Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2306.08624 · Physical review. B./Physical review. B

Multipolar spin liquid in an exactly solvable model for jeff = 32 moments

Vanuildo S. de Carvalho, Hermann Freire, Rodrigo G. Pereira

Published Sep 12, 2023• 4 citations
arXivHTMLPDF

Abstract

We study an exactly solvable model with bond-directional quadrupolar and octupolar interactions between spin-orbital entangled ${j}_{\mathrm{eff}}=\frac{3}{2}$ moments on the honeycomb lattice. We show that this model features a multipolar spin liquid phase with gapless fermionic excitations. In the presence of perturbations that break time-reversal and rotation symmetries, we find Abelian and non-Abelian topological phases in which the Chern number evaluates to 0, $\ifmmode\pm\else\textpm\fi{}1$, and $\ifmmode\pm\else\textpm\fi{}2$. We also investigate quantum phase transitions out of the multipolar spin liquid using a parton mean-field approach and orbital wave theory. In the regime of strong integrability-breaking interactions, the multipolar spin liquid gives way to ferroquadrupolar-vortex and antiferro-octupolar ordered phases that harbor a hidden spin-$\frac{1}{2}$ Kitaev spin liquid. Our work unveils mechanisms for unusual multipolar orders and quantum spin liquids in Mott insulators with strong spin-orbit coupling.

DOI
10.1103/physrevb.108.094418

Related papers

Quadrupolar spin liquid, octupolar Kondo coupling, and odd-frequency superconductivity in an exactly solvable model

Carlene Silva de Farias, Vanuildo S. de Carvalho, E. Miranda +1 more

We propose an exactly solvable model for ${j}_{\text{eff}}=\frac{3}{2}$ local moments on the honeycomb lattice. Our construction is guided by a symmetry analysis and by the requirement of an exact solution in terms of a Majorana fermion representation for multipole operators. The main interaction in the model can be interpreted as a bond-dependent quadrupole-quadrupole interaction. When time reversal symmetry is explicitly broken, we obtain a gapped spin liquid with a single chiral Majorana edge mode. We also investigate another solvable model in which the time-reversal-invariant spin liquid is coupled to conduction electrons in a superconductor. In the presence of a Kondo-like coupling that involves the octupole moment of the localized spins, the itinerant electrons hybridize with the emergent Majorana fermions in the spin liquid. This leads to spontaneous time reversal symmetry breaking and generates odd-frequency pairing. Our results suggest that ${j}_{\text{eff}}=\frac{3}{2}$ systems with strong quadrupole-quadrupole interactions may provide a route towards non-Abelian quantum spin liquids and unconventional superconductivity.

arXiv:2003.13790•Physical review. B./Physical review. B•Aug 7, 2020•15 citations

Microscopic roadmap to a Kitaev-Yao-Lee spin-orbital liquid

Derek Churchill, Emily Z. Zhang, Hae‐Young Kee

The exactly solvable spin-1/2 Kitaev model on a honeycomb lattice has drawn significant interest, as it offers a pathway to realizing the long-sought-after quantum spin liquid. Building upon the Kitaev model, Yao and Lee introduced another exactly solvable model on an unusual star lattice featuring non-abelian spinons. The additional pseudospin degrees of freedom in this model could provide greater stability against perturbations, making this model appealing. However, a mechanism to realize such an interaction in a standard honeycomb lattice remains unknown. Here, we provide a microscopic theory to obtain the Yao-Lee model, on a honeycomb lattice by utilizing strong spin-orbit coupling of anions edge-shared between two e g ions in the exchange processes. This mechanism leads to the desired bond-dependent interaction among spins rather than orbitals, unique to our model, implying that the orbitals fractionalize into gapless Majorana fermions and fermionic octupolar excitations emerge. Since the conventional Kugel-Khomskii interaction also appears, we examine the phase diagram, including these interactions, using classical Monte Carlo simulations and exact diagonalization techniques. Our findings reveal a broad region of disordered states that break rotational symmetry in the bond energy, suggesting intriguing behavior reminiscent of a spin-orbital liquid.

