+1 dimensional -wave superconductors. The first is emergent gravity, where interacting fermions behave effectively as fermions interacting with a form of gravity, which is described by the superconducting order parameter. The second is topological superconductivity, the existence of nontrivial topological phases, characterized by a Chern number, of the effectively free fermions in the emergent gravitational background. The third is the spontaneous breaking of symmetry, which is the statement that the number of effectively free fermions is not conserved. We show that an energy-momentum response of the fermions to the emergent gravitational background reveals their topological phase. This response is encoded in a gravitational Chern-Simons (gCS) term and is intimately related to the existence of chiral Majorana fermions on domain walls between different topological phases, via energy-momentum conservation, or gravitational anomaly inflow. Thus emergent gravity is a natural probe for topological phases of superconductors, and provides a physical interpretation for the boundary gravitational anomaly and bulk gCS term in this context. The spontaneous breaking of symmetry introduces additional bulk responses, encoded in a gravitational pseudo Chern-Simons term. Although not topological in nature, these responses carry surprising similarities to the topological responses encoded in the gCS term.

I Introduction

In this paper we study spin-less -wave superconductors (SC) in 2+1 dimensions. These are superconductors that can be thought of as microscopically comprised of charge 1 spin-less fermions with an attractive two body interaction. The interaction is such that it can efficiently be described as an interaction with a charge 2 spin 1 boson, which is the superconducting order parameter, as in BCS theory. This boson represents a condensate of Cooper pairs, where the fermions in a pair have relative orbital angular momentum 1, as opposed to -wave SC, where the relative orbital angular momentum is 0. -wave pairing has been experimentally observed in thin films of superfluid He-3 Vollhardt and Wölfle (2013), and there are many solid state candidates Sato and Ando (2016). Another notable candidate is the fractional quantum Hall state which has been proposed to be a -wave SC of composite fermions Moore and Read (1991); Read and Green (2000).

One can postpone the treatment of the quantum fluctuations of the order parameter, and think of free fermions coupled to a background boson. This point of view is also appropriate when superconductivity is induced by proximity to an -wave SC, as in some proposals for the realization of -wave SC Sato and Ando (2016). The fermion-boson coupling takes the generic long-wavelength form , where is a fermionic creation operator and is the order parameter. Here runs over spatial coordinates and is the hermitian conjugate.

At this free-fermion level, -wave SC are known to realize gapped topological phases Read and Green (2000), or symmetry protected topological phases (SPT) Wang et al. (2017), and we will deal with the problem of probing and characterizing these phases. The most notable known manifestation of the existence of distinct topological phases, is the formation of chiral Majorana (Majorana-Weyl) fermions on spatial domain walls between different phases, and Majorana bound states, or zero modes, in the cores of vortices Read and Green (2000). These Majorana bound states exhibit non abelian braiding statistics, and this led to the hope of using these as building blocks for topological quantum computation Kitaev (2003); Nayak et al. (2008). This seems to be the main drive behind the intense research of -wave SC in recent years.

Though this paper deals with -wave SC, it will be useful to keep in mind the integer quantum Hall effect (IQHE) as a well understood and closely related example. Both systems are 2+1 dimensional. Depending on convention, one either says that the -wave SC does not have any symmetries and that the IQHE has symmetry Kitaev (2009), or that the -wave SC has particle-hole symmetry (symmetry class D) while the IQHE has no symmetries (symmetry class A) Schnyder et al. (2008); Ryu et al. (2010). Unusually, despite the different symmetries, the SPT phases in both systems are labelled by the same topological invariant which is the integer valued Chern number . Within band theory it can be calculated by where is the Barry curvature on the Brillouin zone Ryu et al. (2010). A more general definition is 111More explicitly, ., where is the single particle propagator Volovik (2009). This formula remains valid in the presence of weak interactions as long as the gap does not close, where the free propagator is replaced by the full propagator, dressed by interactions. Therefore the topological phases of both the -wave SC and the IQHE are robust to weak interactions.

What is the physical meaning of the invariant ? In other words, how can be measured? The main goal of this paper is to provide an answer for this question in the context of the -wave superconductor.

