Orbital Magnetism Induced by Heat Currents in Mott Insulators

Shi-Zeng Lin and Cristian D. Batista

Theoretical Division, T4, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA(Received 17 June 2013; published 18 October 2013)

We derive the effective heat current density operator for the strong-coupling regime of Mott insulators.

Similarly to the case of the electric current density, the leading contribution to this effective operator is

proportional to the local scalar spin chirality �jkl ¼ Sl � ðSj � SkÞ. This common form of the effective

heat and electric current density operators leads to a novel cross response in Mott insulators. A heat

current induces a distribution of orbital magnetic moments in systems containing loops of an odd number

of hopping terms. The relative orientation of the orbital moments depends on the particular lattice of

magnetic ions. This subtle effect arises from the symmetries that the heat and electric currents have in

common.

DOI: 10.1103/PhysRevLett.111.166602 PACS numbers: 72.80.Sk, 72.20.Pa, 73.22.Gk, 75.25.Dk

Mott insulators play a pivotal role in condensed matterphysics. Besides being ’’parent states’’ of exotic emergentphenomena, such as high-Tc superconductivity, they arethe source of most of the known insulating quantum mag-nets. The minimal and paradigmatic model for describingthe electronic degrees of freedom of Mott insulators is thehalf-filled (one electron per atom) single-band HubbardHamiltonian. This model includes an intraatomicCoulomb interaction U and a kinetic energy term thatallows us to move electrons between different atoms witha hopping amplitude t. Electrons become strongly local-ized in the limit U � t because of the very high Coulombenergy barrier for double occupying an atomic orbital.Consequently, the low-energy physics of strongly coupledMott insulators can be entirely described in terms of theremaining spin degree of freedom.

The formal procedure for reducing the original Hubbardmodel to an effective spin Hamiltonian is a canonicaltransformation plus a projection into the lowest energysubspace, which is adiabatically connected with the sub-space of states containing exactly one electron per atom inthe t ! 0 limit. In this way, one can derive the well-knownHeisenberg Hamiltonian that was originally introduced asa phenomenological model for quantum ferromagnets [1].However, a low-energy physics which is entirely describ-able in terms of spin degrees of freedom, does not implythat charge degrees of freedom are completely frozen. Infact, the effective antiferromagnetic (AFM) exchangebetween local moments arises from the combination of afinite electronic localization length and the fermionicstatistics.

As pointed out in Ref. [2], the finite localization lengthcan also lead to nonuniform charge distributions or electricorbital currents. Indeed, both phenomena can occur inequilibrium if the Mott insulator undergoes a symmetrybreaking phase transition. Charge redistributions arisefrom states that spontaneously break the equivalencebetween bonds [3], while orbital currents emerge in states

that exhibit spontaneous scalar spin chirality h�jkli ¼hSl � ðSj � SkÞi � 0 [4,5]. The notion of scalar spin chi-

rality appears in numerous discussions of magnets andsuperconductors [6–14], and its identification with anobservable (electric current density) is crucial for measur-ing this subtle order parameter.The electric charge and current density operators � and

IðcÞ are related by the continuity equation that reflects theconservation of the total charge. Similarly, energy conser-vation leads to a second continuity equation for the energy

and heat current density operators � and IðhÞ. The only

symmetry operation that distinguishes � and IðcÞ from �

and IðhÞ is charge conjugation. This simple observationleads to a subtle connection between the effective electricand heat current density operators, and the main physicalconsequence is the central result of this Letter.We derive the effective heat current density operator

which is also proportional to the local scalar spin chirality.This common nature of the effective electric and heatcurrent density operators leads to a novel effect in Mottinsulators: dc heat current produced by a temperaturegradient can induce an array of orbital magnetic moments,which is different from the spin ordering. Although theproblem of heat conduction in Mott insulators was inves-tigated for many years [15,16], we are not aware of anystudy of heat current-induced orbital magnetic moments.Our predictions can be tested by performing nuclear mag-netic resonance (NMR) measurements in the presence of afinite temperature gradient.We start by considering a half-filled single-band

Hubbard model defined on an arbitrary lattice

H ¼ �Xjk;�

tjkðcyj�ck� þ cyk�cj�Þ þU

2

Xj

ðnj � 1Þ2; (1)

where the operator cyj� (cj�) creates (annihilates) an elec-

tron with spin � ¼" , # on the atom j and tjk denotes the

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hopping between the j and k atoms. nj ¼ P�nj� �P

�cyj�cj� is the electron number operator for the atom j.

