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AIP conference proceedings

Scaling, Entanglement, and Quantum Phase Transitions

Andreas Osterloh

Published Jan 1, 2003• 1 citation
Publisher

Abstract

We study the entanglement near a quantum phase transition in a class of exactly solvable one‐dimensional quantum magnets, the anisotropic XY models. The critical behaviour (scaling) is reflected by the derivatives of the bipartite entanglent, quantified by the concurrence. However, we found a profound difference between classical correlation and non‐local quantum correlation: The range of the concurrence is found to behave in a non‐universal way, being extremely short‐ranged for the Ising model. This range tends to infinity towards the isotropic XY model.

DOI
10.1063/1.1612400

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We study the entanglement near a quantum phase transition in a class of exactly solvable one‐dimensional quantum magnets, the anisotropic XY models. The critical behaviour (scaling) is reflected by the derivatives of the bipartite entanglent, quantified by the concurrence. However, we found a profound difference between classical correlation and non‐local quantum correlation: The range of the concurrence is found to behave in a non‐universal way, being extremely short‐ranged for the Ising model. This range tends to infinity towards the isotropic XY model.
In this paper the entanglement and quantum phase transition of the anisotropic spin-1/21/21/2 XY\mathit{XY}XY model are studied by using the quantum renormalization-group method. By solving the renormalization equations, we get the trivial and nontrivial fixed points, which correspond to the phase of the system and the critical point, respectively. The concurrence between two blocks are calculated and it is found that when the number of iterations of the renormalization tends to infinity, the concurrence develops two saturated values that are associated with two different phases, i.e., Ising-like and spin-fluid phases. We also investigate the first derivative of the concurrence and find that there exists nonanalytic behaviors at the quantum critical point, which are directly associated with the divergence of the correlation length. To gain further insight, the scaling behaviors of the system are analyzed and it is shown that the maximum value of the first derivative of the concurrence reaches infinity and the critical point is approached as the size of the system increases.
We examine the ability of quantum discord (QD) and entanglements (concurrence, EoF and negativity) to detect the critical points associated to quantum phase transitions (QPTs) for XY models, i.e., the isotropic XY model with three-spin interactions at zero temperature, and the anisotropic XY model in a transverse magnetic field h at finite temperatures. For the case of zero temperature, we found that both entanglements and QD can spotlight the critical points of QPTs for these two models. Moreover, QD versus distance M exhibits the long-range behavior of quantum correlation for the anisotropic XY model, while entanglement is short-ranged. For the case of finite temperatures, we found that negativity has the same behaviors with concurrence at or near transition points. Moreover, QD for the anisotropic XY model can increase with temperature even in the absence of a magnetic field.
We investigate entanglement and quantum phase transition in a two-dimensional Heisenberg anisotropic spin-1/2XY1/2\mathit{XY}1/2XY model, using the quantum renormalization-group method (QRG) on a square lattice of $N\ifmmode\times\else\texttimes\fi{}N$ sites. The entanglement through geometric average of concurrences is calculated after each step of the QRG. We show that the concurrence achieves a nonzero value at the critical point more rapidly as compared to the one-dimensional case. The relationship between the entanglement and the quantum phase transition is studied. The evolution of entanglement develops two saturated values corresponding to two different phases. We compute the derivative of the concurrence, which is found to be discontinuous at the critical point $\ensuremath{\gamma}=0$, and indicates a second-order phase transition in the spin system. Further, the scaling behavior of the system is investigated by computing the derivative of the concurrence in terms of the system size.
We provide a study of various quantum phase transitions occurring in the XY Heisenberg chain in a transverse magnetic field using the Meyer-Wallach (MW) measure of (global) entanglement. Such a measure, while being readily evaluated, is a multipartite measure of entanglement as opposed to more commonly used bipartite measures. Consequently, we obtain analytic expression of the measure for finite-size systems and show that it can be used to obtain critical exponents via finite-size scaling with great accuracy for the Ising universality class. We also calculate an analytic expression for the isotropic (XX) model and show that global entanglement can precisely identify the level-crossing points. The critical exponent for the isotropic transition is obtained exactly from an analytic expression for global entanglement in the thermodynamic limit. Next, the general behavior of the measure is calculated in the thermodynamic limit considering the important role of symmetries for this limit. The so-called oscillatory transition in the ferromagnetic regime can only be characterized by the thermodynamic limit where global entanglement is shown to be zero on the transition curve. Finally, the anisotropic transition is explored where it is shown that global entanglement exhibits an interesting behavior in the finite-size limit. In the thermodynamic limit, we show that global entanglement shows a cusp singularity across the Ising and anisotropic transition, while showing non-analytic behavior at the XX multicritical point. It is concluded that global entanglement, despite its relative simplicity, can be used to identify all the rich structure of the ground-state Heisenberg chain.
We investigate a class of one-dimensional, exactly solvable anisotropic XY spin-1/2 models in an alternating transverse magnetic field from an entanglement perspective. We find that a physically motivated Lie-algebraic generalized entanglement measure faithfully portraits the static phase diagram -- including second- and fourth-order quantum phase transitions belonging to distinct universality classes. In the simplest time-dependent scenario of a slow quench across a quantum critical point, we identify parameter regimes where entanglement exhibits universal dynamical scaling relative to the static limit.