Boundary central charge from bulk odd viscosity - chiral superfluids

TL;DR

This paper develops a low energy effective field theory for chiral superfluids, revealing how the boundary chiral central charge influences bulk odd viscosity and proposing measurable bulk observables for its detection.

Contribution

It introduces a novel effective field theory linking boundary chiral central charge to bulk odd viscosity in superfluids, with practical measurement proposals.

Findings

Odd viscosity depends on boundary chiral central charge.

Bulk observables can measure the boundary chiral central charge.

Effective theory matches microscopic model calculations.

Abstract

We derive a low energy effective field theory for chiral superfluids, which accounts for both spontaneous symmetry breaking and fermionic ground-state topology. Using the theory, we show that the odd (or Hall) viscosity tensor, at small wave-vector, contains a dependence on the chiral central charge $c$ of the boundary degrees of freedom, as well as additional non-universal contributions. We identify related bulk observables which allow for a bulk measurement of $c$. In Galilean invariant superfluids, only the particle current and density responses to strain and electromagnetic fields are required. To complement our results, the effective theory is benchmarked against a perturbative computation within a canonical microscopic model.

Equations

$${\eta }_{\text{o}}^{\left(1\right)}=-\left(\hslash /2\right)s\cdot n_0,s\in \mathbb{Q},$$

$$U{\left(1\right)}_N\times SO{\left(2\right)}_L\to U{\left(1\right)}_{L-\left(\ell /2\right)N},$$

$$\delta {\eta }_{\text{o}}^{\left(1\right)}\left(\mathbf{q}\right)=-\frac{c}{24}\frac{1}{4\pi }{q}^2.$$

$$\delta A_i=-{\xi }^k{\partial }_k{A}_i-A_k{\partial }_i{\xi }^k+m{G}_{ij}{\partial }_t{\xi }^j.$$

$$\frac{1}{\sqrt{G}}{\partial }_t\left(\sqrt{G}m{J}^i\right)+{\nabla }_j{T}^{ji}=n{E}_i+{\epsilon }^{ij}{J}_{j}B,$$

$${\nabla }_{\mu }\theta ={\partial }_{\mu }\theta -A_{\mu }-s_{\theta }{\omega }_{\mu },$$

$${\nabla }_{\mu }\psi =\left({\partial }_{\mu }+i{A}_{\mu }+i{s}_{\psi }{\omega }_{\mu }\right)\psi .$$

$${\partial }_{\mu }=O\left(p\right),A_{\mu },G_{ij}=O\left(1\right),\theta =O\left(p^{-1}\right).$$

$${\mathcal{L}}_0=P\left(X\right),X={\nabla }_t\theta -\frac{1}{2m}{G}^{ij}{\nabla }_i\theta {\nabla }_j\theta ,$$

$${\mathcal{L}}_2=$$

$$+F_2\left(X\right){\left[m{K}_i^i-{\nabla }^2\theta \right]}^2$$

$$+F_3\left(X\right)\left[2m\left({\nabla }_i{K}_j^j-{\nabla }^j{K}_{ji}\right){\nabla }^i\theta \right]+\cdots ,$$

$${\mathcal{L}}_3\supset {\text{L}}_{\text{gCS}}=-\frac{c}{48\pi }\omega \text{d}\omega ,$$

$$\frac{\nu }{4\pi }\left(A\text{d}A-2\overline{s}\omega \text{d}A+\overline{s^2}\omega \text{d}\omega \right).$$

$$S_{\text{m}}=\int \text{mbox}{d}^{2}x$$

$$+\left(\frac{1}{2}{\Delta }^j{\psi }^{†}{\nabla }_j{\psi }^{†}+h.c\right)-\frac{1}{2\lambda }{G}_{ij}{\Delta }^{i\ast }{\Delta }^j],$$

$$c=-\left(\ell /2\right)\Theta \left(\mu \right)\in \left\{0,\pm 1/2\right\}.$$

$${\Delta }^j$$

$$\frac{P^{\prime \prime }}{m}=$$

$$m{F}_2=$$

$$c=$$

$${\eta }_{\text{o}}\left(\omega ,\mathbf{q}\right)=$$

$${\eta }_{\text{o}}^{\left(1\right)}\left(q^2\right)=$$

$${\eta }_{\text{o}}^{\left(2\right)}\left(q^2\right)=$$

$${\tilde{\eta }}_{\text{o}}^{\left(1\right)}\left(q^2\right)=$$

$${\tilde{\eta }}_{\text{o}}^{\left(2\right)}\left(q^2\right)=$$

$${\tilde{\chi }}_{TJ,\text{o}}^{ij,k}=-\frac{i}{m}\frac{c}{48\pi }{q}_{\perp }^i{q}_{\perp }^j{q}_{\perp }^k+O\left(q^4\right),$$

$${\tilde{S}}_{\text{ind}}\left[A_{\mu },u_{ij}\right]=S_{\text{ind}}\left[A_{\mu }-\left(\ell /2\right){\omega }_{\mu },u_{ij}\right],$$

$${\tilde{\eta }}_{\text{o}}^{ij,kl}=$$

$$+\frac{i\ell }{4}\left({\kappa }_{\text{e}}^{ij,(k}{q}_{\perp }^{l)}-{\kappa }_{\text{e}}^{kl,(i}{q}_{\perp }^{j)}\right)+\frac{{\ell }^2}{16}{\sigma }_{\text{o}}{q}_{\perp }^{(i}{\epsilon }_{}^{j)(k}{q}_{\perp }^{l)},$$

$${\tilde{\chi }}_{TJ,\text{o}}^{ij,k}=$$

$$+\frac{\ell }{2}i{q}_{\perp }^{(i}{\chi }_{JJ,\text{e}}^{j),k}+\frac{{\ell }^2}{8m}{q}_{\perp }^{(i}{\chi }_{JJ,\text{o}}^{j),t}{q}_{\perp }^k,$$

$$G^{ij}=$$

$$=$$

$$\sqrt{G}=$$

$$\log \sqrt{G}=$$

Full text

I Introduction

The odd (or Hall) viscosity $\eta_{\text{o}}$ is a non-dissipative, time reversal odd, stress response to strain-rate (Avron et al., 1995; Avron, 1998; Bradlyn et al., 2012; Hoyos, 2014; Lapa and Hughes, 2014), which can appear even in superfluids (SFs) and incompressible (or gapped) fluids, where the more familiar dissipative viscosity vanishes. Observable signatures of $\eta_{\text{o}}$ are actively studied in a variety of systems (Lucas and Surówka, 2014; *PhysRevB.94.125427; *PhysRevLett.119.226602; *PhysRevLett.118.226601; *PhysRevB.96.174524; *PhysRevFluids.2.094101; *banerjee2017odd; *bogatskiy2018edge; *holder2019unified; *PhysRevLett.122.128001), and recently led to its measurement in a colloidal fluid (Soni et al., 2018) and in graphene (Berdyugin et al., 2019).

In isotropic 2+1 dimensional fluids, the odd viscosity tensor at zero wave-vector ($\mathbf{q}=\mathbf{0}$) reduces to a single component. In analogy with the celebrated quantization of the odd (or Hall) conductivity in the quantum Hall (QH) effect (Thouless et al., 1982; *avron1983homotopy; *golterman1993chern; *qi2008topological; *Nobel-2016; *mera2017topological), this component obeys a quantization condition

[TABLE]

in incompressible quantum fluids (Avron et al., 1995; Read, 2009; Read and Rezayi, 2011). Here $n_{0}$ is the ground state density, and $s$ is a rational topological invariant labeling the many-body ground state, which corresponds to the average angular momentum per particle (in units of $\hbar$, henceforth set to 1).

Remarkably, Eq.(1) also holds in certain compressible quantum fluids, which are the subject of this paper. These are chiral superfluids (CSFs), where the ground state is a condensate of Cooper pairs of fermions, which are spinning around their centre of mass with an angular momentum $\ell\in\mathbb{Z}$ (Read and Green, 2000; Volovik, 2009; Vollhardt and Wölfle, 2013), see Fig.1(a). Thin films of ${}^{3}\text{He-A}$ are experimentally accessible $p$-wave ($\ell=\pm 1$) CSFs (Levitin et al., 2013; *PhysRevLett.109.215301; *Ikegami59; *Zhelev_2017), and there are proposals for the realization of various CSFs in cold atoms (Zhang et al., 2008; *PhysRevLett.103.020401; *PhysRevA.86.013639; *PhysRevA.87.053609; *PhysRevLett.115.225301; *BOUDJEMAA20171745; *Hao_2017). Closely related chiral superconductors (Kallin and Berlinsky, 2016; *Sato_2017) have recently been realized (Xu et al., 2015; *M_nard_2017), and some of the most debated fractional QH states (Wang et al., 2018; *PhysRevLett.121.026801; *PhysRevB.97.121406; *PhysRevB.98.167401; *PhysRevB.99.085309) are believed to be CSFs of composite fermions (Read and Green, 2000; Son, 2015; *Son_2018).

Computing $\eta_{\text{o}}^{\left(1\right)}$ in an $\ell$-wave CSF, one finds Eq.(1) with the intuitive $s=\ell/2$ (Read, 2009; Read and Rezayi, 2011; Hoyos et al., 2014; Shitade and Kimura, 2014; Moroz and Hoyos, 2015). Thus, a measurement of $\eta_{\text{o}}^{\left(1\right)}$ at $\mathbf{q}=\mathbf{0}$ can be used to obtain the angular momentum of the Cooper pair, but carries no additional information.

An $\ell$-wave pairing involves the spontaneous symmetry breaking (SSB) of time reversal $T$ and parity (spatial reflection) $P$ down to $PT$, and of the symmetry groups generated by particle number $N$ and angular momentum $L$ down to a diagonal subgroup

[TABLE]

which implies a single Goldstone field, charged under the broken generator $N+\left(\ell/2\right)L$, as well as massive Higgs fields (Brusov and Popov, 1981; *Volovik_2013; *Sauls:2017aa; *hsiao2018universal; Golan and Stern, 2018). For CSFs, it is this SSB pattern, rather than ground-state topology, which implies the quantization $s=\ell/2$ (Hoyos et al., 2014) 111As can be seen by considering a mixture of CSFs with different $\ell$s, where $N$ and $L$ are completely broken, and $s\equiv-2\eta_{\text{o}}^{\left(1\right)}/n=\sum_{i}n_{i}\left(\ell_{i}/2\right)/\sum_{i}n_{i}$ is no longer quantized. .

Nevertheless, a CSF with fixed $\ell$ does have a non-trivial ground-state topology - single fermion excitations are gapped, and the fermionic ground state can be assigned a topological invariant. This is the boundary chiral central charge $c\in\left(\ell/2\right)\mathbb{Z}$ (per spin component) (Read and Green, 2000; Volovik, 2009), which counts the net chirality of 1+1 dimensional Majorana spinors present on the boundary between the CSF and vacuum. For example, a $p$-wave CSF comprised of spin-less fermions has a minimal non-vanishing $c=\pm 1/2$, while a $d$-wave ($\ell=\pm 2$) CSF, which requires spin-full fermions, has a minimal non vanishing $c=\pm 2$, or $c=\pm 1$ per spin component.

The invariant $c$ determines the boundary gravitational anomaly (Alvarez-Gaume and Witten, 1984; bertlmann2000anomalies; bastianelli2006path), and the boundary thermal Hall conductance (Kane and Fisher, 1997; Read and Green, 2000; Cappelli et al., 2002), which has been measured in recent experiments on QH and spin systems (Jezouin et al., 2013; *banerjee2017observed; *banerjee2018observation; *Kasahara:2018aa). Based on the fundamental principle of anomaly inflow (Kraus and Larsen, 2006; Ryu et al., 2012; Stone, 2012; Witten, 2016) it is expected that $c$ can be measured in the bulk of a CSF, but whether this is indeed the case, and if so, what should actually be measured, has so far remained unclear. Providing an answer to this question is the main goal of the present paper.

