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arXiv:0805.4062 · Physical Review E

Reduced fidelity susceptibility and its finite-size scaling behaviors in the Lipkin-Meshkov-Glick model

Jian Ma, Lei Xu, Heng-Na Xiong, Xiaoguang Wang

Published Nov 25, 2008• 44 citations
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Abstract

We derive a general formula for the reduced fidelity susceptibility when the reduced density matrix is 2x2 block diagonal. By using this result and a continuous unitary transformation, we study finite-size scaling of the reduced fidelity susceptibility in the Lipkin-Meshkov-Glick model. It is found that it can well characterize quantum phase transitions, that is, we can extract information about quantum phase transitions from only the fidelity of a subsystem, which is of practical use in experiments.

DOI
10.1103/physreve.78.051126

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We derive a general formula for the reduced fidelity susceptibility when the reduced density matrix is 2x2 block diagonal. By using this result and a continuous unitary transformation, we study finite-size scaling of the reduced fidelity susceptibility in the Lipkin-Meshkov-Glick model. It is found that it can well characterize quantum phase transitions, that is, we can extract information about quantum phase transitions from only the fidelity of a subsystem, which is of practical use in experiments.
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In quantum phase transitions induced by continuous level crossing, global-state fidelity drops to zero at every level crossing point, making finite-size scaling not applicable to study the criticality. In this paper, we calculate the partial-state fidelity in two models, i.e., the Lipkin-Meshkov-Glick (LMG) model and one-dimensional Heisenberg model. We show that it can spot the second-order phase transition point in the thermodynamic limit, where continuous level crossing occurs. The finite-size scaling shows the partial-state fidelity susceptibility diverges as NNN for the isotropic LMG model and N3{N}^{3}N3 for the Heisenberg model.
Quantum coherence reflects the origin of quantumness and might be capable of extracting the subtle nature of a system. We investigate the ground-state coherence and steered coherence in the Lipkin-Meshkov-Glick model and show that they detect faithfully the quantum phase transitions of this model. Moreover, we carry out scaling analysis on the coherence and steered coherence by means of the continuous unitary transformation method and find that the scaling exponents are uniquely determined by the phase region of this model. These results may provide useful insights into the mechanism underlying quantum criticality in many-body systems.
We study the reduced fidelity susceptibility ${\ensuremath{\chi}}_{r}$ for an MMM-body subsystem of an NNN-body Lipkin-Meshkov-Glick model with $\ensuremath{\tau}=M/N$ fixed. The reduced fidelity susceptibility can be viewed as the response of subsystem to a certain parameter. In noncritical region, the inner correlation of the system is weak, and ${\ensuremath{\chi}}_{r}$ behaves similar with the global fidelity susceptibility ${\ensuremath{\chi}}_{g}$, the ratio $\ensuremath{\eta}={\ensuremath{\chi}}_{r}/{\ensuremath{\chi}}_{g}$ depends on $\ensuremath{\tau}$ but not on NNN. However, at the critical point, the inner correlation tends to be divergent, and we find ${\ensuremath{\chi}}_{r}$ approaches ${\ensuremath{\chi}}_{g}$ with increasing the NNN. It is interesting to note that, $\ensuremath{\eta}=1$ in the thermodynamic limit, which means the susceptibilities of the local and global system are the same. Finally, we make numerical computations, and they are in perfect agreement with the analytical predictions.
The fidelity, linear entropy and entanglement entropy, concepts of information theory, are used to determine the localization of quantum phase transitions of the Lipkin–Meshkov–Glick (LMG) and the Dicke models for a finite number of particles. Another concept, the so-called fidelity susceptibility, gives us information of the scaling behavior of the quantum phase transitions of the models. In particular universal curves are presented for the specific susceptibility. It is shown that the control parameters of both models have a similar dependence in the number of particles when the systems approach to the thermodynamic limit.