Heat transport as torsional responses and Keldysh formalism in a curved spacetime

We revisit a theory of heat transport in the light of a gauge theory of gravity and find the proper heat current with a corresponding gauge field, which yields the natural definitions of the heat magnetization and the Kubo-formula contribution to the thermal conductivity as torsional responses. We also develop a general framework for calculating gravitational responses by combining the Keldysh and Cartan formalisms. By using this framework, we explicitly calculate these two quantities and reproduce the Wiedemann-Franz law for the thermal Hall conductivity in the clean and non-interacting case. Finally, we discuss an effective action for the quantized thermal Hall effect in $(2 + 1)$-D topological superconductors.

The HM also appears in topological superconductors (TSCs) [23,24]. In contrast to topological insulators where charge responses are active, in TSCs, heat responses are effective due to the gauge symmetry breaking. The important issue is what actions describe the quantized THE in (2 + 1)-D TSCs [25,26] and the heat cross correlation between a temperature gradient and an angular velocity of rotation proposed in (3 + 1)-D TSCs [25]. The spin-connection analogs of the Chern-Simons term and the axion electrodynamics have been proposed [27,28], but cannot describe these heat responses.
The main purposes in this paper are two-fold. One is to establish a unified framework for gravitational responses. The gauge-covariant Keldysh formalism provided one of the most sophisticated perturbation theories with respect to electromagnetic fields [29,30,31], and can be applied to insulators or metals at finite temperature with disorder or interactions. By combining a gauge description of gravity, so called the Cartan formalism, we establish the Keldysh+Cartan formalism within the first-order perturbation with respect to torsion and Riemann tensors. Although the Keldysh formalism in a gravitational potential only was already employed to investigate the Nernst effect [32,33], our framework covers a wider range of gravitational responses including the THE.
The other purpose is to unveil the gauge structure of heat responses. The definition of the heat current has been controversial; the product of the Hamiltonian and the velocity or that of the time derivative and the velocity. Here we propose that the latter has a corresponding gauge potential and is the proper definition. This gauge potential induces a torsional magnetic field, which defines the HM. Also, a torsional electric field defines the Kubo-formula contribution to the thermal conductivity. We can calculate these quantities by using the Keldysh+Cartan formalism, and reproduce the THC satisfying the Wiedemann-Franz law, which is a proof of the proposal above. As a corollary of the gauge structure of heat responses, we propose that the effective action for the quantized THE in (2 + 1)-D TSCs is given by the vielbein analog of the Chern-Simons term. We emphasize that it is incorrect to interpret a "magnetic field" conjugate to the HM as a gravito-magnetic field in the gravito-electromagnetism or an angular velocity [20,28,25].
The organization of this paper is as follows. Section 2 begins with a brief review of Luttinger's idea and the Cartan formalism. From a simple observation in the Dirac Lagrangian density, we propose the proper definition of the heat current and its corresponding gauge potential. We introduce the natural definitions of the HM and the thermal conductivity as torsional responses. In Sec. 3, we derive the Keldysh formalism in a curved spacetime. Here we consider not only a gravitational potential but general gravity with torsion and Riemann tensors. In Sec. 4, we prepare the firstorder perturbation of the Green function with respect to the static and uniform torsion. Based on these preparations, we calculate the HM in Sec. 5 and the Kubo-formula contribution to the thermal conductivity in Sec. 6. We also mention the relevance of our framework compared to the previous works on heat responses. The effective action for the quantized THE in (2 + 1)-D TSCs is discussed in Sec. 7. Section 8 is devoted to the summary of this paper.
Hereafter we assign the Latin (a, b, · · · =0,1, . . . ,d) and Greek (µ, ν, · · · = 0, 1, . . . , d) alphabets to locally flat and global coordinates, respectively. We follow the Einstein convention, which implies summation over the spacetime dimension D = d + 1 when an index appears twice in a single term. The Minkowski metric is taken as η ab = diag(−1, +1, . . . , +1). The Planck constant and the charge are denoted by and q, while the speed of light and the Boltzmann constant are put to c = k B = 1. The upper or lower signs in equations correspond to boson or fermion.

