arXiv subjects

Marcos Rigol

Publications and source records attributed to Marcos Rigol.

At least 73 records · Page 4Linked to original sources

arXiv2017

Entanglement Entropy of Eigenstates of Quantum Chaotic Hamiltonians

In quantum statistical mechanics, it is of fundamental interest to understand how close the bipartite entanglement entropy of eigenstates of quantum chaotic Hamiltonians is to maximal. For random pure states in the Hilbert space, the average entanglement entropy is known to be nearly maximal, with a deviation that is, at most, a constant. Here we prove that, in a system that is away from half filling and divided in two equal halves, an upper bound for the average entanglement entropy of random pure states with a fixed particle number and normally distributed real coefficients exhibits a deviation from the maximal value that grows with the square root of the volume of the system. Exact numerical results for highly excited eigenstates of a particle number conserving quantum chaotic model indicate that the bound is saturated with increasing system size.

cond-mat.stat-mech↗

arXiv2017

Quantum adiabatic protocols using emergent local Hamiltonians

We present two applications of emergent local Hamiltonians to speed up quantum adiabatic protocols for isolated noninteracting and weakly interacting fermionic systems in one-dimensional lattices. We demonstrate how to extract maximal work from initial band-insulating states, and how to adiabatically transfer systems from linear and harmonic traps into box traps. Our protocols consist of two stages. The first one involves a free expansion followed by a quench to an emergent local Hamiltonian. In the second stage, the emergent local Hamiltonian is "turned off" quasistatically. For the adiabatic transfer from a harmonic trap, we consider both zero- and nonzero-temperature initial states.

cond-mat.stat-mech↗

arXiv2017

Optimization of finite-size errors in finite-temperature calculations of unordered phases

It is common knowledge that the microcanonical, canonical, and grand-canonical ensembles are equivalent in thermodynamically large systems. Here, we study finite-size effects in the latter two ensembles. We show that contrary to naive expectations, finite-size errors are exponentially small in grand canonical ensemble calculations of translationally invariant systems in unordered phases at finite temperature. Open boundary conditions and canonical ensemble calculations suffer from finite-size errors that are only polynomially small in the system size. We further show that finite-size effects are generally smallest in numerical linked cluster expansions. Our conclusions are supported by analytical and numerical analyses of classical and quantum systems.

cond-mat.stat-mech↗

arXiv2017

Equilibration Dynamics of Strongly Interacting Bosons in 2D Lattices with Disorder

Motivated by recent optical lattice experiments [J.-y. Choi et al., Science 352, 1547 (2016)], we study the dynamics of strongly interacting bosons in the presence of disorder in two dimensions. We show that Gutzwiller mean-field theory (GMFT) captures the main experimental observations, which are a result of the competition between disorder and interactions. Our findings highlight the difficulty in distinguishing glassy dynamics, which can be captured by GMFT, and many-body localization, which cannot be captured by GMFT, and indicate the need for further experimental studies of this system.

cond-mat.quant-gas↗

arXiv2017

Eigenstate thermalization in the two-dimensional transverse field Ising model: II. Off-diagonal matrix elements of observables

We study the matrix elements of few-body observables, focusing on the off-diagonal ones, in the eigenstates of the two-dimensional transverse field Ising model. By resolving all symmetries, we relate the onset of quantum chaos to the structure of the matrix elements. In particular, we show that a general result of the theory of random matrices, namely, the value 2 of the ratio of variances (diagonal to off-diagonal) of the matrix elements of Hermitian operators, occurs in the quantum chaotic regime. Furthermore, we explore the behavior of the off-diagonal matrix elements of observables as a function of the eigenstate energy differences, and show that it is in accordance with the eigenstate thermalization hypothesis ansatz.

