arXiv:2411.00301

Spin excitations of the Shastry-Sutherland model -- altermagnetism and deconfined quantum criticality

Hongyu Chen, Guijing Duan, Changle Liu, Cui, Yi, Yu, Weiqiang, Zihan Xie, Rong Yu

Published Nov 1, 2024• 0 citations

Abstract

Frustrated quantum magnets can host a variety of exotic spin excitations, including fractionalized spin excitations coupled to emergent gauge fields at deconfined quantum critical points (DQCPs) and chiral magnons in altermagnets. Here, we investigate the spin excitation spectra of the highly frustrated $S=1/2$ antiferromagnetic (AFM) Shastry-Sutherland model, focusing on the evolution of low-energy collective modes from the Néel AFM phase to the plaquette valence bond solid (PVBS). We demonstrate that the AFM state exhibits altermagnetic behavior, characterized by a non-relativistic splitting between two chiral magnon bands. Furthermore, we identify two additional low-energy excitations: a Higgs mode in the longitudinal excitation channel and an $S=0$ excitation with vanishing spectral weight. As the system approaches the AFM-to-PVBS transition, both these modes soften along with the lowest-energy triplet and singlet modes in the PVBS state. The closing gap of the Higgs mode, combined with the nearly degenerate velocities of $S=1$ and $S=0$ excitations, provides spectral evidence that the AFM-to-PVBS transition is proximate to a DQCP with emergent $O(4)$ symmetry. Our results help clarify the spectral signature of a broad class of symmetry enhanced quantum phase transitions including deconfined quantum criticality.

DOI: 10.48550/arxiv.2411.00301

Related papers

Plaquette valence bond solid to antiferromagnet transition and deconfined quantum critical point of the Shastry-Sutherland model

Ning Xi, Hongyu Chen, Z. Y. Xie +1 more

arXiv:2111.07368Physical review. B./Physical review. BJun 29, 202332 citations

Quantum Criticality with Emergent Symmetry in the Extended Shastry-Sutherland Model

Wen-Yuan Liu, Xiaotian Zhang, Zhe Wang +3 more

arXiv:2309.10955Physical Review LettersJul 9, 202417 citations

On the precise boundary of the Néel antiferromagnetic phase in the Shastry-Sutherland model

Rongyi Lv, Xiangjian Qian, Mingpu Qin

arXiv:2609.22907Sep 19, 20260 citations

Deconfined Quantum Critical Point: A Review of Progress

Yi Cui, Rong Yu, Weiqiang Yu

arXiv:2504.10027Chinese Physics LettersMar 19, 202511 citations

Two Plaquette-Singlet Phases and Emergent SO(5) Deconfined Quantum Criticality in SrCu2(BO3)2

Cui, Yi, Kefan Du, Zhanlong Wu +15 more

arXiv:2411.00302Nov 1, 20240 citations

Antiferromagnetism, Spin Liquid, and Plaquette Singlet Phases in the Shastry–Sutherland Model: A Variational Cluster Approximation Approach

Jean Paul Latyr Faye

Jan 1, 20260 citations

arXiv:2411.00301

Spin excitations of the Shastry-Sutherland model -- altermagnetism and deconfined quantum criticality

Hongyu Chen, Guijing Duan, Changle Liu, Cui, Yi, Yu, Weiqiang, Zihan Xie, Rong Yu

Published Nov 1, 2024• 0 citations

arXiv HTML PDF

Abstract

S=1/2

antiferromagnetic (AFM) Shastry-Sutherland model, focusing on the evolution of low-energy collective modes from the Néel AFM phase to the plaquette valence bond solid (PVBS). We demonstrate that the AFM state exhibits altermagnetic behavior, characterized by a non-relativistic splitting between two chiral magnon bands. Furthermore, we identify two additional low-energy excitations: a Higgs mode in the longitudinal excitation channel and an

S=0

excitation with vanishing spectral weight. As the system approaches the AFM-to-PVBS transition, both these modes soften along with the lowest-energy triplet and singlet modes in the PVBS state. The closing gap of the Higgs mode, combined with the nearly degenerate velocities of

S=1

and

S=0

excitations, provides spectral evidence that the AFM-to-PVBS transition is proximate to a DQCP with emergent

O(4)

symmetry. Our results help clarify the spectral signature of a broad class of symmetry enhanced quantum phase transitions including deconfined quantum criticality.

