[Paper Review] Power-law relaxation behavior of an initially localized state in the spin-1/2 Heisenberg chain

Tetsuo Deguchi, Pulak Ranjan Giri+1 more|arXiv (Cornell University)|Jul 27, 2015

Quantum many-body systems1 references3 citations

TL;DR

This paper presents exact time evolution of local magnetizations in the spin-1/2 Heisenberg chain starting from an initially localized state, using determinant formulas for form factors. It demonstrates power-law relaxation of squared deviations of local magnetizations over long times, indicating no definite relaxation time, while fidelity decays exponentially on the Boltzmann time scale, suggesting universal power-law decay for local operators in integrable systems.

ABSTRACT

We present power-law relaxation behavior of the local magnetizations in the equilibration dynamics of the spin-1/2 Heisenberg spin chain as an isolated integrable quantum system. We perform the exact time evolution of the expectation values of the local spin operators by evaluating them with the determinant formula of the form factors. We construct such an initial quantum state that has a localized profile of the local magnetizations, and perform the exact time evolution over a very long period of time. We show that the local magnetization relaxes as some power of the time variable with no definite time scale, while the fidelity relaxes very fast with its relaxation time being proportional to the inverse of the energy width, i.e. the Boltzmann time.

Motivation & Objective

• To investigate the non-equilibrium relaxation dynamics of local observables in an isolated, integrable quantum system.
• To determine whether local magnetizations relax to equilibrium values and with what time dependence.
• To examine the validity of the Generalized Gibbs Ensemble (GGE) conjecture in a finite-size, exact setting.
• To explore the universality of relaxation behavior for local operators in interacting integrable systems.

Proposed method

• Exact time evolution of local spin operators is computed using determinant formulas for form factors in the algebraic Bethe ansatz framework.
• An initial state with a localized magnetization profile is constructed using spinon excitations, particularly the all-spinon state.
• The time evolution is simulated over long timescales to observe the relaxation dynamics of local magnetizations.
• Square deviations of local magnetizations from their spatial average are computed to quantify relaxation behavior.
• The fidelity between the initial state and time-evolved state is evaluated to assess relaxation timescale.
• Power-law fits are applied to the time evolution of squared deviations to extract decay exponents.

Experimental results

Research questions

• RQ1 Does the expectation value of a local operator in an isolated integrable quantum system relax to a constant value over time?
• RQ2 What is the functional form of relaxation for local magnetizations in the spin-1/2 Heisenberg chain?
• RQ3 Is there a characteristic relaxation time for local observables, or does relaxation occur via power-law decay?
• RQ4 How does the relaxation behavior depend on the initial quantum state, particularly for different spinon configurations?
• RQ5 Can power-law relaxation be considered universal for local operators in interacting integrable systems?

Key findings

• The squared deviations of local magnetizations exhibit power-law decay as a function of time, with exponents around 0.68 and 0.89 for N=50.
• For N=30, the decay follows approximately 0.0289/t^1.02 initially, then transitions to a slower decay of 0.0765/t^0.806.
• The relaxation of the fidelity occurs on the Boltzmann time scale, proportional to the inverse of the energy width.
• The time evolution shows a two-stage process: initial separation into counter-propagating waves, followed by collision and merging into a uniformly oscillating profile.
• The power-law decay of squared deviations indicates no definite relaxation time, suggesting persistent oscillations with decaying amplitudes.
• The decay exponent depends on the initial state, with the yrast state showing an exponent close to zero, indicating slower relaxation.