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arXiv:2203.01575 · Physical Review A

Global entanglement in a topological quantum phase transition

Elahe Samimi, Mohammad Hossein Zarei, Afshin Montakhab

Published Mar 23, 2022• 17 citations
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Abstract

A useful approach to characterize and identify quantum phase transitions lies in the concept of multipartite entanglement. In this paper, we consider well-known measures of multipartite (global) entanglement, i.e., average linear entropy of one-qubit and two-qubit reduced density matrices, in order to study topological quantum phase transition (TQPT) in the Kitaev Toric code Hamiltonian with a nonlinear perturbation. We provide an exact mapping from aforementioned measures in the above model to internal energy and energy-energy correlations in the classical Ising model. Accordingly, we find that the global entanglement shows a continuous and sharp transition from a maximum value in the topological phase to zero in the magnetized phase in a sense that its first-order derivative diverges at the transition point. In this regard, we conclude that not only can the global entanglement serve as a reasonable tool to probe quantum criticality at TQPTs, but it also can reveal the highly entangled nature of topological phases. Furthermore, we also introduce a conditional version of global entanglement which becomes maximum at the critical point. Therefore, regarding a general expectation that multipartite entanglement reaches maximum value at the critical point of quantum many-body systems, our result proposes that the conditional global entanglement can be a good measure of multipartite entanglement in TQPTs.

DOI
10.1103/physreva.105.032438

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A useful approach to characterize and identify quantum phase transitions lies in the concept of multipartite entanglement. In this paper, we consider well-known measures of multipartite (global) entanglement, i.e., average linear entropy of one-qubit and two-qubit reduced density matrices, in order to study topological quantum phase transition (TQPT) in the Kitaev Toric code Hamiltonian with a nonlinear perturbation. We provide an exact mapping from aforementioned measures in the above model to internal energy and energy-energy correlations in the classical Ising model. Accordingly, we find that the global entanglement shows a continuous and sharp transition from a maximum value in the topological phase to zero in the magnetized phase in a sense that its first-order derivative diverges at the transition point. In this regard, we conclude that not only can the global entanglement serve as a reasonable tool to probe quantum criticality at TQPTs, but it also can reveal the highly entangled nature of topological phases. Furthermore, we also introduce a conditional version of global entanglement which becomes maximum at the critical point. Therefore, regarding a general expectation that multipartite entanglement reaches maximum value at the critical point of quantum many-body systems, our result proposes that the conditional global entanglement can be a good measure of multipartite entanglement in TQPTs.
Entanglement is known as an important indicator for characterizing different types of quantum phase transitions (QPTs); however, it faces some challenges in the Kosterlitz-Thouless (KT) phase transitions due to an essential singularity which cannot be identified in finite derivatives of the ground-state energy. In this paper, we consider global entanglement (GE) in a KT phase transition and show that while it does not indicate any clear signature of the phase transition, the conditional version of GE is a good indicator with strong signatures of the KT transition. In particular, we study a deformed version of the Zd{Z}_{d}Zd​ Kitaev model which has an intermediate KT phase which separates a Zd{Z}_{d}Zd​ topological phase from a magnetized phase at two different KT transition points. Using a mapping to the classical ddd-state clock model, we consider GE and the generalized GE and show that they do not provide a reliable indicator of transition points. However, their difference, called conditional global entanglement (QQQ), shows a peak at the first KT transition point. Additionally, we show that it can characterize various phases of the model as it behaves substantially differently in each phase. We therefore conclude that QQQ is a useful measure that can characterize various phases of KT QPTs as well as their related critical points.
In this paper we consider the quantum phase transition in the Ising model in the presence of a transverse field in one, two, and three dimensions from a multipartite entanglement point of view. Using exact numerical solutions, we are able to study such systems up to 25 qubits. The Meyer-Wallach measure of global entanglement is used to study the critical behavior of this model. The transition we consider is between a symmetric Greenberger-Horne-Zeilinger--like state to a paramagnetic product state. We find that global entanglement serves as a good indicator of quantum phase transition with interesting scaling behavior. We use finite-size scaling to extract the critical point as well as some critical exponents for the one- and two-dimensional models. Our results indicate that such multipartite measure of global entanglement shows universal features regardless of dimension ddd. Our results also provide evidence that multipartite entanglement is better suited for the study of quantum phase transitions than the much-studied bipartite measures.
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.
We investigate multipartite entanglement and a quantum phase transition for spin chain systems by making use of the density-matrix renormalization-group method in the form of matrix-product states. It is found that multipartite entanglement changes dramatically near the critical region and the residual-to-global entanglement ratio reaches its maximum at the critical point of the quantum phase transition. These results indicate that multipartite entanglement as well as the residual-to-global entanglement ratio can serve as good indicators to detect a quantum phase transition. We also demonstrate a linear relation between the next-nearest-neighbor interaction strength and the critical magnetic field for an XYZXYZXYZ spin chain by using multipartite entanglement and quantum coherence.
Topological order in two-dimensional (2D) quantum matter can be determined by the topological contribution to the entanglement Rényi entropies. However, when close to a quantum phase transition, its calculation becomes cumbersome. Here, we show how topological phase transitions in 2D systems can be much better assessed by multipartite entanglement, as measured by the topological geometric entanglement of blocks. Specifically, we present an efficient tensor network algorithm based on projected entangled pair states to compute this quantity for a torus partitioned into cylinders and then use this method to find sharp evidence of topological phase transitions in 2D systems with a string-tension perturbation. When compared to tensor network methods for Rényi entropies, our approach produces almost perfect accuracies close to criticality and, additionally, is orders of magnitude faster. The method can be adapted to deal with any topological state of the system, including minimally entangled ground states. It also allows us to extract the critical exponent of the correlation length and shows that there is no continuous entanglement loss along renormalization group flows in topological phases.
An exactly solvable Kitaev model in a two-dimensional square lattice exhibits a topological quantum phase transition which is different from the symmetry-breaking transition at zero temperature. When the ground state of a nonlinearly perturbed Kitaev model with different strengths of perturbation taken as the initial state is quenched to a pure Kitaev model, we demonstrate that various features of the dynamical state, such as the Loschmidt echo and time-averaged multipartite entanglement, can determine whether the initial state belongs to the topological phase or not. Moreover, the derivatives of the dynamical quantifiers can faithfully identify the topological quantum phase transition, which is present at equilibrium. When the individual qubits of the lattice interact with the local thermal bath repeatedly, we observe that block entanglement in dissipative dynamics can, nevertheless, distinguish the equilibrium phases from which the system starts evolution.