$Z_{2}$ Topological Order and Topological Protection of Majorana Fermion Qubits

Haq, Rukhsan Ul; Kauffman, Louis H.

doi:10.48550/arxiv.1704.00252

Cited by 3 publications

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“…In this paper, we will focus on the case where the Hamiltonian for the (2N ) Majorana fermions (γ 1 , ..., γ 2N ) actually does not involve the last one γ 2N , but only involves the odd number (2N − 1) of Majorana fermions (γ 1 , ..., γ 2N −1 ), a problem that has attracted a lot of interest recently [33][34][35][36][37][38]. Then the Hamiltonian H commutes both with γ 2N [H, γ 2N ] = 0 (14) and with the Parity of Eq.…”

Section: B Time-reversal-symmetry Tmentioning

confidence: 99%

“…where the N = 2 N −1 pseudo-couplings ω (p) j1j2...jp allow to reproduce the N = 2 N −1 energy levels E n . The first pseudo-Majorana fermion γ1 = ã1 (38) is absent from the Hamiltonian of Eq. 37 and is thus an odd normalized zero mode.…”

Section: Diagonalization In Terms Of Pseudo-majorana Fermionsmentioning

confidence: 99%

“…The present definition will be more convenient technically for our present purposes, and can always be achieved by some appropriate choice of the boundary couplings. As explained in more details in section II C, one can for instance put to zero the couplings involving the last Majorana operator γ 2N , so that the modified Hamiltonian involving an odd number (2N − 1) of Majorana operators has exact odd zero modes [33][34][35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Even and odd normalized zero modes in random interacting Majorana models respecting the parity P and the time-reversal-symmetry T

Monthus¹

2018

J. Phys. A: Math. Theor.

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For random interacting Majorana models where the only symmetries are the Parity P and the Time-Reversal-Symmetry T , various approaches are compared to construct exact even and odd normalized zero modes Γ in finite size, i.e. hermitian operators that commute with the Hamiltonian, that square to the Identity, and that commute (even) or anticommute (odd) with the Parity P . Even Normalized Zero-Modes Γ even are well known under the name of 'pseudo-spins' τ z n in the field of Many-Body-Localization or more precisely 'Local Integrals of Motion' (LIOMs) in the Many-Body-Localized-Phase where the pseudo-spins happens to be spatially localized. Odd Normalized Zero-Modes Γ odd are popular under the name of 'Majorana Zero Modes' or 'Strong Zero Modes'. Explicit examples for small systems are described in detail. Applications to real-space renormalization procedures based on blocks containing an odd number of Majorana fermions are also discussed.I.

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Section: B Time-reversal-symmetry Tmentioning

confidence: 99%

Section: Diagonalization In Terms Of Pseudo-majorana Fermionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Even and odd normalized zero modes in random interacting Majorana models respecting the parity P and the time-reversal-symmetry T

Monthus¹

2018

J. Phys. A: Math. Theor.

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“…Construction of many-body-localized models where all the eigenstates are matrix-product-states [118][119][120][121][122][123][124] instead of the even number (2N) of Majorana operators that are needed to describe a chain of N spins (see the reminder in appendix A).…”

Section: J Stat Mech (2020) 083301mentioning

confidence: 99%

Construction of many-body-localized models where all the eigenstates are matrix-product-states

Monthus¹

2020

J. Stat. Mech.

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The inverse problem of 'eigenstates-to-Hamiltonian' is considered for an open chain of N quantum spins in the context of many-body-localization. We first construct the simplest basis of the Hilbert space made of 2 N orthonormal matrix-product-states (MPS), that will thus automatically satisfy the entanglement area-law. We then analyze the corresponding N local integrals of motions (LIOMs) that can be considered as the local building blocks of these 2 N MPS, in order to construct the parent Hamiltonians that have these 2 N MPS as eigenstates. Finally we study the matrix-product-operator form of the diagonal ensemble density matrix that allows to compute long-time-averaged observables of the unitary dynamics. Explicit results are given for the memory of local observables and for the entanglement properties in operator-space, via the generalized notion of Schmidt decomposition for density matrices describing mixed states.