arXiv:2410.21389•npj Quantum Materials•Mar 1, 2025•12 citations

Field-induced Kitaev multipolar liquid in spin-orbit coupled d2 honeycomb Mott insulators

Ahmed Rayyan, Derek Churchill, Hae‐Young Kee

The Kitaev model, characterized by bond-dependent Ising spin interactions among spin-orbit entangled dipole moments in Mott insulators, offered a new approach to quantum spin liquids. Motivated by another type of bond-dependent interaction among quadrupole moments in $5{d}^{2}$ Mott insulators, we provide a microscopic route to uncover the Kitaev multipolar liquid, featuring fractionalized excitations out of non-Kramers doublets carrying multipole moments. The key ingredient is the magnetic field that allows for bond-anisotropic quadrupole-octupole interactions via mixing with the excited triplet states. The conditions to realize signatures of this phase in real materials are also discussed.

arXiv:2206.10647•Physical review. B./Physical review. B•Jan 23, 2023•13 citations

Emergence of a field-driven U(1) spin liquid in the Kitaev honeycomb model

Ciarán Hickey, Simon Trebst

In the field of quantum magnetism, the exactly solvable Kitaev honeycomb model serves as a paradigm for the fractionalization of spin degrees of freedom and the formation of [Formula: see text] quantum spin liquids. An intense experimental search has led to the discovery of a number of spin-orbit entangled Mott insulators that realize its characteristic bond-directional interactions and, in the presence of magnetic fields, exhibit no indications of long-range order. Here, we map out the complete phase diagram of the Kitaev model in tilted magnetic fields and report the emergence of a distinct gapless quantum spin liquid at intermediate field strengths. Analyzing a number of static, dynamical, and finite temperature quantities using numerical exact diagonalization techniques, we find strong evidence that this phase exhibits gapless fermions coupled to a massless U(1) gauge field. We discuss its stability in the presence of perturbations that naturally arise in spin-orbit entangled candidate materials.

arXiv:1805.05953•Nature Communications•Jan 31, 2019•176 citations

Exact Chiral Spin Liquid with Non-Abelian Anyons

Hong Yao, Steven Allan Kivelson

We establish the existence of a chiral spin liquid (CSL) as the exact ground state of the Kitaev model on a decorated honeycomb lattice, which is obtained by replacing each site in the familiar honeycomb lattice with a triangle. This CSL state spontaneously breaks time reversal symmetry but preserves other symmetries. There are two topologically distinct CSL's separated by a quantum critical point. Interestingly, vortex excitations in the topologically nontrivial (Chern number +/-1) CSL obey non-Abelian statistics.

arXiv:0708.0040•Physical Review Letters•Dec 12, 2007•362 citations

Theory of Dirac Spin-Orbital Liquids: monopoles, anomalies, and applications to $SU(4)$ honeycomb models

Vladimir Calvera, Wang, Chong

Dirac spin liquids represent a class of highly-entangled quantum phases in two dimensional Mott insulators, featuring exotic properties such as critical correlation functions and absence of well-defined low energy quasi-particles. Existing numerical works suggest that the spin-orbital $SU(4)$ symmetric Kugel-Khomskii model of Mott insulators on the honeycomb lattice realizes a Dirac spin-orbital liquid, described at low energy by $(2+1)d$ quantum electrodynamics (QED$_3$) with $N_f=8$ Dirac fermions. We generalize methods previously developed for $SU(2)$ spin systems to analyze the symmetry properties and stability of the Dirac spin-orbital liquid. We conclude that the standard Dirac state in the $SU(4)$ honeycomb system, based on a simple parton mean-field ansatz, is intrinsically unstable at low energy due to the existence of a monopole perturbation that is allowed by physical symmetries and relevant under renormalization group flow. We propose two plausible alternative scenarios compatible with existing numerics. In the first scenario, we propose an alternative $U(1)$ Dirac spin-orbital liquid, which is similar to the standard one except for its monopole symmetry quantum numbers. This alternative $U(1)$ state represents a stable gapless phase. In the second scenario, we start from the standard $U(1)$ Dirac liquid and Higgs the $U(1)$ gauge symmetry down to $\mathbb{Z}_4$. The resulting $\mathbb{Z}_4$ Dirac spin-orbital liquid is stable. We also discuss the continuous quantum phase transitions from the $\mathbb{Z}_4$ Dirac liquids to conventional symmetry-breaking orders, described by the QED$_3$ theory with $N_f=8$ supplemented with a critical charge-$4$ Higgs field. We discuss possible ways to distinguish these scenarios in numerics. We also extend previous calculations of the quantum anomalies of QED$_3$ and match with generalized lattice Lieb-Schultz-Mattis constraints.

