arXiv:2109.10051 · Physical review. B./Physical review. B

Sigma-model analysis of SU(3) antiferromagnetic spins on the triangular lattice

Itsuki Takahashi, Yuya Tanizaki

Published Dec 27, 2021• 2 citations

Abstract

Using field-theoretic techniques, we study the $\mathrm{SU}(3)$ analog of antiferromagnetic Heisenberg spin model on the triangular lattice putting the $p$-box symmetric representation on each site. Taking the large-$p$ limit, we show that the low-energy effective theory is described by a $(2+1)$-dimensional relativistic $\mathrm{SU}(3)/\mathrm{U}{(1)}^{2}$ nonlinear sigma model. Since the target space has a nontrivial homotopy ${\ensuremath{\pi}}_{2}(\mathrm{SU}(3)/\mathrm{U}{(1)}^{2})\ensuremath{\simeq}{\mathbb{Z}}^{2}$, this model has two kinds of magnetic skyrmions, which can be created and annihilated by monopole instantons. By careful analysis of the Wess-Zumino term in the spin coherent path integral, we compute the Berry phase for these monopoles and it produces the destructive interference. This restricts possible perturbations of the effective Lagrangian by monopole operators, and we see that the valence-bond-solid (VBS) phase should have degenerate ground states when $p\ensuremath{\notin}3\mathbb{Z}$. We also compute 't Hooft anomalies to constrain possible phases of this system, and a direct phase transition between N\'eel and VBS phases is supported from the anomaly matching.

DOI: 10.1103/physrevb.104.235152

Related papers

Dimer description of the SU(4) antiferromagnet on the triangular lattice

Anna Keselman, L. Savary, Leon Balents

arXiv:1911.03492SciPost PhysicsMay 13, 20209 citations

Phase diagram of the SU($N$) antiferromagnet of spin $S$ on a square lattice

Jonas Schwab, Francesco Parisen Toldin, Fakher F. Assaad

arXiv:2304.07329Apr 14, 20230 citations

Quantum Electrodynamics in 2+1 Dimensions as the Organizing Principle of a Triangular Lattice Antiferromagnet

Alexander Wietek, Sylvain Capponi, Andreas Martin Läuchli

arXiv:2303.01585Physical Review XApr 15, 202438 citations

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

Sasank Budaraju, Alberto Parola, Yasir Iqbal +2 more

arXiv:2410.18747Physical review. B./Physical review. BMar 21, 202512 citations

Sign-problem-free effective models of triangular lattice quantum antiferromagnets

Henry Shackleton, Subir Sachdev A

arXiv:2311.01572Physical review. B./Physical review. BFeb 3, 20252 citations

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

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

arXiv:2307.12295Jul 23, 20233 citations

arXiv:2109.10051 · Physical review. B./Physical review. B

Sigma-model analysis of SU(3) antiferromagnetic spins on the triangular lattice

Itsuki Takahashi, Yuya Tanizaki

Published Dec 27, 2021• 2 citations

arXiv HTML PDF

Abstract

Using field-theoretic techniques, we study the

\mathrm{SU}(3)

analog of antiferromagnetic Heisenberg spin model on the triangular lattice putting the

p

-box symmetric representation on each site. Taking the large-

p

limit, we show that the low-energy effective theory is described by a

(2+1)

-dimensional relativistic

\mathrm{SU}(3)/\mathrm{U}{(1)}^{2}

nonlinear sigma model. Since the target space has a nontrivial homotopy

{\ensuremath{\pi}}_{2}(\mathrm{SU}(3)/\mathrm{U}{(1)}^{2})\ensuremath{\simeq}{\mathbb{Z}}^{2}

, this model has two kinds of magnetic skyrmions, which can be created and annihilated by monopole instantons. By careful analysis of the Wess-Zumino term in the spin coherent path integral, we compute the Berry phase for these monopoles and it produces the destructive interference. This restricts possible perturbations of the effective Lagrangian by monopole operators, and we see that the valence-bond-solid (VBS) phase should have degenerate ground states when

p\ensuremath{\notin}3\mathbb{Z}

. We also compute 't Hooft anomalies to constrain possible phases of this system, and a direct phase transition between N'eel and VBS phases is supported from the anomaly matching.

