Divergent Thermopower without a Quantum Phase Transition
DDivergent Thermopower without a Quantum Phase Transition
Kridsanaphong Limtragool and Philip W. Phillips
Department of Physics and Institute for Condensed Matter Theory,University of Illinois 1110 W. Green Street, Urbana, IL 61801, U.S.A. (Dated: November 8, 2018)A general principle of modern statistical physics is that divergences of either thermodynamicor transport properties are only possible if the correlation length diverges. We show by explicitcalculation that the thermopower in the quantum XY model d = 1 + 1 and the Kitaev model in d = 2 + 1 can 1) diverge even when the correlation length is finite and 2) remain finite even whenthe correlation length diverges, thereby providing a counterexample to the standard paradigm.Twoconditions are necessary: 1) the sign of the charge carriers and that of the group velocity must beuncorrelated and 2) the current operator defined formally as the derivative of the Hamiltonian withrespect to the gauge field does not describe a set of excitations that have a particle interpretation,as in strongly correlated electron matter. The recent experimental[1] and theoretical[2] findings onthe divergent thermopower of a 2D electron gas are discussed in this context. A truism in modern statistical mechanics is thatdivergences (or more generally non-analyticities) ina thermodynamic quantity or a transport propertyalways signal a transition to a new state of mat-ter. In fact, the very notion of adiabatic continu-ity is based on the intuition that non-analyticitiesresulting from tuning some system parameter can-not emerge without the crossing of a phase bound-ary. More precisely, as long as the correlation lengthremains finite, then no divergences are possible be-cause both transport and thermodynamic propertiesare governed by the singular part of the free energy.We present here a counter example to this rule. Toestablish our result, we consider the quantum XYmodel in 1D and the Kitaev[3] model in 2D, bothof which can be solved[4–6] exactly using a map-ping to fictitious fermionic degrees of freedom. Inboth cases, we show exactly that the thermopower,appropriately defined, diverges at fillings that havenothing to do with the quantum phase transition inthese models. At the spurious divergences, no ther-modynamic quantity experiences a non-analyticity.As we will see, the heart of this problem is a break-down of the particle interpretation of the current-carrying degrees of freedom.This work is motivated by recent measurements[1]on the thermopower in a dilute 2D electron gas.These experiments are the latest in a series of re-markable observations[7] that a dilute 2D electrongas exhibits a resistivity that decreases as the tem-perature is lowered with no apparent upturn (as isexpected from the scaling theory[8]) thereby provid-ing evidence for a low-temperature metallic state.Mokashi et al. reported[1] that the thermopower onthe metallic side of the transition diverges exhibitingscaling of the form S ( T, n ) = eT s ( n ) = T ( n − n c ) − µ (1) with µ = 1 . ± .
1. Consequently, if the thermpowerwere to be measured on the insulating side, it shouldchange sign. As a result, they interpreted[1] such acritical divergence, based on a simple appeal to theadiabatic continuity principle, as definitive evidencethat the transition to the metallic state represents atrue T = 0 quantum phase transition. This wouldthen represent the most important finding since theinitial discovery paper in 1996[9]. More recently,Kirkpatrick and Belitz[2] argued that the divergenceof the thermopower holds crucial implications for thescaling of the specific heat as the exponent µ de-termines the product of dynamical and correlationlength exponents, z and ν , respectively.Hence, while explaining the experimental data iscertainly of interest, our focus is on whether alterna-tive mechanisms exist for a divergent thermopowerother than a quantum phase transition. Althoughthe models in the counterexamples we construct arenot directly applicable to the experiments, the mech-anism for the divergence of the thermopower