Topology of dissipative systems

Summary of main results for two quench protocols from a trivial (T) phase of Hermitian SSH model to a Möbius (M) and a topological (TP) phase of non-Hermitian SSH model. Pancharatnam phase φ^G (k, t) is calculated for single particle with momentum k at time t. Number of DTOPs per Brillouin zone is represented by ν^BZ, which shows different ν^BZ for the final Hamiltonian in the M and TP phase. Adapted from Phys. Rev. B 109, 094311 (2024).

The effective Hamiltonian of an open quantum system is non-Hermitian. In recent years, much research has been carried out to explore topological features in non-Hermitian models. The energies are generally complex-valued in non-Hermitian systems compared to those with real eigenvalues in Hermitian systems. The degeneracy in complex energy spectrum and the coalescence of related eigenvectors lead to the non-analyticities (singularities) coined as exceptional points (EPs). The non-Hermiticity of Hamiltonians paves the way for the ramification and unification of various crucial symmetries in non-Hermitian systems. Such unification and branching out of symmetries have led to intriguing topological phases, both with and without their Hermitian counterparts. For instance, the trivial (T) and topological (TP) phases with a complex-energy gap in a sublattice symmetric non-Hermitian Su-Schrieffer-Heeger (NSSH) model are similar to those with a real-energy gap in the related Hermitian model. We have shown that the NSSH model also hosts a gapless composite topological phase named the Möbius (M) phase in the parameter region between two EPs. The Möbius phase involves the multiple participating complex bands and does not have a Hermitian analog. Such non-Hermitian topological metallic and insulating phases with exciting topology like Penrose triangles can also emerge in different generalizations of the NSSH model with more than two sublattice sites.

We have identified unique physical signatures of these non-Hermitian phases, particularly the gapless Möbius phase, by studying Loschmidt echo from an initial state of the Hermitian SSH model, which is evolved by an NSSH Hamiltonian after a sudden quench in parameters. We show that a quench by the NSSH Hamiltonian in the Möbius phase exhibits dynamical phase transitions from both the trivial and topological phases. Further, we also observe unique dynamical signatures related to the symmetry constraints in the Möbius phase of the sublattice symmetric NSSH model. We observe two dynamical topological order parameters (DTOPs) in the Brillouin zone (BZ) for a topology changing quench by the NSSH model in either the trivial or topological phase and the initial state in any band. However, there is only one DTOP in the positive or negative half of the BZ when the NSSH model is in the Möbius phase.

V. M. Vyas and D. Roy,  Phys. Rev. B 103, 075441 (2021)

R. Nehra and D. Roy, Phys. Rev. B 105, 195407 (2022)

R. Nehra and D. Roy, Phys. Rev. B 109, 094311 (2024)