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:0810.5347 · Physical Review Letters

Algebraic Spin Liquid in an Exactly Solvable Spin Model

Hong Yao, Shoucheng Zhang, Steven Allan Kivelson

Published May 29, 2009• 121 citations
arXivHTMLPDF

Abstract

We have proposed an exactly solvable quantum spin-3/2 model on a square lattice. Its ground state is a quantum spin liquid with a half-integer spin per unit cell. The fermionic excitations, dubbed as "spinons", are gapless with a linear dispersion, while the topological "vison" excitations are gapped. Moreover, these massless fermionic spinon excitations are topologically stable. Thus, this model is, to the best of our knowledge, the first exactly solvable model of half-integer spins whose ground state is an "algebraic spin liquid."

DOI
10.1103/physrevlett.102.217202

Related papers

Symmetry-protected gapless Z2 spin liquids

Yuan-Ming Lu

Despite rapid progress in understanding gapped topological states, much less is known about gapless topological phases of matter, especially in strongly correlated electrons. In this work, we discuss a large class of robust gapless quantum spin liquids in frustrated magnets made of half-integer spins, which are described by gapless fermionic spinons coupled to dynamical ${\mathbb{Z}}_{2}$ gauge fields. Requiring $\text{U}(1)$ spin conservation, time-reversal, and certain space-group symmetries, we show that certain spinon symmetry fractionalization class necessarily leads to a gapless spectrum. These gapless excitations are stable against any perturbations, as long as the required symmetries are preserved. Applying these gapless criteria to spin-$\frac{1}{2}$ systems on square, triangular, and kagome lattices, we show that all gapped symmetric ${\mathbb{Z}}_{2}$ spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped ${\mathbb{Z}}_{2}$ spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.

arXiv:1606.05652•Physical review. B./Physical review. B•Mar 19, 2018•12 citations

Topological Spinon Semimetals and Gapless Boundary States in Three Dimensions

Robert Schaffer, Eric Kin-Ho Lee, Yuan-Ming Lu +1 more

Recently, there has been much effort in understanding topological phases of matter with gapless bulk excitations, which are characterized by topological invariants and protected intrinsic boundary states. Here we show that topological semimetals of Majorana fermions arise in exactly solvable Kitaev spin models on a series of three-dimensional lattices. The ground states of these models are quantum spin liquids with gapless nodal spectra of bulk Majorana fermion excitations. It is shown that these phases are topologically stable as long as certain discrete symmetries are protected. The corresponding topological indices and the gapless boundary states are explicitly computed to support these results. In contrast to previous studies of noninteracting systems, the phases discussed in this work are novel examples of gapless topological phases in interacting spin systems.

arXiv:1409.5125•Physical Review Letters•Mar 17, 2015•51 citations

Quantum orders and symmetric spin liquids

Xiao-Gang Wen

A concept---quantum order---is introduced to describe a new kind of orders that generally appear in quantum states at zero temperature. Quantum orders that characterize the universality classes of quantum states (described by complex ground-state wave functions) are much richer than classical orders that characterize the universality classes of finite-temperature classical states (described by positive probability distribution functions). Landau's theory for orders and phase transitions does not apply to quantum orders since they cannot be described by broken symmetries and the associated order parameters. We introduced a mathematical object---projective symmetry group---to characterize quantum orders. With the help of quantum orders and projective symmetry groups, we construct hundreds of symmetric spin liquids, which have SU(2), U(1), or ${Z}_{2}$ gauge structures at low energies. We found that various spin liquids can be divided into four classes: (a) Rigid spin liquid---spinons (and all other excitations) are fully gapped and may have bosonic, fermionic, or fractional statistics. (b) Fermi spin liquid---spinons are gapless and are described by a Fermi liquid theory. (c) Algebraic spin liquid---spinons are gapless, but they are not described by free fermionic-bosonic quasiparticles. (d) Bose spin liquid---low-lying gapless excitations are described by a free-boson theory. The stability of those spin liquids is discussed in detail. We find that stable two-dimensional spin liquids exist in the first three classes (a)--(c). Those stable spin liquids occupy a finite region in phase space and represent quantum phases. Remarkably, some of the stable quantum phases support gapless excitations even without any spontaneous symmetry breaking. In particular, the gapless excitations in algebraic spin liquids interact down to zero energy and the interaction does not open any energy gap. We propose that it is the quantum orders (instead of symmetries) that protect the gapless excitations and make algebraic spin liquids and Fermi spin liquids stable. Since high-${T}_{c}$ superconductors are likely to be described by a gapless spin liquid, the quantum orders and their projective symmetry group descriptions lay the foundation for a spin liquid approach to high-${T}_{c}$ superconductors.

