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:2211.03008 · Physical review. B./Physical review. B

Quantum phase transitions in the triangular coupled-top model

Liwei Duan, Yanzhi Wang, Qing‐Hu Chen

Published Mar 14, 2023• 6 citations
arXivHTMLPDF

Abstract

We study the coupled-top model with three large spins located on a triangle. Depending on the coupling strength, there exist three phases: disordered paramagnetic phase, ferromagnetic phase, and frustrated antiferromagnetic phase, which can be distinguished by the mean-field approach. The paramagnetic-ferromagnetic phase transition is accompanied by the breaking of the global ${Z}_{2}$ symmetry, whereas the paramagnetic-antiferromagnetic phase transition is accompanied by the breaking of both the global ${Z}_{2}$ symmetry and the translational symmetry. Exact analytical results of higher-order quantum effects beyond the mean-field contribution, such as the excitation energy, quantum fluctuation, and von Neumann entropy, can be achieved by the Holstein-Primakoff transformation and symplectic transformation in the thermodynamic limit. Near the quantum critical point, the energy gap closes, along with the divergence of the quantum fluctuation in certain quadrature and von Neumann entropy. Particular attention should be paid to the antiferromagnetic phase, where geometric frustration takes effect. The critical behaviors in the antiferromagnetic phase are quite different from those in the paramagnetic and ferromagnetic phases, which highlight the importance of geometric frustration. The triangular coupled-top model provides a simple and feasible platform to study the quantum phase transition and the novel critical behaviors induced by geometric frustration.

DOI
10.1103/physrevb.107.094415

Related papers

Quantum Phase Transition in the Coupled-Top Model: From Z2 to U(1) Symmetry Breaking

Wen-Jian Mao, Ye F. Tian, Liwei Duan +1 more

directions. By tuning coupling strengths along distinct directions, the system exhibits different symmetries, ranging from a discrete Z2 to a continuous U(1) symmetry. The anisotropic coupled-top model displays a discrete Z2 symmetry, and the symmetry breaking induced by strong coupling drives a quantum phase transition from a disordered paramagnetic phase to an ordered ferromagnetic or antiferromagnetic phase. In particular, the isotropic coupled-top model possesses a continuous U(1) symmetry, whose breaking gives rise to the Goldstone mode. The phase boundary can be well captured by the mean-field approach, characterized by the distinct behaviors of the order parameter. Higher-order quantum effects beyond the mean-field contribution can be achieved by mapping the large spins to bosonic operators via the Holstein-Primakoff transformation. For the anisotropic coupled-top model with Z2 symmetry, the energy gap closes, and both quantum fluctuations and entanglement entropy diverge near the critical point, signaling the onset of second-order quantum phase transitions. Strikingly, when U(1) symmetry is broken, the energy gap vanishes beyond the critical point, yielding a novel critical exponent of 1, rather than 1/2 for Z2 symmetry breaking. The rich symmetry structure of the coupled-top model underpins its role as a paradigmatic model for studying quantum phase transitions and exploring associated physical phenomena.

arXiv:2502.09394•Entropy•Apr 27, 2025•0 citations

Unconventional quantum ordered and disordered states in the highly frustrated spin- 12 Ising-Heisenberg model on triangles-in-triangles lattices

Jana Čisárová, Jozef Strečka

The spin-$\frac{1}{2}$ Ising-Heisenberg model on two geometrically related triangles-in-triangles lattices is exactly solved through the generalized star-triangle transformation, which establishes a rigorous mapping correspondence with the effective spin-$\frac{1}{2}$ Ising model on a triangular lattice. Basic thermodynamic quantities were exactly calculated within this rigorous mapping method along with the ground-state and finite-temperature phase diagrams. Apart from the classical ferromagnetic phase, both investigated models exhibit several unconventional quantum ordered and disordered ground states. A mutual competition between two ferromagnetic interactions of basically different character generically leads to the emergence of a quantum ferromagnetic phase, in which a symmetric quantum superposition of three up-up-down states of the Heisenberg trimers accompanies a perfect alignment of all Ising spins. In the highly frustrated regime, we have either found the disordered quantum paramagnetic phase with a substantial residual entropy or a similar but spontaneously ordered phase with a nontrivial criticality at finite temperatures. The latter quantum ground state exhibits a striking coexistence of imperfect spontaneous order with partial disorder, which is evidenced by a quantum reduction of the spontaneous magnetization of Heisenberg spins that indirectly causes a small reduction of the spontaneous magnetization of otherwise classical Ising spins.

