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

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.


I. INTRODUCTION
The discovery of Dirac materials, such as topological insulators [1][2][3][4][5][6][7][8][9][10] and Weyl or Dirac [11][12][13][14][15][16] semimetals yields the possibility to measure quantum anomalies in condensed matter systems.In particular, it was shown that two-dimensional Quantum Anomalous Hall (QAH) insulators, such as (Hg,Mn)Te quantum wells 17,18 or magnetically doped (Bi,Sb)Te thin films 19,20 are directly related to the parity anomaly of planar quantum electrodynamics (QED 2+1 ) 21 .In a nutshell, this anomaly implies that it is not possible to quantize an odd number of Dirac cones in a parity symmetric manner 22,23 .As a QAH insulator is characterized by Dirac-like physics in 2+1 space-time dimensions, its mass terms are directly related to the parity anomaly.In general, such an insulator is characterized by two different mass terms: A momentum independent (Dirac) mass, as well as a momentum dependent (Newtonian) mass 24 .While the relation of the Dirac mass to the parity anomaly was initially shown in Ref. [22], the connection of the Newtonian mass to the parity anomaly has not yet been analyzed.To bridge this gap, we derive in this work the effective action of a QAH insulator.In the corresponding calculation for a pure, massless QED 2+1 system, the parity anomaly arises from the particular regularization of its infinite Dirac sea.Within this work, we show that the Newtonian mass acts similar to a parity-breaking regulator in the calculation of the effective action.As such, it is directly related to the parity anomaly.
In particular, we compare the Newtonian mass term to different parity-breaking regulators, such as Pauli-Villars (PV), lattice Wilson fermions, and ζ-function or higher derivative approaches [25][26][27][28] .We prove that the Newtonian mass alone does not remove the UV divergence in the effective action, as it is usually done by higher derivative schemes 28,29 .Instead, it acts like a parity-breaking element of a high-energy regularization scheme, as it leads to an integer quantized DC Hall conductivity.In particular, we show that the Newtonian mass can be seen as a continuum version of a Wilson term, whereas it acts significantly different than a PV mass or the integration contour in a ζ-function regularization.With continuum we mean that this contribution does not vanish during renormalization.Hence, we reveal that the Newtonian mass is related to the regularization of QED 2+1 and, as such, to the parity anomaly itself.
Moreover, we derive the non-quantized finite frequency corrections to the DC Hall conductivity during our calculation of the effective action.We explicitly show that the leading order AC correction contains a term which is proportional to the torsional Hall viscosity.As such, this non-dissipative transport coefficient defines the leading order correction to the Kerr and Faraday angle of our system.In addition, we show that 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.
This work is organized as follows: In Sec.II, we introduce two-dimensional QAH insulators and discuss their mass terms.In Sec.III, we review how to quantize a classical field theory and how the parity anomaly arises in QED 2+1 .In Sec.IV, we derive the effective action of a QAH insulator.We explicitly show its integer quantized DC Hall conductivity and derive its non-quantized AC correction.We show that this correction is directly related to the torsional Hall viscosity and to the Kerr as well as to the Faraday angle.In Sec.V, we compare our findings with differently regularized QED 2+1 systems and connect the Newtonian mass to the parity anomaly.In Sec.VI, we summarize our results and give an outlook.