DOI: 10.48550/arxiv.2411.00301

Related papers

Plaquette valence bond solid to antiferromagnet transition and deconfined quantum critical point of the Shastry-Sutherland model

Ning Xi, Hongyu Chen, Z. Y. Xie +1 more

We study the ground-state phase diagram of the Shastry-Sutherland model by using a variational optimization of the infinite tensor network states, and identify a weakly first-order transition between the plaquette valence bond solid and the antiferromagnetic states. The full plaquette state is found to strongly compete with the empty plaquette ground state, and can be stabilized as the ground state when a staggered ring-exchange interaction preserving the Shastry-Sutherland lattice symmetry is introduced. We propose the triple point where the full plaquette, empty plaquette, and antiferromagnetic phases meet as a deconfined quantum critical point (DQCP). The analysis of susceptibilities provides evidence of an emergent $\text{SO}(5)$ symmetry at this point. These results shed light on the study of DQCP in quantum magnets and provide a way to understand the proximate DQCP signatures in recent experiments on ${\mathrm{SrCu}}_{2}{({\mathrm{BO}}_{3})}_{2}$ .We study the ground-state phase diagram of the Shastry-Sutherland model by using a variational optimization of the infinite tensor network states, and identify a weakly first-order transition between the plaquette valence bond solid and the antiferromagnetic states. The full plaquette state is found to strongly compete with the empty plaquette ground state, and can be stabilized as the ground state when a staggered ring-exchange interaction preserving the Shastry-Sutherland lattice symmetry is introduced. We propose the triple point where the full plaquette, empty plaquette, and antiferromagnetic phases meet as a deconfined quantum critical point (DQCP). The analysis of susceptibilities provides evidence of an emergent $\text{SO}(5)$ symmetry at this point. These results shed light on the study of DQCP in quantum magnets and provide a way to understand the proximate DQCP signatures in recent experiments on ${\mathrm{SrCu}}_{2}{({\mathrm{BO}}_{3})}_{2}$.

arXiv:2111.07368Physical review. B./Physical review. BJun 29, 202332 citations

Quantum Criticality with Emergent Symmetry in the Extended Shastry-Sutherland Model

Wen-Yuan Liu, Xiaotian Zhang, Zhe Wang +3 more

Motivated by the novel phenomena observed in the layered material SrCu_{2}(BO_{3})_{2}, the Shastry-Sutherland model (SSM) has been extensively studied as the minimal model for SrCu_{2}(BO_{3})_{2}. However, the nature of its quantum phase transition from the plaquette valence-bond solid to antiferromagnetic phase is under fierce debate, posing a challenge to understand the underlying quantum criticality. Via the state-of-the-art tensor network simulations, we study the ground state of the SSM on large-scale size up to 20×20 sites. We identify the continuous transition nature accompanied by an emergent O(4) symmetry between the plaquette valence-bond solid and antiferromagnetic phase, which strongly suggests a deconfined quantum critical point (DQCP). Furthermore, we map out the phase diagram of an extended SSM that can be continuously tuned to the SSM, which demonstrates the same DQCP phenomena along a whole critical line. Our results indicate a compelling scenario for understanding the origin of the proposed proximate DQCP in recent experiments of SrCu_{2}(BO_{3})_{2}.Motivated by the novel phenomena observed in the layered material SrCu_{2}(BO_{3})_{2}, the Shastry-Sutherland model (SSM) has been extensively studied as the minimal model for SrCu_{2}(BO_{3})_{2}. However, the nature of its quantum phase transition from the plaquette valence-bond solid to antiferromagnetic phase is under fierce debate, posing a challenge to understand the underlying quantum criticality. Via the state-of-the-art tensor network simulations, we study the ground state of the SSM on large-scale size up to 20×20 sites. We identify the continuous transition nature accompanied by an emergent O(4) symmetry between the plaquette valence-bond solid and antiferromagnetic phase, which strongly suggests a deconfined quantum critical point (DQCP). Furthermore, we map out the phase diagram of an extended SSM that can be continuously tuned to the SSM, which demonstrates the same DQCP phenomena along a whole critical line. Our results indicate a compelling scenario for understanding the origin of the proposed proximate DQCP in recent experiments of SrCu_{2}(BO_{3})_{2}.