cond-mat.stat-mech↗

arXiv2017

Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians

In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)], Page proved that the average entanglement entropy of subsystems of random pure states is $S_{\rm ave}\simeq\ln{\cal D}_{\rm A} - (1/2) {\cal D}_{\rm A}^2/{\cal D}$ for $1\ll{\cal D}_{\rm A}\leq\sqrt{\cal D}$, where ${\cal D}_{\rm A}$ and ${\cal D}$ are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy $\langle S\rangle$ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models $\ln{\cal D}_{\rm A} - (\ln{\cal D}_{\rm A})^2/\ln{\cal D} \leq \langle S \rangle \leq \ln{\cal D}_{\rm A} - [1/(2\ln2)] (\ln{\cal D}_{\rm A})^2/\ln{\cal D}$. Consequently we prove that: (i) if the subsystem size is a finite fraction of the system size then $\langle S\rangle<\ln{\cal D}_{\rm A}$ in the thermodynamic limit, i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal, i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.

cond-mat.stat-mech↗

arXiv2017

Emergent eigenstate solution and emergent Gibbs ensemble for expansion dynamics in optical lattices

Within the emergent eigenstate solution to quantum dynamics [Phys. Rev. X 7, 021012 (2017)], one can construct a local operator (an emergent Hamiltonian) of which the time-evolving state is an eigenstate. Here we show that such a solution exists for the expansion dynamics of Tonks-Girardeau gases in optical lattices after turning off power-law (e.g., harmonic or quartic) confining potentials, which are geometric quenches that do not involve the boost operator. For systems that are initially in the ground state and undergo dynamical fermionization during the expansion, we show that they remain in the ground state of the emergent local Hamiltonian at all times. On the other hand, for systems at nonzero initial temperatures, the expansion dynamics can be described constructing a Gibbs ensemble for the emergent local Hamiltonian (an emergent Gibbs ensemble).

cond-mat.quant-gas↗

arXiv2017

Work extraction in an isolated quantum lattice system: Grand canonical and generalized Gibbs ensemble predictions

We study work extraction (defined as the difference between the initial and the final energy) in noninteracting and (effectively) weakly interacting isolated fermionic quantum lattice systems in one dimension, which undergo a sequence of quenches and equilibration. The systems are divided in two parts, which we identify as the subsystem of interest and the bath. We extract work by quenching the on-site potentials in the subsystem, letting the entire system equilibrate, and returning to the initial parameters in the subsystem using a quasi-static process (the bath is never acted upon). We select initial states that are direct products of thermal states of the subsystem and the bath, and consider equilibration to the generalized Gibbs ensemble (GGE, noninteracting case) and to the Gibbs ensemble (GE, weakly interacting case). We identify the class of quenches that, in the thermodynamic limit, results in GE and GGE entropies after the quench that are identical to the one in the initial state (quenches that do not produce entropy). Those quenches guarantee maximal work extraction when thermalization occurs. We show that the same remains true in the presence of integrable dynamics that results in equilibration to the GGE.

cond-mat.stat-mech↗

arXiv2017

Numerical linked cluster expansions for quantum quenches in one dimensional lattices

We discuss the application of numerical linked cluster expansions (NLCEs) to study one dimensional lattice systems in thermal equilibrium and after quantum quenches from thermal equilibrium states. For the former, we calculate observables in the grand canonical ensemble, and for the latter we calculate observables in the diagonal ensemble. When converged, NLCEs provide results in the thermodynamic limit. We use two different NLCEs - a maximally connected expansion introduced in previous works and a site-based expansion. We compare the effectiveness of both NLCEs. The site-based NLCE is found to work best for systems in thermal equilibrium. However, in thermal equilibrium and after quantum quenches, the site-based NLCE can diverge when the maximally connected one converges. We relate this divergence to the exponentially large number of clusters in the site-based NLCE and the behavior of the weights of observables in those clusters. We discuss the effectiveness of resummations to cure the divergence. Our NLCE calculations are compared to exact diagonalization ones in lattices with periodic boundary conditions. NLCEs are found to outperform exact diagonalization in periodic systems for all quantities studied.

