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Two-dimensional topological phases are characterized by TKNN integers, which classify Bloch energy bands or groups of Bloch bands. However, quantization does not survive thermal averaging or dephasing to mixed states. We show that using Uhlmanns parallel transport for density matrices (Rep. Math. Phys. 24, 229 (1986)), an integer classification of topological phases can be defined for a finite generalized temperature $T$ or dephasing Lindbladian. This scheme reduces to the familiar TKNN classification for $T<T_{{rm c},1}$, becomes trivial for $T>T_{{rm c},2}$, and exhibits a `gapless intermediate regime where topological indices are not well-defined. We demonstrate these ideas in detail, applying them to Haldanes honeycomb lattice model and the Bernevig-Hughes-Zhang model, and we comment on their generalization to multi-band Chern insulators.
We report the discovery of topological magnetism in the candidate magnetic Weyl semimetal CeAlGe. Using neutron scattering we find this system to host several incommensurate, square-coordinated multi-$vec{k}$ magnetic phases below $T_{rm{N}}$. The to
We study the impact of the inter-level energy constraints imposed by Haldane Exclusion Statistics on relaxation processes in 1-dimensional systems coupled to a bosonic bath. By formulating a second-quantized description of the relevant Fock space, we
We show that the topological index of a wavefunction, computed in the space of twisted boundary phases, is preserved under Hilbert space truncation, provided the truncated state remains normalizable. If truncation affects the boundary condition of th
Kramers degeneracy theorem underpins many interesting effects in quantum systems with time-reversal symmetry. We show that the generator of dynamics for Markovian open fermionic systems can exhibit an analogous degeneracy, protected by a combination
We show that the concept of topological order, introduced to describe ordered quantum systems which cannot be classified by broken symmetries, also applies to classical systems. Starting from a specific example, we show how to use pure state density