Topological Sachdev-Ye-Kitaev Model

Pengfei Zhang1 and Hui Zhai1, 2 1Institute for Advanced Study, Tsinghua University, Beijing, 100084, China 2Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China (Dated: March 6, 2018) In this letter we construct a large-N exactly solvable model to study the interplay between interaction and topology, by connecting Sacheve-Ye-Kitaev (SYK) model with constant hopping. The hopping forms a band structure that can exhibit both topological trivial and nontrivial phases. Starting from a topologically trivial insulator with zero Hall conductance, we show that interaction can drive a phase transition to topological nontrivial insulator with quantized non-zero Hall conductance, and a single gapless Dirac fermion emerges when the interaction is fine tuned to the critical point. The finite temperature effect is also considered and we show that the topological phase with stronger interaction is less stable against temperature. Our model provides a concrete example to illustrate interacting topological phases and phase transition, and can shed light on similar problems in physical systems.

Sachdev-Ye-Kitaev (SYK) model [1, 2] has recently drawn a lot of interests from both condensed matter [2– 27] and gravity physics communities [3, 28–39], because it displays an emergent conformal symmetry, holographic duality to AdS2 gravity and maximally chaotic behavior. In addition to its significant impact on the AdS/CFT research, SYK model is also of great interests from a pure condensed matter physics perspective. Let us use SYK dot to refer to a cluster of N Majoarana fermion modes or complex fermion modes with all-to-all random interactions. The Green’s function of such an SYK dot can be exactly solved in the large-N limit. More im- portantly, this large-N limit is strongly in contrast to many other large-N solvable models in condensed mat- FIG. 1: Schematic of the topological Sachdev-Ye-Kitaev ter systems [43, 44], where the leading order action is a model. On each site, different color represents different quadratic one and the resulting quantum state is essen- pseudo-spin degree of freedom. For same color, different point tially a free one. The leading order solution of an SYK denotes N-different flavours. Solid lines with arrow denotes quadratic hopping within the same flavours, and the dashed dot is a strongly correlated state and displays non-Fermi line denotes on-site interaction within the same pseudo-spin. liquid type behavior [3]. Hence, the SYK dot can be used as the building blocks for granule construction of exactly solvable strongly correlated models by connecting SYK dots with tunneling [15–27]. Depending on how tool of group cohomology [46, 52–55], or by constructing to connect them, different types of interacting physics exactly solvable models with specially designed lattice can emerge. These solvable models can be used to shed structures [53–57]. Our topological SYK model provides insight on fundamentally important and open issues in an alternative class of exactly solvable interacting topo- condensed matter physics, such as the non-Fermi liquid logical model. It does not depend on any specific lattice states and phase transitions between two non-Fermi liq- structures and can explicitly show that the interaction uid phases [19], as well as from a non-Fermi liquid phase can drive a transition from a topological trivial phase to another phases [20–23]. with vanishing Hall conductance to a topologically non- arXiv:1803.01411v1 [cond-mat.str-el] 4 Mar 2018 trivial phase with quantized Hall conductance. In this work we consider SYK dots coupled by constant quadratic hopping, and the quadratic hopping it- Model. Our model is schematically shown in Fig. 1. self forms a topological band. Our model, for the first We consider complex fermion in a two-dimensional lat- time, combines the physics of SYK interaction with topo- tice with i being the site (or unit cell) index. Within logical band theory to address the interaction effects in each site, there is spin or pseudo-spin (such as two sublat- topological theory, and is therefore termed as “topolog- tices within each unit cell) degree of freedom denoted by ical Sachdev-Ye-Kitaev model”. Previously, topologi- σ = , and flavour index denoted by λ = 1,...,N. We cal phases with interactions has been classified for both consider↑ ↓ that hopping only takes place between fermions bosons and fermions [45, 46], for instance, by using field with the same flavour index. Thus, the single-particle theory approaches [47–51], or by utilizing mathematical Hamiltonian is simply N decoupled copies of two-band 2

