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arXiv:quant-ph/0512143 · Physical Review B

Exploring quantum phase transitions with a sublattice entanglement scenario

Yan Chen, Z. D. Wang, Fu‐Chun Zhang

Published Jun 9, 2006• 31 citations
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Abstract

We introduce a measure called reduced entropy of sublattice to quantify entanglement in spin, electron, and boson systems. By analyzing this quantity, we reveal an intriguing connection between quantum entanglement and quantum phase transitions in various strongly correlated systems: the local extremes of reduced entropy and its first derivative as functions of the coupling constant coincide, respectively, with the first and second order transition points. Exact numerical studies merely for small lattices reproduce several well-known results, demonstrating that our scenario is quite promising for exploring quantum phase transitions.

DOI
10.1103/physrevb.73.224414

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We introduce a measure called reduced entropy of sublattice to quantify entanglement in spin, electron, and boson systems. By analyzing this quantity, we reveal an intriguing connection between quantum entanglement and quantum phase transitions in various strongly correlated systems: the local extremes of reduced entropy and its first derivative as functions of the coupling constant coincide, respectively, with the first and second order transition points. Exact numerical studies merely for small lattices reproduce several well-known results, demonstrating that our scenario is quite promising for exploring quantum phase transitions.
Entanglement of the ground states in the S = 1/2 XXZ chain, dimerized Heisenberg spin chain, two-leg spin ladders as well as S = 1 anisotropic Haldane chain is analysed using the entanglement entropy between a selected sublattice of spins and the rest of the system. In particular, we reveal that quantum phase transition points/boundaries may be identified based on the analysis on the local extreme of this sublattice entanglement entropy, which is illustrated to be superior over the concurrence scenario and may enable us to explore quantum phase transitions in many other systems including higher dimensional ones.
Spin-orbital entanglement in quantum spin-orbital systems is quantified by a specifically reduced von Neumann entropy and is calculated for the ground state of a coupled spin-orbital chain with $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ symmetry. By analyzing the discontinuity and local extreme of the reduced entropy, we deduce a rich phase diagram describing quantum phase transitions in this system with complex correlations between multiple degrees of freedom.
The entanglement exhibits extremal or singular behavior near quantum critical points (QCPs) in many condensed matter models. These intriguing phenomena, however, still call for a widely accepted understanding. In this paper we study this issue in holographic framework. We investigate the connection between the holographic entanglement entropy (HEE) and the quantum phase transition (QPT) in a lattice-deformed Einstein-Maxwell-Dilaton theory. Novel backgrounds exhibiting metal-insulator transitions (MIT) have been constructed in which both metallic phase and insulating phase have vanishing entropy density in zero temperature limit. We find that the first order derivative of HEE with respect to lattice parameters exhibits extremal behavior near QCPs. We propose that it would be a universal feature that HEE or its derivatives with respect to system parameters can characterize QPT in a generic holographic system. Our work opens a window for understanding the relation between entanglement and the QPT from a holographic perspective.
In this work, we establish a general theory of phase transitions and quantum entanglement in the equilibrium state at arbitrary temperatures. First, we derive a set of universal functional relations between the matrix elements of a two-body reduced density matrix of the canonical density matrix and the Helmholtz free energy of the equilibrium state, which implies that the Helmholtz free energy and its derivatives are directly related to entanglement measures because any entanglement measures are defined as a function of the reduced density matrix. Then we show that the first-order phase transitions are signaled by the matrix elements of the reduced density matrix while the second-order phase transitions are witnessed by the first derivatives of the reduced density matrix elements. Near the second-order phase-transition point, we show that the first derivative of the reduced-density-matrix elements present universal scaling behaviors. Finally, we establish a theorem which connects the phase transitions and bipartite entanglement at arbitrary temperatures. Our general results are demonstrated in an experimentally relevant many-body spin model.
The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.