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arXiv:1912.01545

Spectroscopy of Majorana modes of non-Abelian vortices in Kitaev's chiral spin liquid

Masafumi Udagawa, Roderich Moessner

Published Dec 3, 2019• 2 citations
arXivHTMLPDF

Abstract

We study the temperature (T) dependence of the dynamical structure factor of the chiral spin liquid phase of Kitaev's honeycomb model. We find, using a recently developed analytical approach, direct signatures of the Majorana modes associated with the non-Abelian Z_2 vortices (visons). In particular, a characteristic multiplet of discrete peaks at low but nonzero-T and its field dependence reflect the existence of thermally activated visons whose varying separation yields different splittings of the Majorana `zero' modes. The resonant processes involved are specific to the Majorana modes, and will persist even in the presence of weak non-integrable interactions.

DOI
10.48550/arxiv.1912.01545

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We study the temperature (T) dependence of the dynamical structure factor of the chiral spin liquid phase of Kitaev's honeycomb model. We find, using a recently developed analytical approach, direct signatures of the Majorana modes associated with the non-Abelian Z_2 vortices (visons). In particular, a characteristic multiplet of discrete peaks at low but nonzero-T and its field dependence reflect the existence of thermally activated visons whose varying separation yields different splittings of the Majorana `zero' modes. The resonant processes involved are specific to the Majorana modes, and will persist even in the presence of weak non-integrable interactions.
Kitaev's model of spins interacting on a honeycomb lattice describes a quantum spin liquid, where an emergent static Z2{\mathbb{Z}}_{2}Z2​ gauge field is coupled to Majorana fermions. In the presence of an external magnetic field and for a range of interaction strengths, the system behaves as a gapped, non-Abelian quantum spin liquid. In this phase, the vortex excitations of the emergent Z2{\mathbb{Z}}_{2}Z2​ gauge field have Majorana zero modes bound to them. Motivated by recent experimental progress in measuring and characterizing real materials that could exhibit spin-liquid behavior, we analytically calculate the dynamical spin structure factor in the non-Abelian phase of the Kitaev's honeycomb model. In particular, we treat the case of quenched disorder in the vortex configurations. Our calculations reveal a peak in the low-energy dynamical structure factor that is a signature of the spin-liquid behavior. We map the effective Hamiltonian to that of a chiral ppp-wave superconductor by using the Jordan-Wigner transformation. Subsequently, we analytically calculate the wave functions of the Majorana zero modes, the energy splitting for finite separation of the vortices and, finally, the dynamical structure factor in presence of quenched disorder.
Temperature evolution of the spin correlation and excitation spectrum is investigated for a Kitaev model defined on a decorated honeycomb lattice by using the quantum Monte Carlo simulation in the Majorana fermion representation. The ground state of this quantum spin model is given by two kinds of chiral spin liquids: one is topologically trivial with Abelian anyon excitations, and the other is topologically nontrivial with non-Abelian anyon excitations. While lowering temperature, the model exhibits several crossovers in the paramagnetic state, which originate from the fractionalization of quantum spins into Majorana fermions, in addition to a phase transition associated with time reversal symmetry breaking. We show that the spin correlation develops around the crossover temperatures, whereas it shows a slight change at the critical temperature, as in other Kitaev-type models. We also calculate the excitation spectrum in terms of Majorana fermions, and find that the excitation gap in the non-Abelian phase is fragile against thermal fluctuations of the Z2Z_2Z2​ fluxes, while that in the Abelian phase remains open.
Temperature evolution of the spin correlation and excitation spectrum is investigated for the Kitaev model defined on a decorated honeycomb lattice by using the quantum Monte Carlo simulation in the Majorana fermion representation. The ground state of this quantum spin model is given by two kinds of chiral spin liquids: one is topologically trivial with Abelian anyon excitations, and the other is topologically nontrivial with non-Abelian anyon excitations. While lowering temperature, the model exhibits several crossovers in the paramagnetic state, which originate from the fractionalization of quantum spins into Majorana fermions, in addition to a phase transition associated with time reversal symmetry breaking. We show that the spin correlation develops around the crossover temperatures, whereas it shows a slight change at the critical temperature, as in other Kitaev-type models. We also calculate the excitation spectrum in terms of Majorana fermions, and find that the excitation gap in the non-Abelian phase is fragile against thermal fluctuations of the Z 2 fluxes, while that in the Abelian phase remains open.
We study a frustrated two-dimensional lattice spin model with Kitaev-type interaction. The lattice is obtained from the honeycomb lattice by replacing half of its sites with triangles. Using the SO(3) Majorana representation of spin, the model is exactly mapped into a Z2{Z}_{2}Z2​ lattice gauge theory of complex fermions with standard Gauss law constraint. We show that the ground state of the model is a chiral spin liquid, and it has gapless excitations. The Chern number of the ground state is $\ifmmode\pm\else\textpm\fi{}1$, implying the existence of a chiral edge mode. The vortex excitation of the model is bound with a Majorana zero mode, such mode behaves as non-Abelian anyons.
We study the effect of site dilution in the Kitaev model. We derive an analytical solution of the dynamical spin correlation functions for arbitrary configurations of Z2{Z}_{2}Z2​ fluxes. By incorporating this solution into a classical Monte Carlo scheme, we address how a site vacancy affects the experimental observables, such as the static spin susceptibility and the spin-lattice relaxation rate, 1/T11/{T}_{1}1/T1​. As a result, we found an enhancement of dynamical magnetic response in the vicinity of vacancy, which leads to Friedel-like oscillation in local 1/T11/{T}_{1}1/T1​, in contrast to limited influences on the static susceptibility. Furthermore, we found a sharp zero-energy peak in the magnetic excitation spectrum, which is attributed to the Vison-Majorana zero mode trapped near the site vacancy. This zero mode can be interpreted as a fractionalized spin hole into an Ising triplet with differentiated magnetic axes, which leads to the characteristic temperature and field-orientational dependence of 1/T11/{T}_{1}1/T1​.
We show that spin-spin correlations in a non-Abelian Kitaev spin liquid are associated with a characteristic inhomogeneous charge density distribution in the vicinity of Z_{2} vortices. This density profile and the corresponding local electric fields are observable, e.g., by means of surface probe techniques. Conversely, by applying bias voltages to several probe tips, one can stabilize Ising anyons (Z_{2} vortices harboring a Majorana zero mode) at designated positions, where we predict a clear Majorana signature in energy absorption spectroscopy.