Analysis of the problem has previously been carried out only within the relativistic limit of the $p$-wave CSF, where the non-relativistic kinetic energy of the fermions is neglected (Volovik, 1990; Read and Green, 2000; Wang et al., 2011; Ryu et al., 2012; Bradlyn and Read, 2015a; Golan and Stern, 2018). Within this limit one finds a bulk gravitational Chern-Simons (gCS) term, which implies a $c$-dependent correction to $\eta_{\text{o}}^{\left(1\right)}$ of (1) at small non-zero wave-vector (Abanov and Gromov, 2014; Klevtsov and Wiegmann, 2015; Bradlyn and Read, 2015a),

[TABLE]

One is therefore led to suspect that $c$ can be obtained from the $q^{2}$ correction to $\eta_{\text{o}}$, but the fate of this correction beyond the relativistic limit remains unclear.

In particular, the relativistic limit misses most of the physics of the Goldstone field (Golan and Stern, 2018). Analysis of the Goldstone physics in CSFs was undertaken in (Volovik, 1988a; *goryo1998abelian; *goryo1999observation; *furusaki2001spontaneous; *stone2004edge; *roy2008collective; *lutchyn2008gauge; Ariad et al., 2015). More recently, Refs.(Hoyos et al., 2014; Moroz and Hoyos, 2015) considered CSFs in curved (or strained) space, following the pioneering work (Son and Wingate, 2006) on $s$-wave ($\ell=0$) SFs. These works demonstrated that the Goldstone field, owing to its charge $L+\left(\ell/2\right)N$, produces the $\mathbf{q}=\mathbf{0}$ odd viscosity (1), and it is therefore natural to expect that a $q^{2}$ correction similar to (3) will also be produced. Nevertheless, Refs.(Hoyos et al., 2014; Moroz and Hoyos, 2015) did not consider the derivative expansion to the high order at which $q^{2}$ corrections to $\eta_{\text{o}}$ would appear, nor did they detect any bulk signature of $c$ at lower orders.

In this paper we obtain a low energy effective field theory that captures both SSB and fermionic ground state topology, which extends and unifies the aforementioned results of (Son and Wingate, 2006; Hoyos et al., 2014; Moroz and Hoyos, 2015) and (Volovik, 1990; Read and Green, 2000; Wang et al., 2011; Ryu et al., 2012; Bradlyn and Read, 2015a; Golan and Stern, 2018). Using the theory we compute the $q^{2}$ correction to $\eta_{\text{o}}$, and provide several routes towards the bulk measurement of the boundary central charge in CSFs.

We note that there is an ongoing discussion in the literature regarding a possible bulk thermal Hall conductivity proportional to $c$, including some contradicting results (Qin et al., 2011; *shitade2014heat; *PhysRevLett.114.016802; *nakai2016finite; *nakai2017laughlin; *kapustin2019thermal; Bradlyn and Read, 2015b). This provides further motivation to study the appearance of $c$ in the bulk odd viscosity.

II Building blocks of the effective field theory

In order to probe a CSF, we minimally couple it to two background fields - a time-dependent spatial metric $G_{ij}$, which we use to apply strain $u_{ij}=\left(G_{ij}-\delta_{ij}\right)/2$ and strain-rate $\partial_{t}u_{ij}$, and a $U\left(1\right)_{N}$-gauge field $A_{\mu}=\left(A_{t},A_{i}\right)$, where we absorb a chemical potential $A_{t}=-\mu+\cdots$. The microscopic action $S$ is then invariant under $U\left(1\right)_{N}$ gauge transformations, implying the number conservation $\partial_{\mu}(\sqrt{G}J^{\mu})=0$, where $\sqrt{G}J^{\mu}=-\delta S/\delta A_{\mu}$. It is also clear that $S$ is invariant under spatial diffeomorphisms generated by $\delta x^{i}=\xi^{i}\left(\mathbf{x}\right)$, if $G_{ij}$ transforms as a tensor and $A_{\mu}$ as a 1-form. Less obvious is the fact that a Galilean invariant fluid is additionally symmetric under $\delta x^{i}=\xi^{i}\left(t,\mathbf{x}\right)$, provided one adds to the transformation rule of $A_{i}$ a non-standard mass-dependent piece (Son and Wingate, 2006; Hoyos and Son, 2012; Hoyos et al., 2014; Gromov and Abanov, 2014; Andreev, 2014; Geracie et al., 2015; Andreev, 2015; Geracie, 2017),

[TABLE]

We refer to $\delta x^{i}=\xi^{i}\left(\mathbf{x},t\right)$ as local Galilean symmetry (LGS), as it can be viewed as a local version of the Galilean transformation $\delta x^{i}=v^{i}t$. The LGS implies the momentum conservation law

[TABLE]

where $\sqrt{G}T^{ij}=2\delta S/\delta G^{ij}$ is the stress tensor and the right hand side is the Lorentz force. This fixes the momentum density $P^{i}=mJ^{i}$ - a familiar Galilean relation.

Since CSFs spontaneously break the rotation symmetry in flat space, in order to describe them in curved, or strained, space, it is necessary to introduce a background vielbein. This is a field $E_{\;j}^{A}$ valued in $GL\left(2\right)$, such that $G_{ij}=E_{\;i}^{A}\delta_{AB}E_{\;j}^{B}$, where $A,B\in\left\{1,2\right\}$. For a given metric $G$ the vielbein $E$ is not unique - there an internal $O\left(2\right)_{P,L}=\mathbb{Z}_{2,P}\ltimes SO\left(2\right)_{L}$ ambiguity, or symmetry, acting by $E_{\;j}^{A}\mapsto O_{\;B}^{A}E_{\;j}^{B}$, $O\in O\left(2\right)_{P,L}$. The generators $L,P$ correspond to internal spatial rotations and reflections, and are analogs of angular momentum and spatial reflection (parity) on the tangent space. The inverse vielbein $E_{B}^{\;j}$ is defined by $E_{\;j}^{A}E_{B}^{\;j}=\delta_{B}^{A}$.

The charge $N+\left(\ell/2\right)L$ of the Goldstone field $\theta$ implies the covariant derivative

[TABLE]

with a geometric spin $s_{\theta}=\ell/2$. Here $\omega_{\mu}$ is the non-relativistic spin connection, an $SO\left(2\right)_{L}$-gauge field which is $E^{A}_{\;j}$-compatible, see Appendix A. So far we assumed that the microscopic fermion $\psi$ does not carry a geometric spin, $s_{\psi}=0$, which defines the physical system of interest. It will be useful, however, to generalize to $s_{\psi}\in\left(1/2\right)\mathbb{Z}$, where the covariant derivative of the fermion is

[TABLE]

A non-zero $s_{\psi}$ modifies the geometric spin of $\theta$ to $s_{\theta}=s_{\psi}+\ell/2$, and the unbroken generator in (2) to $L-s_{\theta}N$. In the special case $s_{\psi}=-\ell/2$ the Cooper pair is geometrically spin-less and $L$ is unbroken, as in an $s$-wave SF, see Fig.1(b). This $s_{\theta}=0$ CSF is, however, distinct from a conventional $s$-wave SF, because $P$ and $T$ are still broken down to $PT$, and we therefore refer to it as a geometric $s$-wave (g$s$-wave) CSF, to distinguish the two. In particular, a central charge $c\neq 0$, which is $P,T$-odd, is not forbidden, and is in fact independent of $s_{\psi}$. This makes the g$s$-wave CSF particularly useful for our purposes.

We note that $\omega_{\mu}$ transforms as a 1-form under LGS only if $B/2m$ is added to $\omega_{t}$ (Hoyos et al., 2014; Moroz and Hoyos, 2015), which we do implicitly throughout the paper. For $\psi$, this is equivalent to adding a g-factor $g_{\psi}=2s_{\psi}$ (Geracie et al., 2015).

III Effective field theory

Based on the above characterization of CSFs, the low energy, long wave-length, behavior of the system can be captured by an effective action $S_{\text{eff}}\left[\theta;A,G\right]$, obtained by integrating out all massive degrees of freedom - the single fermion excitations and the Higgs fields. In this section we describe a general expression for $S_{\text{eff}}$, compatible with the symmetries, SSB pattern, and ground state topology of CSFs.

The effective action can be written order by order in a derivative expansion, with the power counting scheme (Son and Wingate, 2006; Hoyos et al., 2014)

[TABLE]

The spin connection is a functional of $G_{ij}$ that involves a single derivative (see Eq.(25)), so $\omega_{\mu}=O\left(p\right)$. Denoting by $\mathcal{L}_{n}$ the term in the Lagrangian which is $O\left(p^{n}\right)$ and invariant under all symmetries, we have $S_{\text{eff}}=\sum_{n=0}^{\infty}\int\text{d}^{2}x\text{d}t\sqrt{G}\mathcal{L}_{n}$. The desired $q^{2}$ corrections to $\eta_{\text{o}}$ are $O\left(p^{3}\right)$, which poses the main technical difficulty.

The leading order Lagrangian

[TABLE]

was studied in (Hoyos et al., 2014), and contains the earlier results of (Volovik, 1988a; Goryo and Ishikawa, 1998, 1999; Furusaki et al., 2001; Stone and Roy, 2004; Roy and Kallin, 2008; Lutchyn et al., 2008). Here $X$ is the unique $O\left(1\right)$ scalar, which reduces to the chemical potential $\mu$ in the ground state(s) $\partial_{\mu}\theta=0$, and $P$ is an arbitrary function of $X$ that physically corresponds to the ground state pressure $P_{0}=P\left(\mu\right)$. The function $P$ also determines the ground state density $n_{0}=P^{\prime}\left(\mu\right)$, and the leading dispersion of the Goldstone mode $\omega^{2}=c_{s}^{2}q^{2}$, where $c_{s}^{2}=\partial_{n_{0}}P_{0}/m=P^{\prime}/P^{\prime\prime}m$ is the speed of sound, squared. For $\ell\neq 0$, the spin connection appears in each $\nabla\theta$ (6), and so $\mathcal{L}_{0}$ includes $O\left(p\right)$ contributions, which produce the leading odd viscosity and conductivity, discussed below. There are no additional terms at $O\left(p\right)$, so that $\mathcal{L}_{1}=0$ (Hoyos et al., 2014).

At $O\left(p^{2}\right)$ one has

[TABLE]

where $K_{ij}=\partial_{t}G_{ij}/2$ and $R$ are the extrinsic curvature and Ricci scalar of the spatial slice at time $t$ (Carroll, 2004), the $F$s are arbitrary functions of $X$, and dots indicate additional terms which do not contribute to $\eta_{\text{o}}$ up to $O\left(p^{2}\right)$, see Appendix C.2 for the full expression. The Lagrangian $\mathcal{L}_{2}$ was obtained in (Son and Wingate, 2006) for $s$-wave SFs. For $\ell\neq 0$ the spin connection in $\nabla\theta$ produces $O\left(p^{3}\right)$ contributions to $\mathcal{L}_{2}$, and, in turn, non-universal $q^{2}$ corrections to $\eta_{\text{o}}$.

The term $\mathcal{L}_{3}$ is the last ingredient required for reliable results at $O\left(p^{3}\right)$. Most importantly, it includes the (non-relativistic) gCS term (Chern and Simons, 1974; Jackiw and Pi, 2003; Kraus and Larsen, 2006; Witten, 2007; Pérez, 2010; Stone, 2012; Bradlyn and Read, 2015a; Gromov et al., 2016)

[TABLE]

where the $c$-dependence is required to match the boundary gravitational anomaly (Kraus and Larsen, 2006; Stone, 2012; Golan and Stern, 2018), and $\omega\text{d}\omega=\varepsilon^{\mu\nu\rho}\omega_{\mu}\partial_{\nu}\omega_{\rho}$. Unlike the lower order terms, $\mathcal{L}_{\text{gCS}}$ is independent of $\theta$, and encodes only the response of the gapped fermions to the background fields. In Appendices B.5 and C.4 we argue that additional terms in $\mathcal{L}_{3}$ do not produce $q^{2}$ corrections to $\eta_{\text{o}}$.

