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Journal of Experimental and Theoretical Physics

Role of Vacancies in the Yao–Lee Model

V. A. Polyakov, Natalia Borisovna Perkins

Published Oct 1, 2023• 2 citations
Publisher

Abstract

Abstract We consider the effect of vacancies on the low-energy excitation spectrum of a quantum spin liquid realized in the exactly solvable Yao–Lee model [H. Yao and D.-H. Lee, Phys. Rev. Lett. 107, 087205 (2011)]. Physically, vacancies can appear for different reasons (e.g., because of zero magnetic moments on the lattice, or the presence of nonmagnetic impurities, or a random reduction of local bonds of magnetic moments with the remaining lattice). It is shown numerically that the finite density of random vacancies in this model leads to the accumulation of states near zero energy, which can be detected from the change of the behavior of heat capacity at low temperatures. Moreover, it is shown that the low-energy modes are localized more strongly than remaining eigenmodes. This effect is illustrated using the inverse participation ratio (IPR). In the case of time reversal symmetry breaking (e.g., due to the presence of a magnetic field), a gap is opened in the fermion spectrum of the model, and vacancy-induced localized states appear. The energies of these states depend on the structure of the interactions responsible for the time inversion symmetry breaking.

DOI
10.1134/s1063776123100084

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Abstract We consider the effect of vacancies on the low-energy excitation spectrum of a quantum spin liquid realized in the exactly solvable Yao–Lee model [H. Yao and D.-H. Lee, Phys. Rev. Lett. 107, 087205 (2011)]. Physically, vacancies can appear for different reasons (e.g., because of zero magnetic moments on the lattice, or the presence of nonmagnetic impurities, or a random reduction of local bonds of magnetic moments with the remaining lattice). It is shown numerically that the finite density of random vacancies in this model leads to the accumulation of states near zero energy, which can be detected from the change of the behavior of heat capacity at low temperatures. Moreover, it is shown that the low-energy modes are localized more strongly than remaining eigenmodes. This effect is illustrated using the inverse participation ratio (IPR). In the case of time reversal symmetry breaking (e.g., due to the presence of a magnetic field), a gap is opened in the fermion spectrum of the model, and vacancy-induced localized states appear. The energies of these states depend on the structure of the interactions responsible for the time inversion symmetry breaking.
We consider the effect of vacancies on the low-energy excitation spectrum of a quantum spin liquid realized in the exactly solvable Yao–Lee model [H. Yao and D.-H. Lee, Phys. Rev. Lett. 107, 087205 (2011)]. Physically, vacancies can appear for different reasons (e.g., because of zero magnetic moments on the lattice, or the presence of nonmagnetic impurities, or a random reduction of local bonds of magnetic moments with the remaining lattice). It is shown numerically that the finite density of random vacancies in this model leads to the accumulation of states near zero energy, which can be detected from the change of the behavior of heat capacity at low temperatures. Moreover, it is shown that the low-energy modes are localized more strongly than remaining eigenmodes. This effect is illustrated using the inverse participation ratio (IPR). In the case of time reversal symmetry breaking (e.g., due to the presence of a magnetic field), a gap is opened in the fermion spectrum of the model, and vacancy-induced localized states appear. The energies of these states depend on the structure of the interactions responsible for the time inversion symmetry breaking.
The Yao-Lee model is an example of exactly solvable spin-orbital models that are generalizations of the original Kitaev honeycomb model with extra local orbital degrees of freedom. Similarly to the Kitaev model, both spin and orbital degrees of freedom are effectively represented using sets of three-flavored Majorana fermions. The Yao-Lee model exhibits a quantum spin liquid ground state with gapped and immobile Z2{Z}_{2}Z2​ fluxes and threefold degenerate itinerant Majorana fermions. Our work demonstrated that by introducing different time-reversal-symmetry- (TRS) breaking fields one can split the degeneracy of Majorana fermions and close the gap for some of the bands, thus changing its topology. We calculated a comprehensive topological phase diagram for the Yao-Lee model by considering various combinations of TRS-breaking fields. This investigation revealed the emergence of distinct topological regions, each separated by nodal lines, signifying an evolution in the model's topological properties. We also investigated the impact of vacancies in the system. Our findings revealed that while vacancies modify the low-energy spectrum of the model, their presence has a limited impact on the topological properties of the model, at least for small-enough concentrations.
Vacancy-induced local moments in quantum paramagnetic phases: An SU(NNN) designer Hamiltonian study