For the IQHE is the quantized Hall conductivity in units of Thouless et al. (1982); Avron et al. (1983); Golterman et al. (1993); Qi et al. (2008); Mera (2017). In more detail, the effective action for a background gauge field (corresponding to electromagnetism) contains a Chern-Simons (CS) term , where we have set .

The CS term entails much of the physics of the IQHE as a gapped topological phase. Its functional derivative gives the bulk response which includes the Hall current in response to an electric field , and the density response to a magnetic field . The CS term is not gauge invariant on a sample with a boundary, and so gauge invariance requires a boundary term in the effective action. Such a boundary term must be nonlocal, and can be interpreted as the effective action corresponding to massless 1+1 dimensional chiral (Weyl) fermions Manes et al. (1985); Bertlmann (2000); Kaplan (1992); Shamir (1993); Chandrasekharan (1994). The action for such fermions is gauge invariant, but the corresponding effective action is not. This is the gauge anomaly Bertlmann (2000). Its physical implication is that the boundary current is conserved classically , but not quantum mechanically, . Here, is the electric field parallel to the boundary and is the net chirality of the boundary fermions. Bulk+boundary charge conservation is encoded in the usual in the bulk and additionally on the boundary. Thus the anomaly corresponds to the inflow of charge from the bulk, and bulk+boundary charge conservation implies (or on a domain wall between phases where jumps by ). The algebraic equation is referred to as bulk-boundary correspondence. This is the anomaly inflow mechanism, which recasts what appears to be charge non-conservation in a 1+1 dimensional system, as charge conservation in a 2+1 dimensional system with a boundary Callan and Harvey (1985); Naculich (1988); Harvey and Ruchayskiy (2001).

The story above answers quite beautifully the question of the physical meaning of for the IQHE by relating it to bulk and boundary observables, and only makes use of the space-time dimension and the symmetry group . The relation between anomaly inflow and topological phases is much more general. It has been suggested, and to a large extent shown, that the existence of anomalies in dimensions is equivalent to the existence of corresponding topological phases in dimensions, related by the anomaly inflow mechanism Ryu et al. (2012); Witten (2015). Moreover, since anomalies are known to be robust to weak interactions, they naturally classify topological phases of weakly interacting fermions Witten (2015). In many instances the anomaly also suggests a natural bulk response that only depends on the topological phase, analogous to the Hall conductivity in the IQHE. We refer to such a response as a topological bulk response.

Let us now go back to the -wave SC. Here, on domain walls where the Chern number jumps by , one finds chiral Majorana fermions with net chirality Read and Green (2000); Stone and Roy (2004); Volovik (2009). A Chiral Majorana fermion in dimensions does not carry charge. Thus there is no gauge anomaly and no corresponding bulk CS term, or quantized Hall conductivity Read and Green (2000); Stone and Roy (2004)222Though there is no quantized Hall conductivity in a -wave SC, there is in fact a Hall conductivity, which we discuss in more detail in our conclusions.. In fact, the only charge such a fermion does carry is energy-momentum, associated with space-time symmetries. The only relevant anomaly is therefore the gravitational anomaly, where energy and momentum are conserved classically, but not quantum mechanically, on a “nontrivial” gravitational background. Chiral Majorana fermions in 1+1 dimensions indeed possess such an anomaly. Here, a gravitational background is ”nontrivial” if it has a non constant curvature. Again, the gravitational anomaly in 1+1 dimensions can be interpreted as the inflow of energy-momentum from a 2+1 dimensional bulk with an appropriate Chern-Simons term. Based on these facts it was argued in Read and Green (2000); Ryu et al. (2012) that a gravitational Chern-Simons (gCS) term

 α∫tr(~Γd~Γ+23~Γ3) (1)

with coefficient should arise when integrating out the bulk fermions in a -wave SC, where is the Christoffel symbol corresponding to the metric (this term will be elaborated on in the main text). A similar statement was made in Wang et al. (2011), regarding the closely related boundaries of dimensional time-reversal invariant SC (class DIII), with half the quantization condition. The gCS term then describes a topological bulk response to the background metric, from which the Chern number can in principle be measured, and bulk-boundary correspondence follows from energy-momentum conservation in the presence of a nontrivial metric. One arrives at the appealing conclusion that a -wave SC is a manifestation of the gravitational anomaly inflow mechanism.

