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This is the accepted manuscript made available via CHORUS. The article has been published as: Thermodynamics of energy magnetizationYinhan Zhang, Yang Gao, and Di Xiao Phys. Rev. B 102, 235161 - Published 29 December 2020

DOI: 10.1103/PhysRevB.102.235161

Thermodynamics of Energy Magnetization

Yinhan Zhang, Yang Gao, and Di Xiao

Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA We construct the thermodynamics of energy magnetization in the presence of gravitomagnetic eld. We show that the free energy must be modied to account for the modication of the energy current operator in the presence of a conning potential. The explicit expression of the energy mag-

netization is derived for a periodic system, and the Streda formula for the thermal Hall conductivity

is rigorously established. We demonstrate our theory of the energy magnetization and the Streda formula in a Chern insulator. Introduction.|Recent years have seen a surge of inter- est in the thermal Hall eect, mainly due to its ability to probe charge neutral excitations in condensed mat- ter systems with broken time-reversal symmetry [ 1 8 These systems have a vanishing (charge) magnetization, but they can still be characterized by an energy magne- tization that arises from the circulating energy currents in thermodynamic equilibrium [ 9 ]. It has been recog- nized that the energy magnetization plays an essential role in the theoretical understanding of thermoelectric transport [ 9 15 ]. It must be properly discounted to ob- tain the correct transport coecients and recover fun- damental relations such as the Onsager relation and the

Einstein relation [

9 12 ]. In this context it is quite sur- prising that the theory of energy magnetization itself, and particularly its thermodynamics, remains in a primitive state.

The main challenge is the lack of a thermodynamic

derivation of the energy magnetization. It has been iden- tied that the conjugate force to the energy magnetiza- tion is the gravitomagnetic eld [ 13 15 17 ]. Therefore one should be able to obtain the energy magnetization as the derivative of the free energy to the latter. However, these studies did not clarify what the exact expression of the free energy is and how the energy magnetization can be evaluated for a general extended system. In fact, explicit calculations of the energy magnetization have re- ferred to the existence of chiral edge states [ 13 16 17 Therefore it is not even clear whether the energy mag- netization is truly a bulk quantity or not. An expres- sion for the energy magnetization has been previously derived [ 12 14 ], but the focus is on thermal transport using linear response theory, not on the thermodynamics of energy magnetization. In this Letter we develop a theory to place the ther- modynamics of the energy magnetization on a rm ba- sis. We rst show that the bulk energy magnetization includes an anomalous contribution from the modied boundary energy current, which survives in the thermo- dynamic limit. Consequently, the free energy should be modied to account for this anomalous contribution. We then derive an explicit expression of the energy magneti- zation using the Maxwell relation, from which the Streda formula for the thermal Hall eect can be obtained [ 13 ].Finally, we demonstrate our theory of the energy magne- tization and the Streda formula in a Chern insulator, and show that our theory is able to capture the contribution from the chiral edge states. General considerations.|The diculty in deriving the energy magnetization is that it is only dened via the relation j

E(r) =r ME(r);(1)

wherejEis the energy current andMEthe energy mag- netization. Naively, one expects thatMEis given by the expectation value of (1=2)^r^jE. However, it is well known that for an extended system the current den- sity alone is not sucient to determine the corresponding magnetization [ 18 21
]. To circumvent this diculty, our strategy is to rst calculate the total magnetic moment of a nite system, then dene the energy magnetization as the thermodynamic limit of the following expression [ 22
M

E= limV!11V

Z V dr12 rjE;(2) whereVis the volume of the system. Physically, a nite system can be realized by adding a conning potential(r). For simplicity, let us consider a two-dimensional system. We assume that the conning potential is constant (0) inside the bulk and gradually increases to innity asr! 1, as shown in Fig.1 (a). We further assume that(r) is suciently smooth such that a local chemical potential(r) =(r) can be dened,ϕ(r)μ

BulkxI

E Energy(a)(b)BoundaryFIG. 1. (a) Side view: The conning potential(r) separates the system into a bulk region and a boundary region. (b) Top view: The boundary energy currentIEgives rise to an energy magnetization. 2 whereis the global chemical potential. At this point, it is crucial to realize that(r) not only changes the local chemical potential, but also modies the energy current operator itself. This is because in addition to its internal energy, a particle also carries the potential energy(r).