arXiv:2103.13405•Mar 24, 2021•8 citations
We study an exactly solvable model with bond-directional quadrupolar and octupolar interactions between spin-orbital entangled jeff=32{j}_{\mathrm{eff}}=\frac{3}{2}jeff​=23​ moments on the honeycomb lattice. We show that this model features a multipolar spin liquid phase with gapless fermionic excitations. In the presence of perturbations that break time-reversal and rotation symmetries, we find Abelian and non-Abelian topological phases in which the Chern number evaluates to 0, \ifmmode±\else\textpm\fi1\ifmmode\pm\else\textpm\fi{}1\ifmmode±\else\textpm\fi1, and \ifmmode±\else\textpm\fi2\ifmmode\pm\else\textpm\fi{}2\ifmmode±\else\textpm\fi2. We also investigate quantum phase transitions out of the multipolar spin liquid using a parton mean-field approach and orbital wave theory. In the regime of strong integrability-breaking interactions, the multipolar spin liquid gives way to ferroquadrupolar-vortex and antiferro-octupolar ordered phases that harbor a hidden spin-12\frac{1}{2}21​ Kitaev spin liquid. Our work unveils mechanisms for unusual multipolar orders and quantum spin liquids in Mott insulators with strong spin-orbit coupling.
We propose an exactly solvable model for jeff=32{j}_{\text{eff}}=\frac{3}{2}jeff​=23​ local moments on the honeycomb lattice. Our construction is guided by a symmetry analysis and by the requirement of an exact solution in terms of a Majorana fermion representation for multipole operators. The main interaction in the model can be interpreted as a bond-dependent quadrupole-quadrupole interaction. When time reversal symmetry is explicitly broken, we obtain a gapped spin liquid with a single chiral Majorana edge mode. We also investigate another solvable model in which the time-reversal-invariant spin liquid is coupled to conduction electrons in a superconductor. In the presence of a Kondo-like coupling that involves the octupole moment of the localized spins, the itinerant electrons hybridize with the emergent Majorana fermions in the spin liquid. This leads to spontaneous time reversal symmetry breaking and generates odd-frequency pairing. Our results suggest that jeff=32{j}_{\text{eff}}=\frac{3}{2}jeff​=23​ systems with strong quadrupole-quadrupole interactions may provide a route towards non-Abelian quantum spin liquids and unconventional superconductivity.
The exactly solvable spin-1/2 Kitaev model on a honeycomb lattice has drawn significant interest, as it offers a pathway to realizing the long-sought-after quantum spin liquid. Building upon the Kitaev model, Yao and Lee introduced another exactly solvable model on an unusual star lattice featuring non-abelian spinons. The additional pseudospin degrees of freedom in this model could provide greater stability against perturbations, making this model appealing. However, a mechanism to realize such an interaction in a standard honeycomb lattice remains unknown. Here, we provide a microscopic theory to obtain the Yao-Lee model, on a honeycomb lattice by utilizing strong spin-orbit coupling of anions edge-shared between two e g ions in the exchange processes. This mechanism leads to the desired bond-dependent interaction among spins rather than orbitals, unique to our model, implying that the orbitals fractionalize into gapless Majorana fermions and fermionic octupolar excitations emerge. Since the conventional Kugel-Khomskii interaction also appears, we examine the phase diagram, including these interactions, using classical Monte Carlo simulations and exact diagonalization techniques. Our findings reveal a broad region of disordered states that break rotational symmetry in the bond energy, suggesting intriguing behavior reminiscent of a spin-orbital liquid.
