Temperature and Chemical Potential Dependence of the Parity Anomaly in Quantum Anomalous Hall Insulators
C. Tutschku, F. S. Nogueira, C. Northe, J. van den Brink, E. M. Hankiewicz
TTemperature and Chemical Potential Dependence of the Parity Anomaly in QuantumAnomalous Hall Insulators
Christian Tutschku,
1, 4
Flavio S. Nogueira,
2, 4
Jeroen van den Brink,
2, 3, 4 and E. M. Hankiewicz
1, 4 Institute for Theoretical Physics, Julius-Maximilians-Universit¨at W¨urzburg, 97074 W¨urzburg, Germany Institute for Theoretical Solid State Physics, IFW Dresden, 01069 Dresden, Germany Institute for Theoretical Physics, TU Dresden, 01069 Dresden, Germany W¨urzburg-Dresden Cluster of Excellence ct.qmat
The low-energy physics of two-dimensional Quantum Anomalous Hall insulators like (Hg,Mn)Tequantum wells or magnetically doped (Bi,Sb)Te thin films can be effectively described by two Cherninsulators, including a Dirac, as well as a momentum-dependent mass term. Each of those Cherninsulators is directly related to the parity anomaly of planar quantum electrodynamics. In thiswork, we analyze the finite temperature Hall conductivity of a single Chern insulator in 2+1 space-time dimensions under the influence of a chemical potential and an out-of-plane magnetic field. Atzero magnetic field, this non-dissipative transport coefficient originates from the parity anomalyof planar quantum electrodynamics. We show that the parity anomaly itself is not renormalizedby finite temperature effects. However, it induces two terms of different physical origin in theeffective action of a Chern insulator, which is proportional to the Hall conductivity. The first termis temperature and chemical potential independent, and solely encodes the intrinsic topologicalresponse. The second term specifies the non-topological thermal response of conduction and valenceband states. In particular, we show that the relativistic mass of a Chern insulator counteractsfinite temperature effects, whereas its non-relativistic mass enhances these corrections. Moreover,we extend our analysis to finite magnetic fields and relate the thermal response of a Chern insulatortherein to the spectral asymmetry, which is a measure of the parity anomaly in orbital fields.
I. INTRODUCTION
Back in the 1980s, Haldane proposed the first solidstate model of a Quantum Anomalous Hall (QAH) insula-tor by adding a parity-breaking [1] Dirac mass term to anotherwise gapless graphene structure [2]. Such a systemfeatures a non-zero Hall conductivity even in the absenceof Landau levels (LLs). From a high-energy perspective,this model is directly related to the parity-anomaly ofplanar quantum electrodynamics, which implies that itis not possible to quantize a single Dirac fermion in aparity symmetric manner [3, 4]. Strictly speaking, theHaldane model contains two Dirac fermions as it is basedon the hexagonal lattice structure of graphene. However,by fine-tuning the Haldane mass, one of the Dirac fermionmass gaps can be closed, whereas the other one remainsopen. In this parameter limit, the band-structure con-tains a single gapless Dirac fermion with a non-zero Hallconductivity. This implies that in the Haldane modelone of the Dirac fermions alone is suitable to realize theparity anomaly in 2+1 dimensions. Hitherto, it was notpossible to experimentally setup the Haldane model in acrystalline structure [5]. Instead, another type of QAHinsulators was predicted in spin-polarized topological in-sulators (TIs) like (Hg,Mn)Te quantum wells [6, 7] ormagnetically doped (Bi,Sb)Te thin films [8, 9]. Theirlow-energy physics is captured by the superposition oftwo Chern insulators [10]. Similar to the Haldane model,the gap of one of these Chern insulators can be closedby magnetic doping of the system, whereas at the sametime the second Chern insulator remains gapped. In thisfine-tuned limit, the gapless Chern insulator realizes theparity anomaly as its contribution to the Hall conduc- tivity is in general non-zero. Hence, the analysis of sin-gle Chern insulators, which is the main purpose