Quantum phase transitions in alternating transverse Ising chains

TL;DR

This paper analyzes quantum phase transitions in a periodically modulated transverse Ising chain, providing exact analytical and numerical results on how alternating interactions influence the critical behavior of the system.

Contribution

It offers the first rigorous analytical and numerical study of quantum phase transitions in an alternating transverse Ising chain with varying bonds and fields.

Findings

Analytical expressions for thermodynamic quantities in alternating chains

Numerical data showing effects of periodic modulation on phase transitions

Detailed analysis of period 2 and 3 modulations on quantum criticality

Abstract

This chapter is devoted to a discussion of quantum phase transitions in regularly alternating spin-1/2 Ising chain in a transverse field. After recalling some generally-known topics of the classical (temperature-driven) phase transition theory and some basic concepts of the quantum phase transition theory I pass to the statistical mechanics calculations for a one-dimensional spin-1/2 Ising model in a transverse field, which is the simplest possible system exhibiting the continuous quantum phase transition. The essential tool for these calculations is the Jordan-Wigner fermionization. The latter technique being completed by the continued fraction approach permits to obtain analytically the thermodynamic quantities for a `slightly complicated' model in which the intersite exchange interactions and on-site fields vary regularly along a chain. Rigorous analytical results for the ground-state and thermodynamic quantities, as well as exact numerical data for the spin correlations computed for long chains (up to a few thousand sites) demonstrate how the regularly alternating bonds/fields effect the quantum phase transition. I discuss in detail the case of period 2, swiftly sketch the case of period 3 and finally summarize emphasizing the effects of periodically modulated Hamiltonian parameters on quantum phase transitions in the transverse Ising chain and in some related models.