arXiv:2309.10955Physical Review LettersJul 9, 202417 citations

On the precise boundary of the Néel antiferromagnetic phase in the Shastry-Sutherland model

Rongyi Lv, Xiangjian Qian, Mingpu Qin

The ground state phase diagram of the Shastry-Sutherland model remains controversial. In particular, the nature of the transition between the plaquette valence bond solid (PVBS) phase and the Néel antiferromagnetic (AFM) phase is still under debate. While some studies suggest a direct transition between the two phases, others propose the existence of a narrow intermediate quantum spin liquid regime. To address this issue, we perform large-scale Density Matrix Renormalization Group calculations to determine the Néel AFM phase boundary in the Shastry-Sutherland model. By applying staggered boundary pinning fields and directly measuring the bulk staggered magnetization, and through systematic extrapolations with respect to truncation error as well as careful finite-size scaling, we find that the staggered magnetization remains finite across the controversial region $0.78 \lesssim J_{1}/J_{2} \leq 0.81$ , which was suggested to host a quantum spin liquid phase in some studies. Together with the previously established consensus placing the PVBS phase boundary at $J_{1}/J_{2} \approx 0.78$ , these results provide evidence against the existence of an intermediate quantum spin liquid phase and support a direct PVBS--Néel AFM transition in the Shastry-Sutherland model.The ground state phase diagram of the Shastry-Sutherland model remains controversial. In particular, the nature of the transition between the plaquette valence bond solid (PVBS) phase and the Néel antiferromagnetic (AFM) phase is still under debate. While some studies suggest a direct transition between the two phases, others propose the existence of a narrow intermediate quantum spin liquid regime. To address this issue, we perform large-scale Density Matrix Renormalization Group calculations to determine the Néel AFM phase boundary in the Shastry-Sutherland model. By applying staggered boundary pinning fields and directly measuring the bulk staggered magnetization, and through systematic extrapolations with respect to truncation error as well as careful finite-size scaling, we find that the staggered magnetization remains finite across the controversial region $0.78 \lesssim J_{1}/J_{2} \leq 0.81$, which was suggested to host a quantum spin liquid phase in some studies. Together with the previously established consensus placing the PVBS phase boundary at $J_{1}/J_{2} \approx 0.78$, these results provide evidence against the existence of an intermediate quantum spin liquid phase and support a direct PVBS--Néel AFM transition in the Shastry-Sutherland model.

