Finite time path field theory perturbative methods for local quantum spin chain quenches

TL;DR

This paper develops perturbative methods within finite time path field theory to analyze local quantum quenches in spin chains, enabling calculation of Loschmidt echo to all orders and resummation of perturbative expansions.

Contribution

It introduces a formalism for perturbative calculations of quantum quenches in spin chains using FTPFT, including resummation techniques for Loschmidt echo.

Findings

Calculated the bubble diagram in LE to any order in perturbation.

Resummed the Schwinger-Dyson equation for fermion two-point functions.

Discussed limitations and potential extensions of the method.

Abstract

We discuss local magnetic field quenches using perturbative methods of finite time path field theory (FTPFT) in the following spin chains: Ising and XY in a transverse magnetic field. Their common characteristics are: (i) they are integrable via mapping to a second quantized noninteracting fermion problem; and (ii) when the ground state is nondegenerate (true for finite chains except in special cases), it can be represented as a vacuum of Bogoliubov fermions. By switching on a local magnetic field perturbation at finite time, the problem becomes nonintegrable and must be approached via numeric or perturbative methods. Using the formalism of FTPFT based on Wigner transforms (WTs) of projected functions, we show how to: (i) calculate the basic ``bubble'' diagram in the Loschmidt echo (LE) of a quenched chain to any order in the perturbation; and (ii) resum the generalized Schwinger--Dyson equation for the fermion two-point retarded functions in the ``bubble'' diagram, hence achieving the resummation of perturbative expansion of LE for a wide range of perturbation strengths under certain analyticity assumptions. Limitations of the assumptions and possible generalizations beyond it and also for other spin chains are further discussed.