There are three topological terms that can be added to $S_{\text{eff}}$ (Ferrari and Klevtsov, 2014; Can et al., 2014; Abanov and Gromov, 2014; Gromov and Abanov, 2014; Gromov et al., 2015; Can et al., 2015; Klevtsov and Wiegmann, 2015; Bradlyn and Read, 2015a; Klevtsov et al., 2016; Klevtsov, 2017; Cappelli and Maffi, 2018). These are the $U\left(1\right)$ Chern-Simons (CS) and first and second Wen-Zee (WZ1, WZ2) terms, which can be added to $\mathcal{L}_{1}$, $\mathcal{L}_{2}$, $\mathcal{L}_{3}$, respectively 222These need to be modified as in (Hoyos and Son, 2012; Andreev, 2014, 2015) for LGS. ,

[TABLE]

As our notation suggests, WZ2 and gCS are identical for the purpose of local bulk responses, of interest here, but the two are globally distinct (Bradlyn and Read, 2015a; Gromov et al., 2016; Cappelli and Maffi, 2018). Based on symmetry, and ignoring boundary physics, the independent coefficients $\nu$, $\nu\overline{s},\nu\overline{s^{2}}$ obey certain quantization conditions (Witten, 2007), but are otherwise unconstrained. The absence of a boundary $U\left(1\right)_{N}$-anomaly then fixes $\nu=0$ (Golan and Stern, 2018), but leaves $\nu\overline{s},\nu\overline{s^{2}}$ undetermined (Bradlyn and Read, 2015a; Gromov et al., 2016; Cappelli and Maffi, 2018). One can argue that a Chern-Simons term can only appear for the unbroken generator $L-s_{\theta}N$, so that $\nu=0$ implies $\nu\overline{s}=\nu\overline{s^{2}}=0$. Moreover, in the following section we will see that a perturbative computation within a canonical model for $\ell=\pm 1$ shows that $\nu\overline{s}=\nu\overline{s^{2}}=0$, which applies to any deformation of the model (which preserves the symmetries, SSB pattern, and single fermion gap), due to the quantization of $\nu\overline{s},\nu\overline{s^{2}}$. Accordingly, we set $\nu\overline{s}=\nu\overline{s^{2}}=0$ in the following.

IV Benchmarking the effective theory against a microscopic model

In this section we take a complementary approach and compute $S_{\text{eff}}$ perturbatively, starting from a canonical microscopic model for a spinless $p$-wave CSF. The perturbative computation verifies the general expression in a particular example, and determines the coefficients of topological terms which are not completely fixed by symmetry. It also gives one a sense of the behavior of the coefficients of non-topological terms as a function of microscopic parameters. Here we will outline the computation and describe its results, deferring many technical details to Appendix E.

The microscopic model is given by

[TABLE]

where $\nabla_{\mu}\psi=\left(\partial_{\mu}+iA_{\mu}\right)\psi$, so $s_{\psi}=0$. Apart from the standard non-relativistic kinetic term, the action includes the simplest attractive two-body interaction (Volovik, 1988b; Quelle et al., 2016), mediated by the complex vector $\Delta^{i}$, the order parameter, with coupling constant $\lambda>0$.

For a given $\Delta^{j}$, the fermion $\psi$ is gapped, unless the chemical potential $\mu$ or chirality $\ell=\text{sgn}\left(\text{Im}\left(\Delta^{x}\Delta^{y*}\right)\right)$ are tuned to 0, and forms a fermionic topological phase characterized by the boundary chiral central charge (Read and Green, 2000; Volovik, 2009; Ryu et al., 2010)

[TABLE]

An effective action $S_{\text{eff},\text{m}}\left[\Delta;A,G\right]$ for $\Delta^{j}$ in the background $A_{\mu},G_{ij}$ is then obtained by integrating over the fermion. The subscript "m" indicates that this is obtained from the particular microscopic model $S_{\text{m}}$. Since Eq.(13) is quadratic in $\psi,\psi^{\dagger}$, obtaining $S_{\text{eff},\text{m}}$ is formally straightforward, and leads to a functional Pfaffian.

To zeroth order in derivatives, the action $S_{\text{eff},\text{m}}$ is given by a potential for $\Delta^{i}$, which is minimized by the $p_{x}\pm ip_{y}$ configurations. In flat space these are given by the familiar $\Delta^{j}\partial_{j}=\Delta_{0}e^{-2i\theta}\left(\partial_{x}\pm i\partial_{y}\right)$. Here $\Delta_{0}$ is a fixed function of $m,\mu$ and $\lambda$, determined by the minimization, while the phase $\theta$ and chirality $\ell=\pm 1$ are undetermined. In order to write down the $p_{x}\pm ip_{y}$ configurations in curved space it is necessary to use a background vielbein Read and Green (2000); Hoyos et al. (2014); Moroz and Hoyos (2015); Moroz et al. (2016); Quelle et al. (2016),

[TABLE]

Fluctuations of $\Delta$ away from these configurations correspond to massive Higgs modes, which should in principle be integrated out to obtain a low energy action $S_{\text{eff},\text{m}}\left[\theta;A,G\right]$ that can be compared with the general $S_{\text{eff}}$ of the previous section. We will simply ignore these fluctuations, and obtain $S_{\text{eff},\text{m}}\left[\theta;A,G\right]$ by plugging Eq.(15) into $S_{\text{eff},\text{m}}\left[\Delta;A,G\right]$. This will suffice as a derivation of $S_{\text{eff}}$ from a microscopic model. A proper treatment of the massive Higgs modes will only further renormalize the coefficients we find, apart from the central charge $c$.

To practically compare the actions $S_{\text{eff}}$ and $S_{\text{eff},\text{m}}$ we expand them in fields, to second order around $\theta=0,A_{\nu}=-\mu\delta_{\nu}^{t},G_{ij}=\delta_{ij}$, and in derivatives, to third order, see Appendices C and E. Equating these two double expansions leads to an overdetermined system of equations for the phenomenological parameters in $S_{\text{eff}}$ in terms of the microscopic parameters in $S_{\text{m}}$, with a unique solution. In particular, we find the dimensionless parameters

[TABLE]

where $\kappa=\left|\mu\right|/m\Delta_{0}^{2}>0$, and the upper and lower values refer to $\mu>0$ and $\mu<0$ respectively. We note that for $\mu>0$ there is a single particle Fermi surface with energy $\varepsilon_{F}=\mu$ and wave-vector $k_{F}=\sqrt{2m\mu}$, which for small $\lambda$ will acquire an energy gap $\varepsilon_{\Delta}=\Delta_{0}k_{F}\ll\varepsilon_{F}$. In this weak-coupling regime, it is natural to parametrize the coefficients in (16) using the small parameter $\varepsilon_{\Delta}/\varepsilon_{F}=\sqrt{2/\kappa}$.

The coefficient $P^{\prime\prime}$ determines the leading odd (or Hall) conductivity and has been computed previously in the literature (Volovik, 1988a; Goryo and Ishikawa, 1998, 1999; Furusaki et al., 2001; Stone and Roy, 2004; Roy and Kallin, 2008; Lutchyn et al., 2008; Ariad et al., 2015), while $F_{1},F_{2}$ and $F_{3}$, to the best of our knowledge, have not been computed previously, even for an $s$-wave SF.

Crucially, Eq.(16) shows that the coefficient $c$ of the bulk gCS term (11) matches the known boundary central charge (14). It follows that there is no WZ2 term in $S_{\text{eff},\text{m}}$, so $\nu\overline{s^{2}}=0$, in accordance with the previous section. We additionally confirm that $\nu=\nu\overline{s}=0$. The direct confirmation of the gCS term and its coefficient within a non-relativistic microscopic model has been anticipated for some time (Volovik, 1990; Read and Green, 2000; Wang et al., 2011; Ryu et al., 2012; Golan and Stern, 2018), and is the main result of the perturbative computation.

A few additional comments regarding Eq.(16) are in order:

The seeming quantization of $P^{\prime\prime}/m$ and $F_{1}^{\prime}$ for $\mu>0$ is a non-generic result, as was shown explicitly for $P^{\prime\prime}/m$ (Ariad et al., 2015). 2. 2.

The free fermion limit $\kappa\rightarrow\infty$, or $\Delta_{0}\rightarrow 0$, of certain coefficients in (16) diverges for $\mu>0$ but not for $\mu<0$. This signals the breakdown of the gradient expansion for a gapless Fermi surface, but not for gapped free fermions. 3. 3.

The opposite limit, $\kappa\rightarrow 0$, or $m\rightarrow\infty$, is the relativistic limit mentioned above, in which the fermionic part of the model reduces to a 2+1 dimensional Majorana spinor with mass $\mu$ and speed of light $\Delta_{0}$, coupled to Riemann-Cartan geometry described by $\Delta^{i},\;A_{\mu}$, and in which $S_{\text{eff},\text{m}}$ was already computed (Golan and Stern, 2018; Hughes et al., 2013). Accordingly, the limit $\kappa\rightarrow 0$ of (16) indeed reproduces the results of (Golan and Stern, 2018; Hughes et al., 2013) in a suitable sense, see Appendix E.

V Induced action and linear response

Having derived and benchmarked the effective theory, we are now in a position to obtain linear response functions, in particular the $q^{2}$ corrections to the odd viscosity, and related observables that allow for the bulk measurement of $c$.

By expanding $S_{\text{eff}}$ to second order in the fields $\theta,A_{t}-\mu,A_{i},u_{ij}$, and performing Gaussian integration over $\theta$, we obtain an induced action $S_{\text{ind}}\left[A_{\mu},u_{ij}\right]$ that captures the linear response of CSFs to the background fields, see Appendix D for explicit expressions. Taking functional derivatives, one obtains the expectation values $J^{\mu}=G^{-1/2}\delta S_{\text{ind}}/\delta A_{\mu}$, $T^{ij}=G^{-1/2}\delta S_{\text{ind}}/\delta u_{ij}$ of the current and stress, and from them the conductivity $\sigma^{ij}=\delta J^{i}/\delta E_{j}$, the viscosity $\eta^{ij,kl}=\delta T^{ij}/\delta\partial_{t}u_{kl}$, and the mixed response function $\kappa^{ij,k}=\delta T^{ij}/\delta E_{k}=\delta J^{k}/\delta\partial_{t}u_{ij}$. We will also need the static susceptibilities $\chi_{JJ}^{\mu,\nu},\;\chi_{TJ}^{ij,\nu}$, defined by restricting to time independent $A_{\mu},u_{ij}$, and computing $\delta J^{\mu}/\delta A_{\nu}$ and $\delta J^{\nu}/\delta u_{ij}$, respectively.

