Momentum-Dependent Mass and AC Hall Conductivity of Quantum Anomalous Hall Insulators and Their Relation to the Parity Anomaly

PHYSICAL REVIEW RESEARCH 2, 033193 (2020)

Momentum-dependent mass and AC Hall conductivity of quantum anomalous Hall insulators and their relation to the parity anomaly

Christian Tutschku, Jan Böttcher , René Meyer, and E. M. Hankiewicz* Institute for Theoretical Physics and Würzburg-Dresden Cluster of Excellence ct.qmat, Julius-Maximilians-Universität Würzburg, 97074 Würzburg, Germany

(Received 9 March 2020; accepted 15 July 2020; published 4 August 2020)

The Dirac mass of a two-dimensional QAH insulator is directly related to the parity anomaly of planar quantum electrodynamics, as shown initially by Niemi and Semenoff [Phys. Rev. Lett. 51, 2077 (1983)]. In this work, we connect the additional momentum-dependent Newtonian mass term of a QAH insulator to the parity anomaly. We reveal that in the calculation of the effective action, before renormalization, the Newtonian mass acts similar to a parity-breaking element of a high-energy regularization scheme. This calculation allows us to derive the ﬁnite frequency correction to the DC Hall conductivity of a QAH insulator. We predict that the leading order AC correction contains a term proportional to the Chern number. This term originates from the Newtonian mass and can be measured via electrical or magneto-optical experiments. Moreover, we prove that the Newtonian mass signiﬁcantly changes the resonance structure of the AC Hall conductivity in comparison to pure Dirac systems like graphene.

DOI: 10.1103/PhysRevResearch.2.033193

I. INTRODUCTION While the contribution of the Dirac mass to the effective action of a QED + systems was initially shown in Refs. [22,23], The discovery of Dirac materials, such as topological 2 1 the effective action of Chern insulator in the presence of a insulators [1–10] and Weyl or Dirac [11–16] semimetals Newtonian mass has not yet been analyzed. To bridge this gap, yields the possibility to measure quantum anomalies in con- we explicitly derive in this work the polarization operator of a densed matter systems. In particular, it was shown that two- Chern insulator. In the corresponding calculation for a pure, dimensional quantum anomalous Hall (QAH) insulators, such massless QED + system, the parity anomaly arises from as (Hg,Mn)Te quantum wells [17,18] or magnetically doped 2 1 the particular regularization of the infinite Dirac sea. Even (Bi,Sb)Te thin films [19,20], are directly related to the parity for a massive Dirac system, each gauge-invariant regulariza- anomaly of planar quantum electrodynamics (QED + )[21]. 2 1 tion scheme necessarily needs to break the parity-symmetry In a nutshell, this anomaly implies that in odd space-time due to the underlying parity anomaly. Consequently, such dimensions it is not possible to quantize an odd number of a regularization provides another 1/2 to the otherwise half- Dirac fermions at the same time in a gauge- and parity- quantized Chern number of a single Dirac fermion. For the symmetric manner [22–24]. Chern insulators introduced above, this requirement is already A two-dimensional QAH insulator consists of two Chern fulfilled due to the presence of the Newtonian mass term. insulators, a topologically trivial, as well as topologically non- From this perspective the Newtonian mass term, provided trivial one. Similar to the Haldane model, each of those Chern by the material, acts similar to a parity-breaking element insulators realizes the parity anomaly if its mass gap is closed of a mathematical regularization scheme [27]. Inspired by at the topological phase transition [24]. In general, each Chern this property, we compare the role of the Newtonian mass insulator is characterized by Dirac-like physics in 2 + 1di- term for the calculation of the effective action of a Chern mensions and, in our system, has two different parity-breaking insulator to common parity-breaking regularization schemes mass terms: A momentum-independent (Dirac) mass, as well of QED + . In particular, we show that for the calculation as a momentum-dependent (Newtonian) mass [25]. As their 2 1 of the Hall conductivity the Newtonian mass can be seen as contribution to the Chern number does not vanish in the parity a continuum version of a Wilson mass term [28], whereas symmetric (zero mass) limit, both of these mass terms are it acts significantly different than a Pauli-Villars (PV) mass directly related to the parity anomaly [18,26]. [29], or the integration contour in a ζ -function regularization [30]. With continuum we mean that the Newtonian mass does not vanish during renormalization when the artificial *[email protected] lattice spacing in the Wilson mass term vanishes. Even though the Newtonian mass implies an integer quantized DC Hall Published by the American Physical Society under the terms of the conductivity, it does not remove the UV divergence in the Creative Commons Attribution 4.0 International license. Further effective action, as it is done by common higher derivative distribution of this work must maintain attribution to the author(s) regularization schemes [31,32]. Consequently, the effective and the published article’s title, journal citation, and DOI. action of a Chern insulator still requires regularization. Due to

