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arXiv:2309.13116 · Physical review. B./Physical review. B

Jordan-Wigner composite fermion liquids in a two-dimensional quantum spin ice

Leonardo Goller, Inti Sodemann

Published Jan 26, 2024• 1 citation
arXivHTMLPDF

Abstract

The Jordan-Wigner map in two dimensions is an exact lattice regularization of the $2\ensuremath{\pi}$-flux attachment to a hard-core boson (or spin $\frac{1}{2}$) leading to a composite fermion particle. When the spin-$\frac{1}{2}$ model obeys ice rules this map preserves locality, namely, local Rohkshar-Kivelson (RK) models of spins are mapped onto local models of Jordan-Wigner composite fermions. Using this composite fermion dual representation of RK models we construct spin-liquid states by projecting Slater determinants onto the subspaces of the ice rules. Interestingly, we find that these composite fermions behave as ``dipolar'' partons for which the projective implementations of symmetries are very different from standard ``pointlike'' partons. We construct interesting examples of these composite Fermi-liquid states that respect all microscopic symmetries of the RK model. In the six-vertex subspace, we constructed a time-reversal and particle-hole-invariant state featuring two massless Dirac nodes, which is a composite fermion counterpart to the classic $\ensuremath{\pi}$-flux state of Abrikosov fermions in the square lattice. This state is a good ground-state candidate for a modified RK--type Hamiltonian of quantum spin ice. In the dimer subspace, we constructed a state featuring a composite Fermi surface but with nesting instabilities towards ordered phases such as the columnar state. We have also analyzed the low-energy emergent gauge structure. If one ignores confinement, the system would feature a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ low-energy gauge structure with two associated gapless photon modes, but with the composite fermion carrying gauge charge only for one photon and behaving as a gauge-neutral dipole under the other. These states are also examples of pseudoscalar $\text{U}(1)$ spin liquids where mirror and time-reversal symmetries act as composite fermion particle-hole conjugations, and the emergent magnetic fields are even under such time-reversal or lattice mirror symmetries.

