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arXiv:1310.4701 · Physical Review B

Majorana edge modes in the Kitaev model

Manisha Thakurathi, K. Sengupta, Diptiman Sen

Published Jun 27, 2014• 76 citations
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Abstract

We study the Majorana modes, both equilibrium and Floquet, which can appear at the edges of the Kitaev model on the honeycomb lattice. We first present the analytical solutions known for the equilibrium Majorana edge modes for both zigzag and armchair edges of a semi-infinite Kitaev model and chart the parameter regimes in which they appear. We then examine how edge modes can be generated if the Kitaev coupling on the bonds perpendicular to the edge is varied periodically in time as periodic $\ensuremath{\delta}$-function kicks. We derive a general condition for the appearance and disappearance of the Floquet edge modes as a function of the drive frequency for a generic $d$-dimensional integrable system. We confirm this general condition for the Kitaev model with a finite width by mapping it to a one-dimensional model. Our numerical and analytical study of this problem shows that Floquet Majorana modes can appear on some edges in the kicked system even when the corresponding equilibrium Hamiltonian has no Majorana mode solutions on those edges. We support our analytical studies by numerics for a finite sized system which show that periodic kicks can generate modes at the edges and the corners of the lattice.

DOI
10.1103/physrevb.89.235434

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We study the Majorana modes, both equilibrium and Floquet, which can appear at the edges of the Kitaev model on the honeycomb lattice. We first present the analytical solutions known for the equilibrium Majorana edge modes for both zigzag and armchair edges of a semi-infinite Kitaev model and chart the parameter regimes in which they appear. We then examine how edge modes can be generated if the Kitaev coupling on the bonds perpendicular to the edge is varied periodically in time as periodic \ensuremathδ\ensuremath{\delta}\ensuremathδ-function kicks. We derive a general condition for the appearance and disappearance of the Floquet edge modes as a function of the drive frequency for a generic ddd-dimensional integrable system. We confirm this general condition for the Kitaev model with a finite width by mapping it to a one-dimensional model. Our numerical and analytical study of this problem shows that Floquet Majorana modes can appear on some edges in the kicked system even when the corresponding equilibrium Hamiltonian has no Majorana mode solutions on those edges. We support our analytical studies by numerics for a finite sized system which show that periodic kicks can generate modes at the edges and the corners of the lattice.
We obtain analytical solutions for zero-energy Majorana edge modes in the three-dimensional Kitaev model on the hyperhoneycomb lattice. We consider three types of edges (zigzag, bearded, and armchair) and find the regions in parameter space where each type of edge mode exists. In the gapless phase, we obtain edge-state solutions which are the drumhead surface states associated with nodal-line semimetals. We also find a correspondence between noninteracting complex fermions and Majorana fermions on bipartite lattices, which explains the equivalence of energy spectrum and eigenmodes of Kitaev model and a corresponding tight-binding model for complex fermions on the same lattice.
z direction. We show that varying the chemical potential (or magnetic field) periodically in time can produce Majorana modes at the ends of a long chain. We discuss two kinds of periodic driving, periodic �-function kicks and a simple harmonic variation with time. We discuss some distinctive features of the end modes such as the inverse participation ratio of their wave functions and their Floquet eigenvalues which are always equal to ±1 for time-reversal symmetric systems. For the case of periodic �-function kicks, we use the effective Hamiltonian of a system with periodic boundary conditions to define two topological invariants. The first invariant is a well-known winding number while the second invariant has not appeared in the literature before. The second invariant is more powerful in that it always correctly predicts the numbers of end modes with Floquet eigenvalues equal to +1 and −1, while the first invariant does not. We find that the number of end modes can become very large as the driving frequency decreases. We show that periodic �-function kicks in the hopping and superconducting terms can also produce end modes. Finally, we study the effect of electron-phonon interactions (which are relevant at finite temperatures) and a random noise in the chemical potential on the Majorana modes.
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 bands 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.