The obvious question that arises is what does it mean to couple a -wave SC to gravity. One must find some probe that couples to the fermions as gravity, at least at low energies. Of course, complex fermions also carry energy-momentum, so the above also applies to the IQHE, with . The difference is that for the IQHE there is also the simpler charge that one can utilize, whereas in the -wave SC the only relevant anomaly is the gravitational anomaly333We note that for spin-full -wave SC, one can exploit spin rotation symmetry, and does not have to resort to a gravitational probe Volovik and Yakovenko (1989); Read and Green (2000); Stone and Roy (2004)..

The most natural approach it to simply use real geometry, by curving the 2 dimensional sample in 3 dimensional space. This works well for the IQHE, at least if one starts with a continuum model, and has led to a remarkable body of work on geometric responses of integer and fractional quantum Hall states Abanov and Gromov (2014); Can et al. (2014); Klevtsov and Wiegmann (2015); Klevtsov et al. (2015); Bradlyn and Read (2015a); Can et al. (2015). For the -wave SC, understanding the effect of real geometry is more complicated, and we will come back to this point in the discussion, section IX.

In previous work, gravity was introduced through a space dependent temperature, and the corresponding bulk response was suggested to be a quantized bulk thermal Hall conductivity Read and Green (2000); Wang et al. (2011); Ryu et al. (2012). The motivation for this suggestion is two fold. First, there is an argument due to Luttinger that shows that the thermal conductivity is essentially given by the response of a system to a gravitational field Luttinger (1964); Cooper et al. (1997). Second, there is a well known derivation of the thermal Hall conductance (as opposed to conductivity) for 2+1 dimensional topological phases with chiral boundaries, where one ignores the bulk, and considers a strip geometry with two counter propagating boundaries, at slightly different temperatures. If the boundaries can be described by a CFT, one finds Cappelli et al. (2002) where is the (average) temperature and is the chiral central charge of the boundary, given by for the IQHE and by for the -wave SC. Using bulk-boundary correspondence one obtains in terms of the bulk Chern number and the temperature, which is analogous to . This thermal Hall conductance was indeed measured recently in several experiments in quantum Hall systems Jezouin et al. (2013); Banerjee et al. (2017a, b). One may then hope to obtain the same result, now for the bulk thermal conductivity, from the gCS term, by using Luttinger’s argument. However, it has been shown that this cannot be the case, because gCS term is third order in derivatives of the metric, as opposed to a single derivative of the temperature required for a thermal conductivity Stone (2012); Bradlyn and Read (2015b). Some authors argue that there is a quantized bulk thermal Hall conductivity, but relate it to other gravitational terms, which are first order in derivatives Qin et al. (2011); Shitade (2014); Nakai et al. (2016), and to global gravitational anomalies Nakai et al. (2017), which will not be discussed in this paper. Other authors argue that there is no bulk thermal conductivity at all Bradlyn and Read (2015b).

We note that on general grounds, the relation between thermal conductivity and conductance is more subtle than the relation between electric conductivity and conductance. First, while there are longitudinal and transverse electric fields, there is no transverse driving force for heat. Second, if one expects a heat current to require the presence of entropy, there cannot be a bulk heat current as long as the temperature is negligible compared with the bulk gap.

Luckily, there is another probe which couples to the fermions in a -wave SC as gravity, which is quite natural. This is the order parameter itself, as was discovered by Volovik (see e.g Volovik (2009)), and refined by Read and Green Read and Green (2000). We refer to this identification as emergent gravity, because the order parameter arises microscopically from a fermionic two-body interaction. In fact, using this observation, Volovik suggested early on the existence of a gCS term in a -wave SC Volovik (1990).

To gain some intuition for the identification of the order parameter with gravity, note that, almost by definition, gravity is a probe that couples to energy-momentum. The pairing term shows that couples to fermionic derivatives, related to fermionic momentum. More accurately, we will see that the operator appears in the energy-momentum tensor of a -wave SC. The mapping of the order parameter onto gravity is the conceptual starting point of our analysis, which is motivated by the search for topological bulk responses in the -wave SC.