Gauge Structure of Heat Responses
Aiming at a systematic framework for heat transport, we have to define the proper heat current which has a corresponding gauge potential. Luttinger's gravitational potential [1] is not enough, because the HM, which contributes to the THC [21,22,20,25], should be defined by a "magnetic field" induced by a "vector potential".
First, let us intuitively review Luttinger's idea which relates a gravitational potential to the nonuniform temperature. We begin with the unperturbed system in the non-uniform chemical potential µ( x) and temperature T ( x) = β −1 ( x). The partition function in the local equilibrium is given by The important point is that the chemical potential and temperature are statistical properties and hence cannot be treated as perturbations. Instead, we have to introduce mechanical forces equivalent to these quantities. Here we define φ( x) and γ( x) by and then the system is equivalent to the perturbed system in the uniform chemical potential µ 0 and temperature T 0 = β −1 0 described by As a scalar potential φ( x) is a mechanical force equivalent to the non-uniform chemical potential, a gravitational potential γ( x) is equivalent to the non-uniform temperature. Next, we briefly introduce the gauge theory for gravity, namely, the Cartan formalism [34,35]. Apart from quantum gravity, a gauge aspect of classical gravity was already established [36,37]. In this formalism, a vielbein h a µ and a spin connection ω ab µ act as gauge potentials corresponding to translations and local Lorentz transformations, and hence are coupled to the momentum and the generator of local Lorentz transformations S ab , respectively. Note that a vielbein is related to a metric by g µν = η ab h a µ h b ν . Thus in a curved spacetime, we have to replace the partial derivative to the covariant one, where h µ a is the inverse of h a µ , and A µ is a vector potential.
Among these gauge potentials, h0 0 is coupled to the time derivative and acts as a gravitational potential which Luttinger introduced [1]. Therefore we expect that h0 i is a gauge potential coupled to the heat current defined by the product of the time derivative and the velocity. This is in parallel to the gauge theory of electromagnetism. The charge conservation law is associated with the global U(1) gauge symmetry according to the Noether theorem, and a vector potential A µ coupled to the charge current is introduced by imposing the local symmetry, which is the gauge principle. Similarly, the energy conservation law is associated with the global temporal translational symmetry, and a vielbein h0 µ coupled to the heat current is introduced by the general covariance principle. Note that the heat current defined by the product of the Hamiltonian and the velocity is obtained by imposing the on-shell condition, but has no corresponding gauge potential. 3/12 To see this, we consider a Dirac fermion in a curved spacetime. The Dirac Lagrangian density is given by in which h = det h a µ = − det g µν is the determinant of a vielbein,ψ ≡ i −1 ψ † γ0 is the Dirac conjugate, and the γ-matrices satisfy the Clifford algebra {γ a , γ b } = 2η ab . When a vielbein is put to h a µ = diag(γ, +1, . . . ), Eq. (4) reads where α and β are the Dirac matrices. Thus h0 0 = γ is a gravitational potential in Eq. (2). On the other hand, when a vielbein has the non-zero off-diagonal component h0 i = A gi , its inverse has h 0 ı = −A gî . Then Eq. (4) is explicitly written as Since α is the velocity of a Dirac fermion, h0 i = A gi is a "vector potential" coupled to the heat current.
We emphasize that h0 i = A gi is essentially different from a rotational vector potential hî 0 = Aî r . The former is coupled to the heat current and leads to the non-trivial metric, while the latter is coupled to the momentum and leads to the metric, If we put A r = Ω × x, Ω is assigned to an angular velocity of rotation. However, these two metrics coincide in the gravito-electromagnetism, namely, when the second-order perturbations to the Minkowski metric are neglected. Here a natural question is whether such second-order perturbations are important when linear responses are concerned. In our formalism, a vielbein is the primary gauge potential but a metric is the secondary quantity. Since the latter is square of the former, i.e., g µν = η ab h a µ h b ν , we cannot drop the second-order perturbations in the metric formalism. Thus the gravito-electromagnetism based on the metric formalism is not even approximation either for Eq. (7) or for Eq. (8). In the previous studies on heat responses [20,28,25], this point was not correctly understood and a gravito-magnetic field was assigned to an angular velocity of rotation.
