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arXiv:1302.6944 · Physical Review B

Reduced density matrix after a quantum quench

Maurizio Fagotti, Fabian H. L. Eßler

Published Jun 12, 2013• 325 citations
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Abstract

We consider the reduced density matrix (RDM) ${\ensuremath{\rho}}_{A}(t)$ for a finite subsystem $A$ after a global quantum quench in the infinite transverse-field Ising chain. It has been recently shown that the infinite time limit of ${\ensuremath{\rho}}_{A}(t)$ is described by the RDM ${\ensuremath{\rho}}_{\mathrm{GGE},A}$ of a generalized Gibbs ensemble. Here, we present some details on how to construct this ensemble in terms of local integrals of motion, and show its equivalence to the expression in terms of mode occupation numbers widely used in the literature. We then address the question of how ${\ensuremath{\rho}}_{A}(t)$ approaches ${\ensuremath{\rho}}_{\mathrm{GGE},A}$ as a function of time. To that end, we introduce a distance on the space of density matrices and show that it approaches zero as a universal power law ${t}^{\ensuremath{-}3/2}$ in time. As the RDM completely determines all local observables within $A$, this provides information on the relaxation of correlation functions of local operators. We then address the issue of how well a truncated generalized Gibbs ensemble with a finite number of local higher conservation laws describes a given subsystem at late times. We find that taking into account only local conservation laws with a range at most comparable to the subsystem size provides a good description. However, excluding even a single one of the most local conservation laws in general completely spoils this agreement.

