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arXiv:2408.02046 · Physical review. B./Physical review. B

Chiral spin liquid in a generalized Kitaev honeycomb model with Z 4 1-form symmetry

Yu-Xin Yang, Meng Cheng, Ji-Yao Chen

Published Jan 14, 2025• 1 citation
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Abstract

We explore a large $N$ generalization of the Kitaev model on the honeycomb lattice with a simple nearest-neighbor interacting Hamiltonian. In particular, we focus on the ${\mathbb{Z}}_{4}$ case with isotropic couplings, which is characterized by an exact ${\mathbb{Z}}_{4}$ 1-form symmetry. Guided by symmetry considerations and an analytical study in the single-chain limit, on infinitely long cylinders, we find the model is gapped with an extremely short correlation length. Combined with the ${\mathbb{Z}}_{4}$ 1-form symmetry, this suggests the model is topologically ordered. To pin down the nature of this phase, we further study the model on both finite and infinitely long strips, where we consistently find a $c=1$ conformal field theory (CFT) description, suggesting the existence of chiral edge modes described by a free boson CFT. Further evidence is found by studying the dimer correlators on infinitely long strips. We find the dimer correlation functions show a power-law decay with the exponent close to 2 on the boundary of the strip, while decaying much faster in the bulk. Combined with the topological entanglement entropy extracted from cylinder geometry, we identify the phase as a chiral spin liquid with a $\mathrm{U}{(1)}_{\ensuremath{-}8}$ chiral topological order. A unified perspective for all ${\mathbb{Z}}_{N}$ type Kitaev models is also discussed.

