Density-matrix Chern insulators: Finite-temperature generalization of topological insulators

Abstract

Thermal noise can destroy topological insulators (TI). However, we demonstrate how TIs can be made stable in dissipative systems. To that aim, we introduce the notion of band Liouvillian as the dissipative counterpart of band Hamiltonian, and show a method to evaluate the topological order of its steady state. This is based on a generalization of the Chern number valid for general mixed states (referred to as density-matrix Chern value), which witnesses topological order in a system coupled to external noise. Additionally, we study its relation with the electrical conductivity at finite temperature, which is not a topological property. Nonetheless, the density-matrix Chern value represents the part of the conductivity which is topological due to the presence of quantum mixed edge states at finite temperature. To make our formalism concrete, we apply these concepts to the two-dimensional Haldane model in the presence of thermal dissipation, but our results hold for arbitrary dimensions and density matrices.

Quantum tight-binding chains with dissipative coupling

“…A number of different physical models can lead to such an equation. For example, similar band structure of dissipators can arise in a graphene-like model based on a honeycomb lattice with nearest-neighbor and next nearest-neighbor couplings (Haldane model) [26]. Now let us discuss some examples to demonstrate feasibility of dissipatively coupled chain and possibility to realize it in practice.…”

Uhlmann Phase as a Topological Measure for One-Dimensional Fermion Systems

“…In addition, the role played by external dissipative effects and thermal baths in topological insulators and superconductors has attracted much interest both in quantum simulations with different platforms and in condensed matter [19][20][21][22][23][24][25][26][27][28][29]. In this Letter, we show that the Uhlmann geometric phase is endowed with a topological structure when applied to one-dimensional fermion systems.…”

“…For more than a decade, there has been a renewed interest in studying geometric phases for mixed states and under dissipative evolutions from the point of view of quantum information [14], and more inequivalent definitions have been introduced [15][16][17]. This has culminated with the first experimental measurement of a geometric phase for mixed quantum states of one system qubit and one ancillary qubit with NMR techniques [18].In addition, the role played by external dissipative effects and thermal baths in topological insulators and superconductors has attracted much interest both in quantum simulations with different platforms and in condensed matter [19][20][21][22][23][24][25][26][27][28][29]. In this Letter, we show that the Uhlmann geometric phase is endowed with a topological structure when applied to one-dimensional fermion systems.…”

Controlling open quantum systems: tools, achievements, and limitations

“…Moreover, the bath spectrum needs to be known. Another example of engineering a spectral separation is given by the protection of so-called edge states in topological insulators via band Liouvillians [139,140].…”