We explore the effects of non-magnetic impurities (vacancy disorder) on the quantum paramagnetic phases stabilized by SU(NNN) designer Hamiltonians on bipartite lattices. Using the results of our quantum Monte Carlo simulations, we demonstrate that isolated vacancies seed emergent spin S=1/2S=1/2S=1/2 moments in their vicinity when the low-temperature state has valence bond solid order. Indeed, our quantum Monte Carlo results for the low-temperature susceptibility in such regimes shows clear evidence of the vacancy-induced Curie tails associated with these emergent moments, and our zero-temperature projector Monte Carlo results on the ground-state wavefunction in the valence bond basis provide additional evidence in support of this picture. Further, for such designer Hamiltonians on the Lieb lattice with two additional sites on each bond of a square lattice, we identify a low-temperature spin liquid-like regime with no sign of spin or valence bond order. This liquid-like regime serves as a test bed for validating a recently-developed argument concerning the effects of vacancy disorder in such low temperature regimes. Consistent with this argument, we find that isolated vacancies do not seed emergent local moments in such spin liquids. Instead, in the presence of vacancy disorder, emergent local moments are associated with the presence of monomers in maximum-density dimer packings of the corresponding diluted lattice.
We study the effect of adding disorder to the Kitaev model on the hyperhoneycomb lattice, which hosts both gapped and gapless spin liquid phases with an emergent Z2{\mathbb{Z}}_{2}Z2​ gauge field. The latter has an unusual gapless spectrum of Majorana fermion excitations, with a co-dimension-two Fermi ring. We thus address the question of the interplay of topological physics and disorder by considering the properties of isolated single and pairs of vacancies. We show that near the vacancies, the local magnetic response to a field hz{h}_{z}hz​ is parametrically enhanced in comparison to the pristine bulk. Unlike the previously studied case of the 2D honeycomb Kitaev model, the vacancies do not bind a flux of the Z2{\mathbb{Z}}_{2}Z2​ gauge field. In the gapped phase, an isolated vacancy gives rise to effectively free spin-half moments with a nonuniversal coupling to an external field. In the gapless phase, the low-field magnetization is suppressed parametrically to ${(\ensuremath{-}ln{h}_{z})}^{\ensuremath{-}1/2}$ because of interactions with the surrounding spin liquid. We also show that a pair of vacancies is subject to a sublattice-dependent interaction on account of coupling through the bulk spin liquid, which is spatially anisotropic even when all Kitaev couplings have equal strength. This coupling is thus exponentially suppressed with distance in the gapped phase. In the gapless phase, two vacancies on the same (opposite) sublattice exhibit an enhanced (suppressed) low-field response, amounting to an effectively (anti-)ferromagnetic interaction.
In this work we consider several exactly solvable models of magnetic impurities in critical quantum antiferromagnetic spin chains and multichannel Kondo impurities. Their ground-state properties are studied, and the finite set of nonlinear integral equations which exactly describe the thermodynamics of the models is constructed. We obtain several analytical low-energy expressions for the temperature, magnetic field, and frequency dependences of important characteristics of exactly solvable disordered quantum spin models and disordered multichannel Kondo impurities with essential many-body interactions. We show that the only low-energy parameter that gets renormalized is the velocity of the low-lying excitations (or the effective crossover scale connected with each impurity); the others appear to be universal. In our study several kinds of strong disorder important for experiments were used. Some of them produce low divergences in certain characteristics of our strongly disordered critical systems (compared with finite values for the homogeneous case or a single impurity). For weak disorder, or for narrow distributions of the local Kondo temperatures, our exact results reveal the presence of Kondo screening of disordered ensembles of magnetic impurities by low-lying excitations of the host. We point out that our results qualitatively coincide with the data of experiments on real disordered quasi-one-dimensional antiferromagnetic systems and with the similar behavior of some heavy metallic alloys.
We study the ground-state properties of the Kitaev-Heisenberg model in a magnetic field and explore the evolution of spin correlations in the presence of nonmagnetic vacancies. By means of exact diagonalizations, the phase diagram without vacancies is determined as a function of the magnetic field and the ratio between Kitaev and Heisenberg interactions. We show that, in the (antiferromagnetic) stripe-ordered phase, the static susceptibility and its anisotropy can be described by a spin canting mechanism. This accounts as well for the transition to the polarized phase when including quantum fluctuations perturbatively. Effects of spin vacancies depend sensitively on the type of the ground state. In the liquid phase, the magnetization pattern around a single vacancy in a small field is determined, and its spatial anisotropy is related to that of nonzero further neighbor correlations induced by the field and/or Heisenberg interactions. In the stripe phase, the joint effect of a vacancy and a small field breaks the sixfold symmetry of the model and stabilizes a particular stripe pattern. Similar symmetry-breaking effects occur even at zero field due to effective interactions between vacancies. This selection mechanism and intrinsic randomness of vacancy positions may lead to spin-glass behavior.