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arXiv:0908.3188 · Physical Review A

Quantum quenches and thermalization in one-dimensional fermionic systems

Marcos Rigol

Published Nov 9, 2009• 243 citations
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Abstract

We study the dynamics and thermalization of strongly correlated fermions in finite one-dimensional lattices after a quantum quench. Our calculations are performed using exact diagonalization. We focus on one- and two-body observables such as the momentum distribution function $[n(k)]$ and the density-density structure factor $[N(k)]$, respectively, and study the effects of approaching an integrable point. We show that while the relaxation dynamics and thermalization of $N(k)$ for fermions is very similar to the one of hardcore bosons, the behavior of $n(k)$ is distinctively different. The latter observable exhibits a slower relaxation dynamics in fermionic systems. We identify the origin of this behavior, which is related to the off-diagonal matrix elements of $n(k)$ in the basis of the eigenstates of the Hamiltonian. More generally, we find that thermalization occurs far away from integrability and that it breaks down as one approaches the integrable point.

DOI
10.1103/physreva.80.053607

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We study the dynamics and thermalization of strongly correlated fermions in finite one-dimensional lattices after a quantum quench. Our calculations are performed using exact diagonalization. We focus on one- and two-body observables such as the momentum distribution function [n(k)][n(k)][n(k)] and the density-density structure factor [N(k)][N(k)][N(k)], respectively, and study the effects of approaching an integrable point. We show that while the relaxation dynamics and thermalization of N(k)N(k)N(k) for fermions is very similar to the one of hardcore bosons, the behavior of n(k)n(k)n(k) is distinctively different. The latter observable exhibits a slower relaxation dynamics in fermionic systems. We identify the origin of this behavior, which is related to the off-diagonal matrix elements of n(k)n(k)n(k) in the basis of the eigenstates of the Hamiltonian. More generally, we find that thermalization occurs far away from integrability and that it breaks down as one approaches the integrable point.
We review exact approaches and recent results related to the relaxation dynamics and description after relaxation of various one-dimensional lattice systems of hard-core bosons after a sudden quench. We first analyze the integrable case, where the combination of analytical insights and computational techniques enable one to study large system sizes. Thermalization does not occur in this regime. However, after relaxation, observables can be described by a generalization of the Gibbs ensemble. We then utilize full exact diagonalization to study what happens as integrability is broken. We show that thermalization does occur in finite nonintegrable systems provided they are sufficiently far away from the integrable point. We argue that the onset of thermalization can be understood in terms of the eigenstate thermalization hypothesis.
We use quantum quenches to study the dynamics and thermalization of hard core bosons in finite one-dimensional lattices. We perform exact diagonalizations and find that, far away from integrability, few-body observables thermalize. We then study the breakdown of thermalization as one approaches an integrable point. This is found to be a smooth process in which the predictions of standard statistical mechanics continuously worsen as the system moves toward integrability. We establish a direct connection between the presence or absence of thermalization and the validity or failure of the eigenstate thermalization hypothesis, respectively.
We present a comparison between the bosonization results for quantum quenches and exact diagonalizations in microscopic models of interacting spinless fermions in a one-dimensional lattice. We show that important features are missed by the bosonization technique, which predicts the persistence of long-wavelength critical properties in the long-time evolution. Instead, numerical analysis provides puzzling evidences: while the momentum distribution appears to be consistent with the presence of a singularity at kFk_FkF​, density-density correlations at small momenta clearly display a thermal-like behavior, namely Nˉq≃const{\bar{N}}_q \simeq {\rm const}Nˉq​≃const (where the overbar indicates the long-time average). This feature at small momenta is preserved in presence of an interaction term that breaks integrability, together with a rounding of the singularities at finite qqq's, showing that the bosonization approach is not able to represent the time evolution of generic one-dimensional models after a quantum quench.
We study the expectation values of observables and correlation functions at long times after a global quantum quench. Our focus is on metallic (``gapless'') fermionic many-body models and small quenches. The system is prepared in an eigenstate of an initial Hamiltonian, and the time evolution is performed with a final Hamiltonian which differs from the initial one in the value of one global parameter. We first derive general relations between time-averaged expectation values of observables as well as correlation functions and those obtained in an eigenstate of the final Hamiltonian. Our results are valid to linear and quadratic order in the quench parameter ggg and generalize prior insights in several essential ways. This allows us to develop a phenomenology for the thermalization of local quantities up to a given order in ggg. Our phenomenology is put to a test in several case studies of one-dimensional models representative of four distinct classes of Hamiltonians: quadratic ones, effectively quadratic ones, those characterized by an extensive set of (quasi-) local integrals of motion, and those for which no such set is known (and believed to be nonexistent). We show that for each of these models, all observables and correlation functions thermalize to linear order in ggg. The more local a given quantity, the longer the linear behavior prevails when increasing ggg. Typical local correlation functions and observables for which the term O(g)\mathcal{O}(g)O(g) vanishes thermalize even to order g2{g}^{2}g2. Our results show that lowest-order thermalization of local observables is an ubiquitous phenomenon even in models with extensive sets of integrals of motion.
We study the expectation values of observables and correlation functions at long times after a global quantum quench. Our focus is on metallic (`gapless') fermionic many-body models and small quenches. The system is prepared in an eigenstate of an initial Hamiltonian, and the time evolution is performed with a final Hamiltonian which differs from the initial one in the value of one global parameter. We first derive general relations between time-averaged expectation values of observables as well as correlation functions and those obtained in an eigenstate of the final Hamiltonian. Our results are valid to linear and quadratic order in the quench parameter g and generalize prior insights in several essential ways. This allows us to develop a phenomenology for the thermalization of local quantities up to a given order in g. Our phenomenology is put to a test in several case studies of one-dimensional models representative of four distinct classes of Hamiltonians: quadratic ones, effectively quadratic ones, those characterized by an extensive set of (quasi-) local integrals of motion, and those for which no such set is known (and believed to be nonexistent). We show that for each of these models, all observables and correlation functions thermalize to linear order in g. The more local a given quantity, the longer the linear behavior prevails when increasing g. Typical local correlation functions and observables for which the term O(g) vanishes thermalize even to order g^2. Our results show that lowest order thermalization of local observables is an ubiquitous phenomenon even in models with extensive sets of integrals of motion.
We implement numerical linked cluster expansions (NLCEs) to study dynamics of lattice systems following quantum quenches, and focus on a hard-core boson model in one-dimensional lattices. We find that, in the nonintegrable regime and within the accessible times, local observables exhibit exponential relaxation. We determine the relaxation rate as one departs from the integrable point and show that it scales quadratically with the strength of the integrability breaking perturbation. We compare the NLCE results with those from exact diagonalization calculations on finite chains with periodic boundary conditions, and show that NLCEs are far more accurate.