A vielbein and a spin connection induce field strengths called torsion and Riemann tensors as In the standard general relativity, we impose the torsion-free condition to determine the spin connection uniquely, but we put ω ab µ = 0 here. With this choice, a torsional electric field T0 j0 is the mechanical force equivalent to a temperature gradient, and a torsional magnetic field T0 ij is a field strength coupled to the HM. Thus the Kubo-formula contribution to the thermal conductivity and 4/12 the HM are naturally defined by where Jî Q is the heat current defined by the time derivative and the velocity, and Ω ≡ E − T 0 S − µ 0 N is the free energy. These definitions are justified below by explicitly deriving the Wiedemann-Franz law in the clean and non-interacting limit.
At the end of this Section, let us comment on the gauge transformations. The Lorentz-temporal components of a vielbein, h0 0 and h0 i , are connected by temporal translations such as x ′0 = x 0 + ξ 0 (x). Since a vielbein is a covariant vector, it transforms as h ′a µ = h a ν ∂x ν /∂x ′µ . For example, if we start from one spacetime with h0 0 = γ( x), we can move to another spacetime with in which, however, the equilibrium temperature cannot be gauged out as seen in Eq. (3). In addition, both spacetimes give the same torsion, T0 j0 = ∂ j γ, which describes a temperature gradient.

Keldysh Formalism in a Curved Spacetime
In this Section, we present a general framework for gravitational responses based on the Keldysh formalism. As far as the thermoelectric conductivity in the Seebeck and Nernst effects and the longitudinal thermal conductivity are concerned, the Keldysh formalism in a gravitational potential only was already established [32,33]. However, the perturbation theory with respect to a torsional magnetic field is necessary to calculate the HM defined above. Furthermore, our framework makes it possible to calculate rotational responses [2,3,4] and viscoelastic responses [5,6,7] on an equal footing, which are out of our scope here.
We begin with the Keldysh formalism in a flat spacetime [38,39] to construct that in a curved spacetime. The Dyson equation in a flat spacetime is well-known, where L is the Lagrangian density, and * is convolution. The Keldysh Green functionĜ and the self-energyΣ contain three independent real-time Green functions and self-energies, and are written in the matrix representation, in which R, A, and < indicate the retarded, advanced, and lesser components.

5/12
There are two effects of gravity. The first effect is to replace the volume element d D x to the covariant one d D xh(x). Owing to this effect, the Dyson equation Eq. (12) is replaced by where convolution and the δ-function in a curved spacetime are defined by [34,35] A However, by introducing a tensor density, this effect can be absorbed, and the Dyson equation can be written in the same form as that in a flat spacetime symbolically, Now that we move to a flat spacetime, just symbolically, we can use the Wigner representation [38,39]. In this representation, we introduce the center-of-mass coordinate X ≡ (x 1 + x 2 )/2 and the relative coordinate x ≡ x 1 − x 2 , and perform the Fourier transformation on the latter, The Wigner representation of convolution is given by the non-commutative Moyal product, Here the left-hand side of ⊗ acts on the Wigner representation in the left-hand side, and vice versa. The Moyal product * is expanded with respect to as * = 1 + i F 0 /2, whose first order F 0 is called the Poisson bracket. Inversely, we can construct the Moyal product from the Poisson bracket by using the deformation quantization [40]. We can derive the commutation relation in the canonical quantization by acting the Moyal product on X a and p b . We pursue this observation that the Moyal product corresponds to some algebras. We introduce a new dynamical variable, i.e., the generator of Lorentz transformations S ab , which satisfies the Poincaré algebra. Then we assume the extended Moyal product * whose Poisson bracket is given by Although the Moyal product may not be represented by a simple form such as e i P0/2 , by definition it is expanded with respect to as * = 1 + i P 0 /2. Since convolution in Eq.
