Hall Conductivity and Hall Viscosity

PHYSICAL REVIEW RESEARCH 3, 013103 (2021)

Time-reversal odd transport in bilayer graphene: Hall conductivity and Hall viscosity

Wei-Han Hsiao Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA

(Received 23 March 2020; accepted 12 January 2021; published 1 February 2021)

We consider the time-reversal odd dynamics of the bilayer graphene at low energies in the quantum Hall regime. A generating functional for the effective action that captures the electromagnetic response to all orders in momentum and frequency is presented and evaluated to the third order in the space-time gradient O(∂3 ). In addition, we calculate the Hall viscosity and derive an explicit relationship with the q2 coefﬁcient of the Hall conductivity. It is reminiscent of the Hoyos-Son relation in Galilean invariant systems, which can be recovered in the limit of large ﬁlling factor N.

DOI: 10.1103/PhysRevResearch.3.013103

I. INTRODUCTION the low-energy excitations in the zero-energy bands under this circumstance were investigated in Refs. [9,10]. The family of multilayer graphite is one of the most This paper concentrates on the time-reversal odd responses intriguing paradigms in the realm of modern condensed mat- of bilayer graphene to dynamical and inhomogeneous external ter. The simplest model in the group, graphene, has drawn perturbations in a strong out-of-plane magnetic field in (2+1) enormous attention from both theoretical and experimental dimensions, which has been partially addressed recently in an communities [1]. Its linear dispersion at low energy makes independent work [11]. Here, we adopt a different approach it a low-dimensional example of a particle-hole symmetric that systematically generates the effective action as a func- ultrarelativistic Dirac fermion, the unconventional electronic tional of external gauge field Aμ to all orders in momentum property of which distinguishes it from other semiconductors and frequency. Using this, we compute the effective action as made up of ordinary nonrelativistic quasiparticles. In par- in the gradient expansion to explore the large-scale dynamics. ticular, when placed in a magnetic field, each Dirac point 1 e2 As a result, we show that for the low-energy model of bilayer possesses the anomalous Hall conductivity 2 h at the filling graphene in a background magnetic field, the Hall conduc- ν = factor 0[2], which is uniquely connected with the filled tivity in an inhomogeneous and time-dependent electric field Fermi sea of Dirac particles. Ei(ω, q) at filling factor N is Even more fruitful physics emerges as one stacks two layers of graphene on top of each other. In the AB-stacked N 1 − 3N2 ω2 2N2 − 1 σ (ω, q) ≈ 1 + (q)2 + , (1) configuration as shown in Fig. 1, the low-energy projection of H π ω2 2 2 4N c 2N the model forms another family without any relativistic ana- ω log: a particle-hole symmetric two-band semiconductor with where and c are the magnetic length and cyclotron fre- parabolic dispersion [4]. It also serves as a model possessing a quency. We also establish a relationship between the Hall 2 Fermi surface with a Berry phase of 2π and is also a candidate viscosity and the coefficient of the (q) term. In particular, ω = for the dual description of the ν = 1 fractional quantum Hall with a high-energy Landau level cutoff Nc, the static ( 0) state in the context of fermion-vortex duality [5]. result reduces to As the background magnetic field is turned on, Landau σ (q) = σ (0) + (q)2[2η − (B) − O(ln N )]. (2) levels form in bilayer graphene as well. The spectrum, nev- H H H c ertheless, possesses two zero-energy bands [6]. This feature reshapes our understanding of low-energy physics in quantum Hall systems [7]. In particular, in a large magnetic field, the conventional wisdom and machinery of the lowest Landau level projection has to be modified since the low-energy sector contains more than holomorphic wave functions in the symmetric gauge and the system can host exotic quantum phases in the lowest Landau level [8]. The dielectric property and

Published by the American Physical Society under the terms of the FIG. 1. The schematic plot for the AB-stacked bilayer graphene. Creative Commons Attribution 4.0 International license. Further The thick bonds represent the upper lattice, while the thin ones repre- distribution of this work must maintain attribution to the author(s) sents the lower layer. The overlapped orange sites have labels A˜ = B. and the published article’s title, journal citation, and DOI. The plot was generated using the PYTHON package PYBINDING [3].

2643-1564/2021/3(1)/013103(9) 013103-1 Published by the American Physical Society WEI-HAN HSIAO PHYSICAL REVIEW RESEARCH 3, 013103 (2021)

