TKNN formula for general Hamiltonian

Topological insulators in odd dimensions are characterized by topological numbers. We prove the well-known relation between the topological number given by the Chern character of the Berry curvature and the Chern-Simons level of the low energy effective action for a general class of Hamiltonians bilinear in the fermion with general U(1) gauge interactions including non-minimal couplings by an explicit calculation. A series of Ward-Takahashi identities are crucial to relate the Chern-Simons level to a winding number, which could then be directly reduced to Chern character of Berry curvature by carrying out the integral over the temporal momenta.


I. INTRODUCTION
Topological insulators in D = 2n+1 dimensions are characterized by topological numbers.
One characterization is given by the Chern character of the Berry connection from the eigenfunctions of the Hamiltonian in the valence band [1,2], the other characterization is given by the level of the Chern-Simons action which appears in the effective action after integrating out the fermion coupled to a smooth external U(1) gauge field, i.e., photon [3][4][5][6][7][8][9]. These two characterization are known to be equivalent because they both arise from the current correlation functions and there are explicit proofs for various cases.
For example, the famous TKNN number [1] describes integer Hall conductivity in spatial two-dimensional systems and it is valid for all kinds of band structures neglecting interactions between electrons. Meanwhile, motivated by the discovery of domain-wall fermion [10] (see also subsequent papers to study the anomaly inflow [11][12][13]) in their paper, Golterman Jansen and Kaplan (GJK) also gave an expression of conductivity of Chern-Simons current for a wide class of fermion propagator on the lattice including Wilson fermion with odddimensional Euclidean spacetime and found out that the topological number is given by the homotopy class for the map T D → S D [9]. They also showed that the topological number one finds correlated perfectly with the number of chiral edge states . The relation is well-known to many people, and in the continuum theory or in some special models there exist studies for the relations [14,15].
In Ref. [16] a proof is given for a large class of models for general odd dimensions, where they consider the most general lattice action for arbitrary free kinetic term on the lattice which is then coupled to U(1) gauge field in a minimal way, i.e. with the gauge interaction in the form of where m, n are the lattice sites h mn are the hopping parameters and A mn is the line integral of the gauge field along the straight line connecting the sites m and n. The advantage of this class of Hamiltonian is that the contact interactions such as fermion-fermion-multi-photon vertices do not contribute to the final expression so that only a set of Feynman diagram which appear also in the continuum theory gives non-vanishing contributions. Of course, this type of gauge interaction is physically motivated since it is based on the famous method of 'Peierls substitution' [17]. However, in more general situation, the gauge interaction may not always be described by such a single straight Wilson-line. It could be a linear combination of various Wilson-lines of arbitrary path, which can give non-minimal coupling. In such cases, one has to include the contribution of contact interaction vertices.
In this paper, we study the equivalence of topological number from TKNN formula and that from the Chern-Simons coupling for the most general lattice fermion Hamiltonian coupled to U(1) gauge field which is bilinear in fermion. The new features of our study is that the Hamiltonian is general enough to include arbitrary non-minimal gauge interactions which has not been considered in the previous studies [3][4][5][6][7][8][9][14][15][16] . We give an explicit proof of the equivalence of the two topological numbers for gapped fermion systems with Hamiltonian on the lattice given by the bilinear form of the fermion coupled to external U(1) fermions. We also discuss how the relation can give the Chern character in higher dimensional case.
The organization of this paper is as follows. In the next section we rewrite the level of the Chern-Simons effective action for the gapped fermion system coupled to a U(1) gauge field using the Feynman rule and relate it to the winding number of a map from T D to the fermion propagator space in D = 2 + 1 and D = 4 + 1 dimensions. In Section 3, we show the equivalence of the winding number to the Chern number for the Berry curvature. Section 4 is devoted to summary and conclusion.

II. GAPPED FERMION SYSTEM ON THE LATTICE
A. General gapped fermion system We consider a gapped fermion system on a lattice (or condensed matter systems on a translational invariant crystal) with the following action in Euclidean space in D = 2n + 1 dimensions. (Note that the time is continuous but the space is discrete as in the condensed matter systems.) where r runs over the 2n dimensional spatial lattice points. We will set x 0 = t in the followings. The Hamiltonian H( A) is given by a summation over all the possible hoppings on the lattice which include gauge interactions with a smooth external U(1) gauge field A µ = (A 0 , A). The fermion fields ψ † (t, r) and ψ(t, r) give creation and annihilation operators of fermions after quantization. We assume that when the gauge field is turned off, the Hamiltonian is translational invariant so that it can allow band structures. We also assume that there are N v bands and N c bands below and above the fermi level, respectively.
Therefore the fermion fields have N v + N c components.