Chern insulator model, that is written as

N ˆ † H0 = cˆpσλhσσ0 (p)ˆcpσ0λ, (1) p λX=1 X Xσσ0 where p is the quasi-momentum. Here we consider a concrete form of h(p) as h(p) =t (m cos(p ) cos(p ))σ 0 − x − y z + t0 sin(px)σx + t0 sin(py)σy. (2) For each copy, it has been shown that for m > 2 the Chern number = 0 and the insulator is a| topological| trivial one, andC = 1 for 0 < m < 2 and 2 < m < 0, respectively, whichC ± leads to two topological− nontrivial insulators. FIG. 2: (a-c). The spectral functions A(ω) for m = 2.5 and Now, following the original proposal of SYK model, J/t0 = 4, 7, 10, respectively, located at marked points in the phase diagram Fig. 3. (d). The evolution of gap ∆ when J/t0 we introduce on-site interaction between the N-different increases with fixed m. flavors but with the same spin index, that is J i,σ λ1λ2λ3λ4 Hînt = cˆ† cˆ† cîσλ ciσλ , (3) and G(iωn, p) is the Fourier transform of G(τ, x), which 4 iσλ1 iσλ2 3 4 iσ can be written as X λ1,λX2,λ3,λ4 where each J iσ are independent Gaussian random 1 λ1λ2λ3λ4 G− (iωn, p) = iωn + h(p) Σ(iωn). (7) variables that satisfies − − Here both Green’s function G and the self-energy Σ is 2J 2 J iσ = 0, J iσ 2 = . (4) a 2 2 matrix written in the spin space. Σ(τ) be the λ1λ2λ3λ4 | λ1λ2λ3λ4 | N 3 × Fourier transfer of Σ(iωn). This choice of Eq.(4) ensures a well-defined large-N limit. Secondly, we shall emphasize some properties of the It is important for later discussion to emphasize that any self-energy that is quite useful for later discussion. Also two J-coefficients are two independent Gaussian vari- because of the disorder average, the incoming and outgo- ables and their correlation after disorder average van- ing legs in the self-energy diagram have to have same spin ishes, if any one of their total six labels is different. and position indices, and therefore, (i) Σ(τ) is diagonal The total Hamiltonian is therefore given by in the spin space, and (ii) Σ(τ) is also a pure local one Hˆ = Hˆ + Hˆ . (5) without any momentum dependence [58]. With these, 0 int Σ(τ) can be written as Here we have set the chemical potential µ = 0. This is ˆ Σ (τ) 0 crucial to ensure that the ground state with H0 alone is Σ(τ) = ↑ , (8) a gapped insulator and with Hˆ alone is a non-Fermi 0 Σ↓(τ) int liquid phase. Spectral function. The single particle spectral function and following the standard procedure of solving the orig- of this model is exactly solvable in the large-N limit. As inal SYK model, it can be show that the self-energy sat- far as the leading order solution in the large-N expansion isfies following equation is concerned, there are a few features that we should Σσ(τ) = J 2(Gσσ(τ))2Gσσ( τ). (9) emphasize here. − First of all, the two-point Green’s function is diagonal Thus, Eq. 7 and Eq. 9 form a coupled self-consistency in the flavor space and is independent of the λ index, de- equation for solving the Green’s function. spite that the SYK interaction does mix different differ- Thirdly, we show that the spectral function is still ent flavors. This is because the disorder average over the generically gapped despite of the SYK interactions. Con- J-coefficient mentioned above forces the incoming and sidering the SYK fixed point at zero tunneling limit, the the outgoing flavor index to be identical. Hence, the scaling dimension of the single fermion operator is 1/4. later discussion of Chern number and Hall conductance Now turning on the quadratic hopping, because the scal- refer to those of each flavor. ing dimension of a bilinear operator is then 1/2, it is a Now we first introduce the two-point Green’s function, relevant perturbation. Hence, the low-energy behavior at ignoring the λ index, as ω 0 is dominated by the quadratic hopping. And be- → σσ0 cause the quadratic hopping generically gives rise to an (G(τ, x)) = cˆ† (τ)ˆc (0) , (6) Tτ i+x,σλ i,σ0λ insulator with a gapped spectral at low-energy, from this D E 3 point of view we expect the system remains generically gapped even when the interaction is turned on. This statement can be verified by numerically solving the self-consistent equation and obtaining the spectral function. This caluclation is most easily done in the real time. We first apply the analytical continuation to the self-consistent equations (7) and (9) to obtain corresponding self-consistent equation for the retarded Green’s function GR(ω, p) [59]. By solving these equations numerically, we could determine the spectral function averaged over momentum and spin as