Outline of this paper: Our main results along with simple examples are given in section II. In section III we start our analysis with a simple lattice model for a -wave SC, which we view as microscopic. We describe the topological phase diagram of the model and also explain some ingredients of the emergent geometry which are visible at this level. In section IV we derive a continuum description of the lattice model, which is an even number of -wave superfluids (SF). In the limit where the order parameter is much larger than the single particle scales, each -wave SF maps precisely to a relativistic Majorana spinor coupled to Riemann-Cartan (RC) geometry, which is a geometry with both curvature and torsion. We discuss the mapping of fields, actions, equations of motion, path integrals, symmetries, conservation laws, and observables in sections V and VI, and in appendices A-F.

The rest of the paper is devoted to the application of the above mapping to the problems described above: finding topological bulk responses of the -wave SC, and relating them to edge anomalies. In section VII we discuss bulk responses. We verify that the effective action obtained by integrating over the bulk fermions contains a gCS term, with coefficient , and we obtain the corresponding topological bulk response of the -wave SC. We also find closely related terms, which do not encode topological bulk responses, and are unrelated to edge anomalies. The first, which we refer to a gravitational pseudo Chern-Simons term, is possible due to the spontaneous breaking of symmetry, or in other words, due to the emergent torsion. The second is a difference of two gCS terms, which appears because the different low energy Majorana spinors do not experience the same order parameter, or in other words, the same gravitational background. The calculation of the effective action within perturbation theory is done in appendix H. In section VIII we describe the edge states, focusing on the physical implication of their gravitational anomaly in the -wave SC, and the relation to the topological bulk response from gCS, via the anomaly inflow mechanism. We conclude and discuss our results in section IX. Tables 1-3 list our notation, and may be useful for the reader. In particular, Tab.1 serves as a quick guide for the mapping of the -wave SF to a Majorana spinor in RC geometry.

Ii Approach and main results

ii.1 Model and approach

As a microscopic starting point, we consider a simple model for a spin-less -wave SC on a square lattice, described in section III. We analyze the model in the regime where the order parameter is much larger than the single particle scales, which we refer to as the relativistic regime. In this regime the model is essentially a lattice regularization of four, generically massive, relativistic Majorana spinors, centered at the particle-hole invariant points in the Brillouin zone. Around each of these four points the low energy description is given by a Hamiltonian , which we refer to as a -wave superfluid (SF) Hamiltonian, with an effective mass 444The effective mass tensor is actually different for the different particle-hole invariant points, but this will not be important in the following., chemical potential , and order parameter , which is in the configuration , where and are constants. In the relativistic regime the effective mass is large, and in the limit one obtains a relativistic Hamiltonian, with mass . This becomes clear in terms of the Nambu spinor , which is a Majorana spinor. We refer to the sign as the orientation, and we note that the different Majorana spinors, associated with the four particle-hole invariant points, have different orientations and masses , where . The th Majorana spinor contributes to the Chern number, and summing over one obtains the Chern number of the lattice model , which gives the topological phase diagram in terms of the low energy data .

In order to probe this topological phase diagram, we perturb the order parameter out of the configuration, and treat as a general space-time dependent field, close to the configuration. This is analogous to applying an electromagnetic field in order to probe the topological phase diagram of the IQHE.

Following Volovik Volovik (2009), and Read and Green Read and Green (2000), we show that fermionic excitations in each -wave SF experience such a general order parameter as a non trivial gravitational background. Some of this gravitational background is described by the (inverse) metric

 gij=−Δ(iΔj)∗, (2)

where brackets denote the symmetrization, and the sign is a matter of convention. We refer to as the Higgs part of the order parameter. Parameterizing with the overall phase and relative phase , the metric is independent of and of the orientation , which splits order parameters into -like and -like. Note that in the configuration the metric is euclidian, . For our purposes it is important that the metric be perturbed out of this form, and in particular it is not enough to take the configuration with a space-time dependent phase .

Before we turn to describe the conclusions that may be drawn from this emergent gravity, we find it instructive to draw analogies to the electromagnetic topological responses of the IQHE.

ii.2 Electromagnetic response in the IQHE and gravitational response in the p-wave SC - analogies and differences

It is illuminating to examine the gravitational response of the -wave SC that we consider with an eye opened to the electromagnetic response of the IQHE. The defining characteristics of the IQHE, the absence of longitudinal conductivity and the quantization of the Hall conductivity, imply a coupling of charge density to magnetic field. A small local increase of the magnetic field from to results in a small local increase of density by . This density accumulates as is turned on, as a consequence of the Hall current that results from the electric field generated when the magnetic field varies. It does not disperse with time, since the bulk is gapped. Since charge is conserved, the density must be supplied by the edges, which forces a correspondence of the bulk and edge responses. As explained in the introduction, this chain of events is encompassed by the bulk CS term, and the corresponding edge anomaly.

