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arXiv:2403.14284 · npj Quantum Materials

Gauge field dynamics in multilayer Kitaev spin liquids

Aprem P. Joy, Achim Rosch

Published Aug 18, 2024• 5 citations
arXivHTMLPDF

Abstract

Abstract The Kitaev spin liquid realizes an emergent static $${{\mathbb{Z}}}_{2}$$ 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.

DOI
10.1038/s41535-024-00673-z

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Emergent gauge theories take a prominent role in the description of quantum matter, supporting deconfined phases with topological order and fractionalized excitations. A common construction of \mathbb{Z}_2 ℤ 2 lattice gauge theories, first introduced by Wegner, involves Ising gauge spins placed on links and subject to a discrete \mathbb{Z}_2 ℤ 2 Gauss law constraint. As shown by Kitaev, \mathbb{Z}_2 ℤ 2 lattice gauge theories also emerge in the exact solution of certain spin systems with bond-dependent interactions. In this context, the \mathbb{Z}_2 ℤ 2 gauge field is constructed from Majorana fermions, with gauge constraints given by the parity of Majorana fermions on each site. In this work, we provide an explicit Jordan-Wigner transformation that maps between these two formulations on the square lattice, where the Kitaev-type gauge theory emerges as the exact solution of a spin-orbital (Kugel-Khomskii) Hamiltonian. We then apply our mapping to study local perturbations to the spin-orbital Hamiltonian, which correspond to anisotropic interactions between electric-field variables in the \mathbb{Z}_2 ℤ 2 gauge theory. These are shown to induce anisotropic confinement that is characterized by emergence of weakly-coupled one-dimensional spin chains. We study the nature of these phases and corresponding confinement transitions in both absence and presence of itinerant fermionic matter degrees of freedom. Finally, we discuss how our mapping can be applied to the Kitaev spin-1/2 model on the honeycomb lattice.

