Exact ground states and topological order in interacting Kitaev/Majorana chains

Katsura, Hosho; Schuricht, Dirk; Takahashi, Masahiro

doi:10.1103/physrevb.92.115137

Cited by 148 publications

(225 citation statements)

References 83 publications

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“…While we use a noninteracting fermionic model (the Kitaev chain (1)) to demonstrate our method, we emphasize that our topological invariants are applicable to interacting models and can be used in numerical simulations, such as quantum Monte Carlo. In Appendix F, we present the calculation of the topological invariant in an interacting Majorana chain, by making use of the known exact expression of the ground state [54]. In addition, throughout this letter, we consider BCS mean-field wave functions which do not preserve the particle number.…”

Section: T1mentioning

confidence: 99%

Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

2017

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We define and compute many-body topological invariants of interacting fermionic symmetryprotected topological phases, protected by an orientation-reversing symmetry, such as time-reversal or reflection symmetry. The topological invariants are given by partition functions obtained by a path integral on unoriented spacetime which, as we show, can be computed for a given ground state wave function by considering a non-local operation, "partial" reflection or transpose. As an application of our scheme, we study the Z8 and Z16 classification of topological superconductors in one and three dimensions.The Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula [1, 2] is the prototype for topological characterization of phases of matter. It relates the quantized Hall conductance to the (first) Chern number defined for Bloch wave functions. At the level of many-body physics, the quantized Hall conductance can also be formulated in terms of ground state wave functions in the presence of twisted boundary conditions ("the many-body Chern number") [3]. In contrast to local order parameters, the TKNN integer distinguishes different quantum phases of matter by focusing on their global topological properties.More recently, the discovery of topological insulators and superconductors [4, 5] has led to a new research frontier, generally referred to as symmetry protected topological (SPT) phases. These phases are adiabatically connected to topologically trivial states, i.e., atomic insulators which can be represented as simple product states without any entanglement. Nevertheless, they are topologically distinct once a symmetry condition, e.g., timereversal symmetry, is imposed. A complete classification of the noninteracting fermionic SPT phases protected by non-spatial discrete symmetries [6][7][8], as well as crystalline SPT phases protected by spatial symmetries [9][10][11][12], were achieved.However, it was later discovered that the noninteracting topological classification is not the full story, and can be dramatically altered once interaction effects are taken into account [13]. Since then, there have been several works which discuss the breakdown of the noninteracting classification in the presence of interactions [14][15][16][17][18][19][20][21][22][23][24][25].There are various topological invariants for noninteracting fermionic SPT phases using single-particle states (e.g., Bloch wave functions). For example, the Z 2 -valued topological invariants have been introduced for topological insulators both in two and three spatial dimensions [26][27][28][29]. For topological superconductors protected by time-reversal symmetry, the integer-valued topological invariants ("the winding number") have been introduced [6]. However, the discovered breakdown of non-interacting classification clearly indicates that the situation at the interacting level is more intricate, and a general framework to distinguish interacting fermionic SPT phases is lacking. This should be contrasted from the quantized Hall conductance, which can be formulate...

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Section: T1mentioning

confidence: 99%

Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

2017

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“…For example, considerable progress has been made towards developing number preserving theories of the Majorana modes, [15][16][17][18][19][20] as well as a growing body of work which examines how free-topological superconducting phases are affected by the addition of interacting electron-electron terms. [21][22][23][24][25][26][27][28][29][30][31][32][33] One aspect of this latter story is concerned with the stability and structure of the Majorana zero-modes themselves and how they are affected by the presence of density-density interaction terms that break the exactly solvable nature of the underlying model. The issue of stable zero-modes has also been addressed in the related context of 1-d parafermionic chains.…”

Section: Introductionmentioning

confidence: 99%

Multiparticle content of Majorana zero modes in the interactingp-wave wire

Kells

2015

Phys. Rev. B

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In the topological phase of p-wave superconductors, zero-energy Majorana quasi-particle excitations can be well-defined in the presence of local density-density interactions. Here we examine this phenomenon from the perspective of matrix representations of the commutator H = [H, •] ,with the aim of characterising the multiparticle content of the many-body Majorana mode. To do this we show that, for quadratic fermionic systems, H can always be decomposed into sub-blocks that act as multi-particle generalisations of the BdG/Majorana forms that encode single-particle excitations. In this picture, density-density like interactions will break this exact excitation-number symmetry, coupling different sub-blocks and lifting degeneracies so that the eigen-operators of the commutator H take the form of individual eigenstate transitions | n m |. However, the Majorana mode is special in that zero-energy transitions are not destroyed by local interactions and it becomes possible to define many-body Majoranas as the odd-parity zero-energy solutions of H that minimise their excitation number. This idea forms the basis for an algorithm which is used to characterise the multi-particle excitation content of the Majorana zero modes of the one-dimensional p-wave lattice model. We find that the multi-particle content of the Majorana zero-mode operators is significant even at modest interaction strengths. This has important consequences for the stability of Majorana based qubits when they are coupled to a heat bath. We will also discuss how these findings differ from previous work regarding the structure of the many-body-Majorana operators and point out that this should affect how certain experimental features are interpreted.

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“…This is similar to the approach in 1D case in Ref. [34]. We modify the bulk HamiltonianK bulk in Eq.…”

Section: Wavefunctions In a Double Wall Geomertymentioning

confidence: 99%

Analytic ground state wave functions of mean-field px+ipy superconductors with vortices and boundaries

Wang

Hazzard

2018

Phys. Rev. B

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We study Read and Green's mean-field model of the spinless px +ipy superconductor [N. Read and D. Green, Phys. Rev. B 61, 10267 (2000)] at a special set of parameters where we find the analytic expressions for the topologically degenerate ground states and the Majorana modes, including in finite systems with edges and in the presence of an arbitrary number of vortices. The wavefunctions of these ground states are similar (but not always identical) to the Moore-Read Pfaffian states proposed for the ν = 5/2 fractional quantum Hall system, which are interpreted as the p-wave superconducting states of composite fermions. The similarity in the long-wavelength universal properties is expected from previous work, but at the special point studied herein the wavefunctions are exact even for short-range, non-universal properties. As an application of these results, we show how to obtain the non-Abelian statistics of the vortex Majorana modes by explicitly calculating the analytic continuation of the ground state wavefunctions when vortices are adiabatically exchanged, an approach different from the previous one based on universal arguments. Our results are also useful for constructing particle number-conserving (and interacting) Hamiltonians with exact projected mean-field states.

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Exact ground states and topological order in interacting Kitaev/Majorana chains

Cited by 148 publications

References 83 publications

Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

Multiparticle content of Majorana zero modes in the interactingp-wave wire

Analytic ground state wave functions of mean-field px+ipy superconductors with vortices and boundaries

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