Roughly speaking, in gravitational response the role of the magnetic field is played by the curvature, while the role of the vector potential is played by the spin connection, which is first order in derivatives of the inverse metric . Thus, the emergent curvature involves two derivatives of the order parameter (see, e.g., (II.3.1) below). The effect of these derivatives becomes evident when considering the gravitational analog to various electromagnetic vector potentials. For example, the vector potential associated with the Aharonov-Bohm effect decays as , with being the distance from the Aharonov-Bohm flux tube. The analogous spin connection requires the perturbation to the order parameter to scale like .

The observable that responds to the spin connection may be the electronic density and current, but it may also be the density and current of momentum, or energy. A crucial difference between the IQHE and the -wave SC, the absence of fermionic charge conservation in the latter, leads to profound differences between the bulk responses of both systems. In the absence of charge conservation, charge accumulation in the bulk does not necessarily involve the edges, and thus the way is opened to bulk Hall-type responses that do not correlate with the edge, and do not have quantized coefficients. There is a known example for such a response: when a weak magnetic field is introduced into a -wave SC, the fermionic density receives a correction Volovik (1988); Goryo and Ishikawa (1998, 1999); Stone and Roy (2004); Roy and Kallin (2008); Hoyos et al. (2014); Ariad et al. (2015), yet with a proportionality constant that is not quantized. In this paper we find an additional example, where the fermionic density receives a correction proportional to the emergent curvature. These responses originate from bulk terms that carry some similarity to Chern-Simons terms, which we refer to as pseudo Chern-Simons terms, see (129) and (131).

ii.3 Bulk responses

ii.3.1 Topological bulk responses from a gravitational Chern-Simons term

We find that the effective action obtained by integrating over the bulk fermions in the presence of a general order parameter contains a gCS term, with coefficient . Although we obtain this result in the limit , we expect it to hold throughout the phase diagram. This is based on known arguments for the quantization of the coefficient due to symmetry, and on the relation with the boundary gravitational anomaly described below.

The gCS term implies a topological bulk response (90), where energy-momentum currents and densities appear due to a space-time dependent order parameter. To gain insight into this result it is best to examine special cases. Assume that the order parameter is time independent, and that the relative phase is , as in the configuration, so that , . Then the metric is time independent, and takes the simple form

 gij=−(|Δx|200|Δy|2). (3)

On this background, we find the following contributions to the expectation values of the fermionic energy current , and momentum density 555 () is the density of the () component of momentum.,

 ⟨JiE⟩gCS = −ν/296π1ℏεij∂j~R, (4) ⟨Pi⟩gCS = −ν/296πℏgikεkj∂j~R.

Here is the curvature, or Ricci scalar, of the metric , which is the inverse of , and . These are written without setting or an effective speed of light to 1 as we do in the bulk of the paper. The curvature for the above metric is given explicitly by

 ~R = −2|Δx||Δy| ×

It is a nonlinear expression in the order parameter, which is second order in derivatives. Thus the responses (4) are third order in derivatives, and start at linear order but include nonlinear contributions as well. The first equation in (4) is analogous to the response of the IQHE. The second equation is analogous to the dual response . Unlike the case of charge density, where the role of the magnetic field is played by the curvature (see Eq.(9) below), for the case of the momentum density it is played by curvature gradients. Note that the dependence on the sign in , which is the orientation of , hides in the Chern number which is an odd function of the orientation. The above responses are odd under time-reversal, which flips the orientation of the order parameter but leaves the metric intact 666The correct notion of time reversal for the -wave SC flips but not , as opposed to the natural time reversal within the relativistic description. This is discussed in appendix E..