1 d2p A(ω) = Im Tr G (ω, p). (10) −2π (2π)2 R Z FIG. 3: The phase diagram of the system in term of sin- Here we need to emphasize that particular attention gle particle mass m and the interaction strength J/t0, where should be paid to the finite size scaling for this nu- the Chern number is calculated for each flavor. The spectral merical calculation [59]. The results for m = 2.5 and function of these three marked points are shown in Fig. 2. J/t0 = 4, 7, 10 are shown in FIG. 2(a-c). We can see from Fig. 2(a) and (c) that a clear U-shaped spectrum near zero-frequency, which reveals a gap state and also ing the Berry curvature using the standard formula allows us to identify the value of gap unambiguously. In A(p) = i α (p) α (p) , (13) Fig. 2(d), we show the gap value as a function of inter- h − |∇| − i action strength for a fixed m. It is clear that the gap is 1 2 = d p (∂xAy(p) ∂yAx(p)) . (14) generically non-zero except at one single point, which we C 2π − Z will later identify as the critical point. This gives different values of C equalling zero, or 1, One notices from Fig. 2(d) another interesting fea- and the phase diagram is shown in Fig. 3. For non-± ture, that is, for large J, the gap also displays a non- interacting case J = 0, the phase diagram is determined monotonic behaviors. This is because the SYK interac- by the parameter m alone. And for interacting case, it tion plays a dual role. On one hands, as we will show only depends on one extra parameter that is J/t0. below, it renormalizes single particle Hamiltonian and In fact, there is an even simple way to determine the drives the topological transition, and the gap shall in- phase diagram. It utilizes two facts: (i) The self-energy crease as moving away from the critical point. On the Σ(ω) is diagonal and is not a function of momentum, as other hand, when the interaction becomes stronger, its discussed above. (ii) Due to the -symmetry defined in effect will become more and more significant once the P Eq. 11, it can be proved that Σ↑(0) = Σ↓(0) [59]. Thus, system is slightly away from the low-energy limit, and − heff(p) only differs from h(p) by a Σ↑(0)σz term and it since the SYK interaction itself favors a non-Fermi liq- renormalizes the mass from m to uid type gapless state, it will eventually make the gap smaller. Σ↑(0) meff = m . (15) Finally we remark that the spectral function A(ω) is − t0 symmetric under ω ω. This is because of a non-local Thus the phase boundary is simply determined by m = symmetry defined→ as − follows: eff P 0, 2. Moreover, the phase diagram of Fig. 3 appears ± 1 to be symmetric with respect to m m, and this is cpσλ − = (σy)σσ c† , (11) → − P P 0 pσ0λ because under an anti-unitary symmetry transformation which can be understood as a combination of particle- 1 1 cp,σλ − = (σy)σσ c(π,π)+p,σ λ i − = i, (16) hole transformation and inversion. T T 0 0 T T − Chern Number and Phase Diagram. Because the sys- with the disorder average, the only changes to the Hamil- tem is always gapped, we can calculate the Chern number tonian is m m, and this anti-unitary transformation → − of this interacting system with the method introduced in changes . C → −C Ref. [48]. Here we define an effective Hamiltonian as The most dramatic feature of the phase diagram is that the interaction actually enlarges the topological nontriv- 1 heff(p) = GR− (ω = 0, p), (12) ial regime. When m > 2, the non-interacting case is a − topological trivial state,| | and the interaction can actually and let α (p) be the eigenstate of heff(p) with negative brings the system into a topological nontrivial regime. eigenvalue.| − Wei can obtain the Chern number by integrat- This appears quite counter-intuitive. In fact, this can be 4

