Supersolid phase in a U(2) symmetric S = 1 magnet on the triangular lattice

doi:10.1088/1674-1056/ae3604

中国物理B ›› 2026, Vol. 35 ›› Issue (4): 47506-047506.doi: 10.1088/1674-1056/ae3604

Supersolid phase in a U(2) symmetric S = 1 magnet on the triangular lattice

Si-Cheng Wang(王思成)^1,2,3 and Zheng-Xin Liu(刘正鑫)^1,4,†

1 Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, China;
2 Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China;
3 University of Chinese Academy of Sciences, Beijing 100049, China;
4 Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education), Renmin University of China, Beijing 100872, China

收稿日期:2025-09-24 修回日期:2026-01-07 接受日期:2026-01-09 发布日期:2026-04-17
通讯作者: Zheng-Xin Liu E-mail:liuzxphys@ruc.edu.cn
基金资助:
We thank Qiang Luo, Han-Jin Shen, Rong Yu and Zhi- Yuan Xie for valuable discussions, and thank Yi Zhou and T. K. Ng for previous collaborations in related topics. This work was supported by the National Key Research and Development Program of China (Grant Nos. 2023YFA1406500 and 2022YFA1405300) and the National Natural Science Foundation of China (Grant Nos. 12374166 and 12134020). Computational resources have been provided by the Physical Laboratory of High Performance Computing at Renmin University of China.

Supersolid phase in a U(2) symmetric S = 1 magnet on the triangular lattice

Si-Cheng Wang(王思成)^1,2,3 and Zheng-Xin Liu(刘正鑫)^1,4,†

1 Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, China;
2 Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China;
3 University of Chinese Academy of Sciences, Beijing 100049, China;
4 Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education), Renmin University of China, Beijing 100872, China

Received:2025-09-24 Revised:2026-01-07 Accepted:2026-01-09 Published:2026-04-17
Contact: Zheng-Xin Liu E-mail:liuzxphys@ruc.edu.cn
Supported by:
We thank Qiang Luo, Han-Jin Shen, Rong Yu and Zhi- Yuan Xie for valuable discussions, and thank Yi Zhou and T. K. Ng for previous collaborations in related topics. This work was supported by the National Key Research and Development Program of China (Grant Nos. 2023YFA1406500 and 2022YFA1405300) and the National Natural Science Foundation of China (Grant Nos. 12374166 and 12134020). Computational resources have been provided by the Physical Laboratory of High Performance Computing at Renmin University of China.

摘要/Abstract

摘要： A spin supersolid is characterized by the simultaneous breaking of lattice translation and continuous spin rotation symmetries. In this work, we study a spin-1 model with $U(2)\cong SU(2)\times U(1)/Z_2$ symmetry on the triangular lattice, and determine the phase diagram with a variational $\mathbb CP^2$ approach. We identify a novel supersolid phase which contains a 3-sublattice solid order and a superfluid order with spontaneous $SU(2)$-symmetry breaking. Unlike usual supersolid phases having only one Goldstone mode, the $SU(2)$-supersolid phase has two Goldstone modes. Another important feature of this supersolid is that the spin excitation spectrum has symmetry protected double degeneracy in the whole Brillouin zone. As by-products, several other ordered phases are obtained, including the ferromagnetic and the antiferromagnetic states breaking the $SU(2)$ symmetry, as well as genuine phases that completely break the $U(2)$ symmetry. Furthermore, the instabilities of $SU(3)$-flavor linear spin-wave theory are consistent with the phase boundaries between different ordered phases.

关键词: supersolid, spontaneous symmetry broken, spin-wave, triangular lattice

Abstract: A spin supersolid is characterized by the simultaneous breaking of lattice translation and continuous spin rotation symmetries. In this work, we study a spin-1 model with $U(2)\cong SU(2)\times U(1)/Z_2$ symmetry on the triangular lattice, and determine the phase diagram with a variational $\mathbb CP^2$ approach. We identify a novel supersolid phase which contains a 3-sublattice solid order and a superfluid order with spontaneous $SU(2)$-symmetry breaking. Unlike usual supersolid phases having only one Goldstone mode, the $SU(2)$-supersolid phase has two Goldstone modes. Another important feature of this supersolid is that the spin excitation spectrum has symmetry protected double degeneracy in the whole Brillouin zone. As by-products, several other ordered phases are obtained, including the ferromagnetic and the antiferromagnetic states breaking the $SU(2)$ symmetry, as well as genuine phases that completely break the $U(2)$ symmetry. Furthermore, the instabilities of $SU(3)$-flavor linear spin-wave theory are consistent with the phase boundaries between different ordered phases.

Key words: supersolid, spontaneous symmetry broken, spin-wave, triangular lattice

中图分类号: (Magnetic phase boundaries (including classical and quantum magnetic transitions, metamagnetism, etc.))

75.30.Kz

75.40.Mg (Numerical simulation studies) 67.80.kb (Supersolid phases on lattices)

Si-Cheng Wang(王思成) and Zheng-Xin Liu(刘正鑫). Supersolid phase in a U(2) symmetric S = 1 magnet on the triangular lattice[J]. 中国物理B, 2026, 35(4): 47506-047506.

Si-Cheng Wang(王思成) and Zheng-Xin Liu(刘正鑫). Supersolid phase in a U(2) symmetric S = 1 magnet on the triangular lattice[J]. Chin. Phys. B, 2026, 35(4): 47506-047506.

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Supersolid phase in a U(2) symmetric S = 1 magnet on the triangular lattice

Supersolid phase in a U(2) symmetric S = 1 magnet on the triangular lattice

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