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arXiv:1801.01363 · Physical review. B./Physical review. B

Dynamical bulk-edge correspondence for degeneracy lines in parameter space

R. Wang, C. Li, X. Z. Zhang, Z. Song

Published Jul 11, 2018• 10 citations
arXivHTMLPDF

Abstract

A degeneracy line in parameter space, at which the energy gap closes up, can be the boundary separating either two topological quantum phases or two conventional phases, depending on the surface containing the line. We study the topological feature of degeneracy line in parameter space via a dimerized Kitaev spin chain with staggered transverse field, which can be mapped onto the system of dimerized spinless fermions with $p$-wave superconductivity. The quantum phase boundaries are straight crossing degeneracy lines in three-dimensional (3D) parameter space. We show that the degeneracy line acts as a vortex filament associated with a vector field, which is generated from the Zak phase of the Bogoliubov--de Gennes band. We also investigate the topological invariant in Majorana fermion representation for open chains. The edge modes are not the zero mode but are still protected by the energy gap. The exact midgap states of the Majorana lattice allow us to obtain the corresponding Majorana probability distribution for the whole 3D parameter space, which can establish a scalar field to identify the topological feature of the degeneracy lines. Furthermore, when we switch on a weak tunneling between two ends of the Majorana lattice, the topological invariants of the degeneracy lines can be obtained by the pumped Majorana probability from an adiabatic passage encircling the lines. Numerical simulations of quasiadiabatic time evolution are performed in a small system. We compute the current to monitor the transport of the Majorana fermion, which exhibits an evident character of topological pumping. Our results link the bulk topological invariant to the dynamical behavior of the edge mode, extending the concept of bulk-edge correspondence.

