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arXiv:1210.6922 · Physical Review A

Quantum Ising dynamics and Majorana-like edge modes in the Rabi lattice model

Brijesh Kumar, Somenath Jalal

Published Jul 8, 2013• 12 citations
arXivHTMLPDF

Abstract

The atomic dipoles in the Rabi lattice model exhibit quantum Ising dynamics in the limit of strong atom-photon interaction. It governs the paraelectric to ferroelectric phase transition in the ground state. On an open chain, it implies the existence of two Majorana-like, albeit topologically unprotected, edge modes in the ordered phase. The relation ${\ensuremath{\rho}}_{1L}^{x}={p}^{8}$ between the end-to-end dipole correlation ${\ensuremath{\rho}}_{1L}^{x}$ and the spontaneous polarization $p$ is proposed as an observable signature of these edge modes. The density-matrix renormalization-group calculations on the one-dimensional Rabi lattice support the strong-coupling quantum Ising behavior and correctly yield the proposed end-to-end dipole correlation. The conditions that protect the edge modes against the adverse perturbations are also identified.

DOI
10.1103/physreva.88.011802

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The atomic dipoles in the Rabi lattice model exhibit quantum Ising dynamics in the limit of strong atom-photon interaction. It governs the paraelectric to ferroelectric phase transition in the ground state. On an open chain, it implies the existence of two Majorana-like, albeit topologically unprotected, edge modes in the ordered phase. The relation \ensuremathρ1Lx=p8{\ensuremath{\rho}}_{1L}^{x}={p}^{8}\ensuremathρ1Lx​=p8 between the end-to-end dipole correlation \ensuremathρ1Lx{\ensuremath{\rho}}_{1L}^{x}\ensuremathρ1Lx​ and the spontaneous polarization ppp is proposed as an observable signature of these edge modes. The density-matrix renormalization-group calculations on the one-dimensional Rabi lattice support the strong-coupling quantum Ising behavior and correctly yield the proposed end-to-end dipole correlation. The conditions that protect the edge modes against the adverse perturbations are also identified.
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The Hubbard and spinless fermion chains are paradigms of strongly correlated systems, very well understood using the Bethe ansatz, density matrix renormalization group (DMRG), and field theory/renormalization group (RG) methods. They have been applied to one-dimensional materials and have provided important insights for understanding higher-dimensional cases. Recently, an interacting fermion model has been introduced, with possible applications to topological materials. It has a single Majorana fermion operator on each lattice site and interactions with the shortest possible range that involve four sites. We present a thorough analysis of the phase diagram of this model in one dimension using field-theory/RG and DMRG methods. It includes a gapped supersymmetric region and a gapless phase with coexisting Luttinger liquid and Ising degrees of freedom. In addition to a first-order transition, three critical points occur: tricritical Ising, Lifshitz, and a generalization of the commensurate-incommensurate transition. We also survey various gapped phases of the system that arise when the translation symmetry is broken by dimerization and find both trivial and topological phases with 0, 1, and 2 Majorana zero modes bound to the edges of the chain with open boundary conditions.
We study the robustness of the Floquet quantum Ising model against integrability-breaking perturbations, focusing on the phase hosting both Majorana zero and πππ modes. A recent work [Phys. Rev. B 110, 075117, (2024)] observed that the Floquet product mode, a composite edge mode constructed from both Majorana operators, is considerably more robust than the individual Majorana edge modes. We analyze these strong modes from the point of view of the eigenphase order present in finite chains with open boundary conditions. As a result of the Majorana modes, all Floquet eigenstates come in quadruplets in the integrable limit. We show that the robustness of the various modes as well as the behavior of the boundary spin correlation functions can be understood in terms of the spectral statistics of these quadruplets in the presence of integrability-breaking perturbations.