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:cond-mat/0404751 · Physical Review B

Stability of U(1) spin liquids in two dimensions

Michael Hermele, T. Senthil, Matthew P. A. Fisher, Patrick A. Lee, Naoto Nagaosa, Xiao-Gang Wen

Published Dec 30, 2004• 317 citations
arXivHTMLPDF

Abstract

We establish that spin liquids described in terms of gapless fermionic (Dirac) spinons and gapless $U(1)$ gauge fluctuations can be stable in two dimensions, at least when the physical $SU(2)$ spin symmetry is generalized to $SU(N)$. Equivalently, we show that compact ${\mathrm{QED}}_{3}$ has a deconfined phase for a large number of fermion fields, in the sense that monopole fluctuations can be irrelevant at low energies. A precise characterization is provided by an emergent global topological $U(1)$ symmetry corresponding to the conservation of gauge flux. Beginning with an $SU(N)$ generalization of the $S=1∕2$ square-lattice Heisenberg antiferromagnet, we consider the $\ensuremath{\pi}$-flux spin liquid and, via a systematic analysis of all operators, show that there are no relevant perturbations (in the renormalization-group sense) about the large-$N$ spin-liquid fixed point, which is thus a stable phase. We provide a further illustration of this conclusion with an approximate renormalization-group calculation that treats the gapless fermions and the monopoles on an equal footing. This approach directly points out some of the flaws in the erroneous ``screening'' argument for the relevance of monopoles in compact ${\mathrm{QED}}_{3}$.

DOI
10.1103/physrevb.70.214437

Related papers

Spin-Peierls instability of the U(1) Dirac spin liquid

Urban F. P. Seifert, Josef Willsher, Markus Drescher +2 more

A complicating factor in the realization and observation of quantum spin liquids in materials is the ubiquitous presence of other degrees of freedom, in particular lattice distortion modes (phonons). These provide additional routes for relieving magnetic frustration, thereby possibly destabilizing spin-liquid ground states. In this work, we focus on triangular-lattice Heisenberg antiferromagnets, where recent numerical evidence suggests the presence of an extended U(1) Dirac spin liquid phase which is described by compact quantum electrodynamics in 2+1 dimensions (QED$_3$), featuring gapless spinons and monopoles as gauge excitations, and believed to flow to a strongly-coupled fixed point with conformal symmetry. Using complementary perturbation theory and scaling arguments, we show that a symmetry-allowed coupling between (classical) finite-wavevector lattice distortions and monopole operators of the U(1) Dirac spin liquid generally induces a spin-Peierls instability towards a (confining) 12-site valence-bond solid state. We support our theoretical analysis with state-of-the-art density matrix renormalization group simulations. Away from the limit of static distortions, we demonstrate that the phonon energy gap establishes a parameter regime where the spin liquid is expected to be stable, and show that the monopole-lattice coupling leads to softening of the phonon in analogy to the Kohn anomaly. We discuss the applicability of our results to similar systems, in particular the Dirac spin liquid on the Kagome lattice.

arXiv:2307.12295•Jul 23, 2023•3 citations

Monopole excitations in the U ( 1 ) Dirac spin liquid on the triangular lattice

Sasank Budaraju, Alberto Parola, Yasir Iqbal +2 more

The $U(1)$ Dirac spin liquid might realize an exotic phase of matter whose low-energy properties are described by quantum electrodynamics in $2+1$ dimensions, where gapless modes exist but spinons and gauge fields are strongly coupled. Its existence was proposed in frustrated Heisenberg models in the presence of frustrating superexchange interactions by the (Abrikosov) fermionic representation of the spin operators [Wen, Phys. Rev. B 65, 165113 (2002)], supplemented by the Gutzwiller projection. Here, we construct charge-$Q$ monopole excitations in the Heisenberg model on the triangular lattice with nearest-neighbor (${J}_{1}$) and next-nearest-neighbor (${J}_{2}$) couplings. In the highly frustrated regime, singlet and triplet monopoles with $Q=1$ become gapless in the thermodynamic limit; in addition, the energies for generic $Q$ agree with field-theoretical predictions, obtained for a large number of gapless fermion modes. Finally, we consider localized gauge excitations, in which magnetic $\ensuremath{\pi}$ fluxes are concentrated in the triangular plaquettes (in analogy with ${\mathbb{Z}}_{2}$ visons), showing that these kinds of states do not play a relevant role at low energies. All our findings lend support to a stable $U(1)$ Dirac spin liquid in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the triangular lattice.