DOI: 10.1103/physrevb.104.235152

Related papers

Dimer description of the SU(4) antiferromagnet on the triangular lattice

Anna Keselman, L. Savary, Leon Balents

In systems with many local degrees of freedom, high-symmetry points in the phase diagram can provide an important starting point for the investigation of their properties throughout the phase diagram. In systems with both spin and orbital (or valley) degrees of freedom such a starting point gives rise to SU(4)-symmetric models. Here we consider SU(4)-symmetric "spin'' models, corresponding to Mott phases at half-filling, i.e. the six-dimensional representation of SU(4). This may be relevant to twisted multilayer graphene. In particular, we study the SU(4) antiferromagnetic "Heisenberg'' model on the triangular lattice, both in the classical limit and in the quantum regime. Carrying out a numerical study using the density matrix renormalization group (DMRG), we argue that the ground state is non-magnetic. We then derive a dimer expansion of the SU(4) spin model. An exact diagonalization (ED) study of the effective dimer model suggests that the ground state breaks translation invariance, forming a valence bond solid (VBS) with a 12-site unit cell. Finally, we consider the effect of SU(4)-symmetry breaking interactions due to Hund's coupling, and argue for a possible phase transition between a VBS and a magnetically ordered state.In systems with many local degrees of freedom, high-symmetry points in the phase diagram can provide an important starting point for the investigation of their properties throughout the phase diagram. In systems with both spin and orbital (or valley) degrees of freedom such a starting point gives rise to SU(4)-symmetric models. Here we consider SU(4)-symmetric "spin'' models, corresponding to Mott phases at half-filling, i.e. the six-dimensional representation of SU(4). This may be relevant to twisted multilayer graphene. In particular, we study the SU(4) antiferromagnetic "Heisenberg'' model on the triangular lattice, both in the classical limit and in the quantum regime. Carrying out a numerical study using the density matrix renormalization group (DMRG), we argue that the ground state is non-magnetic. We then derive a dimer expansion of the SU(4) spin model. An exact diagonalization (ED) study of the effective dimer model suggests that the ground state breaks translation invariance, forming a valence bond solid (VBS) with a 12-site unit cell. Finally, we consider the effect of SU(4)-symmetry breaking interactions due to Hund's coupling, and argue for a possible phase transition between a VBS and a magnetically ordered state.

arXiv:1911.03492SciPost PhysicsMay 13, 20209 citations

Phase diagram of the SU( $N$ ) antiferromagnet of spin $S$ on a square latticePhase diagram of the SU($N$) antiferromagnet of spin $S$ on a square lattice

Jonas Schwab, Francesco Parisen Toldin, Fakher F. Assaad

We investigate the ground state phase diagram of an SU( $N$ )-symmetric antiferromagnetic spin model on a square lattice where each site hosts an irreducible representation of SU( $N$ ) described by a square Young tableau of $N/2$ rows and $2S$ columns. We show that negative sign free fermion Monte Carlo simulations can be carried out for this class of quantum magnets at any $S$ and even values of $N$ . In the large- $N$ limit, the saddle point approximation favors a four-fold degenerate valence bond solid phase. In the large $S$ -limit, the semi-classical approximation points to Néel state. On a line set by $N=8S + 2$ in the $S$ versus $N$ phase diagram, we observe a variety of phases proximate to the Néel state. At $S = 1/2$ and $3/2$ we observe the aforementioned four fold degenerate valence bond solid state. At $S=1$ a two fold degenerate spin nematic state in which the C $_4$ lattice symmetry is broken down to C $_2$ emerges. Finally at $S=2$ we observe a unique ground state that pertains to a two-dimensional version of the Affleck-Kennedy-Lieb-Tasaki state. For our specific realization, this symmetry protected topological state is characterized by an SU(18), $S=1/2$ boundary state, that has a dimerized ground state. These phases that are proximate to the Néel state are consistent with the notion of monopole condensation of the antiferromagnetic order parameter. In particular one expects spin disordered states with degeneracy set by $\text{mod}(4,2S)$ .We investigate the ground state phase diagram of an SU($N$)-symmetric antiferromagnetic spin model on a square lattice where each site hosts an irreducible representation of SU($N$) described by a square Young tableau of $N/2$ rows and $2S$ columns. We show that negative sign free fermion Monte Carlo simulations can be carried out for this class of quantum magnets at any $S$ and even values of $N$. In the large-$N$ limit, the saddle point approximation favors a four-fold degenerate valence bond solid phase. In the large $S$-limit, the semi-classical approximation points to Néel state. On a line set by $N=8S + 2$ in the $S$ versus $N$ phase diagram, we observe a variety of phases proximate to the Néel state. At $S = 1/2$ and $3/2$ we observe the aforementioned four fold degenerate valence bond solid state. At $S=1$ a two fold degenerate spin nematic state in which the C$_4$ lattice symmetry is broken down to C$_2$ emerges. Finally at $S=2$ we observe a unique ground state that pertains to a two-dimensional version of the Affleck-Kennedy-Lieb-Tasaki state. For our specific realization, this symmetry protected topological state is characterized by an SU(18), $S=1/2$ boundary state, that has a dimerized ground state. These phases that are proximate to the Néel state are consistent with the notion of monopole condensation of the antiferromagnetic order parameter. In particular one expects spin disordered states with degeneracy set by $\text{mod}(4,2S)$.