arXiv:cond-mat/0107071•Physical review. B, Condensed matter•Apr 10, 2002•979 citations

Exact chiral spin liquid with stable spin Fermi surface on the kagome lattice

Victor Chua, Hong Yao, Gregory A. Fiete

We study an exactly solvable quantum spin model of Kitaev type on the kagome lattice. We find a rich phase diagram which includes a topological (gapped) chiral spin liquid with gapless chiral edge states, and a gapless chiral spin liquid phase with a spin Fermi surface. The ground state of the current model contains an odd number of electrons per unit cell, which qualitatively distinguishes it from previously studied exactly solvable models with a spin Fermi surface. Moreover, we show that the spin Fermi surface is stable against weak perturbations.

arXiv:1010.1035•Physical Review B•May 17, 2011•65 citations

Exactly solvable models hosting altermagnetic quantum spin liquids

João Augusto Sobral, Pietro M. Bonetti, Subrata Kumar Mandal +1 more

We construct spin- 3 2 and spin- 7 2 models on the square-octagon and checkerboard lattices that are exactly solvable with Majorana representations. They give rise to spin-liquid phases with full spin-rotation and lattice-translational symmetries but broken time-reversal symmetry. Although nonzero on elementary plaquettes, the net orbital magnetic moment is guaranteed to vanish as a result of point symmetries; due to the analogy to long-range ordered altermagnets, these types of phases were dubbed altermagnetic spin liquids in [Sobral ., ]. For the spin- 3 2 model, we find that a g -wave altermagnetic spin liquid emerges as the unique ground state. In contrast, the spin- 7 2 model exhibits a significantly richer phase diagram, involving different types of chiral spin liquids competing with a d -wave altermagnetic spin liquid. Finally, we identify and characterize the topological and nontopological excitations, illustrating the rich physics of altermagnetic spin liquids resulting from the interplay of nontrivial topological and symmetry aspects of this novel phase of matter.

arXiv:2512.24076•Physical review. B./Physical review. B•Oct 9, 2026•0 citations

SU(2)-invariant Majorana spin liquid with stable parton Fermi surfaces in an exactly solvable model

Hsin-Hua Lai, Olexei I. Motrunich

We construct an exactly solvable spin-orbital model on a decorated square lattice that realizes an SU(2)-invariant Majorana spin liquid with parton Fermi surfaces, of the kind discussed recently by Biswas et al. [Phys. Rev. B. 83, 245131 (2011)]. We find power-law spin correlations as well as power-law spin-nematic correlations with the same dominant $1/{|\mathbf{r}|}^{3}$ envelope. The model is solvable also in the presence of Zeeman magnetic field. One fermion species carries ${S}^{z}=0$ quantum number and its Fermi surface is not altered in the field, while the Fermi surfaces of the other species evolve and can disappear. In particular, we find an interesting half magnetization plateau phase in which spin excitations are gapful while there remain spinless gapless excitations that still produce metal-like thermal properties. In the fully magnetized phase, the model reduces to the one proposed by Baskaran et al. [e-print arXiv:0908.1614 (to be published)] in terms of the orbital degrees of freedom.

