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arXiv:2404.16099 · Physical review. B./Physical review. B

Reviving Majorana zero modes in the spin- 1 2 Kitaev ladder model

Haoting Xu, Hae‐Young Kee

Published Nov 14, 2024• 1 citation
arXivHTMLPDF

Abstract

The one-dimensional $p$-wave superconductor, characterized by boundary Majorana modes, has attracted significant interest owing to its potential application in topological quantum computation. Similarly, spin-1/2 Kitaev ladder systems with bond-dependent Ising interactions, featuring Majorana fermions coupled with ${Z}_{2}$ flux, exhibit boundary Majorana modes when in a topological phase. However, the ground-state degeneracy, inherent in these systems, may result in the annihilation of Majorana modes due to the superposition of the degenerate states. To avoid this issue, here we introduce a projective measurement that selects one of the degenerate ${Z}_{2}$ sectors, enabling the revival of Majorana modes. Once the state is selected, we show that the application of the local spin operators on a bond flips the sign of the adjacent ${Z}_{2}$ flux. Repeating such operators enables the system to reach a desired ${Z}_{2}$ flux configuration. The Majorana zero modes can be manipulated and fused by tuning the flux sectors achievable through applying local spin operators. We also discuss the engineering of the Kitaev ladder and open questions for future studies.

