Majorana zero modes in a quantum Ising chain with longer-ranged interactions

Abstract

A one-dimensional Ising model in a transverse field can be mapped onto a system of spinless fermions with p-wave superconductivity. In the weak-coupling BCS regime, it exhibits a zero energy Majorana mode at each end of the chain. Here, we consider a variation of the model, which represents a superconductor with longer ranged kinetic energy and pairing amplitudes, as is likely to occur in more realistic systems. It possesses a richer zero temperature phase diagram and has several quantum phase transitions. From an exact solution of the model these phases can be classified according to the number of Majorana zero modes of an open chain: 0, 1, or 2 at each end. The model posseses a multicritical point where phases with 0, 1, and 2 Majorana end modes meet. The number of Majorana modes at each end of the chain is identical to the topological winding number of the Anderson's pseudospin vector that describes the BCS Hamiltonian. The topological classification of the phases requires a unitary time-reversal symmetry to be present. When this symmetry is broken, only the number of Majorana end modes modulo 2 can be used to distinguish two phases. In one of the regimes, the wave functions of the two phase shifted Majorana zero modes decays exponentially in space but in an oscillatory manner. The wavelength of oscillation is identical to the asymptotic connected spin-spin correlation of the XY -model in a transverse field to which our model is dual.

Entanglement properties of the time-periodic Kitaev chain

“…This corresponds to complex conjugation K for spinless fermions, and is clear in the position space definition of the model (1). While TRS may be a confusing issue for spin-less fermions, note that the model can also be derived from a spin-chain where the spinless fermions of our model are simply the Jordan-Wigner fermions 38,39 . With both PHS and TRS, our model falls into the BDI Altland Zirnbauer (AZ) class 35,36 , which for one dimension, has a topological index of Z.…”

Nonequilibrium quantum phase transitions in the Ising model

“…The idea is based on the fact that Hamiltonian equation (3) conserves momentum and parity. As a consequence, we can use the BCS ansatz for the evolution of the quantum state of the system (4) which implies that for a given quasimomentum k, the quantum evolution is restricted to the Nambu subspace {|1 −k , 1 k , |0 −k , 0 k }, consisting of doubly occupied |1 −k , 1 k and unoccupied |0 −k , 0 k states of ±k fermions [27,28,40]. The matrix representation of the operatorĤ k (t) in the Nambu subspace is given by…”

Topological Invariants for Spin-Orbit Coupled Superconductor Nanowires

“…Note, however, that d = 1 Hamiltonians in class BDI are supposed to be TR-invariant while the spinless p x + ip y superconductor explicitly breaks TR symmetry in d = 2. The key to this difference is that, in d = 1, the Hamiltonian can be made completely real [27] while it is necessarily complex in d = 2. Redefining the time-reversal operator only in terms of the complex conjugation operator K, it follows that in d = 2 this symmetry is broken (class D) but it remains intact in d = 1 (class BDI).…”

“…It is also possible to define a Z 2 invariant which only counts the parity of the number of boundary Majorana modes [24]. Dimensional reduction arguments suggest that the number of possible end MFs in a d = 1 spinless superconductor should also be an integer and this has recently been shown explicitly [27]. Therefore, the Hamiltonian should be in the topological class BDI with a Z invariant in d = 1.…”