The HamiltonianH has 2N degenerate ground states fortjk ¼ 0 (N is the total number of atoms in the lattice)

because the spin of the electron that occupies each atomor site can either be up or down. States in this ground spacewill be denoted by j�i. The massive degeneracy is liftedfor finite tjk=U � 1 and the new low-energy eigenstates

jc i can be obtained by applying a unitary transformationto the states j�i : c ¼ expð�SÞj�i. S is the (anti-Hermitian) generator of the unitary transformation. The

effective low-energy operator ~O for a given observable Ois obtained by projecting it into the low-energy subspacespanned by the states jc i. However, in order to express~O as a function of spin operators only, it is necessaryto work in the basis of j�i states. In this basis we

have ~O ¼ eSPcOPc e�S ¼ P�e

SOe�SP�, where Pc ¼expð�SÞP� expðSÞ is the projector on the subspace gen-

erated by the states jc i, while P� projects on the subspace

generated by the singly occupied states j�i. For O ¼ Hwe obtain the effective AFM Heisenberg spin Hamiltonian

~H ¼ Xhiji

Jjk

�Sj � Sk � 1

4

�(2)

with Jjk ¼ 4t2jk=U, Sj ¼P

�;�cyj����cj� and � the vector

of Pauli matrices. In a similar way, we can obtain the

effective operators for the charge �j ¼ eP

�cyj�cj� and

electric current density

IðcÞjk ¼ � ie

@

X�

tjkðcyk�cj� � cyj�ck�Þejk; (3)

where ejk is a unit vector along the bond jk [2]. Here we are

using the linear dimension of the unit cell as our unit oflength. These two operators are related by the continuity

equation on the lattice @t�þr � IðcÞ ¼ 0 that arises fromthe conservation of the total number of electrons

½H ;P

jnj� ¼ 0, with nj ¼P

�cyj�cj�. Because the small-

est loop in a lattice is a triangle, contributions to theeffective current density operator must involve at leastthree spins. In addition, the electric current density is ascalar under spin rotations and odd under timer reversal.Therefore, three-spin contributions (jkl) must be propor-tional to the scalar spin chirality �jkl [2]:

~IðcÞjk ¼ e

@ejk

Xl

�jklSl � ðSj � SkÞ; (4)

where �jkl ¼ �24tjktkltlj=U2 þOðt5=U4Þ. �j is a scalar

under spin rotations and even under time reversal.Therefore, three-spin contributions (jkl) must consistof a linear combination of scalar products of two spinoperators [2]:

~�j ¼ eþ eXkl

jklðSj � Sk þ Sj � Sl � 2Sk � SlÞ (5)

with jkl ¼ 8tjktkltlk=U3 þOðt4=u4Þ. The sum of the pre-

factors in front of each of the three scalar products must beequal to zero because of the Pauli exclusion principle:~�j ¼ e on a triangle of three fully polarized spins.

It is interesting to note that both ~�j and ~IðcÞjk are odd in the

hopping amplitudes. The reason is that charge conjugation(particle-hole transformation) changes the sign of the hop-ping amplitudes (tjk ! �tjk) in Eq. (1). In other words,

because �j and IðcÞjk are odd under charge conjugation,

contributions to the corresponding effective operatorsmust be odd in the hopping amplitudes. This observationimplies that contributions to these effective operators canonly come from loops of an odd number of hopping terms(see Fig. 1). In the effective Heisenberg model description(2), these are loops of an odd number of AFM exchangeinteractions. Therefore, geometric frustration is a precon-dition for having nontrivial effective charge and electriccurrent density operators in Mott insulators.Equation (5) implies that magnetic configurations that

