Finite-temperature effective boundary theory of the quantized thermal Hall effect

A finite-temperature effective free energy of the boundary of a quantized thermal Hall system is derived microscopically from the bulk two-dimensional Dirac fermion coupled with a gravitational field. In two spatial dimensions, the thermal Hall conductivity of fully gapped insulators and superconductors is quantized and given by the bulk Chern number, in analogy to the quantized electric Hall conductivity in quantum Hall systems. From the perspective of effective action functionals, two distinct types of the field theory have been proposed to describe the quantized thermal Hall effect. One of these, known as the gravitational Chern-Simons action, is a kind of topological field theory, and the other is a phenomenological theory relevant to the St\v{r}eda formula. In order to solve this problem, we derive microscopically an effective theory that accounts for the quantized thermal Hall effect. In this paper, the two-dimensional Dirac fermion under a static background gravitational field is considered in equilibrium at a finite temperature, from which an effective boundary free energy functional of the gravitational field is derived. This boundary theory is shown to explain the quantized thermal Hall conductivity and thermal Hall current in the bulk by assuming the Lorentz symmetry. The bulk effective theory is consistently determined via the boundary effective theory


I. INTRODUCTION
Topology of the energy band structure in insulators and superconductors emerges in transport phenomena, and the topological number can be detected as a transport coefficient 1,2 . The electric Hall conductivity is quantized and given by the Chern number of the occupied wave functions in two-dimensional band insulators 3,4 . The Chern number is nonzero when an electronic system is in time-reversal symmetry broken topological phases. Similarly, two-dimensional insulators and superconductors show the quantized thermal Hall effect in timereversal symmetry broken topological phases 5 . This similarity between electric and thermal responses reflects a parallel between topological insulators in symmetry class A and topological superconductors in symmetry class D, both of which are characterized, in two dimensions, by the Chern number of the filled energy bands [6][7][8] .
Topological quantum field theories are efficient descriptions and characterizations of topological phases. Effective actions for the external electromagnetic and gravitational fields are given in terms of topological terms and attributed to quantum anomalies 9 . They can be used to discuss the classification of topological insulators and superconductors in arbitrary dimensions [10][11][12] . Quantized electromagnetic responses, including the quantum Hall effect in two dimensions, can be viewed as responses resulting from the Chern-Simons action functional. For the case of the electromagnetic field, the Chern-Simons action is given as in the (2+1)-dimensional space-time, where C is the Chern number (here and henceforth natural units with c = = k B = 1 are used). On the other hand, the quantized thermal Hall effect has been predicted to occur in a weak pairing phase of the chiral p-wave superconductor in Ref. 5, which is considered to be a realization of a two-dimensional time-reversal symmetry broken topological superconductor. In analogy with the electromagnetic response, it has been claimed that the quantized thermal response of topological superconductors is described by the gravitational Chern-Simons action 5 where ω µ is the spin connection. In recent years, spacetime curvature is widely used to study characteristic responses of topological materials [13][14][15] . The gravitational Chern-Simons action is associated with the gravitational anomaly in (2+1) dimensions 9,16 , and can be microscopically derived from the (2+1)-dimensional massive Dirac fermion coupled with the background gravitational field at zero temperature 17 . However, it has been noted that a thermal Hall current in the bulk cannot be created by a uniform gravitational field gradient as a response derived from the gravitational Chern-Simons action 18 . Therefore, while the coefficient of the gravitational Chern-Simons term ("the chiral central charge") is related to the quantized thermal Hall conductance, the quantized thermal transport and the gravitational Chern-Simons term appear to be more remotely related then the quantized electromagnetic response and the U (1) Chern-Simons term. An inhomogeneous temperature field driving thermal transport is effectively realized by a gravitational potential