[Paper Review] Translation symmetry restoration under random unitary dynamics

Katja Klobas, Colin Rylands+1 more|arXiv (Cornell University)|Jun 6, 2024

Fractal and DNA sequence analysisBiochemistry, Genetics and Molecular Biology3 citations

TL;DR

This paper extends the observable-independent characterization of symmetry restoration to space-time symmetries, specifically translation symmetry, in generic quantum many-body systems. Using random unitary circuits, it shows that translation symmetry restoration occurs only on timescales linear in subsystem size, with initial-state-independent times for large subsystems and a quantum Mpemba effect for intermediate sizes, where greater initial symmetry breaking accelerates restoration.

ABSTRACT

The finite parts of a large, locally interacting many-body system prepared out-of-equilibrium eventually equilibrate. Characterising the underlying mechanisms of this process and its timescales, however, is particularly hard as it requires to decouple universal features from observable-specific ones. Recently, new insight came by studying how certain symmetries of the dynamics that are broken by the initial state are restored at the level of the reduced state of a given subsystem. This provides a high level, observable-independent probe. Until now this idea has been applied to the restoration of internal symmetries, e.g. U(1) symmetries related to charge conservation. Here we show that that the same logic can be applied to the restoration of space-time symmetries, and hence can be used to characterise the relaxation of fully generic systems. We illustrate this idea by considering the paradigmatic example of "generic" many-body dynamics, i.e. a local random unitary circuit, where our method leads to exact results. We show that the restoration of translation symmetry in these systems only happens on time-scales proportional to the subsystem's volume. In fact, for large enough subsystems the time of symmetry restoration becomes initial-state independent (as long as the latter breaks the symmetry at time zero) and coincides with the thermalisation time. For intermediate subsystems, however, one can observe the so-called "quantum Mpemba effect", where the state of the system restores a symmetry faster if it is initially more asymmetric. We provide the first exact characterisation of this effect in a non-integrable system.

Motivation & Objective

• To extend the observable-independent framework of symmetry restoration from internal symmetries (e.g., U(1)) to space-time symmetries like translation invariance.
• To characterize how translation symmetry is restored in generic, locally interacting quantum systems under random unitary dynamics.
• To investigate whether the quantum Mpemba effect—faster symmetry restoration for more broken initial states—emerges in spatial symmetry restoration.
• To determine the dependence of symmetry restoration time on subsystem size and initial state asymmetry in random unitary circuits.

Proposed method

• Define a distance measure between the reduced density matrix of a subsystem and its symmetrized version over the translation group, quantifying symmetry breaking.
• Use the Frobenius norm of the difference between the evolved state and its translated counterparts to track symmetry restoration dynamics.
• Apply the formalism to random unitary circuits with local Haar-random unitaries, which lack internal symmetries but preserve averaged space-translation invariance.
• Construct initial states that break translation symmetry explicitly, such as spin helices with periodicity ν, to probe the effect of initial asymmetry.
• Compute the time evolution of the symmetry-breaking distance using trace expressions involving shifted density matrices, leveraging rotational invariance in the spin-helix ansatz.
• Analyze the scaling of restoration time with subsystem size ℓ and local Hilbert space dimension q, showing collapse of curves under appropriate rescaling.

Experimental results

Research questions

• RQ1 Does the observable-independent symmetry restoration framework apply to spatial symmetries like translation invariance in generic quantum systems?
• RQ2 What is the timescale for translation symmetry restoration in random unitary circuits, and how does it depend on subsystem size?
• RQ3 Can the quantum Mpemba effect—faster restoration for more broken initial states—occur in the context of spatial symmetry breaking?
• RQ4 How does the local Hilbert space dimension q influence the dynamics of translation symmetry restoration?
• RQ5 Is the symmetry restoration time independent of the initial state for large subsystems, and does it coincide with the thermalization time?

Key findings

• For large subsystems, the time to achieve ε-approximate translation symmetry restoration scales linearly with subsystem size ℓ and becomes independent of the initial state, as long as the initial state breaks the symmetry.
• The symmetry restoration time for large subsystems coincides with the thermalization time, indicating that memory of initial state is erased only at thermalization scales.
• For intermediate subsystems, the quantum Mpemba effect is observed: initial states with greater initial translational symmetry breaking restore symmetry faster.
• The symmetry restoration dynamics collapse onto a single universal curve when rescaled by ν/(ν−1), indicating that the extent of initial symmetry breaking (ν) only rescales the effect without altering the phenomenology.
• As the local Hilbert space dimension q increases, the symmetry restoration approaches a Heaviside step function at t=ℓ/2, indicating faster and sharper restoration in the large-q limit.
• The maximal value of the symmetry-breaking distance is (ν−1)/ν, achieved when all shifted states are orthogonal to the original state at t=0, confirming the role of initial state structure in determining restoration dynamics.