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arXiv:2202.06989 · Physical review. E

Slow dynamics and large deviations in classical stochastic Fredkin chains

Luke Causer, Juan P. Garrahan, Austen Lamacraft

Published Jul 21, 2022• 21 citations
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Abstract

The Fredkin spin chain serves as an interesting theoretical example of a quantum Hamiltonian whose ground state exhibits a phase transition between three distinct phases, one of which violates the area law. Here we consider a classical stochastic version of the Fredkin model, which can be thought of as a simple exclusion process subject to additional kinetic constraints, and study its classical stochastic dynamics. The ground-state phase transition of the quantum chain implies an equilibrium phase transition in the stochastic problem, whose properties we quantify in terms of numerical matrix product states (MPSs). The stochastic model displays slow dynamics, including power-law decaying autocorrelation functions and hierarchical relaxation processes due to exponential localization. Like in other kinetically constrained models, the Fredkin chain has a rich structure in its dynamical large deviations-which we compute accurately via numerical MPSs-including an active-inactive phase transition and a hierarchy of trajectory phases connected to particular equilibrium states of the model. We also propose, via its height field representation, a generalization of the Fredkin model to two dimensions in terms of constrained dimer coverings of the honeycomb lattice.

DOI
10.1103/physreve.106.014128

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•Jul 17, 2019•0 citations
The Fredkin spin chain serves as an interesting theoretical example of a quantum Hamiltonian whose ground state exhibits a phase transition between three distinct phases, one of which violates the area law. Here we consider a classical stochastic version of the Fredkin model, which can be thought of as a simple exclusion process subject to additional kinetic constraints, and study its classical stochastic dynamics. The ground-state phase transition of the quantum chain implies an equilibrium phase transition in the stochastic problem, whose properties we quantify in terms of numerical matrix product states (MPSs). The stochastic model displays slow dynamics, including power-law decaying autocorrelation functions and hierarchical relaxation processes due to exponential localization. Like in other kinetically constrained models, the Fredkin chain has a rich structure in its dynamical large deviations-which we compute accurately via numerical MPSs-including an active-inactive phase transition and a hierarchy of trajectory phases connected to particular equilibrium states of the model. We also propose, via its height field representation, a generalization of the Fredkin model to two dimensions in terms of constrained dimer coverings of the honeycomb lattice.
We study the dynamics and thermalization of the Fredkin spin chain, a system with local three-body interactions, particle conservation, and explicit kinetic constraints. We consider deformations away from its stochastic point in order to tune between regimes where kinetic energy dominates and those where potential energy does. By means of exact diagonalization, perturbation theory, and variational matrix product states, we show there is a sudden change of behavior in the dynamics that occurs, from one of fast thermalization to one of slow metastable (prethermal) dynamics near the stochastic point. This change in relaxation is connected to the emergence of additional kinetic constraints, which lead to the fragmentation of Hilbert space in the limit of a large potential energy. We also show that this change can lead to thermalization being evaded for special initial conditions because of nonthermal eigenstates (akin to quantum many-body scars). We provide clear evidence for the existence of these nonthermal states for large system sizes even when far from the large-potential-energy limit, and explain their connection to the emergent kinetic constraints. Published by the American Physical Society 2024
We study a one-dimensional classical stochastic kinetically constrained model (KCM) inspired by Rydberg atoms in their "facilitated" regime, where sites can flip only if a single of their nearest neighbors is excited. We call this model "XOR-FA" to distinguish it from the standard Fredrickson-Andersen (FA) model. We describe the dynamics of the XOR-FA model, including its relation to simple exclusion processes in its domain wall representation. The interesting relaxation dynamics of the XOR-FA is related to the prominence of large dynamical fluctuations that lead to phase transitions between active and inactive dynamical phases as in other KCMs. By means of numerical tensor network methods we study in detail such transitions in the dynamical large deviation regime.