of thepresent work, allows to study measurable consequencesof the parity anomaly in a solid state material. In con-trast to the Dirac fermions in the Haldane model, eachChern insulator is characterized by two different parity-breaking mass terms: A conventional Dirac mass, as wellas momentum-dependent Newtonian mass. As the Hallconductivity of a single Chern insulator is unaltered ifone takes the parity-symmetric zero-mass limit, both, theDirac as well as the momentum-dependent mass term,are directly related to the parity anomaly. It was re-cently shown that the momentum-dependent mass termacts similar to a Wilson fermion in a lattice regulariza-tion of a pure Dirac system [11]. As such, it ensuresan integer- instead of a half-quantized Hall conductivityassociated to a pure Dirac fermion [12, 13].The parity anomaly is a zero magnetic field effect be-cause an external magnetic field breaks the parity sym-metry at the classical level. However, it can be shownthat even in quantizing magnetic fields the signatures ofthe parity anomaly persist. They remain encoded in thespectral asymmetry [14]. Below a critical magnetic field,the parity anomaly effectively adds one LL to the entireHall response of a Chern insulator. Above this field, themagnetic field closes the Dirac mass gap and the systemexhibits a conventional QH response [15].All these findings do not incorporate thermal effects.So far, finite temperature signatures in parity anomalydriven systems are restricted to pure Dirac models. Cal-culating the quantum effective action of these systems in-duces a temperature dependent and thus large gauge non-invariant Chern-Simons term originating from the par- a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p ity anomaly [16–21]. While it was shown that this non-invariance is absorbed by higher order non-perturbativecorrections to the effective action [22–29], this featurestill gives rise to a fundamental question: Does the parityanomaly get renormalized by thermal effects? Answeringthis question is not only relevant for the QAH effect inthe materials mentioned above. It is especially importantin the case of interfaces between ferromagnetic insula-tors and three-dimensional topological insulators, wherea proximity-induced interface magnetization has been ex-perimentally observed at high temperatures [30, 31]. Inthis case the out-of-plane magnetization causes a gapopening in the interface Dirac spectrum, which inducesa parity anomaly on the TI surface and a concomitantmagnetoelectric torque in the Landau-Lifshitz equation[32–34]. A similar effect is expected to occur on thesurface of the recently discovered antiferromagnetic TIMnBi Te [35, 36], where the gap in the surface Diracspectrum is an intrinsic feature of the system.By definition, the parity anomaly only implies thebreakdown of parity symmetry at the quantum level.This dictates a certain form of the band-structure,which is temperature independent [37]. Hence, theparity anomaly cannot obtain any finite temperaturecorrection. In contrast, the prefactor of the anomalyinduced Chern-Simons term in the effective actioncorresponds to the finite temperature Hall conductivity.We calculate this non-dissipative transport coefficientfor Chern insulators including both, a Dirac as wellas a momentum-dependent mass. We studied thesesystems in the absence and presence of a magneticfield, as well as with and without particle-hole sym-metry. This leads to the following results: ( i ) Theparity anomaly induces a topological part in the Hallconductivity which is temperature as well as chemicalpotential independent and described by the Chernnumber. ( ii ) The non-quantized finite temperature andchemical potential corrections to the Hall conductivityalso originate from the parity anomaly, since they alsodepend on the band-structure. However, they do notdepend on its topology, being rather related to thetemperature-dependent filling of the valence and con-duction bands. As expected, an increasing Dirac masscounteracts finite temperature effects. On