is. Wefind that in strongly correlated systems, the ther-mopwer can diverge simply because the the currentdoes not have a particle interpretation.We treat at first the quantum XY model in 1D.This model can be fermionized[4, 10] H = − (cid:88) i ( c † i c i +1 + c † i +1 c i + Γ c † i c † i +1 + Γ c i +1 c i + h (1 − c † i c i )) , (2)using the Jordan-Wigner transformation schemewith c i a canonical fermionic annihilation operatorfor site i . The hopping between two sites is set to 1 inthe unit of J = ( J x + J y ), Γ = J x − J y J is a measure ofthe exchange anisotropy and h = HJ is the effectivemagnetic field or in the fermionic model − h is a di-mensionless chemical potential. Although Γ (cid:54) = 0 im- a r X i v : . [ c ond - m a t . s t r- e l ] A p r plies an effective particle non-conservation, therebymaking it possible to fix only the average numberof particles, we have shown[11] that a unique ex-pression exists for the thermopower defined as a re-sponse to a longitudinal field. We calculated the ex-act expression for the thermopower[11] and showedthat it diverges at the phase transition, h = ± h = ± H = (cid:88) k ε k γ † k γ k , (3)contains the new fermionic operators, γ k = u k c k − iv k c †− k and γ † k = u k c † k + iv k c − k , whose energies are ε k = ± (cid:113) ( h − cos k ) + Γ sin k with u k = 2 cos θ k and v k = 2 sin θ k and the angle θ k defined throughsin θ k = (Γ sin k ) /ε k and cos θ k = ( h − cos k ) /ε k . Wewill be analyzing the properties of this model as afunction of the average particle density, x = (cid:104) c † i c i (cid:105) = 12 π (cid:90) π dk (cid:18) − cos θ k tanh (cid:18) β | ε k | (cid:19)(cid:19) . The thermodynamic quantity of interest is theheat capacity, CN = k B π π (cid:90) ( ε k k B T ) sech ( βε k . (4)However, our main focus is the thermopower. Tothis end, we write the charge ( ˆ J x ) and thermal cur-rents ( ˆ J Qx ) along the x-direction in terms[12] of theresponses to an electric field and a temperature gra-dient, 1Ω (cid:104) ˆ J x (cid:105) = L E x + L (cid:18) − ∇ x TT (cid:19) (5)1Ω (cid:104) ˆ J Qx (cid:105) = L E x + L (cid:18) − ∇ x TT (cid:19) (6) Although 4 expressions (Eqs. (6a-6d)) are derived in Ref.[11] for the thermopower, Eqs. (6c) and (6d) are valid onlyfor a transverse field, while Eqs. (6a) and (6b) apply strictlyfor a longitudinal field. (6a) follows from (6b) from thecontinuity equation which is not valid here. Since the ther-mopower experimentally is the response to a longitudinalfield, only Eq. (6b) is valid in this context and hence thethermopower has a unique definition. x (particle density) T he r m opo w e r ( i n un i t s o f k B / e ) (a) XY model with Γ = 0 . , t = 0 . x (particle density) T he r m opo w e r( i n un i t s o f k B / e ) (b)Kitaev model with J y = 0 . , t = 0 . FIG. 1. The plots of the thermopower vs. particle den-sity. using the Onsager coefficients, L ij . In these ex-pressions, Ω is the volume of the system. Thethermopower[12], Q = L T L , (7)is the ratio of the voltage generated per gradient oftemperature. An explicit calculation of L ij is possi-ble in frequency and momentum space for the mod-els we consider here. The transport or fast limitcorresponds to lim ω → lim q x → . For the quantumXY model in 1D, the exact expression[11] for thethermopower,lim ω → lim q x → eQk B = π (cid:82) − π dk ε k k B T sin k dndkπ (cid:82) − π dk ( u k − v k ) sin k dndk , (8)involves a simple integral over the first Brillouinzone with an integrand determined by the coher-ence factors and the fermionic occupation, n =1 / ( e ε k /k B T + 1). The numerator of this expression isbounded over integration in the first Brillouin zone.Consequently any divergence arises entirely from thedenominator. We display the results for Γ = 0 . t = 0 . t is the dimensionless temperature and defined as FIG. 2. This is a plot of a number of divergences inthermopower vs. particle filling at given values of Γ and t in the fermionized quantum XY model. t = k B T /J . For these parameters, the particle den-sity at the phase transition, h = ±
1, is x ≈ .
15 orthe particle-hole complement, x ≈ .
