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AIP conference proceedings

Emergence of Glassy States in Lattice Models with no a Priori Disorder

Alberto Petri

Published Jan 1, 2003• 0 citations
Publisher

Abstract

Our investigation concerns the quenching process in systems with several degenerate ordered ground states. In the process the system is initially in thermal equilibrium at high temperature; then a sudden cooling drives the system towards a new equilibrium state. It comes out that when the number of ground states is large, the system evolves into a state made of many coexisting ordered phases, whose mutual interaction energy is not negligible in comparison with the total system energy. This is obtained provided the thermodynamic limit is taken before the infinite time limit.

DOI
10.1063/1.1571301

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Our investigation concerns the quenching process in systems with several degenerate ordered ground states. In the process the system is initially in thermal equilibrium at high temperature; then a sudden cooling drives the system towards a new equilibrium state. It comes out that when the number of ground states is large, the system evolves into a state made of many coexisting ordered phases, whose mutual interaction energy is not negligible in comparison with the total system energy. This is obtained provided the thermodynamic limit is taken before the infinite time limit.
We investigate the out-of-equilibrium physics of monodisperse bosonic ensembles on a square lattice. The effective Hamiltonian description of these systems is given in terms of an extended Hubbard model with cluster-forming interactions relevant to experimental realizations with cold Rydberg-dressed atoms. The ground state of the model, recently investigated in [Phys. Rev. Lett. 123, 045301 (2019)], features, aside from a superfluid and a stripe crystalline phase occurring at small and large interaction strength VVV, respectively, a rare first-order transition between an isotropic and an anisotropic stripe supersolid at intermediate VVV. By means of quantum Monte Carlo calculations we show that the equilibrium crystal may be turned into a glass by simulated temperature quenches and that out-of-equilibrium isotropic (super)solid states may emerge also when their equilibrium counterparts are anisotropic. These out-of-equilibrium states are of experimental interest, their excess energy with respect to the ground state being within the energy window typically accessed in cold atom experiments. We find, after quenching, no evidence of coexistence between superfluid and glassy behavior. Such an absence of superglassiness is qualitatively explained.
Two classical subjects in statistical physics are kinetic theory and phase transitions. The latter has been traditionally studied in the equilibrium framework, where the goal is to characterize the ordered phases that arise below the critical temperature in systems with short-range interactions. A more recent development has been to investigate how this order is formed dynamically . In the present and following chapter, we focus on this dynamics. Phenomenology of coarsening The theory of phase transitions was originally motivated by the goal of understanding ferromagnetism. The Ising model played a central role in this effort and provided a useful framework for investigating dynamics. The basic entity in the Ising model is a spin variable that can take two possible values, s = ±1, at each site of a lattice. A local ferromagnetic interaction between spins promotes their alignment, while thermal noise tends to randomize their orientations. The outcome of this competition is a disordered state for sufficiently high temperature, while below a critical temperature the tendency for alignment prevails and an ordered state arises in which the order parameter – the average magnetization – is non-zero. Suppose that we start with an Ising model in an equilibrium disordered phase and lower the temperature. To understand how the system evolves toward the final state, we must endow this model with a dynamics and we also need to specify the quenching procedure. Quenching is usually implemented as follows: • Start at a high initial temperature T i > T c , where spins are disordered; here T c is the critical temperature. • Instantaneously cool the system to a lower temperature T f .
We now begin our exposition of universal dynamic scaling behavior that emerges under non-equilibrium conditions. We are mainly concerned with the relaxational models A and B here, but also touch on other dynamic universality classes. First, we consider non-equilibrium critical relaxation from a disordered initial state, and compute the associated universal scaling exponents within the dynamic renormalization group framework. Related phenomena in the early-time regime, prior to reaching temporally translationally invariant asymptotics, are ‘aging’ and a non-equilibrium fluctuation-dissipation ratio, both of which have enjoyed prominence in the literature on ‘glassy’ kinetics. In this context, we derive the critical initial slip exponent for the order parameter, and briefly discuss interesting persistence properties. Second, we explore the long-time scaling laws in phase ordering kinetics and coarsening for the relaxational models, following a fast temperature ‘quench’ into the ordered phase, in systems with either non-conserved or conserved order parameters. Aside from a few explicit computations in the spherical model limit (wherein the number of order parameter components n → ∞), we largely employ phenomenological considerations and scaling theory, stressing the importance of topological defects for energy dissipation during coarsening. Next we address the question how violations of Einstein's relation that links relaxation coefficients and noise correlations might affect the asymptotic dynamic critical scaling behavior. Whereas in systems with non-conserved order parameter typically the equilibrium scaling laws are recovered in the vicinity of the critical point, in contrast genuinely novel universality classes may emerge in the case of a conserved parameter, provided it is driven out of equilibrium in a spatially anisotropic fashion.
A variety of recent studies have shed light on the far from equilibrium behavior of quantum systems. Research suggests that this away from equilibrium approach may lead to the realization of unfamiliar dynamical quantum states. For instance, we see ballistic transport at finite temperature, an absence of thermalization, and development of long-range entanglement from initially uncorrelated states. It has been shown that in some cases a time-evolving quantum state is equivalent to the ground state of an "emergent" Hamiltonian where the time enters as a param- eter. This mapping gives us insight into the nature of correlations between particles. For high temperatures, we expect no long-range correlations to develop. However, the emergent model demonstrates that the time-evolved state can be realized as a low energy eigenstate, which naturally possesses long-range correlations. Here, we study the dynamical behavior of both noninteracting and interacting particles in a one-dimensional lattice under various conditions. As the system un- dergoes time evolution, we look at the distribution of particles as well as correlations between particles in the system. We then compare the actual time-evolving state of the system to the ground state of the emergent Hamiltonian to study how long this description is valid. We generally expect that the description is valid in a given region as long as the boundary effects do not propagate into the region. We first verify and reproduce results in the literature (see Vidmar et al. [1]). We proceed to study the effect of interaction strength in a particular interacting model that shows both ballistic and diffusive transport. We then apply this formalism to calculate infinite temperature dynamics of a noninteracting system. Interestingly, this system also develops long-range correlations over time in spite of starting out from an infinite temperature state.
A numerical linked-cluster algorithm was recently introduced to study quantum quenches in the thermodynamic limit starting from thermal initial states [M. Rigol, Phys. Rev. Lett. 112, 170601 (2014)]. Here, we tailor that algorithm to quenches starting from ground states. In particular, we study quenches from the ground state of the antiferromagnetic Ising model to the XXZ chain. Our results for spin correlations are shown to be in excellent agreement with recent analytical calculations based on the quench action method. We also show that they are different from the correlations in thermal equilibrium, which confirms the expectation that thermalization does not occur in general in integrable models even if they cannot be mapped to noninteracting ones.