Topological characterization of chiral models through their long time dynamics

Abstract

We study chiral models in one spatial dimension, both static and periodically driven. We demonstrate that their topological properties may be read out through the long time limit of a bulk observable, the mean chiral displacement. The derivation of this result is done in terms of spectral projectors, allowing for a detailed understanding of the physics. We show that the proposed detection converges rapidly and it can be implemented in a wide class of chiral systems. Furthermore, it can measure arbitrary winding numbers and topological boundaries, it applies to all non-interacting systems, independently of their quantum statistics, and it requires no additional elements, such as external fields, nor filled bands.Topological phases of matter constitute a new paradigm by escaping the standard Ginzburg-Landau theory of phase transitions. These exotic phases appear without any symmetry breaking and are not characterized by a local order parameter, but rather by a global topological order. In the last decade, topological insulators have attracted much interest [1]. These systems are insulators in their bulk but exhibit current carrying edge states protected by the topology. A classification of topological insulators in terms of their discrete symmetries and their spatial dimensionality has been obtained in the celebrated periodic table of topological insulators and superconductors [2]. The topological invariant characterizing these models can be derived from the bulk Hamiltonian and allows one to recover the so called bulk-edge correspondence, namely that the number of topologically protected edge states is proportional to the topological invariant. A famous example of this correspondence can be found in the Quantum-Hall effect where the quantization of the Hall conductance is rooted in the current-carrying protected edge states [3][4][5]. The ensemble of (natural and artificial) topological insulators is steadily growing, and these have been by now synthetically engineered in a multitude of physical systems such as atomic [6][7][8][9][10][11], superconducting [12], photonic [13][14][15][16][17] and acoustic platforms [18][19][20].This work focuses on one-dimensional (1D) topological insulators possessing chiral symmetry. As a consequence of the chiral symmetry, the different sites of the unit cell can always be regarded as part of two sublattices. The topological invariant of the bulk, the winding number  , allows one to predict the number of zero energy edge states. 1D chiral topological insulators have been realized in numerous platforms as ultracold atoms [6,11], photonic crystals [15], photonic quantum walks [21][22][23][24][25]. Let us notice that the 1D chiral Hamiltonian can be static or the effective Hamiltonian of a Floquet system. In the latter case, the topology can be richer than its static counterpart [26][27][28][29]. In both cases, two different approaches to characterize the topology of such systems have been proposed and implemented experimentally. The first one is to look at intrinsic prope...

Topological characterizations of an extended Su–Schrieffer–Heeger model

“…For the SSH4 model, we can choose the unit cell basis {A 1 , B 1 , A 2 , B 2 }, so the total mean chiral displacement operator takes the form Γm = diag(..., 1, −1, 1, −1, 2, −2, 2, −2, ...). At higher dimension, in order to use the mean chiral displacement to measure the winding number, we need to choose an orthogonal and complete basis of a given sub-lattice [77]. In our case, we prepare the initial state at two orthogonal states (1,0,0,0) and (0,0,1,0), and measure the mean chiral displacement C 1 (t) and C 3 (t) respectively.…”

“…One direct extension of the SSH model is the so-called SSH4 model [77]. By changing the site period of the unit cell from two to four, one can transform the standard SSH model into the considerably richer SSH4 model.…”

“…When periodical driving is applied to the SSH4 system, an effectively higher dimensional quantum model can be realized. Reference [77] proposed an example of forming a discrete-time quantum walk with effectively four dimensions by driving the SSH4 model. interesting feature of the SSH4 model relates to the methods for detecting the winding number in such systems.…”

“…The mean chiral displacement has been proposed and experimentally verified as an observable that reveals the winding number in the standard SSH model [50,81]. Reference [77] provides a generalized description for how this observable can be extended to higher dimensional systems, however…”

Topological time crystals

“…The topological time crystals we consider should not be confused with the so-called Floquet topological systems. In the latter, a crystalline structure (usually an optical lattice) is present in space and it is periodically driven so that its effective parameters can be changed and the system can reveal topological properties in space but no crystalline structure can be observed in time [46][47][48][49]. Our systems are also different from Floquet-Bloch systems where time periodicity is considered as an additional synthetic dimensional combined with a crystalline structure in space [50][51][52][53][54][55].…”