arXiv:2410.18747•Physical review. B./Physical review. B•Mar 21, 2025•12 citations

Destabilization of U(1) Dirac spin liquids on two-dimensional nonbipartite lattices by quenched disorder

Santanu Dey

The stability of the Dirac spin liquid on two-dimensional lattices has long been debated. It was recently demonstrated [Nat. Commun. 10, 4254 (2019) and Phys. Rev. B 93, 144411 (2016)] that the staggered $\ensuremath{\pi}$-flux Dirac spin-liquid phase on the nonbipartite triangular lattice may be stable in the clean limit. However, quenched disorder plays a crucial role in determining whether such a phase is experimentally viable. For SU(2) spin systems, the effective zero-temperature low-energy description of Dirac spin liquids in $(2+1)$ dimensions is given by the compact quantum electrodynamics $({\mathrm{cQED}}_{2+1})$ which admits monopoles. It is already known that generic quenched random perturbations to the noncompact version of ${\mathrm{QED}}_{2+1}$ (where monopoles are absent) lead to strong-coupling instabilities. In this paper we study ${\mathrm{cQED}}_{2+1}$ in the presence of a class of time-reversal invariant quenched disorder perturbations. We show that in this model, random non-Abelian vector potentials make the symmetry-allowed monopole operators more relevant. The disorder-induced underscreening of monopoles, thus, generically makes the gapless spin-liquid phase fragile.

arXiv:2008.12307•Physical review. B./Physical review. B•Dec 30, 2020•3 citations

Symmetry-resolved monopoles and stability criteria for an $N_f=12$ Dirac spin liquid on the maple-leaf lattice

Yunchao Zhang, Andreas Feuerpfeil, Subir Sachdev A +2 more

The $\mathrm{U}(1)$ Dirac spin liquid provides a useful organizing framework for frustrated magnets: it offers an algebraic parent state from which competing orders, confinement patterns, and low-energy spectral features can be understood. Whether such a state can occur as a stable ground state of a two-dimensional spin Hamiltonian remains an open question, because monopole operators of the compact gauge field can proliferate and confine the spinons. Here we determine the symmetry quantum numbers of the fundamental monopoles of a maple-leaf-lattice Dirac spin liquid whose generic six-node regime realizes QED$_3$ with $N_f=12$ two-component Dirac fermions. The charge-one monopoles span a $924$-dimensional complex representation. In the spin-singlet, zero-momentum, zero-$C_6$-angular-momentum sector we find five complex charge-one directions, of which three are mapped to their Hermitian conjugates under time reversal. Their real and imaginary Hermitian combinations therefore supply six independent monopole perturbations that are trivial under translations, $C_6$, time reversal, and spin rotation. The Dirac spin liquid phase is thus not protected by symmetry exclusion of charge-one monopoles: its stability depends on whether these allowed perturbations are relevant in the renormalization group sense. Existing large-$N_f$ and numerical estimates of the charge-one monopole scaling dimension lie close to the relevance threshold, making the maple-leaf lattice a concrete platform for testing the stability of compact QED$_3$ in a quantum magnet. Our complete monopole classification also identifies the spin, momentum, and rotation sectors in which monopole excitations should appear, providing symmetry-resolved benchmarks for exact-diagonalization and variational tests of the $N_f=12$ Dirac spin liquid.

arXiv:2605.21587•May 20, 2026•0 citations

Quantum Electrodynamics in 2+1 Dimensions, Confinement, and the Stability of U(1) Spin Liquids