The energy current is thus given by [

9 23
j

E(r) =j0E(r) +(r)jN(r);(3)

wherej0E(r) is the expectation value of the energy cur- rent in the absence of the conning potential andjN(r) is the particle number current. Since bothj0E(r) andjN(r) are equilibrium currents, we can express them in terms of their corresponding magnetizations and writej

E(r) as

j

E(r) =r M0E(r) +(r)r MN(r);(4)

whereM0Eis the magnetization of the unmodied en- ergy current, andMNis the particle number magneti- zation [ 11 19 21
]. Ther-dependence of the magnetiza- tions enters through the local chemical potential(r), e.g.,MN(r) =MN((r)) =MN((r)).

Since(r) is a constant in the bulk,j

E(r) is conned

to the boundary area of the system. Let us assume that the boundary is along theydirection, then the boundary energy current is given by I E=Z 1 bulk dxdM0Edx +(x)dMNdx :(5) It should give rise to a total energy magnetic moment, approximatelyIEA, whereAis the area of the system [Fig. 1 (b)]. The correction comes at the order ofO(pA). Therefore, in the thermodynamic limit the energy mag- netization is simply given byIE. Integrating Eq. (5) by parts and making use of the boundary condition that bothM0EandMNvanish asr! 1, we obtain M

E(0;T) =M0E(0;T) +0MN(0;T)

+Z 0 1 dMN(;T):(6) where in the bulk the local chemical potential becomes a constant00. One can verify that if we let0 depend onr, thenr M E=j E.

The energy magnetizationM

E(0;T) in Eq. (6) still

depends on0, which can be arbitrary. It is more con- venient to remove the0-dependence by introducing the heat magnetizationM QM

EMN, which can be re-

garded as the energy magnetization dened with respect to the chemical potential. It is given by M

Q(0;T) =M0E(0;T)0MN(0;T)

+Z 0 1 dMN(;T):(7)

We see that the heat magnetizationM

Q(0;T) only de-

pends on the local chemical potential0in the bulk.The above analysis seems to suggest that the heat mag-

netizationM

Qis a boundary-dependent quantity. In-

deed, the superscriptindicates that it is calculated us- ing the energy currentj

E, which includes a modication

due to the conning potential(r) [see Eq. (3)]. How- ever, the right-hand side of Eq. ( 7 ) makes no explicit reference to(r). Therefore,M

Q(0;T) should be re-

garded as a genuine bulk quantity and we will drop the superscriptin the following. The modied bulk free energy.|Armed with the insight that a proper calculation of the energy magnetization must include the conning potential(r), we now give a rigorous derivation. To evaluate the energy magnetiza- tion, we introduce an auxiliary vector eldAgthat lin- early couples to the energy current ^j

E(r) in the Hamilto-

nian (throughout this paper we have sete= h=kB= 1) H=Z dr^h0(r) + ^n(r)(r) Z dr^j

E(r)Ag(r);

(8) where ^h0(r) is the hamiltonian density without external elds, ^n(r) is the particle number density operator, and ^j E(r) =^j0E+(r)^jN(r) is the energy current operator. HereAgcan be regarded as a purely mathematical de- vice: as we show below, the static response of the free energy toBg=rAgyields the energy magnetization. Physically,Bgis the gravitomagnetic eld andAgis its vector potential [ 15

Let us expand the free energy

A gwith respect toAg up to rst order, A g 0Z drAg(r)j

E(r):(9)

with 0 A g=0. Herej

E(r) is the statistical expecta-

tion of ^j

E(r) in the absence ofAg, and it is exactly the

current density appearing in Eq. ( 3 ). Inserting Eq. ( 4 into Eq. ( 9 ), and using partial integration, we nd that the linear coupling term in Eq. ( 9 ) becomes Z drAg(r)j

E(r) =Z

drBg(r)M

E(r):(10)

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