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:2409.16344

Gapped and gapless quantum spin liquids on the ruby lattice

Atanu Maity, Rhine Samajdar, Yasir Iqbal

Published Sep 24, 2024• 0 citations
arXivHTMLPDF

Abstract

The ruby lattice has been the subject of much interest recently due its realization in Rydberg atom arrays, where its rich variety of frustrated interactions gives rise to topologically ordered quantum spin liquids. Similarly, numerical studies of ruby-lattice spin models, with both isotropic and anisotropic interactions, have provided evidence of gapped and gapless spin-liquid ground states with different low-energy gauge structures. Motivated by these findings, we perform a projective symmetry group (PSG) classification of U(1) and $\mathbb{Z}_{2}$ fermionic spinon mean-field theories$\unicode{x2014}$respecting space-group and time-reversal symmetries$\unicode{x2014}$for $S=1/2$ spins. We obtain a total of 50 U(1) and 64 $\mathbb{Z}_{2}$ PSGs, and upon restricting their realization via mean-field $\textit{Ansätze}$ with up to second-nearest-neighbor singlet amplitudes (relevant to the models studied here), only 8 U(1) and 18 $\mathbb{Z}_{2}$ distinct states are obtained. We present the singlet fields for all $\textit{Ansätze}$ up to third-nearest-neighboring bonds and discuss their spinon dispersions as well as their dynamical spin structure factors. Building on this information, we also obtain the phase diagram of the Heisenberg model in the presence of first ($J_{1}$), second ($J_{1}'$), and third ($J_{2}$) neighbor antiferromagnetic couplings within a self-consistent mean-field approximation.

DOI
10.48550/arxiv.2409.16344

Related papers

Candidate quantum spin liquids on the maple-leaf lattice

Jonas Sonnenschein, Atanu Maity, Chunxiao Liu +3 more

Motivated by recent numerical studies reporting putative quantum paramagnetic behavior in spin-$1/2$ Heisenberg models on the maple-leaf lattice, we classify Abrikosov fermion mean-field Ans\"atze of fully symmetric U(1) and ${\mathbb{Z}}_{2}$ quantum spin liquids within the framework of projective symmetry groups. We obtain a total of $17\phantom{\rule{4pt}{0ex}}\text{U}(1)$ and $12\phantom{\rule{4pt}{0ex}}{\mathbb{Z}}_{2}$ algebraic PSGs, and, upon restricting their realization via mean-field Ans\"atze with nearest-neighbor amplitudes (relevant to the studied models), only 12 U(1) and 8 ${\mathbb{Z}}_{2}$ distinct phases are obtained. We present both singlet and triplet fields for all Ans\"atze up to third nearest-neighbor bonds and discuss their spinon dispersions as well as their dynamical spin structure factors. We further assess the effects of Gutzwiller projection on the equal-time spin structure factors.

arXiv:2404.05617•Physical review. B./Physical review. B•Jul 10, 2024•10 citations

Projective symmetry groups beyond spin-$1/2$: Classifying integer-spin quantum spin liquids

Atanu Maity, Chunxiao Liu, Rhine Samajdar +1 more

Wen's projective symmetry group (PSG), introduced a quarter of a century ago, reduced the classification of symmetric quantum spin liquids to an algebraic problem and has guided the study of spin-$1/2$ models and materials ever since. Many current candidate materials and quantum simulators, however, have spin-$1$ or larger moments, and no comparable classification exists for them. We formulate the PSG for fermionic partons of arbitrary spin $S$. Integer and half-odd-integer spins already differ in their gauge structure. Half-odd-integer spins give the familiar $\mathrm{SU}(2)$, $\mathrm{U}(1)$, and $\mathbb{Z}_2$ invariant gauge groups, while integer spins give a different hierarchy that includes $\mathrm{U}(1)\rtimes\mathbb{Z}_2$ and $\mathbb{Z}_2\times\mathbb{Z}_2$; the latter has no spin-$1/2$ counterpart. Three lattice-independent results follow for integer spin. Singlet pairing of the partons is odd under site exchange. A lattice symmetry can exchange the particle and hole representations of the local constraint, which forces symmetry-related sites into opposite constraint sectors. And a time-reversal-symmetric $\mathrm{U}(1)$ state with the same constraint on every site can be gapped only if the number of sites per unit cell is a multiple of $2S+1$. We apply the formalism to the spin-$1$ diamond lattice, realized in frustrated A-site spinels, and find ten fully symmetric Ansätze up to second neighbors. In the $J_1$-$J_2$ Heisenberg model at the mean-field level, the ground state depends on how the constraint is imposed. With the particle and hole representations, the lowest states are fully gapped $\mathrm{U}(1)$ states whose two sublattices sit in opposite constraint sectors, and one of them is gauge equivalent to a paired Ansatz. The classification provides a starting point for integer-spin quantum spin liquids on other lattices and in other materials.

