[Paper Review] Ising Quantum Chains

Dragi Karevski|arXiv (Cornell University)|Nov 13, 2006

Theoretical and Computational Physics11 references3 citations

TL;DR

This paper provides a comprehensive review of free-fermionic quantum spin chains, focusing on exact solutions via Jordan-Wigner transformation and canonical diagonalization. It establishes a method to reconstruct the full zero-temperature quantum phase diagram from surface magnetization, demonstrates aging in non-equilibrium dynamics of homogeneous systems, and analyzes critical behavior under aperiodic and disordered perturbations, including Lévy-type disorder and inhomogeneous disorder.

ABSTRACT

The aim of this article is to give a pedagogical introduction to the exact equilibrium and nonequilibrium properties of free fermionic quantum spin chains. In a first part we present in full details the canonical diagonalisation procedure and review quickly the equilibrium dynamical properties. The phase diagram is analysed and possible phase transitions are discussed. The two next chapters are concerned with the effect of aperiodicity and quenched disorder on the critical properties of the quantum chain. The remaining part is devoted to the nonequilibrium dynamical behaviour of such quantum chains relaxing from a nonequilibrium pure initial state. In particular, a special attention is made on the relaxation of transverse magnetization. Two-time linear response functions and correlation functions are also considered, giving insights on the nature of the final nonequilibrium stationnary state. The possibility of aging is also discussed.

Motivation & Objective

• To systematically present the canonical diagonalization method for free-fermionic quantum spin chains, emphasizing analytical tractability and exact solutions.
• To demonstrate how surface magnetization can be used to infer the full zero-temperature quantum phase diagram, offering a low-cost alternative to bulk calculations.
• To analyze non-equilibrium relaxation dynamics, particularly transverse magnetization and two-time correlation functions, revealing aging phenomena in deterministic quantum systems.
• To investigate the effects of aperiodic modulations and quenched disorder (including Lévy-distributed couplings and inhomogeneous disorder) on critical behavior and universality.
• To compare conserved and non-conserved dynamics in the XX and Ising chains, showing similarities in long-time relaxation despite different symmetries.

Proposed method

• Use of the Jordan-Wigner transformation to map spin operators onto non-interacting fermions, enabling exact diagonalization of the Hamiltonian.
• Application of Wick’s theorem and fermionic contractions to compute time-ordered correlation functions, especially transverse spin correlations.
• Derivation of Heisenberg equations of motion for fermionic operators to study non-equilibrium time evolution and relaxation dynamics.
• Employment of a decimation-like renormalization group (RG) approach for random transverse Ising chains, particularly effective near critical points with broad energy scale distributions.
• Development of a relevance criterion for aperiodic modulations based on scaling behavior and anisotropic critical exponents.
• Analytical treatment of two-time correlation functions to detect aging, defined by explicit dependence on both waiting time and measurement time.

Experimental results

Research questions

• RQ1 Can the full zero-temperature quantum phase diagram of free-fermionic spin chains be reconstructed from surface magnetization alone?
• RQ2 How does aging manifest in non-equilibrium dynamics of homogeneous, deterministic quantum spin chains with no quenching or noise?
• RQ3 What is the impact of aperiodic modulation on critical behavior, and does weak universality persist in such systems?
• RQ4 How does Lévy-type disorder or inhomogeneous disorder alter the critical properties of the random transverse Ising chain compared to Gaussian disorder?
• RQ5 To what extent do systems with conserved (XX chain) and non-conserved (Ising chain) dynamics exhibit similar relaxation behavior in the critical regime?

Key findings

• The surface magnetization provides a complete and computationally efficient route to determine the entire zero-temperature phase diagram of free-fermionic spin chains.
• Transverse spin correlations decay algebraically in time at zero temperature and exhibit Gaussian decay at infinite temperature, with exponential decay modulated by power laws in the intermediate regime.
• Aging is observed in the two-time correlation functions of the critical Ising and XX chains, with explicit dependence on both waiting time and measurement time, even in the absence of disorder.
• For the random transverse Ising chain with Lévy-distributed couplings, the broad energy scale distribution supports a strong disorder fixed point, leading to non-universal critical exponents.
• Inhomogeneous disorder with power-law spatial variation leads to marginal or relevant perturbations that alter critical behavior, with distinct surface magnetization scaling compared to homogeneous disorder.
• The long-time relaxation of transverse magnetization in both the XX and Ising chains is described by a convolution of the initial state with an analytically derived response kernel, indicating a conserved-like relaxation process in the Ising model despite non-conserved dynamics.