the otherhand, we show that in the nontrivial phase an increasingNewtonian mass enhances the finite temperature cor-rections. ( iii ) In finite magnetic fields, the thermal LLresponse renormalizes the parity anomalous part of theHall conductivity. In the Dirac mass gap it adds to theotherwise quantized parity anomaly related contribution.This work is structured as follows: In Sec. II, we discussthe relation of magnetically doped two-dimensional TIsto the parity anomaly and compare these systems to theHaldane model. In this context, we analyze the band-structure of Chern insulators in the absence and presenceof an out-of-plane magnetic field and with, as well aswithout particle-hole symmetry. In Sec. III and Sec. IV, we analyze the parity-odd transport of a Chern insulatorfor a finite temperature and chemical potential, as well asfor zero and finite magnetic fields, respectively. In Sec. Vwe summarize our results and give an outlook. II. PARITY ANOMALY IN A QAH SYSTEMBEYOND THE HALDANE MODEL
In this work, we consider 2+1 dimensional Chern in-sulators which are defined by two different mass terms:A momentum independent Dirac mass m , as well as amomentum dependent Newtonian mass term B | k | . TheLagrangian of such an insulator is given by [38] L = ¯ ψ (cid:0) Aγ µ k µ − m + Bk i k i (cid:1) ψ , (1)where ψ and ¯ ψ = ψ † γ are the two-component Diracspinor and its adjoint, γ µ =( σ ,i σ ,i σ ) are the 2+1 di-mensional Dirac matrices, and we consider the met-ric g µν = diag(+ , − , − ). Here and throughout themanuscript, Greek indices run over the space-time coor-dinates { , , } , while roman indices run over the spatialcomponents { , } only. Moreover, the parameter A isproportional to the Fermi velocity and σ , , define thePauli matrices. Notice, that in comparison to a pureDirac Lagrangian, the additional Newtonian mass termin Eq. (1) breaks the Lorentz symmetry as it only involvesspatial momenta.The first-quantized Hamiltonian associated to Eq. (1) canbe derived by a Legendre transformation: H = A ( k σ − k σ ) + ( m − Bα ) σ . (2)Here, we introduced the abbreviation α = k + k . Bothof the mass terms in Eq. (2), m and Bα , break the paritysymmetry of the Hamilton. In 2+1 dimensions, paritysymmetry is defined as invariance of the theory under P : ( x , x , x ) → ( x , − x , x ). Consequently, the Diracas well as the Newtonian mass contribute to the integerChern number [12, 13] [39] C CI = (sgn( m ) + sgn( B )) / . (3)Even in the parity symmetric limit m, B → ± , C CI doesnot vanish for sgn( m/B ) >
0. This effect is known as theparity anomaly of Dirac-like systems in odd space-timedimensions. Initially, the parity anomaly has been pre-dicted for a pure Dirac spectrum in Ref. [4]. Due to theabsence of a Newtonian mass term, the Chern numberof a pure Dirac system is given by C QED = ± sgn( m ) / m/B >
0, the system is topologically nontrivial with C CI = ±
1, while for m/B <
0, the system is topologicallytrivial with C CI = 0. In a solid state system with a crystallattice, Dirac fermions in 2+1 space-time dimensions al-ways come in pairs [40]. The naive lattice discretization E n e r gy k=K'k= Γ Momentum Momentumk=Kk= Γ Haldane ModelBHZ ModelC K = sgn(m K )/2 C K' = - sgn(m K' )/2C =[sgn(m )+sgn(B)]/2 C = - [sgn(m )+sgn(B)]/2 + + - - FIG. 1. Schematic illustration of how a single Dirac fermionor Chern insulator realizes the parity anomaly in the Haldane(blue) or BHZ (red) model. In the Haldane model, both Diracfermions at the K and K (cid:48) points of the hexagonal latticestructure contribute ± / C = ±