field φ through the Luttinger's phenomenological argument using the Tolman-Ehrenfest relation 19 1 Owing to the relation j T = j E − (µ/e)j C , the thermal current j T is identified as the energy current j E when the charge current j C does not contribute at zero chemical potential µ = 0, which is true for superconductors. A thermal current induced by a temperature gradient is then equivalent to an energy current induced by the space-time metric g µν . From the phenomenological point of view, an analogy between the electric and the thermal transport holds. The Wiedemann-Franz law connecting the electric and the thermal Hall conductivity for the Dirac fermion [20][21][22] has been proved to be Note that in the case of topological superconductors, the coefficient is half of that in (4) since a Majorana fermion, a quasi-particle in a topological superconductor, is half of a complex fermion. Based on the phenomenological analogy of the Hall conductivity, an effective free energy for the quantized thermal Hall effect of a Lorentz invariant system has been proposed as 21 where Ω is the angular velocity of the system in a rotating frame, and κ H = Cπ 2 T /6π is the thermal Hall conductivity for insulators whose occupied energy bands have a total Chern number C, and v is the Fermi velocity of the Dirac fermion. This form of the phenomenological effective free energy (6) can be realized when we consider a metric in a rotating frame with a gravitational potential. The effective free energy (6) is considered in the system in equilibrium at a finite temperature in contrast to the effective action (2), which is considered at zero temperature.
In this study, an effective free energy functional of the gravitational field is derived microscopically from the two-dimensional massive Dirac fermion coupled with the static gravitational field. The effective free energy is derived by the following somewhat indirect procedure. First, we consider the gapless boundary fermion, which is a manifestation of the Dirac fermion with nontrivial bulk energy band topology. An effective free energy of the boundary theory is calculated by the field theoretical method. Then the effective free energy in the bulk is deduced from the boundary theory.
This paper is organized as follows. In Sec. II, a microscopic model studied in this paper is introduced. In Sec. III A and III B, the boundary effective free energy is calculated from the boundary fermion. In Sec. III C, we show how the bulk effective free energy is deduced from the boundary effective free energy, and derive an effective free energy for a metric in a rotating frame with a gravitational potential.

II. TWO-DIMENSIONAL DIRAC FERMION UNDER A GRAVITATIONAL FIELD
The Dirac fermion with the Fermi velocity v is invariant under the Lorentz transformation which preserves the line element Introducing the three coordinates x µ = (vt, x, y), a metric representing a system under a gravitational potential φ rotating with an angular velocity Ω is given by 23 up to linear order in φ and Ω. The effective field theory addressed in this paper is the free energy functional of the metric that results from the two-dimensional Dirac fermion coupled with the gravitational field. In the following, we proceed with a general metric form, and finally the metric (8) is substituted into the expression of the effective free energy to show that the effective free energy functional in this paper reproduces the same form as the phenomenological free energy (6). The two-dimensional Dirac fermion Hamiltonian coupled with the gravitational field is separated into the flat and the remaining parts as follows. Deviation of a curved space-time metric from the flat Minkowski space-time is defined by We use the sign convention η µν = diag(+1, −1, −1). The metrics g µν and η µν are related by a triad field e α µ with the equation When a deviation is small enough (h µν ≪ 1), we can write the triad field in terms of h as e α µ ≃ δ α µ + h α µ /2. Subscripts and superscripts of h µν are raised and lowered by η µν and its inverse η µν , like h α µ = h µν η να . The inverse and the determinant of the triad field are given, up to linear order in h, by where h = h µ µ . The inverse of the triad satisfies e α µ e ν α = δ ν µ , e α µ e µ β = δ α β , and e µ α e ν β g µν = η αβ .
Hamiltonian of the two-dimensional Dirac fermion coupled with a gravitational field is written as where γ µ is the 2 × 2 gamma matrix satisfying γ µ γ ν = η µν − iǫ µνρ γ ρ , and ǫ µνρ is the totally anti-symmetric tensor and γ µ = η µν γ ν . Note that the spin connection does not appear in the two-component Dirac fermion theory 24 .
