Transfer matrix approach to the topological phases

Question

The transfer matrix contains all the information. i.e., information about the edges and bulk. What new insight does the transfer matrix approach provide in the study of the topological phases of matter? Is it possible to define new topological invariants using the transfer matrix approach? and are they related to the topological invariants derived from the bulk?

Answer 1

1. The transfer matrix method is convenient for 1D single particle problem. But many interesting topological phases are beyond single particle and beyond 1D. The natural generalization of the transfer matrix is the matrix product state in 1D and tensor network state in higher dimensions. There has been a lot of discussion of how to calculate topological invariants and diagnose topological phases in the tensor network state.

2. For free fermion states in 1D, the transfer matrix is just another way to write the Hamiltonian, so it does not contain more information than the current Hamiltonian approach. One can always back out the Hamiltonian from the transfer matrix and calculate the topological invariant in the bulk.

3. The transfer matrix $T(\epsilon)$ is an on-shell formalism, which is actually inconvenient for the purpose of calculating topological index (but good for calculating edge mode). We can back out the Green's function $$G(k_\mu)=\int\frac{\mathrm{d}\epsilon}{k_0-\epsilon}\sum_xe^{\mathrm{i}k_1 x}T(\epsilon)^x,$$ then the topological index is given by the WZW term $$N=\int\mathrm{d}u\int\mathrm{d}^2k\;\epsilon^{\mu\nu}\mathrm{Tr}G^{-1}\partial_\mu GG^{-1}\partial_\nu GG^{-1}\partial_uG.$$