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arXiv:0707.4667 · Physical Review E

Macroscopic distinguishability between quantum states defining different phases of matter: Fidelity and the Uhlmann geometric phase

Nikola Paunković, Vitor Rocha Vieira

Published Jan 30, 2008• 38 citations
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Abstract

We study the fidelity approach to quantum phase transitions (QPTs) and apply it to general thermal phase transitions (PTs). We analyze two particular cases: The Stoner-Hubbard itinerant electron model of magnetism and the BCS theory of superconductivity. In both cases we show that the sudden drop of the mixed state fidelity marks the line of the phase transition. We conduct a detailed analysis of the general case of systems given by mutually commuting Hamiltonians, where the nonanalyticity of the fidelity is directly related to the nonanalyticity of the relevant response functions (susceptibility and heat capacity), for the case of symmetry-breaking transitions. Further, on the case of BCS theory of superconductivity, given by mutually noncommuting Hamiltonians, we analyze the structure of the system's eigenvectors in the vicinity of the line of the phase transition showing that their sudden change is quantified by the emergence of a generically nontrivial Uhlmann mixed state geometric phase.

DOI
10.1103/physreve.77.011129

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We study the fidelity approach to quantum phase transitions (QPTs) and apply it to general thermal phase transitions (PTs). We analyze two particular cases: The Stoner-Hubbard itinerant electron model of magnetism and the BCS theory of superconductivity. In both cases we show that the sudden drop of the mixed state fidelity marks the line of the phase transition. We conduct a detailed analysis of the general case of systems given by mutually commuting Hamiltonians, where the nonanalyticity of the fidelity is directly related to the nonanalyticity of the relevant response functions (susceptibility and heat capacity), for the case of symmetry-breaking transitions. Further, on the case of BCS theory of superconductivity, given by mutually noncommuting Hamiltonians, we analyze the structure of the system's eigenvectors in the vicinity of the line of the phase transition showing that their sudden change is quantified by the emergence of a generically nontrivial Uhlmann mixed state geometric phase.
We introduce a partial state fidelity approach to quantum phase transitions. We consider a superconducting lattice with a magnetic impurity inserted at its center, and look at the fidelity between partial (either one-site or two-site) quantum states. In the vicinity of the point of the quantum phase transition, we observe a sudden drop of the fidelity between two one-site partial states corresponding either to the impurity location or its close vicinity. This enables us to identify the on-site magnetization as the order parameter for the phase transition studied. In the case of two-site states, the fidelity reveals the transition point as long as one of the two electron sites is located at the impurity, while the other lies elsewhere in the lattice. We also determine the Uhlmann mixed state geometric phase, recently introduced in the study of the structural change of the system state eigenvectors in the vicinity of the lines of thermal phase transitions, and find it to be trivial, both for one- and two-site partial states, except when an electron site is at the impurity. This means that the system partial state eigenvectors do not contribute significantly to the enhanced state distinguishability around the point of this quantum phase transition. Finally, we use the fidelity to analyze the total amount of correlations contained within a composite system, showing that, even for the smallest one-site states, it features an abrupt quantitative change in the vicinity of the point of the quantum phase transition.
The Uhlmann connection is a mixed state generalization of the Berry connection.The latter has a very important role in the study of topological phases at zero temperature.Closely related, the fidelity is an information theoretical measure of distinguishability between quantum states.We show how one can use the fidelity and the Uhlmann connection to study phase transitions at finite temperature.We apply the analysis to free fermion Hamiltonians in 1D exhibiting symmetry protected topological order at zero temperature and also to the BCS theory of superconductivity.We show how one can study finite-temperature dynamical phase transitions by means of the fidelity and interferometric Loschmidt echoes.Moreover, we explain the physical and mathematical origin of the different behaviour of the two Loschmdit echoes by means of the associated susceptibilities.
We study the behavior of the fidelity and the Uhlmann connection in two-dimensional systems of free fermions that exhibit nontrivial topological behavior. In particular, we use the fidelity and a quantity closely related to the Uhlmann factor in order to detect phase transitions at zero and finite temperature for topological insulators and superconductors. We show that at zero temperature both quantities predict quantum phase transitions: a sudden drop of fidelity indicates an abrupt change of the spectrum of the state, while the behavior of the Uhlmann connection signals equally rapid change in its eigenbasis. At finite temperature, the topological features are gradually smeared out, indicating the absence of finite-temperature phase transitions, which we further confirm by performing a detailed analysis of the edge states. Moreover, we performed both analytical and numerical analysis of the fidelity susceptibility in the thermodynamic limit, providing an explicit quantitative criterion for the existence of phase transitions. The critical behavior at zero temperature is further analyzed through the numerical computation of critical exponents.
We study the behavior of the Uhlmann connection in systems of fermions undergoing phase transitions. In particular, we analyze some of the paradigmatic cases of topological insulators and superconductors in one dimension, as well as the BCS theory of superconductivity in three dimensions. We show that the Uhlmann connection signals phase transitions in which the eigenbasis of the state of the system changes. Moreover, using the established fidelity approach and the study of the edge states, we show the absence of thermally driven phase transitions in the case of topological insulators and superconductors. We clarify what is the relevant parameter space associated with the Uhlmann connection so that it signals the existence of order in mixed states. In addition, the study of Majorana modes at finite temperature opens the possibility of applications in realistic stable quantum memories. Finally, the analysis of the different behavior of the BCS model and the Kitaev chain, with respect to the Uhlmann connection, suggested that in realistic scenarios the gap of topological superconductors could also, generically, be temperature dependent.
The Uhlmann connection is a mixed state generalisation of the Berry connection. The latter has a very important role in the study of topological phases at zero temperature. Closely related, the quantum fidelity is an information theoretical quantity which is a measure of distinguishability of quantum states. Moreover, it has been extensively used in the analysis of quantum phase transitions. In this work, we study topological phase transitions in 1D and 2D topological superconductors with long-range hopping and pairing amplitudes, using the fidelity and the quantity [Formula: see text] closely related to the Uhlmann connection. The drop in the fidelity and the departure of [Formula: see text] from zero signal the topological phase transitions in the models considered. The analysis of the ground state fidelity susceptibility and its associated critical exponents are also applied to the study of the aforementioned topological phase transitions.
We extend to finite temperature the fidelity approach to quantum phase transitions (QPTs). This is done by resorting to the notion of mixed-state fidelity that allows one to compare two density matrices corresponding to two different thermal states. By exploiting the same concept we also propose a finite-temperature generalization of the Loschmidt echo. Explicit analytical expressions of these quantities are given for a class of quasifree fermionic Hamiltonians. A numerical analysis is performed as well showing that the associated QPTs show their signatures in a finite range of temperatures.