It is reminiscent of the Hoyos-Son relation [12] for Galilean invariant systems. An explicit formula is derived to compute −1 −1 higher-order corrections in N and Nc . For Galilean invariant systems, the Hoyos-Son relation has been systematically and thoroughly discussed. For instance, Ref. [13] delivers a concrete relationship between S the nonlocal Hall conductivity and the Hall viscosity. This FIG. 2. Diagrammatic expansion of the effective action eff . framework has recently been generalized to anisotropic and Each solid line represents a Feynman propagator D and each wavy lattice-regularized systems in Ref. [14]. As for graphenelike line corresponds to the insertion of the external potential v.Thefirst and the third diagrams correspond to the trace tr[Dv], and the dia- systems, Ref. [15] tackles a similar problem for a single gram in the middle corresponds to the two-point trace − i tr[DvDv]. layer in a strong magnetic field and low temperature, whereas 2 Ref. [16] reports a Hoyos-Son-like relation in the interaction- dominating regime using the hydrodynamic approach in the where j¯ denotes the average charge in the ground state. The absence of a magnetic field. This paper aims to provide a polarization tensor μν (q) encodes the response functions. different generalization by considering the zero-temperature In (2+1) dimensions, the gauge symmetry alone fixes the dynamics of a rotationally invariant and particle-hole symmet- form of the effective Lagrangian Leff . Organized by the num- ric system breaking both Galilean and Lorentz symmetries. ber of derivatives, the Lagrangian is This paper is organized as follows: In Sec. II we first review the methodology of the one-loop effective action from k 1 L = AdA + E2 − B2 a microscopic model and introduce the low-energy two-band eff 4π 2 2μ model for bilayer graphene. We then start presenting this + αE · (∇B) + β ijE ∂ E + O(∂4). (5) work by first deriving the Feynman rules and the generating i t j functional for the polarization tensor. What follows is the Computing the functional determinant for a specific model time-reversal odd effective action computed to cubic order in yields the parameters k, , μ, α, and β. space-time gradients with an emphasis on the coefficient of For a fermionic system, computing Eq. (3) amounts to 2 Hall conductivity at the order of (q ) and an investigation evaluating the functional determinant of the fermion action. of its relationship with the Hall viscosity. In Sec. III,we Formally, one can decompose the microscopic Lagrangian for construct the stress tensor and compute the Hall viscosity and the fermion ψ into the free part iψ†D−1ψ and the potential orbital magnetic susceptibility for our model. This provides part −v(x)ψ†ψ. Such a decomposition is straightforward for numerical support for the observation established in Sec. II. a free fermion system in an external potential. If a two-particle Finally we revisit the conductivity tensor using the Kubo interaction is present, one can first introduce an auxiliary field formula in Sec. IV and derive an exact algebraic relation that by the Hubbard-Stratonovich transformation to decouple the connects the Hall conductivity and the Hall viscosity in the two-particle term. In this way the fermion part of the theory absence of space-time symmetry. We then conclude the paper. can be integrated in the path integral, resulting in a functional Details of the computations and an alternative derivation of of v. Expanding the functional to the second order of v,it the stress tensor are given in the Appendixes for completeness. reads S =− −1 + II. THE EFFECTIVE ACTION eff itr ln[D iv] i A. Methodology =−itr ln[D−1] + tr[Dv] − tr[DvDv] + O(v3). (6) 2 The electromagnetic response can be compactly summa- rized in the effective action as a functional of the U (1) gauge This formula has an intuitive diagrammatic representation potential. Starting with a microscopic model for the matter shown in Fig. 2. The inverse of action D corresponds to the field ψ defined by the action S[ψ], one gauges the charge Feynman propagator of the free theory and the potential v is symmetry by coupling the charge and current densities jμ = the vertex. One can systematically compute Eq. (6)usingthe i (ρ, j ) with the gauge field Aμ. The effective action is defined standard perturbation methods in quantum field theory. as follows: μ † iS[ψ]+i jμA B. The model for bilayer graphene Seff [Aμ] =−i ln Dψ Dψ e . (3) Let us now apply the above machinery to the low-energy If the external electric and magnetic fields E and B take model of AB-stacked bilayer graphene [6]. We depict the constant values, Eq. (3) can be computed and serves as an lattice structure in Fig. 1. The minimal low-energy model for example of Euler-Heisenberg effective action [17]. As far as each valley in the Brillouin zone contains two copies of the the linear response is concerned, S is usually expanded as Dirac fermions and a hopping amplitude 2m bridging sites B eff ˜ ω a multinomial in Aμ.Inthed-dimensional Fourier space, the and A. In the low-energy regime m, the four-band model effective action under a Gaussian approximation assumes the can be projected to the two dominant bands. The model for form valley K is dd q 1 π 2 φ S = ¯μ + − μν , =− 0 ,ψ= A , eff [Aμ] [ j Aμ Aμ( q) (q)Aν (q)] (4) H π † 2 K φ (7) (2π )d 2m ( ) 0 B˜

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with the cyclotron frequency ωc = B/m. The degeneracy of each Landau level, as in a two-dimensional electron gas (2DEG) in the symmetric gauge, is encoded by the other set of ladder operators [b, b†] = 1 which generates the angu- 1 lar momentum z = (r ×p −p × rˆ)z. (See Ref. [19].) Note 2 that in this system we no longer treat z as the canonical angular momentum. The algebraic structure of the second pair of commuting ladder operators still holds. Denoting the Hilbert space with √1 (b†)m|0=|m, the complete basis is m! {| | ≡| } ξ = ε − μ n) m nm) . Writing n n , the inverse of the ker- −1 = ∂ − ε − μ | | nel iD i t n,n( n ) nm)(nm is d i|nm)(nm| D(t, t ) = e−i(t−t ) . (11) 2π − ξ + i sgn(ξ ) n,m n n FIG. 3. The Landau level spectrum of Eq. (10) and the Landau level indices associated with each band. The parabolas depict the Next, we turn on the perturbation on top of A¯μ → A¯μ + Aμ, dispersion in the absence of the background magnetic ﬁeld. The leading to the variation of the Hamiltonian H → H + v(t, x), red dashed line represents the Fermi energy , separating occupied F where the vertex v is black bands and the empty gray one.

