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arXiv:1809.01848 · Journal of Physics Condensed Matter

Critical phase boundaries of static and periodically kicked long-range Kitaev chain

Utso Bhattacharya, Somnath Maity, Amit Dutta, Diptiman Sen

Published Jan 31, 2019• 22 citations
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Abstract

We study the static and dynamical properties of a long-range Kitaev chain, i.e. a p -wave superconducting chain in which the superconducting pairing decays algebraically as [Formula: see text], where l is the distance between the two sites and [Formula: see text] is a positive constant. Considering very large system sizes, we show that when [Formula: see text], the system is topologically equivalent to the short-range Kitaev chain with massless Majorana modes at the ends of the system; on the contrary, for [Formula: see text], there exist symmetry protected massive Dirac end modes. We further study the dynamical phase boundary of the model when periodic [Formula: see text]-function kicks are applied to the chemical potential; we specially focus on the case [Formula: see text] and analyze the corresponding Floquet quasienergies. Interestingly, we find that new topologically protected massless end modes are generated at the quasienergy [Formula: see text] (where T is the time period of driving) in addition to the end modes at zero energies which exist in the static case. By varying the frequency of kicking, we can produce topological phase transitions between different dynamical phases. Finally, we propose some bulk topological invariants which correctly predict the number of massless end modes at quasienergies equal to 0 and [Formula: see text] for a periodically kicked system with [Formula: see text].

DOI
10.1088/1361-648x/ab03b9

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We study the static and dynamical properties of a long-range Kitaev chain, i.e. a p -wave superconducting chain in which the superconducting pairing decays algebraically as [Formula: see text], where l is the distance between the two sites and [Formula: see text] is a positive constant. Considering very large system sizes, we show that when [Formula: see text], the system is topologically equivalent to the short-range Kitaev chain with massless Majorana modes at the ends of the system; on the contrary, for [Formula: see text], there exist symmetry protected massive Dirac end modes. We further study the dynamical phase boundary of the model when periodic [Formula: see text]-function kicks are applied to the chemical potential; we specially focus on the case [Formula: see text] and analyze the corresponding Floquet quasienergies. Interestingly, we find that new topologically protected massless end modes are generated at the quasienergy [Formula: see text] (where T is the time period of driving) in addition to the end modes at zero energies which exist in the static case. By varying the frequency of kicking, we can produce topological phase transitions between different dynamical phases. Finally, we propose some bulk topological invariants which correctly predict the number of massless end modes at quasienergies equal to 0 and [Formula: see text] for a periodically kicked system with [Formula: see text].
In this paper, we have considered the extended version of the Kitaev model in one dimension, i.e., a long-range p-wave superconducting wire. In the long-range Kitaev chain, the superconducting hopping and pairing terms in the Hamiltonian decay, independently, in a power-law fashion , where l is the distance between the two sites and x is some positive constant. We have studied the appearance of Majorana zero-energy edge modes and also, massive Dirac edge modes by exact diagonalization, as well as analytical computations. Exact diagonalization indicates the existence of both kinds of massless and massive edge modes in the energy spectrum. Furthermore, we obtain the phase diagram and the topological phase transitions by calculating the winding number, which is the topological invariant.
We present a detailed study of the topological properties of the Kitaev chain with long-range pairing terms and in the presence of an Aubry-André-Harper on-site potential.Specifically, we consider algebraically decaying superconducting pairing amplitudes; the exponent of this decay is found to determine a critical pairing strength, below which the chain remains topologically trivial.Above the critical pairing, topological edge modes are observed in the central gap.For sufficiently fast decay of the pairing, these modes are identified as Majorana zero modes.However, if the pairing term decays slowly, the modes become massive Dirac modes.Interestingly, these massive modes still exhibit a true level crossing at zero energy, which points towards an intimate relation to Majorana physics.We also observe a clear lack of bulk-boundary correspondence in the long-range system, where bulk topological invariants remain constant, while dramatic changes appear in the behavior at the edge of the system.In addition to the central gap around zero energy, the Aubry-André-Harper potential also leads to other energy gaps at nonzero energy.As for the analogous short-range model, the edge modes in these gaps can be characterized through a 2D Chern invariant.However, in contrast to the short-range model, this topological invariant does not correspond to the number of edge mode crossings anymore.This provides another example of the weakening of the bulk-boundary correspondence occurring in this model.Finally, we discuss possible realizations of the model with ultracold atoms and condensed matter systems.
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.
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.
The power-law profile of hopping and superconductivity introduces rich critical phenomena that are not possible to explore in the nearest-neighbor models. Having this in mind, we construct the Su-Schrieffer-Heeger (SSH)-Kitaev model with mono-, bi-, and tri-partite sub-lattices where superconductivity [hopping] is of power-law Kitaev [SSH] type. The long-range (LR) superconductivity can alone gap out the zero-momentum critical line while LR hopping alters the finite-momentum critical line. There exists open-ended critical line at vanishing superconducting gap on the phase diagram for all versions of the SSH-Kitaev model. Interestingly, LR superconductivity (hopping) promotes massive Dirac (Majorana zero) modes, while in a finite-size system, their emergence in open boundary conditions is caused by the bulk gap of the anti-periodic Hamiltonian in momentum space. This indicates an unconventional bulk-boundary correspondence for power-law models as compared to what is seen in the short-range models. Importantly, the thermodynamic phase boundaries are correctly reproduced by the above unconventional correspondence. Intriguingly, an even (odd) number of sub-lattices yields an integer (half-integer) winding number while the massive Dirac modes continue to survive irrespective of the sub-lattice profile as long as there exists LR pairing only.
In this paper we study the dynamics of long-range pairing and hopping p-wave superconducting chain after a ‎sudden quench of chemical potential. We have shown that the quantum-classical phase transition occurs ‎when the system is quenched to the critical point and the Loschmidt echo shows perfect periodic oscillations. ‎Moreover we have studied the dynamical phase transition for a quench across the critical point. Our analysis ‎shows that the dynamical phase transition occurs for the cases where the system initially prepared in the ‎massless edge modes. While the system does not show dynamical phase transition when the system is ‎prepared in the massive edge modes, even the quench crossing the critical point‎‎.