[Paper Review] Relaxation dynamics of conserved quantities in a weakly non-integrable one-dimensional Bose gas

Giuseppe Piero Brandino, Jean-Sébastien Caux+1 more|arXiv (Cornell University)|Jan 2, 2013

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This study investigates relaxation dynamics of conserved quantities in a weakly non-integrable one-dimensional Bose gas using the Lieb-Liniger model with a quench into a cosine potential that breaks integrability. By combining Bethe ansatz and numerical renormalization group methods, it finds that conserved charges oscillate with a frequency locked to the cosine potential's frequency, independent of its amplitude, indicating a smooth crossover rather than a sharp thermalization time scale in integrability-breaking dynamics.

ABSTRACT

In this work we report preliminary results on the relaxational dynamics of one dimensional Bose gases, as described by the Lieb-Liniger model, upon release from a parabolic trap. We explore the effects of integrability and integrability breaking upon these dynamics by placing the gas post-release in an integrability breaking one-body cosine potential of variable amplitude. By studying the post-quench evolution of the conserved charges that would exist in the purely integrable limit, we begin to quantify the effects of the weak breaking of integrability on the long time thermalization of the gas.

Motivation & Objective

• To understand how weak integrability breaking affects long-time relaxation dynamics in a one-dimensional interacting Bose gas.
• To determine whether integrability-breaking perturbations lead to a sharp thermalization time scale or a smooth crossover between generalized Gibbs ensemble and canonical ensemble behavior.
• To quantify the post-quench evolution of conserved charges in the presence of a one-body cosine potential that breaks integrability.
• To explore the dependence of charge oscillation frequencies and amplitudes on the strength of the integrability-breaking potential.

Proposed method

• Use a quantum quench protocol: prepare the system in the ground state of the Lieb-Liniger Hamiltonian with a parabolic trap, then remove the trap and replace it with a cosine potential at t=0.
• Employ the Bethe ansatz to compute matrix elements and eigenstates of the Lieb-Liniger model, enabling exact calculation of conserved charges in the integrable limit.
• Apply the numerical renormalization group (NRG) method to handle perturbations from the cosine potential, using Lieb-Liniger eigenstates as a computational basis to include strong correlations from the start.
• Compute time evolution of conserved charges Q₄, Q₆, Q₁₀ via their matrix elements in the Lieb-Liniger basis, using Lehmann representations.
• Analyze oscillation frequencies and amplitudes of charges as functions of the cosine potential amplitude A, focusing on the transition from integrable to non-integrable dynamics.
• Use the fact that odd charges vanish due to parity symmetry, focusing analysis on even charges Q₂ₙ to simplify the dynamics.

Experimental results

Research questions

• RQ1 Does integrability breaking lead to a sharp thermalization time scale τ_IB, or a smooth crossover between generalized Gibbs ensemble and canonical ensemble behavior?
• RQ2 How do the oscillation frequencies of conserved charges depend on the amplitude A of the integrability-breaking cosine potential?
• RQ3 How do the amplitudes of charge oscillations scale with the strength A of the cosine potential?
• RQ4 What is the behavior of the mean value of conserved charges as a function of A, and does it vary smoothly from the integrable limit?
• RQ5 Is the frequency of charge oscillations locked to the frequency of the cosine potential, regardless of A?

Key findings

• The oscillation frequencies of conserved charges Q₄, Q₆, and Q₁₀ are independent of the amplitude A of the cosine potential and are instead locked to the frequency of the cosine potential itself (~1 in units used).
• The amplitudes of charge oscillations increase linearly with A, approaching zero as A → 0, consistent with the expectation that charges become constants of motion in the integrable limit.
• The mean value of each conserved charge increases smoothly with A, indicating a continuous transition from integrable to non-integrable dynamics.
• Transients in the early-time evolution of charges decay, followed by a stable, periodic oscillatory regime, suggesting a robust dynamical response to integrability breaking.
• The absence of a dependence of oscillation frequency on A implies that even low-frequency, weakly breaking potentials can sustain long-lived coherent dynamics, challenging the existence of a universal τ_IB.
• The results support a smooth interpolation between generalized Gibbs ensemble and canonical ensemble behavior, consistent with recent theoretical proposals (yur_ols; kollar; canovi), rather than a sharp transition.