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

Rintaro Masaoka, Tomohiro Soejima, Haruki Watanabe

TL;DR

The paper proves a universal lower bound z ≥ 2 on the dynamical exponent for gapless frustration-free systems with power-law ground-state correlations, using the Gosset-Huang inequality to connect ground-state correlations to finite-size scaling of the spectral gap. It unifies quantum and classical dynamics by applying the RK Hamiltonian framework to Markov processes with detailed balance, establishing z ≥ 2 for critical RK Hamiltonians and corresponding stochastic dynamics. The authors extend the GH framework to excitations with localized energy density (hidden correlations), showing z ≥ 2 in broader settings and providing concrete examples such as the kinetic Ising model and boundary-augmented XX models. Together, these results constrain low-energy dynamics in FF systems, highlight the role of ground-state correlations, and suggest limitations on achieving fast stochastic dynamics under locality and detailed balance. The work also clarifies the relationship between quantum critical points, conformal symmetry, and dynamical scaling, indicating that FF gapless critical points cannot be Lorentz invariant with z = 1.

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.

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

TL;DR

The paper proves a universal lower bound z ≥ 2 on the dynamical exponent for gapless frustration-free systems with power-law ground-state correlations, using the Gosset-Huang inequality to connect ground-state correlations to finite-size scaling of the spectral gap. It unifies quantum and classical dynamics by applying the RK Hamiltonian framework to Markov processes with detailed balance, establishing z ≥ 2 for critical RK Hamiltonians and corresponding stochastic dynamics. The authors extend the GH framework to excitations with localized energy density (hidden correlations), showing z ≥ 2 in broader settings and providing concrete examples such as the kinetic Ising model and boundary-augmented XX models. Together, these results constrain low-energy dynamics in FF systems, highlight the role of ground-state correlations, and suggest limitations on achieving fast stochastic dynamics under locality and detailed balance. The work also clarifies the relationship between quantum critical points, conformal symmetry, and dynamical scaling, indicating that FF gapless critical points cannot be Lorentz invariant with z = 1.

Abstract

This work rigorously establishes a universal lower bound 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 for such Markov processes, which is an improvement over existing bounds. Beyond these applications, the quantum analysis of the bound is further broadened to include systems exhibiting hidden correlations, which may not be evident from purely local operators.
Paper Structure (43 sections, 7 theorems, 177 equations, 10 figures, 3 tables)

This paper contains 43 sections, 7 theorems, 177 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. $z \ge 2$ for FF systems with power-law decaying correlation functions
  3. Definitions
  4. Gosset-Huang inequality
  5. Interaction graph
  6. Formulation of the Gosset-Huang inequality
  7. A heuristic proof of $z\ge 2$
  8. A proof of $z \ge 2$ for FF systems with algebraic correlation functions
  9. Example: $XX$ model with fine-tuned magnetic field
  10. Rokhsar-Kivelson Hamiltonians and Markov processes
  11. Review of construction of RK Hamiltonians from Markov processes
  12. Markov processes
  13. Rokhsar-Kivelson Hamiltonians
  14. Critical slowing down and dynamical exponent
  15. Markov chain Monte Carlo methods
  16. ...and 28 more sections

Key Result

Theorem 1

Assume there exist local operators $\hat{\mathcal{O}}$ and $\hat{\mathcal{O}}'$ such that their graph distance satisfies $D(\hat{\mathcal{O}}, \hat{\mathcal{O}}') \ge AL$ for some constant $A>0$, and where $\hat{G}$ denotes the projector onto the ground space of $\hat{H}$, and $\alpha \ge 0$ and $B > 0$ are constants independent of $L$. Then, for sufficiently large $L$, the spectral gap $\epsilon

Figures (10)

  • Figure 1:
  • Figure 2:
  • Figure 4: Correspondence between an RK Hamiltonian and a Markov process. The $d$-dimensional ground state corresponds to the $d$-dimensional Boltzmann distribution as in Eq. \ref{['correspondence of expectation']}. The Hamiltonian and the transition-rate matrix of a Markov process are connected by Eq. \ref{['def of local transition rate']}
  • Figure 5:
  • Figure 6:
  • ...and 5 more figures

Theorems & Definitions (15)

  • Definition 1: Frustration-free Hamiltonians
  • Definition 2: Spectral gap
  • Definition 3: dynamical exponent
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • proof
  • ...and 5 more