arXiv:2609.22907Sep 19, 20260 citations

Deconfined Quantum Critical Point: A Review of Progress

Yi Cui, Rong Yu, Weiqiang Yu

Abstract Deconfined quantum critical points (DQCPs) have been proposed as a class of continuous quantum phase transitions occurring between two ordered phases with distinct symmetry-breaking patterns, beyond the conventional framework of Landau-Ginzburg-Wilson (LGW) theory. At the DQCP, the system exhibits emergent gauge fields, fractionalized excitations, and enhanced symmetries. Here we review recent theoretical and experimental progress on exploring DQCPs in condensed matter systems. We first introduce theoretical advancements in the study of DQCPs over the past twenty years, particularly in magnetic models on square lattices, honeycomb lattices, kagome lattices, and one-dimensional spin chains. We then discuss recent progress on experimental realization of DQCP in quantum magnetic systems. Experimentally, the Shastry-Sutherland model, realized in SrCu2(BO3)2, offers a particularly promising platform for realizing DQCPs. The magnetic frustration inherent to this model drives phase transitions between two distinct symmetry-breaking states: a valence bond solid (VBS) phase and a Néel antiferromagnetic phase. Remarkably, SrCu2(BO3)2 has provided the first experimental evidence of a proximate DQCP through a field-induced Bose-Einstein condensation, transitioning from the VBS state to the Néel state. Nevertheless, the direct experimental realization of a DQCP remains a significant challenge. Despite this, it offers a promising platform for exploring emergent phenomena through quantum phase transition in low-dimensional quantum systems.Abstract Deconfined quantum critical points (DQCPs) have been proposed as a class of continuous quantum phase transitions occurring between two ordered phases with distinct symmetry-breaking patterns, beyond the conventional framework of Landau-Ginzburg-Wilson (LGW) theory. At the DQCP, the system exhibits emergent gauge fields, fractionalized excitations, and enhanced symmetries. Here we review recent theoretical and experimental progress on exploring DQCPs in condensed matter systems. We first introduce theoretical advancements in the study of DQCPs over the past twenty years, particularly in magnetic models on square lattices, honeycomb lattices, kagome lattices, and one-dimensional spin chains. We then discuss recent progress on experimental realization of DQCP in quantum magnetic systems. Experimentally, the Shastry-Sutherland model, realized in SrCu2(BO3)2, offers a particularly promising platform for realizing DQCPs. The magnetic frustration inherent to this model drives phase transitions between two distinct symmetry-breaking states: a valence bond solid (VBS) phase and a Néel antiferromagnetic phase. Remarkably, SrCu2(BO3)2 has provided the first experimental evidence of a proximate DQCP through a field-induced Bose-Einstein condensation, transitioning from the VBS state to the Néel state. Nevertheless, the direct experimental realization of a DQCP remains a significant challenge. Despite this, it offers a promising platform for exploring emergent phenomena through quantum phase transition in low-dimensional quantum systems.

arXiv:2504.10027Chinese Physics LettersMar 19, 202511 citations

Two Plaquette-Singlet Phases and Emergent SO(5) Deconfined Quantum Criticality in SrCu2(BO3)2

Cui, Yi, Kefan Du, Zhanlong Wu +15 more

The deconfined quantum critical point (DQCP) has become a central open concept in the physics of quantum matter, and its proposed presence in the Shastry-Sutherland model was followed by the experimental observation of at least a minimal DQC scenario induced by an applied magnetic field in SrCu $_2$ (BO $_3$ ) $_2$ . However, the nature of the plaquette-singlet phase in SrCu $_2$ (BO $_3$ ) $_2$ remains unresolved, and with it the identification of the DQCP symmetry from among several theoretical scenarios. Here we perform detailed high-pressure $^{11}$ B NMR studies to reveal the presence of both the full-plaquette (FP) and empty-plaquette (EP) phases in SrCu $_2$ (BO $_3$ ) $_2$ , phase-separated at a first-order, pressure-driven transition with a volume-fraction effect. The field-driven transition from the EP to the antiferromagnetic (AFM) phase complements our previous observations of the FP--AFM transition, with both showing deconfined quantum criticality, while the scaling of the spin-lattice relaxation rate near the EP--AFM transition, $1/T_1 \propto T^{0.6}$ , suggests a DQCP governed by a different universality class. We discuss possible extensions to the Shastry-Sutherland model that account for these pressure and field effects. The expanded phase space we discover mandates an SO(5) DQCP symmetry, and hence our results take an important step towards a complete understanding of deconfined quantum criticality in SrCu $_2$ (BO $_3$ ) $_2$ .The deconfined quantum critical point (DQCP) has become a central open concept in the physics of quantum matter, and its proposed presence in the Shastry-Sutherland model was followed by the experimental observation of at least a minimal DQC scenario induced by an applied magnetic field in SrCu$_2$(BO$_3$)$_2$. However, the nature of the plaquette-singlet phase in SrCu$_2$(BO$_3$)$_2$ remains unresolved, and with it the identification of the DQCP symmetry from among several theoretical scenarios. Here we perform detailed high-pressure $^{11}$B NMR studies to reveal the presence of both the full-plaquette (FP) and empty-plaquette (EP) phases in SrCu$_2$(BO$_3$)$_2$, phase-separated at a first-order, pressure-driven transition with a volume-fraction effect. The field-driven transition from the EP to the antiferromagnetic (AFM) phase complements our previous observations of the FP--AFM transition, with both showing deconfined quantum criticality, while the scaling of the spin-lattice relaxation rate near the EP--AFM transition, $1/T_1 \propto T^{0.6}$, suggests a DQCP governed by a different universality class. We discuss possible extensions to the Shastry-Sutherland model that account for these pressure and field effects. The expanded phase space we discover mandates an SO(5) DQCP symmetry, and hence our results take an important step towards a complete understanding of deconfined quantum criticality in SrCu$_2$(BO$_3$)$_2$.