arXiv:1210.0337•Physical Review B•Jan 29, 2013•19 citations

Quantum Phase Transitions in a Generalized Dicke Model

Wen Liu, Liwei Duan

We investigate a generalized Dicke model by introducing two interacting spin ensembles coupled with a single-mode bosonic field. Apart from the normal to superradiant phase transition induced by the strong spin-boson coupling, interactions between the two spin ensembles enrich the phase diagram by introducing ferromagnetic, antiferromagnetic and paramagnetic phases. The mean-field approach reveals a phase diagram comprising three phases: paramagnetic-normal phase, ferromagnetic-superradiant phase, and antiferromagnetic-normal phase. Ferromagnetic spin-spin interaction can significantly reduce the required spin-boson coupling strength to observe the superradiant phase, where the macroscopic excitation of the bosonic field occurs. Conversely, antiferromagnetic spin-spin interaction can strongly suppress the superradiant phase. To examine higher-order quantum effects beyond the mean-field contribution, we utilize the Holstein-Primakoff transformation, which converts the generalized Dicke model into three coupled harmonic oscillators in the thermodynamic limit. Near the critical point, we observe the close of the energy gap between the ground and the first excited states, the divergence of entanglement entropy and quantum fluctuation in certain quadrature. These observations further confirm the quantum phase transition and offer additional insights into critical behaviors.

arXiv:2310.18978•Entropy•Oct 29, 2023•4 citations

Quantum Ising model on the frustrated square lattice

N. Kellermann, M. Schmidt, Fábio M. Zimmer

We investigate the role of a transverse field on the Ising square antiferromagnet with first (J_{1}) and second (J_{2}) neighbor interactions. Using a cluster mean-field approach, we provide a telltale characterization of the frustration effects on the phase boundaries and entropy accumulation process emerging from the interplay between quantum and thermal fluctuations. We found that the paramagnetic (PM) and antiferromagnetic phases are separated by continuous phase transitions. On the other hand, continuous and discontinuous phase transitions, as well as tricriticality, are observed in the phase boundaries between PM and superantiferromagnetic phases. A rich scenario arises when a discontinuous phase transition occurs in the classical limit while quantum fluctuations recover criticality. We also find that the entropy accumulation process predicted to occur at temperatures close to the quantum critical point can be enhanced by frustration. Our results provide a description for the phase boundaries and entropy behavior that can help to identify the ratio J_{2}/J_{1} in possible experimental realizations of the quantum J_{1}-J_{2} Ising antiferromagnet.

arXiv:1811.04111•Physical review. E•Jan 17, 2019•40 citations

Effects of third-neighbor interactions on the frustrated quantum Ising model

M. Roos, I. F. Muhl, M. Schmidt +2 more

We investigate thermal and quantum phase transitions of the J_{1}-J_{2}-J_{3} transverse Ising model on the square lattice. The model is studied within a cluster mean-field decoupling, which allows us to describe phase diagrams and the free-energy landscape in the neighborhood of phase transitions. Our findings indicate that the third-neighbor coupling (J_{3}) can affect the nature of phase transitions of the model. In particular, ferromagnetic third-neighbor couplings favor the onset of continuous order-disorder phase transitions, eliminating the tricritical point of the superantiferromagnetic-paramagnetic (SAFM-PM) phase boundary. On the other hand, the enhancement of frustration introduced by weak antiferromagnetic J_{3} gives rise to the staggered dimer phase favoring the onset of discontinuous classical phase transitions. Moreover, we find that quantum annealed criticality (QAC), which takes place when the classical discontinuous phase transition becomes critical by the enhancement of quantum fluctuations introduced by the transverse magnetic field, is eliminated from the SAFM-PM phase boundary by a relatively weak ferromagnetic J_{3}. Nevertheless, this change in the nature of phase transitions can still be observed in the presence of antiferromagnetic third-neighbor couplings being also found in the staggered-dimer phase boundary. Therefore, our findings support that QAC persists under the presence of frustrated antiferromagnetic third-neighbor couplings and is suppressed when these couplings are ferromagnetic, suggesting that frustration plays a central role in the onset of QAC.