Since there is no time dependence, energy is strictly conserved , and it is meaningful to discuss energy transport. Integrating over any cross section of the sample (a spatial curve that starts and ends on the boundary of the sample) we find the net bulk energy current through the cross section

 ⟨IE⟩gCS = ∫γ⟨JiE⟩gCSdli = ν/296π1ℏ[~R(γ1)−~R(γ0)],

where is a length element perpendicular to the curve, and are its end points.

As an example, consider the order parameter which is a perturbation to the configuration with . The scalar curvature for this order parameter is so there will be an energy current in the direction, . If we assume that the system occupies the strip between to , as depicted in Fig.1, we get the net bulk energy current in the direction,

 ⟨IE⟩gCS=ν/296π1ℏ2ϵΔ0L2+O(ϵ2). (7)

The factor only depends on the Chern number, and thus on the topological phase, and is a quantity that only depends on the order parameter. Note that the non-linear nature of the curvature leads to a dependence of the energy current on both the perturbation scale and the magnitude of the unperturbed order parameter . The topological invariant can then be measured in a thought experiment where one tunes the order parameter as in the example and preforms a measurement of the above contribution to . In this manner a physical meaning is assigned to , while avoiding the conceptual difficulties of thermal responses.

ii.3.2 Additional bulk responses from a gravitational pseudo Chern-Simons term

Apart from the gCS term, the effective action obtained by integrating over the bulk fermions also contains an additional term of interest, which we refer to as a gravitational pseudo Chern-Simons term (gpCS). This term is written explicitly and explained in section VII.2.2. To the best of our knowledge, the gpCS term has not appeared previously in the context of the -wave SC. It is possible because symmetry is spontaneously broken in the -wave SC. In the geometric point of view, this translates to the emergent geometry in the p-wave SC being not only curved but also torsion-full, see section V.

The gpCS term produces bulk responses which are closely related to those of gCS, despite it being fully gauge invariant. This gauge invariance implies that it is not associated with a boundary anomaly, nor does its coefficient need to be quantized. Hence, gpCS does not encode topological bulk responses. Remarkably, we find that is quantized and identical to the coefficient of the gCS term in the limit of , but do not expect this value to hold outside of this limit. We will to put this phenomenon in a broader context in the discussion, section IX.

Let us now describe the bulk responses from gpCS, setting . First, we find the following contributions to the fermionic energy current and momentum density,

 ⟨JiE⟩gpCS = ν/296πεij∂j~R, (8) ⟨Pi⟩gpCS = −ν/296πgikεkj∂j~R.

Up to the sign difference in the first equation, these responses are the same as those from gCS (4).

As opposed to gCS, the gpCS term also contributes to the fermionic charge density . For the bulk responses we have written thus far, every Majorana spinor contributed , and summing over produced the Chern number . For the density response this is not the case. Here, the th Majorana spinor contributes

 ⟨ρ⟩gpCS=onνn/224π√g~R, (9)

where is the emergent volume element. The orientation in Eq. (9) makes the sum over the four Majorana spinors different from the Chern number, . The appearance of can be understood by considering the effect of time reversal. Because both the density and the curvature are time reversal even, the coefficient in (9) must also be even, and cannot be which is odd. The response (9) also holds when the order parameter is time dependent, in which case will also contain time derivatives. One then finds a time dependent density, but there is no corresponding current response, which is due to the non-conservation of fermionic charge in a superconductor. It is instructive to compare (9) to the response of the IQHE. Here is time reversal odd, which is why the coefficient can be the Chern number , and there is also the corresponding current such that , as opposed to the -wave SC.

To gain some insight into the expressions we have written thus far, we write the operators more explicitly. For each Majorana spinor (suppressing the index ),

 Pj = i2ψ†←→Djψ, (10) JjE = gjkPk+o2∂k(1√gεjkρ)+O(1m∗).

These expressions can be understood from the gravitational description of the -wave SC, see section VI.2. The momentum density is the familiar expression for free fermions, but in the energy current we have only written explicitly contributions that survive the limit . These contributions are only possible due to the -wave pairing, and are of order .

