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.

9 pages, 13 figures

References in corpus (6)

• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Geometric phases and criticality in spin chain systems
• Mixed-state fidelity and quantum criticality at finite temperature
• Topological Characterization of Extended Quantum Ising Models
• Characterization of topological phase transitions via topological properties of transition points
• Quantum phase transition induced by real-space topology

Cited by in corpus (15)

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• Quantum Phase Transition in a Quantum Ising Chain at Nonzero Temperatures
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• 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