II. MODEL
Two-dimensional QAH insulators can be described by the Bernevig-Hughes-Zhang (BHZ) Hamiltonian 2 : where particle-hole asymmetry.Within this work, we consider particle-hole symmetric QAH insulators since the D|k| 2 term in Eq. ( 1) is parity-even and does not contribute to the parity anomaly 30 .The mass terms in Eq. ( 1) are m and B|k| 2 .Together, they give rise to an integer quantized Chern number 18,31 C QAH = (sgn(m) + sgn(B)) /2 . ( While the Dirac mass m defines the mass gap at the Γ−point, the momentum dependent B|k| 2 term acts like an effective mass of a non-relativistic fermion system.The spectrum associated to Eq. ( 1) for D = 0 is given by where ± encodes the conduction and the valence band.In Fig. 1, we show the influence of the mass parameters on the band structure.Depending on the values for m, B, and A, the band structure significantly changes.For m/B > 0, the system is topologically non-trivial with C QAH = ±1.However, depending on the absolute values of the input parameters, the minimal gap can either be located at the Γ-point [Fig. 1 ).The minimal gap of the system is therefore either defined by 2|m|, or by In contrast, for m/B < 0, the system is topologically trivial, characterized by C QAH = 0.In this case the minimal gap is always located at the Γ-point.At m = 0, the topological phase transition occurs, which comes along with a gap closing at k = 0.While in our plots we mostly consider positive mass terms, the inverted Dirac mass of an experimental QAH insulator is negative.Changing the overall sign of m and B alters the sign of the Chern number, but not the underlying physics.Notice that according to Eq. ( 3), the Newtonian mass term does not make the spectrum bounded.It is from this perspective expected that the B|k| 2 term does not render the effective action UV finite.
Having discussed the particular influence of the mass terms on the band structure, let us emphasize that both of them explicitly break parity in 2+1 space-time dimensions, defined as P : (x 0 , x 1 , x 2 ) → (x 0 , −x 1 , x 2 ).Therefore, the Dirac as well as the Newtonian mass term need to be directly related to the parity anomaly of massless QED 2+1 .To concretize this statement, we briefly review the concept of quantum anomalies.

III. PARITY ANOMALY
We consider Dirac-like systems in 2+1 space-time dimensions coupled to an external fluctuating abelian U(1) gauge field A µ .Such models can be quantized by calculating the partition function Z[A] via integrating out the fermionic degrees of freedom Here, ψ and ψ are the two-component Dirac spinor and its adjoint.
Hence, S eff [A] is not gauge invariant modulo 2π, implying that the partition function Z[A] lacks gauge invariance.Calculating the fermion determinant explicitly leads to divergences.This requires a regularization scheme, which need to be chosen such that it ensures large gauge invariance of Z[A].In particular, it was shown that each regularization scheme needs to break parity to ensure large gauge invariance.This is known as the parity anomaly 22,25 .The parity anomaly therefore directly results from the particular regularization of QED 2+1 .Even though a massive QED 2+1 system breaks parity on the classical level, the fermion determinant still diverges and lacks gauge invariance.Again, this requires a parity-breaking regularization scheme, which extends the peculiarities of the parity anomaly to the massive case 22,27,33 .We show that adding the Newtonian mass term to a massive QED 2+1 Lagrangian ensures an integer quantized DC Hall conductivity associated to a large gauge invariant Chern-Simons term.In particular, we interpret and compare the Newtonian mass term to different high-energy regularization schemes of QED 2+1 and consequently relate it to the parity anomaly.