85. Fig. 1(a)shows that indeed the thermopower does diverge atthese values of x as we reported earlier[11]. How-ever, there are other divergences, for example at x ≈ . , . T = 0. Hence, non-analyticities inthermodynamics need not affect transport proper-ties and conversely divergences in transport proper-ties are not necessarily accompanied by singularitiesin the thermodynamics. Before we analyze the originof these results, we first show that our findings arenot an artifact of 1-dimensional (d=1+1) physics.To this end, we consider the Kitaev model, H = − J x (cid:88) x − bonds σ xR σ xR (cid:48) − J y (cid:88) y − bonds σ yR σ yR (cid:48) − J z (cid:88) x − bonds σ zR σ zR (cid:48) , (9)on a honeycomb lattice in which the summationsare over all links between site R and R (cid:48) . ThisHamiltonian can be fermionized[5] by the Jordan-Wigner transformation. The result is a model of −2 −1 0 1 200.020.040.060.08 h C / N k B FIG. 3. Heat capacity in the quantum XY model atparameter values Γ = 0 . t = 0 . h = ± x ≈ . , .
85. At h = ± .
27 or x ≈ . , . Dirac fermions, H = J x (cid:88) i ( c † i + c i )( c † i +ˆ x − c i +ˆ x ) + J y (cid:88) i ( c † i + c i ) × ( c † i +ˆ y − c i +ˆ y ) + J z (cid:88) i α i (2 c † i c i − , (10)on a square lattice. At every lattice site there isone conserved quantity, α i , which has the value of-1 or 1. The ground state of this system corre-sponds to having α i equal to 1 everywhere. Sowe choose all α i to be 1. This Hamiltonian canbe solved exactly in the same way as the quantumXY model[5, 6]. The energy spectrum is given by ε k = 2 (cid:112) ( J z − (cid:80) i J i cos k i ) + ( (cid:80) i J i sin k i ) andthe coherence factors defined through the param-eters u k and v k in the Bogoliubov transformationsatisfycos θ k = u k − v k = 2( J z − J x cos k x − J z cos k y ) ε k sin θ k = 2 u k v k = 2( J x sin k x + J y sin k y ) ε k . (11)The sum on i in the energy spectrum above is over x and y . The analogous expression for the ther-mopower, eQk B = π (cid:82) − π dk x π (cid:82) − π dk y ε k k B T sin k x dndk x π (cid:82) − π dk x π (cid:82) − π dk y ( u k − v k ) sin k x dndk x , (12)obtained from an exact calculation of L ij in the fastlimit, is precisely the 2D generalization of Eq. (8).For the Kitaev model, the thermopower, Q = Q ( J x , J y , J z , t ), depends on the average particle den-sity x = x ( J x , J y , J y , t ). We write J y and J z in units FIG. 4. Number of divergences in the thermopower ver-sus particle filling for fixed values of J y and t of thefermionized Kitaev model. Each color region displays adifferent number of divergences. of J x (by setting J x = 1). So for a fixed value of J y and t , we can plot thermopower versus particle den-sity by varying J z . Figs. (1(b)) and (4) demonstratethat the behaviour is identical to that of the quan-tum XY model. Hence, our results are not an arti-fact of 1-dimensional physics. Note that this modelalso exhibits regions in which no divergence obtainsalthough the quantum phase transition is present.The origin of this physics is tied to the denomina-tors of the expressions for the thermopower because L is a completely bounded function for all valuesof k inside the first Brillouin zone. Consider the de-nominator, in the case of the XY modelXY → π (cid:90) − π dk ( u k − v k ) sin k dndk , (13)the Kitaev model being the direct 2D analogue. Thesin k factor arises from the momentum dependenceof the local current operator, J j = − i ( c † j c j +1 − c † j +1 c j ). The quantity q k = u k − v k = cos θ k ∝ h − cos k is the effective charge of the quasiparticles,which is even with respect to k . It is instructive thento rewrite the denominator, I = (cid:90) π − π dkJ ( k ) n k + v d , (14)in a form which lays plain that it is no more thanthe current in response to the applied field with v d = q k E x τ , the drift velocity, and J ( k ) the mo-mentum dependence of the current operator. In theabsence of the drift velocity, I = 0. Taylor ex-panding around v d = 0 yields Eq. (13). Herein −3 −2 −1 0 1 2 3−2−10123 x 10 −7 k I n t eg r and (a)Γ = 0 . , t = 0 . , h = 0 . FIG. 5. Integrand of