2643-1564/2020/2(3)/033193(10) 033193-1 Published by the American Physical Society CHRISTIAN TUTSCHKU et al. PHYSICAL REVIEW RESEARCH 2, 033193 (2020)

the fact that a QAH insulator consists of two Chern insulators, (a) (b) 2 a topologically trivial as well as a topologically nontrivial 1 one, it is possible to regularize the entire system in a gauge 2|m| |Δ| 2|m| |Δ| 0 0 and parity symmetric manner. We choose two Pauli-Villars -1 fields, one for each Chern insulator, which are related by [|m|] Energy -2 [|m|] Energy time-reversal symmetry. Together, these fields compensate the parity anomaly associated to each single Chern insulator. -2 0 2 -1 -|kmin||k0 min| 1 Momentum k [arb. units] Momentum k [arb. units] Moreover, we derive the nonquantized finite frequency 1 1 corrections to the DC Hall conductivity during our calculation FIG. 1. Band structure corresponding to a single (pseudo-)spin of the effective action. We show that the leading order AC block of the BHZ model for k2 = D = 0 and for (a) a topologically correction contains a term which is proportional to the Chern nontrivial phase with m = A = 1andB = 0.1. The minimal gap number. This term originates from the Newtonian mass and 2|m| is located at the point. (b) For a topologically nontrivial can be measured by the leading order AC corrections to phase with m = A = 1andB = 3. The minimal gap || is located the DC Kerr and Faraday angles. In addition, we show that at |k|=±|kmin|. the Newtonian mass fundamentally changes the resonance structure of the AC Hall conductivity in comparison to pure Dirac systems, as it can alter the mass gap in QAH insulators. neglect the (pseudo-)spin subindex ±. Moreover, we consider This work is organized as follows: In Sec. II, we introduce particle-hole symmetric Chern insulators since the D|k|2 term two-dimensional QAH insulators. In Sec. III, we review how in Eq. (2) is parity-even and thus does not contribute to the to quantize a classical field theory and how the parity anomaly parity anomaly [35]. The spectrum associated to Eq. (2)forD = 0 is given by arises in a gapless QED2+1 system. In Sec. IV, we derive the effective action of a two-dimensional QAH insulator. We separately calculate the polarization operators of each of ± =± A2|k|2 + (m − B|k|2 )2. (4) the two Chern insulators, together defining the entire QAH system. In particular, we show the integer quantized DC Hall Here, ± encodes the conduction and the valence band, respec- conductivity of a QAH insulator and derive its nonquantized tively. In Fig. 1, we show the influence of the mass parameters AC correction. In Sec. V, we compare the role of the New- on the band structure. While the Dirac mass m defines the tonian mass term for the calculation of the effective action to mass gap at the point, the momentum-dependent B|k|2 term common QED2+1 regularization schemes, which also ensure acts like an effective mass of a nonrelativistic fermion system. an integer quantized Chern number. In Sec. VI, we summarize Depending on the values for m, B, and A, the band structure our results and give an outlook. significantly changes. For m/B > 0, the system is topologically nontrivial with CQAH =±1. However, depending on II. MODEL the absolute values of the input parameters, the minimal gap can either be located at the point [Fig. 1(a)], driven by Two-dimensional QAH insulators like (Hg,Mn)Te quan- the Dirac√ mass alone,√ or apart from |k|=0 [Fig. 1(b)]at 2 tum wells or magnetically doped (Bi,Sb)Te thin films can be kmin =± 2mB − A /( 2B). The minimal gap is therefore described by the Bernevig-Hughes-Zhang (BHZ) Hamiltonian either defined by 2|m|, or by the absolute value of consisting of the two (pseudo-)spin blocks H±(k)[2,33,34]: = − 2/ . H+(k)0 A 4mB A B (5) HBHZ(k) = . (1) 0 H−(−k) In contrast, for m/B < 0, the system is topologically trivial, On its own, each (pseudo-)spin block corresponds to a single characterized by CQAH = 0. In this case the minimal gap is Chern insulator in 2 + 1 space-time dimensions with always located at the point. At m = 0, the topological phase H =± − | |2 σ − | |2σ + σ − σ . transition occurs, which comes along with a gap closing at ±(k) (m± B k ) 3 D k 0 A(k1 1 k2 2 ) = (2) k 0. While in our plots we mostly consider positive mass terms, the inverted Dirac mass of an experimental QAH As such, a single (pseudo-)spin block of the BHZ model is insulator is negative. Changing the overall sign of m and B directly related to the parity anomaly of QED2+1.InEq.(2), alters the sign of the Chern number, but not the underlying | |2 = 2 + 2 σ k k1 k2 , 1,2,3 represent the Pauli matrices, A is pro- physics. Notice, that according to Eq. (4) the Newtonian mass portional to the Fermi velocity, and D encodes a particle-hole term does not make the spectrum bounded. It is from this 2 asymmetry. The parity-breaking mass terms are m± and B|k| . perspective expected that the B|k|2 term does not render the In the QAH phase, only one of the two (pseudo-)spin blocks is effective action of a Chern insulator UV finite. topologically nontrivial with the finite Chern number [18,26], Having discussed the particular influence of the mass terms on the band structure, let us emphasize one more time that Ci = i[sgn(m ) + sgn(B)]/2, i ∈{+, −}. (3) QAH i both of them explicitly break the parity symmetry in 2 + 1 The trivial (pseudo-)spin block has zero Chern number. There- space-time dimensions, defined as invariance of the theory fore, the entire topological response in the QAH phase is under P :(x0, x1, x2 ) → (x0, −x1, x2 ). Hence, the Dirac and captured by a single Chern insulator in 2 + 1 space-time the Newtonian mass term are directly related to the parity dimensions. If not stated otherwise, then we focus on such anomaly of massless QED2+1. To concretize this statement, a system in the remaining part of this work. In particular, we let us briefly review the concept of quantum anomalies.

