Majorana charges, winding numbers and Chern numbers in quantum Ising models

Abstract

Mapping a many-body state on a loop in parameter space is a simple way to characterize a quantum state. The connections of such a geometrical representation to the concepts of Chern number and Majorana zero mode are investigated based on a generalized quantum spin system with short and long-range interactions. We show that the topological invariants, the Chern numbers of corresponding Bloch band is equivalent to the winding number in the auxiliary plane, which can be utilized to characterize the phase diagram. We introduce the concept of Majorana charge, the magnitude of which is defined by the distribution of Majorana fermion probability in zero-mode states, and the sign is defined by the type of Majorana fermion. By direct calculations of the Majorana modes we analytically and numerically verify that the Majorana charge is equal to Chern numbers and winding numbers.

arXiv comment9 pages, 13 figures

References in corpus (21)

• Topological Insulators
• Topological insulators and superconductors
• Berry Phase Effects on Electronic Properties
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Unpaired Majorana fermions in quantum wires
• Helical liquids and Majorana bound states in quantum wires
• Non-abelian statistics of half-quantum vortices in p-wave superconductors
• Scaling of Entanglement close to a Quantum Phase Transitions
• Majorana fermions in a tunable semiconductor device
• Decay of Loschmidt Echo Enhanced by Quantum Criticality
• $2n$ Quasihole States Realize $2^{n-1}$-Dimensional Spinor Braiding Statistics in Paired Quantum Hall States
• Edge states and topological phases in one-dimensional optical superlattices
• Scaling of geometric phases close to quantum phase transition in the XY chain
• Geometric phases and criticality in spin chain systems
• Mixed-state fidelity and quantum criticality at finite temperature
• Majorana zero modes in a quantum Ising chain with longer-ranged interactions
• Topological phases, Majorana modes and quench dynamics in a spin ladder system
• Topological Characterization of Extended Quantum Ising Models
• Characterization of topological phase transitions via topological properties of transition points
• Topological Majorana and Dirac zero modes in superconducting vortex cores
• Quantum phase transition induced by real-space topology

Cited by in corpus (15)

• Characterization of topological states via dual multipartite entanglement
• Quantum Phase Transition in a Quantum Ising Chain at Nonzero Temperatures
• Quantum phase transition in a non-Hermitian XY spin chain with global complex transverse field
• Ising chain with topological degeneracy induced by dissipation
• One-way deficit and quantum phase transitions in $XY$ model and extended Ising model
• Probing topological states through the exact non-Markovian decoherence dynamics of a spin coupled to a spin bath in real-time domain
• Dynamical bulk-edge correspondence for nodal lines in parameter space
• Coalescing Majorana edge modes in non-Hermitian PT-symmetric Kitaev chain
• Decoherence and relaxation of topological states in extended quantum Ising models
• Dynamic crystallization in a quantum Ising chain
• Maximal distant entanglement in Kitaev tube
• Emerging topological characterization in non-equilibrium states of quenched Kitaev chains
• Quantum Phase Transitions in Long-Range Interacting Hyperuniform Spin Chains in a Transverse Field
• $\eta$-pairing states in the Hubbard model with non-uniform Hubbard interaction
• Topological phases in interacting spin-1 systems