Latin indices in (13) run over the spatial dimensions (j = 1, 2) and Greek indices run over both temporal and spatial dimensions (α = 0, 1, 2). Matrix elements of the Hamiltonian and the perturbation term are given, respectively, by Although the original Lagrangian has the Lorentz invariance, symmetries of the Hamiltonian (14) and (15) are reduced down to the SO(2) rotational invariance in the x 1 -x 2 plane. Time-reversal symmetry and parity symmetry are broken by the mass term.

III. EFFECTIVE FREE ENERGY FUNCTIONAL OF THE GRAVITATIONAL FIELD
In order to derive the finite-temperature effective free energy of the two-dimensional bulk, we, at first, consider the gapless boundary fermion, the existence of which is guaranteed by the nontrivial energy band topology of gapped fermionic systems. The effective free energy of the gravitational field for the one-dimensional boundary modes is calculated by the field theoretical method. Then, the bulk effective free energy is determined so that the energy current incoming from the bulk and that flowing along the boundary are locally conserved near the boundary.
In the (2+1)-dimensional space-time, the Chern-Simons action is induced by the parity breaking mass term of the Dirac fermion. The Chern-Simons action of the U(1) gauge field is derived as an effective action by tracing out the Dirac fermionic degrees of freedom of the action of the (2+1)-dimensional Dirac fermion coupled with an electromagnetic field 25 . Similarly, the gravitational Chern-Simons action, the Chern-Simons action of the spin connection, is derived by tracing out the fermionic degrees of freedom of the action of the Dirac fermion coupled with the gravitational field 17 . For the purpose of deriving the effective free energy (6), we consider the finite temperature path integral. Tracing out fermionic degrees of freedom of the density matrix gives an effective free energy functional of the gravitational field as where G(iω n ) = 1/(−iω n + H 0 + U ) is the temperature Green's function with the Matsubara frequency for fermions ω n = (n + 1/2)2π/β (n ∈ Z), β is the inverse temperature and Det is taken for the Hilbert space on the two-dimensional space and 2 × 2 matrix degrees of freedom. Expanding the temperature Green's function with respect to U , we obtain where G 0 (iω n ) = 1/(−iω n + H 0 ) is the temperature Green's function in the flat space-time. Then expansion of the effective free energy with respect to h µν , that is where the identity ln Det = Tr ln is used, and Tr is taken for the same Hilbert space as Det. F (0) is independent of the gravitational field and gives only a constant, and the second order perturbation term F (2) is the focus of this paper.

A. Boundary fermion
In two-dimensional space, consider a boundary at x 1 = 0 between a gapped bulk at x 1 < 0 with mass m and that at x 1 > 0 with mass −m. The boundary is extended to an entire x 2 space. The masses in the two gapped semiinfinite regions are smoothly connected by introducing an x 1 -dependent mass term, whose sign changes at the boundary (sgn[m(x 1 < 0)] = −sgn[m(x 1 > 0)]). Since the Chern number at both side of the boundary differs by unity, the boundary hosts a single chiral boundary mode of the massless Dirac fermion. The unperturbed Hamiltonian (14) is decoupled into the x 1 -and the x 2dependent parts. The wave function of the boundary mode of the Hamiltonian (14) is a product of a plane wave of the x 2 -coordinate and a two-components spinor wave function of the x 1 -coordinate satisfying Formally the solution of (20) is given by where we have used an identity (γ 1 ) 2 = η 11 = −1. The two component spinor |s corresponding to edge bound states satisfies iγ 1 |s = sgn(m)|s , where sgn(m) indicates the sign of the mass in x 1 < 0, while the other side of the spinor corresponds to states that cannot be normalized. By using the relation γ µ γ ν = g µν − iǫ µνρ γ ρ , the condition on |s can be rewritten in a convenient form, γ 0 γ 2 |s = −sgn(m)|s . Next, we consider the Hamiltonian of the gapless boundary states resulting from the bulk Hamiltonian by projecting the Hilbert space on the x 1 -coordinate onto that of the boundary mode. The projected Hamiltonian is by definition an operator acting on the Hilbert space on the x 2 -coordinate. The unperturbed Hamiltonian (14) is projected asH Also we consider the projection of the perturbation term (15) onto the boundary mode. For convenience, the width of the boundary states in the x 1 -direction is tuned to be narrow enough so that typical length scale of the gravitational field is much longer than the width of the boundary states. This situation allows us to use an assumption that the metric depends only on x 2 near the boundary. Then x 1 and x 2 is completely decoupled in the boundary Hamiltonian. The second term of the perturbation term (15) is projected as The first term in (23) is zero since it contains an integral The second term in (23) is nonzero for α = 0, 2 due to the property of the two-component spinor |s . A projected operator of the perturbation term is written as where ζ(

B. Boundary free energy
Here we calculate the effective free energy functional of the gravitational field of the boundary mode. Tracing out the boundary fermionic degrees of freedom, the second order perturbation term in (19) is given by whereG 0 = 1/(−iω n +H 0 ) is the temperature Green's function of the boundary fermion. Tracing over the Hilbert space on the x 2 -coordinate, the effective free energy becomes where q is the momentum of the gravitational field, and Π(q) is the one-loop integral of the single-component boundary fermion given by The leading contribution of the integral over q in (27) is of order q 0 , where we expand the integral in the powers of q since the long-wave length behavior of the gravitational filed is of interest. Summation over the Matsubara frequencies gives n (−iω n +vp) −2 = βdf (vp)/d(vp), where f (ǫ) is the Fermi-Dirac distribution function at temperature T . The coefficient (28) of the order of q 0 is given by where θ is the Heaviside step function, and is regarded as the zero-temperature Fermi-Dirac distribution function. Here we have replaced vp by p for simplicity. At low temperature, the distribution function in (29) can be expanded with respect to the temperature by the Sommerfeld expansion f (p) ≃ θ(−p) − (π 2 T 2 /6)(dδ(p)/dp) as Then the effective free energy is reduced to a simple form as

C. Bulk free energy
In this subsection we see how the bulk effective free energy is derived from the boundary effective free energy. The energy current j (boundary) E flowing along the boundary can be read off from the boundary effective free energy (31). Using an equivalence between the energy current and the momentum density resulting from the Lorentz invariance, the boundary energy current is j (boundary) E ≡ vT 02 ≃ 2v(δF (boundary) /δh 20 ), which results in If the boundary is considered as an isolated onedimensional system, the energy current can break the energy conservation law when the metric varies spatially: ∂ 2 j (boundary) E = 0. The energy conservation law is retained by including incoming energy flow from the gapped bulk. The bulk effective free energy is determined to precisely describe this compensating bulk energy current. Furthermore, the incoming energy current from one side of the semi-infinite space of the bulk is equal to the other side, since the mass term is inverted by the timereversal operation or the parity inversion, which causes inversion of the thermal Hall coefficient. Thus, the bulk energy current j E near the boundary satisfies the following equation: Since the bulk energy current results from the nontrivial topology of the bulk energy bands, it should not be dependent on the direction of the boundary. To see this, we examine how each term in (31) changes according to the direction of the boundary. The system is rotated by an angle θ anticlockwise to define a new boundary. New coordinate variables perpendicular and parallel to the boundary are introduced as where R j k (θ) is the rotation matrix. Derivatives in the new coordinates are given by where R k j (θ) is the inverse of R j k (θ) (R k j (θ) = R k j (−θ), and R k j (θ)R j l (θ) = δ k l holds). The Hamiltonian (14) and (15) has SO(2) rotational symmetry, that is, introducing new vectors and tensors by and the Hamiltonian is a scalar quantity with respect to the rotation R j k (θ): Note that γ 0 and other scalar quantities contained in the Hamiltonian are unchanged during the rotation. Then the effective free energy of the new boundary perpendicular to x ′ 1 