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arXiv:1108.2703 · Physical Review B

Time and spatially resolved quench of the fermionic Hubbard model showing restricted equilibration

Florian Goth, Fakher F. Assaad

Published Feb 27, 2012• 29 citations
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Abstract

We investigate the quench of half-filled one-dimensional and two-dimensional fermionic Hubbard models to models without Coulomb interaction. Since the time propagation is Gaussian, we can use a variety of time-dependent quantum Monte Carlo methods to tackle this problem without generating a dynamical sign problem. Using a continuous time quantum Monte Carlo method, we achieve a system size of 128 sites in one dimension, and using a Blankenbecler-Scalapino-Sugar type algorithm, we were able to simulate $20\ifmmode\times\else\texttimes\fi{}20$ square lattices. Applying these methods to study the dynamics after the quench, we observe that the final state of the system can be reasonably well described by a thermal single-particle density matrix that takes the initial single-particle conservation laws into account. The characteristic decay toward this limit is found to be oscillatory with an additional power law decay that depends on the dimensionality. This numerically exact result is shown to compare favorably to mean-field approximations as well as to perturbation theory. Furthermore, we observe the information propagation in the one-dimensional case in the charge-charge and spin-spin correlations and find that it is linear with a velocity of roughly $v\ensuremath{\approx}4$ in units of the hopping amplitude.

