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arXiv:2310.11216 · Physical review. B./Physical review. B

Back-reaction and correlation effects on prethermalization in Mott-Hubbard systems

Friedemann Queißer, Christian Kohlfürst, Ralf Schützhold

Published May 13, 2024• 6 citations
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Abstract

For the Fermi-Hubbard model in the strongly interacting Mott insulator state, we study the prethermalization dynamics after a quench (switching on the hopping rate). To this end, we employ the method of the hierarchy of correlations and compare different levels of accuracy. To leading order, the usual free quasiparticle dynamics (as encoded in the two-point correlation functions) yields the standard picture of prethermalization. Taking into account the back-reaction of these quasiparticle fluctuations onto the mean-field background as the first next-to-leading-order effect, we observe a strong degradation of prethermalization, especially in low dimensions. In contrast, the inclusion of three-point correlations enhances prethermalization.

DOI
10.1103/physrevb.109.195140

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•Nov 9, 2013•0 citations
For the Fermi-Hubbard model in the strongly interacting Mott insulator state, we study the prethermalization dynamics after a quench (switching on the hopping rate). To this end, we employ the method of the hierarchy of correlations and compare different levels of accuracy. To leading order, the usual free quasiparticle dynamics (as encoded in the two-point correlation functions) yields the standard picture of prethermalization. Taking into account the back-reaction of these quasiparticle fluctuations onto the mean-field background as the first next-to-leading-order effect, we observe a strong degradation of prethermalization, especially in low dimensions. In contrast, the inclusion of three-point correlations enhances prethermalization.
Via the hierarchy of correlations, we study the strongly interacting Fermi-Hubbard model in the Mott insulator state and couple it to a Markovian environment that constantly monitors the particle numbers ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{\ensuremath{\mu}}^{\ensuremath{\uparrow}}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{\ensuremath{\mu}}^{\ensuremath{\downarrow}}$ for each lattice site $\ensuremath{\mu}$. As expected, the environment induces an imaginary part $\ensuremath{\gamma}$ (i.e., the decay rate) of the quasiparticle frequencies ${\ensuremath{\omega}}_{\mathbf{k}}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{\mathbf{k}}\ensuremath{-}i\ensuremath{\gamma}$, and it tends to diminish the correlations between lattice sites. Surprisingly, the environment also steers the state of the system on intermediate timescales $O(1/\ensuremath{\gamma})$ to a prerelaxed state very similar to the prethermalized state after a quantum quench (i.e., suddenly switching on the hopping rate JJJ). Full relaxation or thermalization occurs via local on-site heating and takes much longer.
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.
We investigate the time evolution of correlations in the Bose-Hubbard model following a quench from the superfluid to the Mott insulator. For large values of the final interaction strength the system approaches a distinctly nonequilibrium steady state that bears strong memory of the initial conditions. In contrast, when the final interaction strength is comparable to the hopping, the correlations are rather well approximated by those at thermal equilibrium. The existence of two distinct nonequilibrium regimes is surprising given the nonintegrability of the Bose-Hubbard model. We relate this phenomenon to the role of quasiparticle interactions in the Mott insulator.
We study a quantum quench in the Bose-Hubbard model where the tunneling rate J is suddenly switched from zero to a finite value in the Mott regime. In order to solve the many-body quantum dynamics far from equilibrium, we consider the reduced density matrices for a finite number of lattice sites and split them up into on-site density operators, i.e., the mean field, plus two-point and three-point correlations etc. Neglecting three-point and higher correlations, we are able to numerically simulate the time-evolution of the on-site density matrices and the two-point quantum correlations (e.g., their effective light-cone structure) for a comparably large number of lattice sites.
Via the hierarchy of correlations, we study the Mott insulator phase of the Fermi-Hubbard model in the limit of strong interactions and derive a quantum Boltzmann equation describing its relaxation dynamics. In stark contrast to the weakly interacting case, we find that the scattering cross sections strongly depend on the momenta of the colliding quasiparticles and holes. Therefore, the relaxation towards equilibrium crucially depends on the spectrum of excitations. For example, for particle-hole excitations directly at the minimum of the (direct) Mott gap, the scattering cross sections vanish such that these excitations can have a very long lifetime.
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.