B. Effective gauge action
Since the fermion system is gapped with a gap size ∆ > 0 , the effective gauge action obtained by integrating out fermions can be expanded in terms of gauge invariant local actions as Here, S eff (A) is defined as and S k (A) are the gauge invariant actions given by the local Lagrangian L k (A) and a k are the coefficients. By dimensional analysis, if the Lagragian L k (A) has a mass dimension d k the coefficient a k is suppressed by the d k − (2n + 1)-powers in 1 ∆ or lattice spacing a. Many of the Lagrangians are given in terms of gauge invariant field F µν = ∂ µ A ν − ∂ ν A µ , (e.g. S F 2 µν,ρσ ≡ d 2n+1 r F µν F ρσ with a coefficent a F 2 µν,ρσ ). Since we do not have the Lorentzinvariance on the lattice, the structure of the coefficients a k in the effective action heavily depend the geometry of the lattice.
However, there is a very special parity-violating term called Chern-Simons action S cs (A) given by This action is topological and always takes this form no matter what the geometry of the lattice is. Topological information of the fermion system is contained in the effective action through the coefficient c cs as S eff (A) = ic cs S cs + other gauge invariant terms.
Here the gauge invariance of the action requires that the coefficient is quantized as (See [14] for example.) Since the Chern-Simons action is of the lowest dimension in the parity-violating sector, the coefficient c cs can be obtained by the following quantity In the continuum theory or in simple lattice fermions such as Wilson fermion, this can be rewritten in terms of fermion path-integral as Here, BZ stands for the Brillioune zone, q n+1 ≡ q 1 + · · · + q n , S F (p) is the Fourier transform of the free fermion propagator 1 with momentum p, and Γ (1) [q, α; p] is the fermion-fermion-photon vertex with in-coming fermion momentum p and in-coming photon momentum q. However, the situation is not simple in general, since there are also contributions from contact interactions such as fermion-fermion-multi-photon vertices, which can naturally arise from non-minimal gauge couplings or generic lattice artifacts. In the following, we will explicitly show that these contributions automatically cancel against the contributions from the momentum derivative of vertex functions due to the Ward-Takahashi identities.
In the next subsection, we formulate how to evaluate Eq.(8) using fermion propagators and vertex functions for general Hamiltonian system.

C. Fermion propagator representation of Eq.(8)
The effective action can be given by the log of the fermion determinant as where D 0 = ∂ ∂x 0 +iA 0 . Splitting the kinetic operator D 0 +H(A) into free part and interaction part as where H 0 is the free fermion part defined as Simple algebra shows that the following equation holds: From Eq.(13), we find that the Chern-Simons coupling for D = 2 + 1 dimension is given by where S F (p) is the fermion propagator 1 ip 0 +H 0 ( p) and Γ (1) [q 1 , α 1 ; p] and Γ (2) [q 1 , α 1 ; q 2 , α 2 ; p] are fermion-fermion-photon and fermion-fermion-photon-photon vertices with in-coming fermion momentum p and in-coming photon momenta q i (i = 1, 2) with Lorentz index α i (i = 1, 2) Note that the contributions with multi-photon vertices vanishes for the class of Hamiltonians From a similar calculation, we find that the Chern-Simons level for D = 4 + 1 dimension is given by where Γ (3) [q 1 , α 1 ; q 2 , α 2 ; q 3 , α 3 ; p] is the fermion-fermion-photon-photon-photon vertex with in-coming fermion momentum p and in-coming photon momenta q i (i = 1, 2, 3) with Lorentz index α i (i = 1, 2, 3) , which is given by D. The case for D = 2 + 1 dimension (n = 1) In D = 2 + 1 dimension, i.e. n = 1 case, the Chern-Simons coupling c cs has contributions from the loop involving two fermion-fermion-photon vertices and the loop involving a single fermion-fermion-photon-photon vertex (contact interaction) as where X is given as The first term on the right hand side is the one-loop contribution with contact interaction and the second term is the usual one-loop contribution with simple fermion-fermion-photon vertices.
Carrying out the momentum derivative with q, In Appendix A , we derive the Ward-Takahashi identities as follows: Using these identities, we obtain The first and the second terms on the right hand side can be combined to give a total divergence which vanishes when we integrate over the momentum. Therefore, one finds that the Chern-Simons coupling is given by the winding number as E. The case for D = 4 + 1 dimension (n = 2) The Chern-Simons coupling can be written as where X 1 , X 2 , X 3 , X 4 are defined as follows we obtain The total divergence term will vanish after integrating over the spatial momenta due to the periodicity in BZ.
Thus, we finally get Therefore, Chern-Simons coupling is given by the winding number with fermion propagator also for D = 4 + 1 case.
We expect that the relation of Chern-Simons coupling and the winding number for general Hamiltonian including non-minimal coupling holds for arbitrary odd dimensions (D = 2n + 1). This will be left for future studies.

