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arXiv:1707.05777 · Physical review. B./Physical review. B

Edge insulating topological phases in a two-dimensional superconductor with long-range pairing

Luca Lepori, Domenico Giuliano, Simone Paganelli

Published Jan 17, 2018• 25 citations
arXivHTMLPDF

Abstract

We study the zero-temperature phase diagram of a two-dimensional square lattice loaded by spinless fermions, with nearest-neighbor hopping and algebraically decaying pairing. We find that for sufficiently long-range pairing, new phases occur, not continuously connected with any short-range phase and not belonging to the standard families of topological insulators/superconductors. These phases are signaled by the violation of the area law for the von Neumann entropy, by semi-integer Chern numbers, and by edge modes with nonzero mass. The latter feature results in the absence of single-fermion edge conductivity, present instead in the short-range limit. The definition of a bulk-topology and the presence of a bulk-boundary correspondence is suggested also for the long-range phases. Recent experimental proposals and advances open the possibility to probe the described long-range effects in near-future realistic setups.

DOI
10.1103/physrevb.97.041109

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We study the zero-temperature phase diagram of a two-dimensional square lattice loaded by spinless fermions, with nearest-neighbor hopping and algebraically decaying pairing. We find that for sufficiently long-range pairing, new phases occur, not continuously connected with any short-range phase and not belonging to the standard families of topological insulators/superconductors. These phases are signaled by the violation of the area law for the von Neumann entropy, by semi-integer Chern numbers, and by edge modes with nonzero mass. The latter feature results in the absence of single-fermion edge conductivity, present instead in the short-range limit. The definition of a bulk-topology and the presence of a bulk-boundary correspondence is suggested also for the long-range phases. Recent experimental proposals and advances open the possibility to probe the described long-range effects in near-future realistic setups.
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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.
We study the combined effects of disorder and range of the couplings on the phase diagram of one-dimensional topological superconductors. We consider an extended version of the Kitaev chain where hopping and pairing terms couple many sites. Deriving the conditions for the existence of Majorana zero modes, we show that either the range and the on-site disorder can greatly enhance the topological phases characterized by the appearance of one or many Majorana modes localized at the edges. We consider both a discrete and a continuous disorder distribution. Moreover, we discuss the role of correlated disorder which might further widen the topological regions. Finally, we show that in the purely long-range regime and in the presence of disorder, the spatial decay of the edge modes remains either algebraic or exponential, with eventually a modified localization length, as in the absence of disorder.
We study the two-dimensional topological superconductors of spinless fermions in a checkerboard-lattice Chern-insulator model. With the short-range p-wave superconducting pairing, multifarious topological quantum phase transitions have been found and several phases with high Chern numbers have been observed. We have established a rich phase diagram for these topological superconducting states. A finite-size checkerboard-lattice cylinder with a harmonic trap potential has been further investigated. Based upon the self-consistent numerical calculations of the Bogoliubov-de Gennes equations, various phase transitions have also been identified at different regions of the system. Multiple pairs of Majorana fermions are found to be well-separated and localized at the phase boundaries between the phases characterized by different Chern numbers.
We discover novel topological effects in the one-dimensional Kitaev chain modified by long-range Hamiltonian deformations in the hopping and pairing terms. This class of models display symmetry-protected topological order measured by the Berry/Zak phase of the lower-band eigenvector and the winding number of the Hamiltonians. For exponentially decaying hopping amplitudes, the topological sector can be significantly augmented as the penetration length increases, something experimentally achievable. For power-law decaying superconducting pairings, the massless Majorana modes at the edges get paired together into a massive nonlocal Dirac fermion localized at both edges of the chain: a new topological quasiparticle that we call topological massive Dirac fermion. This topological phase has fractional topological numbers as a consequence of the long-range couplings. Possible applications to current experimental setups and topological quantum computation are also discussed.
Higher-order topological insulators are materials with a bulk energy gap featuring gapless modes at (d−n)(d-n)(d−n)-dimensional edges, where n>1n > 1n>1. Despite this modified bulk-boundary correspondence, their nontrivial topology can nevertheless be observed through the bipartite entanglement spectrum. Inspired by works suggesting fluctuations in conserved quantities could be used to study topology in one-dimensional systems, which likewise feature zero-dimensional edge modes, we extend this treatment to a two-dimensional second-order topological insulator where corner states play an analogous role. We show that the standard bipartite entanglement entropy and particle number fluctuation lack striking signals of topological phase transitions, both in the static case and in the time evolution after a quench, obscured by an area-law term from the subsystem edges. By introducing a quadripartite construction of the subsystem, we show that it is possible to isolate the topological contributions, producing sharp transition peaks in both quantities, observable by studying subsystems much smaller than the full system. As particle number fluctuations are accessible in existing experiments, these results establish a concrete and scalable path to experimental detection of the entanglement in higher-order topological insulators.