π = π − iπ π = p + A 1 where x y and i i i is the kinematic mo- v(t, x) = A0 − { i, Ai} mentum. ψK is the spinor storing the dominant two bands. For 2m this model, k and the dielectric constant were discovered in 1 − 2 Ref. [9]. The focus of this paper is the computation of α and β. − 0(Ax iAy ) , + 2 (12) We first derive the propagator D for model (7) in a magnetic 2m (Ax iAy ) 0 field. Turning on a finite background magnetic field in the ¯ , ¯ = B − , symmetric gauge (Ax Ay ) 2 ( y x), the spectrum of the with the momentum operators system consists of nontrivial Landau levels. It also introduces = −1/2 the length scale of magnetic length B .Tosolvethe 0 π spectrum, it is convenient to introduce the ladder operators x = , (13a) π † 0 i a = √ π, (8a) 2 0 −iπ y = . (13b) i iπ † 0 a† =−√ π †, (8b) 2 In order to perform a gradient expansion, the Fourier trans- which satisfies [a, a†] = 1. The Hilbert space of the (r,p) form needs to be introduced properly, since the coordinates operators is then organized partly using the eigenstates of the x = (x, y) are treated as operators on Hilbert space. Given a operatorn = a†a, {|n|n ∈ Z+}. The Hamiltonian (7) can then c-valued vector k, we recognize that be shown to have the eigenstates [18] · −| |2 † ¯ † ¯ 0 0 eik x = e k eia keiakeib keibk, (14) |0) = , |1) = (9a) |0 |1 1 || |− where the dimensionless complex momenta are k = √ (k − | = √ sgn(n) n 2 , | | , 2 x n) || | n 2 (9b) ∗ 2 n iky ) and k¯ = k . Each operator in v(t, x) is Fourier expanded associated with the spectrum (Fig. 3) as an integral of these exponential operators weighted by the Fourier coefficients. For concrete instances, we Fourier εn = sgn(n)ωc |n|(|n|−1) (10) transform terms which are linear in the external field:

3 d k 2 † ¯ † ¯ A = e−iωt−|k| A (k)eia keiakeib keibk, (15a) 0 (2π )3 0 3 −|k|2−iωt −1 d k ie † ¯ 0 a † ¯ { , A }= √ A (k)eib keibk , eia keiak , (15b) x x 3 x − † 2m (2π ) 2m a 0 3 −|k|2−iωt −1 d k e † ¯ 0 a † ¯ { , A }= √ A (k)eib keibk , eia keiak . (15c) y y 3 y † 2m (2π ) 2m a 0

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† Because of the ±i involved in the definitions of (a, a ), perturbation in the x direction now resembles σy and vice versa. This twist will lead to an extra minus in the effective action. We have built the machinery for computing the traces in Eq. (6). As far as the response functions are concerned, the second diagram in Fig. 2 is the only nontrivial one. The first diagram gives rise to the ground-state charge density, while in the third diagram, the contact term or diamagnetic current vanishes exactly for this model. For the vertices v in Eqs. (15a)–(15c), the trace assumes the general form i i d d i(nm|v(t )|n m )i(n m |v(t )|nm) − =− −i(t −t ) −i (t−t ) . tr[DvDv] dt dt e e (16) 2 2 (2π )2 [ − ξ + i sgn(ξ )][ − ξ + i sgn(ξ )] n,m,n,m n n n n The denominator does not depend on the angular momenta m and m. Thus, the trace over {|m, |m} space can be computed separately. To this end, we decompose the vertices (15a)–(15c)into 3 d k 2 † ¯ A = A (k)e−iωt e−|k| γ 0(k)eib keibk, (17a) 0 (2π )3 0 − 3 1 d k −iωt −|k|2 x ib†k¯ ibk { x, Ax}= Ax (k)e e γ (k)e e , (17b) 2m (2π )3 − 3 1 d k −iωt −|k|2 y ib†k¯ ibk { y, Ay}= Ay(k)e e γ (k)e e . (17c) 2m (2π )3 | ib†k¯ ibk| | ib†k¯ ibk | 2π |k|2 δ(2) + In this way, we trace out m e e m m e e m , obtaining a delta function 2 e (k k ). The time and frequency integrals for (t, t ) and (, ) can be performed with δ functions and residue calculus. The nontrivial summands remaining incorporate only the virtual processes between filled and empty Landau levels [sgn(ξn )sgn(ξn ) < 0]. Suppose the Fermi energy is pinned between the Nth and the (N + 1)th Landau levels as shown in Fig. 3. We then have 3 γ μ − γ ν γ μ − γ ν i 1 d k −|k|2 nn ( k) nn(k) nn( k) nn (k) − tr[DvDv] =− e Aμ(−k)Aν (k) + , (18) 2 2π2 (2π )3 ξ − ξ + ω + i ξ − ξ − ω + i n>N,nN n n n n γ μ |γ μ| with nn denoting the matrix element (n n ). Equation (18) is an exact result of the trace to all orders in frequency and wave γ μ α | |2 numbers. The matrix elements nn (k) can be written as linear combinations of the associated Laguerre polynomials Ln ( k ). Some useful identities are documented in Appendix A. The long-wavelength and low-frequency effective theory is derived upon expanding the μν (k)insmallω and |k|. By comparing Eqs. (18) and (3), the time-ordered polarization tensor is identified as −|k|2 μ ν μ ν e γ (−k)γ (k) γ (−k)γ (k) μν (ω, k) =− nn n n + n n nn . (19) π2 ξ − ξ + ω + i ξ − ξ − ω + i n>N,nN n n n n