arXiv:2610.30048•Oct 21, 2026•0 citations

Characterization of quantum spin liquids and their spinon band structures via functional renormalization

Max Hering, Jonas Sonnenschein, Yasir Iqbal +1 more

We combine the pseudofermion functional renormalization group (PFFRG) method with a self-consistent Fock-like mean-field scheme to calculate low-energy effective theories for emergent spinon excitations in spin-1/2 quantum spin liquids. Using effective spin interactions from PFFRG as an input for the Fock equation and allowing for the most general types of free spinon Ans\"atze as classified by the projective symmetry group (PSG) method, we are able to systematically determine spinon band structures for spin-liquid candidate systems beyond mean-field theory. We apply this approach to the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice and to the antiferromagnetic nearest-neighbor Heisenberg model on the kagome lattice. For the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, we find that in the regime of maximal frustration a SU(2) $\ensuremath{\pi}$-flux state with Dirac spinons yields the largest mean-field amplitudes. For the kagome model, we identify a gapless ${\mathbb{Z}}_{2}$ spin liquid with a small circular spinon Fermi surface and approximate Dirac cones at low but finite energies.

arXiv:1806.05021•Physical review. B./Physical review. B•Mar 28, 2019•49 citations

Z 2 spin liquids in the S = 1 2 Heisenberg model on the kagome lattice: A projective symmetry-group study of Schwinger fermion mean-field states

Yuan-Ming Lu, Ying Ran, Patrick A. Lee

Due to strong geometric frustration and quantum fluctuation, the $S=1/2$ quantum Heisenberg antiferromagnet on the $\text{kagome}$ lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg $S=1/2,\text{kagome}$ lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of ${\mathbb{Z}}_{2}$ topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of ${\mathbb{Z}}_{2}$ SLs on the $\text{kagome}$ lattice. Among them we found only one gapped ${\mathbb{Z}}_{2}$ SL (which we call the ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state to be the numerically discovered SL ground state of the HKLM.

arXiv:1104.1432•Physical Review B•Jun 22, 2011•147 citations

Projective symmetry group classifications of quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic lattices

Jonas Sonnenschein, Aishwarya Chauhan, Yasir Iqbal +1 more

We perform extensive classifications of ${\mathbb{Z}}_{2}$ quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic (fcc) lattices using a spin-rotation-invariant fermionic projective symmetry group approach. Taking into account that all three lattices share the same point group ${O}_{h}$, we apply an efficient gauge where the classification for the simple cubic lattice can be partially carried over to the other two lattices. We identify hundreds of projective representations for each of the three lattices, however, when constructing short-range mean-field models for the fermionic partons (spinons) these phases collapse to only very few relevant cases. We self-consistently calculate the corresponding mean-field parameters for frustrated Heisenberg models on all three lattices with up to third-neighbor spin interactions and discuss the spinon dispersions, ground-state energies, and dynamical spin structure factors. Our results indicate that phases with nonuniform spinon hopping or pairing amplitudes are energetically favored. An unusual situation is identified for the fcc lattice where the spinon dispersion minimizing the mean-field energy features a network of symmetry-protected linelike zero modes in reciprocal space. We further discuss characteristic fingerprints of these phases in the dynamical spin structure factor which may help to identify and distinguish them in future numerical or experimental studies.

arXiv:2007.02589•Physical review. B./Physical review. B•Sep 22, 2020•13 citations

Classification of spin- 1 2 fermionic quantum spin liquids on the trillium lattice

