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arXiv:1107.5931 · Physical Review A

Fidelity spectrum and phase transitions of quantum systems

Pedro D Sacramento, Nikola Paunković, Vitor Rocha Vieira

Published Dec 21, 2011• 15 citations
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Abstract

Quantum fidelity between two density matrices $F({\ensuremath{\rho}}_{1},{\ensuremath{\rho}}_{2})$ is usually defined as the trace of the operator $\mathcal{F}=\sqrt{\sqrt{{\ensuremath{\rho}}_{1}}{\ensuremath{\rho}}_{2}\sqrt{{\ensuremath{\rho}}_{1}}}$. We study the logarithmic spectrum of this operator, which we denote by the fidelity spectrum, in the cases of the $XX$ spin chain in a magnetic field, a magnetic impurity inserted in a conventional superconductor, and a bulk superconductor at finite temperature. When the density matrices are equal, ${\ensuremath{\rho}}_{1}={\ensuremath{\rho}}_{2}$, the fidelity spectrum reduces to the entanglement spectrum. We find that the fidelity spectrum can be a useful tool in giving a detailed characterization of the different phases of many-body quantum systems.

DOI
10.1103/physreva.84.062318

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Quantum fidelity between two density matrices F(\ensuremathρ1,\ensuremathρ2)F({\ensuremath{\rho}}_{1},{\ensuremath{\rho}}_{2})F(\ensuremathρ1​,\ensuremathρ2​) is usually defined as the trace of the operator F=\ensuremathρ1\ensuremathρ2\ensuremathρ1\mathcal{F}=\sqrt{\sqrt{{\ensuremath{\rho}}_{1}}{\ensuremath{\rho}}_{2}\sqrt{{\ensuremath{\rho}}_{1}}}F=\ensuremathρ1​​\ensuremathρ2​\ensuremathρ1​​​. We study the logarithmic spectrum of this operator, which we denote by the fidelity spectrum, in the cases of the XXXXXX spin chain in a magnetic field, a magnetic impurity inserted in a conventional superconductor, and a bulk superconductor at finite temperature. When the density matrices are equal, \ensuremathρ1=\ensuremathρ2{\ensuremath{\rho}}_{1}={\ensuremath{\rho}}_{2}\ensuremathρ1​=\ensuremathρ2​, the fidelity spectrum reduces to the entanglement spectrum. We find that the fidelity spectrum can be a useful tool in giving a detailed characterization of the different phases of many-body quantum systems.
We review the definition and properties of fidelity as a measure of distinguishability of quantum states, presenting some of its geometric aspects. We apply then fidelity to the study of phase transitions in condensed matter systems, both at zero and finite temperatures. It does not rely on any already known order parameter, being useful not only in the usual Landau type phase transitions but also in phase transitions displaying topological order. The so called fidelity susceptibility and its relation to the usual thermodynamic susceptibilities and response functions are discussed. Finally, we consider the fidelity between different reduced density matrices for a subsystem of a larger system.
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Topological insulators and topological superconductors display various topological phases that are characterized by different Chern numbers or by gapless edge states. In this work we show that various quantum information methods such as the von Neumann entropy, entanglement spectrum, fidelity, and fidelity spectrum may be used to detect and distinguish topological phases and their transitions. As an example we consider a two-dimensional ppp-wave superconductor, with Rashba spin-orbit coupling and a Zeeman term. The nature of the phases and their changes are clarified by the eigenvectors of the kkk-space reduced density matrix. We show that in the topologically nontrivial phases the highest weight eigenvector is fully aligned with the triplet pairing state. A signature of the various phase transitions between two points on the parameter space is encoded in the kkk-space fidelity operator.
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In this paper we analyze the ground-state phase diagram of a class of fermionic Hamiltonians by looking at the fidelity of ground states corresponding to slightly different Hamiltonian parameters. The Hamiltonians under investigation can be considered as the variable range generalization of the fermionic Hamiltonian obtained by the Jordan-Wigner transformation of the XYXYXY spin chain in a transverse magnetic field. Under periodic boundary conditions, the matrices of the problem become circulant and the models are exactly solvable. Their free-ends counterparts are instead analyzed numerically. In particular, we focus on the long-range model corresponding to a fully connected directed graph, providing asymptotic results in the thermodynamic limit, as well as the finite-size scaling analysis of the second-order quantum phase transitions of the system. A strict relation between fidelity and single particle spectrum is demonstrated, and a peculiar gapful transition due to the long-range nature of the coupling is found. A comparison between fidelity and another transition marker borrowed from quantum information, i.e., single site entanglement, is also considered.
The reduced density fidelity is a measure of distance between two reduced density matrix, which can be used to characterize quantum phase transitions in quantum many-body systems. In this paper, we use the multi-scale entanglement reorganization ansatz (MERA) algorithm to simulate the spin 1 quantum Blume-Capel model and determine its ground-state phase diagram through calculating the reduced density fidelity. The qualitative relevant information contained in one site reduced density matrix is different from that contained two-site reduced density matrix, which can be detected by using the reduced density fidelity. In addition, we also characterize quantum phase transitions in quantum many-body systems by using the local parameters and energy gaps.