arXiv:2304.07329Apr 14, 20230 citations

Quantum Electrodynamics in 2+1 Dimensions as the Organizing Principle of a Triangular Lattice Antiferromagnet

Alexander Wietek, Sylvain Capponi, Andreas Martin Läuchli

Quantum electrodynamics in 2+1 dimensions ( QED3 ) has been proposed as a critical field theory describing the low-energy effective theory of a putative algebraic Dirac spin liquid or of quantum phase transitions in two-dimensional frustrated magnets. We provide compelling evidence that the intricate spectrum of excitations of the elementary but strongly frustrated J1−J2 Heisenberg model on the triangular lattice is in one-to-one correspondence to a zoo of excitations from QED3 , in the quantum spin liquid regime. This evidence includes a large manifold of explicitly constructed monopole and bilinear excitations of QED3 , which is thus shown to serve as an organizing principle of phases of matter in triangular lattice antiferromagnets and their low-lying excitations. Moreover, we observe signatures of emergent valence-bond solid (VBS) correlations, which can be interpreted either as evidence of critical VBS fluctuations of an emergent Dirac spin liquid or as a transition from the 120° Néel order to a VBS whose quantum critical point is described by QED3 . Our results are obtained by comparing ansatz wave functions from a parton construction to exact eigenstates obtained using large-scale exact diagonalization up to N=48 sites. Published by the American Physical Society 2024Quantum electrodynamics in 2+1 dimensions ( QED3 ) has been proposed as a critical field theory describing the low-energy effective theory of a putative algebraic Dirac spin liquid or of quantum phase transitions in two-dimensional frustrated magnets. We provide compelling evidence that the intricate spectrum of excitations of the elementary but strongly frustrated J1−J2 Heisenberg model on the triangular lattice is in one-to-one correspondence to a zoo of excitations from QED3 , in the quantum spin liquid regime. This evidence includes a large manifold of explicitly constructed monopole and bilinear excitations of QED3 , which is thus shown to serve as an organizing principle of phases of matter in triangular lattice antiferromagnets and their low-lying excitations. Moreover, we observe signatures of emergent valence-bond solid (VBS) correlations, which can be interpreted either as evidence of critical VBS fluctuations of an emergent Dirac spin liquid or as a transition from the 120° Néel order to a VBS whose quantum critical point is described by QED3 . Our results are obtained by comparing ansatz wave functions from a parton construction to exact eigenstates obtained using large-scale exact diagonalization up to N=48 sites. Published by the American Physical Society 2024

arXiv:2303.01585Physical Review XApr 15, 202438 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.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:2410.18747Physical review. B./Physical review. BMar 21, 202512 citations

Sign-problem-free effective models of triangular lattice quantum antiferromagnets

Henry Shackleton, Subir Sachdev A

The triangular lattice antiferromagnet with $S=1/2$ spins and nearest-neighbor interactions is known to have long-range antiferromagnetic order, with nearest-neighbor spins at an angle of $120{}^{\ensuremath{\circ}}$. Numerical studies of quantum phases proximate to this state have been limited to small systems because of the sign problem in Monte Carlo simulations in imaginary time. We propose an effective lattice model for quantum fluctuations of the antiferromagnetic order and a sign-problem-free Monte Carlo algorithm enabling studies in large systems sizes. The model is a ${\mathbb{Z}}_{2}$ gauge theory coupled to gauge-charged scalars which have a relativistic dispersion in the continuum limit. Crucially, the gauge theory is odd, i.e., there is a static, background ${\mathbb{Z}}_{2}$ gauge charge on each site, accounting for the Berry phases of the half-odd-integer spins on each site. We present results of simulations on lattices of sizes up to $36\ifmmode\times\else\texttimes\fi{}36\ifmmode\times\else\texttimes\fi{}36$. Along with the antiferromagnetically ordered phase, our phase diagram has a valence bond solid state with a $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ unit cell and a gapped ${\mathbb{Z}}_{2}$ spin liquid. Deconfined critical points or phases in intermediate regions are not ruled out by our present simulations.The triangular lattice antiferromagnet with $S=1/2$ spins and nearest-neighbor interactions is known to have long-range antiferromagnetic order, with nearest-neighbor spins at an angle of $120{}^{\ensuremath{\circ}}$. Numerical studies of quantum phases proximate to this state have been limited to small systems because of the sign problem in Monte Carlo simulations in imaginary time. We propose an effective lattice model for quantum fluctuations of the antiferromagnetic order and a sign-problem-free Monte Carlo algorithm enabling studies in large systems sizes. The model is a ${\mathbb{Z}}_{2}$ gauge theory coupled to gauge-charged scalars which have a relativistic dispersion in the continuum limit. Crucially, the gauge theory is odd, i.e., there is a static, background ${\mathbb{Z}}_{2}$ gauge charge on each site, accounting for the Berry phases of the half-odd-integer spins on each site. We present results of simulations on lattices of sizes up to $36\ifmmode\times\else\texttimes\fi{}36\ifmmode\times\else\texttimes\fi{}36$. Along with the antiferromagnetically ordered phase, our phase diagram has a valence bond solid state with a $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ unit cell and a gapped ${\mathbb{Z}}_{2}$ spin liquid. Deconfined critical points or phases in intermediate regions are not ruled out by our present simulations.

