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.
We determine the phase diagrams of anisotropic Kitaev-Heisenberg models on the honeycomb lattice using parton mean-field theories based on different Majorana fermion representations of the S=1/2 spin operators. First, we use a two-dimensional Jordan-Wigner transformation (JWT) involving a semi-infinite snake string operator. To ensure that the fermionized Hamiltonian remains local, we consider the limit of extreme Ising exchange anisotropy in the Heisenberg sector. Second, we use the conventional Kitaev representation in terms of four Majorana fermions subject to local constraints, which we enforce through Lagrange multipliers. For both representations, we self-consistently decouple the interaction terms in the bond and magnetization channels and determine the phase diagrams as a function of the anisotropy of the Kitaev couplings and the relative strength of the Ising exchange. While both mean-field theories produce identical phase boundaries for the topological phase transition between the gapless and gapped Kitaev quantum spin liquids, the JWT fails to correctly describe the magnetic instability and finite-temperature behavior. Our results show that the magnetic phase transition is first order at low temperatures but becomes continuous above a certain temperature. At this energy scale, we also observe a finite-temperature crossover on the quantum-spin-liquid side, from a fractionalized paramagnet at low temperatures, in which gapped flux excitations are frozen out, to a conventional paramagnet at high temperatures. Published by the American Physical Society 2024
We explore the properties of the SO(3) Majorana representation of spin. Based on its nonlocal nature, it is shown that there is an equivalence between the SO(3) Majorana representation and the Jordan-Wigner transformation in one and two dimensions. From the relation between the SO(3) Majorana representation and one-dimensional Jordan-Wigner transformation, we show that application of the SO(3) Majorana representation usually results in
Z2 gauge structure. Based on lattice Chern-Simons gauge theory, it is shown that the anticommuting link variables in the SO(3) Majorana representation make it equivalent to an operator form of compact
U(1)1 Chern-Simons Jordan-Wigner transformation in 2D. As examples of its application, we discuss two spin models, namely the quantum XY model on a honeycomb lattice and the ${90}^{\ensuremath{\circ}}$ compass model on a square lattice. It is shown that under the SO(3) Majorana representation both spin models can be exactly mapped into
Z2 gauge theory of spinons, with the standard form of
Z2 Gauss law constraint.
The honeycomb $K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model is known to have both deconfined spin liquid and confined phases. We study here a confinement transition between the Kitaev spin liquid and a dimerized phase in the limit of strong bond anisotropy. By partially projecting out Majorana states we are able to map the model onto a model of weakly coupled Ising chains in a transverse field. Within this mapping the ordered Ising phase corresponds to the condensation of
Z2 fluxes, or confinement. Our results may improve our understanding of the extensively studied spin liquid candidate material $\ensuremath{\alpha}\ensuremath{-}{\mathrm{RuCl}}_{3}$, where $K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ interactions are dominant.
The Kitaev model on a honeycomb lattice with bond-dependent Ising interactions offers an exactly solvable model of a quantum spin liquid (QSL) with fractionalized excitations: gapped
Z2 fluxes and gapless linearly dispersing Majorana fermions in the isotropic limit (
Kx=Ky=Kz). We explore the phase diagram along two axes, an external magnetic field
h, applied out of plane of the honeycomb, and anisotropic interactions
Kz larger than the other two (${K}_{x}={K}_{y}\ensuremath{\equiv}K$). For ${K}_{z}/K\ensuremath{\gg}2$ and
h=0, the matter Majorana fermions have the largest gap, and the system is described by a gapped
Z2 toric code. One of the central questions we address is whether the fractionalized excitations in the different phases have sharp signatures that can be detected in experiments. We show that while the response to single-spin excitations is broad, the spectral function corresponding to two-spin excitations across a bond has sharp signatures that can be attributed to specific anyons. In the toric code regime, the $\ensuremath{\epsilon}=e\ifmmode\times\else\texttimes\fi{}m$ fermion, formed from the bosonic Ising electric (
e) and magnetic (
m) charges, disperses along a specific one-dimensional direction that provides a fingerprint of fractionalization. At lower
Kz in the center of the Abelian phase, in a regime we dub the primordial fractionalized regime, the field generates a hybridization between the $\ensuremath{\epsilon}$ fermion and the Majorana matter fermion, resulting in a $\ensuremath{\psi}$ fermion which too has a distinct quasi-one-dimensional dispersion. All the other phases in the field-anisotropy plane are naturally obtained from this primordial soup. These highly constrained fractonlike dispersions can be observable by inelastic light and neutron scattering, thereby providing ``smoking gun'' signatures of fractionalization in the QSL phase. Our analysis is based on calculations of susceptibilities, topological entanglement entropy, and excitation dynamics, obtained using exact diagonalization and density matrix renormalization group, and supported by perturbation theory.
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 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.
These lecture notes focus on the Kitaev honeycomb model, a two-dimensional quantumsystem with three distinct Ising-type interactions, depending on the direction of eachbond. This model has proven particularly important for describing quantum spin liquids,as well as for applications in topological quantum computing, due to its exotictopological properties. It constitutes the first exactly solvable theoretical model thatfully captures the behavior of a quantum spin liquid. We begin by presenting the geometryof the model, the Hamiltonian that describes the system, and Alexei Kitaev's effort toobtain an exact analytical solution, based on the decomposition of the Hilbert spaceinto flux sectors. We then present Ettore Majorana's observation that the Dirac equationalso admits real solutions, leading to the theoretical prediction of Majorana fermions-- particles that are identical to their antiparticles. We examine how the spatialseparation of Majorana operators leads to the rise of unbound Majorana fermions, andshowcase the exotic topological properties that manifest in physical systems such assuperconducting wires. Next, we represent the Hamiltonian in the basis of Majoranafermions, by expressing spin operators using four Majorana operators. The transitionfrom the extended space
L~ to the physical subspace
Lis achieved through a gauge transformation of the
Z2 group. We thenreformulate the Hamiltonian in the basis of complex bond and matter fermions, transitionto momentum space, and assume that the system lies in the zero-flux sector. Through theBogoliubov--de Gennes transformation, we arrive at the final diagonalized Hamiltonianin the quasiparticle basis. Finally, we represent the possible phases -- the gaplessphase (phase
B) and the gapped phases (phase
Aαi) -- compute thecorresponding energy spectrum, and present the complete phase diagram. By employing thecorrelation function, we show that all phases are characterized by strong quantumentanglement and spin fractionalization, classifying the system as a quantum spin liquid.