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arXiv:2604.23014 · Physical review. B./Physical review. B

Visons in Kitaev spin liquids with Majorana Fermi surfaces

Anonymous, Rodrigo G. Pereira

Published Aug 11, 2026• 1 citation
arXivHTMLPDF

Abstract

The excitation spectrum of Kitaev quantum spin liquids consists of itinerant Majorana fermions, which can be gapless or gapped, and vortices of a $\mathbb{Z}_2$ gauge field, known as visons, which are \new{gapped within a stable $\mathbb Z_2$ spin liquid phase}. In this work, we investigate visons in Kitaev-type models where the Majorana fermions form a Fermi surface. In this case, the creation of a vison pair is analogous to introducing a local impurity potential in a metal. Since the gapless modes lead to strong finite-size effects, we compare the numerical calculation of the \new{two-vison gap} on finite lattices with the result from an analytical approach based on Green's function techniques. We find that the \new{two-vison gap} decreases as the size of the Fermi surface increases, signalling an instability of the quantum spin liquid ground state. We also show that larger Fermi surfaces tend to suppress the change in local spin correlations due to the Majorana-vison scattering potential.

DOI
10.1103/tgtq-j6r6

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The excitation spectrum of Kitaev quantum spin liquids consists of itinerant Majorana fermions, which can be gapless or gapped, and vortices of a Z2\mathbb{Z}_2Z2​ gauge field, known as visons, which are \new{gapped within a stable Z2\mathbb Z_2Z2​ spin liquid phase}. In this work, we investigate visons in Kitaev-type models where the Majorana fermions form a Fermi surface. In this case, the creation of a vison pair is analogous to introducing a local impurity potential in a metal. Since the gapless modes lead to strong finite-size effects, we compare the numerical calculation of the \new{two-vison gap} on finite lattices with the result from an analytical approach based on Green's function techniques. We find that the \new{two-vison gap} decreases as the size of the Fermi surface increases, signalling an instability of the quantum spin liquid ground state. We also show that larger Fermi surfaces tend to suppress the change in local spin correlations due to the Majorana-vison scattering potential.
In quantum magnetism, Kitaev's honeycomb model is well known for its gapped and gapless spin liquid ground states in which the elementary spin degrees of freedom fractionalize into Majorana fermions and a Z2{\mathbb{Z}}_{2}Z2​ gauge field. In this paper, the authors discuss the gapless spin liquids arising in three-dimensional Kitaev models and show that the gapless Majorana fermions form metallic states which, depending on the underlying lattice structure, exhibit Fermi surfaces, nodal lines, or Weyl points.
The formation of quantum spin liquids in frustrated magnets represents an exciting possibility due to the rather exotic features they harbor, including fractionalized excitations and emergent gauge fields. \nUnfortunately, they are notoriously difficult to study as there are often no good analytical methods available and quantum Monte Carlo simulations are hindered by the negative sign problem. \nThe Kitaev honeycomb model is a notable exception of a frustrated quantum model which is exactly solvable and which hosts a number of distinct quantum spin liquid ground states. \nAs such, it allows for a rare opportunity to study the physics of spin liquids with full analytical control. \n \nIn this thesis, we study the fractionalization of spin-1/2 moments into Majorana fermions and an emergent Z2 gauge field in a generalization of the Kitaev honeycomb model to a number of three-dimensional lattices. \nWhile the excitations of the gauge field are always gapped, the fermionic quasiparticles may exhibit a gapless dispersion, forming fully two-dimensional Fermi surfaces, symmetry protected nodal lines, or topological Weyl nodes. \nWe show that one can deduce rather general constraints on the possible gapless excitations by making use of an object called the projective symmetry group. \nIn doing so we provide a scheme for classifying the various gapless Kitaev spin liquids. \nA thorough analysis is carried out for a number of these spin liquids, primarily investigating the stability of the gapless modes and the novel features resulting from their sometimes non-trivial topology, as well as their effects on certain equal-time correlation functions.
Although the ground-state energy of the Kitaev spin liquid can be calculated exactly, the associated vison gap energy has to date only been calculated numerically from finite size diagonalization. Here we show that the phase shift for scattering Majorana fermions off a single bond flip can be calculated analytically, leading to a closed-form expression for the vison gap energy $\mathrm{\ensuremath{\Delta}}=0.2633J$. Generalizations of our approach can be applied to Kitaev spin liquids on more complex lattices such as the three-dimensional hyperoctagonal lattice.
Abstract The Kitaev spin liquid realizes an emergent static Z2{{\mathbb{Z}}}_{2}Z2​ Z 2 gauge field with vison excitations coupled to Majorana fermions. We consider Kitaev models stacked on top of each other, weakly coupled by Heisenberg interaction ∝ J⊥. This inter-layer coupling breaks the integrability of the model and makes the gauge fields dynamic. Conservation laws and topology keep single visons immobile. However, an inter-layer vison pairs can hop with a hopping amplitude linear in J⊥ confined to the layer, but their motion is strongly influenced by the type of stacking. For AA stacking, an interlayer pair has a two-dimensional motion but for AB or ABC stacking, sheet conservation laws restrict its motion to a one-dimensional channel within the plane. For all stacking types, an intra-layer vison-pair is constrained to move out-of-plane only. Depending on the anisotropy of the Kitaev couplings Kx, Ky, Kz, the intra-layer vison pairs can display either coherent tunnelling or purely incoherent hopping. When a magnetic field opens a gap for Majorana fermions, there exist two types of intra-layer vison pairs - a bosonic and a fermionic one. Only the bosonic pair obtains a hopping rate linear in J⊥. We use our results to identify the leading instabilities of the spin liquid phase induced by the inter-layer coupling.
A three-dimensional Kitaev model on a hyperhoneycomb lattice is investigated numerically at finite temperature. The Kitaev model is one of the solvable quantum spin models, where the ground state is given by gapped and gapless spin liquids, depending on the anisotropy of the interactions. This model can be rewritten as a free Majorana fermion system coupled with Z 2 variables. The density of states of Majorana fermions shows an excitation gap in the gapped region, while it is semimetallic in the gapless region reflecting the Dirac node. Performing the Monte Carlo simulation, we calculate the temperature dependence of the Majorana spectra. We find that the semimetallic dip is filled as temperature increases in the gapless region, but surprisingly, the spectrum develops an excitation gap in the region near the gapless-gapped boundary. Such changes of the low-energy spectrum appear sharply at the transition temperature from the spin liquid to the paramagnetic state. The results indicate that thermal fluctuations of the Z 2 fields significantly influence the low-energy state of Majorana fermions, especially in the spin liquid formation.
We propose a mechanism to explain the emergence of an intermediate gapless spin liquid phase in the antiferromagnetic Kitaev model in an externally applied magnetic field, sandwiched between the well-known gapped chiral spin liquid and the gapped partially polarized phase. We propose that, in moderate fields, π-fluxes nucleate in the ground state and trap Majorana zero modes. As these fluxes proliferate with increasing field, the Majorana zero modes overlap creating an emergent Z2{{\mathbb{Z}}}_{2}Z2​ quantum Majorana metallic state with a “Fermi surface” at zero energy. We further show that the Majorana spectral function captures the dynamical spin and dimer correlations obtained by the infinite Projected Entangled Pair States method, thereby validating our variational approach. The authors study the formation and characteristics of a Majorana metallic quantum spin liquid state in Kitaev’s honeycomb model under a moderate external magnetic field. This state represents a novel class of gapless quantum spin liquids stabilized by magnetic field.