Ming-Hao Li, Sounak Biswas, S. A. Parameswaran

We study fermionic quantum spin liquids (QSLs) on the three-dimensional trillium lattice of corner-sharing triangles. We are motivated by recent experimental and theoretical investigations that have explored various classical and quantum spin liquid states on similar networks of triangular motifs with strong geometric frustration. Using the framework of projective symmetry groups (PSG), we obtain a classification of all symmetric Z 2 and U ( 1 ) QSLs on the trillium lattice. We find two Z 2 spin-liquids, and a single U ( 1 ) spin-liquid that is proximate to one of the Z 2 states. The small number of solutions reflects the constraints imposed by the nonsymmorphic symmetries in the space group of the trillium lattice. Using self-consistency conditions of the mean-field equations, we obtain the spinon band-structure and spin structure factors corresponding to these states. All three of our spin liquids are gapless at their saddle points: one of the two Z 2 QSLs is nodal, while the U ( 1 ) case hosts a spinon Fermi surface. One of our Z 2 spin liquids hosts a stable gapless nodal star that is protected by projective symmetries against additions of further neighbor terms in the mean-field ansatz. We comment on directions for further work.

arXiv:2409.02898•Physical review. B./Physical review. B•Jul 17, 2025•2 citations
The ruby lattice has been the subject of much interest recently due its realization in Rydberg atom arrays, where its rich variety of frustrated interactions gives rise to topologically ordered quantum spin liquids. Similarly, numerical studies of ruby-lattice spin models, with both isotropic and anisotropic interactions, have provided evidence of gapped and gapless spin-liquid ground states with different low-energy gauge structures. Motivated by these findings, we perform a projective symmetry group (PSG) classification of U(1) and Z2\mathbb{Z}_{2}Z2​ fermionic spinon mean-field theories\unicodex2014\unicode{x2014}\unicodex2014respecting space-group and time-reversal symmetries\unicodex2014\unicode{x2014}\unicodex2014for S=1/2S=1/2S=1/2 spins. We obtain a total of 50 U(1) and 64 Z2\mathbb{Z}_{2}Z2​ PSGs, and upon restricting their realization via mean-field Ansa¨tze\textit{Ansätze}Ansa¨tze with up to second-nearest-neighbor singlet amplitudes (relevant to the models studied here), only 8 U(1) and 18 Z2\mathbb{Z}_{2}Z2​ distinct states are obtained. We present the singlet fields for all Ansa¨tze\textit{Ansätze}Ansa¨tze up to third-nearest-neighboring bonds and discuss their spinon dispersions as well as their dynamical spin structure factors. Building on this information, we also obtain the phase diagram of the Heisenberg model in the presence of first (J1J_{1}J1​), second (J1′J_{1}'J1′​), and third (J2J_{2}J2​) neighbor antiferromagnetic couplings within a self-consistent mean-field approximation.
Motivated by recent numerical studies reporting putative quantum paramagnetic behavior in spin-1/21/21/2 Heisenberg models on the maple-leaf lattice, we classify Abrikosov fermion mean-field Ans\"atze of fully symmetric U(1) and Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquids within the framework of projective symmetry groups. We obtain a total of 17U(1)17\phantom{\rule{4pt}{0ex}}\text{U}(1)17U(1) and 12Z212\phantom{\rule{4pt}{0ex}}{\mathbb{Z}}_{2}12Z2​ algebraic PSGs, and, upon restricting their realization via mean-field Ans\"atze with nearest-neighbor amplitudes (relevant to the studied models), only 12 U(1) and 8 Z2{\mathbb{Z}}_{2}Z2​ distinct phases are obtained. We present both singlet and triplet fields for all Ans\"atze up to third nearest-neighbor bonds and discuss their spinon dispersions as well as their dynamical spin structure factors. We further assess the effects of Gutzwiller projection on the equal-time spin structure factors.
Projective symmetry groups beyond spin-1/21/21/2: Classifying integer-spin quantum spin liquids
Wen's projective symmetry group (PSG), introduced a quarter of a century ago, reduced the classification of symmetric quantum spin liquids to an algebraic problem and has guided the study of spin-1/21/21/2 models and materials ever since. Many current candidate materials and quantum simulators, however, have spin-111 or larger moments, and no comparable classification exists for them. We formulate the PSG for fermionic partons of arbitrary spin SSS. Integer and half-odd-integer spins already differ in their gauge structure. Half-odd-integer spins give the familiar SU(2)\mathrm{SU}(2)SU(2), U(1)\mathrm{U}(1)U(1), and Z2\mathbb{Z}_2Z2​ invariant gauge groups, while integer spins give a different hierarchy that includes U(1)⋊Z2\mathrm{U}(1)\rtimes\mathbb{Z}_2U(1)⋊Z2​ and Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2Z2​×Z2​; the latter has no spin-1/21/21/2 counterpart. Three lattice-independent results follow for integer spin. Singlet pairing of the partons is odd under site exchange. A lattice symmetry can exchange the particle and hole representations of the local constraint, which forces symmetry-related sites into opposite constraint sectors. And a time-reversal-symmetric U(1)\mathrm{U}(1)U(1) state with the same constraint on every site can be gapped only if the number of sites per unit cell is a multiple of 2S+12S+12S+1. We apply the formalism to the spin-111 diamond lattice, realized in frustrated A-site spinels, and find ten fully symmetric Ansätze up to second neighbors. In the J1J_1J1​-J2J_2J2​ Heisenberg model at the mean-field level, the ground state depends on how the constraint is imposed. With the particle and hole representations, the lowest states are fully gapped U(1)\mathrm{U}(1)U(1) states whose two sublattices sit in opposite constraint sectors, and one of them is gauge equivalent to a paired Ansatz. The classification provides a starting point for integer-spin quantum spin liquids on other lattices and in other materials.