cond-mat.stat-mech↗

arXiv2017

Correlations and diagonal entropy after quantum quenches in XXZ chains

We study quantum quenches in the XXZ spin-$1/2$ Heisenberg chain from families of ferromagnetic and antiferromagnetic initial states. Using Bethe ansatz techniques, we compute short-range correlators in the complete generalized Gibbs ensemble (GGE), which takes into account all local and quasi-local conservation laws. We compare our results to exact diagonalization and numerical linked cluster expansion calculations for the diagonal ensemble finding excellent agreement and thus providing a very accurate test for the validity of the complete GGE. Furthermore, we compute the diagonal entropy in the post-quench steady state. By careful finite-size scaling analyses of the exact diagonalization results, we show that the diagonal entropy is equal to one half the Yang-Yang entropy corresponding to the complete GGE. Finally, the complete GGE is quantitatively contrasted with the GGE built using only the local conserved charges (local GGE). The predictions of the two ensembles are found to differ significantly in the case of ferromagnetic initial states. Such initial states are better suited than others considered in the literature to experimentally test the validity of the complete GGE and contrast it to the failure of the local GGE.

cond-mat.stat-mech↗

arXiv2016

Multiband Effects and the Bose-Hubbard Model in One-Dimensional Lattices

We study phase diagrams of one-dimensional bosons with contact interactions in the presence of a lattice. We use the worm algorithm in continuous space and focus on the incommensurate superfluid Mott-insulator transition. Our results are compared to those from the one-band Bose-Hubbard model. When Wannier states are used to determine the Bose-Hubbard model parameters, the comparison unveils an apparent breakdown of the one-band description for strong interactions, even for the Mott-insulating state with an average of one particle per site ($n=1$) in deep lattices. We introduce an inverse confined scattering analysis to obtain the ratio $U/J$, with which the Bose-Hubbard model provides correct results for strong interactions, deep lattices, and $n=1$.

cond-mat.quant-gas↗

arXiv2016

Universal scaling of density and momentum distributions in Lieb-Liniger gases

We present an exact numerical study of the scaling of density and momentum distribution functions of harmonically trapped one-dimensional bosons with repulsive contact interactions at zero and finite temperatures. We use path integral quantum Monte Carlo with worm updates in our calculations at finite interaction strengths, and the Bose-Fermi mapping in the Tonks-Girardeau regime. We discuss the homogeneous case and, within the local density approximation, use it to motivate the scaling in the presence of a harmonic trap. For the momentum distribution function, we pay special attention to the high momentum tails and their $k^{-4}$ asymptotic behavior.

cond-mat.quant-gas↗

arXiv2016

From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics

This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We introduce the concept of the generalized Gibbs ensemble, and discuss its connection with ideas of prethermalization in weakly interacting systems.

cond-mat.stat-mech↗

arXiv2016

Dynamical preparation of Floquet Chern insulators

Realizing topological insulators is of great current interest because of their remarkable properties and possible future applications. There are recent proposals, based on Floquet analyses, that one can generate topologically nontrivial insulators by periodically driving topologically trivial ones. Here we address what happens if one follows the dynamics in such systems. Specifically, we present an exact study of the time evolution of a graphene-like system subjected to a circularly polarized electric field. We prove that, for infinite (translationally invariant) systems, the Chern number is conserved under unitary evolution. For systems with boundaries, on the other hand, we show that a properly defined topological invariant, the Bott index, can change. Hence, it should be possible to experimentally prepare topological states starting from non-topological ones. We show that the chirality of the edge current in such systems can be controlled by adjusting the filling.