C. Polarization tensors and transport coefficients For nonrelativistic fermions, the level corresponds pre- Equation (19) contains complete information about trans- cisely to the number of filled Landau levels. The level here, N, port in our model of bilayer graphene. To avoid the ambiguity is the index of the largest filled Landau level. As opposed to of doubly degenerate bands, we fill them simultaneously by the bilayer graphene model, the number of filled Landau lev- taking N > 2 and compute the polarization tensor to the els, and thus the interpretation of N, could be ambiguous due specified limit or desired order in momentum. The transport to the filled negative energy bands. This ambiguity is resolved observables can be extracted after properly modifying the ep- by understanding the net contribution from the Fermi sea. By redoing the computation for N =−1, it is straightforward silon prescription. For instance, with a homogeneous external 1 −iωt 00 0i to confirm that negative energy bands contribute to AdA. electric field of form E = Eωe , = = 0 and the re- 4π + tarded xy can be computed exactly to all orders in frequency, Therefore, N 1 is understood as the number of filled bands R with non-negative energies. ω 3ω4 − ω2ω2 1 xy i 4N c 2N c A ∂ A − A ∂ A = , (20) We refer to the coefficient of the term 2 ( x t y y t x ) R π ω4 − 2ω2 ω2 − ω2 2 2 2 + 4N c + c as the Hall conductivity σH [20]. To order (q) and ω ,the Hall conductivity is with ω+ = ω + i . In the presence of a spatial fluctuation, at large scale, the ∂ ∂ /ω 2 2 2 effective action is organized by powers of ( i ) and t c.By σH N + 1 ω 2N − 1 ∂3 = 1 + (q)2 − N + , (22) expanding and evaluating the sum to order O( ), the targeted σ (0) 4N ω2 2N2 parity-odd part of the Lagrangian is H c N 3N2 − 1 2 L (odd) =− AdA− E ·∇B with σ (0) = N/(2π ). To make more sense out of this result, eff π π H 4 4 2 we can look at the static large N limit, in which the hole 2N2 − 1 1 band becomes insignificant and we should recover the physics − ijE ∂ E . (21) π ω2 i t j →∞ 4 N 2 c of nonrelativistic 2DEG. Taking N ,Eq.(22) can be

