TY - JOUR

T1 - Majorana zero modes in a quantum Ising chain with longer-ranged interactions

AU - Niu, Yuezhen

AU - Chung, Suk Bum

AU - Hsu, Chen Hsuan

AU - Mandal, Ipsita

AU - Raghu, S.

AU - Chakravarty, Sudip

PY - 2012/1/13

Y1 - 2012/1/13

N2 - 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, we find that these phases can be classified according to the number of Majorana zero modes of an open chain: zero, one, or two at each end. The model possesses a multicritical point where phases with zero, one, and two 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 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 decay exponentially in space but in an oscillatory manner. The wavelength of oscillation is identical to that in the asymptotic connected spin-spin correlation of the XY model in a transverse field, to which our model is dual.

AB - 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, we find that these phases can be classified according to the number of Majorana zero modes of an open chain: zero, one, or two at each end. The model possesses a multicritical point where phases with zero, one, and two 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 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 decay exponentially in space but in an oscillatory manner. The wavelength of oscillation is identical to that in the asymptotic connected spin-spin correlation of the XY model in a transverse field, to which our model is dual.

UR - http://www.scopus.com/inward/record.url?scp=84856482693&partnerID=8YFLogxK

U2 - 10.1103/PhysRevB.85.035110

DO - 10.1103/PhysRevB.85.035110

M3 - Article

AN - SCOPUS:84856482693

SN - 1098-0121

VL - 85

JO - Physical Review B - Condensed Matter and Materials Physics

JF - Physical Review B - Condensed Matter and Materials Physics

IS - 3

M1 - 035110

ER -