arXiv:2306.09405•SciPost Physics•Jun 5, 2024•4 citations
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 vison is an excitation of the Kitaev spin liquid which carries a Z 2 gauge flux.While immobile in the pure Kitaev model, it becomes a dynamical degree of freedom in the presence of perturbations.We study an isolated vison in the isotropic Kitaev model perturbed by a small external magnetic field h, an off-diagonal exchange interaction Γ, and a Heisenberg coupling J.In the ferromagnetic Kitaev model, the dressed vison obtains a dispersion linear in Γ and h and a fully universal low-T mobility, μ ¼ 6ℏv 2 m =ðk B TÞ 2 , where v m is the velocity of Majorana fermions.In contrast, in the antiferromagnetic (AFM) Kitaev model interference effects suppress coherent propagation and an incoherent Majorana-assisted hopping leads to a T-independent mobility.The motion of a single vison due to Heisenberg interactions is strongly suppressed for both signs of the Kitaev coupling.Vison bands in AFM Kitaev models can be topological and may lead to characteristic features in the thermal Hall effects in Kitaev materials.Furthermore, we argue that vison diffusion leads to universal signatures in quench experiments.
Kitaev's honeycomb-lattice spin-1/21/21/2 model has become a paradigmatic example for Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquids, both gapped and gapless. Here we study the fate of these spin-liquid phases in differently stacked bilayer versions of the Kitaev model. Increasing the ratio between the interlayer Heisenberg coupling ${J}_{\ensuremath{\perp}}$ and the intralayer Kitaev couplings Kx,y,z{K}^{x,y,z}Kx,y,z destroys the topological spin liquid in favor of a paramagnetic dimer phase. We study phase diagrams as a function of ${J}_{\ensuremath{\perp}}/K$ and Kitaev coupling anisotropies using Majorana-fermion mean-field theory, and we employ different expansion techniques in the limits of small and large ${J}_{\ensuremath{\perp}}/K$. For strongly anisotropic Kitaev couplings, we derive effective models for the different layer stackings that we use to discuss the quantum phase transition out of the Kitaev phase. We find that the phase diagrams depend sensitively on the nature of the stacking and anisotropy strength. While in some stackings and at strong anisotropies we find a single transition between the Kitaev and dimer phases, other stackings are more involved. Most importantly, we prove the existence of two novel macrospin phases, which can be understood in terms of Ising chains that can be either coupled ferromagnetically or remain degenerate, thus realizing a classical spin liquid. In addition, our results suggest the existence of a flux phase with spontaneous interlayer coherence. We discuss prospects for experimental realizations.
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.
Understanding the nonequilibrium relaxation dynamics of quantum spin liquids is an important and relatively unexplored frontier, where fractionalized quasiparticles and emergent gauge fields can give rise to dynamical behaviors far beyond those of conventional magnets. Here we study the relaxation dynamics of the Kitaev honeycomb model following a thermal quench, focusing on visons—gapped Z2{{\mathbb{Z}}}_{2}Z2​ flux excitations of the emergent gauge field in the Kitaev spin liquid. The long-time evolution is governed by the diffusion and pair annihilation of visons, with both the activation barrier for vison motion and the effective vison-vison interactions mediated by fractionalized Majorana fermions. The resulting open-system quantum dynamics, induced by coupling to a thermal bath, can be efficiently simulated using an effective kinetic Monte Carlo approach. We uncover a rich set of temperature-dependent dynamical regimes. With decreasing temperature, the relaxation crosses over from diffusion-limited annihilation to terminal-velocity–limited kinetics, and ultimately to pronounced dynamical arrest and freezing. We show that this freezing behavior is closely tied to the formation of long-lived, metastable 3×3\sqrt{3}\times \sqrt{3}3​×3​ vison crystals, which strongly suppress further annihilation. Remarkably, even in the absence of true long-range crystalline order, the system exhibits a hidden coarsening dynamics of vison superclusters associated with an emergent Z3{{\mathbb{Z}}}_{3}Z3​ symmetry.
In the field of frustrated magnetism, Kitaev models provide prototypical examples of exactly solvable quantum spin liquids, in which the spin degrees of freedom fractionalize into Majorana fermions coupled to an emergent Z2 gauge field. While the ground states of these models are already well understood as exhibiting different kinds of Majorana (semi)metals, it has still been an open task to achieve a detailed understanding of the thermodynamics, and, namely, the low-temperature ordering of the gauge field into different Z2 flux configurations. In this thesis, we investigate the thermodynamics of Kitaev systems in two and three spatial dimensions with sign-problem-free quantum Monte Carlo simulations. In a first study of elementary 3D Kitaev models, we verify that the ground state Z2 flux sectors of these systems are entirely determined by their elementary plaquette length – a result which shows the validity of Lieb’s theorem for lattice geometries which lack the geometric requirements for its proof. We closely investigate the low-temperature phase transition associated with gauge-ordering, which is a particular realization of an inverted Ising phase transition, as it occurs in general lattice Z2 gauge theories. Our results corroborate the understanding of this transition as separating different regimes in terms of vison-loop excitations, by showing a clear correlation between the critical temperature and the vison gap. We also introduce the concept of “gauge frustration”, which, for a particular 3D Kitaev model, leads to the suppression of the phase transition and a more complex physical behavior at low temperatures. A second quantum Monte Carlo study focuses on a generalized 2D Kitaev model on a five-coordinated lattice system. Here, the gauge ordering is accompanied by the spontaneous breaking of time-reversal symmetry, a scenario which not only allows for a variety of topological ground states, but also for the occurrence of a phase transition in two spatial dimensions. We show that this model exhibits such a transition at a particularly high temperature scale. Moreover, the system possesses a number of phases where the Z2 fluxes are only partially ordered.
Emergent gauge theories take a prominent role in the description of quantum matter, supporting deconfined phases with topological order and fractionalized excitations. A common construction of \mathbb{Z}_2 ℤ 2 lattice gauge theories, first introduced by Wegner, involves Ising gauge spins placed on links and subject to a discrete \mathbb{Z}_2 ℤ 2 Gauss law constraint. As shown by Kitaev, \mathbb{Z}_2 ℤ 2 lattice gauge theories also emerge in the exact solution of certain spin systems with bond-dependent interactions. In this context, the \mathbb{Z}_2 ℤ 2 gauge field is constructed from Majorana fermions, with gauge constraints given by the parity of Majorana fermions on each site. In this work, we provide an explicit Jordan-Wigner transformation that maps between these two formulations on the square lattice, where the Kitaev-type gauge theory emerges as the exact solution of a spin-orbital (Kugel-Khomskii) Hamiltonian. We then apply our mapping to study local perturbations to the spin-orbital Hamiltonian, which correspond to anisotropic interactions between electric-field variables in the \mathbb{Z}_2 ℤ 2 gauge theory. These are shown to induce anisotropic confinement that is characterized by emergence of weakly-coupled one-dimensional spin chains. We study the nature of these phases and corresponding confinement transitions in both absence and presence of itinerant fermionic matter degrees of freedom. Finally, we discuss how our mapping can be applied to the Kitaev spin-1/2 model on the honeycomb lattice.