Flavio S. Nogueira, Hagen M. Kleinert

Compact quantum electrodynamics in 2 + 1 dimensions often arises as an effective theory for a Mott insulator, with the Dirac fermions representing the low-energy spinons. An important and controversial issue in this context is whether a deconfinement transition takes place. We perform a renormalization group analysis to show that deconfinement occurs when N > Nc = 36/pi3 approximately to 1.161, where N is the number of fermion replica. For N < Nc, however, there are two stable fixed points separated by a line containing a unstable nontrivial fixed point: a fixed point corresponding to the scaling limit of the noncompact theory, and another one governing the scaling behavior of the compact theory. The string tension associated with the confining interspinon potential is shown to exhibit a universal jump as N --> Nc-. Our results imply the stability of a spin liquid at the physical value N = 2 for Mott insulators.

arXiv:cond-mat/0501022•Physical Review Letters•Oct 20, 2005•34 citations

Deconfined criticality and a gapless Z 2 spin liquid in the square-lattice antiferromagnet

Henry Shackleton, Alex Thomson, Subir Sachdev A

The theory for the vanishing of N\'eel order in the spin $S=1/2$ square lattice antiferromagnet has been the focus of attention for many decades. A consensus appears to have emerged in recent numerical studies on the antiferromagnet with first and second neighbor exchange interactions (the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model): A gapless spin liquid is present for a narrow window of parameters between the vanishing of the N\'eel order and the onset of a gapped valence bond solid state. We propose a deconfined critical SU(2) gauge theory for a transition into a stable ${\mathbb{Z}}_{2}$ spin liquid with massless Dirac spinon excitations; on the other side of the critical point, the SU(2) spin liquid (the `$\ensuremath{\pi}$-flux' phase) is presumed to be unstable to confinement to the N\'eel phase. We identify a dangerously irrelevant coupling in the critical SU(2) gauge theory, which contributes a logarithm-squared renormalization. This critical theory is also not Lorentz invariant and weakly breaks the SO(5) symmetry which rotates between the N\'eel and valence bond solid order parameters. We also propose a distinct deconfined critical U(1) gauge theory for a transition into the same gapless ${\mathbb{Z}}_{2}$ spin liquid; on the other side of the critical point, the U(1) spin liquid (the `staggered flux' phase) is presumed to be unstable to confinement to the valence bond solid. This critical theory has no dangerously irrelevant coupling, dynamic critical exponent $z\ensuremath{\ne}1$, and no SO(5) symmetry. All of these phases and critical points are unified in a SU(2) gauge theory with Higgs fields and fermionic spinons which can naturally realize the observed sequence of phases with increasing ${J}_{2}/{J}_{1}$: N\'eel, gapless ${\mathbb{Z}}_{2}$ spin liquid, and valence bond solid.