DOI
10.1103/physrevb.111.035129

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We study a Z3{\mathbb{Z}}_{3}Z3​ Kitaev model on the honeycomb lattice with nearest neighbor interactions. Based on matrix product state simulations and symmetry considerations, we find evidence that, with ferromagnetic isotropic couplings, the model realizes a chiral spin liquid, characterized by a possible U(1)12\mathrm{U}{(1)}_{12}U(1)12​ chiral topological order. This is supported by simulations on both cylinder and strip geometries. On infinitely long cylinders with various widths, scaling analysis of entanglement entropy and maximal correlation length suggests that the model has a gapped two-dimensional bulk. The topological entanglement entropy is extracted and found to be in agreement with the U(1)12\mathrm{U}{(1)}_{12}U(1)12​ topological order. On infinitely long strips with moderate widths, we find the model is critical with a central charge consistent with the chiral edge theory of the U(1)12\mathrm{U}{(1)}_{12}U(1)12​ topological phase. We conclude by discussing several open questions.
A spin fractionalizes into matter and gauge fermions in the Kitaev spin liquid on the honeycomb lattice, which is topologically ordered for certain parameters. To what extent does only one constituent of a fractionalized topological system signal the existence of topological order? To answer this, the authors focus on the reduced density matrix of the gauge sector on a cylinder after tracing out the matter degrees of freedom. The ensuing gauge entanglement Hamiltonian contains infinitely long-range correlations along the symmetry axis of the cylinder. Rather small correctly chosen gauge partitions can still account for the topological entanglement entropy in spite of long-range correlations in the gauge entanglement Hamiltonian.
The possible existence of a quantum spin liquid (QSL) phase, an exotic state of matter with long-range quantum entanglement and fractionalized excitations, in ααα-RuCl3_33​ has sparked widespread interests in exploring QSLs in various Kitaev models under magnetic fields. Recently, a KKK-JJJ-ΓΓΓ-Γ′Γ'Γ′ model has been proposed to accurately describe the compound by fitting the measured thermodynamic data, where JJJ is Heisenberg interaction, and ΓΓΓ, Γ′Γ'Γ′ are off-diagonal exchanges on top of the dominant Kitaev coupling KKK. Based on this effective model, an intermediate QSL phase in presence of an out-of-plane fields along the [111][1 1 1][111] direction, between the low-field zigzag order and high-field polarized phase, has been predicted. By combining density matrix renormalization group (DMRG), exponential tensor renormalization group (XTRG), and variational Monte Carlo (VMC) calculations, we address the nature of this QSL phase in the honeycomb KKK-JJJ-ΓΓΓ-Γ′Γ'Γ′ model under the [111][1 1 1][111]-direction field. Our DMRG calculations find the algebraic-like decay of spin correlation function, the finite spin scalar chiral order, and lattice nematic order. Together with the XTRG results of power-law specific heat at low temperature, our findings naturally suggest a gapless nematic chiral spin liquid. On the other hand, our VMC study finds a gapped nematic chiral spin liquid with the variational energy very close to that obtained in DMRG. As VMC finds a very small gap that is beyond the current resolution of both ground-state DMRG and finite-temperature XTRG calculations on finite-width cylinders, we resort the full clarification for the nature of the intermediate QSL to future studies. Lastly, we discuss the implications of our results to the recent experiment on ααα-RuCl3_33​ and the QSL-like phase in a similar KKK-ΓΓΓ-Γ′Γ'Γ′ model.
In the physics of the fractional quantum Hall (FQH) effect, a zoo of Abelian topological phases can be obtained by varying the magnetic field. Aiming to reach the same phenomenology in spin like systems, we propose a family of SU(N)\mathrm{SU}(N)SU(N)-symmetric models in the fundamental representation, on the square lattice with short-range interactions restricted to triangular units, a natural generalization for arbitrary NNN of an SU(3) model studied previously where time-reversal symmetry is broken explicitly. Guided by the recent discovery of SU(2)1{\mathrm{SU}(2)}_{1}SU(2)1​ and SU(3)1{\mathrm{SU}(3)}_{1}SU(3)1​ chiral spin liquids (CSL) on similar models we search for topological SU(N)1\mathrm{SU}({N)}_{1}SU(N)1​ CSL in some range of the Hamiltonian parameters via a combination of complementary numerical methods such as exact diagonalizations (ED), infinite density matrix renormalization group (iDMRG) and infinite Projected Entangled Pair State (iPEPS). Extensive ED on small (periodic and open) clusters up to N=10N=10N=10 and an innovative SU(N)\mathrm{SU}(N)SU(N)-symmetric version of iDMRG to compute entanglement spectra on (infinitely long) cylinders in all topological sectors provide unambiguous signatures of the SU(N)1\mathrm{SU}({N)}_{1}SU(N)1​ character of the chiral liquids. An SU(4)-symmetric chiral PEPS, constructed in a manner similar to its SU(2) and SU(3) analogs, is shown to give a good variational ansatz of the N=4N=4N=4 ground state, with chiral edge modes originating from the PEPS holographic bulk-edge correspondence. Finally, we discuss the possible observation of such Abelian CSL in ultracold atom setups where the possibility of varying NNN provides a tuning parameter similar to the magnetic field in the physics of the FQH effect.
The frustrated XY model on the honeycomb lattice has drawn lots of attention because of the potential emergence of chiral spin liquid (CSL) with the increasing of frustrations or competing interactions. In this paper, we study the extended spin-12\frac{1}{2}21​ XY model with nearest-neighbor (J1)({J}_{1})(J1​), and next-nearest-neighbor (J2)({J}_{2})(J2​) interactions in the presence of a three-spins chiral $({J}_{\ensuremath{\chi}})$ term using density matrix renormalization group methods. We obtain a quantum phase diagram with both conventionally ordered and topologically ordered phases. In particular, the long-sought Kalmeyer-Laughlin CSL is shown to emerge under a small ${J}_{\ensuremath{\chi}}$ perturbation due to the interplay of the magnetic frustration and chiral interactions. The CSL, which is a nonmagnetic phase, is identified by the scalar chiral order, the finite spin gap on a torus, and the chiral entanglement spectrum described by chiral SU(2)1SU{(2)}_{1}SU(2)1​ conformal field theory.
The Kitaev honeycomb model has attracted significant interest due to its quantum spin liquid ground state, fractionalized Majorana excitations, and topological properties. Motivated by these features, we introduce a quasi-one-dimensional Kitaev-like spin chain derived from a truncated honeycomb geometry. The resulting Majorana band structure contains both dispersive and flat bands, with a gap closing at a critical anisotropy that separates trivial and chiral topological phases. Under open boundary conditions, the topological regime hosts zero-energy edge modes protected by chiral symmetry, placing the system in the BDI symmetry class with a quantized winding number. In the excited spectrum, localized plaquette modes emerge near zero energy and can be tuned by introducing domains of negative plaquettes. The dynamical spin correlation function reveals broad continua associated with fractionalized excitations, together with characteristic low-energy spectral weight from edge Majorana modes. These features distinguish the present system from conventional Heisenberg spin chains and provide experimentally relevant fingerprints of chiral topological order. Our model thus establishes a conceptual bridge between the two-dimensional Kitaev spin liquid and Kitaev's one-dimensional quantum wire.