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

Abstract

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.

Beyond Kitaev physics in strong spin-orbit coupled magnets

“…In the following, we briefly review two notable examples. The influence of magnetic fields on the spin-1/2 Kitaev chain has been explored in several works [256][257][258][259][260]. Here we shall briefly expand on the field-induced chiral soliton phase reported in [260].…”

“…Similar to the spin-1/2 and spin-1 chains, these studies show clear differences in the ground states between half-integer and integer spin chain systems. Another line of investigations concerns the response of these 1D models to various perturbations, such as external magnetic fields [215,[256][257][258][259][260]. In the following, we briefly review two notable examples.…”

Quantum criticality in interacting bosonic Kitaev-Hubbard models

“…We note that the hard-core bosonic Kitaev-Hubbard chain with periodic boundary conditions is exactly equivalent to the fermionic counterpart with antiperiodic boundary conditions on a finite system. Interestingly, two Ising transition lines merge at µ = 0, θ = π/2, implying that this point might be a multi-critical point, which corresponds to the celebrated compass model with 2 L/2−1 -fold degenerate ground states [77][78][79][80]. To be more explicit, the Hamiltonian of the hard-core bosonic Kitaev-Hubbard chain with t = ∆, θ = π/2 and µ = 0 is given by…”

Lifshitz phase transitions in one-dimensional Gamma model

“…For α = 0 and β=0, H o turns out to vanish and one can notice the ground state is 2 N/2−1 -fold degenerate with PBCs and 2 N/2 -fold degenerate with open boundary conditions. The macroscopic degeneracy can be interpreted either by the intermediate symmetry operators [39][40][41] or by the free Majorana modes [42]. Once a finite local transverse magnetic field g l (1 ≤ l ≤ N ) is turned on, an lth tooth will hybridize the even Majorana chain with a Majorana mode in the odd Majorana chain since g l σ z l = ig l γ 2l−1 γ 2l .…”