Topological Superconductors and Category Theory

Bernevig, Andrei; Neupert, Titus

doi:10.48550/arxiv.1506.05805

Cited by 17 publications

(33 citation statements)

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“…When ∆ 1 = ∆ 2 = µ = 0 the system becomes the SSH model that belongs to the same topological class as the Kitaev model, as is well known [7,11].…”

Section: Anisotropic Modelmentioning

confidence: 98%

“…In condensed matter physics, non-local Majorana fermions are quasi-particle excitations that arise when a single electronic mode fractionalize into two halves [1]. The Kitaev chain has been used to study the properties of non-local Majorana zero-energy bound states that reside at edges of the chain [1][2][3][4][5][6][7]. The Kitaev model belongs to the BDI topological class [7].…”

Section: Introductionmentioning

confidence: 99%

“…The Kitaev chain has been used to study the properties of non-local Majorana zero-energy bound states that reside at edges of the chain [1][2][3][4][5][6][7]. The Kitaev model belongs to the BDI topological class [7]. In this class, the non-trivial topological phase depends on time-reversal symmetry, particle-hole symmetry and chiral symmetry [12,13].…”

Section: Introductionmentioning

confidence: 99%

“…In this class, the non-trivial topological phase depends on time-reversal symmetry, particle-hole symmetry and chiral symmetry [12,13]. The special particle-hole symmetry ensures that the Majorana zero-energy bound states behave like a superposition of particle and hole and have no well-defined charge [1,7]. However, besides the particle-hole symmetry, the number of zero-energy bound states at the edges is crucial to obtain one isolated Majorana zero-energy bound state.…”

Section: Introductionmentioning

confidence: 99%

“…Any chain with an even number of zeroenergy bound states per edge does not have an isolated Majorana at the edges. To obtain an isolated Majorana zero-energy bound state is necessary an odd number of zero energy at the ends of the chain [7].…”

Section: Introductionmentioning

confidence: 99%

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Effects of anisotropic correlations in fermionic zero-energy bound states of topological phases

Griffith,

Mamani,

Nunes

et al. 2019

Preprint

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Topological phases of matter have been used as a fertile realm of intensive discussions about fermionic fractionalization. In this work, we study the effects of anisotropic superconducting correlations in the fermionic fractionalization on the topological phases. We consider a hybrid version of the SSH and Kitaev models with an anisotropic superconducting order parameter to investigate the unusual states with zero energy that emerges in a finite chain. To obtain these zero energy solutions, we built a chain with a well-defined domain wall at the middle of the chain. Our solutions indicates an interesting dynamic between the zero-energy state around the domain wall and the superconducting correlation parameters. Finally, we find that the presence of an isolated Majorana at the ends of the chain is strongly depending on the existence of the solitonic excitation at the middle of the chain.

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“…When ∆ 1 = ∆ 2 = µ = 0 the system becomes the SSH model that belongs to the same topological class as the Kitaev model, as is well known [7,11].…”

Section: Anisotropic Modelmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Effects of anisotropic correlations in fermionic zero-energy bound states of topological phases

Griffith,

Mamani,

Nunes

et al. 2019

Preprint

View full text Add to dashboard Cite

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On 2-group global symmetries and their anomalies

Benini

Córdova

Hsin

2019

J. High Energ. Phys.

243

353

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In general quantum field theories (QFTs), ordinary (0-form) global symmetries and 1-form symmetries can combine into 2-group global symmetries. We describe this phenomenon in detail using the language of symmetry defects. We exhibit a simple procedure to determine the (possible) 2-group global symmetry of a given QFT, and provide a classification of the related 't Hooft anomalies (for symmetries not acting on spacetime). We also describe how QFTs can be coupled to extrinsic backgrounds for symmetry groups that differ from the intrinsic symmetry acting faithfully on the theory. Finally, we provide a variety of examples, ranging from TQFTs (gapped systems) to gapless QFTs. Along the way, we stress that the "obstruction to symmetry fractionalization" discussed in some condensed matter literature is really an instance of 2-group global symmetry.

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No-go theorem for boson condensation in topologically ordered quantum liquids

Neupert

Keyserlingk

et al. 2016

New J. Phys.

Self Cite

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Certain phase transitions between topological quantum field theories (TQFTs) are driven by the condensation of bosonic anyons. However, as bosons in a TQFT are themselves nontrivial collective excitations, there can be topological obstructions that prevent them from condensing. Here we formulate such an obstruction in the form of a no-go theorem. We use it to show that no condensation is possible in SO(3) k TQFTs with odd k. We further show that a 'layered' theory obtained by tensoring SO(3) k TQFT with itself any integer number of times does not admit condensation transitions either. This includes (as the case k = 3) the noncondensability of any number of layers of the Fibonacci TQFT.Topological order, a fundamental concept in quantum many-body physics, is best understood in twodimensional gapped quantum liquids, such as the fractional quantum Hall effect and certain spin liquids [1-9]. In these systems, quasiparticle excitations with anyonic quantum-statistical properties emerge [10]. Their fusion and braiding behavior at large distances define a topological quantum field theory (TQFT), which characterizes the universal properties of the phase [11][12][13][14].The phase transitions between topological phases are, most of the times, driven by the condensation of bosons [11,[15][16][17][18][19][20][21][22][23][24][25]. In the context of TQFTs, a boson is an emergent quasiparticle in the topologically ordered phase with bosonic self-statistics, but which could have nontrivial fusion and braiding relations with the other anyons. Such a quasiparticle can potentially undergo Bose-Einstein condensation, causing a phase transition to another topologically ordered phase. The topological data of the new phase can be inferred from those of the initial topological order [25].One motivation to study condensation transitions is to classify topological order. An important example are the 16 types of gauged chiral superconductors introduced by Kitaev [3]. Kiteav showed that while twodimensional superconductors are classified by an integer , only 16 bulk phases are topologically distinct. This construction can be understood by considering ℓ layers of initially disconnected chiral p-wave superconductors, i.e., elementary (Ising) TQFTs. Upon introducing generic couplings between these layers, one obtains a single layer of a chiral ℓ-wave superconductor, which corresponds to a specific TQFT in Kitaev's classification. This physical process of coupling the layers (by condensing inter-layer cooper pairs), corresponds to a condensation transition on the level of the TQFTs. For every 16 < ℓ , there is a unique condensation possible and one obtains exactly 16 distinct TQFTs including Ising, the toric code and the double semion model. They determine the nature of the topologically protected excitations in the vortices of each superconductor, including their braiding statistics. In essence, this 16  classification can be seen as a property of the Ising TQFT.It is imperative to ask whether multi-layer systems of other TQFTs show a simil...

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Topological Superconductors and Category Theory

Cited by 17 publications

References 0 publications

Effects of anisotropic correlations in fermionic zero-energy bound states of topological phases

Effects of anisotropic correlations in fermionic zero-energy bound states of topological phases

On 2-group global symmetries and their anomalies

No-go theorem for boson condensation in topologically ordered quantum liquids

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