DOI
10.1103/physrevb.109.035162

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The Jordan-Wigner map in two dimensions is an exact lattice regularization of the 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ-flux attachment to a hard-core boson (or spin 12\frac{1}{2}21​) leading to a composite fermion particle. When the spin-12\frac{1}{2}21​ model obeys ice rules this map preserves locality, namely, local Rohkshar-Kivelson (RK) models of spins are mapped onto local models of Jordan-Wigner composite fermions. Using this composite fermion dual representation of RK models we construct spin-liquid states by projecting Slater determinants onto the subspaces of the ice rules. Interestingly, we find that these composite fermions behave as dipolar'' partons for which the projective implementations of symmetries are very different from standard pointlike'' partons. We construct interesting examples of these composite Fermi-liquid states that respect all microscopic symmetries of the RK model. In the six-vertex subspace, we constructed a time-reversal and particle-hole-invariant state featuring two massless Dirac nodes, which is a composite fermion counterpart to the classic \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux state of Abrikosov fermions in the square lattice. This state is a good ground-state candidate for a modified RK--type Hamiltonian of quantum spin ice. In the dimer subspace, we constructed a state featuring a composite Fermi surface but with nesting instabilities towards ordered phases such as the columnar state. We have also analyzed the low-energy emergent gauge structure. If one ignores confinement, the system would feature a U(1)\ifmmode×\else\texttimes\fiU(1)\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)U(1)\ifmmode×\else\texttimes\fiU(1) low-energy gauge structure with two associated gapless photon modes, but with the composite fermion carrying gauge charge only for one photon and behaving as a gauge-neutral dipole under the other. These states are also examples of pseudoscalar U(1)\text{U}(1)U(1) spin liquids where mirror and time-reversal symmetries act as composite fermion particle-hole conjugations, and the emergent magnetic fields are even under such time-reversal or lattice mirror symmetries.
A variety of analytical approaches have been developed for the study of quantum spin systems in two dimensions, the notable ones being spin-waves, slave boson/fermion parton constructions, and for lattices with one-to-one local correspondence of faces and vertices, the two-dimensional (2D) Jordan-Wigner (JW) fermionization. Field-theoretically, JW fermionization is implemented through Chern-Simons (CS) flux attachment. For a correct fermionization of lattice quantum spin-1/21/21/2 magnets, it is necessary that the fermions obey mutual bosonic (anyonic) statistics under exchange---this is not possible to implement on arbitrary 2D lattices if fermionic matter couples only to the lattice gauge fields. Enlarging the gauge degrees of freedom to include the dual lattice allows the construction of consistent mutual Chern-Simons field theories. Here we propose a mutual CS theory where the microscopic (spin) degrees of freedom are represented as lattice fermionic matter additionally coupled to specific combinations of dual lattice gauge fields that depend on the local geometry. We illustrate the use of this method for understanding the properties of a honeycomb Kitaev model subjected to a strong Zeeman field in the zzz-direction. Our CS gauge theory framework provides an understanding of why the topological phase is degraded at lower (higher) critical fields for the ferromagnetic (antiferromagnetic) Kitaev interaction. Additionally, we observe an effectively one-dimensional character of the low excitations at higher fields in the zzz-direction, which we also confirm by spin-wave calculations.
We study possible spin liquids on square lattice that respect all lattice symmetries and time-reversal symmetry within the framework of Schwinger boson (mean-field) theory. Such spin liquids have spin gap and emergent Z2{Z}_{2}Z2​ gauge field excitations. We classify them by the projective symmetry group method, and find six spin-liquid states that are potentially relevant to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. The properties of these states are studied under mean-field approximation. Interestingly we find a spin-liquid state that can go through continuous phase transitions to either the N\'eel magnetic order or magnetic orders of the wave vector at the Brillouin zone edge center. We also discuss the connection between our results and the Abrikosov fermion spin liquids.
Recent developments have established exact bosonization and fermionization with a Z2\mathbb{Z}_2Z2​ symmetry as dualities through gauging. In this work, we apply fermionic gauging, which realizes generalized Jordan-Wigner transformations, to the Bose-Hubbard (BH) model with a global U(1)U(1)U(1) symmetry and derive an exact dual description in terms of fermionic composites, built from bosons and fermions. In 1D, this duality generalizes the exact mapping between the extended hard-core BH model and the spinless Fermi-Hubbard model to include soft-core bosons. At low energies, the mapping reduces to the well-known equivalence between the sine-Gordon model and the Thirring model. The oscillation wave vector of the fermionic composite correlation function in the gapless phase is fixed by their density, providing a novel manifestation of Luttinger's theorem. We verify the exact duality using density matrix renormalization group (DMRG) calculations and demonstrate that the gapless phase and the phase transition are governed by the compact boson conformal field theory. Our construction naturally extends to generic bosonic systems and higher dimensions, opening new avenues for studying Bose-Fermi mixtures in optical lattices and other strongly correlated quantum systems.
Projective symmetry group classification of fermionic Z2Z_2Z2​ spin liquids on the dipolar-octupolar pyrochlore magnets
We construct and classify Z2Z_2Z2​ quantum spin liquids (QSLs) of fermionic partons on the dipolar-octupolar (DO) pyrochlore lattice. The DO pseudospin shares the space-group and time-reversal algebra of the effective spin-1/2 Kramers doublet, its projective symmetry group (PSG) yields the same 48 symmetric Z2Z_2Z2​ classes as the generic fermionic classification. We show, the distinct DO symmetry action, encoded in a set of sublattice-independent operators, selects a physically different set of symmetry-allowed mean-field channels, so that the resulting ansatze, spinon spectra, and observables differ from both the effective spin-1/2 case and the bosonic DO classification. Decomposing the anisotropic XYZ exchange model into all symmetric singlet and triplet hopping and pairing channels and minimizing the mean-field energy self-consistently, we obtain phase diagrams in the (Jx~,Jz~\tilde{J_x}, \tilde{J_z}Jx​~​,Jz​~​) plane in which only a few Z2Z_2Z2​ states, several supporting gapless or flat-band spinon spectra, are stabilized. For these states we compute the spinon band structures, the low-temperature specific heat, and the dynamical spin structure factor, and we argue that the power-law specific-heat exponent is a diagnostic of the spinon nodal structure. A near-linear-in-temperature specific heat together with broad continua in the structure factor gives a fermionic-parton account of the gapless, fractionalized thermodynamics reported in DO materials such as Nd2ScNbO7Nd_2ScNbO_7Nd2​ScNbO7​, complementing the bosonic quantum-spin-ice picture of the Ce-based compounds. The work extends the fermionic-parton classification of DO pyrochlore QSLs beyond the U(1)U(1)U(1) sector, where Z2Z_2Z2​ topological order is realized equally naturally by fermionic and bosonic spinons.
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I construct a Jordan-Wigner transformation for spin-one-half quantum systems on two-dimensional lattices. I show that the spin-one-half XY (i.e., a hard-core Bose system) is equivalent (on any two-dimensional Bravais lattice) to a system of spinless fermions and gauge fields satisfying the constraint that the gauge flux on a plaquette must be proportional to the spin (particle) density on site. The constraint is enforced by the addition of a Chern-Simons term of strength \ensuremath{\theta} to the Lagrangian of the theory. For the particular value \ensuremath{\theta}=1/2\ensuremath{\pi}, the resulting particles are fermions. In general they are anyons. The implications of these results for quantum spin liquids are briefly discussed.