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arXiv:2410.18747 · Physical review. B./Physical review. B

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

Sasank Budaraju, Alberto Parola, Yasir Iqbal, Federico Becca, Didier Poilblanc

Published Mar 21, 2025• 12 citations
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Abstract

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.

DOI
10.1103/physrevb.111.125150

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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 \ensuremathπ\ensuremath{\pi}\ensuremathπ 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 J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ Heisenberg model on the triangular lattice.
The parton approach for quantum spin liquids gives a transparent description of low-energy elementary excitations, e.g., spinons and emergent gauge-field fluctuations. The latter ones are directly coupled to the hopping/pairing of spinons. By using the fermionic representation of the U(1)U(1)U(1) Dirac state on the kagome lattice and variational Monte Carlo techniques to include the Gutzwiller projection, we analyze the effect of modifying the gauge fields in the spinon kinematics. In particular, we construct low-energy monopole excitations, which are shown to be gapless in the thermodynamic limit. States with a finite number of monopoles or with a finite density of them are also considered, with different patterns of the gauge fluxes. We show that these chiral states are not stabilized in the Heisenberg model with nearest-neighbor superexchange couplings, and the Dirac state corresponds to the lowest-energy ansatz within this family of variational wave functions. Our results support the idea that spinons with a gapless conical spectrum coexist with gapless monopole excitations, even for the spin-1/2 case.
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.
It has been recently demonstrated [Song et al., Nature Comm. 10, 4254 (2019), Iqbal et al., Phys. Rev. B 93, 144411 (2016)] that the staggered π\piπ-flux Dirac spin liquid phase on the non-bipartite triangular lattice may be stable in the clean limit. However, quenched disorder plays a crucial role in determining whether such a phase can be experimentally viable. The effective low-energy description of Dirac spin liquids in (2+1)(2+1)(2+1) dimensions is given by the compact quantum electrodynamics (cQED2+1\rm cQED_{2+1}cQED2+1​) which admits monopoles. It is already known that generic quenched random perturbations to the non-compact version of QED2+1\rm QED_{2+1}QED2+1​ (where monopoles are absent) lead to strong-coupling instabilities. In this work we study cQED2+1\rm 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 under-screening of monopoles thus generically makes the gapless spin liquid phase fragile.
Understanding the nature of the excitation spectrum in quantum spin liquids is of fundamental importance, in particular for the experimental detection of candidate materials.However, current theoretical and numerical techniques have limited capabilities, especially in obtaining the dynamical structure factor, which gives a crucial characterization of the ultimate nature of the quantum state and may be directly assessed by inelastic neutron scattering.In this work, we investigate the low-energy properties of the S ¼ 1=2 Heisenberg model on the triangular lattice, including both nearest-neighbor J 1 and next-nearestneighbor J 2 superexchanges, by a dynamical variational Monte Carlo approach that allows accurate results on spin models.For J 2 ¼ 0, our calculations are compatible with the existence of a well-defined magnon in the whole Brillouin zone, with gapless excitations at K points (i.e., at the corners of the Brillouin zone).The strong renormalization of the magnon branch (also including rotonlike minima around the M points, i.e., midpoints of the border zone) is described by our Gutzwiller-projected state, where Abrikosov fermions are subject to a nontrivial magnetic π flux threading half of the triangular plaquettes.When increasing the frustrating ratio J 2 =J 1 , we detect a progressive softening of the magnon branch at M, which eventually becomes gapless within the spin-liquid phase.This feature is captured by the band structure of the unprojected wave function (with two Dirac points for each spin component).In addition, we observe an intense signal at low energies around the K points, which cannot be understood within the unprojected picture and emerges only when the Gutzwiller projection is considered, suggesting the relevance of gauge fields for the low-energy physics of spin liquids.
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.
We present the dynamical spin structure factor of the antiferromagnetic spin-$\frac{1}{2} {J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on a triangular lattice obtained from large-scale matrix-product state simulations. The high frustration due to the combination of antiferromagnetic nearest- and next-nearest-neighbor interactions yields a rich phase diagram. We resolve the low-energy excitations both in the ${120}^{\ensuremath{\circ}}$ ordered phase and in the putative spin-liquid phase at J2/J1=0.125{J}_{2}/{J}_{1}=0.125J2​/J1​=0.125. In the ordered phase, we observe an avoided decay of the lowest magnon branch, demonstrating the robustness of this phenomenon in the presence of gapless excitations. Our findings in the spin-liquid phase chime with the field-theoretical predictions for a gapless Dirac spin liquid, in particular the picture of low-lying monopole excitations at the corners of the Brillouin zone. We comment on possible practical difficulties of distinguishing proximate liquid and solid phases based on the dynamical structure factor.