FIG. 4: The momentum resolved spectral function A(ω, p) for m = 2.5, J/t0 = 7. Three different contours correspond to peaks of A(ω, p) with ω/t0 = 0, 0.3 and 0.6, respectively. understood by perturbative calculation in term of inter- 2 FIG. 5: The longitudinal (a) and Hall conductance (b) as a action strength J, which shows that Σ↑ = αmJ with function of temperature at different interaction J with the α > 0 [59]. This means that, independent of the sign same vaue of meff 2 . of m, the interaction always renormalizes the absolute | − | value of m toward a smaller value, which a topological nontrivial state is favored. Our model can also clearly illustrate the critical the- Summary. In summary, we construct an exact solvable ory. Fig. 2(b) shows that the spectral function displays interaction model to show an interaction driven quantum a V-shape at zero-energy right at the phase boundary. In Hall plateau transition. We show that interaction plays fact, a momentum resolved plot of the spectral function two roles in this model. On one hand, it normalizes the A(ω, p) can clearly show a single Dirac point in the mo- band parameter and drives a transition from topological mentum space, as shown in Fig. 4. Therefore, we believe trivial phase to a nontrivial phase. By calculating the the critical theory is a free Dirac theory. single particle spectral function, we show clearly that Conductance. At zero-temperature, it can be shown a single Dirac fermion emerges at the critical line. On the other hand, a very large interaction will eventually that the Hall conductance σxy obeys 2πσxy = [47, 48, 60]. Here, we focus on the conductance at finiteC tem- make the gap smaller because the SYK interaction alone perature to see how stable the topological phases here favors a gapless non-Fermi liquid phase. By calculat- against temperature. By implementing the Keldysh for- ing the conductance at finite temperature, we find that mula combined with large-N expansion, we can explore the topological phase at larger interaction is more sen- how the U(1) phase response to the external electric field, sitive to temperature. Though the model itself is hard and eventually we can obtain both the longitudinal and to realize, the physics illustrated here can shed light on the Hall conductance directly from the two-point Green’s other related physical systems, for instance, recent exper- function obtained above [59]. Here we directly show the iments have realized topological Haldane model with cold numerical results in Fig. 5. atomic gases [61–63] where the on-site repulsive interaction is tunable, and the interaction effect in the Haldane- Since the phase boundary is determined by meff = 2, here we choose three points in the phase diagram that± Hubbard model is a subject of considerable research interests [64–75]. Our model can be viewed as a special has the same values of meff 2 , i.e. nearly equal dis- tance away from the phase| boundary.− | But the interac- (disordered interaction) large-N version of this model. tion strength of these three cases are different. We can Moreover, our model can also be extended to other topo- clearly see that for larger interaction strength (the case logical phases, such as three-dimensional topological insulator, with the idea of granule construction based on with m = 2.2 and J = 10t0), it demands lower temperature in order that the longitudinal conductance increases SYK interaction. significantly above zero and the Hall conductance drops Acknowledgment. We thank Hong Yao, Zhong Wang, below quantized value. That means that in this model, Cenke Xu, Chao-Ming Jian and Shao-Kai Jian for help- although increasing interaction always drives the system ful discussions. This work is supported by MOST un- into a topological nontrivial regime, the topological phase der Grant No. 2016YFA0301600 and NSFC Grant No. with strong interaction is less stable against temperature. 11734010. 5

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SELF-CONSISTENT EQUATION IN THE REAL TIME.