DOI
10.1103/physrevb.87.245107

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We consider the reduced density matrix (RDM) \ensuremathρA(t){\ensuremath{\rho}}_{A}(t)\ensuremathρA​(t) for a finite subsystem AAA after a global quantum quench in the infinite transverse-field Ising chain. It has been recently shown that the infinite time limit of \ensuremathρA(t){\ensuremath{\rho}}_{A}(t)\ensuremathρA​(t) is described by the RDM \ensuremathρGGE,A{\ensuremath{\rho}}_{\mathrm{GGE},A}\ensuremathρGGE,A​ of a generalized Gibbs ensemble. Here, we present some details on how to construct this ensemble in terms of local integrals of motion, and show its equivalence to the expression in terms of mode occupation numbers widely used in the literature. We then address the question of how \ensuremathρA(t){\ensuremath{\rho}}_{A}(t)\ensuremathρA​(t) approaches \ensuremathρGGE,A{\ensuremath{\rho}}_{\mathrm{GGE},A}\ensuremathρGGE,A​ as a function of time. To that end, we introduce a distance on the space of density matrices and show that it approaches zero as a universal power law t\ensuremath−3/2{t}^{\ensuremath{-}3/2}t\ensuremath−3/2 in time. As the RDM completely determines all local observables within AAA, this provides information on the relaxation of correlation functions of local operators. We then address the issue of how well a truncated generalized Gibbs ensemble with a finite number of local higher conservation laws describes a given subsystem at late times. We find that taking into account only local conservation laws with a range at most comparable to the subsystem size provides a good description. However, excluding even a single one of the most local conservation laws in general completely spoils this agreement.
We present a numerical study of subsystem distance decay following a global quantum quench in the infinite one-dimensional transverse-field Ising chain, using the mathematically rigorous Bures distance BA(t)B_A(t)BA​(t) to quantify the deviation of the time-evolved reduced density matrix from its stationary generalized Gibbs ensemble state. We show that the late-time decay follows a discrete power law BA(t)∼t−λB_A(t) \sim t^{-λ}BA​(t)∼t−λ, with the exponent λλλ confined to discrete values: 111, 5/45/45/4, 3/23/23/2, 7/47/47/4, 222, 5/25/25/2, and potentially further values. The specific exponent is jointly determined by the pre- and post-quench transverse fields, as well as by properties of the symmetric excitation-fraction function mS(φ)m_S(\varphi)mS​(φ), defined on φ∈[0,π]\varphi\in[0,π]φ∈[0,π] to characterize the pre-quench Hamiltonian eigenstates, including continuity, boundary values, and first-derivative boundary values, among others. The previously established t−3/2t^{-3/2}t−3/2 decay for the initial ground state of the pre-quench Hamiltonian is naturally recovered as a special case of this general classification. Our results reveal a universal discrete structure governing local equilibration dynamics in integrable quantum systems.
We consider the time evolution after sudden quenches of global parameters in translational invariant Hamiltonians and study the time average expectation values and entanglement entropies in finite chains. We show that in noninteracting models the time average of spin correlation functions is asymptotically equal to the infinite time limit in the infinite chain, which is known to be described by a generalized Gibbs ensemble. The equivalence breaks down considering nonlocal operators, and we establish that this can be traced back to the existence of conservation laws common to the Hamiltonian before and after the quench. We develop a method to compute the leading finite-size corrections. We find that the finite-size corrections are generally large in observables with large relaxation timescales.
We consider dynamic (non-equal-time) correlation functions of local observables after a quantum quench. We show that, in the absence of long-range interactions in the final Hamiltonian, the dynamics is determined by the same ensemble that describes static (equal-time) correlations. For many integrable models, static correlation functions of local observables after a quantum quench relax to stationary values, which are described by a generalized Gibbs ensemble. The same generalized Gibbs ensemble then determines dynamic correlation functions, and the basic form of the fluctuation dissipation theorem holds, although the absorption and emission spectra are not simply related as in the thermal case. For quenches in the transverse field Ising chain, we derive explicit expressions for the time evolution of dynamic order parameter correlators after a quench.
Abstract We revisit the question of whether and how the steady states arising after non-equilibrium time evolution in integrable models (and in particular in the XXZ spin chain) can be described by the so-called generalized Gibbs ensemble (GGE). Whereas it is known that the micro-canonical ensemble built on a complete set of charges correctly describes the long-time limit of local observables, it has been shown recently by Ilievski et al that the corresponding canonical ensemble is not well defined, and instead a different canonical ensemble was proposed in terms of particle occupation number operators. Here we provide an alternative construction by considering truncated GGEs (tGGEs) that include only a finite number of local and quasi-local conserved operators. It is shown that the tGGEs can approximate the steady states with arbitrary precision, i.e. all physical observables are exactly reproduced in the infinite truncation limit. We trace back the problems encountered in defining an untruncated GGE to the dependence of the associated Lagrange multipliers on the truncation index. Conversely, we show that this problem may be circumvented by considering a new set of (quasi)local charges which are linear combinations of the standard ones, and whose associated Lagrange multipliers are well-defined state functions. Our general arguments are applied to concrete quench situations in the XXZ chain, where the initial states are simple two-site or four-site product states. Depending on the quench we find that numerical results for the local correlators can be obtained with remarkable precision using truncated GGEs with only 10–100 charges.
Abstract We study the quench dynamics in continuous relativistic quantum field theory, more specifically the locality properties of the large time stationary state. After a quantum quench in a one-dimensional integrable model, the expectation values of local observables are expected to relax to a generalized Gibbs ensemble (GGE), constructed out of the conserved charges of the model. Quenching to a free bosonic theory, it has been shown that the system indeed relaxes to a GGE described by the momentum mode occupation numbers. We first address the question whether the latter can be written directly in terms of local charges and we find that, in contrast to the lattice case, this is not possible in continuous field theories. We then investigate the less stringent requirement of the existence of a sequence of truncated local GGEs that converges to the correct steady state, in the sense of the expectation values of the local observables. While we show that such a sequence indeed exists, in order to unequivocally determine the so-defined GGE, we find that information about the expectation value of the recently discovered quasi-local charges is in the end necessary, the latter being the suitable generalization of the local charges while passing from the lattice to the continuum. Lastly, we study the locality properties of the GGE and show that the latter is completely determined by the knowledge of the expectation value of a countable set of suitably defined quasi-local charges.