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III. PARITY ANOMALY the calculation of the effective action of a Chern insulator to common parity-breaking regularization schemes of QED + . In what follows, we consider Chern insulators in 2 + 1 2 1 space-time dimensions coupled to an external fluctuating abelian U(1) gauge field Aμ [cf. Eq. (2)]. Such models can IV. EFFECTIVE ACTION be quantized by calculating the partition function Z[A]via integrating out the fermionic degrees of freedom, In what follows, we perturbatively evaluate the effective action corresponding to a single Chern insulator. Therefore, 1 S ψ,ψ,¯ A 1 A Z[A] = dψ¯ dψ ei [ ] = eiSeff [ ]. (6) we Taylor expand the fermion determinant in Eq. (9)to Z[0] Z[0] second order in the external background field Aμ. The free ψ ψ¯ Lagrangian associated to Eq. (2) can be obtained by a Legen- Here, and are the two-component Dirac spinor and dre transformation. For A = h¯ = 1 and D=0, this Lagragian its adjoint. The bare and the effective action are defined by S ψ,ψ,¯ A S A S A is equivalent to a pure QED2+1 Lagrangian except for an [ ] and eff [ ]. If eff [ ] has less symmetries than additional correction, which is quadratic in spatial derivatives S[ψ,ψ,¯ A], then a quantum anomaly is present. For instance, massless QED2+1 is described by the parity-symmetric bare L = ψ¯ (i∂/ − m)ψ − Bγ 0(∂ ψ)†(∂iψ). (10) action, 0 i Here and in the following, we use the properties of the Dirac S[ψ,ψ,¯ A] = d3xiψ¯ (∂/ + ieA/ )ψ. (7) matrices given in Appendix A. Coupling L0 covariantly to the A μ U(1) gauge field μ, leads to the Lagrangian Here, we use the Feynman slash notation ∂/ = γμ∂ with the + γ = σ , σ , σ 2 1-dimensional Dirac matrices μ ( 3 i 2 i 1), and L = ψ¯ (iD/ − m)ψ − Bγ 0(D ψ)†(Diψ) consider the metric tensor gμν = diag(+, −, −). The associ- i / i ated effective action is given by = L0 − eψ¯ (A+eBAAi )ψ / / / + Ai ψ¯ ∂ ψ − ∂ ψ¯ ψ , Seff [A] =−i Ln det[(∂ + ieA)/∂]. (8) ieB [ ( i ) ( i ) ] (11)

As it was shown initially in Ref. [23], the fermion determinant where Dμ = ∂μ + ieAμ is the covariant derivative. From the in Eq. (8) changes sign under large gauge transformations [36] interaction terms in L, we can read off the vertex contribu- ω = + ∈ , of odd winding 2n 1 with n Z tions. For an incoming electron of momentum k, incoming det[(∂/ + ieA/ )/∂/] → (−1)|ω| det[(∂/ + ieA/ )/∂/] photons of momentum p, and an outgoing electron of momentum k+p, we find the vertices ⇒ Seff → Seff ± π|ω|. (9) μ μ V (2k+p) =−ie γ μ − B δ (2k+p)i , (12) Hence, as it stands Seff [A] is not a well-defined object. This (1) i can also be seen by calculating the fermion determinant ex- μν =− 2 δμδiν . plicitly, which leads to a divergent expression. Consequently, V(2) (k) 2ie B i (13) the theory requires a regularization scheme which needs to be chosen such that it ensures infinitesimal—as well as large Here, the subscript defines the number of involved pho- | |2 gauge invariance of Z[A]. In particular, it was shown that tons. In comparison to pure QED2+1,theextraB k term each regularization scheme needs to break parity symmetry in the Lagrangian renormalizes the gauge-matter coupling. to ensure gauge invariance. This is known as the parity The original QED vertex in Eq. (12) obtains a momentum- anomaly [22,23]. The parity anomaly therefore results from dependent correction. Additionally, Eq. (13) defines a new vertex structure, which is of second order in gauge fields. the particular regularization of QED2+1. Even though a mas- This vertex encodes the diamagnetic response of the Chern sive QED2+1 system breaks parity on the classical level, the fermion determinant still diverges and lacks gauge invariance. insulator in Eq. (2)[38]. The fermion propagator associated L Again, this requires a parity-breaking regularization scheme, to the Lagrangian is given by which ensures an integer quantized DC Hall conductivity i k/ + M(k) associated to a gauge-invariant Chern-Simons term in the S(k) = = i . (14) effective action. This property extends the peculiarities of k/ − (m−B|k|2 ) + i k2 − M(k)2 + i the parity anomaly to the massive case [22,23,30,37]. In this work, we show that adding the Newtonian mass term to a mas- Here, we defined the momentum-dependent mass term = − | |2 sive QED + Lagrangian also leads to an integer quantized M(k) m B k and used the Feynman prescription with 2 1 → + DC Hall conductivity associated to a gauge-invariant Chern- 0 . To perturbatively obtain the second order effective Simons term. From this perspective the Newtonian mass term, action in external fields, we need to calculate the vacuum μν provided by the material, acts similar to a parity-breaking polarization operator (p)[39], element of a mathematical regularization scheme [27]. Its 3 contribution to the Chern number does not vanish in the 1 d p μν Seff = Aμ(−p) (p)Aν (p). (15) parity-symmetric limit m, B → 0. Consequently, beside the 2 (2π )3 Dirac mass also the Newtonian mass term is directly related to the parity anomaly. Inspired by this property, we compare Due to the vertex structure in Eqs. (12) and (13), the vacuum in the following the role of the Newtonian mass term for polarization operator is obtained by the sum of two one-loop

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Feynman integrals, μν = μν + μν i (p) i 2a (p) i 2b (p) 3 d k μ μν =− Tr S(k)V (2k+p)S(k+p)V ν (2k+p) + S(k)V (k) , (16) (2π )3 (1) (1) (2) μν which are diagrammatically illustrated in Fig. 2. We start with the calculation of the ﬁrst term in Eq. (16), i 2a (p), which is the usual QED2+1 vacuum polarization operator with renormalized vertex and propagator structure. As shown in Fig. 2(a),this tensor is given by μ 2 γ μ − δ + i σ / + γ ν − δν + j σ / + + + μν e Tr B (2k p) 0 i[k M(k)] B (2k p) 0 i[(k /p) M(k p)] i (p) = d3k i j . (17) 2a (2π )3 [k2 − M(k)2 + i][(k+p)2 − M(k+p)2 + i]

