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Journal of Physics A Mathematical and Theoretical

Nonzero-temperature dynamics near quantum phase transition in the isotropic Lipkin–Meshkov–Glick model: an open quantum system approach

Yamen Hamdouni

Published Mar 17, 2008• 2 citations
Publisher

Abstract

The dynamics of a central qubit system coupled to an isotropic ferromagnetic Lipkin–Meshkov–Glick spin bath at nonzero temperature is studied. We derive exactly the reduced density matrix and investigate the pairwise thermal entanglement, the coherence and the concurrence of the central spins. Through the behavior of these quantities, we show that at very low temperatures the dynamics is sensitive to the presence of the critical point of the environment.

DOI
10.1088/1751-8113/41/13/135302

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The dynamics of a central qubit system coupled to an isotropic ferromagnetic Lipkin–Meshkov–Glick spin bath at nonzero temperature is studied. We derive exactly the reduced density matrix and investigate the pairwise thermal entanglement, the coherence and the concurrence of the central spins. Through the behavior of these quantities, we show that at very low temperatures the dynamics is sensitive to the presence of the critical point of the environment.
We explore a dynamic signature of quantum phase transition (QPT) in an isotropic Lipkin-Meshkov-Glick (LMG) model by studying the time evolution of a central qubit coupled to it. We evaluate exactly the time-dependent purity, which can be used to measure quantum coherence, of the central qubit. It is found that distinctly different behaviors of the purity as a function of the parameter reveal clearly the QPT point in the system. It is also clarified that the present model is equivalent to an anti--Jaynes-Cummings model under certain conditions.
We study the non-Markovianity of the single qubit system coupled with an isotropic Lipkin–Meshkov–Glick (LMG) model by an effective method proposed by Breuer et al . (Breuer H P and Piilo J 2009 Europhys. Lett. 85 5004). It is discovered that the non-Markovianity is concerned with the quantum phase transitions (QPTs). In the open system, we present that the strong coupling inside the bath and the strong interaction between the system and bath can enhance the degree of non-Markovianity. Moreover, the non-Markovianity is stronger and more sensitive for the bath in the symmetric phase than the symmetry broken phase.
We investigate the Lipkin-Meshkov-Glick model coupled to a thermal bath. Since the isolated model itself exhibits a quantum phase transition, we explore the critical signatures of the open system. Starting from a system-reservoir interaction written in positive-definite form, we find that the position of the critical point remains unchanged, in contrast to the popular mean-field prediction. Technically, we employ the polaron transform to be able to study the full crossover regime from the normal to the symmetry-broken phase, which allows us to investigate the fate of quantum-critical points subject to dissipative environments. The signatures of the phase transition are reflected in observables such as magnetization, stationary mode occupation, or waiting-time distributions.
We investigate the dynamics of quantum discord and entanglement for two central spin qubits coupled to an isotropic Lipkin-Meshkov-Glick bath. It is found that both quantum discord and entanglement have quite distinct behaviors with respect to the two different phases of the bath. In the case of the symmetry broken phase bath, quantum discord and entanglement can remain as constant. In the case of the symmetric phase bath, quantum discord and entanglement always periodically oscillate with time. The critical point of quantum phase transition of the bath can be revealed clearly by the distinct behaviors of quantum correlations. Furthermore, it is observed that quantum discord is significantly enhanced during the evolution while entanglement periodically vanishes.
Based on conditional past-future (CPF) correlations, we study the non-Markovianity of a central spin coupled to an isotropic Lipkin-Meshkov-Glick (LMG) bath. Although the dynamics of a system is always non-Markovian, it is found that some measurement time intervals considering a specific process, with respect to a particular set of CPF measurement operators, can be zero, which means that in this case the non-Markovianity of the system could not be detected. Furthermore, the initial system-bath correlations only slightly influence the non-Markovianity of the system in our model. Significantly, it is also found that the dynamics of the system for LMG baths, initially in the ground states corresponding to the symmetric phase and symmetry broken phase, exhibit different properties, and the maximal value of the CPF at the critical point is the smallest, independent of the measurement operator, which means that the criticality can manifest itself by the CPF. Moreover, the effect of bath temperature on the quantum criticality of the CPF depends on the measurement operator.
Abstract We introduce a new class of generalized isotropic Lipkin–Meshkov–Glick models with spin and long-range non-constant interactions, whose non-degenerate ground state is a Dicke state of type. We evaluate in closed form the reduced density matrix of a block of L spins when the whole system is in its ground state, and study the corresponding von Neumann and Rényi entanglement entropies in the thermodynamic limit. We show that both of these entropies scale as when L tends to infinity, where the coefficient a is equal to ( m − k )/2 in the ground state phase with k vanishing magnon densities. In particular, our results show that none of these generalized Lipkin–Meshkov–Glick models are critical, since when their Rényi entropy R q becomes independent of the parameter q . We have also computed the Tsallis entanglement entropy of the ground state of these generalized Lipkin–Meshkov–Glick models, finding that it can be made extensive by an appropriate choice of its parameter only when . Finally, in the case we construct in detail the phase diagram of the ground state in parameter space, showing that it is determined in a simple way by the weights of the fundamental representation of . This is also true in the case; for instance, we prove that the region for which all the magnon densities are non-vanishing is an ( m + 1)-simplex in whose vertices are the weights of the fundamental representation of .