013103-4 TIME-REVERSAL ODD TRANSPORT IN BILAYER … PHYSICAL REVIEW RESEARCH 3, 013103 (2021) organized into the form In Appendix B, we present another distinct way to derive the above results, where a natural conjecture for the model 2 2 σH N /(8π ) τ = 1 + (q)2 − N . on a general curved manifold is postulated and ij is obtained N/(2π ) N/(2π2 ) by varying the Hamiltonian with respect to the vielbein field. A recent work [11] derives the stress tensor by constructing Formally, it produces the Hoyos-Son relation [12,15]for strain generators and computing their commutators with the 2DEG, model Hamiltonian [13]. This approach does not evade the 2 difficulty discussed above. Taking the pseudospin degree of σH ηH 2π = 1 + (q)2 − B2 (B) , (23) freedom as an example, it is not obvious that the pseudospin σ (0) n ν ω H c matrices generate physical rotations and should be included in if we identify the number density as N/(2π2 ). the strain generators. τ τ The Hoyos-Son result establishes a relation between the Plugging xx and xy into Eq. (24) yields the Hall viscosity ω → Hall viscosity ηH (ω) at long wavelengths, orbital magnetic to all orders in frequency. Particularly in the static limit − ∂2 0, we have susceptibility ∂B2 (B), and the coefficient of Hall conductivity at order (q)2 based on the Galilean symmetry of the 1 2 microscopic physics. In addition, the 2DEG result is physi- ηH (ω = 0) = (N + 1). (26) 8π2 cally understood as the ratio of ηH to the charge density n. Starting from the model Eq. (7), we would not expect this Caution is required to compute the orbital magnetic sus- relation to persist at finite N because it lacks Galilean sym- ceptibility − (B), which is the negative of the second metry and the charge density depends on the regularization derivative of the energy density as a function of the back- of the bottom of the Fermi sea. Nonetheless, motivated by ground field. Suppose we naively evaluate the energy density observations at large N, we explore if a similar or approximate per Landau level, (B): relation exists without any particular space-time symmetry. In particular, we wish to clarify the roles of Hall viscosity and N ωc orbital magnetic susceptibility. (B) = sgn(n) |n|(|n|−1). (27a) 2π2 n=−∞ III. HALL VISCOSITY AND ORBITAL MAGNETIC After canceling the summands from n =−√N to n = N,the SUSCEPTIBILITY OF BILAYER GRAPHENE ∞ − remaining sum is proportional to n=N+1 n(n 1), which To proceed with the last observation, our goal is to compute is severely divergent. To extract a finite result, we regu- the Hall viscosity η and the orbital magnetic susceptibility of larize√ the sum by introducing a natural cutoff Nc obeying H ω − = the model Eq. (7). Within the framework of linear response, c Nc(Nc 1) m, which constrains the validity of the ηH is given by the correlation function of the stress tensors low-energy model. The regularized sum now reads τxxτxy, which can be expressed using the Kubo formula as [15,21] ω Nc (B) =− c n(n − 1). (27b) π2 1 (τ ) (τ ) 2 = + η (ω) = Im xx ba xy ab , (24) n N 1 H 2 2 2 π ω+ − (ξa − ξb) a,b The sum can be evaluated approximately with the Euler- where a, b label bands above and below the Fermi energy. Maclaurin formula. In a double expansion of large N and ω / Defining the stress tensor nevertheless requires some caution small c m, the leading contribution is for the following reason: The stress tensor can be defined as N2 1 Nω the response of the Hamiltonian with respect to the variation − B ≈− + c . ( ) π π ln (28) of the metric gij, but there is no obvious covariant way of 2 16 m coupling (7) to a general curved manifold. We circumvent ∼ /ω this conceptual obstacle as follows: Instead of the projected Since N is bounded from above by Nc m c,atlargeN the logarithmic term is subleading. Consequently, both Eqs. (26) low-energy Hamiltonian (7), we revisit the original model →∞ consisting of two Dirac spinors, where one consistently de- and (28) coincide with the 2DEG results in the limit N fines the action over a curved manifold [22], and hence the [24]. In the rest of this paper, we exploit these observations and establish a concrete algebraic identity. stress tensor τij. We derive the stress tensor in this manner and again project the components to the low-energy bands [23]. τ The components of ij are found to be IV. AN ALGEBRAIC RELATION FROM 1 1 THE KUBO FORMULA τ =− π 2σ − σ {π ,π }, xx x x y x y (25a) m 2m To better understand the relationship, we reexamine the 1 1 Hall conductivity from the more conventional perspective of τ = π 2σ − σ {π ,π }, yy y x y x y (25b) the Kubo formula. It is algebraically equivalent to the com- m 2m putation of the two-point functions μν . Nonetheless, as we 1 † will show in a moment, it can make our speculation more τxy =− σy{π ,π}. (25c) 4m transparent. The current operator in the first quantized form

