REDUCED DENSITY MATRIX AFTER A QUANTUM QUENCH

By M. Fagotti, F. Essler. 2013

222 citations recorded by the source corpus.

We consider the reduced density matrix (RDM) \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 \rho_A(t) is described by the RDM \rho_{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, how \rho_A(t) approaches \rho_{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^{-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.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1f8907495033920ec52dc04a875d21695265e401

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. Pasquale Calabrese
2. J. Cardy
3. M. Fagotti
4. V. Alba
5. F. Essler
6. J. Caux
7. J. Dubail
8. L. Vidmar

In Condensed Matter Physics

1. Pasquale Calabrese
2. J. Cardy
3. M. Fagotti
4. J. Caux
5. J. Dubail
6. V. Alba
7. F. Essler
8. Marcos Rigol

In Statistical and Nonlinear Physics

1. Pasquale Calabrese
2. J. Cardy
3. V. Alba
4. J. Caux
5. M. Fagotti
6. F. Essler
7. L. Vidmar
8. Marcos Rigol

Citations in the corpus

• Quench dynamics and relaxation in isolated integrable quantum spin chains (2016, methodology): “Spin-flip symmetry The TFIC Hamiltonian (6) is invariant under rotations in spin space around the z-axis by 180 degrees Uσα` U † = −σα` , α = x, y, Uσz`U† = σz` .”
• Generalized Gibbs ensemble in integrable lattice models (2016, background): “Since then, the validity of the GGE to describe observables in integrable systems after equilibration has been demonstrated in a wide range of 1D models including: Luttinger liquids [39, 40, 41], the 1/r fermionic Hubbard model [42], the sine-Gordon model [43], the transverse field Ising model [44, ”
• The Quench Action (2016, methodology): “The post-quench reduced density matrix has been computed in [89], giving ρ(t) = |ρsp〉〈ρsp| ∝ e 1 4 aWa with a being a vector of Majorana modes a2n−1 =”
• Entanglement dynamics after quantum quenches in generic integrable systems (2017, methodology): “For all these models, the GGE built with local conservation laws is equivalent to the one built with the mode occupation numbers n̂k = b † kbk since they are linearly related [46, 50].”
• Aspects of non-equilibrium in classical and quantum systems: Slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics (2017, background): “initial energy. All other initial details are forgotten. Thermalisation is thus the general setup for quantum ergodicity where the system can act as its own thermal reservoir.For references see e.g., [60, 61, 62, 63, 64, 65]. Quantum thermalisation can be seen as a consequence of the spectrum of a m”
• Quantum quenches in 1  +  1 dimensional conformal field theories (2016, background): “Let us now discuss the behaviour of correlation functions and let us again start by reviewing some exact results for the Ising chain (28), whose correlation functions after a quench have been determined analytically in [92, 93, 94, 95, 96, 97] (see [98, 99, 100, 101, 102] for earlier numerical works”
• Entanglement scaling of operators: a conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1  +  1d (2016, background): “a sum of algebraically decaying terms [62], and yet the mutual information satisfies an area law according to the discussion in the conclusion of Ref.”