arXiv:quant-ph/0506169 · Physical Review Letters

Entanglement and Criticality in Translationally Invariant Harmonic Lattice Systems with Finite-Range Interactions

Razmik G. Unanyan, Michael Fleischhauer

Published Dec 27, 2005• 14 citations

Abstract

We discuss the relation between entanglement and criticality in translationally invariant harmonic lattice systems with nonrandom, finite-range interactions. We show that the criticality of the system as well as validity or breakdown of the entanglement area law are solely determined by the analytic properties of the spectral function of the oscillator system, which can easily be computed. In particular, for finite-range couplings we find a one-to-one correspondence between an area-law scaling of the bipartite entanglement and a finite correlation length. This relation is strict in the one-dimensional case and there is strong evidence for the multidimensional case. We also discuss generalizations to couplings with infinite range. Finally, to illustrate our results, a specific 1D example with nearest and next-nearest-neighbor coupling is analyzed.

DOI: 10.1103/physrevlett.95.260604

Related papers

Entanglement-area law for general bosonic harmonic lattice systems

M. Cramer, Jens Eisert, Martin Bodo Plenio +1 more

We demonstrate that the entropy of entanglement and the distillable entanglement of regions with respect to the rest of a general harmonic-lattice system in the ground or a thermal state scale at most as the boundary area of the region. This area law is rigorously proven to hold true in noncritical harmonic-lattice systems of arbitrary spatial dimension, for general finite-ranged harmonic interactions, regions of arbitrary shape, and states of nonzero temperature. For nearest-neighbor interactions---corresponding to the Klein-Gordon case---upper and lower bounds to the degree of entanglement can be stated explicitly for arbitrarily shaped regions, generalizing the findings of Phys. Rev. Lett. 94, 060503 (2005). These higher-dimensional analogs of the analysis of block entropies in the one-dimensional case show that under general conditions, one can expect an area law for the entanglement in noncritical harmonic many-body systems. The proofs make use of methods from entanglement theory, as well as of results on matrix functions of block-banded matrices. Disordered systems are also considered. We moreover construct a class of examples for which the two-point correlation length diverges, yet still an area law can be proven to hold. We finally consider the scaling of classical correlations in a classical harmonic system and relate it to a quantum lattice system with a modified interaction. We briefly comment on a general relationship between criticality and area laws for the entropy of entanglement.

arXiv:quant-ph/0505092Physical Review AJan 10, 2006223 citations

Entropy, Entanglement, and Area: Analytical Results for Harmonic Lattice Systems

Martin Bodo Plenio, Jens Eisert, J. Dreißig +1 more

We revisit the question of the relation between entanglement, entropy, and area for harmonic lattice Hamiltonians corresponding to discrete versions of real free Klein-Gordon fields. For the ground state of the d-dimensional cubic harmonic lattice we establish a strict relationship between the surface area of a distinguished hypercube and the degree of entanglement between the hypercube and the rest of the lattice analytically, without resorting to numerical means. We outline extensions of these results to longer ranged interactions, finite temperatures, and for classical correlations in classical harmonic lattice systems. These findings further suggest that the tools of quantum information science may help in establishing results in quantum field theory that were previously less accessible.

arXiv:quant-ph/0405142Physical Review LettersFeb 17, 2005392 citations

Entanglement Dynamics in Harmonic Oscillator Chains

Razmik G. Unanyan, Michael Fleischhauer

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.

arXiv:1011.4838Nov 22, 20100 citations

Correlations, spectral gap and entanglement in harmonic quantum systems on generic lattices

M. Cramer, Jens Eisert

We investigate the relationship between the gap between the energy of the ground state and the first excited state and the decay of correlation functions in harmonic lattice systems. We prove that in gapped systems, the exponential decay of correlations follows for both the ground state and thermal states. Considering the converse direction, we show that an energy gap can follow from algebraic decay and always does for exponential decay. The underlying lattices are described as general graphs of not necessarily integer dimension, including translationally invariant instances of cubic lattices as special cases. Any local quadratic couplings in position and momentum coordinates are allowed for, leading to quasi-free (Gaussian) ground states. We make use of methods of deriving bounds to matrix functions of banded matrices corresponding to local interactions on general graphs. Finally, we give an explicit entanglement-area relationship in terms of the energy gap for arbitrary, not necessarily contiguous regions on lattices characterized by general graphs.

arXiv:quant-ph/0509167New Journal of PhysicsMay 22, 200688 citations

Ground state entanglement in one-dimensional translationally invariant quantum systems

Sandy Irani

We examine whether it is possible for one-dimensional translationally invariant Hamiltonians to have ground states with a high degree of entanglement. We present a family of translationally invariant Hamiltonians {Hn} for the infinite chain. The spectral gap of Hn is Ω(1/poly(n)). Moreover, for any state in the ground space of Hn and any m, there are regions of size m with entanglement entropy Ω(min{m,n}). A similar construction yields translationally invariant Hamiltonians for finite chains that have unique ground states exhibiting high entanglement. The area law proven by Hastings [“An area law for one dimensional quantum systems,” J. Stat. Mech.: Theory Exp. 2007 (08024)] gives a constant upper bound on the entanglement entropy for one-dimensional ground states that is independent of the size of the region but exponentially dependent on 1/Δ, where Δ is the spectral gap. This paper provides a lower bound, showing a family of Hamiltonians for which the entanglement entropy scales polynomially with 1/Δ. Previously, the best known such bound was logarithmic in 1/Δ.

arXiv:0901.1107Journal of Mathematical PhysicsFeb 1, 201043 citations

Entropic area law for Fermions

Wolf, M M

We investigate the scaling of the entanglement entropy in an infinite translational invariant Fermionic system of any spatial dimension. The states under consideration are ground states and excitations of tight-binding Hamiltonians with arbitrary interactions. We show that the entropy of a finite region typically scales with the area of the surface times a logarithmic correction. The latter can be traced back to the criticality of the models. The relation between the entanglement entropy and the structure of the Fermi surface is discussed, and it is proven, that the presented scaling law holds whenever the Fermi surface is finite. This is in particular true for all ground states of Hamiltonians with finite range interactions.

Mar 29, 20051 citation