arXiv:2411.00302Nov 1, 20240 citations

Antiferromagnetism, Spin Liquid, and Plaquette Singlet Phases in the Shastry–Sutherland Model: A Variational Cluster Approximation Approach

Jean Paul Latyr Faye

We study the half-filled single-band Hubbard model on the Shastry–Sutherland lattice with nearest-neighbor hopping $t$ , incorporating (i) a frustrating diagonal hopping $t'$ to probe the emergence of spin-liquid behavior, and (ii) an additional Heisenberg exchange $J$ that explicitly stabilizes plaquette-singlet correlations. Using the variational cluster approximation (VCA) with exact diagonalization at zero temperature on an eight-site cluster, we map the phase diagrams in the $(U/t,t'/t)$ and $(U/t,J/t)$ parameter planes. In the $(U/t,t'/t)$ plane, our analysis uncovers a competition between N\'eel antiferromagnetic (AFM) order with staggered magnetization and a nonmagnetic regime lacking conventional order, consistent with an algebraic spin-liquid (ASL) phase characterized by a gapped excitation spectrum and vanishing staggered magnetization. In contrast, the $(U/t,J/t)$ phase diagram exhibits a competition between the N\'eel AFM and a solid valence-bonding valence bond plaquette (PS), where spins form localized singlet plaquettes. Most notably, we observe that the transition from the PS phase to the N\'eel AFM is a first-order phase transition, with a discontinuous jump of the order parameters at the phase boundary, in quantitative agreement with high-pressure heat-capacity measurements on SrCu $_2$ (BO $_3$ ) $_2$ [Guo \emph{et al.}, Commun. Phys. 8, 75 (2025)]. Thus, our VCA analysis underscores the subtle interplay of geometric frustration, quantum fluctuations, and dimensionality, and elucidates the crucial role of short-range correlations in stabilizing and competing ground states of frustrated quantum magnets.We study the half-filled single-band Hubbard model on the Shastry–Sutherland lattice with nearest-neighbor hopping $t$, incorporating (i) a frustrating diagonal hopping $t'$ to probe the emergence of spin-liquid behavior, and (ii) an additional Heisenberg exchange $J$ that explicitly stabilizes plaquette-singlet correlations. Using the variational cluster approximation (VCA) with exact diagonalization at zero temperature on an eight-site cluster, we map the phase diagrams in the $(U/t,t'/t)$ and $(U/t,J/t)$ parameter planes. In the $(U/t,t'/t)$ plane, our analysis uncovers a competition between N\'eel antiferromagnetic (AFM) order with staggered magnetization and a nonmagnetic regime lacking conventional order, consistent with an algebraic spin-liquid (ASL) phase characterized by a gapped excitation spectrum and vanishing staggered magnetization. In contrast, the $(U/t,J/t)$ phase diagram exhibits a competition between the N\'eel AFM and a solid valence-bonding valence bond plaquette (PS), where spins form localized singlet plaquettes. Most notably, we observe that the transition from the PS phase to the N\'eel AFM is a first-order phase transition, with a discontinuous jump of the order parameters at the phase boundary, in quantitative agreement with high-pressure heat-capacity measurements on SrCu$_2$(BO$_3$)$_2$ [Guo \emph{et al.}, Commun. Phys. 8, 75 (2025)]. Thus, our VCA analysis underscores the subtle interplay of geometric frustration, quantum fluctuations, and dimensionality, and elucidates the crucial role of short-range correlations in stabilizing and competing ground states of frustrated quantum magnets.