From the relation (10) between , and we can understand that the equality expressed in equation (4) is a result of the vanishing contribution of gCS to the density . We can also understand the sign difference between the first and second line of (8) as a result of (9). The important point is that a measurement of the charge density can be used to fix the value of the coefficient , which is generically unquantized, and thus separate the contributions of gpCS to , from those of gCS. In this manner, one can overcome the obscuring of gCS by gpCS. A more detailed analysis is given in section VII.4.

ii.4 Bulk-boundary correspondence from gravitational anomaly

Among the two terms in the bulk effective action which we described above only gCS is related to the boundary gravitational anomaly. This relation can be fully analyzed in the case where is a perturbation of the configuration with small , and there is a domain wall (or boundary) at where the value of jumps. For simplicity, assume for and for . This situation is illustrated in Fig.1. In section VIII we derive the action for the boundary, or edge mode,

 Se=i2∫dtdx~ξ(∂t−|Δx(t,x)|∂x)~ξ, (11)

which describes a chiral Majorana fermion localized on the boundary, with a space-time dependent velocity . Classically, the edge fermion conserves energy-momentum in the following sense,

 ∂βtβeα+∂αLe=0. (12)

Here is the canonical energy-momentum tensor for , with indices , and is the edge Lagrangian, , see VI.1.2. For (), equation (12) describes the sense in which the edge fermion conserves energy (momentum) classically. The source term follows from the space-time dependence of through . Quantum mechanically, the action is known to have a gravitational anomaly, which means that energy-momentum is not conserved at the quantum level Bertlmann (2000). In the context of emergent gravity, this implies that equation (12) is violated for the expectation values,

 (13)

This equation is written with and for simplicity. Since depends on time, is not the curvature of the spatial metric , but of a corresponding space-time metric (V), and is given by in this case. Note that time dependence in this example is crucial. From gCS we find for the bulk energy-momentum tensor

 ⟨tyα⟩gCS=−ν/296πgαγεγβy∂β~R, (14)

which explains the anomaly as the inflow of energy-momentum from the bulk to the boundary,

 (15)

Since jumps from 1 to 0 at the energy-momentum current (14) stops at the boundary and does not extend to the region. The gravitationally anomalous boundary mode is then essential for the conservation of total energy-momentum to hold. As this example shows, bulk-boundary correspondence follows from bulk+boundary conservation of energy-momentum in the presence of a space-time dependent order parameter.

Iii Lattice model

In this section we review and slightly generalize a simple lattice model for a -wave SC Bernevig and Hughes (2013), which will serve as our microscopic starting point. We describe its band structure and its symmetry protected topological phases, and also explain some of the basics of the emergent geometry which can be seen in this setting.

The hamiltonian is given in real space by

 H= − 12∑l[tψ†lψl+x+tψ†lψl+y+μψ†lψl (16) + δxψ†lψ†l+x+δyψ†lψ†l+y+h.c].

Here the sum is over all lattice sites of a 2 dimensional square lattice , with a lattice spacing . are creation and annihilation operators for spin-less fermions on the lattice, with the canonical anti commutators . denotes the nearest neighboring site to in the direction. The hopping amplitude is real and is the chemical potential. Apart from the single particle terms , there is also the pairing term , with the order parameter . We think of as resulting from a Hubbard-Stratonovich decoupling of interactions, in which case we refer to it as intrinsic, or as being induced by proximity to an -wave SC. In both cases we treat as a bosonic background field that couples to the fermions.

The generic order parameter is charged under a few symmetries of the single particle terms. The order parameter has charge 2 under the global group generated by , in the sense that , which physically represents the electromagnetic charge of Cooper pairs777Since has charge 2, commutes with the fermion parity . The Ground state of will therefore be labelled by a fermion parity eigenvalue , in addition to the topological label which is the Chern number Read and Green (2000); Kitaev (2009). Fermion parity is a subtle quantity in the thermodynamic limit, and will not be important in the following.. The order parameter is also charged under time reversal , which is an anti unitary transformation satisfying , that acts as the complex conjugation of coefficients in the Fock basis corresponding to . The equation shows under time reversal. Finally, is also charged under the point group symmetry of the lattice, which for the square lattice is the Dihedral group . The continuum analog of this is that the order parameter is charged under spatial rotations and reflections, and more generally, under space-time transformations (diffeomorphisms), which is due to the orbital angular momentum 1 of Cooper pairs in a -wave SC. This observation will be important for our analysis, and will be discussed further below.