In this section, we would like to present the self-consistent equaton used in our numerics. The Hamiltonian is given by the Eq. (5) of the main text. Following the standard large-N analysis for the SYK model [1], the only non-vaninshing diagram for the self-energy is shown in FIG. 1. The Schwinger-Dyson equation in the imaginary time is given by:

1 G− (iω , p) = iω + h(p) Σ(iω ), (1) n − n − n Σ (τ) 0 Σ(τ) = ↑ , (2) 0 Σ (τ) ↓ with

Σσ(τ) = J 2(Gσσ(τ))2Gσσ( τ). (3) − The propagator G here is deﬁned as

(G(τ, x))σσ0 δ = c (τ)c (0) , λλ0 hTτ i+x,σλ i,σ0λ0 i and we have used a simpliﬁed notion G (τ) G(τ, 0). Following similar derivations as in Ref.[1–5] by doing an τ ≡ analytical continuation, we obtain the self-consistent equation for the retarded Green’s function GR(x, t) in the real time as

1 G− (p, ω) = ω h(p) Σ (ω), (4) R − − R ΣR↑ (ω) 0 ΣR(ω) = , (5) 0 Σ↓ (ω) R where

∞ Σσ (ω) = iJ 2 dteiωt((nσ(t))2nσ(t) + (nσ(t))2nσ(t)), (6) R − 1 2 3 4 Z0 and

σ σ iωt σ n (t) = A (ω)n ( ω)e− = n (t)∗, (7) 1 F − 4 Z σ σ iωt σ n2 (t) = A (ω)nF (ω)e = n3 (t)∗. (8)

Z 2 σ 1 d p σσ A (ω) = Im GR (p, ω). (9) arXiv:1803.01411v1 [cond-mat.str-el] 4 Mar 2018 −π (2π)2 Z Here Aσ is the spectral function of fermions. This set of equations can be directly solved numerically by iteration. To get correct zero-temperature result, one should be careful about the discretization of integrals, which will be discussed in the next section.

FINITE SIZE SCALING.

When solving the self-consistent equation in the real time at T = 0, the integration over time t and frequency ω are approximated by a summation over finite number of points. This sets a cutoff which serves as an effective 2

FIG. 1: The self-energy diagram at the leading order of the large N expansion for the Green’s function of fermions. The dashed line means the disorder average and the solid line is the full Green’s function of fermions. temperature. To obtain physical results at the zero temperature, we need to perform a finite size scaling to send the discretization L to infinity. In this section we explain the details of this procedure. In the numerics we take constant hopping t0 to be the unit and the cutoff of the integration over frequency ω (time t) is given by 20t0 (20/t0). We check that the result is robust against the change of this cutoff. The most sensitive parameter is the discretization L when we fix the cutoff. Instead of a uniform discretization, to increase the accuracy, we implement the Gauss-Kronrod rule to approximate the integration to the L-th order and perform a finite size scaling in terms of L. A typical result is shown in FIG. 2. We set J/t0 = 8 and m = 2.5. After a finite size scaling, it is clear from Fig. 2(a) that for sufficiently small ω, A(ω) always scales to zero and the system is gapped. From Fig. 2(b) the peak also becomes sharper which indicates the effective temperature is lowered. We have also checked the sum rule of the 4 spectral function that dωA(ω) = 1 is always satisfied with an error of 10− order. R SYMMETRY AND ITS CONSTRAIN TO THE SELF-ENERGY.

In this section we show that because of the symmetry deﬁned as the Eq. (11) of the main text, there is a relation P Σ↑(0) = Σ↓(0) when the self-energy at ω = 0 is real. Recalling the deﬁnition of the symmetry: − P 1 cpσλ − = (σy)σσ c† , (10) P P 0 pσ0λ now we study the constraint that it can impose on the Green’s function:

G(ω = 0, p) = dτ cp,λ(τ)cp† ,λ(0) (11) Z D E 1 1 1 = dτ − c (τ) − c† (0) − (12) PP p,λ PP p,λ PP Z D E = dτ σ c† (τ)c (0) ( σ ) (13) y p,λ p,λ − y Z D E =σyG(ω = 0, p)σy (14)

