TKNN formula for general lattice Hamiltonian in odd dimensions

Topological insulators in $D=2n+1$ dimensions are characterized either by the Chern character of the Berry connection from the eigenfunctions of the Hamiltonian in the valence band [1, 2] or by the coefficient of the effective action as a functino of an external U(1) gauge field, i.e., photon [3, 4, 5, 6]. 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 [7, 8]. In Ref. [9] 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’ [10]. However, in more general situation, the gauge interaction may not always be described by such a single straight Wilson-line. In such cases, one has to include the contribution of contact interaction vertices. In this proceedings, we report our recent study on the equivalence for generalized lattice Hamiltonian so as to include arbitrary non-minimal gauge interactions [11].

We consider a gapped fermion system (with a gap $\Delta$) on a lattice with the following action in Euclidean space in $D=2n+1$ dimensions.

where $\vec{r}$ runs over the $2n$ dimensional spatial lattice points. We will set $x^{0}=t$ in the followings. The Hamiltonian $H(\vec{A})$ 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 $A_{\mu}=(A_{0},\vec{A})$. The fermion fields $\psi^{\dagger}(t,\vec{r})$ and $\psi(t,\vec{r})$ give creation and annihilation operators of fermions after quantization. We 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.

Since the fermion system is gapped with a gap size $\Delta>0$ , the effective gauge action obtained by integrating out fermions can be expanded in terms of gauge invariant local actions as $S_{\rm eff}=\sum_{k}a_{k}S_{k}(A)$. Here, $S_{\rm eff}(A)$ is defined as $e^{S_{\rm eff}}=\int\mathcal{D}\psi\mathcal{D}\psi^{\dagger}e^{-S_{\rm E}}$, 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 $\frac{1}{\Delta}$ or lattice spacing $a$. Many of the Lagrangians are given in terms of gauge invariant field $F_{\mu\nu}$ Since we do not have the Lorentz-invariance 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 $S_{\rm cs}(A)$ 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 $c_{\rm cs}$ as $S_{\rm eff}(A)=ic_{\rm cs}S_{\rm cs}+\cdots$ where ”$\cdots$ ” stands for other gauge invariant terms . Here the gauge invariance of the action requires that the coefficient is quantized as

Since the Chern-Simons action is of the lowest dimension in the parity-violating sector, the coefficient $c_{\rm cs}$ can be obtained by the following quantity

The effective action can be given by the log of the fermion determinant as $S_{\rm eff}(A)=\mbox{Tr}\left[\ln\left(D_{0}+H(A)\right)\right]$, where $D_{0}=\frac{\partial}{\partial x^{0}}+iA_{0}$. Splitting the kinetic operator $D_{0}+H(A)$ into free part and interaction part as $D_{0}+H(A)=\frac{\partial}{\partial x_{0}}+H_{0}-\Gamma(A$), where $H_{0}$ is the free fermion part defined as $H_{0}\equiv\left.H(A)\right|_{A=0}$ and $\Gamma(A)$ is the interaction part defined as $\Gamma(A)\equiv-iA_{0}-H(A)+H_{0}$. Expanding in $\Gamma(A)$, we obtain

From Eq.(5), we find that the Chern-Simons coupling for $D=2+1$ dimension is given by

where $S_{F}(p)$ is the fermion propagator $\frac{1}{ip_{0}+H_{0}(\vec{p})}$ and $\Gamma^{(1)}[q_{1},\alpha_{1};p]$ and $\Gamma^{(2)}[q_{1},\alpha_{1};q_{2},\alpha_{2};p]$ 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 $\alpha_{i}~{}(i=1,2)$

Note that the contributions with multi-photon vertices vanishes for the class of Hamiltonians with gauge interactions given by a single straight Wilson-line because the multi-photon vertices are symmetric under the interchange of Lorentz indices of photons. When contracted with the antisymmetric tensor, such contributions vanish. However, in general Hamiltonian we must consider these contributions.

In addition to the usual Ward-Takahashi identity $\Gamma^{(1)}[0,\alpha;p]=\textcolor{blue}{-}\frac{\partial S^{-1}_{F}(p)}{\partial p_{\alpha}}$, we derive the Ward-Takahashi identities in Appendix  A as

Using these identities, the integrand of Eq. (7) can be rewritten as

The first term is 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

For the case for $D=4+1$ dimension ($n=2$), we also have the following Ward-Takahashi identitiy given in AppendixA

with which one can show extra terms add up to total derivatives so we obtain

Therefore, Chern-Simons coupling is given by the winding number with fermion propagator also for $D=4+1$ case.

We now 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. [9], 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 $c_{\rm cs}$ 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 $p_{\mu}$ as $\partial_{\mu}\equiv\frac{\partial}{\partial p_{\mu}}$. 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 $S(p)=\frac{1}{ip^{0}+H}$ and inserting a complete set of energy eigenstates $\displaystyle{\sum_{\alpha}|\alpha\rangle\langle\alpha|}$, where $\alpha$ is the label of energ, the Chern-Simons coupling $c_{\rm cs}$ is given as

where $i_{1},\cdots,i_{2n}$ stand for the spatial indices.

All we have to do is to integrate over $p^{0}$ using Cauchy’s theorem. Here, we use a trick to simplify the integration. It is easy to see that the expression Eqs. (11), (14) are invariant under continuous deformation of $S_{F}$ (or $H$) provided that the integrand remains to have no singularities. Therefore, under a continuous change of the Hamilitonian, the winding number remains unchanged from its original value as long as the enegry spectrum is kept gapped throughout the deformation. Now, the most general Hamiltonian with $N_{v}$ valence bands and $N_{c}$ conduction bands is expressed as

where $|a(\vec{p})\rangle$ labeled by $a$ is the energy eigenstate in the valence band with spatial momentum $\vec{p}$ and negative energy eigenvalue $E_{a}(\vec{p})<0$. The state $|\dot{b}(\vec{p})\rangle$ labeled by $\dot{b}$ is the energy eigenstate in the conduction band with spatial momentum $\vec{p}$ and positive energy eigenvalue $E_{\dot{b}}(\vec{p})>0$. One can continuously deform the Hamiltonian without hitting the singularity of $S(p)$ ( i.e. keeping the system gapped ) so that all energy eigenvalues in the conduction bands and all energy eigenvalues in the valence bands are degenerate and momentum independent (i.e. flat band ) respectively. Then the deformed Hamiltonian $H_{\rm new}$ which gives the same winding number becomes

where $E_{v}<0$, $E_{c}>0$ are the momentum independent constant. Here the eigenstates are identical to those with the original Hamiltonian. Since there are only two poles, we can easily carrry our $p^{0}$ integral to obtain

Let us define the Berry connection using the negative energy eigenstates as

Then it is straightforward to show that the Berry curvature $\mathcal{F}^{ab}$ is

Using Eq. (19), the Chern-Simons coupling can be expressed as

where $\mbox{ch}_{n}(\mathcal{A})$ is the 2nd Chern character defined by $\mbox{ch}_{n}(\mathcal{A})=\frac{1}{n!}\frac{1}{(2\pi)^{n}}\mbox{tr}(\mathcal{F}^{n})$ . Comparing this expression with Eq. (4) $c_{\rm cs}=\frac{k}{(n+1)!(2\pi)^{n}}$,

We have shown the Chern-Simons level and the topological number in terms of the Berry connection is identical. On the other hand, while the Berry connection approach is limited to the free theory case, the effective theory approach can be applied also to interacting theories.