L (denominator of the ther-mopower) showing the cancellation which leads to a di-vergence in the thermopower. lies the crux of the problem. In a non-interactingsystem, the local definition of the current opera-tor used here and that arising from the continuityequation both yield the same result, namely that J ( k ) = q k dε k /dk = dH/dk , in which case the in-tegrand is positive definite and cannot integrate tozero. However, for the problem at hand, the cur-rent operator arising from the continuity equation,namely q k dε k /dk , is non-local in space, possessingsink and source terms, and hence is not tenable.Such non-locality typifies most strongly correlatedsystems because the entities which carry the cur-rent are not simply determined by the kinetic partof the Hamiltonian. Consequently, the current op-erator, defined from the continuity equation is non-local and lacks a particle interpretation. In suchcases, I can vanish. The vanishing of I here takesplace because the group velocity, dε k /dk , is an oddfunction of k , while q k is even. Consequently, themomenta at which q k and dε k /dk change sign neednot be correlated. Because the overall integrand isan even function of k , it will have positive and neg-ative contributions on the interval [0 , π ], which forcertain system parameters could yield a cancellationas illustrated in Fig. (5).Classic examples in which the operators in the lo-cal current operator do not coincide with the chargecarriers are the insulating state of the Hubbardmodel at half-filling for sufficiently large U . In thisproblem, there is no divergent length scale as thereis no order parameter for the Mott insulating state.It is entirely likely that the insulator in the dilute2D electron gas[1] is induced by the correlations aswell as it obtains in the large r s regime. Hence, cau-tion must be taken in using standard scaling argu-ments to relate the thermopower to divergent cor-relation lengths as has been done recently[2]. Un-less the charge carriers are local degrees of freedom,naive scaling with the correlation length is insuffi-cient to describe transport properties such as thethermopower. Acknowlegements
We thank Taylor Hughes,Jeffrey Teo, Mike Stone, Brandon Langley, TonyHegg, Wei-cheng Lee, Ted Kirkpatrick, and NigelGoldenfeld for sustained commentary throughoutthe completion of this work and NSF DMR-1104909for partial funding of this project. KL is supportedby the Department of Physics at the University ofIllinois and a scholarship from the Ministry of Sci-ence and Technology, Royal Thai Government. [1] A. Mokashi, S. Li, B. Wen, S. V. Kravchenko,A. A. Shashkin, V. T. Dolgopolov, and M. P.Sarachik, Phys. Rev. Lett. , 096405 (Aug2012), http://link.aps.org/doi/10.1103/PhysRevLett.109.096405 [2] T. R. Kirkpatrick and D. Belitz, Phys. Rev. Lett. , 035702 (Jan 2013), http://link.aps.org/doi/10.1103/PhysRevLett.110.035702 [3] A. Kitaev, Annals of Physics , 2 (2006),ISSN 0003-4916, ¡ce:title¿January Special Is-sue¡/ce:title¿, [4] S. Katsura, Phys. Rev. , 1508 (Sep 1962), http://link.aps.org/doi/10.1103/PhysRev.127.1508 [5] H.-D. Chen and Z. Nussinov, Journal of PhysicsA: Mathematical and Theoretical , 075001(2008), http://stacks.iop.org/1751-8121/41/i=7/a=075001 [6] Z. Nussinov and G. Ortiz, Phys. Rev. B , 214440(Jun 2009), http://link.aps.org/doi/10.1103/PhysRevB.79.214440 [7] E. Abrahams, S. V. Kravchenko, and M. P.Sarachik, Rev. Mod. Phys. , 251 (Mar 2001), http://link.aps.org/doi/10.1103/RevModPhys.73.251 [8] E. Abrahams, P. W. Anderson, D. C. Licciardello,and T. V. Ramakrishnan, Phys. Rev. Lett. , 673(Mar 1979), http://link.aps.org/doi/10.1103/PhysRevLett.42.673 [9] S. V. Kravchenko, D. Simonian, M. P. Sarachik,W. Mason, and J. E. Furneaux, Phys. Rev. Lett. , 4938 (Dec 1996), http://link.aps.org/doi/10.1103/PhysRevLett.77.4938 [10] S. Sachdev, Quantum Phase Transitions , 2nd ed.(Cambridge University Press, 2012)[11] A. Garg, B. S. Shastry, K. B. Dave, and P. Phillips,New Journal of Physics , 083032 (2011), http://stacks.iop.org/1367-2630/13/i=8/a=083032 [12] B. S. Shastry, Reports on Progress in Physics , 016501 (2009),, 016501 (2009),