We study in detail the properties of the quantum East model, an interacting quantum spin chain inspired by simple kinetically constrained models of classical glasses.Through a combination of analytics, exact diagonalization, and tensor-network methods, we show the existence of a transition, from a fast to a slow thermalization regime, which manifests itself throughout the spectrum.On the slow side, by exploiting the localization of the ground state and the form of the Hamiltonian, we explicitly construct a large (exponential in size) number of nonthermal states that become exact finite-energy-density eigenstates in the large size limit, as expected for a true phase transition.A "superspin" generalization allows us to find a further large class of area-law states proved to display very slow relaxation.These states retain memory of their initial conditions for extremely long times.Our numerical analysis reveals that the localization properties are not limited to the ground state and that many eigenstates have large overlap with product states and can be approximated well by matrix product states at arbitrary energy densities.The mechanism that induces localization to the ground state, and hence the nonthermal behavior of the system, can be extended to a wide range of models including a number of simple spin chains.We discuss implications of our results for slow thermalization and nonergodicity more generally in disorder-free systems with constraints, and we give numerical evidence that these results may be extended to two-dimensional systems.
The dynamical behavior of a quantum many-particle system is characterized by the lifetime of its excitations. When the system is perturbed, observables of any nonconserved quantity decay exponentially, but those of a conserved quantity relax to equilibrium with a power law. Such processes are associated with a dynamical exponent zzz that relates the spread of correlations in space and time. We present numerical results for the Fredkin model, a quantum spin chain with a three-body interaction term, which exhibits an unusually large dynamical exponent. We discuss our efforts to produce a reliable estimate $z\ensuremath{\doteq}3.16(1)$ through direct simulation of the quantum evolution and to explain the slow dynamics in terms of an excited bond that executes a constrained random walk in Monte Carlo time.
We study in detail the properties of the quantum East model, an interacting quantum spin chain inspired by simple kinetically-constrained models of classical glasses. Through a combination of analytics, exact diagonalization and tensor-network methods we show the existence of a transition, from a fast to a slow thermalization regime, which manifests itself throughout the spectrum. On the slow side, by exploiting the localization of the ground state and the form of the Hamiltonian, we explicitly construct a large (exponential in size) number of non-thermal states which become exact finite-energy-density eigenstates in the large-size limit, as expected for a true phase transition. A ``super-spin'' generalization allows us to find a further large class of area-law states proved to display very slow relaxation. These states retain memory of their initial conditions for extremely long times. Our numerical analysis reveals that the localization properties are not limited to the ground state and that} many eigenstates have large overlap with product states and can be approximated well by matrix product states at arbitrary energy densities. The mechanism that induces localization to the ground state, and hence the non-thermal behavior of the system, can be extended to a wide range of models including a number of simple spin chains. We discuss implications of our results for slow thermalization and non-ergodicity more generally in disorder-free systems with constraints and we give numerical evidence that these results may be extended to two dimensional systems.
The work in this thesis is split into two main parts; the dynamical study of the fully packed classical dimer model, and the development of advanced large deviation techniques applied to the classical dimer model, highlighting the connection to the quantum dimer model. The classical dimer model on the square lattice is a paradigmatic example of a system subject to strong local constraints. We study its behavior under local stochastic dynamics, by means of Monte Carlo simulations and theoretical arguments. We observe clear signatures of correlated dynamics in both global and local observables and over a broad range of time scales, indicating a breakdown of the simple continuum description that approximates well the statics. We show that this collective dynamics can be understood in terms of one-dimensional “strings” of high mobility. We introduce a coarse-grained description of the strings, which leads to exact results in the limit of low string density and provides a detailed qualitative understanding of the dynamics in all flux sectors. We then go on, still using the fully packed dimers on the square lattice as a paradigmatic system, to study the connection between the phase behaviour of the ground state of quantum dimers and the dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK) point, a quantum dimer Hamiltonian is equivalent to the Markov generator of the dynamics of classical dimers. A less well understood fact is that away from the RK point the quantum-classical connection persists: in this case the quantum Hamiltonian maps to a non-stochastic "tilted" master operator of the classical stochastic problem. This implies a direct relation between the phase behaviour of quantum dimers and properties of ensembles of stochastic trajectories of classical dimers. Using transition path sampling (TPS) -- supplemented by trajectory umbrella sampling -- we obtain the large deviation statistics of dynamical activity in the classical problem, and show the correspondence between the phase behaviour of the classical and quantum systems. Finally, we take these ideas further by developing a feedback-based modified dynamics used for umbrella sampling alongside TPS. This feedback approach was first used in the cloning (or population dynamics) method. We compare different modified dynamics within TPS and show how much more efficient the feedback-modified dynamics can be than the original dynamics, for the purpose of rare event sampling. We also discuss how our approach may generalise to other problems with slow complex dynamics.