cond-mat.quant-gas↗

arXiv2016

Generalized Gibbs ensemble in integrable lattice models

The generalized Gibbs ensemble (GGE) was introduced ten years ago to describe observables in isolated integrable quantum systems after equilibration. Since then, the GGE has been demonstrated to be a powerful tool to predict the outcome of the relaxation dynamics of few-body observables in a variety of integrable models, a process we call generalized thermalization. This review discusses several fundamental aspects of the GGE and generalized thermalization in integrable systems. In particular, we focus on questions such as: which observables equilibrate to the GGE predictions and who should play the role of the bath; what conserved quantities can be used to construct the GGE; what are the differences between generalized thermalization in noninteracting systems and in interacting systems mappable to noninteracting ones; why is it that the GGE works when traditional ensembles of statistical mechanics fail. Despite a lot of interest in these questions in recent years, no definite answers have been given. We review results for the XX model and for the transverse field Ising model. For the latter model, we also report original results and show that the GGE describes spin-spin correlations over the entire system. This makes apparent that there is no need to trace out a part of the system in real space for equilibration to occur and for the GGE to apply. In the past, a spectral decomposition of the weights of various statistical ensembles revealed that generalized eigenstate thermalization occurs in the XX model (hard-core bosons). Namely, eigenstates of the Hamiltonian with similar distributions of conserved quantities have similar expectation values of few-spin observables. Here we show that generalized eigenstate thermalization also occurs in the transverse field Ising model.

cond-mat.stat-mech↗

arXiv2016

Observation of Spatial Charge and Spin Correlations in the 2D Fermi-Hubbard Model

Strong electron correlations lie at the origin of transformative phenomena such as colossal magneto-resistance and high-temperature superconductivity. Already near room temperature, doped copper oxide materials display remarkable features such as a pseudo-gap and a "strange metal" phase with unusual transport properties. The essence of this physics is believed to be captured by the Fermi-Hubbard model of repulsively interacting, itinerant fermions on a lattice. Here we report on the site-resolved observation of charge and spin correlations in the two-dimensional (2D) Fermi-Hubbard model realized with ultracold atoms. Antiferromagnetic spin correlations are maximal at half-filling and weaken monotonically upon doping. Correlations between singly charged sites are negative at large doping, revealing the Pauli and correlation hole\textemdash a suppressed probability of finding two fermions near each other. However, as the doping is reduced below a critical value, correlations between such local magnetic moments become positive, signaling strong bunching of doublons and holes. Excellent agreement with numerical linked-cluster expansion (NLCE) and determinantal quantum Monte Carlo (DQMC) calculations is found. Positive non-local moment correlations directly imply potential energy fluctuations due to doublon-hole pairs, which should play an important role for transport in the Fermi-Hubbard model.

cond-mat.quant-gas↗

arXiv2016

Eigenstate thermalization in the two-dimensional transverse field Ising model

We study the onset of eigenstate thermalization in the two-dimensional transverse field Ising model (2D-TFIM) in the square lattice. We consider two non-equivalent Hamiltonians: the ferromagnetic 2D-TFIM and the antiferromagnetic 2D-TFIM in the presence of a uniform longitudinal field. We use full exact diagonalization to examine the behavior of quantum chaos indicators and of the diagonal matrix elements of operators of interest in the eigenstates of the Hamiltonian. A finite size scaling analysis reveals that quantum chaos and eigenstate thermalization occur in those systems whenever the fields are nonvanishing and not too large.

cond-mat.stat-mech↗

arXiv2016

Fundamental Asymmetry in Quenches Between Integrable and Nonintegrable Systems

We study quantum quenches between integrable and nonintegrable hard-core boson models in the thermodynamic limit with numerical linked cluster expansions. We show that while quenches in which the initial state is a thermal equilibrium state of an integrable model and the final Hamiltonian is nonintegrable (quantum chaotic) lead to thermalization, the reverse is not true. While this might appear counterintuitive given the fact that the eigenstates of both Hamiltonians are related by a unitary transformation, we argue that it is generic. Hence, the lack of thermalization of integrable systems is robust against quenches starting from stationary states of nonintegrable ones. Nonintegrable systems thermalize independently of the nature of the initial Hamiltonian.

cond-mat.stat-mech↗