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

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.

💡 Research Summary

This paper establishes a rigorous universal lower bound of z ≥ 2 for the dynamical exponent in gapless, frustration-free (FF) quantum many-body systems whose ground states exhibit power-law decaying correlations.

The core achievement is a mathematical proof that circumvents limitations of previous partial proofs, which often relied on open boundary conditions. The derivation is unified and independent of specific lattice structures, spatial dimensions, or boundary conditions (including periodic boundaries), making it widely applicable.

The proof leverages the Gosset-Huang inequality, a key tool for FF systems. This inequality bounds the connected correlation function between two local operators by an expression that decays exponentially with the graph distance D between them, but with a decay rate proportional to the square root of the dimensionless spectral gap, √ε. Assuming the ground state has power-law correlations, the authors show that if z were less than 2, the gap would scale as ε ~ L^{-z} with system size L. For large L, this would make √ε * L vanish, forcing correlations at distance L to decay faster than any power law—contradicting the initial assumption. Therefore, z cannot be less than 2.

This result has profound implications. First, it constrains the low-energy physics of FF systems, indicating they cannot describe Lorentz-invariant (z=1) conformal quantum critical points. Instead, they are restricted to non-relativistic critical points with z ≥ 2. This quadratic or softer dispersion has known consequences, such as enabling spontaneous continuous symmetry breaking in 1+1 dimensions.

A major application of the bound is to classical stochastic processes. Using a well-established mapping, the transition rate matrix of a local Markov process satisfying detailed balance (e.g., Glauber dynamics for the Ising model) can be mapped to an FF quantum Hamiltonian known as a Rokhsar-Kivelson (RK) Hamiltonian. Consequently, the quantum bound z ≥ 2 directly implies the same lower bound for the dynamical exponent of these classical processes. This provides a rigorous “no-go” theorem for the performance of standard local-update Markov Chain Monte Carlo (MCMC) algorithms like Gibbs sampling or Metropolis-Hastings at criticality: under the conditions of detailed balance and local updates, one can never achieve z < 2. This theoretical foundation explains why faster algorithms require breaking detailed balance or employing non-local updates.

The paper further extends the analysis to systems with “hidden correlations.” These are systems where correlations may not be evident from local operators but arise from non-local excitations with localized energy density. The authors demonstrate that even in such cases, if power-law correlations (in system size) exist for appropriate operators, the z ≥ 2 bound continues to hold, showcasing the robustness and generality of the fundamental connection between the FF condition, slow correlation decay, and a large dynamical exponent.

In summary, this work provides a rigorous and general foundation for the observed atypical dynamical exponents in gapless FF systems, with significant consequences for understanding quantum criticality and the fundamental limits of classical stochastic simulation algorithms.