Robust Majorana edge modes with low frequency multiple time periodic driving

Abstract

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...

Topological properties of a periodically driven Creutz ladder

“…Note that, the modified Hamiltonian ( Hk (t)) shares the same periodicity as that of the original one (H(t)). With Hk (t) being periodic in time, the Fourier decomposition, Hk (t) = p e ipωt Hp,k , with p = 0, ±1, ±2, .. and so on, allows us to write an expansion in powers of the inverse of the driving frequency, namely, 1/ω, which is known as the Magnus expansion [29,30,42]. This yields an effective Hamiltonian, H eff given by,…”

“…Whereas, in the case of a sinusoidal drive, the Hamiltonian at different times do not commute, and hence the time evolution becomes non-trivial. However, the time dependence of the Hamiltonian for the sinusoidal drive can be eliminated by a similarity transformation, following which the factorization of the evolution operator is nothing but a rotating frame transformation followed by an expansion in powers of the inverse of the driving frequency, namely, 1/ω (Floquet-Magnus expansion) [29][30][31].…”

Inducing Shifted Edge Modes by the Spin‐Dependent Time‐Periodical Driving and the Corresponding Topological Phase Transitions

“…One pivotal method in realizing the dynamical synthetic gauge field lies in driving quantum systems periodically, which is also known as Floquet engineering. [1][2][3][4][5][6][7][8][9] The discrete time translational symmetry in Floquet engineering signifies that the systems periodically exchange energy (or particles) with the external driving field, and hence there exist no well defined ground states. In consequence, in contrast to the static cases, the topological characterizations of periodically driven systems are based on the Floquet evolving operator U(T), of which the diagonalization may give birth to not only zero quasi-energy edge modes, but also the Floquet edge modes pinned at quasi-energy ± 𝜔 2 .…”

Novel topological end modes and breaking of bulk boundary correspondence in skin-effect absent superconductive chain

“…Topological nontrivial systems feature isolated gapless edge modes [1][2][3][4][5][6] , and play a key role in advancing our understanding of quantum matter [7][8][9][10][11][12][13][14][15][16][17][18] . A most profound way to characterize edge modes above is through bulk topological invariants, which is known as bulk boundary correspondence [19][20][21][22][23] .…”

“…However, results above are broken for a junction with combined η − γ modes. To illustrate, we denote states at the left (right) end of the junction to be ψ 1(2) , and assume 2) . The two end states are gov-…”