There are four different contributions to the Dirac trace Tr = Notice, that the off-diagonal Chern-Simons contribution in Trγγ + Trγ 0 + Tr0γ + Tr00, where the subscript deﬁnes the Eq. (19) gets shifted by the renormalized vertex structure in Dirac and identity part of the vertex structure in Eq. (12). Eq. (18). As argued above, this will lead to an integer quan- Since all physical response functions are given as functional tized DC Hall conductivity associated to a gauge-invariant derivatives of the effective action at zero external spatial Chern-Simons term. momentum, from now on we focus on the calculation of μν , = 2 = 2 μνλ = i (p0 p 0). With this assumption, p p0, pλ μν0 A. Off-diagonal response p0, and M(k) = M(k+p). Next, we introduce the Feyn- man parameter x ∈ [0, 1] and shift the loop momentum ac- To prove this statement, we evaluate the integral cording to k =l −px. This gives the denominator in Eq. (17) μν , = a quadratic form, allowing us to drop all linear terms in l i CS(p0 p 0) in the numerator, due to an antisymmetric integration over e2 1 −2iμν0(M(α)+2Bα)p0 α =||2 = dx d3l symmetric boundaries [40]. With l , this leads to [cf. 3 2 (2π ) 0 2 −α− α 2 + − 2 + Appendix A] l0 M( ) x(1 x)p0 i μν μ ν 2 1 ∞ = 2 − − − 2 e μν m + Bα Tr 2g l 4l l 2x(1 x)p0 = 0 α . p0 dx d / π 2 3 2 μν 2 2 μν0 8 0 0 α+M(α)2 −x(1−x)p − i − 2g M(α) +x(1−x)p −2i[M(α) + 2Bα] p0 0 0 (20) + αδν δμ α − 2 −α− − 2 + α 2 . 4B m m 2M( ) B l0 x(1 x)p0 M( ) Here, we used the Feynman parametrization and solved the (18) complex time-integration via the residue theorem. For Chern → In the QED2+1 limit B 0, the Dirac trace in Eq. (18) reduces insulators, time and spatial momenta need to be integrated to the well-known result [40–42] separately since the B|k|2 term breaks the Lorentz symmetry. Hence, it is not possible to Wick-rotate and integrate over an Tr = 2gμν l2 − 4lμlν + 4x(1−x)p2 0 Euclidean three-sphere. Integrating the Feynman parameter x − μν 2 + − 2 − μν0 . implies 2g m x(1 x)p0 2im p0 (19)

1 1

dx / < 3 2 0 α + α 2 − − 2 − (a) M( ) x(1 x)p0 i μ ν + 4 V(1) k p V(1) = , (21) α + α 2 − α + α 2 − 2 − M( ) i 4 4M( ) p0 4i ppk where we kept the i-prescription to circumvent the poles for

α>0, appearing if p exceeds the gap. Finally, we perform << 0 the remaining α-integration for an arbitrary driving frequency → + (b) p0 and subsequently set 0 . Due to its lengthy form, we present the general Chern-Simons contribution and its AC Hall conductivity σxy(p0 ) in Appendix C, k μν , = = σ μν0 . i CS(p0 p 0) xy(p0 ) p0 (22) μν Instead, let us ﬁrst analyze the Taylor expansion of the AC ppV(2) Hall conductivity in terms of the frequency p0, FIG. 2. Feynman diagrams for the vacuum polarization operator μ,ν μν σ (p )| = of a Chern insulator with vertices V and V . External momenta xy 0 p0 0 2 4 (1) (2) σxy(p0 ) = σxy(0) + p + O p , (23) are denoted by p, loop momenta by k. 2! 0 0

033193-4 MOMENTUM-DEPENDENT MASS AND AC HALL … PHYSICAL REVIEW RESEARCH 2, 033193 (2020) with the coefficients (reintroducing A andh ¯) Newtonian mass. Moreover, σxy(p0 ) is related to the Fara- day and the Kerr angle of two-dimensional QAH insulators e2 e2 [45,46]. The first term in Eq. (25) can be therefore resolved σ (0) = [sgn(m) + sgn(B)] = C , (24) xy QAH by magneto-optical experiments, as well. Let us consider a 2h h 2 4 linearly polarized electric field, which incidents normally on e 2 CQAH A sgn(m) σ (p )| = = − . (25) the QAH system. For frequencies much smaller than the gap, xy 0 p0 0 h 32 2B2 24m2 which justify Eq. (23), one finds [46]

