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arXiv:0910.3868 · Physical Review A

Short-time versus long-time dynamics of entanglement in quantum lattice models

Razmik G. Unanyan, Dominik Muth, Michael Fleischhauer

Published Feb 24, 2010• 8 citations
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Abstract

We study the short-time evolution of the bipartite entanglement in quantum lattice systems with local interactions in terms of the purity of the reduced density matrix. A lower bound for the purity is derived in terms of the eigenvalue spread of the interaction Hamiltonian between the partitions. Starting from an initially separable state the purity decreases as $1\ensuremath{-}(t/\ensuremath{\tau}){}^{2}$ (i.e., quadratically in time, with a characteristic timescale $\ensuremath{\tau}$ that is inversely proportional to the boundary size of the subsystem, that is, as an area law). For larger times an exponential lower bound is derived corresponding to the well-known linear-in-time bound of the entanglement entropy. The validity of the derived lower bound is illustrated by comparison to the exact dynamics of a one-dimensional spin lattice system as well as a pair of coupled spin ladders obtained from numerical simulations.

DOI
10.1103/physreva.81.022119

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We study the long-time evolution of the bipartite entanglement in translationally invariant gapped harmonic lattice systems with finite-range interactions. A lower bound for the von Neumann entropy is derived in terms of the purity of the reduced density matrix. It is shown that starting from an initially Gaussian state the entanglement entropy increases at least linearly in time. This implies that the dynamics of gapped (non-critical) harmonic lattice systems cannot be efficiently simulated by algorithms based on matrix-product decompositions of the quantum state.
We study the long-time evolution of the bipartite entanglement in translationally invariant one-dimensional harmonic lattice systems. We show that for Gaussian states, in quadratic interactions with periodic boundary conditions, there exists a lower bound for the von Neumann entropy which increases linearly in time. This implies that the dynamics of harmonic lattice systems can in general not efficiently be simulated by algorithms based on matrix-product decompositions of the quantum state, and interactions are needed to suppress the entanglement growth with time.
We prove that the entanglement entropy of any state evolved under an arbitrary 1/r^{α} long-range-interacting D-dimensional lattice spin Hamiltonian cannot change faster than a rate proportional to the boundary area for any α>D+1. We also prove that for any α>2D+2, the ground state of such a Hamiltonian satisfies the entanglement area law if it can be transformed along a gapped adiabatic path into a ground state known to satisfy the area law. These results significantly generalize their existing counterparts for short-range interacting systems, and are useful for identifying dynamical phase transitions and quantum phase transitions in the presence of long-range interactions.
We study the dynamics of the entanglement spectrum, that is the time evolution of the eigenvalues of the reduced density matrices after a bipartition of a one-dimensional spin chain. Starting from the ground state of an initial Hamiltonian, the state of the system is evolved in time with a new Hamiltonian. We consider both instantaneous and quasi adiabatic quenches of the system Hamiltonian across a quantum phase transition. We analyse the Ising model that can be exactly solved and the XXZ for which we employ the time-dependent density matrix renormalisation group algorithm. Our results show once more a connection between the Schmidt gap, i.e. the difference of the two largest eigenvalues of the reduced density matrix and order parameters, in this case the spontaneous magnetisation.
We analyze the dynamical generation of entanglement in systems of two interacting spins initially prepared in a product of spin coherent states. For arbitrary time-independent Hamiltonians, we derive a semiclassical expression for the purity of the reduced density matrix as a function of time. The final formula, subsidiary to the linear entropy, shows that the short-time dynamics of entanglement depends exclusively on the stability of trajectories governed by the underlying classical Hamiltonian. In addition, this semiclassical measure is shown to reproduce the general properties of its quantum counterpart and give the expected result in the large spin limit. The accuracy of the semiclassical formula is further illustrated in a problem of phase exchange for two particles of spin jjj.
We investigate bipartite entanglement in spin-1∕21∕21∕2 systems on a generic lattice. For states that are an equal superposition of elements of a group GGG of spin flips acting on the fully polarized state ${\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}n}$, we find that the von Neumann entropy depends only on the boundary between the two subsystems AAA and BBB. These states are stabilized by the group GGG. A physical realization of such states is given by the ground state manifold of the Kitaev's model on a Riemann surface of genus g\mathfrak{g}g. For a square lattice, we find that the entropy of entanglement is bounded from above and below by functions linear in the perimeter of the subsystem AAA and is equal to the perimeter (up to an additive constant) when AAA is convex. The entropy of entanglement is shown to be related to the topological order of this model. Finally, we find that some of the ground states are absolutely entangled, i.e., no partition has zero entanglement. We also provide several examples for the square lattice.