Rigorous Lower Bound on Dynamical Exponents in Gapless Frustration-Free Systems

This work rigorously establishes a universal lower bound <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>z</a:mi> <a:mo>≥</a:mo> <a:mn>2</a:mn> </a:math> 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 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>z</c:mi> <c:mo>≥</c:mo> <c:mn>2</c:mn> </c:math> for such Markov processes, which is an improvement over existing bounds. Beyond these applications, the quantum analysis of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>z</e:mi> <e:mo>≥</e:mo> <e:mn>2</e:mn> </e:math> bound is further broadened to include systems exhibiting hidden correlations, which may not be evident from purely local operators.