arXiv:2104.09537•Physical review. B./Physical review. B•Jul 6, 2021•27 citations
We establish that spin liquids described in terms of gapless fermionic (Dirac) spinons and gapless U(1)U(1)U(1) gauge fluctuations can be stable in two dimensions, at least when the physical SU(2)SU(2)SU(2) spin symmetry is generalized to SU(N)SU(N)SU(N). Equivalently, we show that compact QED3{\mathrm{QED}}_{3}QED3​ has a deconfined phase for a large number of fermion fields, in the sense that monopole fluctuations can be irrelevant at low energies. A precise characterization is provided by an emergent global topological U(1)U(1)U(1) symmetry corresponding to the conservation of gauge flux. Beginning with an SU(N)SU(N)SU(N) generalization of the S=1∕2S=1∕2S=1∕2 square-lattice Heisenberg antiferromagnet, we consider the \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux spin liquid and, via a systematic analysis of all operators, show that there are no relevant perturbations (in the renormalization-group sense) about the large-NNN spin-liquid fixed point, which is thus a stable phase. We provide a further illustration of this conclusion with an approximate renormalization-group calculation that treats the gapless fermions and the monopoles on an equal footing. This approach directly points out some of the flaws in the erroneous ``screening'' argument for the relevance of monopoles in compact QED3{\mathrm{QED}}_{3}QED3​.
A complicating factor in the realization and observation of quantum spin liquids in materials is the ubiquitous presence of other degrees of freedom, in particular lattice distortion modes (phonons). These provide additional routes for relieving magnetic frustration, thereby possibly destabilizing spin-liquid ground states. In this work, we focus on triangular-lattice Heisenberg antiferromagnets, where recent numerical evidence suggests the presence of an extended U(1) Dirac spin liquid phase which is described by compact quantum electrodynamics in 2+1 dimensions (QED3_33​), featuring gapless spinons and monopoles as gauge excitations, and believed to flow to a strongly-coupled fixed point with conformal symmetry. Using complementary perturbation theory and scaling arguments, we show that a symmetry-allowed coupling between (classical) finite-wavevector lattice distortions and monopole operators of the U(1) Dirac spin liquid generally induces a spin-Peierls instability towards a (confining) 12-site valence-bond solid state. We support our theoretical analysis with state-of-the-art density matrix renormalization group simulations. Away from the limit of static distortions, we demonstrate that the phonon energy gap establishes a parameter regime where the spin liquid is expected to be stable, and show that the monopole-lattice coupling leads to softening of the phonon in analogy to the Kohn anomaly. We discuss the applicability of our results to similar systems, in particular the Dirac spin liquid on the Kagome lattice.
The U(1)U(1)U(1) Dirac spin liquid might realize an exotic phase of matter whose low-energy properties are described by quantum electrodynamics in 2+12+12+1 dimensions, where gapless modes exist but spinons and gauge fields are strongly coupled. Its existence was proposed in frustrated Heisenberg models in the presence of frustrating superexchange interactions by the (Abrikosov) fermionic representation of the spin operators [Wen, Phys. Rev. B 65, 165113 (2002)], supplemented by the Gutzwiller projection. Here, we construct charge-QQQ monopole excitations in the Heisenberg model on the triangular lattice with nearest-neighbor (J1{J}_{1}J1​) and next-nearest-neighbor (J2{J}_{2}J2​) couplings. In the highly frustrated regime, singlet and triplet monopoles with Q=1Q=1Q=1 become gapless in the thermodynamic limit; in addition, the energies for generic QQQ agree with field-theoretical predictions, obtained for a large number of gapless fermion modes. Finally, we consider localized gauge excitations, in which magnetic $\ensuremath{\pi}$ fluxes are concentrated in the triangular plaquettes (in analogy with Z2{\mathbb{Z}}_{2}Z2​ visons), showing that these kinds of states do not play a relevant role at low energies. All our findings lend support to a stable U(1)U(1)U(1) Dirac spin liquid in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the triangular lattice.
The stability of the Dirac spin liquid on two-dimensional lattices has long been debated. It was recently demonstrated [Nat. Commun. 10, 4254 (2019) and Phys. Rev. B 93, 144411 (2016)] that the staggered $\ensuremath{\pi}$-flux Dirac spin-liquid phase on the nonbipartite triangular lattice may be stable in the clean limit. However, quenched disorder plays a crucial role in determining whether such a phase is experimentally viable. For SU(2) spin systems, the effective zero-temperature low-energy description of Dirac spin liquids in (2+1)(2+1)(2+1) dimensions is given by the compact quantum electrodynamics (cQED2+1)({\mathrm{cQED}}_{2+1})(cQED2+1​) which admits monopoles. It is already known that generic quenched random perturbations to the noncompact version of QED2+1{\mathrm{QED}}_{2+1}QED2+1​ (where monopoles are absent) lead to strong-coupling instabilities. In this paper we study cQED2+1{\mathrm{cQED}}_{2+1}cQED2+1​ in the presence of a class of time-reversal invariant quenched disorder perturbations. We show that in this model, random non-Abelian vector potentials make the symmetry-allowed monopole operators more relevant. The disorder-induced underscreening of monopoles, thus, generically makes the gapless spin-liquid phase fragile.