Physical review. E•Jan 30, 2024•7 citations

Unconventional spin frustration due to two competing ferromagnetic interactions of a spin-1/2 Ising-Heisenberg model on martini and martini-diced lattices

Hamid Arian Zad, Jozef Strečka

The spin-1/2 Ising-Heisenberg model on martini and martini-diced lattices is exactly solved using a star-triangle transformation, which affords an exact mapping correspondence to an effective spin-1/2 Ising model on a triangular lattice. The ground-state phase diagram of both investigated quantum spin models display two spontaneously ordered ferromagnetic phases and one macroscopically degenerate disordered phase. In contrast to a classical ferromagnetic phase where the spontaneous magnetization of the Ising as well as Heisenberg spins acquire fully saturated values, the spontaneous magnetization of the Heisenberg spins is subject to a quantum reduction to one-third of its saturated value within a quantum ferromagnetic phase. The spontaneous magnetization and logarithmic divergence of the specific heat as the most essential features of both ferromagnetic phases disappear whenever the investigated quantum spin model is driven to the highly degenerate disordered phase. The disordered phase with nonzero residual entropy originates either from a geometric spin frustration caused by antiferromagnetic interactions or more strikingly it may also alternatively arise from a competition of the ferromagnetic Ising and Heisenberg interactions of easy-axis and easy-plane type, respectively. All three available ground states coexist together at a single triple point, around which anomalous magnetic and thermodynamic behavior can be detected.

