Rigorous lower bound on dynamical exponents in gapless frustration-free systems

TL;DR

This paper establishes a universal lower bound of 2 on the dynamical exponent in frustration-free quantum systems with power-law correlations, using the Gosset-Huang inequality, and extends the result to classical stochastic processes.

Contribution

It provides a rigorous, universal lower bound on the dynamical exponent for a broad class of frustration-free systems, including classical stochastic processes, using a unified mathematical framework.

Findings

Proves z ≥ 2 for frustration-free quantum systems with power-law correlations.

Extends the bound to classical Markov processes via a quantum-classical mapping.

Applicable across various lattice structures and dimensions, independent of boundary conditions.

Abstract

This work rigorously establishes a universal lower bound $z\ge2$ for the dynamical exponent in frustration-free quantum many-body systems whose ground states exhibit power-law decaying correlations. The derivation relies on the Gosset-Huang inequality, providing a unified framework applicable across various lattice structures and spatial dimensions, independent of specific boundary conditions. Remarkably, our result can be applied to prove new bounds for dynamics of classical stochastic processes. Specifically, we utilize a well-established mapping from the time evolution of local Markov processes with detailed balance to that of frustration-free quantum Hamiltonians, known as Rokhsar-Kivelson Hamiltonians. This proves $z \ge 2$ for such Markov processes, which is an improvement over existing bounds. Beyond these applications, the quantum analysis of the $z\ge2$ bound is further broadened to include systems exhibiting hidden correlations, which may not be evident from purely local operators.