Exact zero modes in twisted Kitaev chains

Kawabata, Ken‐ichi; Kobayashi, Ryohei; Wu, Ning; Katsura, Hosho

doi:10.1103/physrevb.95.195140

Cited by 42 publications

(63 citation statements)

References 79 publications

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“…We insert a flux term into the closed chain that plays the role of switching the boundary conditions from periodic to anti-periodic. Such a system was previously studied in [40]. The Hamiltonian is…”

Section: Flux Insertion and Z 2 -Valued Spectral Flowmentioning

confidence: 99%

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On ℤ2-indices for ground states of fermionic chains

Bourne

Schulz-Baldes

2020

Rev. Math. Phys.

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For parity-conserving fermionic chains, we review how to associate Z 2 -indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the Z 2 -valued spectral flow provides a topological obstruction for two systems to have the same Z 2 -index. A rudimentary definition of a Z 2 -phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.

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Section: Flux Insertion and Z 2 -Valued Spectral Flowmentioning

confidence: 99%

“…Our analysis closely follows [40,Appendix D], who considered the interacting Kitaev chain with twisted boundary conditions. We add periodic boundary conditions to the Hamiltonian with local flux, H int Λ (α) = −w(e −iα a * 1 a 2 + e iα a * 2 a 1 ) + w(e iα a 1 a 2 + e −iα a * 2 a * 1 )…”

Section: Flux Insertion and Gap Closing In The Closed Chainmentioning

confidence: 99%

On ℤ2-indices for ground states of fermionic chains

Bourne

Schulz-Baldes

2020

Rev. Math. Phys.

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“…Nevertheless, simple equally weighted linear superpositions of the two yield two orthogonal states with distinct fermion parities [26]. It was further shown in [30] that the aforementioned two orthogonal states, one of which having odd fermion parity and the other having even parity, are also the ground states of the interacting Kitaev chain under periodic and antiperiodic boundary conditions, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Actually, it was shown in Ref. [30] that |Ψ (e) (|Ψ (o) ) is exactly the ground state of the fermion version of the homogeneous periodic XYZ chain…”

mentioning

confidence: 99%

Longitudinal magnetization dynamics in the quantum Ising ring: A Pfaffian method based on correspondence between momentum space and real space

2020

Phys. Rev. E

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As perhaps the most studied paradigm for a quantum phase transition, the periodic quantum Ising chain is exactly solvable via the Jordan-Wigner transformation followed by a Fourier transform that diagonalizes the model in the momentum space of spinless fermions. Although the above procedures are well-known, there remain some subtle points to be clarified regarding the correspondence between the real-space and momentum-space representations of the finite-size quantum Ising ring, especially those related to fermion parities. In this work, we establish the relationship between the two fully aligned ferromagnetic states in real space and the two degenerate momentum-space ground states of the classical Ising ring, with the former being a special case of the factorized ground states of the more general XYZ model on the frustration-free hypersurface. Based on this observation, we then provide a Pfaffian formula for calculating real-time dynamics of the parity-breaking longitudinal magnetization with the system initially prepared in one of the two ferromagnetic states and under translationally invariant drivings. The formalism is shown to be applicable to large systems with the help of online programs for the numerical computation of the Pfaffian, thus providing an efficient method to numerically study, for example, the emergence of discrete time crystals in related systems.

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“…In general, these two modes are "almost" Majorana zero modes (AMZM) in the sense that they decay exponentially away from the edges and possesses an exponentially small excitation energy in the thermodynamic limit. However, exact Majorana zero modes (EMZMs) that strictly commute with the Hamiltonian can emerge [11,24] in the special case with equal hopping and pairing strength. These two unpaired EMZMs are spatially separated and lead to two-fold degenerate ground states robust under fermionparity-preserving perturbations.…”

Section: Introductionmentioning

confidence: 99%

Exact zero modes in a quantum compass chain under inhomogeneous transverse fields

You

2019

Phys. Rev. B

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We study the emergence of exact Majorana zero modes (EMZMs) in a one-dimensional quantum transverse compass model with both the nearest-neighbor interactions and transverse fields varying over space. By transforming the spin system into a quadratic Majorana-fermion model, we derive an exact formula for the number of the emergent EMZMs, which is found to depend on the partition nature of the lattice sites on which the magnetic fields vanish. We also derive explicit expressions for the wavefunctions of these EMZMs and show that they indeed depend on fine features of the foregoing partition of site indices. Based on the above rigorous results about the EMZMs, we provide an interpretation for the interesting dependence of the eigenstate-degeneracy on the transverse fields observed in prior literatures. As a special case, we employ a plane-wave ansatz to exactly solve an open compass chain with alternating nearest-neighbor interactions and staggered magnetic fields. Explicit forms of the canonical Majorana modes diagonalizing the model are given even for finite chains. We show that besides the possibly existing EMZMs, no almost Majorana zero modes exist unless the fields on both the two sublattices are turned off. Our results might shed light on the control of ground-state degeneracies by solely tuning the external fields in related systems.

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Exact zero modes in twisted Kitaev chains

Cited by 42 publications

References 79 publications

On ℤ2-indices for ground states of fermionic chains

On ℤ2-indices for ground states of fermionic chains

Longitudinal magnetization dynamics in the quantum Ising ring: A Pfaffian method based on correspondence between momentum space and real space

Exact zero modes in a quantum compass chain under inhomogeneous transverse fields

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