013103-5 WEI-HAN HSIAO PHYSICAL REVIEW RESEARCH 3, 013103 (2021) is given by the symmetrized velocity operator restoration of Galilean symmetry, manifest asymptotically in the limit of a large filling factor. Explicit examples are the i = 1 i ,δ − . j (r) k (r rk ) (29) forms of Hall conductivity, Hall viscosity, orbital magnetic 2m k susceptibility, poles of transport coefficients, and the Hoyos- Applying the Kubo formula for Hall conductivity to the Son relation, although, in terms of hydrodynamic relations current (29), we obtain [25], it is not yet clear in what sense the charge current and momentum density approach each other in the same limit. −q q¯ q −q¯ x √ π † √ π y √ π † √ π , e 2 e 2 , e 2 e 2 Established exact formulae can be utilized for interpolating σ = Im ba ab . (30) H 2 2 |q|2 2 2 between the asymmetric model and the Galilean symmetric 4m π e [ω+ − (ξ − ξ ) ] a,b a b paradigm. To move forward, this paper opens various directions. Em- This formula can be quickly justified by considering 1 ij, iω pirical ones include the real-time effective theory at finite which reduces to the conductivity tensor in the temporal gauge temperature using the Schwinger-Keldysh formalism [26], A = 0. The above formula can be straightforwardly expanded 0 and interaction-generated transport properties owing to either in small q = (q). To proceed, let us exploit rotational invari- an instantaneous Coulomb interaction or a mixed-dimensional ance and take q = (0, q): Maxwell term [27,28]. These developments will be critical in −q q¯ x √ π † √ π order to connect the single-particle toy model in the quantum , e 2 e 2 Hall regime with experimental investigations of graphene ma- iqτ q24 ≈ x + 2/ − xx − π 2, x . terials [29], which are usually conducted in a hydrodynamic (1 (q ) 4) x ωc 4 regime with strong disorder or interactions [30,31]. Equally q −q¯ interesting are the inclusion of lattice effects in the generalized y √ π † √ π , e 2 e 2 Hoyos-Son relation [32], generalization of the linear response 2iqτ qσ τ q22 theory to graphite multilayers as in Ref. [33], and clarifying ≈ y + q 2/ + xy + z yy − π 2, . ν = (1 ( ) 4) ω ω x y the distinction between dual descriptions of the 1 frac- c c 4 tional quantum Hall state [5,34]. To organize the products, we first observe the current op- To conclude, we have determined the time-reversal odd erators map the nth Landau level to (n ± 1)th, whereas the electromagnetic response for the low-energy model of bilayer stress tensor operators only generate transitions between n → graphene to quadratic order in momentum and frequency, n, n ± 2. Their products thus have no finite summand and as endowed a precise definition of the stress tensor to the a result, there is no term linear in q. Another nontrivial obser- low-energy projected model, and established a conductivity- vation is at the level of the matrix elements (τxx)ba (τxy)ab = viscosity relationship in the absence of obvious space-time (τxx)ba (iσzτyy )ab even though τxy = (iσzτyy)ab in general. Con- symmetry. We investigated the limit in which the symmetry is sequently, expanding the numerator of (30) to the second order restored and provided support from concrete computation at in (q) yields the following identity, the operator level. In addition to the conclusions in the above, the effective action derived and the vielbein formulation intro- 2 2 duced in this paper can be further applied to the exploration σH (ω, q) ≈ σH (ω) + (q) ηH (ω) of unknown facets of this model. x 2 y 2 x y ( )ba π , + π , ( )ab −Im x ab x ba , (31) 2 2 2 ACKNOWLEDGMENTS 4πm[ω+ − (ξ − ξ ) ] a,b a b The author thanks Yu-Ping Lin, M. Lapa, and F Setiawan which unambiguously identifies the role of Hall viscosity as for comments on the manuscript. This work is supported by a 2 part of the coefficient of (q ) . The sum in the second line of Simons Investigator Grant from the Simons Foundation. Eq. (31) is not expressed directly with a physical observable ω − N2 =− − at finite . In the static limit, it sums to 2π (B) 1 ln N , and it reduces to Eq. (2) in the static limit ω → 0. APPENDIX A: COHERENT STATE ALGEBRAS 16 Nc At finite frequency, despite its lack of lucid physical inter- 1. Computing traces involving b operators pretation, Eq. (31) provides a decomposition that generates corrections to the Hoyos-Son relation in powers of ω2 and In the two-point function, the trace over angular mo- N−1. mentum subspace can be performed separately, since the denominator of the propagator does not involve the angular momentum quantum number. The quantity we wish to com- V. DISCUSSION AND CONCLUSION pute is The model Eq. (7) is often considered a hybridization of † ¯ † ¯ Dirac and nonrelativistic fermions, capturing the particle-hole m|eib keibk|m m|eib k eibk |m structure of the former and the massive parabolic dispersion m,m of the latter. The results derived in the main text entail in ib†k¯ ibk ib†k¯ ibk many ways that the features of nonrelativistic fermions, or the = m|e e e e |m. m

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From here we replace the sum over m with an integral over all coherent states: † ¯ † ¯ 2 † ¯ † ¯ † dμ(q) q|eib keibkeib k eibk |q= dμ(q) e−|q| 0|eqb¯ eib keibkeib k eibk eqb |0. (A1)

Since [b, b†] = 1, we can use the identity eAeB = eBeAe[A,B] and simplify the brackets to get

2 2π 2 πe|k| δ(2)(k + k ) = e|k| δ(2)(k + k). (A2) 2

γμ 2. Matrix elements nn (k) Here, we document the matrix elements for the vertices used in the main text. The fundamental ingredient is n|eia†keiak¯ |n. Let us evaluate

† s ¯ s n n s ¯ s √ ia†k iak¯ (ia k) (iak) (ik) (ik) n|e e |n = n| |n = δ − , − n!n !. (A3) s! s! s!s!(n − s)! n s n s s,s s s m < = The sum can be expressed in terms of generalized Laguerre polynomials Ln (x). If n n , we have to evaluate s at s s + n − n, † ¯ n! n|eia keiak|n=(ik¯ )n −n Ln −n(|k|2 ). (A4a) n! n On the other hand, if n n,wehavetoevaluateats = s + n − n, ia†k iak¯ n−n n ! n−n 2 n|e e |n =(ik) L (|k| ). (A4b) n! n γ μ γ μ = γ μ − ∗ Together with the eigenstates (9a) and (9b), it is then straightforward to compute nn (k) and nn(k) [ nn ( k)] . For deﬁniteness, we consider the case n > 1 and n < n. γ 0 = a. nn (k). For n 0 and 1, 1 γ 0 (k) = √ (ik)|n|, (A5) n0 2(|n|!)