1. More explanations are given in the text. of a pure Dirac fermion leads to a phenomenon calledfermion-doubling, which predicts the existence of a sec-ond Dirac fermion of opposite Chern number at the edgeof the lattice Brillouin zone. Thus, the entire system hasChern number zero and the parity anomaly of a singleDirac fermion cannot be measured. However, in his sem-inal work [2], Haldane found a way to circumvent thisdifficulty in a condensed matter system. He proposed away of how to realize a single Dirac fermion in the bulkspectrum of graphene by separately manipulating the twoDirac gaps at the K and K (cid:48) points of the hexagonal lat-tice structure via a complex hopping parameter. In par-ticular, this parameter allows to close only one of theDirac gaps, whereas the other one remains open. Hence,his model suggests a way of how to realize a solid statesystem which has a single gapless Dirac fermion in 2+1dimensions but still a non-zero, integer Chern number C HM = (sgn(m K ) − sgn(m K (cid:48) )) /
2. Here, m K and m K (cid:48) arethe Dirac mass terms at the K and K (cid:48) points in graphene,respectively.While so far the Haldane model has not yet been re-alized in a solid state material, a closely related QAHeffect has been predicted in two-dimensional systemslike (Hg,Mn)Te quantum wells or magnetically doped(Bi,Sb)Te thin films. In the vicinity of the bulk gap, thesesystems can be effectively described by the Bernevig-Hughes-Zhang (BHZ) model H BHZ = (cid:18) H +CI ( k ) 00 H − (cid:63) CI ( − k ) (cid:19) , (4)which consists of two copies of the Chern insulators de-fined in Eq. (2). They are decoupled [41] and the index ± defines their (pseudo-)spin polarization H ± CI = A ( k σ − k σ ) ± ( m ± − Bα ) σ − Dασ . (5)In comparison to Eq. (2), Eq. (5) also includes a particle-hole asymmetry Dασ . Since this term is parity-even, it neither contributes to the parity anomaly nor changes theChern number in Eq. (3) [42]. Therefore, let us first con-sider particle-hole symmetric systems with D = 0. TheDirac masses m ± of each (pseudo-)spin block in Eq. (5)can be tuned by magnetic doping of the system. It isin particular possible to drive one of the Chern insula-tors in the topologically trivial regime and to close, atthe same time, the gap of the second non-trivial Cherninsulator. Analogously to the Haldane model, in this sce-nario the single gapless Chern insulator alone realizes theparity anomaly of a Dirac-like system in 2+1 dimensions.Schematically, this limit is illustrated in Fig. 1.However, while in the Haldane model both Diracfermions contribute ± / C CI = ±
1. The other one istopologically trivial with C CI = 0. Hence, studying thesingle Chern insulator in Eq. (2) is sufficient to analyzethe consequences of the parity anomaly in experimen-tally realizable systems like (Hg,Mn)Te quantum wellsor magnetically doped (Bi,Sb)Te thin films.The spectrum associated to Eq. (2) is given by (cid:15) ± ( α ) = ± (cid:112) A α + ( m − Bα ) , (6)where ± encodes the conduction and the valence band,respectively. In Fig. 2, we show the influence of the massparameters on the band-structure. Depending on the val-ues for m , B , and A , the band-structure changes signif-icantly. For m/B >
0, the system is topologically non-trivial with C CI = ±
1. The minimal gap can be eitherlocated at the Γ-point or at α min = (2 mB − A ) / (2 B ),corresponding to a camel-back structure. Thus, it is de-fined by 2 | m | or by the absolute value of∆ = A (cid:112) mB − A /B . (7)Increasing | m | or | B | in the nontrivial phase leads to acamel-back structure if 2 mB > A , associated to α min > | ∆ | increases with m but decreaseswith B . For 4 mB = A , ∆ vanishes and the spectrumsimplifies to, (cid:15) ± ( α ) = ±| ( m + Bα ) | . (8)For m/B <
0, the system is topologically trivial. In thiscase the minimal gap is always located at the Γ-point.If we include an out-of-plane magnetic field H , a LLspectrum forms if the magnetic length l H = (cid:112) (cid:126) / | eH | is smaller than the system size [43]. For s = sgn( eH ) anda finite particle-hole asymmetry D , one obtains (cid:15) ± n (cid:54) =0 = − sβ/ − nδ ± λ n , (9) (cid:15) = s ( m − β/ − δ/ . (10)Here, we defined α = √ A/l H , β = 2 B/l H , δ = 2 D/l H ,and λ n = (cid:112) α n + ( m − nβ − sδ/ with n ∈ N + . (11) FIG. 2. Band-structure of a Chern insulator for zero magneticfield and D = 0. Red and blue curves encode topologicallynon-trivial phases with m = A = 1, and B = 5 (red, camel-back), or m = 1, A = 3, and B = 0 . m = 1, A = 5,and B = − .