The second effect of gravity is to replace the partial derivative to the covariant one [34,35], ∂ a → D a ≡ h µ a (∂ µ − iqA µ / − iω ab µ S ab /2 ). Therefore, by introducing the quantity associated 6/12 with the covariant derivative, the Dyson equation in gauge fields is represented as Since π(X, p, S) is a complicated function of (X, p, S) as seen in Eq. (22), it is convenient to change the set of variables from (X, p, S) to (X, π, S). Correspondingly, owing to the chain rule, the Poisson bracket Eq. (20) is replaced to the new Poisson bracket, and the Dyson equation in a curved spacetime can be rewritten in a simple form as The fourth term in Eq. (24) describes the emergent commutation relation between the mechanical momenta in the presence of gauge fields. A set of Eqs. (24) and (25) is one of our main results to establish a general framework for gravitational responses, and includes that for electromagnetic responses previously established [29,30,31]. The star product ⋆, which is the Moyal product perturbed by gauge potentials, is expanded with respect to as ⋆ = 1 + i P/2. It is an important future problem to construct the star product ⋆ from the Poisson bracket P by the deformation quantization [40] in order to go beyond the first-order perturbation theory with respect to field strengths. Note that construction of the star products was completed in the presence of a vector potential [30] and in the presence of both a vector potential and a spin connection [31].

Perturbation Theory with respect to Torsion
Let us proceed the linear-response theory with respect to the static and uniform torsion T a cd to focus on heat responses. We drop the X-dependence in the Green function and the self-energy, and impose the S-dependence to the band indexes. Now the star product, the Green function, and the self-energy can be expanded as Note that Eq. (26a) is not semi-classical, namely, the first order with respect to , but exact and fully quantum-mechanical up to the first order with respect to the uniform and static torsion. The exact form of the star product is given by the infinite-order expansion with respect to , and may be a couple of exponentials of the Poisson bracket [30,31]. Even if we restrict ourselves to the first order with respect to torsion, infinite terms containing the spacetime derivatives of torsion arise from combination of the first and fourth terms in Eq (24). Only in the uniform and static torsion where the spacetime derivatives of torsion vanish, we do not suffer from this infinite-order problem with respect to . The subscript 0 in the Green function and the self-energy indicates zero field, and the capital letter G indicates that the effects of disorder or interactions are taken into account. Eq. (26) 7/12 is expansion with respect to torsion but not with respect to the disorder strengths or interactions. By substituting these into the Dyson equation Eq. (25), we obtainĜ 0 = (L −Σ 0 ) −1 and Among real-time Green functions, the lesser component is the most important to calculate thermal expectation values. We use the equilibrium condition in zero field, , decompose the lesser components of the Green function and the self-energy as and finally obtain The self-energy Σ

Heat Magnetization
In this Section, we explicitly calculate the HM. Since it is difficult to directly calculate the HM defined in Eq. (10b), let us calculate the auxiliary HM defined by the total-energy perturbation with respect to a torsional magnetic field. The total energy K ≡ E − µ 0 N is given by Owing to symmetrization, the star product is reduced to the ordinary product, and the auxiliary HM is obtained asM To translate the auxiliary HM Eq. (31) to the proper one Eq. (10b), it is necessary to solve the differential equation [20], Practically, we can calculate the HM from a set of Eqs. (29), (31), and (32). Below we restrict ourselves to consider the clean and non-interacting limit,Σ = 0. In this limit, Eq. (31) is written as where g R/A 0 = (−π0 − H + µ 0 + iη) −1 is the retarded/advanced Green function, and vî = −∂ πî g R/A−1 0 is the velocity. By expanding the trace with respect to the Bloch basis and employing 8/12 the integral over −π0 with the residue theorem, we obtaiñ [m n πk (2f n π + f ′ n π (ǫ n π − µ 0 ))(ǫ n π − µ 0 ) − 2Ω n πk f n π (ǫ n π − µ 0 ) 2 ], (34a) m n πk ≡i 2 ǫîk m n π|vî|m π m π|v|n π ǫ n π − ǫ m π , (34b) Ω n πk ≡i 2 ǫîk m n π|vî|m π m π|v|n π (ǫ n π − ǫ m π ) 2 , where m n πk and Ω n πk are the magnetic moment and the Berry curvature, respectively. By solving Eq. (32), the proper HM is given by m n πk f n π (ǫ n π − µ 0 ) + 2Ω n πk

Thermal Conductivity
Next we calculate the thermal conductivity based on its definition Eq. (10a). The thermal expectation value of the heat current is given by Here (−π0) is the Wigner representation of the covariant time derivative, and vî is the renormalized velocity in disordered or interacting systems. Owing to symmetrization in Eq. (36), the star product is reduced to the ordinary product up to the first order, and the Kubo-formula contribution to the thermal conductivity is obtained as Again, we focus on the clean and non-interacting limit,Σ = 0. In this limit, Eq. (37) is written by After straightforward calculations, we obtain T 0κ (1b)î = 1 2 n π ǫîkm n πk f ′ n π (ǫ n π − µ 0 ) 2 , T 0κ (1a)î = −1 2 n π ǫîkm n πk f ′ n π (ǫ n π − µ 0 ) 2 , where the first two contributions arise from Eq. (38a) and the third one arises from Eq. (38b). In the clean and non-interacting limit, T 0κ (1a)î and T 0κ (1b)î are exactly canceled, and T 0κ (0)î gives the Kubo formula, as well-known in the context of the anomalous Hall effect [9]. 9/12 It is known that the proper THC consists not only of the Kubo formula but of the HM [21,22,20,25], In the context of the Nernst and Ettingshausen effects, the origins of the magnetization correction were already discussed [32,33]. In the Nernst effect where the charge current flows perpendicular to a temperature gradient, the magntization correction arises from combination of the magnetization current ∇ × M and the X-dependence in the Green function. On the other hand, in the Ettingshausen effect where the heat current flows perpendicular to an electric field, the magnetization energy M · B in the heat density translates into E × M in the heat current density through the dynamics of electromagnetic fields. In the THE, these two mechanisms exist, which leads to the coefficient 2 in Eq. (40). After all, the proper THC in the clean and non-interacting limit consists of Eqs. (39) and (35), Note that both the Kubo-formula contribution Eq. (39) and the HM Eq. (35) are different from those in Ref. [20], but the proper THC Eq. (41) coincides with that in Ref. [20] and hence satisfies the Wiedemann-Franz law. Since observable in transport measurements is the proper THC only, such difference is not important.
Here let us comment on the relevance of our framework compared to the previous works on heat transport [21,32,33,20]. First, Ref. [21] is not suitable for calculations in periodic systems because the magnetization corrections explicitly contain the position operator. Qin et al. [20] overcame this problem and correctly obtained the THC, in which, however, the scaling relations on the charge and heat currents were assumed without any microscopic explanations, and the non-trivial current corrections had to be derived in each model. On the other hand, our framework is applicable even to disordered or interacting systems without any assumptions or complicated calculations. Apart from this practical reason, more fundamentally, the "magnetization" should be defined by the free-energy perturbation with respect to a "magnetic field", as the orbital magnetization was defined by that with respect to a magnetic field [41,42,43]. The Keldysh formalism in a gravitational potential [32,33] is quite intriguing, and enough to investigate the thermoelectric effect but not the THE since a torsional magnetic field is not taken into account.

