Eigenstate thermalization in ( 1 + 1 )-dimensional SU(2) lattice gauge theory coupled with dynamical fermions

We study the matrix elements of local and nonlocal operators in the single-particle eigenstates of two paradigmatic quantum-chaotic quadratic Hamiltonians; the quadratic Sachdev-Ye-Kitaev (SYK2) model and the three-dimensional Anderson model below the localization transition. We show that they display eigenstate thermalization for normalized observables. Specifically, we show that the diagonal matrix elements exhibit vanishing eigenstate-to-eigenstate fluctuations, and a variance proportional to the inverse Hilbert space dimension. We also demonstrate that the ratio between the variance of the diagonal and the off-diagonal matrix elements is $2$, as predicted by the random matrix theory. We study distributions of matrix elements of observables and establish that they need not be Gaussian. We identify the class of observables for which the distributions are Gaussian.

A nonlocal observation operator has been developed to assimilate GPS radio occultation (RO) refractivity with WRF 3DVAR. For simplicity, in the past GPS RO refractivity was often assimilated using a local observation operator with the assumption that the GPS RO observation was representative of a model local point. Such an operator did not take into account the effects of horizontal inhomogeneity on the derived GPS RO refractivity. In order to more accurately model the observables, Sokolovskiy et al. (2005a) developed a nonlocal observation operator; which would take into account the effects of horizontal inhomogeneity on GPS RO measurements. This nonlocal observation operator calculates the integrated amount of the model refractivity along the ray paths centered at the perigee points. For comparative purposes, the nonlocal observation operator can be simplified by limiting the length of integration near the RO point. This is called the “local operator variant”, which is equivalent to the original local operator except that the original one is performed with fixed tangent points at observation levels. For computational efficiency, assimilation using both the nonlocal operator and local operator variant now is performed with smear tangent points at the mean height of each model vertical level. In this study, the statistics of observation errors using both local and nonlocal operators were estimated based on WRF simulations. The observation errors produced by the nonlocal operator are about two times smaller than those generated by the local operator and in agreement with Sokolovskiy et al. (2005b). Each of the three operators is used to assimilate GPS RO refractivity soundings from the FORMOSAT-3/COSMIC mission using the WRF 3DVAR system. The WRF model then is applied to simulate Typhoons Kaemi (July 2006) which struck Taiwan with significant torrential rainfall. The analysis increments produced by the nonlocal operator and local operator variant are quite similar in horizontal and vertical distributions; whereas, the former is slightly stretched along the ray's direction, as a result of the longer integration length. The simulated typhoon tracks prior to landfall are quite similar for the three operators. Both the nonlocal operator and local operator variant improve the detoured track after landfall as predicted by the local operator. The nonlocal operator outperforms the two local operators in rainfall prediction at later times. The performances of the nonlocal operator in general are promising and can replace the local operator at no marked cost of computational efficiency.

The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.

Finite size (Fisher) scaling is investigated for four dimensional SU(2) and SU(3) lattice gauge theories without quarks. It allows to disentangle violations of (asymptotic) scaling and finite volume corrections. Mass spectrum, string tension, deconfinement temperature and lattice β-function are considered. For appropriate volumes, Monte Carlo investigations seem to be able to control the finite volume continuum limit. Contact is made with Lüscher’s small volume expansion and possibly also with the asymptotic large volume behaviour.

The eigenstate thermalization hypothesis (ETH) is one of the cornerstones of contemporary quantum statistical mechanics. The extent to which ETH holds for nonlocal operators is an open question that we partially address in this Letter. We report on the construction of highly nonlocal operators, behemoths, that are building blocks for various kinds of local and nonlocal operators. The behemoths have a singular distribution and width w∼D^{-1} (D being the Hilbert space dimension). From there, one may construct local operators with the ordinary Gaussian distribution and w∼D^{-1/2} in agreement with ETH. Extrapolation to even larger widths predicts sub-ETH behavior of typical nonlocal operators with w∼D^{-δ}, 0<δ<1/2. This operator construction is based on a deep analogy with random matrix theory and shows striking agreement with numerical simulations of nonintegrable many-body systems.

We study the entanglement entropy of Hamiltonian SU(2) lattice gauge theory in 2+1 dimensions on linear plaquette chains and show that the entanglement entropies of both ground and excited states follow Page curves. The transition of the subsystem size dependence of the entanglement entropy from the area law for the ground state to the volume law for highly excited states is found to be described by a universal crossover function. Quantum many-body scars in the middle of the spectrum, which are present in the electric flux truncated Hilbert space, where the gauge theory can be mapped onto an Ising model, disappear when higher electric field representations are included in the Hilbert space basis. This suggests the continuum (2+1)-dimensional SU(2) gauge theory does not have such scarred states. Published by the American Physical Society 2024

Using numerical exact diagonalization, we study matrix elements of a local spin operator in the eigenbasis of two different nonintegrable quantum spin chains. Our emphasis is on the question to what extent local operators can be represented as random matrices and, in particular, to what extent matrix elements can be considered as uncorrelated. As a main result, we show that the eigenvalue distribution of band submatrices at a fixed energy density is a sensitive probe of the correlations between matrix elements. We find that, on the scales where the matrix elements are in a good agreement with all standard indicators of the eigenstate thermalization hypothesis, the eigenvalue distribution still exhibits clear signatures of the original operator, implying correlations between matrix elements. Moreover, we demonstrate that at much smaller energy scales, the eigenvalue distribution approximately assumes the universal semicircle shape, indicating transition to the random-matrix behavior, and in particular that matrix elements become uncorrelated.

