Entanglement and spontaneous symmetry breaking in quantum spin models

Abstract

It is shown that spontaneous symmetry breaking does not modify the ground-state entanglement of two spins, as defined by the concurrence, in the XXZ-and the transverse field Ising-chain. Correlation function inequalities, valid in any dimensions for these models, are presented outlining the regimes where entanglement is unaffected by spontaneous symmetry breaking.PACS numbers: 03.67. Mn, 75.10.Jm Entanglement is a property of a quantum state shared between two or more parties. It is defined with the aim of capturing the essential quantum non-locality encoded in the state. While for a long time interests in entanglement stemmed from the opportunity to understand fundamental concepts in quantum mechanics such as the EPR-paradox and violation of Bell-inequalities, recent interest in entanglement comes from its use as a resource for performing tasks not possible by classical means.From an applied viewpoint it is thus worth quantifying the degree of entanglement in natural systems, such as solid-state materials. A number of studies have been devoted to quantifying such "natural" entanglement in states of simple models describing idealized quantum magnets, such as the XXZ-and transverse field Isingmodels [1,2,3,4,5,6]. However the possible alteration of entanglement by spontaneous symmetry breaking (SSB) were not discussed in any of these works, although the need for such a study was mentioned in Ref. [5]. SSB happens invariably in real materials described by a Hamiltonian possessing a global symmetry, thus it is important to investigate whether or not the entanglement calculated in the symmetric ground state(s) is changed by SSB. Here we find that entanglement is not modified by SSB for the XXZ-and transverse field Ising-chains.It is a fundamental requirement of entanglement that it cannot on average be created by mixing classically two quantum states. Classical mixing is incoherent, and so one should not gain more entanglement from this mixing than what is already encoded in the states that are being mixed. Thus if we consider the density matrix ρ = (ρ + + ρ − )/2 put together by an equal mixture of the broken symmetry states, ρ ± (not necessarily pure), the entanglement E(ρ) must satisfywhere we have assumed in the last equality that the entanglement in the two different symmetry-broken ground states are equal, the states being related by a global change of basis states. Thus the entanglement in the broken state cannot be smaller than in the symmetric state.Here we focus on the entanglement of two specific spins (or qubits) in the ground-state of a quantum system. The state of two spins i and j in the ground state of a quantum system is described in terms of the reduced density matrix ρ ij obtained by tracing over all spins in the ground state except the two spins i and j. Writing it out explicitly in the standard basis {| ↑↑ , | ↑↓ , | ↓↑ , | ↓↓ } one has ρ ij =

Reexamination of entanglement and the quantum phase transition

“…It is found that C i,i+1 and ∂C i,i+1 /∂∆ are both continuous at ∆ = 1, and ∂C i,i+1 /∂∆| ∆=1 = 0 (or C i,i+1 reaches its maximum value at ∆ = 1) [5,6]. It is interesting to see how these results can be realized in the present framework.…”

“…From Refs. [6,7] and our result, it is clear that QPTs in general can not be distinctly characterized through the analysis of the analyticity properties of concurrence. Moreover, we show that, for the model considered in this paper, the von Neumann entropy remains constant even crossing the critical point.…”

“…That is, it is not always possible to infer the existence of QPTs from concurrence. Furthermore, for the one-dimensional XXZ model at the critical point of the isotropic ferromagnetic case, it is found that the first derivative of the concurrence is discontinuous [6]. However, it is a 1QPT, instead of 2QPT.…”

“…Thus the results in Refs. [5,6] are reproduced. In summary, although many examples indicate that QPTs can be distinctly characterized through the analyticity properties of concurrence, we stress in this paper that this viewpoint is not true in general.…”

“…Quantum entanglement, as one of the most fascinating feature of quantum theory, has attracted much attention over the past decade, mostly because its nonlocal connotation [1] is regarded as a valuable resource in quantum communication and information processing [2]. Recently a great deal of effort has been devoted to the understanding of the connection between quantum entanglement and quantum phase transitions (QPTs) [3,4,5,6,7,8,9,10,11,12,13,14,15]. Quantum phase transitions [16] are transitions between qualitatively distinct phases of quantum many-body systems, driven by quantum fluctuations.…”