1. The minimal gap is either defined by 2 | m | atthe Γ-point or by | ∆ | at α = α min , indicated by the black orgreen arrow, respectively. As shown in Ref. [15], H renormalizes the zero-fieldChern number C CI in Eq. (3) to C CI ( H ) = (cid:2) sgn (cid:0) m − B/l H (cid:1) + sgn ( B ) (cid:3) / . (12)Hence, a magnetic field counteracts the parity anomalyrelated contribution to the Chern number, and closes theDirac mass gap at H crit = sgn( eH ) m/B . Beyond thiscritical magnetic field the parity anomaly vanishes.This highly nontrivial statement deserves further clarifi-cation. The parity anomaly of a single Chern insulatoris a zero magnetic field effect. It is directly related to theparity breaking elements of the zero-field Hamiltonianin Eq.(2) and its band-structure in Fig. 2. In quantizingmagnetic fields, the Chern number of each LL onlyresults from the magnetic field as it only depends on themagnetic length l H . Nevertheless, the parity anomalystill has significant consequences in magnetic fields.Namely, it defines the Chern number in the Dirac massgap, Eq. (12), resulting from the spectral asymmetry ofthe entire LL spectrum [15]. Since for | H | > | H crit | , thespectral asymmetry vanishes, there are no measurableconsequences of the parity anomaly beyond this criticalvalue.Above, we have introduced the concept of Chern insu-lators and have discussed their concrete relation to theparity anomaly in 2+1 space-time dimensions. Next, westudy finite temperature and density effects on the parityanomaly induced transport by calculating the Hall con-ductivity in zero, as well as in finite out-of-plane magneticfields. III. ANOMALY INDUCED TRANSPORT INZERO MAGNETIC FIELD
In what follows, we calculate the finite temperatureHall conductivity σ xy corresponding to a particle-holesymmetric Chern insulator at zero magnetic field. Thisparity-odd and non-dissipative transport coefficient is di-rectly related to the parity anomaly in 2+1 space-timedimensions as it does not vanish in the parity symmet-ric limit m, B → ± . In our calculation, we disentangletopological from non-topological contributions to σ xy , thelatter originating from thermal effects. For D = 0, the fi-nite temperature Hall conductivity of the Chern insulatorin Eq. (2) is given by [44] [11] σ xy ( T, µ ) = − e h ∞ (cid:90) d α A ( m + Bα ) [ f v ( T, µ ) − f c ( T, µ )]4( A α + ( m − Bα ) ) / , (13)where f c , v ( T, µ ) = [e ( (cid:15) ± ( α ) − µ ) / ( k B T ) + 1] − are the con-duction and valence band Fermi functions [20]. To disen-tangle topological from thermal contributions to σ xy , weuse that f v ( T, µ ) = 1 − Θ( − (cid:15) ) e ( (cid:15) − µ ) / ( k B T ) e ( (cid:15) − µ ) / ( k B T ) + 1 , (14)where (cid:15) ( α ) encodes the entire spectrum and Θ is theHeaviside step function. With this identity, Eq. (13) de-composes into two building blocks, σ xy ( T, µ ) = σ xy + σ xy ( T, µ ) , (15)with σ xy = − e h (sgn( m ) + sgn(B)) , (16) σ xy ( T, µ ) = e h (cid:90) d α A ( m + Bα ) sgn( (cid:15) )4 π(cid:15) (17) × (cid:18) Θ( (cid:15) )e ( (cid:15) − µ ) / ( k B T ) + 1 + Θ( − (cid:15) )e − ( (cid:15) − µ ) / ( k B T ) + 1 (cid:19) . Equation (16) encodes the topological part of the Hallconductivity. In contrast, Eq. (17) defines