1 γ 0 (k) = √ (ik)n−1Ln−1(|k|2 ). (A6) n1 2(n!) 1 For |n| > 1, ⎧ ⎪ sgn(n ) n−|n| (|n|−2)! n−|n | | |2 + 1 n−|n| |n|! n−|n | | |2 , | |, ⎨ 2 (ik) (n−2)! L|n|−2 ( k ) 2 (ik) n! L|n| ( k ) n n γnn = (A7) ⎪ ⎩ 1 ¯ |n |−n n! |n |−n | |2 + sgn(n ) ¯ |n |−n (n−2)! |n |−n | |2 , < | |. 2 (ik) |n|! Ln ( k ) 2 (ik) (|n|−2)! Ln−2 ( k ) n n

γ i b. nn (k). It is convenient to deﬁne γ + = 1 σ + σ { , ia†k iak¯ },γ− = γ + − †, (k) 2 ( x i y ) a e e (k) [ ( k)] (A8) in terms of which i γ x (k) = √ [γ +(k) − γ −(k)], (A9a) 2 m 1 γ y(k) = √ [γ +(k) + γ −(k)]. (A9b) 2 m For n > 1, ⎧ ⎪ 1 n−|n|−1 |n|! n−|n |−1 | |2 + n−|n |−1 | |2 , | |+ , ⎨ 2 (ik) (n−2)! L|n| ( k ) L|n|−1 ( k ) n n 1 γ + = nn (k) (A10) ⎩⎪ | |− + | |− + 1 ¯ |n |−n+1 (n−2)! − n n 1 | |2 +| | n n 1 | |2 , < | |+ , 2 (ik) |n|! (n 1)Ln−1 ( k ) n Ln−2 ( k ) n n 1

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⎧ ⎨⎪ sgn(n ) |n |−n−1 n! |n |−1−n | |2 + |n |−n−1 | |2 , | | + , 2 (ik) (|n|−2)! Ln ( k ) Ln−1 ( k ) n n 1 γ + = nn(k) (A11) ⎩⎪ −| |+ −| |+ sgn(n ) ¯ n−|n |+1 (|n |−2)! | |− n n 1 | |2 + n n 1 | |2 , | | < + . 2 (ik) n! ( n 1)L|n|−1 ( k ) nL|n|−2 ( k ) n n 1 Using these matrix elements, the generating functional (18) can be computed in a straightforward manner.

APPENDIX B: ANOTHER DERIVATION OF STRESS TENSOR We derive here the stress tensor with a conjectured curved space generalization of model (7)[35]. Let us consider a curved a = δ a b space endowed with the metric gij(x). A matrix field vielbein e j (x) can be introduced via the relation gij abei e j, where , = , i i b = δb ij = δab i j a b 1 2 are local SO(2) rotation indices [36]. The inverse field Ea fulfills Eaei a and g EaEb . With the vielbeins, ∂ ∂ → i ∂ i = i ± i the spatial gradient operator a can be promoted to a curved manifold following a Ea i. Let us further denote E± E1 iE2. The natural generalization of Hamiltonian (7) in the curved space reads 1 2 i † j H = d x [(E−πiψ )(E−π jχ ) + H.c.], (B1) 2m ψ χ φ φ where and are the Grassmannian density of A and B˜ and “H.c.” refers to the Hermitian conjugate. To derive the stress tensor relevant for viscosity computations, we turn on a slightly curved manifold with the metric gij = δij + δgij and assume δgij to be homogeneous. Under this circumstance, τij assumes the following form, 1 ∂H ∂H τ = ea + ea . (B2) ij i j j ∂ i 2 ∂Ea Ea Applying this formula to (B1), we arrive at identical results, Eqs. (25a)–(25c).