break the equivalence between different bonds lead toelectric charge redistributions. This simple observationhas multiple consequences. For instance, the charge redis-tribution induced by certain spin orderings can lead to a netelectric polarization [2]. This magnetically driven ferro-electricity is observed in type-II multiferroic materials andEq. (5) allows us to compute the electronic contribution tothe electric polarization [17]. For example, the chargeeffects that have been recently observed in the Mott insu-lator Cu3MoO9 can be explained by applying this equation[18,19]. Topological defects provide another example ofspin configurations that typically break the equivalencebetween bonds. According to Eq. (5), if the underlyingspin model is frustrated, these defects must induce anelectric charge redistribution. This observation wasrecently exploited by D. Khomskii to demonstrate thatmagnetic monoples in spin ice carry a net electricdipole [20].After introducing the effective charge and current den-

sity operators, we are ready to connect the latter one withthe effective heat current density operator. The total energyis conserved by H because @tH ¼ 0, and this conserva-tion law is expressed by a second continuity equation:

@t�j þr � IðhÞ ¼ 0. �j is the energy density and

j k lj k

l(a) (b)

FIG. 1 (color online). Leading order contributions to theeffective (a) electric and (b) heat current density operators.

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IðhÞjk ¼�tjk2ejk

X�

ðcyj� _ck�� _cyj�ck��cyk� _cj�þ _cyk�cj�Þ (6)

is the heat current density [21]. The time derivative of thecreation and annihilation operators is obtained from the

Heisenberg equation _cðyÞj� ¼ i½H ; cðyÞj� �=@. By replacing

_cðyÞj� in Eq. (6), we obtain the following contributions to

the heat current density operator IðhÞjk ¼ Iðh;UÞjk þ Iðh;tÞjk :

Iðh;UÞjk ¼ itjkejk

2@U�jk

X�

½cyk�cj� � cyj�ck��;

Iðh;tÞjk ¼ itjkejk2@

Xl�

ðtlkcyl�cj� þ tjlcyk�cl� � H:c:Þ;

(7)

where �jk ¼ nj þ nk � 2.

The electric and heat current operators have the samesymmetry properties except for the parity under charge

conjugation (both �j and IðhÞjk are even under charge con-

jugation). Therefore, the leading order contribution to theeffective heat current density operator, from trimers con-taining the bond jk, is also proportional to the scalar spinchirality �jkl. However, the proportionality constant must

be even in the hopping amplitude. By performing the

canonical transformation for O ¼ IðhÞjk , we obtain

~IðhÞjk ¼ 1

@ejk

Xl;l0ð�j;klSl þ �k;jl0Sl0 Þ � ðSj � SkÞ (8)

with �j;kl ¼ �8t2jkt2kl=U

2 and �k;jl0 ¼ �8t2jkt2jl0=U

2. A

similar expression was derived for a Heisenberg spin chainfrom the continuity equation in terms of effective operators[22]. The expression for�j;kl implies that, in contrast to the

case of the electric current density, loops are not needed to

get finite contributions to ~IðhÞjk (see Fig. 1). In other words,

the effective heat current density operator is nonzero for aone-dimensional system with only nearest-neighbor hop-ping. This additional difference between the effective heatand electric current density operators arises from the fact

that the net electric current hPjk~IðcÞjk i is always zero in the

low-energy sector of a Mott insulator (electrons are local-

ized), while the net heat current hPjk~IðhÞjk i can be finite

(spin excitations can transport energy).

At this point it is important to emphasize that ~IðcÞjk and

~IðhÞjk cannot be obtained in general from the continuity

equations for the effective operators @t ~�þr � ~IðcÞ ¼ 0

and @t~�þr � ~IðhÞ ¼ 0. The effective current densityoperators on adjacent bonds jk and kl have a commoncontribution if the hopping tjl is nonzero (triangular loop).

