Path Integral analogue for discrete systems

Question

What is the analogue for path integral formulation for a discrete system? e.g. chain of typical 2-level excitonic systems: \begin{equation} H=\sum_k \sigma^+_k\sigma^-_k+\sum_{k<l}(\sigma_k^+\sigma^-_l+\sigma_k^-\sigma^+_l). \end{equation} Any connections to random or quantum walks?

Answer 1

I can think of a few versions of discrete path integral.

One is to take $\exp(itH)$ and expand it order-by-order in $t$. You can a large number of matrix products, and by inserting a resolution of the identity between each of these matrix products, you get a sum over somethings that you could think of as discrete paths hopping around a basis set of Hilbert space.

Another way to say this is we attempt to approximate $\exp(itH)$ by a finite-depth quantum circuit and then take the limit as the depth goes to infinity.

A more covariant version of all of this is to work with tensor networks rather than circuits (there is a direct transformation from the latter to the former). Then evaluating the tensor network starts to look like a stat mech model in one higher dimension. If the tensors have a conservation law, you will see discrete random walks appearing.

All of this is rather familiar in TQFT and integrable stat mech models, but I don't know any references that try to do it seriously for some simple spin models like you suggest.

Answer 2

On the textbook of Atland-Simons, they have one section which uses path integral to treat a spin system with the following simple Hamiltonian $$H=-\sigma_z$$ in any representation of SU(2). The treatment is very much like a baby version of the coherent state treatment mentioned above. I believe for more complicated systems, a similar coherent state treatment is possible, but I also hope that some expert will elaborate on this.