III. EQUIVALENCE OF WINDING NUMBER AND CHERN NUMBER
In this section, we show the equivalence of the Chern-Simons coupling given by the winding number expression and the Chern character given by the Berry connection for the energy eigenstates in the valence bands.
The proof of this part is already given in Ref. [16], but since the proof is simple, we give it here for completeness. We give the calculation for arbitrary odd (D = 2n + 1) dimensions, even though we have shown that the Chern-Simons coupling c cs can be written by the winding number using S F only for D = 2 + 1 and D = 4 + 1 dimensions.
In order to simplify the notation, hereafter we abbreviate the derivative with respect to the momentum p µ as ∂ µ ≡ ∂ ∂pµ .
A. Winding number in D = 2n + 1 dimension The result of the previous section for D = 2 + 1 and D = 4 + 1 can be unified to the following results: In the expression using the fermion propagator S(p), the Chern-Simons coupling c cs in D = 2n + 1 dimensions is given as Next we insert a complete set α |α α|, where α is the label of energy. Then we have: Here J is defined as where i 1 , · · · , i 2n stand for the spatial indices and summation over these indices are implicitly assumed following the Einstein's contraction rule. All we have to do is to integrate over p 0 using Cauchy's theorem. In order to discuss it in detail let us define the key integral J as follows.

B. p 0 integration
Here, we use a trick to simplify the integration. It is easy to see that the expression where |a( p) labeled by a is the energy eigenstate in the valence band with spatial momentum p and negative energy eigenvalue E a ( p) < 0. The state |ḃ( p) labeled byḃ is the energy eigenstate in the conduction band with spatial momentum p and positive energy eigenvalue where ǫ i 1 j 1 ···i 2n j 2n dp 0 2π 1 (ip 0 + E v ) n+1 (ip 0 + E c ) n a 1 |∂ i 1 H|ȧ 1 ȧ 1 |∂ j 1 H|a 2 × · · · × a n |∂ in H|ȧ n ȧ n |∂ jn H|a 1 + dp 0 2π a 1 |∂ i 1ȧ 1 ȧ 1 |∂ j 1 a 2 × · · · × a n |∂ inȧn ȧ n |∂ jn a 1 , where I [m,n] (E 1 , E 2 ) is defined as The expression of I [m,n] (E 1 , E 2 ) after p 0 integration is given in Appendix C.
where ch n (A) is the 2nd Chern character defined by Comparing this expression with Eq. (7) we arrive at the relation This means that the Chern-Simons level and the topological number in terms of the Berry connection is shown to be identical.

IV. SUMMARY AND CONCLUSION
We derived general TKNN formula from Chern-Simons level in the effective action for lattice system with general Hamiltonian bilinear in fermion in (2+1)-and (4+1)-dimensions.
We have shown that Chern-Simons level is given by the winding number of a map from T D to the fermion propagator space. For this relation Ward-Takahashi identities including higher order relations are crucial.
There has been an understanding that for the field theory approach to work, there should be a low energy mode which can be described by a relativistic field theory. Therefore, people had the impression that the field theory approach works only a special type of systems with emergent relativistic spectrum. The interesting point to note in our proof is that one does not need to assume anything but gauge invariance. The detailed structure of minimal or non-minimal gauge coupling is irrelevant. Also one does not need to assume that there exists an effective theory described by the relativistic field theory and it applies to any system including arbitrary bands which may be far away from the Fermi level.
Since we have found that the two methods can equally work well, we are now certain that we can use field theory approach to study the topological properties for arbitrary condensed matter systems which include interactions where one can fully utilize the power of field theory.
There are topological materials for systems with additional symmetries and in other dimensions Whether a complete equivalence holds for those systems remains an open problem.
We hope to extend our study to those systems in the future.
where summation over µ 1 , · · · , µ n are implicit. M µ 1 ···µn are some N × N matrix where N = N c + N v is the number of fermion degrees of freedom per site.
Expanding this action in terms of gauge fields and making Fourier transformations, one can obtain the formal expressions of the inverse propagator and the vertex functions in the momentum space. In the following, let us denote the inverse fermion propagator with momentum p as S −1 F (p) and the vertex functions with incoming fermion momentum p and n photons with incoming momentum k i and µ i components (i = 1, · · · , n) and outgoing fermion with momentum p + n i=1 k i as Γ (n) [k 1 , µ 1 ; · · · ; k n , µ n ; p]. Then the formal expression gives Γ (3) [k, µ; l, ν, r, λ; p] (ip + r µ i ) +[ other 5 terms obtained from the permutaion of (µ, k), (ν, l), (λ, r)].
Differentiating Eq.(A4) with respect to p µ and taking soft photon limit (k → 0) in Eq. (A5), one obtains This is the well-known Ward-Takahashi identity in QED, which is generalized for the lattice fermion system.

First order Ward-Takahashi identities
It is interesting to note that we could also obtain Ward-Takahashi identities for quantities involving higher order terms in photon momenta and multi-photon vertex functions. In order to see that, let us take the second derivatives of Eq. (A4). One obtains Let us also differentiate Eq.(A5) with k ν or p ν and take the soft photon limit. We obtain Taking also the soft photon limit of Eq.(A6), we obtain We then obtain the following identities:
Let us consider differentiation with respect to p µ . Here we introduce the simplified notation Then differentiating Eq.(B2), we obtain Also, differentiating Eq.(B3) and making a little algebra, we have This means the matrix element of the momentum derivative of the Hamiltonian is given as α|∂ µ H|β = −(E α − E β ) α|∂ µ β (α = β).