The common contributions cancel out in the divergence ofthe effective current density operator. Therefore, knowing

r � ~I is not enough to get ~I.Our next goal is to demonstrate that the common nature

of ~IðcÞjk and ~IðhÞjk (both are proportional to the local scalar

spin chirality) leads to a novel effect in Mott insulators on aparticular lattice. To illustrate this point we will assumethatH is defined on a honeycomb lattice with only nearestand next nearest hopping amplitudes t and t0, respectively,with t0 � t. We will also assume that both sides of thesystem are connected to different thermal baths with tem-peratures Th and Tl [see Fig. 2(a)]. If the highest tempera-ture Th is much lower than the charge gap of the Mottinsulator (kBTh � U), we can use our low-energy effective

model ~H and operators ~O to describe the electronicproperties of the system under consideration.The finite temperature difference�T ¼ Th � Tl induces

a heat current density h~IðhÞjk i ¼ ihx on the horizontal bonds

jk and h~IðhÞkl i ¼ �ihe�=2, on the oblique bonds kl, where

e� ¼ x=2� ffiffiffi3

py=2 with x (y) the unit vector along the x

(y) direction. According to Eq. (8), this heat current densitydistribution must arise from a nonzero distribution of�jkl � h�jkli. We will adopt the convention that the three

sites jkl are oriented clockwise. Because the system istranslationally invariant, there are only six types of tri-angles that are depicted in Fig. 2(b). The labels of the sixtriangles, right (R), left (L), top right (TR), top left (TL),bottom right (BR), and bottom left (BL), are relative to ahorizontal bond. The system remains translationally invari-ant and preserves its mirror symmetry plane perpendicularto the y axis in the presence of the uniform heat currentdensity along x. The other symmetry that survives is theproduct of a reflection in the plane perpendicular to the xaxis and time reversal. These remaining symmetries implythat the mean value of scalar spin chirality is zero for the Rand L triangles, �R;L ¼ 0, while it is finite and of opposite

signs for the T and B triangles: �TM ¼ ��BM ¼ �, withM ¼ L;R. Knowing h�jkli on each triangle, we can obtain

the mean value of the heat current density on the horizontalbonds from Eq. (8), and the electric current density ic onthe oblique dashed bonds shown in Fig. 2(b), from Eq. (4):

ih ¼ �32t4

@U2�; ic ¼ �24et2t0

@U2�: (9)

Here ih does not depend on t0 because we are neglectingcontributions of order t02=t2. The electric orbital currentcirculates around the top and bottom triangles and thecorresponding orbital magnetic moments are� ¼ �icA ¼�3et0Aih=4t2, where the þ (� ) sign holds for the top(bottom) triangles [see Fig. 2(c)] and A is the area of atriangle. These orbital magnetic moments are originated bythe local scalar spin chirality induced by the heat current.An even simpler example is provided by the Hubbard

model defined on the sawtooth chain depicted in Fig. 3.The dominant contribution to the heat current comesfrom the nearest-neighbor hopping t and it is ih ¼�16t4�0=@U

2. Here �0 � hSjþ1 � ðSj�1 � SjÞi does not

depend on the site index j because the system is transla-tionally invariant by one lattice parameter (j ! jþ 1)

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when t0 ¼ 0 (we are neglecting contributions of ordert02=t2 � 1). The finite �0 leads to orbital currents

ic ¼ 3et0

2t2ih (10)

that circulate around the triangles. In contrast to the pre-vious case, the orbital currents are all oriented in the samedirection (see Fig. 3); i.e., the heat current induces orbitalferromagnetism and the magnitude of each orbital momentis � ¼ 3et0ihA=2t2. To understand the origin of this

uniform orbital magnetization induced by a thermalcurrent, it is convenient to go back to Eqs. (3) and (7) tonotice that

Iðh;UÞjk ¼ U

2e�jkI

ðcÞjk ; (11)

where Iðh;UÞjk is the Coulomb contribution to the heat current

density. Multiplying both sides of Eq. (11) by �jkRjk,

where Rjk ¼ ðrj þ rkÞ=2 is the coordinate of the bond jk,

we obtain

Pjk � Iðh;UÞjk ¼ U

2e�2

jkMjk: (12)