The design of quantum many body systems, which have to fulfill an extensive number of constraints, appears as a formidable challenge within the field of quantum simulation. Lattice gauge theories are a particular important class of quantum systems with an extensive number of local constraints and play a central role in high energy physics, condensed matter and quantum information. Whereas recent experimental progress points towards the feasibility of large-scale quantum simulation of abelian gauge theories, the quantum simulation of non-abelian gauge theories appears still elusive. In this paper we present minimal non-abelian lattice gauge theories, whereby we introduce the necessary formalism in well-known abelian gauge theories, such as the Jaynes–Cumming model. In particular, we show that certain minimal non-abelian lattice gauge theories can be mapped to three or four level systems, for which the design of a quantum simulator is standard with current technologies. Further we give an upper bound for the Hilbert space dimension of a one dimensional SU(2) lattice gauge theory, and argue that the implementation with current digital quantum computer appears feasible.

Matrix elements of observables in eigenstates of generic Hamiltonians are described by the Srednicki ansatz within the eigenstate thermalization hypothesis (ETH). We study a quantum chaotic spin-fermion model in a one-dimensional lattice, which consists of a spin-1/2 XX chain coupled to a single itinerant fermion. In our study, we focus on translationally invariant observables including the charge and energy current, thereby also connecting the ETH with transport properties. Considering observables with a Hilbert-Schmidt norm of one, we first perform a comprehensive analysis of ETH in the model taking into account latest developments. A particular emphasis is on the analysis of the structure of the offdiagonal matrix elements $|\ensuremath{\langle}\ensuremath{\alpha}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}{|\ensuremath{\beta}\ensuremath{\rangle}|}^{2}$ in the limit of small eigenstate energy differences $\ensuremath{\omega}={E}_{\ensuremath{\beta}}\ensuremath{-}{E}_{\ensuremath{\alpha}}$. Removing the dominant exponential suppression of $|\ensuremath{\langle}\ensuremath{\alpha}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}{|\ensuremath{\beta}\ensuremath{\rangle}|}^{2}$, we find that (1) the current matrix elements exhibit a system-size dependence that is different from other observables under investigation and (2) matrix elements of several other observables exhibit a Drude-like structure with a Lorentzian frequency dependence. We then show how this information can be extracted from the autocorrelation functions as well. Finally, our study is complemented by a numerical analysis of the fluctuation-dissipation relation for eigenstates in the bulk of the spectrum. We identify the regime of $\ensuremath{\omega}$ in which the well-known fluctuation-dissipation relation is valid with high accuracy for finite systems.

In this study, an Observing System Simulation Experiment (OSSE) is performed to evaluate the performance of a nonlocal excess phase operator and a local refractivity operator for a GPS radio occultation (RO) sounding that passes through the eye of Hurricane Katrina as simulated by a high-resolution model, with significant horizontal refractivity gradients. Both observation operators are tested on the NCEP gridpoint statistical interpolation (GSI) data assimilation system at 12- and 36-km horizontal resolution. It is shown that the shape and magnitude of the analysis increments for sea level pressure, temperature, and water vapor mixing ratio exhibit significant differences between the use of local and nonlocal operators. The nonlocal operator produces more accurate analyses when verified against the “truth” derived from the ground truth run. It is found that the improvements of the analysis with the use of the nonlocal operator over that of the local operator are essentially the same at 12- and 36-km horizontal resolution. An additional experiment is performed over a region with small horizontal gradients. As expected, the use of both nonlocal and local operators produces similar results over such a region.

The dynamic relaxation process for the (2+1)--dimensional SU(2) lattice gauge theory at critical temperature is investigated with Monte Carlo methods. The critical initial increase of the Polyakov loop is observed. The dynamic exponents $\theta$ and $z$ as well as the static critical exponent $\beta/\nu$ are determined from the power law behaviour of the Polyakov loop, the auto-correlation and the second moment at the early stage of the time evolution. The results are well consistent and universal short-time scaling behaviour of the dynamic system is confirmed. The values of the exponents show that the dynamic SU(2) lattice gauge theory is in the same dynamic universality class as the dynamic Ising model.

Short-time scaling in SU(2) lattice gauge theory at finite temperature

We report on progress in full quantum understanding of thermalization in non-Abelian gauge theories. Specifically, we test the eigenstate thermalization hypothesis for (2+1)-dimensional SU(2) lattice gauge theory.

We explore the origin of the arrow of time in an isolated quantum system described by the Schr\"odinger equation. We provide an explanation from weak values in the configuration space, which are understood as operational properties obtained in the laboratory following a well-defined protocol. We show that quantum systems satisfying the eigenstate thermalization hypothesis can simultaneously provide thermalized ensemble expectation values and nonthermalized weak values of the momentum, both from the same operational probability distribution. The reason why weak values of the momentum may escape from the eigenstate thermalization hypothesis is because they are linked only to off-diagonal elements of the density matrix in the energy representation. Therefore, we state that the origin of the arrow of time in isolated quantum systems described by the Schr\"odinger equation comes from dealing with properties obtained by averaging (tracing out) some degrees of freedom of the configuration space. We then argue that thermalization does not occur in the properties defined in the configuration space, and our argument is compatible with defending that thermalization is a real phenomenon in the properties defined in the physical space. All of these conclusions are testable in the laboratory through many-body weak values.

Diagonal and off-diagonal hyperfine structure matrix elements in KCs within the relativistic Fock space coupled cluster theory