IV. EFFECTIVE ACTION
In what follows, we perturbatively evaluate the effective action of a QAH insulator.Therefore, we Taylor expand the fermion determinant in Eq. ( 8) to second order in the external background field A µ .The free Lagrangian of a QAH insulator is obtained by a Legendre transformation of Eq. ( 1).For A = = 1 and D = 0, this Lagragian is equivalent to a pure QED 2+1 Lagrangian except for an additional correction, which is quadratic in spatial derivatives Here and in the following, we use the properties of the Dirac matrices given in App. A. Coupling L 0 covariantly to the U(1) gauge field A µ , leads to the Lagrangian where D µ = ∂ µ + ieA µ is the covariant derivative.From the interaction terms in L, we can read off the vertex contributions.For an incoming electron of momentum k, incoming photons of momentum p, and an outgoing electron of momentum k+p, we find the vertices Here, the subscript defines the number of involved photons.In comparison to pure QED 2+1 , the extra B|k| 2 term in the Lagrangian renormalizes the gauge-matter coupling.The original QED vertex in Eq. ( 11) obtains a momentum dependent correction.Additionally, Eq. ( 12) defines a new vertex structure, which is of second order in gauge fields.This vertex encodes the diamagnetic response of the QAH insulator in Eq. ( 1) 34 .The fermion propagator associated to the Lagrangian L is given by Feynman diagrams for the vacuum polarization operator of a QAH insulator with vertices V µ,ν (1) and V µν (2) .External momenta are denoted by p, loop momenta by k.
Here, we defined the momentum dependent mass term M (k) = m − B|k| 2 and used the Feynman prescription with → 0 + .To perturbatively obtain the second order effective action in external fields, we need to calculate the vacuum polarization operator Π µν (p) 35 , Due to the vertex structure in Eqs. ( 11) and ( 12), the vacuum polarization operator is obtained by the sum of two one-loop Feynman integrals which are diagrammatically illustrated in Fig. 2. We start with the calculation of the first term in Eq. ( 15), iΠ µν 2a (p), which is the usual QED 2+1 vacuum polarization operator with renormalized vertex and propagator structure.As shown in Fig. 2(a), this tensor is given by There are four different contributions to the Dirac trace Tr = Tr γγ +Tr γ0 +Tr 0γ +Tr 00 , where the subscript defines the Dirac and identity part of the vertex structure in Eq. (11).Since all physical response functions are given as functional derivatives of the effective action at zero external spatial momentum, from now on we focus on the calculation of iΠ µν (p 0 , p = 0).With this assumption, p 2 = p 2 0 , µνλ p λ = µν0 p 0 , and M (k) = M (k +p).Next, we introduce the Feynman parameter x ∈ [0, 1] and shift the loop momentum according to k = l −px.This gives the denominator in Eq. ( 16) a quadratic form, allowing us to drop all linear terms in l in the numerator, due to an anti-symmetric integration over symmetric boundaries 36 .With α = |l| 2 , this leads to [cf.App.A] In the QED 2+1 limit B → 0, the Dirac trace in Eq. ( 17) reduces to the well-known result [36][37][38] Notice, that the off-diagonal Chern-Simons contribution in Eq. ( 18) gets shifted by the renormalized vertex structure in Eq. ( 17).As argued above, this will lead to an integer quantized DC Hall conductivity associated to a large gauge invariant Chern-Simons term.