DOI
10.1103/physrevb.98.014303

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A degeneracy line in parameter space, at which the energy gap closes up, can be the boundary separating either two topological quantum phases or two conventional phases, depending on the surface containing the line. We study the topological feature of degeneracy line in parameter space via a dimerized Kitaev spin chain with staggered transverse field, which can be mapped onto the system of dimerized spinless fermions with ppp-wave superconductivity. The quantum phase boundaries are straight crossing degeneracy lines in three-dimensional (3D) parameter space. We show that the degeneracy line acts as a vortex filament associated with a vector field, which is generated from the Zak phase of the Bogoliubov--de Gennes band. We also investigate the topological invariant in Majorana fermion representation for open chains. The edge modes are not the zero mode but are still protected by the energy gap. The exact midgap states of the Majorana lattice allow us to obtain the corresponding Majorana probability distribution for the whole 3D parameter space, which can establish a scalar field to identify the topological feature of the degeneracy lines. Furthermore, when we switch on a weak tunneling between two ends of the Majorana lattice, the topological invariants of the degeneracy lines can be obtained by the pumped Majorana probability from an adiabatic passage encircling the lines. Numerical simulations of quasiadiabatic time evolution are performed in a small system. We compute the current to monitor the transport of the Majorana fermion, which exhibits an evident character of topological pumping. Our results link the bulk topological invariant to the dynamical behavior of the edge mode, extending the concept of bulk-edge correspondence.
In this article, we address the possibility of topological quantum spin liquids with only short-range (spin) correlations in the whole phase diagram in a ladder system, through an exactly solvable model which represents a generalization of the two-dimensional Kitaev honeycomb spin model to a ribbon geometry. This system gives rise to Z2\mathbb{Z}_2Z2​ gauge fields emerging in a static way in the ground state through Majorana fermions which then produce either 000 or πππ flux. Through a quantum field theory perspective, mathematical and Density Matrix Renormalization Group results, we present our progress on the geometrical, topological and physical classification of this model through observables navigating on the whole phase diagram presenting three spin-liquid phases with a spin gap and a metallic phase in the sense of Majorana fermions. We propose a bulk-edge correspondence revealing the presence of protected zero-energy Majorana fermions at the edge in the phases with a gap and revealing also the presence of itinerant Majorana fermions in the bulk in the topological liquid phase. The proposed approach allows us to distinguish between properties of the brickwall ladder giving rise to a special line and those of the square ladder allowing for the gapless Majorana liquid phase. This model gives some hope on finding solutions for topological quantum spin models, e.g. with πππ flux, on a square lattice geometry. This approach also allows for a correspondence onto models of topological superconductivity.
We investigate the spectral and topological properties of a zigzag Kitaev chain constructed from two diagonally coupled one dimensional Kitaev chains with a zero and finite superconducting pairing phase difference. Using a Bogoliubov-de Gennes formulation, we analyze the energy spectrum, distribution of Majorana zero modes (MZMs), the quasi-particle dispersion, and the winding number, respectively. For a zero phase difference, the resulting energy spectrum shows topological phases with two, four MZMs, and trivial regions. The phases of gap closure determine the topological phase boundaries. In particular, for the phase difference between φ=πφ=πφ=π, the degeneracy of MZMs is partially lifted, leading to modified topological phases compared to the case φ=0φ=0φ=0. The topological and trivial phase boundaries are further confirmed by evaluating the quasi-particle dispersion and the topological invariant, namely the winding number. We show that the zigzag Kitaev chain contributes independently to the total winding number ν=1ν= 1ν=1 and 222, giving rise to distinct topological phases that support two and four MZMs. The ν=0ν= 0ν=0 gives rise to a trivial region. The energy spectrum of systems corroborates the analytical phase boundaries and reveals characteristics associated with hybridization, enabling us to obtain the complete phase diagram of the zigzag model. Our results establish the zigzag Kitaev chain as a minimal platform for engineering MZM quantum computations, with potential applications in the study of topological phases and Majorana based qubit physics.
The influence of boundary conditions on bulk properties is generally expected to vanish in the thermodynamic limit. Here we show that this expectation can fail in an open transverse field Ising chain with modified boundary fields. By removing only the transverse fields acting on the two edge spins while leaving the bulk Hamiltonian unchanged, we uncover a boundary-induced switching of the topological spectral degeneracy. In contrast to the conventional open chain, where the topological degeneracy occurs in the ferromagnetic phase (∣g∣<1\left\vert g\right\vert <1∣g∣<1), the modified boundary condition shifts it to the paramagnetic phase (∣g∣>1\left\vert g\right\vert >1∣g∣>1). We demonstrate that the underlying mechanism originates from two exact local conserved quantities associated with the edge spins, which partition the Hilbert space into invariant sectors. Based on this structure, we establish an exact description in terms of domain-wall excitations, under which the Hamiltonian decomposes into two independent Kitaev chains. Complementarily, the Majorana representation reveals that the spectral switching originates from different unit-cell conventions and boundary terminations of the corresponding effective SSH lattice, leading to distinct boundary zero-mode structures without altering the bulk Hamiltonian. Furthermore, we show that the boundary-induced topological degeneracy remains observable at finite temperatures through non-Hermitian quench dynamics, providing a dynamical manifestation of the bulk-boundary correspondence. These results demonstrate that boundary engineering, through the introduction of local conserved boundary degrees of freedom, provides an effective route for controlling topological degeneracy in quantum spin chains without modifying bulk interactions.
Majorana zero modes are well studied in the gapped phases of topological systems. We investigate Majorana zero modes at the topological quantum criticality in one dimensional topological superconducting model with longer range interaction. We identify stable localized Majorana zero modes appearing at criticality under certain conditions. Topological invariant number for these non-trivial criticalities is obtained from zeros of a complex function associated with the Hamiltonian. Behavior of parametric curve at criticalities validate the invariant obtained and account for the appearance of Majorana zero modes at criticality. Trivial and non-trivial topological nature of criticality due to the presence of multicritical point cause an unusual topological transition along the critical line. We observe and investigate this unique transition in terms of eigenvalue spectrum. Appearance of MZMs at criticality demands integer value of topological invariant number in order to validate the concept of bulk-boundary correspondence. Hence we propose a scheme to separate the invariant number into fractional and integer contribution to establish bulk-boundary correspondence at criticality.
We investigate Majorana modes in a quantum spin chain with bond-dependent exchange interactions by studying its dynamics. Specifically, we consider two-time correlations for the anisotropic Kitaev-Heisenberg (KH) Hamiltonian close to the so-called Kitaev critical point. Here the model coincides with a phase boundary of two uncoupled instances of Kitaev's model for ppp-wave superconductors, together supporting a degenerate ground state characterized by multiple Majorana modes. In this regime, the real-time dynamics of local spins reveal a set of strong zero modes, corresponding to a set of protruding frequencies in the two-time correlation function. We derive perturbative interactions that map the KH spin chain onto the topological regime of Kitaev's fermionic model, thus opening up a bulk gap while retaining almost degenerate modes in the mesoscopic regime, i.e., for finite system sizes. This showcases the emergence of Majorana modes in a chain of effective dimers. Here, the binding energy within each unit cell competes with the interdimer coupling to generate a finite-size energy gap, in analogy with local energy terms in the transverse-field Ising model. These modes give rise to long coherence times of local spins located at the system edges. By breaking the local symmetry in each dimer, one can also observe a second class of Majorana modes in terms of a beating frequency in the two-time correlations function of the edge spin. Furthermore, we develop a scenario for realizing these model predictions in ion-trap quantum simulators with collective addressing of the ions.
The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and Z2 topological invariant of the bulk spectrum. This model can be obtained from a transverse field Ising model(TFIM) using the Jordan–Wigner transformation. TFIM has neither topological degeneracy nor any edge modes. Topological degeneracy associated with topological order is central to topological quantum computation. In this paper, we explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit Z2 topological order. We show that there are at least two different ways to understand this topological protection of Majorana fermion qubits: one way is based on fermionic mode operators and the other is based on anti-commuting symmetry operators. We also show how these two different ways are related to each other. We provide a very general approach to understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians. We then show how in topological phases in Majorana fermion models gives rise to new braid group representations. So, we give a unifying and broad perspective of topological phases in Majorana fermion models based on anti-commuting symmetry operators and braid group representations of Majorana fermions as anyons.