Dynamic winding number for exploring band topology

Abstract

Topological invariants play a key role in the characterization of topological states. Due to the existence of exceptional points, it is a great challenge to detect topological invariants in both Hermitian and non-Hermitian systems via a unified approach. Here, we put forward a dynamic winding number, the winding of realistic observables in long-time average, for exploring band topology in generic two-band models. We build a concrete relation between dynamic winding numbers and equilibrium topological invariants. In one dimension, the dynamical winding number directly gives the conventional winding number. In two dimension, the Chern number relates to the weighted sum of dynamic winding numbers of all phase singularity points, in which the weight is +1 for the north pole and −1 for the south pole. This work opens a new avenue to detecting topological aspects of both Hermitian and non-Hermitian systems via general dynamic evolution.

Dissipative Floquet Dynamical Quantum Phase Transition

“…In the next section we will study the topological properties of the effective Hamiltonian in Eq. ( 7) using the winding numbers of the non-Hermitian Hamiltonians [63].…”

“…Finding the dynamical signatures of these non-equilibrium topological matter has become a fascinating area for more experimental and theoretical research. In recent works, several dynamical probes to the topological invariants of non-Hermitian phases in one and two dimensions have been introduced, such as the non-Hermitian extension of dynamical winding numbers [61][62][63][64][65][66] and mean chiral displacements [67,68]. Further, the dynamical quantum phase transitions (DQPTs) following a quench across the EPs of a non-Hermitian lattice model is studied in Refs.…”

Non-Hermitian Floquet Phases with Even-Integer Topological Invariants in a Periodically Quenched Two-Leg Ladder

“…Non-Hermitian states of matter have attracted great attention in recent years due to their intriguing dynamical and topological properties (see [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 ] for reviews). Theoretically, a wide range of non-Hermitian topological phases and phenomena have been classified and characterized according to their symmetries [ 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 ] and dynamical signatures [ 18 , 19 , 20 , 21 , 22 , 23 ]. Experimentally, non-Hermitian topological matter have also been realized in cold atom [ 24 , 25 ], photonic [ 26 , 27 , 28 , 29 ], acoustic [ 30 , 31 , 32 ], electrical circuit [ 33 , 34 , 35 ] systems, and nitrogen-vacancy-center in diamond [ 36 ], leading to potential applications such as topological lasers [ 37 , 38 , 39 ] and high-performance sensors [ 40 , 41 , 42 , 43 ].…”

Untying links through anti-parity-time-symmetric coupling

“…The creation, observation, and investigation on the static and dynamical properties of quantum knots in Bose-Einstein condensates are reported [2]. Besides, topological phases are extremely stable [3][4][5][6][7][8][9][10][11]. In topological systems, zero-energy Fermi surfaces can form knotted or linked nodal lines [12][13][14][15][16][17][18]; alternatively, the fictitious magnetic field of a topological system with topological defects, associated with vortex or antivortex textures, reflects nontrivial topology [19,20].…”