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arXiv:1209.3697 · New Journal of Physics

Relaxation timescales and decay of correlations in a long-range interacting quantum simulator

Mauritz van den Worm, Brian C. Sawyer, J. J. Bollinger, Michael Kästner

Published Aug 2, 2013• 79 citations
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Abstract

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.

DOI
10.1088/1367-2630/15/8/083007

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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,
Obtaining quantitative ground-state behavior for geometrically-frustrated quantum magnets with long-range interactions is challenging for numerical methods. Here, we demonstrate that the ground states of these systems on two-dimensional lattices can be efficiently obtained using state-of-the-art translation-invariant variants of matrix product states and density-matrix renormalization-group algorithms. We use these methods to calculate the fully-quantitative ground-state phase diagram of the long-range interacting triangular Ising model with a transverse field on six-leg infinite-length cylinders and scrutinize the properties of the detected phases. We compare these results with those of the corresponding nearest neighbor model. Our results suggest that, for such long-range Hamiltonians, the long-range quantum fluctuations always lead to long-range correlations, where correlators exhibit power-law decays instead of the conventional exponential drops observed for short-range correlated gapped phases. Our results are relevant for comparisons with recent ion-trap quantum simulator experiments that demonstrate highly-controllable long-range spin couplings for several hundred ions.
Motivated by recent experiments with a Penning ion trap quantum simulator, we perform numerically exact Stochastic Series Expansion quantum Monte Carlo simulations of long-range transverse-field Ising models on a triangular lattice for different decay powers $\ensuremath{\alpha}$ of the interactions. The phase boundary for the ferromagnet is obtained as a function of $\ensuremath{\alpha}$. For antiferromagnetic interactions, there is strong indication that the transverse field stabilizes a clock ordered phase with sublattice magnetization $(M,\ensuremath{-}\frac{M}{2},\ensuremath{-}\frac{M}{2})$ with unsaturated M<1M<1M<1 in a process known as ``order by disorder'' similar to the nearest-neighbor antiferromagnet on the triangular lattice. Connecting the known limiting cases of nearest-neighbor and infinite-range interactions, a semiquantitative phase diagram is obtained. Magnetization curves for the ferromagnet for experimentally relevant system sizes and with open boundary conditions are presented.
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.
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.