is given by 0 (x ′ 2 )]/2, and thus the boundary energy current flowing along the x ′ 2 -coordinate is The second and third terms of the right-hand side of (40) explicitly depend on the boundary angle θ through (37) and (38), while the first term does not. The bulk energy current, which is proportional to the Chern number of the bulk energy band, should not be dependent on the details of the boundary, such as the boundary angle. Therefore the first term of the right-hand side of (40) is related to the bulk energy current by (33). The same conclusion can be given by considering the number of indices in (33). The right-hand side of the (33) must be a vector quantity with respect to rotation since the lefthand side is a vector quantity. This condition is satisfied when j (boundary) E is a scalar quantity, which results in the first term of the right-hand side of (40). The bulk energy current corresponding to the boundary energy current j (boundary) E = −sgn(m)(πT 2 /12)h is given by the form of which is independent of the direction of the boundary. The equation (41) exactly represents the quantized thermal Hall effect (5) by substituting h = h µ µ = −2φ from the metric (8), identifying the energy current with the thermal current, and using the Tolman-Ehrenfest relation (3). The thermal Hall coefficient is quantized by the Chern number C = sgn(m)/2 as κ H = sgn(m)(πT /12). The bulk free energy that realizes the energy current (41) is given by Total energy conservation can be seen by the translation symmetry of the bulk free energy together with the boundary free energy in the temporal direction. Under an infinitesimal coordinate transformation the metric varies as δh k0 = ∂ k ζ 0 , where we assume ζ 0 is a function of x 1 and x 2 since we consider a static theory in this study. The translation (43) leaves the bulk free energy (42) unchanged except the boundary term given by On the other hand, by defining the boundary parallel to x 2 and the bulk in x 1 < 0, the translation (43) makes a change in the boundary free energy as which precisely cancels the boundary term in (44). Therefore the total free energy is invariant under the translation (43), and it indicates the energy of the bulk and the boundary is totally conserved. The boundary free energyF (boundary) considered here is a part of (31) that represents the scalar boundary energy current shown above (41). Furthermore the boundary free energy in the parenthesis of the left-hand side of (45) is a half of it, since the other half corresponds to the other side of the bulk (x 1 > 0).
In the presence of a gravitation potential and a rotation (8), h and h j0 in (42) are given by h = −2φ and h j0 = (−Ωx 2 /v, Ωx 1 /v). Therefore the integrand of (42) together with that of half of the boundary free energy is ǫ kl h∂ k h l0 = −4φΩ/v. The effective free energy under the metric (8) is given by The free energy (46) realizes the coupling of the gravitational potential and the angular momentum, which is the same form as the phenomenologically derived free energy presented in (6). However, it turns out that, from the field theoretical method, the coefficient in (46) is half of that predicted in (6).

IV. CONCLUSION
Using the finite-temperature field theoretical method, we have derived an effective free energy functional of the gravitational field that accounts for the quantized thermal Hall effect of the two-dimensional massive Dirac fermion. At first, we consider the gapless chiral boundary fermion coupled with the gravitational field, existence of which is guaranteed by the bulk Chern number of the Dirac fermion. The boundary effective free energy is calculated by tracing out the boundary Dirac fermion. The bulk energy current is determined from the boundary energy current to ensure that the total energy is conserved in the whole system. The resulting bulk energy current precisely represents the thermal Hall effect whose coefficient is quantized by the Chern number of the bulk energy bands. The effective free energy functional (42) for the two-dimensional massive Dirac fermion is then deduced to realize the bulk energy current. The bulk effective free energy reproduces the same form as the phenomenologically derived effective free energy (6) by substituting a metric of a gravitational potential in a rotating frame.