DOI
10.1103/physrevb.110.195133

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The one-dimensional ppp-wave superconductor, characterized by boundary Majorana modes, has attracted significant interest owing to its potential application in topological quantum computation. Similarly, spin-1/2 Kitaev ladder systems with bond-dependent Ising interactions, featuring Majorana fermions coupled with Z2{Z}_{2}Z2​ flux, exhibit boundary Majorana modes when in a topological phase. However, the ground-state degeneracy, inherent in these systems, may result in the annihilation of Majorana modes due to the superposition of the degenerate states. To avoid this issue, here we introduce a projective measurement that selects one of the degenerate Z2{Z}_{2}Z2​ sectors, enabling the revival of Majorana modes. Once the state is selected, we show that the application of the local spin operators on a bond flips the sign of the adjacent Z2{Z}_{2}Z2​ flux. Repeating such operators enables the system to reach a desired Z2{Z}_{2}Z2​ flux configuration. The Majorana zero modes can be manipulated and fused by tuning the flux sectors achievable through applying local spin operators. We also discuss the engineering of the Kitaev ladder and open questions for future studies.
One-dimensional systems with topological order are intimately related to the appearance of zero-energy modes localized on their boundaries. The most common example is the Kitaev chain, which displays Majorana zero-energy modes and it is characterized by a twofold ground-state degeneracy related to the global Z2{\mathbb{Z}}_{2}Z2​ symmetry associated with fermionic parity. By extending the symmetry to the ZN{\mathbb{Z}}_{N}ZN​ group, it is possible to engineer systems hosting topological parafermionic modes. In this work, we address one-dimensional systems with a generic discrete symmetry group GGG. We define a ladder model of gauge fluxes that generalizes the Ising and Potts models and displays a symmetry broken phase. Through a non-Abelian Jordan-Wigner transformation, we map this flux ladder into a model of dyonic operators, defined by the group elements and irreducible representations of GGG. We show that the so-obtained dyonic model has topological order, with zero-energy modes localized at its boundary. These dyonic zero-energy modes are in general weak topological modes, but strong dyonic zero modes appear when suitable position-dependent couplings are considered.
We explore the salient features of the 'Kitaev ladder', a two-legged ladder version of the spin-1/2 Kitaev model on a honeycomb lattice, by mapping it to a one-dimensional fermionic p-wave superconducting system. We examine the connections between spin phases and topologically non-trivial phases of non-interacting fermionic systems, demonstrating the equivalence between the spontaneous breaking of global Z 2 symmetry in spin systems and the existence of isolated Majorana modes. In the Kitaev ladder, we investigate topological properties of the system in different sectors characterized by the presence or absence of a vortex in each plaquette of the ladder. We show that vortex patterns can yield a rich parameter space for tuning into topologically non-trivial phases. We introduce and employ a new topological invariant for explicitly determining the presence of zero energy Majorana modes at the boundaries of such phases. Finally, we discuss dynamic quenching between topologically non-trivial phases in the Kitaev ladder and, in particular, the post-quench dynamics governed by tuning through a quantum critical point.
We explore the salient features of the `Kitaev ladder', a two-legged ladder version of the spin-1/2 Kitaev model on a honeycomb lattice, by mapping it to a one-dimensional fermionic p-wave superconducting system. We examine the connections between spin phases and topologically non-trivial phases of non-interacting fermionic systems, demonstrating the equivalence between the spontaneous breaking of global Z(2) symmetry in spin systems and the existence of isolated Majorana modes. In the Kitaev ladder, we investigate topological properties of the system in different sectors characterized by the presence or absence of a vortex in each plaquette of the ladder. We show that vortex patterns can yield a rich parameter space for tuning into topologically non-trivial phases. We introduce and employ a new topological invariant for explicitly determining the presence of zero energy Majorana modes at the boundaries of such phases. Finally, we discuss dynamic quenching between topologically non-trivial phases in the Kitaev ladder and, in particular, the post-quench dynamics governed by tuning through a quantum critical point.
A time-reversal invariant Kitaev-type model is introduced in which spins (Dirac matrices) on the square lattice interact via anisotropic nearest-neighbor and next-nearest-neighbor exchange interactions. The model is exactly solved by mapping it onto a tight-binding model of free Majorana fermions coupled with static Z2{\mathbb{Z}}_{2}Z2​ gauge fields. The Majorana fermion model can be viewed as a model of time-reversal-invariant superconductor and is classified as a member of symmetry class DIII in the Altland-Zirnbauer classification. The ground-state phase diagram has two topologically distinct gapped phases which are distinguished by a Z2{\mathbb{Z}}_{2}Z2​ topological invariant. The topologically nontrivial phase supports both a Kramers' pair of gapless Majorana edge modes at the boundary and a Kramers' pair of zero-energy Majorana states bound to a 0-flux vortex in the $\ensuremath{\pi}$-flux background. Power-law decaying correlation functions of spins along the edge are obtained by taking the gapless Majorana edge modes into account. The model is also defined on the one-dimension ladder, in which case again the ground-state phase diagram has Z2{\mathbb{Z}}_{2}Z2​ trivial and nontrivial phases.
One promising avenue to study one-dimensional (1D) topological phases is to realize them in synthetic materials such as cold atomic gases. Intriguingly, it is possible to realize Majorana boundary modes in a 1D number-conserving system consisting of two fermionic chains coupled only by pair-hopping processes [C. V. Kraus et al., Phys. Rev. Lett. 111, 173004 (2013)]. It is commonly believed that significant interchain single-particle tunneling necessarily destroys these Majorana modes, as it spoils the Z2{\mathbb{Z}}_{2}Z2​ fermion-parity symmetry that protects them. In this Rapid Communication, we present a mechanism to overcome this obstacle, by piercing a (synthetic) magnetic $\ensuremath{\pi}$ flux through each plaquette of the Fermi ladder. Using bosonization, we show that in this case there exists an exact leg-interchange symmetry that is robust to interchain hopping, and acts as fermion parity at long wavelengths. We utilize density matrix renormalization group and exact diagonalization to verify that the resulting model exhibits Majorana boundary modes up to large single-particle tunnelings, comparable to the intrachain hopping strength. Our work highlights the unusual impacts of different topologically trivial band structures on these interaction-driven topological phases, and identifies a distinct route to stabilizing Majorana boundary modes in 1D fermionic ladders.
We investigate Majorana modes in a quantum spin chain with bond-dependent exchange interactions by studying its dynamics. Specifically, we consider two-time correlations for the anisotropic Kitaev-Heisenberg (KH) Hamiltonian close to the so-called Kitaev critical point. Here the model coincides with a phase boundary of two uncoupled instances of Kitaev's model for ppp-wave superconductors, together supporting a degenerate ground state characterized by multiple Majorana modes. In this regime, the real-time dynamics of local spins reveal a set of strong zero modes, corresponding to a set of protruding frequencies in the two-time correlation function. We derive perturbative interactions that map the KH spin chain onto the topological regime of Kitaev's fermionic model, thus opening up a bulk gap while retaining almost degenerate modes in the mesoscopic regime, i.e., for finite system sizes. This showcases the emergence of Majorana modes in a chain of effective dimers. Here, the binding energy within each unit cell competes with the interdimer coupling to generate a finite-size energy gap, in analogy with local energy terms in the transverse-field Ising model. These modes give rise to long coherence times of local spins located at the system edges. By breaking the local symmetry in each dimer, one can also observe a second class of Majorana modes in terms of a beating frequency in the two-time correlations function of the edge spin. Furthermore, we develop a scenario for realizing these model predictions in ion-trap quantum simulators with collective addressing of the ions.