Chern-Simons Term in TSCs
In analogy to the quantum Hall effect in (2 + 1)-D time-reversal-broken topological insulators [23,24], the quantized THE is expected in (2 + 1)-D time-reversal-broken TSCs [25,26]. Now that the gauge structure of heat responses is revealed, we can derive the effective action for the quantized THE.
To see this, we check the heat analog of the Středa formula already obtained in Ref. [25]. From Eq. (35), we obtain in which the first term is equal to Eq. (41) after the partial integral over z, and the second term can be dropped in insulators at sufficiently low temperature. As a result, the Středa formula [25] can be 10/12 reproduced as In addition, in (2 + 1)-D, the THC is quantized through the Wiedemann-Franz law [25,26], where C3 is the first Chern number defined by In combination with the Středa formula Eq. (43), the temperature dependence of the HM at low temperature is given by Equations (44) and (46) is encoded with the effective action, besides the ground-state HM M Q3 (T 0 = 0). Since h0 0 is a gravitational potential, the µ = 0 part describes the free-energy perturbation in a torsional magnetic field, namely, the HM Eq. (46). On the other hand, h0 i is a gauge potential coupled to the heat current, and hence the µ = i part describes the heat-current perturbation in a torsional electric field, i.e., the quantized THE Eq. (44). This is a corollary of the gauge structure of heat responses. If the heat current is defined by the product of the Hamiltonian and the velocity, the Wiedemann-Franz law and the Středa formula may be obtained [20,25], but the effective action is never available. Equation (47) is assigned to the Lorentztemporal part in the vielbein analog of the Chern-Simons term, with the dimensional parameter l −1 ∝ T 0 . Note that the Lorentz-spatial part, which may have a different dimensional parameter, describes the topological viscoelastic responses [5,6,7]. Since we do not assume the Lorentz symmetry, the temporal and spatial dimensional parameters do not necessarily coincide. We also note that the (3 + 1)-D analog of Eq. (48) is the Nieh-Yan term [44], which predicts that the HM is induced by a torsional electric field, i.e., a temperature gradient, in (3 + 1)-D time-reversal-invariant topological insulators. This kind of cross correlation response was originally proposed in Ref. [25], in which, however, a gravito-magnetic field was assigned to an angular velocity and hence the HM was not distinguished from the angular momentum. In general, these two are different, and hence the latter is not induced by a temperature gradient.

Summary
To summarize, we established a general framework for gravitational responses to unveil the gauge structure of heat responses. As the heat density is coupled to a gravitational potential h0 0 , we found that the heat current is coupled to a vielbein h0 i , only if it is defined by the product of the time 11/12 derivative and the velocity. This gauge potential induces a torsional magnetic field T0 ij , which naturally defines the HM. On the other hand, a torsional electric field T0 j0 defines the Kubo-formula contribution to the thermal conductivity. Such heat responses, as well as rotational, torsional, and the other responses, can be calculated by using our Keldysh+Cartan formalism. It consists of six steps; (1) to start from the Dyson equation in a flat spacetime, (2) to introduce a tensor density to absorb the non-uniform weight, (3) to introduce the Wigner representation and the Moyal product, (4) to introduce gauge potentials such as a vielbein and a spin connection, which leads to the star product, (5) to expand the Dyson equation in a curved spacetime with respect to a given gravitational field, and (6) to calculate the thermal expectation value of a given physical quantity. The obtained Green-function formulas can be easily extended to disordered or interacting systems.
In the clean and non-interacting limit, we correctly reproduced the thermal conductivity without any assumptions or complicated calculations, which satisfies the Wiedemann-Franz law at low temperature and the Středa formula in insulators. In combination with the gauge structure of heat responses, we proposed the effective action for the heat responses in (2 + 1)-D time-reversal-broken TSCs.
Note added. After the first submission of this paper, we wrote another paper on the heat polarization [45]. Although polarization is coupled to a kind of electric field, it cannot be defined in crystals by the field derivative of the free energy. We employed the gradient expansion by using the first term in the Poisson bracket Eq. (24) to define the heat polarization.