DOI
10.1103/physrevb.85.085129

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•Nov 9, 2013•0 citations
We investigate the quench of half-filled one-dimensional and two-dimensional fermionic Hubbard models to models without Coulomb interaction. Since the time propagation is Gaussian, we can use a variety of time-dependent quantum Monte Carlo methods to tackle this problem without generating a dynamical sign problem. Using a continuous time quantum Monte Carlo method, we achieve a system size of 128 sites in one dimension, and using a Blankenbecler-Scalapino-Sugar type algorithm, we were able to simulate 20\ifmmode×\else\texttimes\fi2020\ifmmode\times\else\texttimes\fi{}2020\ifmmode×\else\texttimes\fi20 square lattices. Applying these methods to study the dynamics after the quench, we observe that the final state of the system can be reasonably well described by a thermal single-particle density matrix that takes the initial single-particle conservation laws into account. The characteristic decay toward this limit is found to be oscillatory with an additional power law decay that depends on the dimensionality. This numerically exact result is shown to compare favorably to mean-field approximations as well as to perturbation theory. Furthermore, we observe the information propagation in the one-dimensional case in the charge-charge and spin-spin correlations and find that it is linear with a velocity of roughly v\ensuremath≈4v\ensuremath{\approx}4v\ensuremath≈4 in units of the hopping amplitude.
Ultracold-atom simulations of the Hubbard model provide insights into the character of charge and spin correlations in and out of equilibrium. The corresponding numerical simulations, on the other hand, remain a significant challenge. We build on recent progress in the quantum Monte Carlo (QMC) simulation of electrons in continuous space and apply similar ideas to the square-lattice Hubbard model. We devise and benchmark two discrete-time QMC methods, namely the fermionic-propagator QMC (FPQMC) and the alternating-basis QMC (ABQMC). In FPQMC, the time evolution is represented by snapshots in real space, whereas the snapshots in ABQMC alternate between real and reciprocal space. The methods may be applied to study equilibrium properties within the grand-canonical or canonical ensemble, external field quenches, and even the evolution of pure states. Various real-space/reciprocal-space correlation functions are also within their reach. Both methods deal with matrices of size equal to the number of particles (thus independent of the number of orbitals or time slices), which allows for cheap updates. We benchmark the methods in relevant setups. In equilibrium, the FPQMC method is found to have an excellent average sign and, in some cases, yields correct results even with poor imaginary-time discretization. ABQMC has a significantly worse average sign, but also produces good results. Out of equilibrium, FPQMC suffers from a strong dynamical sign problem. On the contrary, in ABQMC, the sign problem is not time-dependent. Using ABQMC, we compute survival probabilities for several experimentally relevant pure states.
The Lieb–Robinson bound implies that the unitary time evolution of an operator can be restricted to an effective light cone for any Hamiltonian with short-range interactions. Here we present a very efficient renormalization group algorithm based on this light cone structure to study the time evolution of prepared initial states in the thermodynamic limit in one-dimensional quantum systems. The algorithm does not require translational invariance and allows for an easy implementation of local conservation laws. We use the algorithm to investigate the relaxation dynamics of double occupancies in fermionic Hubbard models as well as a possible thermalization. For the integrable Hubbard model, we find a pure power-law decay of the number of doubly occupied sites towards the value in the long-time limit, while the decay becomes exponential when adding a nearest-neighbor interaction. In accordance with the eigenstate thermalization hypothesis, the long-time limit is reasonably well described by a thermal average. We point out, however, that such a description naturally requires the use of negative temperatures. Finally, we study a doublon impurity in a Néel background and find that the excess charge and spin spread at different velocities, providing an example of spin–charge separation in a highly excited state.
Thermalization in strongly correlated fermionic systems remains a central open problem in quantum many-body physics. In this work, we investigate the real-time dynamics and the approach to thermalization in the two-dimensional Hubbard model, a paradigmatic framework for correlated electrons, relevant to high-temperature superconductivity and ultracold quantum simulation. Focusing on the half-filled square lattice, we monitor the time evolution of the double occupancy following a quench in the on-site interaction UUU, and assess whether its long-time value is captured by a canonical thermal ensemble. We employ time-dependent variational Monte Carlo methods combined with transformer-based Neural-Network Quantum States to accurately describe the nonequilibrium dynamics of fermions, especially for the behavior at long times, thereby accessing regimes that were previously inaccessible to numerical simulations. Our results reveal two dynamical behaviors: for weak to intermediate interactions, the double occupancy rapidly approaches the thermal prediction, consistent with ergodic evolution; beyond a critical interaction UCU_{C}UC​, the dynamics remains distinct from the thermal expectation on the timescales investigated, revealing signatures of a prethermal plateau delaying fast relaxation. These results establish numerical simulation as a powerful tool to probe nonequilibrium quantum phenomena in correlated fermionic matter.
We study the Bose-Hubbard and Fermi-Hubbard models in the (formal) limit of large coordination numbers $Z\ensuremath{\gg}1$. Via an expansion into powers of 1/Z1/Z1/Z, we establish a hierarchy of correlations which facilitates an approximate analytical derivation of the time evolution of the reduced density matrices for one and two sites, etc. With this method, we study the quantum dynamics (starting in the ground state) after a quantum quench, i.e., after suddenly switching the tunneling rate JJJ from zero to a finite value, which is still in the Mott regime. We find that the reduced density matrices approach a (quasi)equilibrium state after some time. For one lattice site, this state can be described by a thermal state (within the accuracy of our approximation). However, the (quasi)equilibrium state of the reduced density matrices for two sites including the correlations can not be described by a thermal state. Thus, real thermalization (if it occurs) should take a much longer time. This behavior has already been observed in other scenarios and is sometimes called ``prethermalization''. Finally, we compare our results to numerical simulations for finite lattices in one and two dimensions and find qualitative agreement.
The Lieb–Robinson bound implies that the unitary time evolution of an operator can be restricted to an effective light cone for any Hamiltonian with short-range interactions. Here we present a very efficient renormalization group algorithm based on this light cone structure to study the time evolution of prepared initial states in the thermodynamic limit in one-dimensional quantum systems. The algorithm does not require translational invariance and allows for an easy implementation of local conservation laws. We use the algorithm to investigate the relaxation dynamics of double occupancies in fermionic Hubbard models as well as a possible thermalization. For the integrable Hubbard model, we find a pure power-law decay of the number of doubly occupied sites towards the value in the long-time limit, while the decay becomes exponential when adding a nearest-neighbor interaction. In accordance with the eigenstate thermalization hypothesis, the long-time limit is reasonably well described by a thermal average. We point out, however, that such a description naturally requires the use of negative temperatures. Finally, we study a doublon impurity in a Néel background and find that the excess charge and spin spread at different velocities, providing an example of spin–charge separation in a highly excited state.
We study the Bose and Fermi Hubbard model in the (formal) limit of large coordination numbers Z≫1Z\gg1Z≫1. Via an expansion into powers of 1/Z1/Z1/Z, we establish a hierarchy of correlations which facilitates an approximate analytical derivation of the time-evolution of the reduced density matrices for one and two sites etc. With this method, we study the quantum dynamics (starting in the ground state) after a quantum quench, i.e., after suddenly switching the tunneling rate JJJ from zero to a finite value, which is still in the Mott regime. We find that the reduced density matrices approach a (quasi) equilibrium state after some time. For one lattice site, this state can be described by a thermal state (within the accuracy of our approximation). However, the (quasi) equilibrium state of the reduced density matrices for two sites including the correlations cannot be described by a thermal state. Thus, real thermalization (if it occurs) should take much longer time. This behavior has already been observed in other scenarios and is sometimes called ``pre-thermalization.'' Finally, we compare our results to numerical simulations for finite lattices in one and two dimensions and find qualitative agreement.