[Paper Review] Impact of local integrals of motion to metastable non-equilibrium states

Peter Barmettler, Corinna Kollath|arXiv (Cornell University)|Dec 19, 2013

Quantum many-body systems3 citations

TL;DR

This paper introduces a non-equilibrium Mazur-type approach using local integrals of motion to predict long-time correlation functions in integrable quantum systems after a quench. It shows that thermal-like behavior emerges at strong interactions due to negligible overlaps with conserved quantities, while non-thermal quasi-stationary states appear at weaker interactions—accurately captured by projecting onto just the first few integrals of motion, enabling efficient computation via equilibrium methods like DMRG.

ABSTRACT

We analyse the stationary behaviour of correlations in a strongly correlated Bose gas out of equilibrium. The dynamics are triggered by a quench of the interaction starting from the strongly interacting limit where the system is in a perfect Mott state. Despite the complete integrability of our theoretical description, we find seemingly thermal behaviour for the experimentally measurable correlations at large interactions. Quite opposed, away from the strongly interacting regime these correlation functions show highly non-thermal stationary values. This behaviour is explained by overlaps of the integrals of motion with the observable and the initial state in an effective thermal ensemble. These results suggest that non-equilibrium Mazur equalities are an efficient way to calculate short range correlations for arbitrary integrable models.

Motivation & Objective

• To understand the emergence of metastable non-thermal states in integrable quantum systems after a quench, particularly in strongly correlated ultracold atoms.
• To develop a practical method for computing long-time correlation functions in integrable models beyond quadratic systems.
• To explain the crossover from thermal to non-thermal behavior in terms of overlaps between observables, initial states, and local integrals of motion.
• To demonstrate that a truncated series of projections onto conserved quantities can accurately approximate non-equilibrium steady states without full time evolution.

Proposed method

• Adapt the non-equilibrium Mazur equality to approximate long-time expectation values of local observables using overlaps with local integrals of motion.
• Use the generalized Gibbs ensemble (GGE) framework but replace full GGE averaging with a truncated series of projections onto the most relevant conserved quantities.
• Compute matrix elements of observables and integrals of motion using equilibrium methods such as DMRG and exact diagonalization.
• Define a modified Mazur equation (Eq. 9) that measures the deviation of diagonal ensemble results from an approximate ensemble, using only the first few integrals of motion.
• Focus on parity-parity correlations in the Bose-Hubbard model after an interaction quench, with the first non-trivial integral of motion (Eq. 10) capturing most of the non-thermal deviation.
• Validate the method by comparing predictions with time-dependent DMRG simulations and experimental data from Cheneau et al. (2012).

Experimental results

Research questions

• RQ1 Why do strongly interacting Bose gases exhibit thermal-like correlation functions after a quench, despite being integrable?
• RQ2 What causes the transition from thermal to non-thermal behavior in correlation functions as interaction strength is reduced?
• RQ3 Can a few local integrals of motion accurately describe non-equilibrium steady states in non-quadratic integrable models?
• RQ4 How do overlaps between observables, initial states, and conserved quantities determine the long-time behavior of correlations?
• RQ5 To what extent can equilibrium methods like DMRG be used to predict non-equilibrium steady-state properties via a truncated Mazur series?

Key findings

• At strong interactions (large U), parity-parity correlations exhibit thermal-like behavior due to negligible overlaps between the observable and the first non-trivial integral of motion.
• At weaker interactions, the first integral of motion (a fermionic two-point correlator) dominates the deviation from thermal behavior, explaining the emergence of non-thermal quasi-stationary states.
• The first-order contribution in the Mazur expansion (Eq. 10) captures nearly all the deviation from the extended ensemble for short distances (d ≤ 3), with higher-order terms becoming relevant only at d ≥ 4.
• The method achieves good agreement with time-dependent DMRG and experimental data, validating its use for long-time predictions without direct time evolution.
• The approach is efficient and extendable to non-quadratic integrable models, such as the 1D Hubbard model, where the energy current operator plays a similar role to the first integral of motion.
• The results show that the standard $J^2/U$ perturbation theory is insufficient to describe the observed non-thermal behavior, indicating the need for exact treatment of conserved quantities.