The Kitaev model, characterized by bond-dependent Ising spin interactions among spin-orbit entangled dipole moments in Mott insulators, offered a new approach to quantum spin liquids. Motivated by another type of bond-dependent interaction among quadrupole moments in 5d25{d}^{2}5d2 Mott insulators, we provide a microscopic route to uncover the Kitaev multipolar liquid, featuring fractionalized excitations out of non-Kramers doublets carrying multipole moments. The key ingredient is the magnetic field that allows for bond-anisotropic quadrupole-octupole interactions via mixing with the excited triplet states. The conditions to realize signatures of this phase in real materials are also discussed.
In the field of quantum magnetism, the exactly solvable Kitaev honeycomb model serves as a paradigm for the fractionalization of spin degrees of freedom and the formation of [Formula: see text] quantum spin liquids. An intense experimental search has led to the discovery of a number of spin-orbit entangled Mott insulators that realize its characteristic bond-directional interactions and, in the presence of magnetic fields, exhibit no indications of long-range order. Here, we map out the complete phase diagram of the Kitaev model in tilted magnetic fields and report the emergence of a distinct gapless quantum spin liquid at intermediate field strengths. Analyzing a number of static, dynamical, and finite temperature quantities using numerical exact diagonalization techniques, we find strong evidence that this phase exhibits gapless fermions coupled to a massless U(1) gauge field. We discuss its stability in the presence of perturbations that naturally arise in spin-orbit entangled candidate materials.
We establish the existence of a chiral spin liquid (CSL) as the exact ground state of the Kitaev model on a decorated honeycomb lattice, which is obtained by replacing each site in the familiar honeycomb lattice with a triangle. This CSL state spontaneously breaks time reversal symmetry but preserves other symmetries. There are two topologically distinct CSL's separated by a quantum critical point. Interestingly, vortex excitations in the topologically nontrivial (Chern number +/-1) CSL obey non-Abelian statistics.
Theory of Dirac Spin-Orbital Liquids: monopoles, anomalies, and applications to SU(4)SU(4)SU(4) honeycomb models
Dirac spin liquids represent a class of highly-entangled quantum phases in two dimensional Mott insulators, featuring exotic properties such as critical correlation functions and absence of well-defined low energy quasi-particles. Existing numerical works suggest that the spin-orbital SU(4)SU(4)SU(4) symmetric Kugel-Khomskii model of Mott insulators on the honeycomb lattice realizes a Dirac spin-orbital liquid, described at low energy by (2+1)d(2+1)d(2+1)d quantum electrodynamics (QED3_33​) with Nf=8N_f=8Nf​=8 Dirac fermions. We generalize methods previously developed for SU(2)SU(2)SU(2) spin systems to analyze the symmetry properties and stability of the Dirac spin-orbital liquid. We conclude that the standard Dirac state in the SU(4)SU(4)SU(4) honeycomb system, based on a simple parton mean-field ansatz, is intrinsically unstable at low energy due to the existence of a monopole perturbation that is allowed by physical symmetries and relevant under renormalization group flow. We propose two plausible alternative scenarios compatible with existing numerics. In the first scenario, we propose an alternative U(1)U(1)U(1) Dirac spin-orbital liquid, which is similar to the standard one except for its monopole symmetry quantum numbers. This alternative U(1)U(1)U(1) state represents a stable gapless phase. In the second scenario, we start from the standard U(1)U(1)U(1) Dirac liquid and Higgs the U(1)U(1)U(1) gauge symmetry down to Z4\mathbb{Z}_4Z4​. The resulting Z4\mathbb{Z}_4Z4​ Dirac spin-orbital liquid is stable. We also discuss the continuous quantum phase transitions from the Z4\mathbb{Z}_4Z4​ Dirac liquids to conventional symmetry-breaking orders, described by the QED3_33​ theory with Nf=8N_f=8Nf​=8 supplemented with a critical charge-444 Higgs field. We discuss possible ways to distinguish these scenarios in numerics. We also extend previous calculations of the quantum anomalies of QED3_33​ and match with generalized lattice Lieb-Schultz-Mattis constraints.