This work rigorously establishes a universal lower bound z ≥ 2 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 z ≥ 2 for such Markov processes, which is an improvement over existing bounds. Beyond these applications, the quantum analysis of the z ≥ 2 bound is further broadened to include systems exhibiting hidden correlations, which may not be evident from purely local operators.
In this work we study the Glauber dynamics of the one-dimensional Ising model with nearest-neighbor and next-nearest-neighbor interactions, for which an approximate solution of the magnetization per site is obtained. When the dynamical critical exponent z is investigated following the treatment of Cordery, Sarker, and Tobochnik [Phys. Rev. B 24, 5402 (1981)], our observation shows that its upper-bound value is the same as the known value, thus implying that z is independent of the range of the interaction. We also suggest a high-temperature expansion approximation which is then used to solve the two-dimensional Glauber dynamics governed by a master equation; this solution is compared with that of the decoupling method. The time-delayed correlation function is also calculated.
We present a new lower bound on the spectral gap of the Glauber dynamics for the Gibbs distribution of a spectrally independent
q-spin system on a graph
G=(V,E) with maximum degree
Δ. Notably, for several interesting examples, our bound covers the entire regime of
Δ excluded by arguments based on coupling with the stationary distribution. As concrete applications, by combining our new lower bound with known spectral independence computations and known coupling arguments: (1) We show that for a triangle-free graph
G=(V,E) with maximum degree
Δ≥3, the Glauber dynamics for the uniform distribution on proper
k-colorings with
k≥(1.763⋯+δ)Δ colors has spectral gap
Ω~δ(∣V∣−1). Previously, such a result was known either if the girth of
G is at least
5 [Dyer et.~al, FOCS 2004], or under restrictions on
Δ [Chen et.~al, STOC 2021; Hayes-Vigoda, FOCS 2003]. (2) We show that for a regular graph
G=(V,E) with degree
Δ≥3 and girth at least
6, and for any
ε,δ>0, the partition function of the hardcore model with fugacity
λ≤(1−δ)λc(Δ) may be approximated within a
(1+ε)-multiplicative factor in time
O~δ(n2ε−2). Previously, such a result was known if the girth is at least
7 [Efthymiou et.~al, SICOMP 2019]. (3) We show for the binomial random graph
G(n,d/n) with
d=O(1), with high probability, an approximately uniformly random matching may be sampled in time
Od(n2+o(1)). This improves the corresponding running time of
O~d(n3) due to [Jerrum-Sinclair, SICOMP 1989; Jerrum, 2003].
We obtain the exact critical relaxation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}$(\ensuremath{\xi}), where \ensuremath{\xi} is the bulk correlation length, for the Glauber kinetic Ising model of spins on a one-dimensional lattice of finite length L for both periodic and free boundary conditions (BC's). We show that, independent of the BC's, the dynamic critical exponent has the well-known value z=2, and we comment on a recent claim that z=1 for this model. The ratio ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}$(\ensuremath{\xi})/${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\infty}}}$(\ensuremath{\xi}), in the double limit L,\ensuremath{\xi}\ensuremath{\rightarrow}\ensuremath{\infty} for fixed x=L/\ensuremath{\xi}, approaches a limiting functional form, ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(L/\ensuremath{\xi}), the finite-size scaling function. For free BC's we derive the exact scaling function ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)=[1+(\ensuremath{\omega}(x)/x
)2${]}^{\mathrm{\ensuremath{-}}1}$, where \ensuremath{\omega}(x) is the smallest root of the transcendental equation \ensuremath{\omega}tan(\ensuremath{\omega}/2)=x. We provide expansions of \ensuremath{\omega}(x) in powers of x and ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ for the regimes of small and large x, respectively, and establish their radii of convergence. The scaling function shows anomalous behavior at small x, ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)\ensuremath{\approxeq}x, instead of the usual ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)\ensuremath{\approxeq}
xz, as x\ensuremath{\rightarrow}0. This is because, even for finite L, the lifetime of the slowest dynamical mode diverges for T\ensuremath{\rightarrow} 0 K. For periodic BC's, with the exception of one system, ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}$ is independent of L, and hence ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$=1. The exceptional system, that with an odd number of spins and antiferromagnetic couplings, exhibits frustration at T=0 K, and the scaling function is given by ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)=[1+(\ensuremath{\pi}/x
)2${]}^{\mathrm{\ensuremath{-}}1}$. \textcopyright{} 1996 The American Physical Society.
We consider power spectra for fluctuating quantities at a critical point, P( omega ) varies as omega - phi , where phi is determined by the dynamic exponent z and static exponents specific to the quantity considered. From the magnetization spectra obtained with simulations, z is found to be 2.13+or-0.03 for the two-dimensional Glauber kinetic Ising model, hence in accord with recent Monte Carlo relaxational studies, but at odds with those obtained by series expansion and damage spreading methods. Excellent dynamic finite-size scaling for system sizes L=10 to 128 supports our estimate.
Hastings established exponential decay of correlations for ground states of gapped quantum many-body systems. A ground state of a (geometrically) local Hamiltonian with spectral gap ε has correlation length ξ upper bounded as ξ=O(1/ε). In general this bound cannot be improved. Here we study the scaling of the correlation length as a function of the spectral gap in frustration-free local Hamiltonians, and we prove a tight bound ξ=O(1/sqrt[ε]) in this setting. This highlights a fundamental difference between frustration-free and frustrated systems near criticality. The result is obtained using an improved version of the combinatorial proof of correlation decay due to Aharonov, Arad, Vazirani, and Landau.