Before computing $\eta_{\text{o}}$, it is useful to restrict its form based on dimensionality and symmetries: space-time translations, spatial rotations, and $PT$. The analysis is performed in Appendices B.1-B.4, and results in the expression

[TABLE]

written in the basis $\sigma^{ab}=2\sigma^{[a}\otimes\sigma^{b]}$ of anti-symmetrized tensor products of the symmetric Pauli matrices (Avron, 1998). As components of the strain tensor, the matrices $\sigma^{x},\sigma^{z}$ correspond to shears, while the identity matrix $\sigma^{0}$ corresponds to a dilatation. The details of the system are encoded in two independent coefficients $\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)}\in\mathbb{C}$, which are functions of $\omega,q^{2}$. At $\mathbf{q}=\mathbf{0}$ the odd viscosity tensor reduces to a single component, $\eta_{\text{o}}\left(\omega,\mathbf{0}\right)=\eta_{\text{o}}^{\left(1\right)}\left(\omega\right)\sigma^{xz}$, as is well known (Avron et al., 1995; Avron, 1998; Bradlyn et al., 2012; Hoyos, 2014; Lapa and Hughes, 2014). The additional component $\eta_{\text{o}}^{\left(2\right)}$ has not been discussed much in the literature (Abanov and Gromov, 2014; Hoeller and Read, 2018), and also appears in the presence of (pseudo-)vector anisotropy (Peña Benitez et al., 2019; Offertaler and Bradlyn, 2019), in which case $\mathbf{q}$ should be replaced by a background (pseudo-)vector $\mathbf{b}$. The expression (17) applies at finite temperature, out of equilibrium, and in the presence of disorder that preserves the symmetries on average. For clean systems at zero temperature, $\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)}$ are both real, even functions of $\omega$. In gapped systems $\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)}$ will usually be regular at $\omega=0=q^{2}$, though exceptions to this rule have recently been found (Rose et al., ).

For the CSF, we find the $\omega=0$ coefficients

[TABLE]

where $C^{\left(1\right)},C^{\left(2\right)}\in\mathbb{R}$ are generically non-zero, and are given by particular linear combinations of the dimensionless coefficients $F_{1}^{\prime}\left(\mu\right),mF_{2}\left(\mu\right)$ and $mF_{3}\left(\mu\right)$ defined in (10), see Appendix D for more details.

The leading term in $\eta_{\text{o}}^{\left(1\right)}$ is the familiar Eq.(1), which also appears in gapped states, while the non-analytic leading term in $\eta_{\text{o}}^{\left(2\right)}$ occurs because the superfluid is gapless, and does not appear when $q\rightarrow 0$ at $\omega\neq 0$ (Hoyos et al., 2014). Both leading terms obey the same quantization condition due to SSB, and are independent of $c$. The sub-leading corrections to both $\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)}$ contain the quantized gCS contributions proportional to $c$, but also the non-universal coefficients $C^{\left(1\right)},C^{\left(2\right)}$. Thus $c$ cannot be extracted from a measurement of $\eta_{\text{o}}$ alone.

Noting that the non-universal sub-leading corrections to $\eta_{\text{o}}$ originate from the geometric spin $s_{\theta}=\ell/2$ of the Goldstone field, one is naturally led to consider the g$s$-wave CSF, where $s_{\theta}=0$ and the odd viscosity is, to leading order in $q$, purely due to $\mathcal{L}_{\text{gCS}}$,

[TABLE]

Here and below we use $O$ and $\tilde{O}$, for the quantity $O$ in the CSF and in the corresponding $g$s-wave CSF, respectively. Equation (19) follows from (18) by setting $s_{\theta}=0$, but can be understood directly from $S_{\text{eff}}$. Indeed, for the g$s$-wave CSF, $S_{\text{eff}}$ is identical to that of the conventional $s$-wave SF to $O\left(p^{2}\right)$, but contains the additional $\mathcal{L}_{\text{gCS}}$ at $O\left(p^{3}\right)$, which produces (19).

Due to the LGS (4)-(5), the viscosity (19) implies also

[TABLE]

where $q_{\perp}^{i}=\varepsilon^{ij}q_{j}$, and the subscript “o” (“e”) refers to the $P,T$-odd (even) part of an object. Thus, a steady $P,T$-odd current $\tilde{J}_{\text{o}}^{k}=-\frac{1}{m}\frac{c}{96\pi}\partial_{\perp}^{k}R+O\left(q^{4}\right)$ flows perpendicularly to gradients of curvature $R=-2\partial_{\perp}^{i}\partial_{\perp}^{j}u_{ij}$. We conclude that, in the g$s$-wave CSF, $c$ can be extracted from a measurement of $\tilde{\eta}_{\text{o}}$, and in the Galilean invariant case, also from a measurement of the current $\tilde{J}$ in response to strain.

Though the simple results above do not apply to the physical system of interest, the CSF, there is a relation between the observables of the CSF and the corresponding $g$s-wave CSF, which we can utilize. At the level of induced actions, it is given by

[TABLE]

where $\omega_{\mu}$ is expressed through $u_{ij}$ as in Appendix A, and by taking functional derivatives one obtains relations between response functions (Geracie et al., 2015). In particular,

[TABLE]

where the response functions $\eta_{\text{o}},\sigma_{\text{o}},\kappa_{\text{e}}$ depend on $\omega,\mathbf{q}$. In a Galilean invariant system one further has

[TABLE]

and we note the relations $\chi_{TJ,\text{e}}^{ij,t}=\kappa_{\text{e}}^{ij,k}iq_{k},\;\chi_{JJ,\text{o}}^{j,t}=\sigma_{\text{o}}q_{\bot}^{j},\;\chi_{JJ,\text{e}}^{j,k}=\rho_{\text{e}}q_{\bot}^{j}q_{\bot}^{k}$, between the above susceptibilities, the response functions $\kappa_{\text{e}},\sigma_{\text{o}}$, and the London diamagnetic response $\rho_{\text{e}}$.

VI Discussion

Equations (19) and (22) are the main results of this paper. They rely on the SSB pattern (2), but not on Galilean symmetry. Equation (22) expresses $\tilde{\eta}_{\text{o}}$ as a bulk observable of CSFs, which we refer to as the improved odd viscosity. According to (19), the leading term in the expansion of $\tilde{\eta}_{\text{o}}\left(0,\mathbf{q}\right)$ around $\mathbf{q}=\mathbf{0}$ is fixed by $c$. Since this leading term occurs at second order in $\mathbf{q}$, in order to extract $c$ one needs to measure $\sigma_{\text{o}},\chi_{\text{e}}$, and $\eta_{\text{o}}$, at zeroth, first, and second order, respectively. In a Galilean invariant system, (19) and (22) imply (20) and (23) respectively, which, in turn, show that $c$ can be extracted in an experiment where $U\left(1\right)_{N}$ fields and strain are applied, and the resulting number current and density are measured. In particular, a measurement of the stress tensor is not required. Since $U\left(1\right)_{N}$ fields can be applied in Galilean invariant fluids by tilting and rotating the sample, we believe that a bulk measurement of the boundary central charge, through (20) and (23), is within reach of existing experimental techniques.

Finally, we comment on the implications of our results to QH physics. The problem of obtaining $c$ from a bulk observable has been previously studied in QH states, described by (11)-(12) (Ferrari and Klevtsov, 2014; Can et al., 2014; Abanov and Gromov, 2014; Gromov and Abanov, 2014; Gromov et al., 2015; Can et al., 2015; Klevtsov and Wiegmann, 2015; Bradlyn and Read, 2015a; Klevtsov et al., 2016; Gromov et al., 2016; Klevtsov, 2017; Cappelli and Maffi, 2018). It was found that $c$ can only be extracted if $\text{var}s=\overline{s^{2}}-\overline{s}^{2}=0$, in which case the response to strain, at fixed $A_{\mu}-\overline{s}\omega_{\mu}$, depends purely on $c$ (Bradlyn and Read, 2015a; Gromov et al., 2016). This is a useful theoretical characterization, which seems challenging experimentally in light of the need to maintain the fine tuned relation $A_{\mu}=\overline{s}\omega_{\mu}$ while the strain $u_{ij}$, and therefore $\omega_{\mu}$, vary in time and space. The improved odd viscosity (22), constructed here, applies also to $\text{var}s=0$ QH states, with $\ell$ replaced by $-2\overline{s}$, and defines a bulk observable which is determined by $c$, and whose measurement does not require such fine tuning.

Acknowledgements.

The authors thank Daniel Ariad for sharing and explaining the Mathematica code used in (Ariad et al., 2015). We also benefited from discussions with Ady Stern, Andrey Gromov, Barry Bradlyn, David Mross, Micha Berkooz, Ryan Thorngren, Semyon Klevtsov, Tobias Holder, and Vatsal Dwivedi. O.G. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, CRC/Transregio 183, EI 519/7-1), the Israel Science Foundation (ISF), and the European Research Council (ERC, Project LEGOTOP). C.H. is partially supported by the Spanish grant MINECO-16-FPA2015-63667-P, the Ramon y Cajal fellowship RYC-2012-10370 and GRUPIN18-174 research grant from Principado de Asturias. The work of S.M. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Emmy Noether Programme grant no. MO 3013/1-1 and under Germany’s Excellence Strategy - EXC-2111 - 390814868.

Appendix A Geometric quantities and their perturbative expansion

We write $E_{A}^{\;i}=\delta_{A}^{\;i}+H_{A}^{\;i}$ for the inverse vielbein, and expand the relevant geometric quantities in $H$. For the inverse metric $G^{ij}=E_{A}^{\;i}\delta^{AB}E_{B}^{\;j}$ and volume element $\sqrt{G}=\left|E\right|=\left|\text{det}\left(E_{A}^{\;i}\right)\right|$ we find

[TABLE]

where, in expanded expressions, all index manipulations are trivial, and in particular, there is no difference between coordinate indices $i,j$ and $SO\left(2\right)_{L}$ indices $A,B$. Note that the strain used in the main text is given by $u_{ij}=\left(G_{ij}-\delta_{ij}\right)/2=-H_{(ij)}+O\left(H^{2}\right)$. We use the notation $\varepsilon^{\mu\nu\rho}$ for the totally anti-symmetric (pseudo) tensor, normalized such that $\varepsilon^{xyt}=1/\sqrt{G}$, as well as $\varepsilon^{ij}=\varepsilon^{ijt}$.

The non-relativistic spin connection used in the main text is the $SO\left(2\right)_{L}$ connection

[TABLE]

where $\partial_{\bot}^{l}=\varepsilon^{lk}\partial_{k}$, which is obtained naturally within Newton-Cartan geometry Moroz and Hoyos (2015); Gromov et al. (2016). This connection is torsion-full, but has a vanishing “reduced torsion” Bradlyn and Read (2015b). In the main text, a term $B/2m$ was implicitly added to $\omega_{t}$, but here we will add it explicitly when writing expressions for $S_{\text{eff}}$ and $S_{\text{ind}}$. Such a term appears in the presence of an additional background field $E_{0}^{\;i}$ which couples to momentum density $P_{i}$ Bradlyn and Read (2015b); Moroz and Hoyos (2015), and can be identified with $G^{ij}A_{j}/m$ in a Galilean invariant system, where $P_{i}=mG_{ij}J^{j}$. The Ricci scalar is given by

[TABLE]

Appendix B Odd viscosity at non-zero wave-vector: generalities

B.1 Definition and $T$ symmetry

We define the viscosity tensor as the linear response of stress to strain rate

[TABLE]

where

[TABLE]

In a translationally invariant system we can pass to Fourier components $T^{ij}\left(\omega,\mathbf{q}\right)=i\omega\eta^{ij,kl}\left(\omega,\mathbf{q}\right)H_{kl}\left(\omega,\mathbf{q}\right)$. By definition, $\eta^{ij,kl}\left(t,\mathbf{x},t^{\prime},\mathbf{x}^{\prime}\right)$ is real, and therefore

[TABLE]

Under time reversal $T$,

[TABLE]

The even and odd viscosities are then defined by $\eta_{\text{e},\text{o}}=\left(\eta\pm\eta_{T}\right)/2$, and satisfy $\left(\eta_{\text{e},\text{o}}\right)_{T}=\pm\eta_{\text{e},\text{o}}$. More explicitly,

[TABLE]

We will see below that in isotropic (or $SO\left(2\right)$ invariant) systems $\eta$ is even in $\mathbf{q}$, so that

[TABLE]

which is identical to the definition of $\eta_{\text{e},\text{o}}$ at $\mathbf{q}=0$ Avron et al. (1995); Avron (1998); Bradlyn et al. (2012); Hoyos (2014); Lapa and Hughes (2014).