the correc-tions originating from a finite temperature and chemicalpotential. These non-topological and thus non-quantizedcorrections are based on particle-hole excitations of theconduction and valence band. To solve Eq. (17), we usethe assumption of a particle-hole symmetric Chern insu-lator with D = 0. In particular, this implies σ xy ( T, µ ) = σ corr xy ( T, µ ) + σ corr xy ( T, − µ ) . (18)To determine σ corr xy ( T, µ ) in the energy space, we needto solve Eq. (6) for α . Due to the possible camel-backstructure, this leads to two solutions α ± = α min ± √ (cid:15) − ∆ | B | with d α ± d (cid:15) = ± (cid:15) | B |√ (cid:15) − ∆ . With these identities, we find the correction σ corr xy ( T, µ ) = e h Θ[ α min ] √ m (cid:90) | ∆ | A + 2 | B | ∆ √ (cid:15) − ∆ B(cid:15) (cid:0) e ( (cid:15) − µ ) /T + 1 (cid:1) d (cid:15) + e h ∞ (cid:90) √ m A + 2 | B | ∆ √ (cid:15) − ∆ B(cid:15) (cid:0) e ( (cid:15) − µ ) /T + 1 (cid:1) d (cid:15) . (19)While in Eq. (19) the second term captures the correctionfrom a monotonic band-structure, the first term encodesa possible camel-back correction. For 4 mB = A with∆ = 0, σ corr xy ( T, µ ) reduces to σ corr xy ( T, µ ) = me h ∞ (cid:90) √ m A (cid:15) (cid:0) e ( (cid:15) − µ ) / ( k B T ) +1 (cid:1) d (cid:15) , (20)which is twice the QED result with B = 0. Analo-gously to Eq. (16), the Newtonian mass provides a factorof two to the thermal corrections of the QED con-ductivity. For the solution in Eq. (20), we can define thecorrections in terms of the Gamma function Γ( x ) and thereduced Fermi-Dirac integral F j ( x, b ) [App. A]. In total,this leads to σ xy ( T, µ ) = e h (cid:88) s = ± A Γ( − F − (cid:16) sµk B T , | m | k B T (cid:17) k B T T (cid:28)| m | = e h A Γ (cid:16) − , | m | k B T (cid:17) k B T . (21)In Eq. (21), we approximate the result for low temper-atures in comparison to the gap and for zero chemicalpotential. Γ( s, b ) is the incomplete Gamma function[App. A].The general correction in Eq. (19) cannot be expressedvia the integral functions above since ∆ (cid:54) = 0. In Fig. 3, weplot the functional dependence of σ xy ( µ, T ) for differentchoices of m and B . While increasing the Dirac mass al-ways counteracts the temperature, increasing B enhancestemperature effects in the topologically nontrivial phase.As discussed below Eq. (7), this originates from the prop-erty that B decreases the camel-back gap. Thus, bothmasses contribute equally to the topological part of theHall conductivity in Eq. (16), while they counteract eachother in the thermal corrections, Eq.(17), for m/B > m/B < m/B ). However, in the topologi-cally trivial phase the Newtonian mass cannot generatea camel-back structure. In this case both, the Dirac andthe Newtonian mass term counteract the finite tempera-ture broadening of the Fermi-Dirac distribution. σ xy [ e / h ] k B T [|m |] k B T [|m |] σ xy [ e / h ] m =1m =2m =1.5 B =0.1B =3B =1 (a) (b) σ xy [ e / h ] k B T [|m |] k B T [|m |] σ xy [ e / h ] -0.03 0.00-0.030.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 m =1m =2m =1.5 B =-0.1B =-1 (c) (d) B =-3 FIG. 3. Finite temperature Hall conductivity of a Chern insu-lator with A = 1 and D = 0. In (a) and (c) , we vary the Diracmass while B = ± .