We combine the pseudofermion functional renormalization group (PFFRG) method with a self-consistent Fock-like mean-field scheme to calculate low-energy effective theories for emergent spinon excitations in spin-1/2 quantum spin liquids. Using effective spin interactions from PFFRG as an input for the Fock equation and allowing for the most general types of free spinon Ans\"atze as classified by the projective symmetry group (PSG) method, we are able to systematically determine spinon band structures for spin-liquid candidate systems beyond mean-field theory. We apply this approach to the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice and to the antiferromagnetic nearest-neighbor Heisenberg model on the kagome lattice. For the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, we find that in the regime of maximal frustration a SU(2) $\ensuremath{\pi}$-flux state with Dirac spinons yields the largest mean-field amplitudes. For the kagome model, we identify a gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid with a small circular spinon Fermi surface and approximate Dirac cones at low but finite energies.
Due to strong geometric frustration and quantum fluctuation, the S=1/2S=1/2S=1/2 quantum Heisenberg antiferromagnet on the kagome\text{kagome}kagome lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg S=1/2,kagomeS=1/2,\text{kagome}S=1/2,kagome lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of Z2{\mathbb{Z}}_{2}Z2​ topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of Z2{\mathbb{Z}}_{2}Z2​ SLs on the kagome\text{kagome}kagome lattice. Among them we found only one gapped Z2{\mathbb{Z}}_{2}Z2​ SL (which we call the ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state to be the numerically discovered SL ground state of the HKLM.
We perform extensive classifications of Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic (fcc) lattices using a spin-rotation-invariant fermionic projective symmetry group approach. Taking into account that all three lattices share the same point group Oh{O}_{h}Oh​, we apply an efficient gauge where the classification for the simple cubic lattice can be partially carried over to the other two lattices. We identify hundreds of projective representations for each of the three lattices, however, when constructing short-range mean-field models for the fermionic partons (spinons) these phases collapse to only very few relevant cases. We self-consistently calculate the corresponding mean-field parameters for frustrated Heisenberg models on all three lattices with up to third-neighbor spin interactions and discuss the spinon dispersions, ground-state energies, and dynamical spin structure factors. Our results indicate that phases with nonuniform spinon hopping or pairing amplitudes are energetically favored. An unusual situation is identified for the fcc lattice where the spinon dispersion minimizing the mean-field energy features a network of symmetry-protected linelike zero modes in reciprocal space. We further discuss characteristic fingerprints of these phases in the dynamical spin structure factor which may help to identify and distinguish them in future numerical or experimental studies.
We study fermionic quantum spin liquids (QSLs) on the three-dimensional trillium lattice of corner-sharing triangles. We are motivated by recent experimental and theoretical investigations that have explored various classical and quantum spin liquid states on similar networks of triangular motifs with strong geometric frustration. Using the framework of projective symmetry groups (PSG), we obtain a classification of all symmetric Z 2 and U ( 1 ) QSLs on the trillium lattice. We find two Z 2 spin-liquids, and a single U ( 1 ) spin-liquid that is proximate to one of the Z 2 states. The small number of solutions reflects the constraints imposed by the nonsymmorphic symmetries in the space group of the trillium lattice. Using self-consistency conditions of the mean-field equations, we obtain the spinon band-structure and spin structure factors corresponding to these states. All three of our spin liquids are gapless at their saddle points: one of the two Z 2 QSLs is nodal, while the U ( 1 ) case hosts a spinon Fermi surface. One of our Z 2 spin liquids hosts a stable gapless nodal star that is protected by projective symmetries against additions of further neighbor terms in the mean-field ansatz. We comment on directions for further work.