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Dynamics of long-range interacting quantum spin systems

Mauritz van den Worm

Published Dec 1, 2015• 0 citations
OpenAlex

Abstract

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,

Related papers

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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.

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

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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.

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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 $α$. Using a dynamical version of the strong disorder renormalization group (SDRG) we find for $α>α_c$ that the entanglement entropy grows logarithmically with time and becomes smaller with larger $α$ as $S(t) = S_p \ln(t)/(2α)$. Here, $S_p= 2 \ln2 -1$. 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 $α> α_c \approx 1.8$. For $α<α_c$, we find that the entanglement entropy grows as a power-law with time, $S(t)\sim t^{γ(α)}$ with $0<γ(α)<1$ a decaying function of the interaction exponent $α$.

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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,
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.
By applying complementary analytic and numerical methods, we investigate the dynamics of spin-12XXZ\frac{1}{2} XXZ21​XXZ models with variable-range interactions in arbitrary dimensions. The dynamics we consider is initiated from uncorrelated states that are easily prepared in experiments; it can be equivalently viewed as either Ramsey spectroscopy or a quantum quench. Our primary focus is 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. In one spatial dimension, we show that the time evolution of correlation functions manifests a nonperturbative dynamic singularity. This singularity is characterized by a universal power-law exponent that is insensitive to small perturbations. Explicit realizations of these models in current experiments using polar molecules, trapped ions, Rydberg atoms, magnetic atoms, and alkaline-earth and alkali-metal atoms in optical lattices, along with the relative merits and limitations of these different systems, are discussed.
In this thesis we present our results on the propagation of correlations in long-range interacting quantum systems. The dynamics of local observables in these systems cannot be described with the standard methods used in equilibrium statistical physics and completely new methods have to be developed. Several bounds on the time evolution of correlations have been derived for these systems. However the propagation found in experimental and numerical results is completely different and several regimes are present depending on the long-range character of the interactions. Here we present analytical expressions to describe the time evolution of generic observables in systems where the Hamiltonian takes a quadratic form with long- and short-range interactions. These expressions describe the spreading of local observables as the spreading of the fundamental excitations of the system. We apply these expressions to a spin model finding three different propagation regimes. They can be described qualitatively et quantitatively by the divergences in the energy spectrum. The most important result is that the propagation is at most ballistic, but it can be also significantly slower, where the general bounds predict a propagation faster than ballistic. This points out that the bounds are not able to describe properly the propagation, but a more specific approach is needed. We then move to a system of lattice bosons interacting via long-range interactions. In this case we study two different observables finding completely different results for the same interactions: the spreading of two-body correlations is always ballistic while the one of the one-body correlations ranges from faster-than-ballistic to ballistic. Using our general analytic expressions we find that different parts of the spectrum contribute differently to different observables determining the previous differences. This points out that an observable-dependent notion of locality, missing in the general bounds, have to be developed to correctly describe the time evolution.
In this PhD thesis we investigate some properties of one-dimensional quantum systems, focusing on two important aspects of integrable models: Their entanglement properties at equilibrium and their dynamical correlators after a quantum quench. The first part of the thesis will therefore be devoted to the study of the entanglement entropy in one-dimensional integrable systems, with a special focus on the XYZ spin-1/2 chain, which, in addition to being integrable, is also an interacting theory with non perturbative solutions. We derive its bipartite Renyi entropies in the thermodynamic limit and its behaviour in different phases and for different values of the mass-gap is analysed, both analytically and numerically. In particular it is worth mentioning that the numerical analysis of the entropies of the XYZ model presented in this dissertation has never been published in literature. In the second part of the thesis we study the dynamics of correlators after a quantum quench, preparing the system in a squeezed coherent initial state. The emphasis will be on the Transverse Field Ising Chain and the O(3) non-linear sigma model, the latter studied by means of the semi-classical theory, the former by a form-factor approach. Moreover in the last chapter we outline a general result about the dynamics of correlation functions of local observables after a quantum quench. In particular we show that if there are not long-range interactions in the final Hamiltonian, then the dynamics of the model at long times (non equal- time correlations) is described by the same statistical ensemble that describes its statical properties (equal-time correlations). For the Transverse Field Ising Chain this result means that its dynamics is described by a Generalized Gibbs Ensemble.
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.