A. Off-Diagonal Response
To prove this statement, we evaluate the integral Here, we used the Feynman parametrization and solved the complex time-integration via the residue theorem.
For QAH insulators, time and spatial momenta need to be integrated separately since the B|k| 2 term breaks the Lorentz symmetry.Hence, it is not possible to Wickrotate and integrate over a Euclidean three-sphere.Integrating the Feynman parameter x implies where we kept the i -prescription to circumvent the poles for α > 0, appearing if p 0 exceeds the gap of the system.Last, we perform the remaining α-integration for an arbitrary driving frequency p 0 and subsequently set → 0 + .Due to its lengthy form, we present the general Chern-Simons contribution and its AC Hall conductivity σ xy (p 0 ) in App.C, Instead, let us first analyze the Taylor expansion of the AC Hall conductivity in terms of the frequency p 0 , with the coefficients (reintroducing A and ) Equation ( 23) defines the well-known DC Hall conductivity of a QAH insulator 18,31 .In comparison to pure QED 2+1 with a half-quantized σ xy (0), the Newtonian mass term ensures integer quantization.Hence, the associated Chern-Simons term is large gauge invariant 39 .In contrast, Eq. ( 24) defines the leading order AC corrections to σ xy (0), which are quadratically suppressed by the ratio of p 0 over the minimal energy gap of the system.As discussed in Sec.II, this gap is either defined by 2|m|, or by ∆, depending on values chosen for m, B, and A 40 .Due to the quadratic suppression, the AC signal stays close to quantized values for small driving frequencies.Most remarkably, Eq. (24) shows that even the leading order AC correction contains a part proportional to the integer quantized Chern number C QAH , encoded in terms of the torsional Hall viscosity [41][42][43] In the trivial phase m/B < 0, this term vanishes and the first order AC correction is solely given by the second term in Eq. (24).Instead, for m/B > 0, ζ H contributes to the first order AC correction.For instance in (Hg,Mn)Te quantum wells with m = −10meV, B = −1075meVnm 2 , and A = 365meVnm, the torsional Hall viscosity defines ≈ 10% of the entire signal in Eq. ( 24).Moreover, let us predict another signature of Eq. ( 24) in (Hg,Mn)Te quantum wells.In these systems the topological phase transition originates from a sign change of the Dirac mass.Due to the parameters above, this transition is associated to an overall sign change of σ xy (p 0 )| p0=0 .For Galilean invariant systems, it is known that the AC Hall conductivity includes a Hall viscous correction proportional to the squared external spatial momentum 44,45 .Our system neither fulfills Lorentz-, nor Galilean invariance.Therefore, the second order frequency correction related to the torsional Hall viscosity is unexpected.As an experimental consequence of Eq. ( 24), measuring the AC conductivity in QAH insulators is a direct way to access the associated torsional Hall viscosity.
Since σ xy (p 0 ) is directly related the Faraday and the Kerr angle of two-dimensional QAH insulators 46,47 , we predict that the torsional Hall viscosity can be resolved by magneto-optical experiments.Let us consider a linearly polarized electric field, which incidents normally on the QAH system.For frequencies much smaller than the gap, which justify Eq. ( 22), one finds 47 Here, c is the vacuum speed of light and 0 the vacuum permittivity.While these identities imply quantized values of the Faraday and Kerr angles in the DC limit, they carry the information of how these angles change due to the contribution of the torsional Hall viscosity in Eq. ( 24).
To resolve the effects of ζ H in one of these experiments, |p 0 | Min(2|m|, ∆).For instance in inverted (Hg,Mn)Te/CdTe quantum wells, the gap is of the order of several meV 48 , depending on the particular manganese concentration.This corresponds to frequencies in the THz regime.For such frequencies, the first order correction to the DC Faraday and Kerr angles is on the order of milli-rad, which can be resolved by recent Faraday polarimeters 49 .
Before we discuss the general solution of the AC Hall conductivity, note that the non-quantized value for σ xy (p 0 = 0) makes the associated Chern-Simons term large gauge non-invariant.Since analogously to thermal effects, an AC driving field excites non-topological degrees of freedom, this effect corresponds to the non-large gauge invariance of finite temperature Chern-Simons terms.For these theories, it was shown that the full effective action contains non-perturbative corrections in A µ , absorbing this non-invariance 50,51 .These terms cannot be found by the Taylor expansion of the fermion determinant Eq. ( 8).However, due to the fact that they are higher order in gauge fields, they do not contribute to the conductivity.
Let us now analyze the general solution of the AC Hall conductivity.Figure 3 shows the real and the imaginary part of σ xy (p 0 ) according to its general form in App. C. To study the influence of the Newtonian mass term, Fig. 3(a) shows σ xy (p 0 ) for a pure QED 2+1 system.At p 0 = 0, one observes the characteristic half-quantization.Moreover, the real part of σ xy (p 0 ) shows a resonance at p 0 = ±2|m| and tends to zero for larger frequencies.For |p 0 | ≥ 2|m|, the AC field excites particle-hole pairs which can propagate unhindered for p 0 = ±2|m|.This is the origin of the resonance [52][53][54] .For large frequencies, the AC field dominates the mass gap that protects the topological phase.This leads to a vanishing AC Hall conductivity.The imaginary part of σ xy (p 0 ) satisfies the

Kramers-Kronig relation
Re It is zero in the mass gap, becomes finite for |p 0 | ≥ 2|m| and decreases afterwards.Since Im σ xy (p 0 ) results from interband absorptions 46,52 , it is only non-zero if the external frequency is able to excite a finite density of states.Figure 3(b) and 3(c) show the corresponding plots for a non-trivial and trivial BHZ model with minimal gap size 2|m|.The AC Hall conductivity shows the same features as a pure Dirac system, except for the integer quantization of its DC Hall conductivity.However, for a minimal gap apart from the Γ-point, the situation differs, as shown in Fig. 3(d).Here, the first resonance of the real part occurs at p 0 = ±∆.The p 0 = ±2|m| resonance persists, but peaks in opposite direction since the density of states now decreases at p 0 = ±2|m|.This property can also be seen in Im σ xy (p 0 ), which resolves the Van Hove singularity at p 0 = ±∆ and drops at p 0 = ±2|m|.
Consequently, measuring the AC conductivity in QAH insulators provides the information if the minimal gap is defined by the Dirac mass at the Γ-point, or rather by an interplay between the Dirac and the Newtonian mass apart from k = 0.