Here we have used the matrix representation by keeping the spin index implicit and we assume the ground state do not break the symmetry. By taking the inverse of above identity, one obtains P 1 1 G− (ω = 0, p) = σyG− (ω = 0, p)σy. (15)

1 Knowing that G− (ω = 0, p) = h(p) Σ(0) and when the self energy is purely real, one reaches Σ(0) = σ Σ(0)σ − y y and thus Σ(0) σ . Consequently, Σ↑(0) = Σ↓(0). ∝ z −

UNDERSTANDING THE PHASE BOUNDARY: A TOY MODEL FOR WEAK INTERACTION.

In this section we aim at understanding the reason for why SYK interaction favors a topological nontrivial phase by a simple toy model. We approximate the original model by a simpliﬁed two-sites complex SYK model with single-particle mixing as

1 σ H = t0 c† (mσz + δσx)σσ cσ λ + J c† c† cσλ cσλ , (16) σλ 0 0 4 λ1λ2λ3λ4 σλ1 σλ2 3 4 σ Xσσ0 X λ1λX2λ3λ4 3

FIG. 2: The numerical results for finite size scaling. (a). Spectral function at some frequency as a function of 1/L and the fitting to a straight line. (b). Spectral function for different 1/L. where J σ satisfies the Gaussian distribution as the on-site random interaction in the original model. This toy λ1λ2λ3λ4 model mimics the original model when m 1 and if we could neglect the momentum dependence of the mixing between spins. Without random interaction, the retarded Green’s function is given by: ω + mt σ + δt σ ω + mt σ + δt σ G (ω) = 0 z 0 x = 0 z 0 x . (17) 0 ω2 t2m2 t2δ2 ω2 M 2t2 − 0 − 0 − 0 Here we define M = √m2 + δ2. Focusing on fermions with spin , without interaction, the spectral function is given by: ↑ 1 M + m M m A↑(ω) = ImG↑↑(ω) = δ(ω Mt ) + − δ(ω + Mt ). (18) −π 0 2M − 0 2M 0 As given in Eq. (6), the self-energy for the retarded Green’s function is given by:

∞ Σ(ω) = iJ 2 dt n (t)2n (t) + n (t)2n (t) eiωt. (19) − 1 2 3 4 Z0 Here ni(t) is given by:

iωt M + m iMt t n (t) = A(ω)n ( ω)e− dω = e− 0 = n (t)∗, (20) 1 F − 2M 4 Z iωt M m iMt t n (t) = A(ω)n (ω)e dω = − e− 0 = n (t)∗. (21) 2 F 2M 3 Z As a result one has: mJ 2δ2 meﬀ = m 2 2 2 2 . (22) − 12t0(m + δ ) Since random interaction shifts m to be smaller, this eﬀect stabilizes the topological phase. Physically, one could understand this by considering a| single| particle excitation with spin . This single excitation, with energy t m , can ↑ 0| | be scattered to a state with two particles with spin and one hole with spin , whose energy is 3t0 m . As a result, the energy of a single excitation, m , is lowered. ↑ ↓ | | | |

CALCULATION OF CONDUCTANCE.

In this section we explain our method for calculating the conductance. We use the Keldysh contour to formulate the field theory directly in the real time to avoid the analytical continuation [8]. In the standard Keldysh approach 4 there are two time contours +/ and thus two copies of fields ci,+ and ci, (we drop flavor and spin indexes for − − simplicity). The phase fluctuation is introduced by ci, exp( iφi, )ci, with an assumption that φi is a smooth fluctuation approximated by φ (x) [4, 5]. We take the± convention→ − [8]:± ± ± 1 1 c1 = (c+ + c ), c2 = (c+ c ), (23) √2 − √2 − − 1 1 c1 = (c+ c ), c2 = (c+ + c ), (24) √2 − − √2 − 1 1 φcl/q = (φ+ φ ),Aα,cl/q = (Aα,+ Aα, ). (25) 2 ± − 2 ± −

