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arXiv:1702.07289 · Physical Review Letters

Uhlmann Connection in Fermionic Systems Undergoing Phase Transitions

Bruno Mera, Chrysoula Vlachou, Nikola Paunković, Vitor Rocha Vieira

Published Jul 7, 2017• 37 citations
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Abstract

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.

DOI
10.1103/physrevlett.119.015702

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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.
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.
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 introduce the Uhlmann geometric phase as a tool to characterize symmetry-protected topological phases in one-dimensional fermion systems, such as topological insulators and superconductors. Since this phase is formulated for general mixed quantum states, it provides a way to extend topological properties to finite temperature situations. We illustrate these ideas with some paradigmatic models and find that there exists a critical temperature Tc at which the Uhlmann phase goes discontinuously and abruptly to zero. This stands as a borderline between two different topological phases as a function of the temperature. Furthermore, at small temperatures we recover the usual notion of topological phase in fermion systems.
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.
I will introduce the Uhlmann geometric phase as a tool to characterise density matrices of 1D and 2D topological insulators and superconductors. We achieve this goal by constructing new topological invariants called Topological Uhlmann numbers. Since this phase is formulated for general mixed quantum states, it provides a way to extend topological properties to finite temperature situations. New effects appear such as the existence of critical temperatures, novel thermal-topological transitions in models with high Chern numbers, breakdown of the usual bulk-edge correspondence, … Moreover, as the Uhlmann phase is an observable itself, we analyse potential measurement schemes that could be applicable to current experimental setups like cold atoms in optical lattices.
Abstract We perform the fidelity analysis for Boltzmann–Gibbs-like states in order to investigate whether the topological order of 1D fermionic systems at zero temperature is maintained at finite temperatures. We use quantum walk protocols that are known to simulate topological phases and the respective quantum phase transitions for chiral symmetric Hamiltonians. Using the standard approaches of the fidelity analysis and the study of edge states, we conclude that no thermal-like phase transitions occur as temperature increases, i.e. the topological behaviour is washed out gradually. We also show that the behaviour of the Uhlmann geometric factor associated to the considered fidelity exhibits the same behaviour as the latter, thus confirming the results obtained using the previously established approaches.