B. Diagonal Response
Having discussed the off-diagonal response, we are still left with the calculation of the diagonal parts in Eq. ( 17).Since our system is a bulk insulator, we physically expect that these terms vanish for p = 0.The diagonal contributions can be calculated via the same techniques as used above.This leads to [cf.App.A] where Λ is a hard momentum cutoff in α.So far, we focused on the contributions of the first Feynman diagram in Fig. (2).Using analog techniques and the quadratic vertex in external fields, Eq. ( 12), the second Feynman diagramm in Fig. ( This expression exactly cancels the finite and logarithmic divergent terms in Eq. ( 29).Nevertheless, the full effective action still contains a divergent term, which needs to be subtracted by a counterterm in the bare Lagrangian.
Physically, this renormalization corresponds to a proper definition of the particle density.As discussed, the second term in Eq. ( 15), which corresponds to the quadratic vertex in gauge fields Eq. ( 12), encodes the diamagnetic response of our system.This response is proportional to the particle density and as such needs to vanish in the gap.Due to the fact that we did not renormalize the Dirac sea contribution to the particle density, e.g. by anti-symmetrization, the divergence in Eq. ( 30) persists.

V. NEWTONIAN MASS IN THE CONTEXT OF REGULARIZATION
In the context of quantum field theories there are plenty of different regularization schemes, each breaking different symmetries.As discussed in Sec.III, the regularization scheme associated to an odd number of 2+1 dimensional Dirac fermions 55 needs to break parity to ensure large gauge invariance of the effective action.Manifestly parity breaking regularization schemes are for example PV regularization, lattice regularization with Wilson fermions, and ζ-function regularization, which we briefly review in App.B 27,33,56,57 .All these schemes induce a parity odd Chern-Simons term in the effective action, directly proportional to the sign of the regularization parameter which breaks parity.For the schemes mentioned, this is the PV mass, the Wilson parameter, as well as the integration contour in the ζ-function regularization.Together with the Chern-Simons contribution induced by the finite Dirac mass m, this leads to an integer quantized DC Hall conductivity 26,27 .
In what follows, let us analyze the properties of the Newtonian mass in the context of regularization.By construction, each regularization needs to render the effective action finite 36 .As shown in Eq. (30), adding the Newtonian mass to a massive Dirac Lagrangian does not exhibit this property.We still had to introduce a hard momentum cutoff Λ.
Instead, Eq. (23) shows that the Newtonian mass ensures an integer quantized DC Hall conductivity.Ac-cording to the discussion above, it therefore acts similar to the parity breaking element of a certain regularization scheme.While most regularization schemes rely on the introduction of additional particles or change the bosonic part of the bare action, there are only a few ways to renormalize a single Dirac Lagrangian alone.In the context of QAH insulators, regularization schemes which add terms of higher order derivatives to the bare Lagrangian are of particular interest, since these schemes are related to the extra B|k| 2 term in Eq. (10).Such approaches are for example lattice regularization with Wilson fermions 26 [App.B 1] and higher derivative regularization 28 [App.B 2].However, except for the property that these schemes yield also an integer quantized Chern number, there are several key differences to the B|k| 2 approach with a hard momentum cut-off.
To reduce the superficial degree of divergence, the higher derivative regularization multiplies the entire non-interacting Dirac Lagrangian by a term ∝ ∂ 2 .This has two implications.In contrast to the B|k| 2 term, it circumvents the vertex renormalizations in Eqs. ( 11) and ( 12), but as a price manifestly breaks local gauge invariance 58 .Moreover, by construction this approach has a parity even and a parity odd contribution and therefore also influences the diagonal response.
In a lattice approach, the inverse lattice spacing a −1 , which corresponds to our hard momentum cut-off Λ, makes the theory finite.To avoid fermion doubling and to break the parity, the lattice QED 2+1 Lagrangian comes along with an additional Wilson mass term ∝ sak 2 .Here, s = ±1 is the Wilson parameter and k is the lattice threemomentum 26,59,60 .Clearly, the Wilson mass is directly related to the B|k| 2 term.However, by construction the Wilson mass is Lorentz invariant, while the Newtonian mass breaks this symmetry.Further, the Wilson mass vanishes during renormalization, a → 0, which is not the case for the B|k| 2 term.
According to our discussion, the Newtonian mass term in summary can be interpreted as the parity breaking element of a certain regularization scheme and is as such directly related to the parity anomaly [cf.Sec.III].

