Floquet Majorana edge modes capture the topological features of periodically driven superconductors. We present a Kitaev chain with multiple time periodic driving and demonstrate how the avoidance of band crossing is altered, which gives rise to new regions supporting Majorana edge modes. A one dimensional generalized method was proposed to predict Majorana edge modes via the Zak phase of the Floquet bands. We also study the time independent effective Hamiltonian at high frequency limit and introduce diverse index to characterize topological phases with different relative phase between the multiple driving. Our work enriches the physics of driven system and paves the way for locating Majorana edge modes in larger parameter space.Topolgical state of matter is an intriguing topic, and has been studied intensively for years. Topological superconductivity is one of the most attractive theme in this subject for its topological excitation, Majorana fermion [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], which is its own antiparticle. The non-abelian statistics [16][17][18][19] of Majorana edge modes empower it as a novel prospect for quantum computation [20,21]. Thus, methods on how to generate Majorana edge modes are gaining increasing attention [22][23][24][25][26][27]. Conventional techniques of creating Majorana edge modes are via unpaired spin polarized fermions in quantum nanowires [28][29][30][31][32] or superconductors in proximity with topological insulators or semiconductors [33,34]. However, topological nontrivial regions with Majorana edge modes gained above are quite limited in parameter space.In recent years, Floquet engineering emerges as a new protocol for designing topological states of matter [35][36][37], which correspondingly brings the concept Floquet Majorana edge modes. The main idea of Floquet engineering lies in driving the physical parameter periodically with time. In contrast to the adiabatic limit, where the system remains in the eigenstate at each instantaneous time, driven system may absorb quantized energy from external fields, featuring non-equilibrium properties. Hence, the Floquet version of system may exhibit fruitful topological properties [38][39][40][41][42][43][44][45][46][47][48][49]. For a driven Kitaev chain, results have shown that Majorana edge modes can be sustained within a larger parameter space [50][51][52][53]. Besides, multiple Majorana edge modes are discovered with low driving frequency [54]. However, all results above are obtained within a frame of single time periodic driving. A natural question appears to us, how will the multiple time periodic driving affect the Majorana edge modes?In this Letter, we demonstrate that due to the avoidance of bands crossing, multiple time periodic driving are capable of generating gaps at different regions, leading to extraordinary different topological phase transitions, and larger parameter spaces with Majorana edge modes. To identify the Majorana edge modes, we propose a one dimensional method based on the Zak phase of Floqu...

We study a non-Hermitian chiral topological superconductor system on two dimensional square lattice, from which we obtained a rich topological phase diagram and established an exact relationship between topological charge flow of exceptional points in generalized Brillouin zone and change of topological properties. Its rich topological phase diagram is the result of competition between anisotropy and non-Hermitian effect. This system belongs to class D according to AZ classification of non-Hermitian systems. Each topological phase can be characterized by a 2D Z number, which indicates the number of chiral edge modes, and two 1D Z2numbers, which indicate the existence of zero modes at edge dislocations.

Floquet engineering plays a key role in realizing novel dynamical topological states. The conventional Floquet engineering, however, only applies to time-periodic non-dissipative Hermitian systems, and for the open quantum systems, non-Hermitian processes usually occur. So far, it remains unclear how to characterize the topological phases of time-periodic open quantum systems via the frequency space Floquet Hamiltonian. Here, we propose the non-Floquet theory to solve the problem and illustrate it by a continuously time-periodic non-Hermitian bipartite chain. In non-Floquet theory, a temporal non-unitary transformation is exercised on the Floquet states, and the transformed Floquet spectrum restores the form of the Wannier-Stark ladder. Besides, we also show that different choices of the starting points of the driving period can result in different localization behavior, effects of which can reversely be utilized to design quantum detectors of phases in dissipative oscillating fields. Our methods are capable of describing topological features in dynamical open quantum systems with various driving types and can find its applications to construct new types of dynamical topological materials.