B.2 $SO\left(2\right)$ and $P$ symmetries

Complex tensors satisfying (28) and (33), in $2$ spatial dimensions, form a vector space $V\cong\mathbb{C}^{3}$ which can be spanned by Avron (1998)

[TABLE]

where $\sigma^{x},\sigma^{z}$ are the symmetric Pauli matrices, and $\sigma^{0}$ is the identity matrix. Thus every odd viscosity tensor can be written as

[TABLE]

with complex coefficients $\eta_{ab}\left(\omega,\mathbf{q}\right)$. Under a rotation $R=e^{i\alpha\left(i\sigma^{y}\right)}\in SO\left(2\right)$ the metric perturbation and stress tensor transform as

[TABLE]

where $\left(R\cdot\mathbf{q}\right)^{i}=R_{\;j}^{i}q^{j}$. The same transformation rules apply for $R\in O\left(2\right)$, which defines the parity transformation $P$, in flat space. It follows that

[TABLE]

under $O\left(2\right)$, which is compatible with (28), and the decomposition \eta=\text{\eta}_{\text{o}}+\eta_{\text{e}}. In particular, equation (37) shows that the viscosity tensor is $P$-even, or more accurately, a tensor under $P$ rather than a pseudo-tensor. In an $SO\left(2\right)$-invariant system, the viscosity tensor will also be $SO\left(2\right)$-invariant

[TABLE]

Note that this holds even when $SO\left(2\right)$ symmetry is spontaneously broken, as in $\ell$-wave SFs. At $\mathbf{q}=\mathbf{0}$, there is a unique tensor satisfying (38), namely

[TABLE]

leaving a single odd viscosity coefficient $\eta_{xz}\left(\omega\right)=\eta_{\text{o}}^{\left(1\right)}\left(\omega\right)$ Avron et al. (1995); Avron (1998); Bradlyn et al. (2012); Hoyos (2014); Lapa and Hughes (2014).

A non-zero $\mathbf{q}$, however, along with the tensors $\delta^{ij}$ and $\varepsilon^{ij}$, can be used to construct additional $SO\left(2\right)$-invariant odd viscosity tensors, beyond $\sigma^{xz}$. From the data $\mathbf{q},\delta^{ij},\varepsilon^{ij}$, three linearly independent, symmetric, rank-2 tensors can be constructed, which we take to be

[TABLE]

where $q_{\bot}^{i}=\varepsilon^{ij}q_{j}$. The notation above is due to the relation

[TABLE]

where $\theta=\text{arg}\left(\mathbf{q}\right)$, so that $\tau^{x},\tau^{z}$ are a rotated version of $\sigma^{x},\sigma^{z}$. Moreover, all three $\tau$s are $SO\left(2\right)$-invariant, $\tau^{ij}\left(\mathbf{q}\right)=R_{\;i^{\prime}}^{i}R_{\;j^{\prime}}^{j}\tau^{i^{\prime}j^{\prime}}\left(R^{-1}\cdot\mathbf{q}\right)$, and can therefore be used to construct three $SO\left(2\right)$-invariant odd viscosity tensors

[TABLE]

which form a basis for $V$. Any odd viscosity tensor (at $\mathbf{q}\neq\mathbf{0}$) can then be written as

[TABLE]

Furthermore, for an $SO\left(2\right)$-invariant $\eta_{\text{o}}$, the coefficients $\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)},\eta_{\text{o}}^{\left(3\right)}$ depend on $\mathbf{q}$ through its norm, owing to the $SO\left(2\right)$-invariance of $\tau^{ab}$. We therefore arrive at the general form of an $SO\left(2\right)$-invariant odd viscosity tensor,

[TABLE]

In particular, we see that $\eta_{\text{o}}$ is even in $\mathbf{q}$ (and the same applies also to the even viscosity $\eta_{\text{e}}$). To determine the small $\omega,\mathbf{q}$ behavior of the coefficients we change to the $\mathbf{q}$-independent basis of $\sigma$s,

[TABLE]

In gapped systems (such as QH states) $\eta_{\text{o}}$ will be regular around $\omega=0=q$, and so will the coefficients $\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)},\eta_{\text{o}}^{\left(3\right)}$. In gapless systems (such as $\ell$-wave SFs) there will be a singularity at $\omega=0=q$, but the limit $q\rightarrow 0$ at $\omega\neq 0$ will be regular. In both cases, the limit $q\rightarrow 0$ at $\omega\neq 0$ of (45) reduces to the known result $\eta_{\text{o}}\left(\omega,\mathbf{0}\right)=\eta_{\text{o}}^{\left(1\right)}\left(\omega,0\right)\sigma^{xz}$ Avron et al. (1995); Avron (1998); Bradlyn et al. (2012); Hoyos (2014); Lapa and Hughes (2014).

B.3 $PT$ symmetry

The combination $PT$ of parity and time-reversal is a symmetry in any system in which $T$ is broken (perhaps spontaneously) due to some kind of angular momentum, as in QH states, $\ell$-wave SFs, and active chiral fluids Banerjee et al. (2017a). Here we consider the implications of $PT$ symmetry on (45).

From the definition (40) it is clear that $\tau^{0}$ and $\tau^{z}$ are $P$-even, while $\tau^{x}$ is $P$-odd. Therefore, $\tau^{xz},\tau^{0x}$ are $P$-odd while $\tau^{0z}$ is $P$-even (and all three are $T$-even). Since $\eta_{\text{o}}$ is $T$-odd and $P$-even, and using (44), it follows that $\eta_{\text{o}}^{\left(1\right)}$ and $\eta_{\text{o}}^{\left(2\right)}$ are $P,T$-odd, while $\eta_{\text{o}}^{\left(3\right)}$ is $T$-odd but $P$-even. In particular, $\eta_{\text{o}}^{\left(3\right)}$ is $PT$-odd, and must vanish in $PT$-symmetric systems. The odd viscosity tensor in $SO\left(2\right)$ and $PT$ symmetric systems is therefore given by

[TABLE]

This form is confirmed by previous results for QH states Abanov and Gromov (2014), and by the results presented in Sec.V for CSFs. The same form is obtained at $\mathbf{q}=\mathbf{0}$, but in the presence of vector, or pseudo-vector, anisotropy $\mathbf{b}$, in which case we find

[TABLE]

which explains the tensor structure found in Peña Benitez et al. (2019); Offertaler and Bradlyn (2019).

B.4 Frequency dependence and reality conditions

In closed and clean systems, like the $\ell$-wave SFs discussed in this paper, the viscosity can be obtained from an induced action

[TABLE]

As a result, $\eta$ satisfies the additional property,

[TABLE]

which, along with (33)-(32) and the fact that $\eta$ is even in $\mathbf{q}$, implies that $\eta_{\text{o}}$ ($\eta_{\text{e}}$) is even (odd) in $\omega$,

[TABLE]

This result, along with (29) and the fact that $\eta$ is even in $\mathbf{q}$, implies that $\eta_{\text{o}}$ ($\eta_{\text{e}}$) is real (imaginary),

[TABLE]

These general properties are satisfied by the odd viscosity tensor computed in this paper. These are also compatible with the examples worked out in Bradlyn et al. (2012), as well with viscosity-conductivity relations that hold in Galilean invariant systems (in conjugation with known properties of the conductivity) Hoyos and Son (2012); Bradlyn et al. (2012); Hoyos et al. (2014).

We note that some care is required when interpreting (50)-(51) around singularities of $\eta$. For example, the first equation in (50) naively implies that $\eta_{\text{e}}\left(0,\mathbf{q}\right)=0$, which in particular implies that the bulk and shear viscosities $\eta_{\text{e}}\left(0,\mathbf{0}\right)=\zeta\sigma^{0}\otimes\sigma^{0}+\eta^{\text{s}}\left(\sigma^{x}\otimes\sigma^{z}+\sigma^{z}\otimes\sigma^{x}\right)$ vanish in the closed, clean, case. This however, is not quite correct, due to a possible singularity of $\eta_{\text{e}}$ at $\omega=0$, as well as the usual infinitesimal imaginary part of $\omega$ required to obtain the retarded response. For example, for free fermions, reference Bradlyn et al. (2012) finds $\eta^{\text{s}}\left(\omega,\mathbf{0}\right)\sim\frac{i}{\omega+i\epsilon}=\pi\delta\left(\omega\right)+i\text{PV}\frac{1}{\omega}$ (where PV is the principle value), which has an infinite real part at $\omega=0$, in analogy with the Drude behavior of the conductivity.

B.5 Odd viscosity from Gaussian integration: a technical result

We now restrict attention to CSFs. The effective Lagrangian, perturbatively expanded to second order, and in the absence of the $U\left(1\right)$ background, takes the form

[TABLE]

where the Green’s function $\mathcal{G}$ is independent of $H$, the vertex $\mathcal{V}$ is linear in $H$, and the contact term $\mathcal{C}$ is quadratic in $H$. Performing Gaussian integration over $\theta$ yields the induced Lagrangian

[TABLE]

and comparing with (48) one can read off $\eta_{\text{o}}$. In Appendix C we write explicit expressions for a Galilean invariant $\mathcal{L}_{\text{eff}}$, which we then expand to obtain explicit expressions for $\mathcal{G}^{-1},\mathcal{V},\mathcal{C}$. Appendix D then describes the resulting $\mathcal{L}_{\text{ind}}$. Here we take a complementary approach and obtain the general form of $\eta_{\text{o}}$ from (53), using the formalism developed above, based only on $SO\left(2\right)$ and $PT$ symmetries.

The motivation for the analysis in this appendix is the following. The power counting (8) is designed such that the $O\left(p^{n}\right)$ Lagrangian $\mathcal{L}_{n}\subset\mathcal{L}_{\text{eff}}$ produces $O\left(p^{n}\right)$ contributions to $\mathcal{L}_{\text{ind}}$. Therefore, naively, one expects the $O\left(q^{2}\right)$ odd viscosity to depend on $\mathcal{L}_{0},\mathcal{L}_{2},$ and $\mathcal{L}_{3}$ (since $\mathcal{L}_{1}=0$). Using the notation $\eta_{\text{o}}=\eta_{\mathcal{V}}+\eta_{\mathcal{C}}$ for the parts of $\eta_{\text{o}}$ due to $-\mathcal{V}\mathcal{G}\mathcal{V}/2$ and $\mathcal{C}$, respectively, the result of this appendix is that $\eta_{\mathcal{V}}$, to $O\left(q^{2}\right)$, is actually independent of $\mathcal{L}_{3}$.