Here Pjk ¼ �jkRjk andMjk ¼ Rjk � IðcÞjk are the electric

and magnetic polarization densities. Equation (12) impliesthat a uniform thermal current density induces a net mag-netization only if the Mott insulator has a net electricpolarization: hMi / hPi � hIhi, where P ¼ P

hjkiPjk and

M ¼ PhjkiMjk are the macroscopic electric and magnetic

polarizations. In other words, the Mott insulator must beferroelectric or the lattice must break inversion symmetry,like the sawtooth chain of Fig. 3, for the heat current toinduce a net orbital magnetization. Indeed, by takingmean values in Eq. (12) and using that jhPjkij / t2t0=U3

and h�2jki / t4=U4 for U � jtj, we obtain jhMjkij /

et0jhIðhÞij=t2, in agreement with Eq. (10).For a magnetic contribution to the thermal conductivity

of the order of 100 W=ðm � KÞ and an exchange constant of1000 K [23], the orbital moments induced by a thermalgradient of 10 K=�m are of order 10�4�B, where �B isthe Bohr magneton (we are assuming that U=t 10). Thismagnetic moment corresponds to a magnetic field value atthe center of each triangle of the order of 1 G [24].Although these are small magnetic moments, the magni-tude of the effect should be significantly larger in theintermediate coupling regime because jhPjkij=Uh�2

jkiremains of the same order but the electronic contributionto the thermal current becomes much larger. Indeed, mag-netoelectric effects measured in the intermediate-couplingorganic Mott insulator �-ðBEDT-TTFÞ2Cu2ðCNÞ3 indicatea rather strong spin-charge coupling that is relevant forunderstanding the low temperature properties of this spinliquid candidate [25].

t ih

ic

j

j-1 j+1

FIG. 3 (color online). Sawtooth chain with hoppings t and t0,where t0 � t. Black arrows indicate the circulation of the heatcurrent ih, while red arrows denote the electric current. Themagnetic moments induced by orbital electric currents areindicated with crossed circles.

Hot Th

(a)

ttih

ih/2 Tl

Cold

(b)

ih/2 TL

ic ic

ih/2 TR

ih/2

L R

BL BR ih/2

ic ic

ih

Tl

Hot Cold Th

(c)

FIG. 2 (color online). (a) Hubbard model on a honeycomblattice with nearest- and next-nearest-neighbor hoppings t (fulllines) and t0 (dashed lines), and different temperatures Tl and Th

on both sides of the system. (b) The heat current ih induces afinite scalar spin chirality with opposite signs on the top (TR andTL) and bottom (BU and BL) triangles. Because t0 is finite, thelocal scalar spin chirality produces orbital currents [see Eq. (4)],which generate antiferromagnetically ordered orbital magneticmoments. (c) Orientation of the orbital magnetic momentsinduced by the heat current. Circled dots (crosses) denote up(down) moments.

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In summary, we have derived the effective heat currentoperator for the strong-coupling limit of the half-filledHubbard model and demonstrated that, as in the case ofthe electric current density, the leading order contributionis proportional to the scalar spin chirality. This commonproperty of both current density operators is dictated bysymmetry considerations. The physical consequence ofthis commonality is a novel thermomagnetic effect: heatcurrents induce orbital magnetic moments in frustratedMott insulators. These moments can be measured withNMR if a large enough temperature gradient can be appliedto the Mott insulator (temperature gradients of 50 K=�mcan be applied to nanodevices [26]). Moreover, we haveshown that the orbital moments produce a net magnetiza-tion, which is much easier to measure with conventionalmethods, if a net electric polarization is present. This spin-charge effect should be much stronger in the intermediate-coupling regime that is relevant for several frustrated Mottinsulating materials. We emphasize that this thermomag-netic effect does not rely on any (low-energy) quasiparticledescription, such as spin waves for magnetically orderedstates, and remains valid in the diffusive high temperatureregime, i.e., for temperatures higher than the magneticordering temperature.

This work was carried out under the auspices of theNNSA of the U.S. DOE at LANL under Award No. DE-AC52-06NA25396, and was supported by the U.S.Department of Energy, Office of Basic Energy Sciences,Division of Materials Sciences and Engineering.

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