1, respectively. In (b) and (d) , we varythe Newtonian mass for m = 1. In all sub-figures we considerzero chemical potential. Sub-figures (a) and (b) correspondto the the topologically non-trivial regime, while sub-figures (c) and (d) correspond to the topologically trivial regime. IV. ANOMALY INDUCED TRANSPORT INFINITE MAGNETIC FIELD
Having analyzed a particle-hole symmetric Chern in-sulator at zero magnetic field, we now include a particle-hole asymmetry and an out-of-plane magnetic field H ,where the latter gives rise to the LL spectrum in Eq. (9).The Hall conductivity can be calculated by means ofStreda’s formula via the expectation value of the chargeoperator (cid:104) N (cid:105) T,µ [16, 45], yielding σ xy ( T, µ ) = − ∂ (cid:104) eN (cid:105) T,µ ∂H = e ∂η H ∂H − ∂ (cid:104) eN (cid:105) ∂H . (22)Here, (cid:15) z = − mD/B is the charge neutrality point, and η H = (cid:88) n sgn( (cid:15) n − (cid:15) z ) = − eHh [sgn( m − β/
2) + sgn( B )] , (cid:104) N (cid:105) = (cid:88) n sgn( (cid:15) n − (cid:15) z ) (cid:34) Θ( (cid:15) n − (cid:15) z )e (cid:15)n − µk B T +1 + Θ( (cid:15) z − (cid:15) n )e − (cid:15)n − µk B T +1 (cid:35) . (23)The spectral asymmetry η H counts the difference in thenumber of conduction and valence band states. There-fore, as long as the band structure is not changed, itis temperature and chemical potential independent andsolely carries the information of the topological contri-bution of the parity anomaly to σ xy in magnetic fields.This enables the connection between the Hall conductiv-ity and the parity anomaly even at finite magnetic fields.In contrast, (cid:104) N (cid:105) encodes the thermal LL response, asit defines the thermal occupation of the valence and theconduction band. Due to the associated flat dispersionrelation, this response entirely originates from the mag-netic field topology and no more from the parity anomaly.All LLs with n ∈ N + come in pairs. With the degeneracy | eH | /h , their contribution to the charge operator is givenby (cid:104) N (cid:105) n (cid:54) =0 = | eH | h (cid:88) n (cid:54) =0 ,s = ± s e s ( (cid:15)sn − µ ) k B T + 1 . (24)Additionally to this conventional LL response for finite µ and T , the zero LL also needs to compensate its contribu-tion to η H outside of the Dirac mass gap. In particular,it needs to cancel the term ∝ sgn( m − β/
2) in Eq. (23)for | µ + δ/ | > | m − β/ | [15]. Since the zero LL can eitherbe part of the conduction or of the valence band, we cansimplify its contribution to (cid:104) N (cid:105) . By using the propertiesof the hyperbolic tangent, we find for the zero LL with n = 0 (cid:104) N (cid:105) = − | eH | sgn( (cid:15) − (cid:15) z )2 h (cid:34) Θ( (cid:15) − (cid:15) z ) (cid:20) tanh (cid:18) (cid:15) − µ k B T (cid:19) − (cid:21) − Θ( (cid:15) z − (cid:15) ) (cid:20) tanh (cid:18) (cid:15) − µ k B T (cid:19) +1 (cid:21)(cid:35) . This expression can be simplified further via the identitiessgn( (cid:15) − (cid:15) z ) = sgn( (cid:15) + δ/
2) = sgn( eH ) sgn( m − β/ (cid:15) − (cid:15) z ) − Θ( − (cid:15) + (cid:15) z ) = sgn( (cid:15) − (cid:15) z )Θ( (cid:15) − (cid:15) z ) + Θ( − (cid:15) + (cid:15) z ) = 1 . (25)Eventually, this implies the zero LL contribution (cid:104) N (cid:105) = | eH | h (cid:20) sgn( eH ) sgn( m − β/ − tanh (cid:18) (cid:15) − µ k B T (cid:19)(cid:21) , (26)which reduces for T → µ = µ + D/l H to [App. A] (cid:104) N (cid:105) = | eH | h Θ( | ¯ µ |−| m − β/ | ) × [ sgn( eH ) sgn(m − β/
2) + sgn(¯ µ ) ] . (27)For T = 0, the zero LL contribution to η H clearlygets compensated outside of the Dirac mass gap. Asexpected, finite temperature effects soften this property.In Fig. 4, we used Eq. (22) to connect the charge operatorto σ xy , and plotted Hall conductivity corresponding tothe parity anomaly and to each LL, separately. Moreover,we show the entire signal as a function of the chemicalpotential. While the Hall conductivity contribution re-lated to the parity anomaly is T and µ independent, eachLL comes along with an exponentially suppressed tem-perature broadening. Consequently, all LLs contribute tothe Hall conductivity in the Dirac mass gap. This renor-malizes the zero temperature violation of the Onsagerrelation [46] discussed recently in Ref. [15]. Last but notleast, let us emphasize that the Hall plateau originatingfrom the parity anomaly is much more robust than LLplateaus with respect to finite temperature effects. Due σ xy [ e / h ] Chemical Potential µ [|m|] k B T =0 k B T ≠ σ xy [ e / h ] FIG. 4. Finite temperature Hall conductivity of a Chern in-sulator in a magnetic field H = 3 T with: A = 1, m = − B = − .