We now describe the details. For $\eta_{\mathcal{C}}$, we cannot do better than the general discussion thus far - it is given by (45), with $\eta_{\text{o}}^{\left(3\right)}=0$, and both $\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)}$ are real and regular at $\omega=0=q$, since $\mathcal{L}_{\text{eff}}$, and $\mathcal{C}$ in particular, are obtained by integrating out gapped degrees of freedom (the Higgs modes and the fermion $\psi$). For $\eta_{\mathcal{V}}$, however, we can do better. We first write more explicitly

[TABLE]

Based on $SO\left(2\right)$ and $PT$ symmetries, the objects $\mathcal{G}^{-1},V^{ij}$ take the forms

[TABLE]

where

[TABLE]

are, in this context, more convenient than the $\tau$s (40), and $a,b,c,D$ are general functions of their arguments which are $P,T$-even, real, and regular at $\omega=0=q$, as follows from the same properties of $\mathcal{L}_{\text{eff}}$. In particular, we will use the following expansions

[TABLE]

where $D_{0}=0$ because $\theta$ enters $\mathcal{L}_{\text{eff}}$ only through its derivatives. The odd viscosity $\eta_{\mathcal{V}}$ is then given by

[TABLE]

which is of the form (45), with $\eta_{\text{o}}^{\left(3\right)}=0$ and

[TABLE]

Setting $\omega=0$ and expanding in $q$, we find

[TABLE]

Having identified the coefficients $a_{0},a_{1},b_{0},c_{0},c_{1},D_{1},D_{2}$ that determine $\eta_{\mathcal{V}}$ to $O\left(q^{2}\right)$, we now determine the order in the derivative expansion of $\mathcal{L}_{\text{eff}}$ in which these enter. Explicitly, the above coefficients are defined by

[TABLE]

We see that $c_{1}$ enters $\mathcal{L}_{\text{eff}}$ at $O\left(p^{3}\right)$, while all other coefficients enter at a lower order, and come from $\mathcal{L}_{0},\mathcal{L}_{2}$. In particular, \eta_{\text{\mathcal{V}}}^{\left(1\right)} in (60) is independent of $\mathcal{L}_{3}$. Even though $c_{1}$ is the coefficient of an $O\left(p^{3}\right)$ term, it is actually due to $\mathcal{L}_{2}$. Using (25) we identify $c_{0}\theta q^{2}q^{i}q_{\bot}^{j}H_{ij}=-s_{\theta}c_{1}\partial^{i}\theta\partial^{2}\omega_{i}$, which must be a part of

[TABLE]

This is an $O\left(p^{2}\right)$ term, and in fact comes from $\mathcal{L}_{2}^{\left(2\right)}\subset\mathcal{L}_{2}$, see (69). Thus, both \eta_{\text{\mathcal{V}}}^{\left(1\right)},\eta_{\text{\mathcal{V}}}^{\left(2\right)} in (60) are completely independent of $\mathcal{L}_{3}$.

Appendix C Effective action and its perturbative expansion

C.1 Zeroth order

It is useful to write the zeroth order scalar $X$ as

[TABLE]

where

[TABLE]

We will also use $\mathcal{B}=B+\frac{s_{\theta}}{2}R,\;\mathcal{E}_{i}=E_{i}+s_{\theta}E_{\omega,i}$ for the magnetic and electric fields obtained from $\mathcal{A}_{\mu}$, where $E_{\omega,i}=\partial_{t}\omega_{i}-\partial_{i}\omega_{t}$. Expanding $\mathcal{L}_{0}=P\left(X\right)$ to second order in the fields, one finds (up to total derivatives)

[TABLE]

where $\partial^{2}=\partial^{i}\partial_{i},\;\mathcal{A}^{2}=\mathcal{A}_{i}\mathcal{A}^{i}$, and we defined the inverse Green’s function $\mathcal{G}^{-1}$, vertex $\mathcal{V}$, and contact terms $\mathcal{C}$, respectively. These are used in Appendix D below to obtain $S_{\text{ind}}$.

In (65), the geometric objects $\sqrt{G}$ and $\omega_{\mu}$ should be interpreted as expanded to the required order according to (24)-(25). In particular, the term $-n_{0}\sqrt{G}\mathcal{A}_{t}$ includes $-s_{\theta}n_{0}\sqrt{G}\omega_{t}$, which produces the leading contribution to $\eta_{\text{o}}^{\left(1\right)}$. To see this, we expand

[TABLE]

which is identical to the expansion (24) of $\omega_{t}$, apart from $H_{iA}\leftrightarrow H_{Ai}$. Ignoring total derivatives, this reduces to

[TABLE]

Comparing with (39) and (48), the second term in the second line corresponds to $\eta_{\text{o}}^{\left(1\right)}=-s_{\theta}n_{0}/2$. The first term in the second line depends on the anti-symmetric part of $H$, and shows that the full expression (67) actually corresponds to a torsional Hall (or odd) viscosity Hughes et al. (2011, 2013) $\zeta_{H}=-s_{\theta}n_{0}$, which can be read off from the third line. The appearance of the torsional Hall viscosity at the level of $S_{\text{eff}}$ (but not at the level of $S_{\text{ind}}$, see Appendix D) can be understood from the mapping of Golan and Stern (2018) of the $p$-wave SF to a Majorana spinor in Riemann-Cartan space-time.

C.2 Second order

The full expression for $\mathcal{L}_{2}$ is given by $\mathcal{L}_{2}=\sum_{i=1}^{6}\mathcal{L}_{2}^{\left(i\right)}$ , where Son and Wingate (2006)

[TABLE]

The terms $\mathcal{L}_{2}^{\left(5\right)}$ and $\mathcal{L}_{2}^{\left(6\right)}$ were not written explicitly in Son and Wingate (2006) because, on shell (on the equation of motion for $\theta$), they are proportional to $\mathcal{L}_{2}^{\left(4\right)}$ up to $O\left(p^{4}\right)$ corrections, and can therefore be eliminated by a redefinition of $F_{4}$. However, for the purpose of comparing the general $S_{\text{eff}}$ with the microscopic expression (104), it is convenient to work off shell and keep all terms explicit.

Specializing to 2+1 dimensions and expanding to second order in fields, one finds

[TABLE]

from which one can easily extract the second order corrections to $\mathcal{G}^{-1},\mathcal{V},\mathcal{C}$, of (65). Note that $\mathcal{L}_{2}^{\left(3\right)}$ includes a term $\propto\varepsilon^{ij}\omega_{i}\partial_{t}\mathcal{A}_{j}=\varepsilon^{ij}\omega_{i}\partial_{t}\left(A_{j}+s_{\theta}\omega_{j}\right)$. Comparing with (70) below, it is clear that distinguishing $\mathcal{L}_{2}^{\left(3\right)}$ from $\mathcal{L}_{\text{gCS}}$ is non-trivial. This is in fact the same problem of extracting the central charge from the Hall viscosity addressed in the main text, but at the level of $S_{\text{eff}}$ (where $\theta$ is viewed as a background field) rather than $S_{\text{ind}}$ (where $\theta$ has been integrated out). Accordingly, the central charge can be computed by applying Eq.(22) to the response functions obtained from $S_{\text{eff}}$. Additionally, relying on LGS, one can extract $F_{3}$ as the coefficient of $H^{(ij)}\partial_{t}^{2}H_{\left(ij\right)}$. Both approaches produce the same central charge (14) in the perturbative computation of Appendix E.5.

C.3 Gravitational Chern-Simons term

The gCS Lagrangian is given explicitly by

[TABLE]

Its expansion to second order in fields, using (24)-(25), is

[TABLE]

As opposed to $\sqrt{G}\omega_{t}$ in Eq.66, the gCS term is (locally) $SO\left(2\right)_{L}$ gauge invariant, and accordingly depends only on the metric, or, within the perturbative expansion, on the symmetric part $H_{(ij)}$. From this expansion one can read off the gCS contributions to the odd viscosity $\eta_{\text{o}}$ (Eq.(19)), and to the odd, mixed, static susceptibility $\chi_{TJ,\text{o}}$ (Eq.(20)).

C.4 Additional terms at third order

To obtain reliable results at $O\left(p^{3}\right)$ we, in principle, need the full Lagrangian $\mathcal{L}_{3}$, which includes, but is not equal to, $\mathcal{L}_{\text{gCS}}$. Nevertheless, we argue that $\mathcal{L}_{3}-\mathcal{L}_{\text{gCS}}$ does not contribute to the quantity of interest in this paper - $\eta_{\text{o}}$ to $O\left(q^{2}\right)$. We already demonstrated in Appendix B.5 that the vertex part of the odd viscosity $\eta_{\mathcal{V}}$ is independent of $\mathcal{L}_{3}$, and it remains to show that the contact term part $\eta_{\mathcal{C}}$ is independent of $\mathcal{L}_{3}-\mathcal{L}_{\text{gCS}}$. We do not have a general proof, but we address this issue in two ways:

Within the microscopic model (13), the perturbative computation of Appendix E.5 provides an explicit expression for $\eta_{\mathcal{C}}$, which is completely saturated by the effective action presented thus far. Thus $\eta_{\mathcal{C}}$ is independent of $\mathcal{L}_{3}-\mathcal{L}_{\text{gCS}}$ in the particular realization (13). 2. 2.

The term $\mathcal{L}_{3}$ is $P,T$-odd, and therefore vanishes in an $s$-wave SF. On the other hand, it suffices to consider the $g$s-wave SF where $s_{\theta}=0$ (but $\ell\neq 0$), since for $s_{\theta}\neq 0$ the spin connection included in $\nabla_{\mu}\theta$ will only produce $O\left(p^{4}\right)$ corrections. By contracting Galilean vectors, we were able to construct four $P,T$-odd terms in $\mathcal{L}_{3}-\mathcal{L}_{\text{gCS}}$ for the $g$s-wave SF,

[TABLE]

where $\tilde{E}_{i}$ is the electric field of the improved $U\left(1\right)$ connection $\tilde{A}_{t}=A_{t}+\frac{1}{2m}\nabla^{i}\theta\nabla_{i}\theta,\;\tilde{A}_{i}=\partial_{i}\theta-s_{\theta}\omega_{i}$ Hoyos et al. (2014). Perturbatively expanding these, we do not find any $O\left(q^{2}\right)$ contributions to $\eta_{\mathcal{C}}$ (or to \eta_{\text{\mathcal{V}}}, in accordance with Appendix B.5).

Appendix D Induced action

The arguments presented in the main text suffice to establish the quantization of $\tilde{\eta}_{\text{o}}$ and $\tilde{\chi}_{TJ,\text{o}}$ directly from $S_{\text{eff}}$ - an explicit expression for $S_{\text{ind}}$ is not required. Nevertheless, it is instructive to compute certain contributions in $S_{\text{ind}}$ to demonstrate these results explicitly, and also to reproduce simpler properties of $\ell$-wave SFs. Here we will compute the contribution of $\mathcal{L}_{0}+\mathcal{L}_{2}^{\left(1\right)}\subset\mathcal{L}_{\text{eff}}$ to the induced Lagrangian $\mathcal{L}_{\text{ind}}$, and, along the way, demonstrate explicitly the relation between $\text{var}s=0$ QH states and CSFs alluded to in the discussion Sec.VI.

The starting point is the induced action due to $\mathcal{L}_{0}=P\left(X\right)$, obtained from (65). It is given by

[TABLE]

This expression contains, rather compactly, the entire linear response of the $\ell$-wave SF to $O\left(p\right)$ in the derivative expansion, as well as certain $O\left(p^{2}\right)$ contributions Hoyos et al. (2014), and should be interpreted as expanded to second order using (24)-(25). In using (25), one can set $H_{[AB]}=0$, since $S_{\text{ind}}$ is $SO\left(2\right)_{L}$ invariant and the anti-symmetric part $H_{[AB]}$ corresponds to the $SO\left(2\right)_{L}$ phase of the vielbein $E_{A}^{\;i}$. Technically, $H_{[AB]}$ always appears in the combination $\partial_{\mu}\left(\theta+s_{\theta}\varepsilon^{AB}H_{AB}/2\right)\subset\nabla_{\mu}\theta$, so that integrating out $\theta$ eliminates $H_{[AB]}$.

Note that, diagrammatically, equation (73) corresponds to linear response at tree-level. Higher orders in $\theta$ will generate diagrams with $\theta$ running in loops, which can be shown to produce $O\left(p^{3}\right)$ corrections above the leading order to any observable Son and Wingate (2006), and are therefore irrelevant for the purpose of $q^{2}$ corrections to $\eta_{\text{o}}$.

