Topology of density matrices

Abstract

We investigate topological properties of density matrices motivated by the question to what extent phenomena like topological insulators and superconductors can be generalized to mixed states in the framework of open quantum systems. The notion of geometric phases has been extended from pure to mixed states by Uhlmann in [Rep. Math. Phys. 24, 229 (1986)], where an emergent gauge theory over the density matrices based on their pure-state representation in a larger Hilbert space has been reported. However, since the uniquely defined square root $\sqrt{\rho }$ of a density matrix $\rho$ provides a global gauge, this construction is always topologically trivial. Here, we study a more restrictive gauge structure which can be topologically non-trivial and is capable of resolving homotopically distinct mappings of density matrices subject to various spectral constraints. Remarkably, in this framework, topological invariants can be directly defined and calculated for mixed states. In the limit of pure states, the well known system of topological invariants for gapped band structures at zero temperature is reproduced. We compare our construction with recent approaches to Chern insulators at finite temperature.

References in corpus (5)

• Classification of topological insulators and superconductors in three spatial dimensions
• Dissipative preparation of Chern insulators
• Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers
• Topological indices for open and thermal systems via Uhlmann's phase
• Manifestly gauge independent formulations of the Z2 invariants

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• Fate of fractional quantum Hall states in open quantum systems: characterization of correlated topological states for the full Liouvillian
• Universalities of thermodynamic signatures in topological phases
• Symmetry-protected Topological Phases at Finite Temperature
• Topological order of mixed states in quantum many-body systems
• R-DMFT study of a non-Hermitian skin effect for correlated systems: analysis based on a pseudo-spectrum
• Ubiquity of zeros of Loschmidt amplitude for mixed states in different physical processes and their implications
• Preparing Atomic Topological Quantum Matter by Adiabatic Nonunitary Dynamics
• Temperature dependence of the topological phase transition of BiTeI from first principles
• Topological characterization of one-dimensional open fermionic systems
• Role of Bath-Induced Many-Body Interactions in the Dissipative Phases of the Su-Schrieffer-Heeger Model
• Holonomy in Quantum Information Geometry
• Quantized single-particle Thouless pump induced by topology transfer from a Chern insulator at finite temperature
• Comparison of finite-temperature topological indicators based on Uhlmann connection