arXiv:2311.01572Physical review. B./Physical review. BFeb 3, 20252 citations

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.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:2307.12295Jul 23, 20233 citations

Using field-theoretic techniques, we study the

\mathrm{SU}(3)

analog of antiferromagnetic Heisenberg spin model on the triangular lattice putting the

p

-box symmetric representation on each site. Taking the large-

p

limit, we show that the low-energy effective theory is described by a

(2+1)

-dimensional relativistic

\mathrm{SU}(3)/\mathrm{U}{(1)}^{2}

nonlinear sigma model. Since the target space has a nontrivial homotopy

{\ensuremath{\pi}}_{2}(\mathrm{SU}(3)/\mathrm{U}{(1)}^{2})\ensuremath{\simeq}{\mathbb{Z}}^{2}

p\ensuremath{\notin}3\mathbb{Z}

. We also compute 't Hooft anomalies to constrain possible phases of this system, and a direct phase transition between N'eel and VBS phases is supported from the anomaly matching.

Phase diagram of the SU(

N

) antiferromagnet of spin

S

on a square lattice

We investigate the ground state phase diagram of an SU(

N

)-symmetric antiferromagnetic spin model on a square lattice where each site hosts an irreducible representation of SU(

N

) described by a square Young tableau of

N/2

rows and

2S

columns. We show that negative sign free fermion Monte Carlo simulations can be carried out for this class of quantum magnets at any

S

and even values of

N

. In the large-

N

limit, the saddle point approximation favors a four-fold degenerate valence bond solid phase. In the large

S

-limit, the semi-classical approximation points to Néel state. On a line set by

N=8S + 2

in the

S

versus

N

phase diagram, we observe a variety of phases proximate to the Néel state. At

S = 1/2

and

3/2

we observe the aforementioned four fold degenerate valence bond solid state. At

S=1

a two fold degenerate spin nematic state in which the C

_4

lattice symmetry is broken down to C

_2

emerges. Finally at

S=2

we observe a unique ground state that pertains to a two-dimensional version of the Affleck-Kennedy-Lieb-Tasaki state. For our specific realization, this symmetry protected topological state is characterized by an SU(18),

S=1/2

boundary state, that has a dimerized ground state. These phases that are proximate to the Néel state are consistent with the notion of monopole condensation of the antiferromagnetic order parameter. In particular one expects spin disordered states with degeneracy set by

\text{mod}(4,2S)

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.

The triangular lattice antiferromagnet with

S=1/2

spins and nearest-neighbor interactions is known to have long-range antiferromagnetic order, with nearest-neighbor spins at an angle of $120{}^{\ensuremath{\circ}}$. Numerical studies of quantum phases proximate to this state have been limited to small systems because of the sign problem in Monte Carlo simulations in imaginary time. We propose an effective lattice model for quantum fluctuations of the antiferromagnetic order and a sign-problem-free Monte Carlo algorithm enabling studies in large systems sizes. The model is a

{\mathbb{Z}}_{2}

gauge theory coupled to gauge-charged scalars which have a relativistic dispersion in the continuum limit. Crucially, the gauge theory is odd, i.e., there is a static, background

{\mathbb{Z}}_{2}

gauge charge on each site, accounting for the Berry phases of the half-odd-integer spins on each site. We present results of simulations on lattices of sizes up to $36\ifmmode\times\else\texttimes\fi{}36\ifmmode\times\else\texttimes\fi{}36$. Along with the antiferromagnetically ordered phase, our phase diagram has a valence bond solid state with a $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ unit cell and a gapped

{\mathbb{Z}}_{2}

spin liquid. Deconfined critical points or phases in intermediate regions are not ruled out by our present simulations.

_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.