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arXiv:1707.02400 · Journal of Statistical Mechanics Theory and Experiment

The boundary effects of transverse field Ising model

Yan He, Hao Guo

Published Sep 1, 2017• 18 citations
arXivHTMLPDF

Abstract

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 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.

DOI
10.1088/1742-5468/aa85b0

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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 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.
When studying the quantum Ising chain in a transverse field, we usually transform the spin operators to fermion operators by applying the Jordan-Wigner transformation,i.e.we take it for granted that the spin model and corresponding fermionic model are equivalent naively. But deliberate treatment could reveal that the equivalence accompanies some delicate conditions.Redundant degrees of freedom should be projected out carefully, especially for finite length of chain. In this paper, we compare the two models in detail so as to find the conditions of equivalence of them,including the interaction, the boundary conditions, and odevity of the number of lattice sites.We also exemplify in detail how to project out the redundant degrees of freedom for the fermionic representation.
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.
We study a finite spin-1/2 Ising chain with a spatially alternating transverse field of period 2. By means of a Jordan-Wigner transformation for even and odd sites, we are able to map it into a one-dimensional model of free fermions. We determine the ground-state energies in the positive- and negative-parity subspaces (subspaces with an even or odd total number of down spins, respectively) and compare them in order to establish the ground-state energy for the entire Hamiltonian. We derive closed-form expressions for this energy gap between the different parity subspaces and analyze its behavior and dependence on the system size in the various regimes of the applied field. Finally, we suggest an expression for the correlation length of such a model that is consistent with the various values found in the literature for its behavior in the vicinity of critical points.
The inhomogeneous transverse field Ising models mainly impurity based and the joint chain are analysed analytically using Jordan-Wigner transformations. The effects of inhomogeneities on the phase transition have been discussed in detail. We constructed an ansatz to diagonalize the two models which are taken into consideration. The inhomogeneity is quantified by a coupling parameter, which can be tuned to control the occurrence of quantum phase transition in these models. We have shown a systematic setup using which we can generalise the solution to a system with an arbitrary number of impurity sites and junctions, which are separated by at least two lattice sites. We have analysed the quantum critical point by calculating the correlation functions, transverse magnetization and the gap between the ground state and the first excited state.
Quantum phase transitions in the one-dimensional extended quantum compass model in a transverse field are studied by using the Jordan-Wigner transformation. This model is always gapful, except at the critical surfaces where the energy gap disappears. We obtain the analytic expressions of all critical fields which drive quantum phase transitions. This model shows a rich phase diagram which includes spin-flop, strip antiferromagnetic, and saturated ferromagnetic phases in addition to the phase with antiparallel ordering of the yyy spin component on odd bonds. We also study also the universality and scaling properties of the transverse susceptibility and nearest-neighbor correlation function derivatives in different regions to confirm the results obtained using the energy gap analysis.
Phase transitions in Ising models are a broadly studied subject within physics.In this thesis we will study the quantum mechanical transverse-field Ising model in order to determine analytical critical points for quantum phase transitions.This will be done by introducing the concept of fermionic operators and the Jordan-Wigner transformation.Hereafter it is possible to Fourier transform the transverse Ising Hamiltonian and derive an exact eigenenergy spectrum.With these eigenenergies it is possible to derive the condition(s) for which a phase transition can occur.This is done by looking when the eigenenergy solutions are equal to zero.This causes the system to become unstable and thus indicates a point of phase transition.Further in this thesis we will cover two different variations of the transverse-field Ising model which are the di-atomic transverse Ising chain and the frustrated Ising chain.The frustrated Ising chain will be a modification of the mono-atomic Ising chain where there is an extra interaction term introduced which will require an assumption of lowest order approximation of perturbation theory to solve analytically.I would like to thank Lars Fritz for the great learning experience as a physicist within my bachelor thesis.