Symmetry-resolved monopoles and stability criteria for an Nf=12N_f=12Nf​=12 Dirac spin liquid on the maple-leaf lattice
The U(1)\mathrm{U}(1)U(1) Dirac spin liquid provides a useful organizing framework for frustrated magnets: it offers an algebraic parent state from which competing orders, confinement patterns, and low-energy spectral features can be understood. Whether such a state can occur as a stable ground state of a two-dimensional spin Hamiltonian remains an open question, because monopole operators of the compact gauge field can proliferate and confine the spinons. Here we determine the symmetry quantum numbers of the fundamental monopoles of a maple-leaf-lattice Dirac spin liquid whose generic six-node regime realizes QED3_33​ with Nf=12N_f=12Nf​=12 two-component Dirac fermions. The charge-one monopoles span a 924924924-dimensional complex representation. In the spin-singlet, zero-momentum, zero-C6C_6C6​-angular-momentum sector we find five complex charge-one directions, of which three are mapped to their Hermitian conjugates under time reversal. Their real and imaginary Hermitian combinations therefore supply six independent monopole perturbations that are trivial under translations, C6C_6C6​, time reversal, and spin rotation. The Dirac spin liquid phase is thus not protected by symmetry exclusion of charge-one monopoles: its stability depends on whether these allowed perturbations are relevant in the renormalization group sense. Existing large-NfN_fNf​ and numerical estimates of the charge-one monopole scaling dimension lie close to the relevance threshold, making the maple-leaf lattice a concrete platform for testing the stability of compact QED3_33​ in a quantum magnet. Our complete monopole classification also identifies the spin, momentum, and rotation sectors in which monopole excitations should appear, providing symmetry-resolved benchmarks for exact-diagonalization and variational tests of the Nf=12N_f=12Nf​=12 Dirac spin liquid.
Compact quantum electrodynamics in 2 + 1 dimensions often arises as an effective theory for a Mott insulator, with the Dirac fermions representing the low-energy spinons. An important and controversial issue in this context is whether a deconfinement transition takes place. We perform a renormalization group analysis to show that deconfinement occurs when N > Nc = 36/pi3 approximately to 1.161, where N is the number of fermion replica. For N < Nc, however, there are two stable fixed points separated by a line containing a unstable nontrivial fixed point: a fixed point corresponding to the scaling limit of the noncompact theory, and another one governing the scaling behavior of the compact theory. The string tension associated with the confining interspinon potential is shown to exhibit a universal jump as N --> Nc-. Our results imply the stability of a spin liquid at the physical value N = 2 for Mott insulators.
The theory for the vanishing of N\'eel order in the spin S=1/2S=1/2S=1/2 square lattice antiferromagnet has been the focus of attention for many decades. A consensus appears to have emerged in recent numerical studies on the antiferromagnet with first and second neighbor exchange interactions (the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model): A gapless spin liquid is present for a narrow window of parameters between the vanishing of the N\'eel order and the onset of a gapped valence bond solid state. We propose a deconfined critical SU(2) gauge theory for a transition into a stable Z2{\mathbb{Z}}_{2}Z2​ spin liquid with massless Dirac spinon excitations; on the other side of the critical point, the SU(2) spin liquid (the `$\ensuremath{\pi}$-flux' phase) is presumed to be unstable to confinement to the N\'eel phase. We identify a dangerously irrelevant coupling in the critical SU(2) gauge theory, which contributes a logarithm-squared renormalization. This critical theory is also not Lorentz invariant and weakly breaks the SO(5) symmetry which rotates between the N\'eel and valence bond solid order parameters. We also propose a distinct deconfined critical U(1) gauge theory for a transition into the same gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid; on the other side of the critical point, the U(1) spin liquid (the `staggered flux' phase) is presumed to be unstable to confinement to the valence bond solid. This critical theory has no dangerously irrelevant coupling, dynamic critical exponent $z\ensuremath{\ne}1$, and no SO(5) symmetry. All of these phases and critical points are unified in a SU(2) gauge theory with Higgs fields and fermionic spinons which can naturally realize the observed sequence of phases with increasing J2/J1{J}_{2}/{J}_{1}J2​/J1​: N\'eel, gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid, and valence bond solid.