1, and D = − .
05. The response of each valence (blue)and conduction band (red) LL is shown separately. The zeroLL response is illustrated in black, the parity anomaly re-lated contribution is depicted in green. The combined signalis shown in orange. Dashed lines correspond to k B T = 0,solid lines are associated to k B T = 0 .
01. The Dirac mass gapis shown in grey. to the lack of a zero-LL partner, the parity anomaly re-sponse is approximately unaltered [47] until the chemicalpotential comes close to the n = 1 conduction or valenceband LL, depending on the sign of the magnetic field.Quantitatively, this means | (cid:15) +1 − (cid:15) | > | (cid:15) ± n +1 − (cid:15) ± n | ∀ n ∈ N + , (28)assuming that the zero-LL is part of the valence band[cf. Fig. 4]. Therefore, finite temperature effects firstlysmear out the LL steps before they eventually preventany quantization for the finite temperature Hall conduc-tivity. V. SUMMARY AND OUTLOOK
In this work we analyzed the finite temperature Hallconductivity of two-dimensional Chern insulators underthe influence of a chemical potential and an out-of-planemagnetic field. At zero magnetic field, this quantityoriginates from the parity anomaly. As such, we wereable to show that the parity anomaly is not renormalizedby finite temperature effects. Instead, it induces twoterms of different physical origin in the effective actionof a Chern insulator, which are proportional to itsHall conductivity. The first term is temperature andchemical potential independent, solely encoding theintrinsic topological response. The second term specifiesthe non-topological thermal response of conduction andvalence band states. We showed that in the topologicallynontrivial phase, an increasing relativistic mass termof a Chern insulator counteracts finite temperatureeffects, whereas an increasing non-relativistic mass termenhances these corrections. In contrast, both massterms counteract the finite temperature broadeningof the Fermi-Dirac distribution in the topologicallytrivial phase, as the Newtonian mass cannot causea camel-back gap in this case. In magnetized II-VIQAH insulators, like (Hg,Mn)Te quantum wells, theseparameters can be tuned by changing the quantum wellwidth, or by changing the concentration of the magneticdopants. Moreover, we derived the thermal response ofa Chern insulator in a magnetic field and clarified itsrelation to the spectral asymmetry η H . This quantityis a measure of the parity anomaly in magnetic fields.In particular, we derived in which way the thermal LLresponse renormalizes the parity anomalous part of theHall conductivity in magnetic fields. Especially in theDirac mass gap, this response adds to the otherwisequantized and temperature independent part of theHall conductivity arising from the parity anomaly. Weshowed that the anomalous part of the Hall responsein the Dirac mass gap is much more robust than thecommon LL contributions with respect to finite tem-perature effects. Our findings should be experimentallyverifiable in QAH insulators such as (Hg,Mn)Te quan-tum wells, magnetically doped (Bi,Sb)Te thin films, orbilayer structures of three-dimensional topological and ferromagnetic insulators.In the future, it would be interesting to extend this anal-ysis to different anomalies in various space-time dimen-sions. For instance, the chiral, gravitational, and con-formal anomaly should not depend on thermal effects,whereas they necessarily induce a temperature depen-dence at the effective action level. ACKNOWLEDGMENTS
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The reduced Fermi-Dirac integral is defined by F j ( x, b ) = 1Γ( j + 1) ∞ (cid:90) b d t t j e t − x + 1 . (A1)The incomplete Γ-function is defined byΓ( s, b ) = (cid:90) ∞ b d t t s − e − t . (A2)Let us comment on how to derive Eq. (27) from Eq. (26).In the zero temperature limit, the hyperbolic tangent inEq. (27) becomes a sign-function, lim T → tanh( x/T ) =sgn( x ). Due to this property, we need to distinguish twocases. The chemical potential is either located inside ( i )or outside ( ii ) of the Dirac mass gap:( i ) | µ + δ/ | < | m − β/ | (A3)( ii ) | µ + δ/ | > | m − β/ | . (A4)For case ( i ), the hyperbolic tangent in Eq. (27) reduces tosgn( eH )sgn( m − β/