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arXiv:2010.06629 · Physical review. B./Physical review. B

Interferometric geometry from symmetry-broken Uhlmann gauge group with applications to topological phase transitions

Héctor Silva, Bruno Mera, Nikola Paunković

Published Feb 19, 2021• 11 citations
arXivHTMLPDF

Abstract

We provide a natural generalization of a Riemannian structure, i.e., a metric, recently introduced by Sj\"oqvist for the space of nondegenerate density matrices, to the degenerate case, i.e., the case in which the eigenspaces have dimension greater than or equal to 1. We present a physical interpretation of the metric in terms of an interferometric measurement. We apply this metric, physically interpreted as an interferometric susceptibility, to the study of topological phase transitions at finite temperatures for band insulators. We compare the behaviors of this susceptibility and the one coming from the well-known Bures metric, showing them to be dramatically different. While both infer zero-temperature phase transitions, only the former predicts finite-temperature phase transitions as well. The difference in behaviors can be traced back to a symmetry-breaking mechanism, akin to Landau-Ginzburg theory, by which the Uhlmann gauge group is broken down to a subgroup determined by the type of the system's density matrix (i.e., the ranks of its spectral projectors).

DOI
10.1103/physrevb.103.085127

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We provide a natural generalization of a Riemannian structure, i.e., a metric, recently introduced by Sj"oqvist for the space of nondegenerate density matrices, to the degenerate case, i.e., the case in which the eigenspaces have dimension greater than or equal to 1. We present a physical interpretation of the metric in terms of an interferometric measurement. We apply this metric, physically interpreted as an interferometric susceptibility, to the study of topological phase transitions at finite temperatures for band insulators. We compare the behaviors of this susceptibility and the one coming from the well-known Bures metric, showing them to be dramatically different. While both infer zero-temperature phase transitions, only the former predicts finite-temperature phase transitions as well. The difference in behaviors can be traced back to a symmetry-breaking mechanism, akin to Landau-Ginzburg theory, by which the Uhlmann gauge group is broken down to a subgroup determined by the type of the system's density matrix (i.e., the ranks of its spectral projectors).
We study finite-temperature dynamical quantum phase transitions (DQPTs) by means of the fidelity and the interferometric Loschmidt echo (LE) induced metrics. We analyze the associated dynamical susceptibilities (Riemannian metrics), and derive analytic expressions for the case of two-band Hamiltonians. At zero temperature, the two quantities are identical, nevertheless, at finite temperatures they behave very differently. Using the fidelity LE, the zero-temperature DQPTs are gradually washed away with temperature, while the interferometric counterpart exhibits finite-temperature phase transitions. We analyze the physical differences between the two finite-temperature LE generalizations, and argue that, while the interferometric one is more sensitive and can therefore provide more information when applied to genuine quantum (microscopic) systems, when analyzing many-body macroscopic systems, the fidelity-based counterpart is a more suitable quantity to study. Finally, we apply the previous results to two representative models of topological insulators in one and two dimensions.
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.
Geometric phases play a fundamental role in understanding the geometric structure of quantum states, yet extending the Uhlmann phase to non-Hermitian systems poses significant challenges due to parameter-dependent inner product structures. In this work, we develop a comprehensive theory of the Uhlmann phase for quasi-Hermitian systems, where the physical Hilbert space metric varies with external parameters. By constructing a generalized purification that respects the quasi-Hermitian inner product, we derive the corresponding parallel transport condition and Uhlmann connection. Our analysis reveals that the parameter-dependent metric modifies the Uhlmann connection and leads to a finite-temperature distribution of Uhlmann-phase regions that differs from the standard Hermitian case. Applying this formalism to solvable two-level models, we uncover tunable finite-temperature Uhlmann-phase diagrams, where the parameter-dependent metric shifts and deforms the zeros of the Uhlmann amplitude, thereby reshaping the temperature intervals in which nontrivial Uhlmann phases occur. Furthermore, by extending established interferometric protocols originally developed for Hermitian systems, the geometric amplitude can be recast as a measurable Loschmidt amplitude between purified states, providing a practical and experimentally accessible pathway to investigate quasi-Hermitian mixed-state geometric phases and their finite-temperature transitions. This work establishes a unified framework for understanding mixed-state geometric phases in quasi-Hermitian quantum systems and, through its natural relation to the mixed-state quantum geometric tensor, opens new avenues for exploring local geometry in quasi-Hermitian thermal states.
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.
For a long time, it was believed that the Ginzburg-Landau formalism was able to classify all different types of phase transitions. This view changed with the discovery of the quantum Hall effect and topological insulators. Topological insulators (superconductors) are materials that host symmetry-protected metallic edge states in an insulating (superconducting) bulk. Although topological insulators are well understood by now, a thermodynamic description of their behavior remained elusive, firstly because the edges are lost in the thermodynamic limit, when the system size goes to infinity, and secondly because topological quantum field theories involve non-local order parameters, and hence cannot be captured by the conventional Ginzburg-Landau formalism. Recently, this challenge has been overcome: by using Hill thermodynamics to describe the paradigmatic Bernevig-Hughes-Zhang model in two dimensions, it has been shown that at the topological phase change the thermodynamic potential does not scale extensively due to the boundary effects.\n\nHere, we extend this thermodynamic approach firstly to the Kitaev chain in one-dimension and then to different topological models in various dimensions (the Su-Schrieffer-Heeger model in one dimension, the Kane-Mele model in two dimensions and the Bernevig-Hughes-Zhang model in three dimensions) at zero temperature. Surprisingly, we find that all these models exhibit the same universal behavior in the order of the topological-phase transition, which depends on the dimensionality of the system. Next, we use this thermodynamic approach to describe topological phases at finite-temperatures. We calculate the entropy and heat capacity for the Kitaev chain and verify its behavior in comparison with conformal field theory. Furthermore, we extend the topological phase transition to finite-temperatures, by numerically deriving the topological phase diagram for the bulk of the system, and show that it displays a good agreement with the one calculated analytically from the Uhlmann phase. Finally, we calculate measurable quantities such as the density of states and the heat capacity for different topological insulators. Our work reveals unexpected universalities and opens the path to a thermodynamic description of systems with a non-local order parameter, provided that nonlinear terms are appropriately taken into account for the thermodynamic extensive variables.
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.