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“…They have been considered both in random systems in relation with Many-Body-Localization models [56] or in non-random models like the integrable XYZ chain [112] where they were called 'Strong Zero Mode', with various consequences for the long coherence time of edge spins [113,114], for the phenomenon of prethermalization [115], and for their fate in the presence of dissipation [116], while generalization to ladders can be found in [117]. In the Majorana formulation, these exact odd zero modes appear whenever the Hamiltonian involves an odd number (2N − 1) of Majorana operators [118][119][120][121][122][123][124] instead of the even number (2N ) of Majorana operators that are needed to describe a chain of N spins (See the reminder in Appendix A).…”

Section: B Parent Hamiltonians In Terms Of the Local Integrals Of Mot...mentioning

confidence: 99%

Construction of Many-Body-Localized Models where all the eigenstates are Matrix-Product-States

Monthus

2019

Preprint

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The inverse problem of 'eigenstates-to-Hamiltonian' is considered for an open chain of N quantum spins in the context of Many-Body-Localization. We first construct the simplest basis of the Hilbert space made of 2 N orthonormal Matrix-Product-States (MPS), that will thus automatically satisfy the entanglement area-law. We then analyze the corresponding N Local Integrals of Motions (LIOMs) that can be considered as the local building blocks of these 2 N MPS, in order to construct the parent Hamiltonians that have these 2 N MPS as eigenstates. Finally we study the Matrix-Product-Operator form of the Diagonal Ensemble Density Matrix that allows to compute long-time-averaged observables of the unitary dynamics. Explicit results are given for the memory of local observables and for the entanglement properties in operator-space, via the generalized notion of Schmidt decomposition for density matrices describing mixed states. I. INTRODUCTIONMany-Body-Localization for quantum interacting disordered systems is a fascinating phase of matter with very unusual properties with respect to the standard thermalization scenario of statistical physics (see the reviews [1-8] and references therein). In particular, excited eigenstates display an entanglement area-law [9-13] instead of the entanglement volume-law of thermalized eigenstates. As a consequence, they can be efficiently approximated by Matrix Product States or DMRG-X algorithms [14-20] that generalize to excited states the Density-Matrix-RG algorithm concerning ground-states [21][22][23]. Another proposal is to construct them via the RSRG-X procedure [24][25][26][27][28][29][30][31][32][33][34][35][36] that generalizes to excited states the Strong Disorder Real-Space RG approach [37] introduced initially by Ma-Dasgupta-Hu [38] and Daniel Fisher [39] to construct the ground states of random quantum spin chains. Another surprising property is that the Many-Body-Localized phase can be characterized by an extensive number of Local Integrals of Motions (LIOMs) [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57]. The emergence of these LIOMs can be for instance understood within the RSRG-t procedure [58][59][60][61][62] that generalizes to the unitary dynamics the Strong Disorder Real-Space RG approach already mentioned above. These LIOMs can be interpreted as the building blocks of the whole set of eigenstates.The main activity in the field of Many-Body-Localization has been focused on the 'direct problem', where one analyzes the properties of a given disordered interacting Hamiltonian H in order to determine if a Many-Body-Localized phase exists in a certain region of parameters, usually numerically or via approximate analytical methods, although some mathematically rigorous results also exist [44,63]. In the present paper, we will instead consider the 'inverse problem' : we will first build an orthonormal basis of the Hilbert space made of Matrix-Product-States, that will thus automatically satisfy the entanglement area-law; we will then construct the p...

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$Z_{2}$ Topological Order and Topological Protection of Majorana Fermion Qubits

Cited by 3 publications

References 0 publications

Even and odd normalized zero modes in random interacting Majorana models respecting the parity P and the time-reversal-symmetry T

Even and odd normalized zero modes in random interacting Majorana models respecting the parity P and the time-reversal-symmetry T

Construction of many-body-localized models where all the eigenstates are matrix-product-states

Construction of Many-Body-Localized Models where all the eigenstates are Matrix-Product-States

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