Physical review. E•Apr 13, 2022•1 citation
We study the coupled-top model with three large spins located on a triangle. Depending on the coupling strength, there exist three phases: disordered paramagnetic phase, ferromagnetic phase, and frustrated antiferromagnetic phase, which can be distinguished by the mean-field approach. The paramagnetic-ferromagnetic phase transition is accompanied by the breaking of the global Z2{Z}_{2}Z2​ symmetry, whereas the paramagnetic-antiferromagnetic phase transition is accompanied by the breaking of both the global Z2{Z}_{2}Z2​ symmetry and the translational symmetry. Exact analytical results of higher-order quantum effects beyond the mean-field contribution, such as the excitation energy, quantum fluctuation, and von Neumann entropy, can be achieved by the Holstein-Primakoff transformation and symplectic transformation in the thermodynamic limit. Near the quantum critical point, the energy gap closes, along with the divergence of the quantum fluctuation in certain quadrature and von Neumann entropy. Particular attention should be paid to the antiferromagnetic phase, where geometric frustration takes effect. The critical behaviors in the antiferromagnetic phase are quite different from those in the paramagnetic and ferromagnetic phases, which highlight the importance of geometric frustration. The triangular coupled-top model provides a simple and feasible platform to study the quantum phase transition and the novel critical behaviors induced by geometric frustration.
directions. By tuning coupling strengths along distinct directions, the system exhibits different symmetries, ranging from a discrete Z2 to a continuous U(1) symmetry. The anisotropic coupled-top model displays a discrete Z2 symmetry, and the symmetry breaking induced by strong coupling drives a quantum phase transition from a disordered paramagnetic phase to an ordered ferromagnetic or antiferromagnetic phase. In particular, the isotropic coupled-top model possesses a continuous U(1) symmetry, whose breaking gives rise to the Goldstone mode. The phase boundary can be well captured by the mean-field approach, characterized by the distinct behaviors of the order parameter. Higher-order quantum effects beyond the mean-field contribution can be achieved by mapping the large spins to bosonic operators via the Holstein-Primakoff transformation. For the anisotropic coupled-top model with Z2 symmetry, the energy gap closes, and both quantum fluctuations and entanglement entropy diverge near the critical point, signaling the onset of second-order quantum phase transitions. Strikingly, when U(1) symmetry is broken, the energy gap vanishes beyond the critical point, yielding a novel critical exponent of 1, rather than 1/2 for Z2 symmetry breaking. The rich symmetry structure of the coupled-top model underpins its role as a paradigmatic model for studying quantum phase transitions and exploring associated physical phenomena.
The spin-12\frac{1}{2}21​ Ising-Heisenberg model on two geometrically related triangles-in-triangles lattices is exactly solved through the generalized star-triangle transformation, which establishes a rigorous mapping correspondence with the effective spin-12\frac{1}{2}21​ Ising model on a triangular lattice. Basic thermodynamic quantities were exactly calculated within this rigorous mapping method along with the ground-state and finite-temperature phase diagrams. Apart from the classical ferromagnetic phase, both investigated models exhibit several unconventional quantum ordered and disordered ground states. A mutual competition between two ferromagnetic interactions of basically different character generically leads to the emergence of a quantum ferromagnetic phase, in which a symmetric quantum superposition of three up-up-down states of the Heisenberg trimers accompanies a perfect alignment of all Ising spins. In the highly frustrated regime, we have either found the disordered quantum paramagnetic phase with a substantial residual entropy or a similar but spontaneously ordered phase with a nontrivial criticality at finite temperatures. The latter quantum ground state exhibits a striking coexistence of imperfect spontaneous order with partial disorder, which is evidenced by a quantum reduction of the spontaneous magnetization of Heisenberg spins that indirectly causes a small reduction of the spontaneous magnetization of otherwise classical Ising spins.
We investigate a generalized Dicke model by introducing two interacting spin ensembles coupled with a single-mode bosonic field. Apart from the normal to superradiant phase transition induced by the strong spin-boson coupling, interactions between the two spin ensembles enrich the phase diagram by introducing ferromagnetic, antiferromagnetic and paramagnetic phases. The mean-field approach reveals a phase diagram comprising three phases: paramagnetic-normal phase, ferromagnetic-superradiant phase, and antiferromagnetic-normal phase. Ferromagnetic spin-spin interaction can significantly reduce the required spin-boson coupling strength to observe the superradiant phase, where the macroscopic excitation of the bosonic field occurs. Conversely, antiferromagnetic spin-spin interaction can strongly suppress the superradiant phase. To examine higher-order quantum effects beyond the mean-field contribution, we utilize the Holstein-Primakoff transformation, which converts the generalized Dicke model into three coupled harmonic oscillators in the thermodynamic limit. Near the critical point, we observe the close of the energy gap between the ground and the first excited states, the divergence of entanglement entropy and quantum fluctuation in certain quadrature. These observations further confirm the quantum phase transition and offer additional insights into critical behaviors.
We investigate the role of a transverse field on the Ising square antiferromagnet with first (J_{1}) and second (J_{2}) neighbor interactions. Using a cluster mean-field approach, we provide a telltale characterization of the frustration effects on the phase boundaries and entropy accumulation process emerging from the interplay between quantum and thermal fluctuations. We found that the paramagnetic (PM) and antiferromagnetic phases are separated by continuous phase transitions. On the other hand, continuous and discontinuous phase transitions, as well as tricriticality, are observed in the phase boundaries between PM and superantiferromagnetic phases. A rich scenario arises when a discontinuous phase transition occurs in the classical limit while quantum fluctuations recover criticality. We also find that the entropy accumulation process predicted to occur at temperatures close to the quantum critical point can be enhanced by frustration. Our results provide a description for the phase boundaries and entropy behavior that can help to identify the ratio J_{2}/J_{1} in possible experimental realizations of the quantum J_{1}-J_{2} Ising antiferromagnet.
We investigate thermal and quantum phase transitions of the J_{1}-J_{2}-J_{3} transverse Ising model on the square lattice. The model is studied within a cluster mean-field decoupling, which allows us to describe phase diagrams and the free-energy landscape in the neighborhood of phase transitions. Our findings indicate that the third-neighbor coupling (J_{3}) can affect the nature of phase transitions of the model. In particular, ferromagnetic third-neighbor couplings favor the onset of continuous order-disorder phase transitions, eliminating the tricritical point of the superantiferromagnetic-paramagnetic (SAFM-PM) phase boundary. On the other hand, the enhancement of frustration introduced by weak antiferromagnetic J_{3} gives rise to the staggered dimer phase favoring the onset of discontinuous classical phase transitions. Moreover, we find that quantum annealed criticality (QAC), which takes place when the classical discontinuous phase transition becomes critical by the enhancement of quantum fluctuations introduced by the transverse magnetic field, is eliminated from the SAFM-PM phase boundary by a relatively weak ferromagnetic J_{3}. Nevertheless, this change in the nature of phase transitions can still be observed in the presence of antiferromagnetic third-neighbor couplings being also found in the staggered-dimer phase boundary. Therefore, our findings support that QAC persists under the presence of frustrated antiferromagnetic third-neighbor couplings and is suppressed when these couplings are ferromagnetic, suggesting that frustration plays a central role in the onset of QAC.
The spin-1/2 Ising-Heisenberg model on martini and martini-diced lattices is exactly solved using a star-triangle transformation, which affords an exact mapping correspondence to an effective spin-1/2 Ising model on a triangular lattice. The ground-state phase diagram of both investigated quantum spin models display two spontaneously ordered ferromagnetic phases and one macroscopically degenerate disordered phase. In contrast to a classical ferromagnetic phase where the spontaneous magnetization of the Ising as well as Heisenberg spins acquire fully saturated values, the spontaneous magnetization of the Heisenberg spins is subject to a quantum reduction to one-third of its saturated value within a quantum ferromagnetic phase. The spontaneous magnetization and logarithmic divergence of the specific heat as the most essential features of both ferromagnetic phases disappear whenever the investigated quantum spin model is driven to the highly degenerate disordered phase. The disordered phase with nonzero residual entropy originates either from a geometric spin frustration caused by antiferromagnetic interactions or more strikingly it may also alternatively arise from a competition of the ferromagnetic Ising and Heisenberg interactions of easy-axis and easy-plane type, respectively. All three available ground states coexist together at a single triple point, around which anomalous magnetic and thermodynamic behavior can be detected.