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arXiv:2011.07110 · Physical review. B./Physical review. B

Slow dynamics of the Fredkin spin chain

Khagendra Adhikari, K. S. D. Beach

Published Sep 23, 2021• 9 citations
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Abstract

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 $z$ 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.

DOI
10.1103/physrevb.104.115149

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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≐3.16(1)z\ensuremath{\doteq}3.16(1)z\ensuremath≐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.
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
Abstract The Motzkin and Fredkin quantum spin chains are described by frustration-free Hamiltonians recently introduced and studied because of their anomalous behaviors in the correlation functions and in the entanglement properties. In this paper we analyze their quantum dynamical properties, focusing in particular on the time evolution of the excitations driven by a quantum quench, looking at the correlations functions of spin operators defined along different directions, and discussing the results in relation with the cluster decomposition property. We show that, for such models, one can have absence of cone-like structure in the propagation of the excitations after a quantum quench, together with a violation of the cluster decomposition property for the ground state characterized by only a logarithmic violation of the area law for the entanglement entropy.
We investigate the scaling relation between the physical entanglement time τent and the modular time τmod in the dynamics of the integrable Heisenberg XXX spin chain. Contrary to the hypothesis of a stable universal exponent γ ≈ 2, we find no evidence for a time-independent power-law scaling. Instead, the effective exponent γ(t) exhibits a systematic drift toward smaller values at longer evolution times, following a robust power-law relaxation γ(t) −γ∞ ∼ t−α with α ≈ 0.65, supported by an excellent data collapse with relative spread below 6%. Remarkably, the amplitude of this relaxation is independent of system size N within the accessible window N ∈ {14,16,18,20}. We interpret this behavior as an effective renormalization group flow of the scaling exponent in time, suggesting that γ is not a fixed scaling dimension but a running quantity controlled by a marginally irrelevant operator. Our results demonstrate that previously reported values γ ≈ 2 originate from preasymptotic regimes rather than a true scaling law, and reveal a dynamical mechanism by which simple scaling breaks down in quantum entanglement evolution.
The non-equilibrium dynamics of disordered many-body quantum systems after a global quantum quench unveils important insights about the competition between interactions and disorder, yielding in particular an insightful perspective on many body localization (MBL). Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on the quantum quench dynamics have so far not been investigated. In this letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power ααα. Using a dynamical version of the strong disorder renormalization group (SDRG) we find for α>αcα>α_cα>αc​ that the entanglement entropy grows logarithmically with time and becomes smaller with larger ααα as S(t)=Spln⁡(t)/(2α)S(t) = S_p \ln(t)/(2α)S(t)=Sp​ln(t)/(2α). Here, Sp=2ln⁡2−1S_p= 2 \ln2 -1Sp​=2ln2−1. We use numerical exact diagonalization (ED) simulations to verify our results for system sizes up to $ N\sim 16$ spins, yielding good agreement for sufficiently large α>αc≈1.8α> α_c \approx 1.8α>αc​≈1.8. For α<αcα<α_cα<αc​, we find that the entanglement entropy grows as a power-law with time, S(t)∼tγ(α)S(t)\sim t^{γ(α)}S(t)∼tγ(α) with 0<γ(α)<10<γ(α)<10<γ(α)<1 a decaying function of the interaction exponent ααα.
Abstract We study gapless quantum spin chains with spin 1/2 and 1: the Fredkin and Motzkin models. Their entangled groundstates are known exactly but not their excitation spectra. We first express the groundstates in the continuum which allows for the calculation of spin and entanglement properties in a unified fashion. Doing so, we uncover an emergent conformal-type symmetry, thus consolidating the connection to a widely studied family of Lifshitz quantum critical points in 2d. We then obtain the low lying excited states via large-scale DMRG simulations and find that the dynamical exponent is z = 3.2 in both cases. Other excited states show a different z , indicating that these models have multiple dynamics. Moreover, we modify the spin-1/2 model by adding a ferromagnetic Heisenberg term, which changes the entire spectrum. We track the resulting non-trivial evolution of the dynamical exponents using DMRG. Finally, we exploit an exact map from the quantum Hamiltonian to the non-equilibrium dynamics of a classical spin chain to shed light on the quantum dynamics.