Generating many Majorana modes via periodic driving: A superconductor model

Tong, Qingjun; An, Jun-Hong; Gong, Jiangbin; Luo, Hong‐Gang; Oh, C. H.

doi:10.1103/physrevb.87.201109

Cited by 165 publications

(101 citation statements)

References 65 publications

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“…21 Theoretical proposals have already seen several such states at a single edge if the driving also ensures CS. 22 The bulk topological invariant controlling the number of these edge states is as yet unknown, but it could be derived using the approach of this Rapid Communication.…”

Section: Physical Review B 88 121406(r) (2013)mentioning

confidence: 99%

Bulk-boundary correspondence for chiral symmetric quantum walks

2013

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Discrete-time quantum walks (DTQW) have topological phases that are richer than those of time-independent lattice Hamiltonians. Even the basic symmetries, on which the standard classification of topological insulators hinges, have not yet been properly defined for quantum walks. We introduce the key tool of time frames, i.e., we describe a DTQW by the ensemble of time-shifted unitary time-step operators belonging to the walk. This gives us a way to consistently define chiral symmetry (CS) for DTQW's. We show that CS can be ensured by using an "inversion symmetric" pulse sequence. For one-dimensional DTQW's with CS, we identify the bulk Z × Z topological invariant that controls the number of topologically protected 0 and π energy edge states at the interfaces between different domains, and give simple formulas for these invariants. We illustrate this bulk-boundary correspondence for DTQW's on the example of the "4-step quantum walk," where tuning CS and particle-hole symmetry realizes edge states in various symmetry classes. The realization that band insulators can have nontrivial topological properties that determine the low-energy physics at their boundary has been a rich source of new physics in the last decade. The general theory of topological insulators and superconductors 1,2 classifies gapped Hamiltonians according to their dimension and their symmetries.3 As very few real-life materials are topological insulators, there is a strong push to develop model systems, "artificial materials," that simulate topological phases. 4 One of the promising approaches is to use discrete-time quantum walks (DTQW), 5-8 which can simulate topological insulators from all symmetry classes in 1D and 2D.9-11 DTQW's with particle-hole symmetry (PHS) go beyond simulating topological insulating Hamiltonians: they have topological phases with no counterpart in standard solid-state setups. In 1D DTQW's with PHS, edge states, "Majorana modes" can have two protected quasienergies: ε = 0 or π (time is measured in units of the time step andh = 1). Building on the results for periodically driven systems, 12 one of us has defined the corresponding Z 2 × Z 2 topological invariant.13 Both 0 and π energy Majorana edge states have been experimentally observed in a quantum walk. 14 The situation of chiral symmetry (CS) of DTQW's is much less clear. Even for the simplest one-dimensional DTQW, it is disputed whether it even has CS 9 or not. 13 Although it is expected that CS should imply a Z × Z bulk topological invariant, this has not yet been found for DTQW's. As opposed to the case of PHS, there is also not much to draw on from periodically driven systems. What DTQW's have CS? How can the bulk "winding number" be expressed for DTQW's with CS? These are the problems we tackle in this Rapid Communication.A DTQW concerns the dynamics of a particle, "walker," whose wave function is given by a vector, | = N x=1 s=−1,1 (x,s)|x,s . Here, x = 1, . . . ,N is the discrete position, and s = ±1 indexes the two orthogonal internal states of the walker, t...

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Section: Physical Review B 88 121406(r) (2013)mentioning

confidence: 99%

Bulk-boundary correspondence for chiral symmetric quantum walks

2013

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“…[51][52][53] , the appearance of new Dirac cones was due to increasing either the hopping strength or hopping range in a static Hamiltonian. We point out that by increasing J 1 and J 2 here, we are effectively simulating a static Hamiltonian with long-range hopping 18 . To see this, note the effective Hamiltonian is defined via…”

Section: B Topological Invariants For the Bulk Spectrum Of Dklmentioning

confidence: 99%

Topological effects in chiral symmetric driven systems

Gong

2014

Phys. Rev. B

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107

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Recent years have seen a strong interest in topological effects within periodically driven systems.In this work, we explore topological effects in two closely related 2-dimensional driven systems described by Floquet operators possessing chiral symmetry (CS). Our numerical and analytical results suggest the following. Firstly, the CS is associated with the existence of the anomalous counter-propagating (ACP) modes reported recently. Specifically, we show that a particular form of CS protects the ACP modes occurring at quasienergies of ±π. We also find that these modes are only present along selected boundaries, suggesting that they are a weak topological effect. Secondly, we find that CS can give rise to protected 0 and π quasienergy modes, and that the number of these modes may increase without bound as we tune up certain system parameters. Like the ACP modes, these 0 and π modes also appear only along selected boundaries and thus appear to be a weak topological effect. To our knowledge, this work represents the first detailed study of weak topological effects in periodically driven systems. Our findings add to the still-growing knowledge on driven topological systems.

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“…While every technique brings its own technical challenges, the Floquet approaches appear to be simpler. Similar proposals appear in the solid state and photonics literature [9][10][11][12]. Recent cold atoms experiments have successfully demonstrated uniform 1D gauge fields [13] and band hybridization [14] using shaken optical lattices.…”

Section: Introductionmentioning

confidence: 72%

Floquet edge states with ultracold atoms

Reichl

Mueller

2014

Phys. Rev. A

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We describe an experimental setup for imaging topologically protected Floquet edge states using ultracold bosons in an optical lattice. Our setup involves a deep two dimensional optical lattice with a time dependent superlattice that modulates the hopping between neighboring sites. The finite waist of the superlattice beam yields regions with different topological numbers. One can observe chiral edge states by imaging the real-space density of a bosonic packet launched from the boundary between two topologically distinct regions.

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Generating many Majorana modes via periodic driving: A superconductor model

Cited by 165 publications

References 65 publications

Bulk-boundary correspondence for chiral symmetric quantum walks

Bulk-boundary correspondence for chiral symmetric quantum walks

Topological effects in chiral symmetric driven systems

Floquet edge states with ultracold atoms

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