Proliferation of effective interactions: decoherence-induced equilibration in a closed many-body system

TL;DR

This paper investigates how weak perturbations in large many-body quantum systems lead to decoherence and irreversibility, emphasizing the proliferation of effective interactions and their role in polarization equilibration and Loschmidt echo decay.

Contribution

It demonstrates that proliferated effective interactions cause Loschmidt echo decay governed by an effective Fermi golden rule in large systems, revealing a fundamental mechanism for irreversibility.

Findings

Loschmidt echo decay is dominated by proliferated effective interactions.

The lower bound for perturbation strength shrinks with system size.

Polarization remains equilibrated despite reversal when decay exceeds FGR time.

Abstract

We address the question on how weak perturbations, that are quite ineffective in small many-body systems, can lead to decoherence and hence to irreversibility when they proliferate as the system size increases. This question is at the heart of solid state NMR. There, an initially local polarization spreads all over due to spin-spin interactions that conserve the total spin projection, leading to an equilibration of the polarization. In principle, this quantum dynamics can be reversed by changing the sign of the Hamiltonian. However, the reversal is usually perturbed by non reversible interactions that act as a decoherence source. The fraction of the local excitation recovered defines the Loschmidt echo (LE), here evaluated in a series of closed $N$ spin systems with all-to-all interactions. The most remarkable regime of the LE decay occurs when the perturbation induces proliferated effective interactions. We show that if this perturbation exceeds some lower bound, the decay is ruled by an effective Fermi golden rule (FGR). Such a lower bound shrinks as $ N $ increases, becoming the leading mechanism for LE decay in the thermodynamic limit. Once the polarization stayed equilibrated longer than the FGR time, it remains equilibrated in spite of the reversal procedure.