Equation (24) defines the DC Hall conductivity of the πσ p = xy( 0 ) , single Chern insulator given in Eq. (2). In the QAH phase, F(p0 ) Arctan (28) 0c this value matches the DC Hall conductivity of the entire 0c system, as only one Chern insulator contributes to the topo- K (p0 ) =−Arctan . (29) πσxy(p0 ) logical response [18,26]. In comparison to a pure QED2+1 σ system with a half-quantized xy(0), the Newtonian mass term Here, c is the vacuum speed of light and 0 the vacuum per- ensures integer quantization. Hence, the associated Chern- mittivity. While these identities imply quantized values of the Simons term is gauge invariant [43]. Faraday and Kerr angles in the DC limit, they carry the infor- In contrast, Eq. (25) defines the leading order AC correc- mation of how these angles change due to the contribution of σ tion to xy(0), which contains two terms of different origin. thefirstterminEq.(25). To resolve this effect in one of these One the one hand, there is a term proportional to the Chern experiments, |p | Min(2|m|, ||). For instance, in inverted C / < 0 number QAH. In the trivial phase m B 0, this term vanishes (Hg,Mn)Te/CdTe quantum wells, the gap is of the order of and the first-order AC correction is solely given by the second several meV [47], depending on the particular manganese term in Eq. (25). On the other hand, for m/B > 0, this term concentration. This corresponds to frequencies in the THz contributes to the first-order AC correction. In experimental regime. For such frequencies, the first-order correction to the systems like (Hg,Mn)Te quantum wells with m+ =−10 meV, DC Faraday and Kerr angles is on the order of millirad, which 2 B =−1075 meVnm , and A = 365 meVnm, the term propor- can be resolved by recent Faraday polarimeters [48]. C ≈ tional to QAH defines 10% of the entire signal in Eq. (25). Before we discuss the general solution of the AC Hall From a theoretical point of view, this term is induced by a conductivity, note that the nonquantized value for σxy(p0 =0) finite Newtonian mass, which breaks the Lorentz symmetry in makes the associated Chern-Simons term (large) gauge non- Eq. (11). Consequently, it vanishes in the QED limit B→0, invariant. Since analogously to thermal effects, an AC driving since field excites nontopological degrees of freedom, this effect corresponds to the nongauge invariance of finite temperature 1 A4 lim = 0 ∧ lim =−1. (26) Chern-Simons terms. For these theories, it was shown that the B→0 2 B→0 2B2 full effective action contains nonperturbative corrections in Aμ, absorbing this noninvariance [49,50]. These terms cannot In this limit, Eq. (25) reduces to the QED result be found by the Taylor expansion of the fermion determinant 2 Eq. (9). However, due to the fact that they are higher order in e sgn(m) lim σ (p )| = = . (27) gauge fields, they do not contribute to the conductivity. xy 0 p0 0 2 B→0 h 24m Let us now analyze the general solution of the AC Hall conductivity corresponding to a single Chern insulator. Figure 3 Due to its unique relation to the Newtonian mass, the term σ proportional to the Chern number in Eq. (25) is quadratically shows the real and the imaginary part of xy(p0 ) according to its general form in Appendix C. To study the influence of suppressed by the ratio of p0 over the gap ||. In contrast, the Newtonian mass term, Fig. 3(a) shows σxy(p0 ) for a pure the second term in Eq. (25), which is the first-order QED = correction to the DC Hall conductivity, is quadratically sup- QED2+1 system. At p0 0, one observes the characteristic half-quantization. Moreover, the real part of σxy(p0 )showsa pressed by p0 over the Dirac mass. According to the quadratic resonance at p0 =±2|m| and tends to zero for larger frequen- suppression of both terms, the AC signal in Eq. (25) stays | | | | close to quantized values and matches the AC Hall response cies. For p0 2 m , the AC field excites particle-hole pairs which can propagate unhindered for p0 =±2|m|.Thisisthe of the entire QAH system for small driving frequencies p0. Let us briefly comment on how to experimentally disen- origin of the resonance [51–53]. For large frequencies, the AC tangle the QED from the Newtonian part in the first-order AC field dominates the gap which protects the topological phase. This leads to a vanishing AC Hall conductivity. The imaginary Hall correction. In QAH insulators like (Hg,Mn)Te quantum σ wells the topological phase transition originates from a sign part of xy(p0 ) satisfies the Kramers-Kronig relation change of the Dirac mass of one of the two Chern insulators. Im σ (p ) σ = 1 xy 0 . m+ Re xy(p0 ) P dp (30) In what follows, let us assume it is , meaning that the π 0 p − p topological phase transition takes place in the (pseudo-)spin 0 0 up block of the BHZ model [cf. Eq. (1)]. Due to the parameters It is zero in the mass gap, becomes finite for |p0| 2|m| above, this transition is associated to an overall sign change of and decreases afterwards. Since Im σxy(p0 ) results from in- σ | xy(p0 ) p0=0, which is mainly driven by the QED correction in terband absorptions [45,51], it is only nonzero if the external Eq. (25). Consequently, measuring the AC Hall signal at ±m+ frequency is able to excite a finite density of states. [44], allows to substract the QED correction and, therefore, to Figures 3(b) and 3(c) show the corresponding plots for isolate the contribution to Eq. (25) which is induced by the a nontrivial and a trivial Chern insulator with minimal gap

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(a) Re (b) Re to [cf. Appendix A] 2 Im 3 Im 2δμδiν μν ie i /h] /h] 2 2 2 i , (p = 0) = 1−2mB−2|m||B| 1 D 2a π| | [e [e 4 B xy xy 1