In an intrinsic SC, the configuration of which minimizes the ground state energy is given by , where is determined by the minimization, but the sign and the phase (which dynamically corresponds to a goldstone mode) are left undetermined. See Volovik (2009) for a pedagogical discussion of a closely related model within mean field theory. A choice of and corresponds to a spontaneous symmetry breaking of the group including both the and time reversal transformations. More accurately, in the SC, the group is spontaneously broken down to a certain diagonal subgroup. We discuss the continuum analog of this and its implications in section VI.1.2.

Crucially, we do not restrict to the configuration, and treat it as a general two component complex vector . In the following we will take to be space time dependent, , and show that this space time dependence can be thought of as a perturbation to which there is a topological response, but for now we assume is constant.

iii.1 Band structure and phase diagram

Writing the Hamiltonian (16) in Fourier space, and in the BdG form in terms of the Nambu spinor we find

 H =12∫BZd2q(2π)2Ψ†q(hqδqδ∗q−hq)Ψq+const =12∫BZd2q(2π)2Ψ†q(dq⋅σ)Ψq+const, (17)

with real and symmetric, and complex and anti-symmetric. Here is the vector of Pauli matrices and is the Brillouin zone . By definition, the Nambu spinor obeys the reality condition , and is therefore a Majorana spinor, see appendix E.1. Accordingly, the BdG Hamiltonian is particle-hole (or charge conjugation) symmetric, , and therefore belongs to symmetry class D of the Altland-Zirnbauer classification of free fermion Hamiltonians Ryu et al. (2010). The constant in (17) is where is the infinite volume. This operator ordering correction is important as it contributes to physical quantities such as the energy density and charge density, but we will mostly keep it implicit in the following. The BdG band structure is given by where

 Eq=|dq|=√h2q+|δq|2. (18)

For the configuration , and therefore can only vanish at the particle-hole invariant points , which happens when . Representative band structures are plotted in Fig.4. For the spectrum takes the form of a gapped single particle Fermi surface with gap , while for one obtains Four regulated relativistic fermions centered at the points with masses , speed of light , bandwidth and momentum cutoff .

With generic the spectrum is gapped, and the Chern number labeling the different topological phases is well defined. For two band Hamiltonians such as (17) , defined in the introduction, reduces to the homotopy type of the map from (which is a flat torus) to the sphere,

 ν=a2(2π)2∫BZd2q^dk⋅(∂qy^dq×∂qy^dq)∈Z. (19)

One obtains for , for and for . The topological phase diagram is plotted in Fig.7.

Away from the configuration, the topological phase diagram is essentially unchanged. For , gap closings happen at the same points and the same values of described above. takes the same values, with the orientation , described below, generalizing the sign that characterizes the configuration. For the spectrum is always gapless. The topological phase diagram, plotted in Fig.7, is most easily understood from the formula where are orientations associated with the relativistic fermions which we describe below Sticlet (2012).

It will also be useful consider a slight generalization of the single particle part of the lattice model, with un-isotropic hopping . This changes the masses to . In particular, the degeneracy between the masses breaks, and additional trivial phases appear around . See Fig.7.

iii.2 Basics of the emergent geometry

A key insight which we will extensively use, originally due to Volovik, is that the order parameter is in fact a vielbein. In the present space-time independent situation, this vielbein is just a matrix which generically will be invertible

 ejA=(Re(δx)Re(δy)Im(δx)Im(δy))∈GL(2), (20)

where . More accurately, is invertible if . We refer to an order parameter as singular if . From the vielbein one can calculate a metric, which in the present situation is a general symmetric positive semidefinite matrix

 gij=eiAδABejB =δ(iδj)∗ (21)

Every vielbein determines a metric uniquely, but the converse is not true. Vielbeins that are related by an internal reflection and rotation with give rise to the same metric. By diagonalization, it is also clear that any metric can be written in terms of a vielbein. Therefore the set of (constant) metrics can be parameterized by the coset . To see this explicitly we parameterize with the overall phase and relative phase . Then

 gij=(|δx|2|δx||δy|cosϕ|δx||δy|cosϕ|δy|2) (22)

is independent of which parametrizes and which parametrizes . Note that the group of internal rotations and reflections is just acting on . In more detail, (or ) corresponds to with

 L=(cos2αsin2α−sin2αcos2α)(