The $O\left(p^{0}\right)$ part of (73) is obtained by setting $s_{\theta}=0$, as in an $s$-wave SF,

[TABLE]

The first line contains the ground state pressure and density $P_{0},n_{0}$, as well as the London diamagnetic function $\rho_{\text{e}}=\frac{n_{0}}{m}\frac{1}{q^{2}-c_{s}^{-2}\omega^{2}}$ and the ideal Drude longitudinal conductivity $\sigma_{\text{e}}=-\frac{n_{0}}{m}\frac{i\omega c_{s}^{-2}}{q^{2}-c_{s}^{-2}\omega^{2}}$ of the SF Hoyos et al. (2014). The second line contains the mixed response and mixed static susceptibility

[TABLE]

defined in Sec.V, as well as the inverse compressibility $K^{-1}=-n_{0}m\frac{\omega^{2}}{q^{2}-c_{s}^{-2}\omega^{2}}$ (which agrees with the thermodynamic expression $K^{-1}=n_{0}^{2}\frac{\partial\mu}{\partial n_{0}}=n_{0}mc_{s}^{2}$ at $q=0$). In particular, the $\ell$-wave SF is indeed a superfluid

• the even viscosity $\eta_{\text{e}}$ vanishes to zeroth order in derivatives (see Bradlyn et al. (2012) for a subtlety in separating $K^{-1}$ from $\eta_{\text{e}}$).

The $O\left(p\right)$ part of the (73) is $P,T$-odd and vanishes when $s_{\theta}=0$. It is given by

[TABLE]

The first and third lines produce the following odd viscosity Hoyos et al. (2014),

[TABLE]

and setting $\omega=0$ one obtains the leading terms in Eq.(18). By using the identity (up to a total derivative)

[TABLE]

the second line of Eq.(76) can be written as a non-local CS term

[TABLE]

with the odd (or Hall) conductivity $\sigma_{\text{o}}\left(\omega,q\right)=\sigma_{\text{o}}^{0}q^{2}/\left(q^{2}-c_{s}^{-2}\omega^{2}\right)$, $\sigma_{\text{o}}^{0}=s_{\theta}n_{0}/2m^{2}c_{s}^{2}$ Volovik (1988a); Goryo and Ishikawa (1998, 1999); Furusaki et al. (2001); Stone and Roy (2004); Roy and Kallin (2008); Lutchyn et al. (2008); Hoyos et al. (2014), with $\sigma_{\text{o}}\left(0,q\right)=\sigma_{\text{o}}^{0}$ unquantized, and $\sigma_{\text{o}}\left(\omega,0\right)=0$, in accordance with the boundary $U\left(1\right)_{N}$-neutrality Golan and Stern (2018).

To demonstrate explicitly that $c$ cannot be extracted from the odd viscosity alone, it suffices to add the $O\left(p^{2}\right)$ term $\mathcal{L}_{2}^{\left(1\right)}=F_{1}\left(X\right)R\subset\mathcal{L}_{2}$. The situation is particularly simple for the special case $F_{1}\left(X\right)=-s_{\theta}^{2}P^{\prime}\left(X\right)/4m$. Then

[TABLE]

which shows that $F_{1}\left(X\right)R$ can be absorbed into $P\left(X\right)$ by a modification of $X$. The scalar $X-\frac{s_{\theta}^{2}}{4m}R$ is useful because, unlike $X$, it depends on $A_{\mu}$ and $\omega_{\mu}$ only through the combination $\mathcal{A}_{\mu}=A_{\mu}+s_{\theta}\omega_{\mu}$. This is evident in (63), where $B$ rather than $\mathcal{B}=B+\frac{s}{2}R$ appears. It is then clear that, to $O\left(p^{3}\right)$, adding $\mathcal{L}_{2}^{\left(1\right)}=F_{1}\left(X\right)R=-\frac{s_{\theta}^{2}}{4m}P^{\prime}\left(X\right)R$ to $\mathcal{L}_{0}=P\left(X\right)$ amounts to changing $B$ to $\mathcal{B}$ in the induced Lagrangian (73),

[TABLE]

The only contribution to $\eta_{\text{o}}$, beyond (77), comes from the term proportional to $\mathcal{E}^{i}\partial_{i}\mathcal{B}$. By using the identity (78) for $\mathcal{A}_{\mu}$, this term can be written as the sum of non-local CS, WZ1, and WZ2 terms, which generalizes (79) to

[TABLE]

Most importantly, this includes a non-local version of WZ2, which is indistinguishable from $\mathcal{L}_{\text{gCS}}$ at $\omega=0$, where $\sigma_{\text{o}}\left(0,q\right)=\sigma_{\text{o}}^{0}$ is a constant. Noting that $F_{1}^{\prime}=-s_{\theta}^{2}P^{\prime\prime}/4m=-\left(s_{\theta}/2\right)\sigma_{\text{o}}^{0}$, and comparing to (70), it follows that $c$ and $F_{1}^{\prime}$ will enter the $\omega=0$ odd viscosity only through the combination $c+48\pi s_{\theta}F_{1}^{\prime}$. In more detail, the odd viscosity tensor due to $\mathcal{L}_{0}+\mathcal{L}_{2}^{\left(1\right)}+\mathcal{L}_{\text{gCS}}$, is given by

[TABLE]

which, at $\omega=0$, is a special case of equation (12) of the main text.

Equation (83) remains valid away from the special point $F_{1}=-s_{\theta}^{2}P^{\prime}/4m$, even though (82) does not. Examining the perturbatively expanded $\mathcal{L}_{0}$ (65) and $\mathcal{L}_{2}^{\left(1\right)}$ (69), we see that a general $F_{1}$ amounts to replacing $B$ in (73) with $B+\alpha\frac{s_{\theta}}{2}R$, where $\alpha=-\frac{4mF_{1}^{\prime}}{s_{\theta}^{2}P^{\prime\prime}}\neq 1$ generically (as well as in the microscopic model (16)). The general induced Lagrangian due to $\mathcal{L}_{0}+\mathcal{L}_{2}^{\left(1\right)}$, valid to $O\left(p^{3}\right)$, is then given by

[TABLE]

and, along with the $\mathcal{L}_{\text{gCS}}$ , produces the odd viscosity (83). This expression does not depend on $A_{\mu},\omega_{\mu}$ only through $\mathcal{A}_{\mu}$, but the terms contributing to (83) still vanish $s_{\theta}=0$, which is why the improved odd viscosity due to (84) vanishes. In addition to $\mathcal{L}_{2}^{\left(1\right)}$, the second order terms $\mathcal{L}_{2}^{\left(2\right)},\mathcal{L}_{2}^{\left(3\right)}$ (68) also produce $q^{2}$ corrections to the odd viscosity, but not to the improved odd viscosity.

Though Eq.(82) describes only a part of $\mathcal{L}_{\text{ind}}$, and is non-generic, it does reveal the analogy between CSFs and $\text{var}s=0$ QH states, described in the discussion Sec.VI in a very simple setting. Indeed, comparing Eq.(82) with Eq.(12) we see that CSFs are analogous to $\text{var}s=0$ QH states, with $\overline{s}=-s_{\theta}=-\ell/2$, but with a non-local, non-quantized, Hall conductivity, in place of the filling factor $\nu/2\pi$. Additionally, both QH states and CSFs have the same gCS term Eq.(70), with $c$ the boundary chiral central charge.

Appendix E Detailed analysis of the microscopic model (13)

E.1 Symmetry

The action $S_{\text{m}}$ is invariant under $U\left(1\right)_{N}$ gauge transformations,

[TABLE]

which implies the current conservation $\partial_{\mu}(\sqrt{G}J^{\mu})=0$, where $\sqrt{G}J^{\mu}=-\delta S/\delta A_{\mu}$. It is also clear that $S_{\text{m}}$ is invariant under time-independent spatial diffeomorphisms, generated by $\delta x^{i}=\xi^{i}\left(\mathbf{x}\right)$, if $\psi$ transforms as a function, $A_{\mu}$ as a 1-form, $\Delta^{j}$ as a vector, and $G_{ij}$ as a rank-2 tensor. As described in section II, due to its Galilean symmetry in flat space, $S_{\text{m}}$ is also invariant under time-dependent spatial diffeomorphisms $\delta x^{i}=\xi^{i}\left(\mathbf{x},t\right)$, provided one modifies the transformation rule of $A_{i}$ to Eq.(4).

E.2 Effective action and fermionic Green’s function

Starting with the microscopic action (13), the effective action for the order parameter $\Delta$ in the $A,G$ background is obtained by integrating out the (generically) gapped fermion $\psi$,

[TABLE]

where $G^{1/4}=\left(\text{det}G_{ij}\right)^{1/4}$ is the square root of the volume element $\sqrt{G}$. The form of the measure is fixed by the fact that the fundamental fermionic degree of freedom is the fermion-density $\tilde{\psi}=G^{1/4}\psi$, which satisfies the usual canonical commutation relation $\left\{\tilde{\psi}^{\dagger}\left(\mathbf{x}\right),\tilde{\psi}\left(\mathbf{y}\right)\right\}=\delta^{\left(2\right)}\left(\mathbf{x}-\mathbf{y}\right)$ as an operator Hawking (1977); Fujikawa (1980); Abanov and Gromov (2014); Golan and Stern (2018). This is to be contrasted with $\left\{\psi^{\dagger}\left(\mathbf{x}\right),\psi\left(\mathbf{y}\right)\right\}=\delta^{\left(2\right)}\left(\mathbf{x}-\mathbf{y}\right)/\sqrt{G\left(\mathbf{x}\right)}$ which ties the fermion to the background metric.

In terms of $\tilde{\psi}$ the action (13) takes the form

[TABLE]

where $\nabla_{\mu}=\partial_{\mu}+iA_{\mu}-\frac{1}{4}\partial_{\mu}\log G$ is the covariant derivative for densities, and $\mathcal{U}=\frac{1}{2\lambda}\sqrt{G}G_{ij}\Delta^{i*}\Delta^{j}$. Passing to the BdG form of the fermionic part of the action, in terms of the Nambu spinor-density $\tilde{\Psi}^{\dagger}=\left(\tilde{\psi}^{\dagger},\tilde{\psi}\right)$ (which is a Majorana spinor-density Golan and Stern (2018)), one finds

[TABLE]

where derivatives act on all fields to the right; $\tilde{A}=1,2$ is an index for $U\left(1\right)_{N}$, viewed as a copy of $SO\left(2\right)$; the gamma matrices are $\gamma^{0}=\sigma^{z},\;\gamma^{1}=-i\sigma^{x},\;\gamma^{2}=i\sigma^{y}$, satisfying $\left\{\gamma^{\mu},\gamma^{\nu}\right\}=2\eta^{\mu\nu}$ with $\eta^{\mu\nu}=\text{diag}\left[1,-1,-1\right]$, and $\text{tr}\left(\gamma^{0}\gamma^{1}\gamma^{2}\right)=2i$; and

[TABLE]

is the emergent vielbein Volovik (1990); Golan and Stern (2018), to be distinguished from the background vielbein $E_{A}^{\;i}$ (with an $SO\left(2\right)_{L}$ index $A=1,2$) that appeared in the main text and that will be used momentarily. We also defined the inverse Green’s function $\mathcal{G}^{-1}$. The effective action (86) is then given by the logarithm of the Pfaffian

[TABLE]

E.3 Fermionic ground state topology

For a given $\Delta^{j}$, the fermion $\psi$ is gapped, unless the chemical potential $\mu$ or chirality $\ell=\text{sgn}\left(\text{Im}\left(\Delta^{x*}\Delta^{y}\right)\right)$ are tuned to 0, and forms a fermionic topological phase characterized by the bulk Chern number. Assuming $A_{\mu}=0$ and space-time independent $\Delta^{i},G^{ij}$, it is given by Volovik (2009)

[TABLE]

and determines the boundary chiral central charge $c=C/2$ Read and Green (2000); Kitaev (2006); Volovik (2009); Ryu et al. (2010). Here the fermionic Green’s function $\mathcal{G}$ is Fourier transformed to Euclidian 3-momentum $q=\left(iq_{0},\mathbf{q}\right)$ (see (101)). For the particular model (13) one finds

[TABLE]

see Read and Green (2000); Volovik (2009); Golan and Stern (2018) for similar expressions. Note that the central charge is well defined for both $m>0$ and $m<0$, even though the single particle dispersion is not bounded from below in the latter, and many physical quantities naively diverge (we will see below that certain physical quantities diverge also with $m>0$). A negative mass can occur as an effective mass in lattice models, in which case the lattice spacing provides a natural cutoff (which must be smooth in momentum space for (91) to hold). In any case, a negative mass makes it possible to obtain both fundamental central charges $c=\pm 1/2$, for fixed $\ell$, within the model (13). All possible $c\in\left(1/2\right)\mathbb{Z}$ can then be obtained by stacking layers of the model (13) with the same $\ell$ but different $m,\mu$. Thus the model (13) suffices to generate a representative for all topological phases of the $p$-wave CSF. For concreteness, below we will work only with $m>0$, in which case $c$ is given by Eq.(14).