σ σ 4B2 0 0 − Ln , (31) -4 -2 0 24 -4 -2 0 24 1 − 2Bm + 2|m||B| Frequency [|m|] Frequency [|m|] where is a hard momentum cutoff in α =|l|2.Sofar,we (c) Re (d) Re 3 focused on the contributions of the first Feynman diagram 1 Im Im 2 in Fig. 2. Using analog techniques and the quadratic vertex /h] /h] 2 2 1 in external fields, Eq. (13), the second Feynman diagram in [e [e xy xy σ 0 σ 0 Fig. 2 yields [cf. Appendix A] 2 μ iν -1 μν ie |B|δ δ μν = =− i − = . -4 -2 0 24 -4 -2 -|Δ| 0 |Δ| 24 i D,2b(p 0) i D,2a (p 0) (32) Frequency [|m|] Frequency [|m|] 2π This expression exactly cancels the finite and logarithmic FIG. 3. Hall conductivity σ (p )for(a)aQED+ system with xy 0 2 1 divergent terms in Eq. (31). Nevertheless, the full effec- m = A = 1andB = 0, (b) a nontrivial Chern insulator with m = A = = . = = = tive action still contains a term proportional to the UV 1andB 0 1, (c) a trivial Chern insulator with m A 1andB →∞ −0.1 and (d) a nontrivial Chern insulator with m = A = 1andB = 3. cutoff which diverges during renormalization . While the spectrum associated to (a–c) has the minimal gap 2|m| However, as stated above, the diagonal contribution should / = at the point, the spectrum related to (d) has the minimal gap || be regularized renormalized such that it vanishes for p 0. Physically, such a renormalization corresponds to a proper at kmin [cf. Sec. II]. Notice that all discontinuities/singularities arise from the assumption of zero temperature and disorder. Taking into definition of the particle density. The second term in Eq. (16), account these ingredients makes all curves continuous. which corresponds to the quadratic vertex in gauge fields Eq. (13), encodes the diamagnetic response of our system. This response is proportional to the particle density and as size 2|m|, respectively. The AC Hall conductivity shows the such needs to vanish in the gap. Due to the fact that we did same features as a pure Dirac system, except for the integer not renormalize the Dirac sea contribution to the particle den- quantization of its DC Hall conductivity. However, for a sity, e.g., by antisymmetrization, the divergence in Eq. (32) minimal gap apart from the -point, the situation differs, as persists. shown in Fig. 3(d). Here, the first resonance of the real part As a consequence of the underlying parity anomaly each occurs at p0 =±||.Thep0 =±2|m| resonance persists, but regularization scheme for a single Chern insulator does either peaks in opposite direction since the density of states now break parity or gauge symmetry. The naive introduction of the decreases at p0 =±2|m|. This property can also be seen in hard momentum cutoff in Eq. (31) breaks the gauge sym- Im σxy(p0 ), which resolves the Van Hove singularity at p0 = metry. To preserve this symmetry, we should rather choose a ±|| and drops at p0 =±2|m|. Consequently, measuring the parity-breaking regulator, such as a PV field with mass terms AC conductivity provides the information if the minimal gap M and B [54]. While this field by construction ensures is defined by the Dirac mass at the -point, or rather by an μν = + μν = = , interplay between the Dirac and the Newtonian mass apart i D,2a (p 0) i D,2b(p 0) 0 (33) from k = 0. it contributes to the AC Hall response, as it breaks parity. For Above, we have discussed the AC Hall response of a the entire system, this can be resolved by introducing a second single Chern insulator which is either in the topologically PV field for the regularization of the second (pseudo-)spin trivial phase with m ≡ m−, or in the topologically nontrivial block of the BHZ model. If this field is constructed such phase with m ≡ m+. The entire QAH response of the two that both PV fields are Kramers (time-reversal) partners, then (pseudo-)spin blocks H±(k) in the BHZ model [cf. Eq. (1)] their combined contribution to the AC Hall conductivity of the corresponds to the superposition of both of these signals. In entire QAH system vanishes. contrast to the DC Hall conductivity, the AC Hall conductivity contains corrections of nontopological origin. As such, V. NEWTONIAN MASS IN THE CONTEXT OF QED also the topologically trivial (pseudo-)spin block of the BHZ 2+1 REGULARIZATION SCHEMES model significantly contributes to the entire QAH response if the frequency |p0| is not much smaller than the trivial gap Albeit the Newtonian mass term in the Hamilton of a single 2|m−|. Chern insulator [cf. Eq. (2)] is a physical parameter provided by the material, before renormalization it acts similar to a parity-breaking regulator of a pure QED2+1 system in the B. Diagonal response calculation of the effective action [55]. In what follows, let Having discussed the off-diagonal response, we are still left us concretize this statement. with the calculation of the diagonal parts in Eq. (18). Since In the context of quantum field theories there are plenty our system is a bulk insulator, we physically expect that these of different regularization schemes, each breaking differ- terms vanish for p = 0. The diagonal contributions can be ent symmetries. As discussed in Sec. III, the regularization calculated via the same techniques as used above. This leads scheme associated to an odd number of 2 + 1-dimensional

033193-6 MOMENTUM-DEPENDENT MASS AND AC HALL … PHYSICAL REVIEW RESEARCH 2, 033193 (2020)

