  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.