E.4 Symmetry breaking and bosonic ground state in the presence of a background

metric

For time independent fields $A,G,\Delta$ the effective action reduces to

[TABLE]

where $\varepsilon_{0}$ is the ground-state energy-density as a function of the fields. In flat space $G_{ij}=\delta_{ij}$, with $A_{t}=-\mu$ and $A_{i}=0$, and assuming $\Delta$ is constant, it is given by Volovik (2009); Golan and Stern (2018)

[TABLE]

where

[TABLE]

is the single particle dispersion, and $g^{ij}=\Delta^{(i}\Delta^{j)*}=\delta^{\tilde{A}\tilde{B}}e_{\tilde{A}}^{\;i}e_{\tilde{B}}^{\;j}$ is the emergent metric - a dynamical metric to be distinguished from the background metric $G^{ij}$. The ground state configuration of $g^{ij}$ is determined by minimizing $\varepsilon_{0}$, while the overall phase $\theta$ of the order parameter and the chirality $\ell$, of which $g^{ij}$ is independent, are left undetermined. Thus $g^{ij}$ corresponds to a massive Higgs field, while $\theta$ is a Goldstone field. The energy-density (94) is UV divergent, and requires regularization. We do this in the simplest manner, by introducing a momentum cutoff $q^{2}<\Lambda^{2}$. Since the divergence disappears for $g^{ij}=0$ (assuming $m>0$), this can be thought of as a small, but non-vanishing, range $1/\Lambda$ for the interaction mediated by $\Delta$. With a finite $\Lambda$, the energy-density is well defined and has a unique global minimum at $g^{ij}=\Delta_{0}^{2}\delta^{ij}$, with $\Delta_{0}$ determined by the self-consistent equation

[TABLE]

For $\mu>0$ the non-interacting system has a Fermi surface, and a solution exists for all $\lambda>0$, which is the statement of the BCS instability. For $\mu<0$, the non-interacting system is gapped, and a solution exists if the interaction is large enough compared with the gap, $\lambda\Lambda^{-4}\gtrsim\left|\mu\right|$.

Consider now the case of a general constant metric $G_{ij}$, and let us introduce a constant vielbein $E$ such that $G_{ij}=E_{\;i}^{A}\delta_{AB}E_{\;j}^{B}$. The inverse transpose $E^{-T}=\left(E^{-1}\right)^{T}$ is given in coordinates by $E_{A}^{\;i}$. We also introduce the internal order parameter $\Delta^{A}=E_{\;i}^{A}\Delta^{i}$. The action (87) then reduces to

[TABLE]

This is identical to the flat space case, with $\partial_{i}$ replaced by $E_{A}^{\;i}\partial_{i}$. We also need to change the UV cutoff to $\delta^{AB}E_{A}^{\;i}q_{i}E_{B}^{\;j}q_{j}=G^{ij}q_{i}q_{j}<\Lambda^{2}$. This in natural since we interpret $\Lambda^{2}$ as a range of the interaction mediated by $\Delta$, which should be defined in terms of the geodesic distance rather than the Euclidian distance. It follows that the flat space result (94) is modified to

[TABLE]

where $\mathbf{k}=E^{-T}\mathbf{q}$, or $k_{A}=E_{A}^{\;i}q_{i}$, and $g^{AB}=\Delta^{(A}\Delta^{B)*}=\delta^{\tilde{A}\tilde{B}}e_{\tilde{A}}^{\;A}e_{\tilde{B}}^{\;B}$ is the internal emergent metric. This is identical to the $G_{ij}=\delta_{ij}$ result (94), apart from the volume element $\sqrt{G}$, and the fact that it is the internal metric $g^{AB}$ that appears, rather than $g^{ij}$. It is then clear that minimizing (98) with respect to $g^{AB}$ gives

[TABLE]

with the same $\Delta_{0}$ of (96), which is $G$ independent. Thus, the emergent metric is proportional to the background metric in the ground state. This solution corresponds to emergent vielbeins $e_{\tilde{A}}^{\;A}\in O\left(2\right)$, or order parameters $\Delta^{A}=\Delta_{0}e^{2i\theta}\left(1,\pm i\right)$, which is the $p_{x}\pm ip_{y}$ configuration, and implies the SSB pattern

[TABLE]

described less formally in the main text. Note that fermion parity $\mathbb{Z}_{2,\left(-1\right)^{N}}$ is the $\mathbb{Z}_{2}$ subgroup of $U\left(1\right)_{L-\frac{\ell}{2}N}$ for odd $\ell$. For $\Delta^{j}$, we find the ground state configuration (15)

• a result that was stated previously in the literature Read and Green (2000); Hoyos et al. (2014); Moroz and Hoyos (2015); Moroz et al. (2016); Quelle et al. (2016), and is derived here to zeroth order in derivatives.

As described in Sec.E, we will ignore the massive Higgs fluctuations, and obtain $S_{\text{eff}}\left[\theta;A,G\right]$ by plugging the ground state configuration (15) into the functional Pfaffian (90).

E.5 Perturbative expansion

We now write $E_{A}^{\;i}=\delta_{A}^{i}+H_{A}^{\;i}$ and $e_{\tilde{A}}^{\;A}=\Delta_{0}\delta_{\tilde{A}}^{A}$ (which corresponds to $\Delta^{A}=\Delta_{0}\left(1,i\right)^{A}$) and expand (88) to second order in $H,A$. Due to $SO\left(2\right)_{L}$ gauge symmetry, the anti-symmetric part of $H$ can be interpreted as the Goldstone field, $\theta=\left(s_{\theta}/2\right)\varepsilon_{AB}H^{AB}$, as explained in Appendix D. The $p_{x}-ip_{y}$ configuration $\Delta^{A}=\Delta_{0}\left(1,-i\right)^{A}$ can be incorporated by changing the sign of one of the gamma matrices $\gamma^{\tilde{A}}$. The expansion in $H,A$ produces a splitting of the the propagator into an unperturbed propagator and vertices, $\mathcal{G}^{-1}=\mathcal{G}_{0}^{-1}+\mathcal{V}$, where $\mathcal{V}$ further splits as $\mathcal{V}=\mathcal{V}_{1}+\mathcal{V}_{2}$, where $\mathcal{V}_{1}$ ($\mathcal{V}_{2}$) is first (second) order in the fields. The terms in $\mathcal{V}_{2}$ are often referred to as contact terms. Using (24) we find the explicit form of $\mathcal{G}_{0}^{-1},\mathcal{V}_{1},\mathcal{V}_{2}$ in Fourier components,

[TABLE]

Here $\left(\cdots\right)_{p}$ denotes the $p$ Fourier component of the field $\left(\cdots\right)$, and we set $p=0$ in $\mathcal{V}_{2}$ since only this component will be relevant. The unperturbed Greens’s function is given explicitly by

[TABLE]

The perturbative expansion of $S_{\text{eff}}$ is obtained from (90) by using $\text{log}\left[\text{Det}\left(\cdot\right)\right]=\text{Tr}\left[\log\left(\cdot\right)\right]$, and expanding the logarithm in $\mathcal{V}$,

[TABLE]

where in the last line we kept explicit only terms at first and second order in $H,A$ (the term of zeroth order was described in the previous section). Writing the functional traces as integrals over Fourier components and traces over spinor indices, we then find

[TABLE]

where $\int_{q}=\int\frac{\text{d}^{2}q\text{d}q_{0}}{\left(2\pi\right)^{3}}$. We are interested in $S_{\text{eff}}$ to third order in derivatives, which amounts to expanding the above expression to $O\left(p^{3}\right)$, and evaluating the resulting traces and integrals. These computations are performed in the accompanying Mathematica notebook.

The result, focusing on terms relevant for $\eta_{\text{o}},\tilde{\eta}_{\text{o}}$ to $O\left(q^{2}\right)$, is compatible with the general effective action of Sec.III and Appendix C, as confirmed by comparing (104) to the perturbatively expanded $S_{\text{eff}}$. This comparison provides explicit expressions for all of the coefficients that appear in $S_{\text{eff}}$, as we now describe. The ground state pressure $P\left(\mu\right)$ diverges logarithmically, and is given by

[TABLE]

Directly computing the ground state density $n_{0}$ and leading odd viscosity $\eta_{\text{o}}^{\left(1\right)}$ one finds

[TABLE]

so the relations $n_{0}=P^{\prime}\left(\mu\right)$, and $\eta_{\text{o}}^{\left(1\right)}=-\left(\ell/4\right)n_{0}$, described in the main text, are maintained to leading order in the cutoff.

As explained in Appendix E.4, the cutoff $\Lambda$ corresponds to a non-vanishing interaction range, which softens the contact interaction in the model (13). With a space-independent metric, a smooth cutoff can easily be implemented by replacing

[TABLE]

for example, in the Fourier transformed Eq.(97), and should lead to the *exact *relations $n_{0}=P^{\prime}\left(\mu\right)$, $\eta_{\text{o}}^{\left(1\right)}=-\left(\ell/4\right)n_{0}$. However, a computation of the $q^{2}$ correction to $\eta_{\text{o}}$ requires a space-dependent metric, where a non-vanishing interaction range involves the geodesic distance and complicates the vertex $\mathcal{V}$ in (101) considerably. Moreover, all other coefficients in $S_{\text{eff}}$ converge, and we can therefore work with the simple contact interaction, $\Lambda=\infty$.

The coefficients $P^{\prime\prime},F_{1}^{\prime},F_{2},F_{3}$ were presented in Sec.E. The remaining coefficients $F_{4},F_{5},F_{6}$, are irrelevant for the quantities discussed in the main text, and are presented here for completeness,

[TABLE]

As stated in Sec.E, there is a sense in which the relativistic limit $\kappa\rightarrow 0$, or $m\rightarrow\infty$ reproduces the effective action of a massive Majorana spinor in Riemann-Cartan space-time Golan and Stern (2018); Hughes et al. (2013). In particular, in the limit $\kappa\rightarrow 0$ the dimensionless coefficients (16) are all quantized, as follows from dimensional analysis. Apart from $c$, only the coefficient $F_{1}^{\prime}$ is discontinuous at $\mu=0$ within this limit, with a quantized discontinuity $-\left(\ell/4\right)\left[F^{\prime}_{1}\left(0^{+}\right)-F^{\prime}_{1}\left(0^{-}\right)\right]=\left(\ell/2\right)/96\pi$ that matches the coefficient $\beta$ of the gravitational pseudo Chern-Simons term of Golan and Stern (2018). As anticipated in Golan and Stern (2018), the coefficient $c$ remains quantized away from the relativistic limit, while $F_{1}^{\prime}$ does not. Taking the relativistic limit of the dimensionful coefficients (109), one finds $F_{6}=0$, while $F_{4}=-\Delta_{0}^{2}F_{5}\neq 0$ describe a relativistic term which is second order in torsion, and was not written explicitly in Golan and Stern (2018); Hughes et al. (2013).

Finally, we note that our perturbative computation of the gCS term is analogous to the computations of Goni and Valle (1986); Van der Bij et al. (1986); Vuorio (1986a, b); Kurkov and Vassilevich (2018) for relativistic fermions, and reduces to these as $\kappa\rightarrow 0$.