Dirac fermions needs to break parity symmetry to ensure we showed that before renormalization the Newtonian mass gauge invariance of the effective action. Manifestly par- term acts similar to a parity-breaking regulator in the calcula- ity breaking regularization schemes are for example PV tion of the effective action. Hence, this mass is directly related regularization, lattice regularization with Wilson fermions, to the regularization of QED2+1 and, as such, to the parity and ζ -function regularization, which we briefly review in anomaly. More precisely, we showed that the Newtonian mass Appendix B [30,37,56,57]. All these schemes induce a parity alone does not render the effective action UV finite, but odd Chern-Simons term in the effective action of a QED2+1 ensures an integer quantized DC Hall conductivity. Before system, directly proportional to the sign of the regularization renormalization, it acts similar to a Wilson mass term, which parameter which breaks parity. For the schemes mentioned, avoids the fermion doubling in a lattice approach. this is the PV mass, the Wilson parameter, as well as the Moreover, we derived the AC Hall conductivity of a QAH integration contour in the ζ -function regularization. Together insulator during the calculation of the effective action. We with the Chern-Simons contribution induced by the finite showed that the leading order AC correction to the DC Hall Dirac mass m, this leads to an integer quantized DC Hall conductivity contains a term proportional to the Chern num- conductivity [28,30]. ber. This term originates from the Newtonian mass and can be Adding the Newtonian mass term to a pure QED2+1 system measured by purely electrical means, or by determining the also ensures this property [cf. Eq. (24)]. Even if one would leading order frequency correction to the DC Faraday or Kerr remove the physical parameter B→0 in the end of the cal- angles of our system. Further, we revealed that the Newtonian culation, its contribution to the DC Hall conductivity persists mass significantly changes the resonance structure of the AC and the Chern number stays integer quantized. From this per- Hall conductivity in comparison to pure QED2+1 systems. spective the Bk2 term acts similar to a parity breaking element The classification of three-dimensional topological insula- of a certain regularization scheme. To concretize what we tors is also given by the interplay between their Dirac and mean with “similar”, let us compare the role of the Newtonian Newtonian mass term. The effective action of those systems mass term in the calculation of the effective action to common can be calculated by the parity-anomaly induced Chern- parity-breaking regulators before renormalization [55]. Since Simons term in 4 + 1 space-time dimensions [18]. Conse- the B|k|2 term provides a momentum-dependent Dirac mass quently, a natural extension of our work would be to derive the correction, it is natural to compare its role in the calculation effective action of a 4 + 1-dimensional QED system including of the effective action to regularization schemes which add both, a Dirac as well as a momentum-dependent term. terms of higher order derivatives to the bare Lagrangian. Such approaches are for example lattice regularization with ACKNOWLEDGMENTS Wilson fermions [28] (Appendix B1) and higher derivative regularization [31] (Appendix B2). However, except for the We thank F. A. Schaposnik, F. Wilczek, G. W. Semenoff, property that these schemes yield also an integer quantized T. Kießling, B. Trauzettel, and L. W. Molenkamp for useful Chern number, there are several key differences to the B|k|2 discussions. We acknowledge financial support through the term. By construction, each regularization needs to render Deutsche Forschungsgemeinschaft (DFG, German Research the effective action finite [40]. As shown in Eq. (32), adding Foundation), project id 258499086—SFB 1170 “ToCoTron- the Newtonian mass to a massive Dirac Lagrangian does ics,” the ENB Graduate School on Topological Insulators, the not exhibit this property. To reduce the superficial degree Würzburg-Dresden Cluster of Excellence on Complexity and of divergence, the higher derivative regularization multiplies Topology in Quantum Matter—ct.qmat (EXC 2147, Project the entire noninteracting Dirac Lagrangian by (1 + ∂2/M2 ). No. 39085490), and in part through National Science Founda- This has two implications. In contrast to the B|k|2 term, it tion Grant No. NSF PHY-1748958. E. M. Hankiewicz thanks circumvents the vertex renormalizations in Eqs. (12) and (13), the KITP institute for its hospitality. but as a price manifestly breaks local gauge invariance [58]. Moreover, by construction this approach also regularizes the APPENDIX A: CALCULATION OF THE kinetic part of the Lagrangian. EFFECTIVE ACTION In a lattice approach, the inverse lattice spacing a−1 makes the theory finite. To avoid fermion doubling and to break the For the calculation of the trace in Eq. (17), we used the + parity, the lattice QED2+1 Lagrangian comes along with an properties of the 2 1-dimensional Dirac matrices [61]: additional Wilson mass term ∝sak2. Here, s =±1 is the Wil- son parameter and k is the lattice three-momentum [28,59,60]. Tr[γμ] = 0, {γμ,γν }=2gμν , | |2 Clearly, the Wilson mass is directly related to the B k Tr[γμγν ] = 2gμν , term. However, by construction the Wilson mass is Lorentz γ γ γ = =− , invariant, while the Newtonian mass breaks this symmetry. Tr[ μ ν λ] 2i μνλ with 012 1 Further, the Wilson mass vanishes during renormalization, Tr[γμγν γλγρ] = 2(gμν gλρ − gμλgνρ + gμρ gνλ). (A1) a → 0, which is not the case for the B|k|2 term since it is a μ =− δμ + i material parameter. With C B i (2k p) , this in particular leads to

1 μ ν μ ν ν μ − Trγγ = 2k k + k p + k p VI. SUMMARY AND OUTLOOK 2 + gμν [M(k)M(k+p) − k2 − kp] In this work, we connected the Newtonian mass term of a − μνλ{ + − − }, QAH insulator to the parity anomaly of QED2+1. In particular, i [M(k p) M(k)]kλ M(k)pλ

