The boundary effects of transverse field Ising model

TL;DR

This paper investigates how boundary effects influence the transverse field Ising model, especially in finite systems with odd sites and anti-ferromagnetic interactions, revealing significant boundary impacts at low temperatures.

Contribution

The study develops a method to incorporate fermion-number parity constraints in finite TFIM systems and compares boundary effects under different boundary conditions.

Findings

Boundary effects are significant in anti-ferromagnetic TFIM with odd sites.

Fermion-number parity constraints affect free energy and correlations.

Boundary effects are prominent at low temperatures.

Abstract

Advance in quantum simulations using trapped ions or superconducting elements allows detailed analysis of the transverse field Ising model (TFIM), which can exhibit a quantum phase transition and has been a paradigm in exactly solvable quantum systems. The Jordan-Wigner transformation maps the one-dimensional TFIM to a fermion model, but additional complications arise in finite systems and introduce a fermion-number parity constraint when periodic boundary condition (PBC) is imposed. By constructing the free energy and spin correlations with the fermion-number parity constraint and comparing the results to the TFIM with open boundary condition, we show that the boundary effects can become significant for the anti-ferromagnetic TFIM with odd number of sites at low temperature.