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arXiv:cond-mat/0606126 · Journal of Physics A Mathematical and Theoretical

Two-point entanglement near a quantum phase transition

Han-Dong Chen

Published Aug 1, 2007• 13 citations
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Abstract

In this work, we study the two-point entanglement S ( i , j ), which measures the entanglement between two separated degrees of freedom ( ij ) and the rest of system, near a quantum phase transition. Away from the critical point, S ( i , j ) saturates with a characteristic length scale ξ E , as the distance | i − j | increases. The entanglement length ξ E agrees with the correlation length. The universality and finite size scaling of entanglement are demonstrated in a class of exactly solvable one-dimensional spin model. By connecting the two-point entanglement to correlation functions in the long range limit, we argue that the prediction power of a two-point entanglement is universal as long as the two involved points are separated far enough.

DOI
10.1088/1751-8113/40/33/017

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In this work, we study the two-point entanglement S ( i , j ), which measures the entanglement between two separated degrees of freedom ( ij ) and the rest of system, near a quantum phase transition. Away from the critical point, S ( i , j ) saturates with a characteristic length scale ξ E , as the distance | i − j | increases. The entanglement length ξ E agrees with the correlation length. The universality and finite size scaling of entanglement are demonstrated in a class of exactly solvable one-dimensional spin model. By connecting the two-point entanglement to correlation functions in the long range limit, we argue that the prediction power of a two-point entanglement is universal as long as the two involved points are separated far enough.
The role of two-point and multipartite entanglement at quantum phase transitions (QPTs) in correlated electron systems is investigated. We consider a bond-charge extended Hubbard model exactly solvable in one dimension which displays various QPTs---with two (qubit) as well as more (qudit) on-site degrees of freedom involved. The analysis is carried out by means of appropriate measures of bipartite/multipartite quantum correlations. It is found that all transitions ascribed to two-point correlations are characterized by an entanglement range which diverges at the transition points. The exponent coincides with that of the correlation length at the transitions. We introduce the correlation ratio, namely, the ratio of quantum mutual information and single-site entanglement. We show that at T=0T=0T=0, it captures the relative role of two-point and multipartite quantum correlations at transition points, generalizing to qudit systems the entanglement ratio. Moreover, a finite value of quantum mutual information between infinitely distant sites is seen to quantify the presence of off-diagonal long-range order induced by multipartite entanglement.
The entanglement in quantum XY spin chains of arbitrary length is investigated via the geometric measure of entanglement. The emergence of entanglement is explained intuitively from the perspective of perturbations. The model is solved exactly and the energy spectrum is determined and analyzed in particular for the lowest two levels for both finite and infinite systems. The overlaps for these two levels are calculated analytically for arbitrary number of spins. The entanglement is hence obtained by maximizing over a single parameter. The corresponding ground-state entanglement surface is then determined over the entire phase diagram, and its behavior can be used to delineate the boundaries in the phase diagram. For example, the field-derivative of the entanglement becomes singular along the critical line. The form of the divergence is derived analytically and it turns out to be dictated by the universality class controlling the quantum phase transition. The behavior of the entanglement near criticality can be understood via a scaling hypothesis, analogous to that for free energies. The entanglement density vanishes along the so-called disorder line in the phase diagram, the ground space is doubly degenerate and spanned by two product states. The entanglement for the superposition of the lowest two states is also calculated. The exact value of the entanglement depends on the specific form of superposition. However, in the thermodynamic limit the entanglement density turns out to be independent of the superposition. This proves that the entanglement density is insensitive to whether the ground state is chosen to be the spontaneously Z2Z_2Z2​ symmetry broken one or not. The finite-size scaling of entanglement at critical points is also investigated from two different view points. First, the maximum in the field-derivative of the entanglement density is computed and fitted to a logarithmic dependence of the system size, thereby deducing the correlation length exponent for the Ising class using only the behavior of entanglement. Second, the entanglement density itself is shown to possess a correction term inversely proportional to the system size, with the coefficient being universal (but with different values for the ground state and the first excited state, respectively).
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Motivated by recent development in quantum entanglement, we study relations among concurrence C, SUq(2){\mathrm{SU}}_{q}(2)SUq​(2) algebra, quantum phase transition and correlation length at the zero temperature for the XXZ\mathrm{XXZ}XXZ chain. We find that at the SU(2) point, the ground state possesses the maximum concurrence. When the anisotropic parameter $\ensuremath{\Delta}$ is deformed, however, its value decreases. Its dependence on $\ensuremath{\Delta}$ scales as ${C=C}_{0}\ensuremath{-}{C}_{1}(\ensuremath{\Delta}\ensuremath{-}{1)}^{2}$ in the XY\mathrm{XY}XY metallic phase and near the critical point (i.e., $1<\ensuremath{\Delta}<1.3)$ of the Ising-like insulating phase. We also study the dependence of C on the correlation length $\ensuremath{\xi},$ and show that it satisfies ${C=C}_{0}\ensuremath{-}1/2\ensuremath{\xi}$ near the critical point. For different sizes of the system, we show that there exists a universal scaling function of C with respect to the correlation length $\ensuremath{\xi}.$
We study the two-point correlation functions and the bipartite entanglement in the ground state of the exactly-solvable variable-range extended Ising model of qubits in the presence of a transverse field on a one-dimensional lattice. We introduce the variation in the range of interaction by varying the coordination number, Z\mathcal{Z}Z, of each qubit, where the interaction strength between a pair of qubits at a distance rrr varies as ∼r−α\sim r^{-α}∼r−α. We show that the algebraic nature of the correlation functions is present only up to r=Zr=\mathcal{Z}r=Z, above which it exhibits short-range exponential scaling. We also show that at the critical point, the bipartite entanglement exhibits a power-law decrease (∼Z−γ\sim\mathcal{Z}^{-γ}∼Z−γ) with increasing coordination number irrespective of the partition size and the value of ααα for α>1α>1α>1. We further consider a sudden quench of the system starting from the ground state of the infinite-field limit of the system Hamiltonian via turning on the critical Hamiltonian, and demonstrate that the long-time averaged bipartite entanglement exhibits a qualitatively similar variation (∼Z−γ\sim\mathcal{Z}^{-γ}∼Z−γ) with Z\mathcal{Z}Z.