[1] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, [12] C. Hoyos and D. T. Son, Hall Viscosity and Electromagnetic and A. K. Geim, The electronic properties of graphene, Rev. Response, Phys.Rev.Lett.108, 066805 (2012). Mod. Phys. 81, 109 (2009). [13] B. Bradlyn, M. Goldstein, and N. Read, Kubo formulas for [2] V. P. Gusynin and S. G. Sharapov, Unconventional Integer viscosity: Hall viscosity, Ward identities, and the relation with Quantum Hall Effect in Graphene, Phys. Rev. Lett. 95, 146801 conductivity, Phys.Rev.B86, 245309 (2012). (2005). [14] P. Rao and B. Bradlyn, Hall Viscosity in Quantum Systems With [3] D. Moldovan, M. Andelkovic, and F. Peeters, pybinding v0.9.5: Discrete Symmetry: Point Group and Lattice Anisotropy, Phys. a Python package for tight-binding calculations (2020). Rev. X 10, 021005 (2020). [4] E. McCann and M. Koshino, The electronic properties of bi- [15] M. Sherafati, A. Principi, and G. Vignale, Hall viscosity and layer graphene, Rep. Prog. Phys. 76, 056503 (2013). electromagnetic response of electrons in graphene, Phys. Rev. [5] D. F. Mross, J. Alicea, and O. I. Motrunich, Bosonic Analogue B 94, 125427 (2016). of Dirac Composite Fermi Liquid, Phys.Rev.Lett.117, 136802 [16] U. Briskot, M. Schütt, I. V. Gornyi, M. Titov, B. N. Narozhny, (2016). and A. D. Mirlin, Collision-dominated nonlinear hydrodynam- [6] E. McCann and V. I. Fal’ko, Landau-Level Degeneracy and ics in graphene, Phys.Rev.B92, 115426 (2015). Quantum Hall Effect in a Graphite Bilayer, Phys.Rev.Lett.96, [17] M. Katsnelson, G. Volovik, and M. Zubkov, Euler-Heisenberg 086805 (2006). effective action and magnetoelectric effect in multilayer [7]K.S.Novoselov,E.McCann,S.V.Morozov,V.I.Fal’ko,M.I. graphene, Ann. Phys. 331, 160 (2013). Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and A. K. Geim, [18] We use the notation |·) to denote the doublet wave functions. Unconventional quantum Hall effect and Berry’s phase of 2π in [19] J. K. Jain, Composite Fermions (Cambridge University Press, bilayer graphene, Nat. Phys. 2, 177 (2006). Cambridge, UK, 2007). [8] Z. Papic´ and D. A. Abanin, Topological Phases in the Ze- [20] This is not the most conventional definition, yet it is consistent roth Landau Level of Bilayer Graphene, Phys.Rev.Lett.112, when the extra minus sign from the electron charge is accounted 046602 (2014). for. [9] T. Misumi and K. Shizuya, Electromagnetic response and [21] I. V. Tokatly and G. Vignale, Lorentz shear modulus of a two- pseudo-zero-mode Landau levels of bilayer graphene in a mag- dimensional electron gas at high magnetic field, Phys. Rev. B netic field, Phys.Rev.B77, 195423 (2008). 76, 161305(R) (2007). [10] K. Shizuya, Pseudo-zero-mode Landau levels and collective [22] D. X. Nguyen and A. Gromov, Exact electromagnetic response excitations in bilayer graphene, Phys. Rev. B 79, 165402 of Landau level electrons, Phys.Rev.B95, 085151 (2017). (2009). [23] D. X. Nguyen, G. Wagner, and S. H. Simon, Quantum Boltz- [11] M. Imran, Quantizing momentum transport in bilayer graphene, mann equation for bilayer graphene, Phys.Rev.B101, 035117 arXiv:1909.09608. (2020).

013103-8 TIME-REVERSAL ODD TRANSPORT IN BILAYER … PHYSICAL REVIEW RESEARCH 3, 013103 (2021)

[24] We note that the cutoff-dependent logarithm is not an artifact [30] B. N. Narozhny and M. Schütt, Magnetohydrodynamics in of the sharp cutoff, as it also appears in other regularization graphene: Shear and Hall viscosities, Phys.Rev.B100, 035125 schemes. (2019). [25] M. Greiter, F. Wilczek, and E. Witten, Hydrodynamic relations [31] B. N. Narozhny, Optical conductivity in graphene: Hydrody- in superconductivity, Mod. Phys. Lett. B 03, 903 (1989). namic regime, Phys.Rev.B100, 115434 (2019). [26] C. Fräßdorf, Abelian Chern-Simons theory for the fractional [32] F. Harper, D. Bauer, T. S. Jackson, and R. Roy, Finite- quantum Hall effect in graphene, Phys. Rev. B 97, 115123 wave-vector electromagnetic response in lattice quantum Hall (2018). systems, Phys. Rev. B 98, 245303 (2018). [27] E. C. Marino, L. O. Nascimento, V. S. Alves, and C. M. Smith, [33] W.-H. Hsiao, Landau quantization of multilayer graphene Interaction Induced Quantum Valley Hall Effect in Graphene, on a Haldane sphere, Phys.Rev.B101, 155310 Phys. Rev. X 5, 011040 (2015). (2020). [28] W.-H. Hsiao and D. T. Son, Duality and universal transport [34] W.-H. Hsiao and D. T. Son, Self-dual ν = 1 bosonic quan- in mixed-dimension electrodynamics, Phys.Rev.B96, 075127 tum Hall state in mixed-dimensional QED, Phys.Rev.B100, (2017). 235150 (2019). [29] A. I. Berdyugin, S. G. Xu, F. M. D. Pellegrino, R. Krishna [35] The author thanks Dam Thanh Son for suggesting a derivation Kumar, A. Principi, I. Torre, M. Ben Shalom, T. Taniguchi, from this perspective. K. Watanabe, I. V. Grigorieva, M. Polini, A. K. Geim, and [36] M. Nakahara, Geometry, Topology and Physics, 2nd ed., D. A. Bandurin, Measuring Hall viscosity of graphene’s elec- Graduate Student Series in Physics (Taylor & Francis, London, tron ﬂuid, Science 364, 162 (2019). 2003).

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