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1 ν μλ ∇ − Trγ 0 = C g {[M(k) + M(k+p)]kλ + M(k)pλ} is the massless three-dimensional lattice Dirac operator, μ 2 γ E =σ ν μλσ is the lattice covariant derivative, and μ μ are the Eu- + iC kλ(k+p)σ , clidean Dirac matrices. In comparison to continuum QED2+1, 1 μ νλ Slatt (ψ,ψ,¯ A) includes the so-called Wilson term, propor- − Tr0γ = C g {[M(k) + M(k+p)]kλ + M(k)pλ} 2 tional to the Wilson parameter s =±1. This term acts like a μ νλσ − iC kλ(k+p)σ , momentum-dependent fermion mass and therefore explicitly 1 μ ν λσ breaks the parity symmetry of the system. As a consequence, − Tr = C C [g kλ(k+p) + M(k)M(k+p)]. (A2) 2 00 σ it induces a Chern-Simons term in the effective action, pro- μν μν Moreover, for the calculation of i and i ,we portional to sgn(s). Hence, this term acts very similar to the Diag,2a Diag,2b | |2 used the following identities for the integration of the spatial Newtonian B k term in Eq. (2). However, there are two loop momentum [62]: significant differences. On the one hand, the Wilson mass is Lorentz covariant, while the Newtonian mass breaks this 1 symmetry. On the other hand, the Wilson mass vanishes as dx (ax2 + bx + c)3/2 a → 0, which is not the case for the Newtonian mass term as b + ax it is a real parameter of the system. =− 2( √ 2 ) (b2 − 4ac) c + x(b + ax) 2. Higher derivative regularization x dx (ax2 + bx + c)3/2 The Lagrangian associated to the higher derivative regularization of QED + is given by [32,65] c + bx 2 1 = 4 √ 2 ∂2 (b2 − 4ac) c + x(b + ax) μ μ LHD = ψ¯ (iγ ∂μ − m) 1 + ψ + e ψγ¯ Aμψ, (B3) x M2 dx√ ax2 + bx + c where M is a parameter, allowing to remove the higher deriva- 1 tive correction during the renormalization process, M →∞. = ax2 + bx + c Notice, that by construction, the higher derivative term breaks a local gauge invariance. However, this property is fixed during b 1 − dx√ , renormalization [31]. If the higher derivative correction would 2a ax2 + bx + c come with covariant derivatives, then it would not reduce the superficial degree of divergence. Hence, the higher derivative √ 1 dx regulator differs significantly form the Newtonian mass in ax2 + bx + c Eq. (2). While the Newtonian mass term is parity-odd, breaks √1 Ln|2 a(ax2 + bx + c) + 2ax + b| for a > 0 Lorentz symmetry and renormalizes the Dirac mass in a = a . − √1 √2ax+b < gauge-invariant fashion, the higher derivative term is Lorenz − Arcsin 2 for a 0 a b −4ac invariant, breaks local gauge symmetry and multiplicates the (A3) full noninteracting QED2+1 part. As such, it contains a parity even as well as a parity odd contribution. Moreover, it van- APPENDIX B: REGULARIZATION SCHEMES ishes during renormalization as M →∞. In what follows, we briefly review the high-energy regularization schemes which we discuss in the main text. 3. Pauli-Villars regularization In contrast to the two schemes above, the Pauli-Villars reg- 1. Lattice regularization ularization adds additional bosonic particles χ to the classic QED + Lagrangian: A common way to regularize a quantum field theory is the 2 1 μ μ introduction of a space-time lattice. In this method, the finite LPV = ψ¯ (iγ Dμ − m)ψ + χ¯ (iγ Dμ − M)χ. (B4) −1 lattice spacing a introduces a momentum cut-off lattice ∝a . Lattice regularization explicitly breaks the parity symmetry of Their mass term M breaks parity and therefore induces a classical QED2+1. This is not a property of the lattice itself, Chern-Simons term in the effective action, proportional to but happens due the mandatory introduction of additional sgn(M). During renormalization the Pauli-Villars field decou- terms in the Lagrangian which prevent artificial gap closings ples from the theory, M →∞. However, it still leaves its trace at the high symmetry points of the lattice Brillouin zone. In in the Chern number [23,29]. Since this regularization scheme particular, the Euclidean lattice action Slatt (ψ,ψ,¯ A)isgiven includes an additional particle to ensure an integer quantized by [28,63,64] DC conductivity, it significantly differs from the Newtonian mass term in Eq. (2). 3 Slatt (ψ,ψ,¯ A) =−a ψ¯ (x)(D − m)ψ(x), (B1) x 4. ζ-function regularization = , , ∈ where x (an1 an2 an3) with nμ Z. Moreover, The ζ -function regularization completely differs from the

1 E 1 schemes introduced above [30]. It regularizes the generating D = γ ∇ + ∇μ + sa∇ ∇μ 2 μ ( μ ) 2 μ (B2) functional via a certain calculation scheme for the fermion

033193-8 MOMENTUM-DEPENDENT MASS AND AC HALL … PHYSICAL REVIEW RESEARCH 2, 033193 (2020) determinant: APPENDIX C: AC HALL CONDUCTIVITY μν The general solution of i (p , p = 0) in Eq. (20), is CS 0 − d ζ (D[A],s) given by the AC Hall conductivity Z[A]reg = detD[A] = e ds . (B5) reg s=0 e2 (1 − 4mB − )(Ln + Ln ) σ (p ) =− p0 1 2 xy 0 2h 4B |p | s=±1 p0 0

E Here, the Euclidean Dirac operator with γμ = σμ and the ζ (1 − 4mB + )(Ln + Ln ) + p0 3 4 , function are deﬁned via | | (C1) 4B p0 p0 = − + 2 2 where we deﬁned p0 s 1 4mB B p0 and abbreviate E μ μ D[A] = γμ (i∂ + eA ) + im, the four logarithms −s 2 ζ (D[A], s) = Tr(D [A]). (B6) Ln1 = Ln[−2sB ], Ln = Ln s(1 − 4mB − 2B2|m||p |) − s(1 − 2mB)2 , 2 0 p0 p0

By construction, this scheme is gauge invariant but breaks Ln3 = Ln[s(1 − 2mB + p )], 0 parity symmetry, related to peculiarities during the associated 2 2 Ln4 = Ln 2sB − p |B||p0| . (C2) contour integration (explicit path). It therefore also leads to p0 0 = an integer quantized DC conductivity where one part comes Notice, that exactly at p0 , p0 evaluates to zero. Hence, from sgn(m) and an additional contribution (±1) stems from this quantity encodes the physics stemming from the mass gap the choice of the integration contour [30]. apart from the point.

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