Here Ai, are gauge ﬁelds introduced by minimal coupling to extract the current-current correlation function. In the low-energy± limit, the coupling is given by: 1 S [A] =A c† (∂ h)c + A A c† (∂ ∂ h)c K α,+ + α + 2 α,+ β,+ + α β + 1 Aα, c† (∂αh)c Aα, Aβ, c† (∂α∂βh)c . (26) − − − − − 2 − − − − Here the partial derivative is for the corresponding component of momentums and we leave out the label of momentum for simplicity. Here we start with the standard Green’s function in Keldysh formalism:

GR(x, t) GK (x, t) ci+x,α(t)ci,β(0) = , h i 0 GA(x, t) where G /G /G is the retarded/advanced/Keldysh Green’s function for fermions. We have G (p, ω) = (1 R A K K − 2nF (ω))(GR(p, ω) GA(p, ω)) in thermal equilibrium which is called the ﬂuctuation-dissipation theorem. We have omitted the spin index− and all the correlation function are matrix. The current and the retarded Green’s function for current operators are then given by:

2 i ∂ ln Z ∂Jα,cl i ∂ ln Z Jα,cl = , ΠR,αβ = = . (27) −2 ∂Aα,q ∂Aβ,cl −2 ∂Aβ,clAα,q

The longitudinal conductance and the Hall conductance are then given by: 1 σxx = lim ΠR,xx(0, ω). (28) ω 0 → iω 1 σxy = lim (ΠR,xy(0, ω) ΠR,yx(0, ω)). (29) ω 0 → 2iω − As any well-deﬁned large-N model, after integrating out the fermions, the eﬀective action is proporational to N:

Seff [φ] = NSeff0 [φ], because of which the Green’s function of φ is suppressed by 1/N [9]. As a result, only the Gaussian fluctuation of φ is taken into account in the leading order of large-N [4, 5], and therefore it only requires the information of two-point correlation function of fermions computed in the previous sections. Following similar procedures as in Ref. [4], we could derive

αβ αβ iSeﬀ0 [φ, Aα] =(Ddδ + Dpp )(∂αφq + Aα,q)(∂βφcl + Aβ,cl) + Dωω∂tφcl∂tφq α α +Dωp(∂αφcl + Aα,cl)∂tφq + Dωp0 ∂tφcl(∂αφq + Aα,q), (30) dq d2q ∂h ∂h ∂h ∂h Dαβ(ω) = 0 (tr G (q + ω, q) G (q , q) + tr G (q + ω, q) G (q , q) ), (31) pp − (2π)3 ∂q R 0 ∂q K 0 ∂q K 0 ∂q A 0 Z α β α β dq d2q D (ω) = 0 (tr [G (q + ω, q)G (q , q)] + tr [G (q + ω, q)G (q , q)]), (32) ωω − (2π)3 R 0 K 0 K 0 A 0 Z dq d2q ∂h ∂h Dα (ω) = 0 (tr G (q + ω, q)G (q , q) + tr G (q + ω, q)G (q , q) ), (33) ωp − (2π)3 ∂q R 0 K 0 ∂q K 0 A 0 Z α α 2 α dq0d q ∂h ∂h D0 (ω) = (tr G (q + ω, q) G (q , q) + tr G (q + ω, q) G (q , q) ). (34) ωp − (2π)3 R 0 ∂q K 0 K 0 ∂q A 0 Z α α 5

αα Here we keep the label of freqency implicit and only show the retarded part of the action, and Dd = Dpp (0) is the α α − contribution from the diamagnetic term. In the zero momentum limit, one could show Dωp = Dωp0 = 0, we ﬁnd now:

1 xx xx σxx = lim (Dpp (ω) Dpp (0)), (35) ω 0 − → 2ω − 1 yx xy σxy = lim (Dpp (ω) Dpp (ω)). (36) ω 0 − → 4ω − Eq.(36) is consistent with the exact result derived in [7] by using the Wald identity.

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