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arXiv:1403.4187 · Physica Scripta

Relaxation timescales and prethermalization ind-dimensional long-range quantum spin models

Michael Kästner, Mauritz van den Worm

Published Oct 1, 2015• 9 citations
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Abstract

We report analytic results for the correlation functions of long-range quantum Ising models in arbitrary dimension. In particular, we focus on the long-time evolution and the relevant timescales on which correlations relax to their equilibrium values. By deriving upper bounds on the correlation functions in the large-system limit, we prove that a wide separation of timescales, accompanied by a pronounced prethermalization plateau, occurs for sufficiently long-ranged interactions.

DOI
10.1088/0031-8949/2015/t165/014039

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We report analytic results for the correlation functions of long-range quantum Ising models in arbitrary dimension. In particular, we focus on the long-time evolution and the relevant timescales on which correlations relax to their equilibrium values. By deriving upper bounds on the correlation functions in the large-system limit, we prove that a wide separation of timescales, accompanied by a pronounced prethermalization plateau, occurs for sufficiently long-ranged interactions.
We study the time evolution of correlation functions in long-range interacting quantum Ising models. For a large class of initial conditions, exact analytic results are obtained in arbitrary lattice dimension, both for ferromagnetic and antiferromagnetic coupling, and hence also in the presence of geometric frustration. In contrast to the nearest-neighbour case, we find that correlations decay like stretched or compressed exponentials in time. Provided the long-range character of the interactions is sufficiently strong, pronounced prethermalization plateaus are observed and relaxation timescales are widely separated. Specializing to a triangular lattice in two spatial dimensions, we propose to utilize these results for benchmarking a recently developed ion-trap-based quantum simulator.
In this thesis we study the time evolution of correlation functions in quantum lattice models in the presence of long-range interactions or hopping decaying asymptotically as a power law.For a large class of initial conditions, exact analytic results are obtained in arbitrary lattice dimension for the longrange Ising model.In contrast to the nearest-neighbour case, we find that correlations decay like stretched or compressed exponentials in time.Provided the long-range character of the interactions is sufficiently strong, pronounced prethermalization plateaus are observed and relaxation timescales are widely separated.Starting from uncorrelated states that are easily prepared in experiments, we show the dynamical emergence of correlations and entanglement in these far-from-equilibrium interacting quantum systems.We characterize these correlations by the entanglement entropy, concurrence, and squeezing, which are inequivalent measures of entanglement corresponding to different quantum resources.I am grateful for financial support form NITheP over the past six years, not only for the yearly stipend, but also for supporting me in attending two international workshops, one in Boulder Colorado and the other in Trieste Italy.With the support from the NITheP staff I have explored the frontiers of quantum physics, however they also taught me how to communicate scientific ideas and results in the best possible way, for this I am extremely grateful.A word of thanks is due to Dawid Storch and Kaden Hazzard for many helpful discussions.Lastly, but most importantly, I would like to thank my friends,
Abstract Nonequilibrium dynamics in an isolated quantum spin chain with long-range Ising interactions that decay as ( ) with the distance r is studied. It turns out that long-range interactions give rise to a big timescale separation, which causes prethermalization for all . This conclusion is deduced by comparing two important timescales relevant for relaxation dynamics; one is the relaxation time of local permutation operators, which are quasi-conserved quantities in this system, and the other is the timescale of the initial relaxation of local quantities due to the growth of quantum fluctuations. We also explore the entire nonequilibrium dynamics by using the discrete truncated Wigner approximation, which is consistent with the result mentioned above.
Prethermalization describes long-lived intermediate regimes that precede equilibrium and can dominate experimentally accessible dynamics. Here, we show that a separation of spectral energy scales, despite giving rise to a hierarchy of dynamical timescales, does not by itself guarantee the appearance of a prethermal plateau. Under the same Hamiltonian, some observables may exhibit prethermal behavior, while others relax directly toward equilibrium. We uncover the mechanism governing this selectivity, showing that the emergence of a prethermal plateau depends not only on the Hamiltonian, but also on the observable and the initial state. Our results apply broadly to systems close to a fully permutation-symmetric limit and are illustrated with a long-range interacting spin model. We further prove that the Loschmidt echo provides a lower bound on the prethermal lifetime of any bounded observable whenever the prethermal and exact dynamics are unitary.
Although statistical mechanics describes thermal equilibrium states, these states may or may not emerge dynamically for a subsystem of an isolated quantum many-body system. For instance, quantum systems that are near-integrable usually fail to thermalize in an experimentally realistic time scale, and instead relax to quasi-stationary prethermal states that can be described by statistical mechanics, when approximately conserved quantities are included in a generalized Gibbs ensemble (GGE). We experimentally study the relaxation dynamics of a chain of up to 22 spins evolving under a long-range transverse-field Ising Hamiltonian following a sudden quench. For sufficiently long-range interactions, the system relaxes to a new type of prethermal state that retains a strong memory of the initial conditions. However, the prethermal state in this case cannot be described by a standard GGE; it rather arises from an emergent double-well potential felt by the spin excitations. This result shows that prethermalization occurs in a broader context than previously thought, and reveals new challenges for a generic understanding of the thermalization of quantum systems, particularly in the presence of long-range interactions.
The approach to equilibrium is studied for long-range quantum Ising models where the interaction strength decays like r(-α) at large distances r with an exponent α not exceeding the lattice dimension. For a large class of observables and initial states, the time evolution of expectation values can be calculated. We prove analytically that, at a given instant of time t and for sufficiently large system size N, the expectation value of some observable (t) will practically be unchanged from its initial value (0). This finding implies that, for large enough N, equilibration effectively occurs on a time scale beyond the experimentally accessible one and will not be observed in practice.