arXiv:1106.0028•Physical Review B•Aug 30, 2011•27 citations
We have proposed an exactly solvable quantum spin-3/2 model on a square lattice. Its ground state is a quantum spin liquid with a half-integer spin per unit cell. The fermionic excitations, dubbed as "spinons", are gapless with a linear dispersion, while the topological "vison" excitations are gapped. Moreover, these massless fermionic spinon excitations are topologically stable. Thus, this model is, to the best of our knowledge, the first exactly solvable model of half-integer spins whose ground state is an "algebraic spin liquid."
Despite rapid progress in understanding gapped topological states, much less is known about gapless topological phases of matter, especially in strongly correlated electrons. In this work, we discuss a large class of robust gapless quantum spin liquids in frustrated magnets made of half-integer spins, which are described by gapless fermionic spinons coupled to dynamical Z2{\mathbb{Z}}_{2}Z2​ gauge fields. Requiring U(1)\text{U}(1)U(1) spin conservation, time-reversal, and certain space-group symmetries, we show that certain spinon symmetry fractionalization class necessarily leads to a gapless spectrum. These gapless excitations are stable against any perturbations, as long as the required symmetries are preserved. Applying these gapless criteria to spin-12\frac{1}{2}21​ systems on square, triangular, and kagome lattices, we show that all gapped symmetric Z2{\mathbb{Z}}_{2}Z2​ spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.
Recently, there has been much effort in understanding topological phases of matter with gapless bulk excitations, which are characterized by topological invariants and protected intrinsic boundary states. Here we show that topological semimetals of Majorana fermions arise in exactly solvable Kitaev spin models on a series of three-dimensional lattices. The ground states of these models are quantum spin liquids with gapless nodal spectra of bulk Majorana fermion excitations. It is shown that these phases are topologically stable as long as certain discrete symmetries are protected. The corresponding topological indices and the gapless boundary states are explicitly computed to support these results. In contrast to previous studies of noninteracting systems, the phases discussed in this work are novel examples of gapless topological phases in interacting spin systems.
A concept---quantum order---is introduced to describe a new kind of orders that generally appear in quantum states at zero temperature. Quantum orders that characterize the universality classes of quantum states (described by complex ground-state wave functions) are much richer than classical orders that characterize the universality classes of finite-temperature classical states (described by positive probability distribution functions). Landau's theory for orders and phase transitions does not apply to quantum orders since they cannot be described by broken symmetries and the associated order parameters. We introduced a mathematical object---projective symmetry group---to characterize quantum orders. With the help of quantum orders and projective symmetry groups, we construct hundreds of symmetric spin liquids, which have SU(2), U(1), or Z2{Z}_{2}Z2​ gauge structures at low energies. We found that various spin liquids can be divided into four classes: (a) Rigid spin liquid---spinons (and all other excitations) are fully gapped and may have bosonic, fermionic, or fractional statistics. (b) Fermi spin liquid---spinons are gapless and are described by a Fermi liquid theory. (c) Algebraic spin liquid---spinons are gapless, but they are not described by free fermionic-bosonic quasiparticles. (d) Bose spin liquid---low-lying gapless excitations are described by a free-boson theory. The stability of those spin liquids is discussed in detail. We find that stable two-dimensional spin liquids exist in the first three classes (a)--(c). Those stable spin liquids occupy a finite region in phase space and represent quantum phases. Remarkably, some of the stable quantum phases support gapless excitations even without any spontaneous symmetry breaking. In particular, the gapless excitations in algebraic spin liquids interact down to zero energy and the interaction does not open any energy gap. We propose that it is the quantum orders (instead of symmetries) that protect the gapless excitations and make algebraic spin liquids and Fermi spin liquids stable. Since high-Tc{T}_{c}Tc​ superconductors are likely to be described by a gapless spin liquid, the quantum orders and their projective symmetry group descriptions lay the foundation for a spin liquid approach to high-Tc{T}_{c}Tc​ superconductors.
We study an exactly solvable quantum spin model of Kitaev type on the kagome lattice. We find a rich phase diagram which includes a topological (gapped) chiral spin liquid with gapless chiral edge states, and a gapless chiral spin liquid phase with a spin Fermi surface. The ground state of the current model contains an odd number of electrons per unit cell, which qualitatively distinguishes it from previously studied exactly solvable models with a spin Fermi surface. Moreover, we show that the spin Fermi surface is stable against weak perturbations.
We construct spin- 3 2 and spin- 7 2 models on the square-octagon and checkerboard lattices that are exactly solvable with Majorana representations. They give rise to spin-liquid phases with full spin-rotation and lattice-translational symmetries but broken time-reversal symmetry. Although nonzero on elementary plaquettes, the net orbital magnetic moment is guaranteed to vanish as a result of point symmetries; due to the analogy to long-range ordered altermagnets, these types of phases were dubbed altermagnetic spin liquids in [Sobral ., ]. For the spin- 3 2 model, we find that a g -wave altermagnetic spin liquid emerges as the unique ground state. In contrast, the spin- 7 2 model exhibits a significantly richer phase diagram, involving different types of chiral spin liquids competing with a d -wave altermagnetic spin liquid. Finally, we identify and characterize the topological and nontopological excitations, illustrating the rich physics of altermagnetic spin liquids resulting from the interplay of nontrivial topological and symmetry aspects of this novel phase of matter.
We construct an exactly solvable spin-orbital model on a decorated square lattice that realizes an SU(2)-invariant Majorana spin liquid with parton Fermi surfaces, of the kind discussed recently by Biswas et al. [Phys. Rev. B. 83, 245131 (2011)]. We find power-law spin correlations as well as power-law spin-nematic correlations with the same dominant 1/∣r∣31/{|\mathbf{r}|}^{3}1/∣r∣3 envelope. The model is solvable also in the presence of Zeeman magnetic field. One fermion species carries Sz=0{S}^{z}=0Sz=0 quantum number and its Fermi surface is not altered in the field, while the Fermi surfaces of the other species evolve and can disappear. In particular, we find an interesting half magnetization plateau phase in which spin excitations are gapful while there remain spinless gapless excitations that still produce metal-like thermal properties. In the fully magnetized phase, the model reduces to the one proposed by Baskaran et al. [e-print arXiv:0908.1614 (to be published)] in terms of the orbital degrees of freedom.