Jan 1, 20260 citations

S=1/2

S=0

S=1

and

S=0

excitations, provides spectral evidence that the AFM-to-PVBS transition is proximate to a DQCP with emergent

O(4)

symmetry. Our results help clarify the spectral signature of a broad class of symmetry enhanced quantum phase transitions including deconfined quantum criticality.

\text{SO}(5)

symmetry at this point. These results shed light on the study of DQCP in quantum magnets and provide a way to understand the proximate DQCP signatures in recent experiments on

{\mathrm{SrCu}}_{2}{({\mathrm{BO}}_{3})}_{2}

0.78 \lesssim J_{1}/J_{2} \leq 0.81

, which was suggested to host a quantum spin liquid phase in some studies. Together with the previously established consensus placing the PVBS phase boundary at

J_{1}/J_{2} \approx 0.78

, these results provide evidence against the existence of an intermediate quantum spin liquid phase and support a direct PVBS--Néel AFM transition in the Shastry-Sutherland model.

_2

(BO

_3

)

_2

. However, the nature of the plaquette-singlet phase in SrCu

_2

(BO

_3

)

_2

remains unresolved, and with it the identification of the DQCP symmetry from among several theoretical scenarios. Here we perform detailed high-pressure

^{11}

B NMR studies to reveal the presence of both the full-plaquette (FP) and empty-plaquette (EP) phases in SrCu

_2

(BO

_3

)

_2

, phase-separated at a first-order, pressure-driven transition with a volume-fraction effect. The field-driven transition from the EP to the antiferromagnetic (AFM) phase complements our previous observations of the FP--AFM transition, with both showing deconfined quantum criticality, while the scaling of the spin-lattice relaxation rate near the EP--AFM transition,

1/T_1 \propto T^{0.6}

, suggests a DQCP governed by a different universality class. We discuss possible extensions to the Shastry-Sutherland model that account for these pressure and field effects. The expanded phase space we discover mandates an SO(5) DQCP symmetry, and hence our results take an important step towards a complete understanding of deconfined quantum criticality in SrCu

_2

(BO

_3

)

_2

We study the half-filled single-band Hubbard model on the Shastry–Sutherland lattice with nearest-neighbor hopping

t

, incorporating (i) a frustrating diagonal hopping

t'

to probe the emergence of spin-liquid behavior, and (ii) an additional Heisenberg exchange

J

that explicitly stabilizes plaquette-singlet correlations. Using the variational cluster approximation (VCA) with exact diagonalization at zero temperature on an eight-site cluster, we map the phase diagrams in the

(U/t,t'/t)

and

(U/t,J/t)

parameter planes. In the

(U/t,t'/t)

plane, our analysis uncovers a competition between N\'eel antiferromagnetic (AFM) order with staggered magnetization and a nonmagnetic regime lacking conventional order, consistent with an algebraic spin-liquid (ASL) phase characterized by a gapped excitation spectrum and vanishing staggered magnetization. In contrast, the

(U/t,J/t)

phase diagram exhibits a competition between the N\'eel AFM and a solid valence-bonding valence bond plaquette (PS), where spins form localized singlet plaquettes. Most notably, we observe that the transition from the PS phase to the N\'eel AFM is a first-order phase transition, with a discontinuous jump of the order parameters at the phase boundary, in quantitative agreement with high-pressure heat-capacity measurements on SrCu

_2

(BO

_3

)

_2

[Guo \emph{et al.}, Commun. Phys. 8, 75 (2025)]. Thus, our VCA analysis underscores the subtle interplay of geometric frustration, quantum fluctuations, and dimensionality, and elucidates the crucial role of short-range correlations in stabilizing and competing ground states of frustrated quantum magnets.