We show how Majorana end modes can be generated in a one-dimensional system by varying some of the parameters in the Hamiltonian periodically in time. The specific model we consider is a chain containing spinless electrons with a nearest-neighbor hopping amplitude, a ppp-wave superconducting term, and a chemical potential; this is equivalent to a spin-12\frac{1}{2}21​ chain with anisotropic XYXYXY couplings between nearest neighbors and a magnetic field applied in the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ direction. We show that varying the chemical potential (or magnetic field) periodically in time can produce Majorana modes at the ends of a long chain. We discuss two kinds of periodic driving, periodic $\ensuremath{\delta}$-function kicks, and a simple harmonic variation with time. We discuss some distinctive features of the end modes such as the inverse participation ratio of their wave functions and their Floquet eigenvalues which are always equal to $\ifmmode\pm\else\textpm\fi{}1$ for time-reversal-symmetric systems. For the case of periodic $\ensuremath{\delta}$-function kicks, we use the effective Hamiltonian of a system with periodic boundary conditions to define two topological invariants. The first invariant is a well-known winding number, while the second invariant has not appeared in the literature before. The second invariant is more powerful in that it always correctly predicts the numbers of end modes with Floquet eigenvalues equal to +1+1+1 and $\ensuremath{-}1$, while the first invariant does not. We find that the number of end modes can become very large as the driving frequency decreases. We show that periodic $\ensuremath{\delta}$-function kicks in the hopping and superconducting terms can also produce end modes. Finally, we study the effect of electron-phonon interactions (which are relevant at finite temperatures) and a random noise in the chemical potential on the Majorana modes.
Recent progress in programmable quantum simulators motivates exact benchmarks for nonequilibrium dynamics in topological quantum matter. We study the Kitaev honeycomb model driven through its critical regions by a linear ramp of the bond coupling JzJ_zJz​ over a ramp time τττ, comparing zigzag and armchair ribbons with fully periodic tubes. The production of excitations exhibits a geometry-dependent finite-size crossover controlled by N2/τN_2/\sqrtτN2​/τ​, the ratio of the ribbon width to the Kibble--Zurek length. Periodic tubes obey the expected critical-line Kibble--Zurek scaling and serve as the boundary-free benchmark. Zigzag ribbons follow the same scaling until defect production is exponentially suppressed by a finite-width algebraic gap floor associated with the formation of Majorana edge modes, whose occupation is pinned by inversion parity. Armchair ribbons instead support transient Majorana edge modes with exponentially small gaps, producing width-parity-dependent defect plateaus: exactly one defect per unit length for odd widths and an approximate, transient plateau near one half for even widths. The quench-generated entanglement entropy and fermionic logarithmic negativity inherit the same geometry-dependent structure, and the negativity further distinguishes the coherence-borne entanglement of the Kibble--Zurek regime from the population-borne entanglement of the edge-mode plateaus. These results identify boundary Majorana modes as the mechanism controlling departures from ideal Kibble--Zurek dynamics and provide geometry-resolved benchmarks for programmable simulations of topological matter.
Abstract One-dimensional (1D) Floquet topological superconductors possess two types of degenerate Majorana edge modes at zero andπquasienergies, leaving more room for the design of boundary time crystals and quantum computing schemes than their static counterparts. In this work, we discover Floquet superconducting phases with large topological invariants and arbitrarily many Majorana edge modes in periodically driven Kitaev chains (KCs). Topological winding numbers defined for the Floquet operator and Floquet entanglement Hamiltonian are found to generate consistent predictions about the phase diagram, bulk-edge correspondence and numbers of zero andπMajorana edge modes of the system under different driving protocols. The bipartite entanglement entropy further shows non-analytic behaviors around the topological transition point between different Floquet superconducting phases. These general features are demonstrated by investigating the KC with periodically kicked pairing or hopping amplitudes. Our discovery reveals the rich topological phases and many Majorana edge modes that could be brought about by periodic driving fields in 1D superconducting systems. It further introduces a unified description for a class of Floquet topological superconductors from their quasienergy bands and entanglement properties.