VI. SUMMARY AND OUTLOOK
In this work, we connected the Newtonian mass term of a QAH insulator to the parity anomaly of QED 2+1 .In particular, we showed that in the calculation of the effective action the Newtonian mass term acts similar to a parity-breaking regulator.Hence, this mass is directly related to the regularization of QED 2+1 and, as such, to the parity anomaly.More precisely, we showed that the Newtonian mass alone does not render the effective action UV finite, but ensures an integer quantized DC Hall conductivity.Therefore, it can be seen as the parity breaking element of a certain regularization scheme.For instance, it can be interpreted as a continuum Wilson term in a lattice approach.
Moreover, we derived the AC Hall conductivity of a QAH insulator during the calculation of the effective action.We showed that the leading order AC correction to the integer quantized DC value contains a term proportional to the torsional Hall viscosity.This paves the way to measure this non-dissipative transport coefficient.This can be done by purely electrical means, or by measuring the leading order frequency correction to the DC Faraday or Kerr angle of our system.Further, we revealed that the Newtonian mass significantly changes the resonance structure of the AC Hall conductivity in comparison to pure QED 2+1 systems.As a next step, it would be interesting to include finite temperature effects in our calculation of the effective action.This analysis would clarify in which way the Newtonian mass renormalizes the finite temperature AC Hall conductivity of pure QED 2+1 systems.
Lorentz covarianz, while the Newtonian mass breaks this symmetry.On the other hand, the Wilson mass vanishes as a → 0, which is not the case for the Newtonian mass term.

Higher Derivative Regularization
The Lagrangian associated to the higher derivative regularization of QED 2+1 is given by 29,65 : where M is a parameter, allowing to remove the higher derivative correction during the renormalization process, M → ∞.Notice, that by construction, the higher derivative term breaks local gauge invariance.However, this property is fixed during renormalization 28 .If the higher derivative correction would come with covariant derivatives, it would not reduce the superficial degree of divergence.Hence, the higher derivative regulator differs significantly form the Newtonian mass in Eq. ( 1).While the Newtonian mass term is parity-odd, breaks Lorentz symmetry and renormalizes the Dirac mass in a gauge fashion, the higher derivative term is Lorenz invariant, breaks local gauge symmetry and multiplicates the full non-interacting QED 2+1 part.As such, it contains a parity even as well as a parity odd contribution.Moreover, it vanishes during renormalization as M → ∞.

Pauli-Villars Regularization
In contrast to the two schemes above, the Pauli-Villars regularization adds additional bosonic particles χ to the classic QED 2+1 Lagrangian: Their mass term M breaks parity and therefore induces a Chern-Simons term in the effective action, proportional to sgn(M ).During renormalization the Pauli-Villars field decouples from the theory, M → ∞.However, it still leaves its trace in the Chern number 22,25 .Since this regularization scheme introduces a new particle to ensure an integer quantized DC conductivity, it significantly differs from the Newtonian mass term in Eq. (1).

Zeta-Function Regularization
The ζ-function regularization completely differs from the schemes introduced above 27 .It regularizes the generating functional via a certain calculation scheme for the fermion determinant: By construction, this scheme is gauge invariant, but breaks parity symmetry, related to peculiarities during the associated contour integration (explicit path).It therefore also leads to an integer quantized DC conductivity where one part comes from sgn(m) and an additional contribution (±1) stems from the choice of the integration contour 27 .

Appendix C: AC Hall Conductivity
The general solution of iΠ µν CS (p 0 , p = 0) in Eq. ( 19), is given by the AC Hall conductivity Notice, that exactly at p 0 = ∆, ∆ p0 evaluates to zero.Hence, this quantity encodes the physics stemming from the mass gap apart from the Γ−point.

FIG. 1 .
FIG. 1. Band structure of the BHZ model for k2 = D = 0 and for (a) a topologically nontrivial phase with m = A = 1 and B = 0.1.The minimal gap 2|m| is located at the Γ-point.(b) For a topologically nontrivial phase with m = A = 1 and B = 3.The minimal gap ∆ is located at |k| = ±|kmin|.