Quantum Phase Transition in the Coupled-Top Model: From <i>Z</i><sub>2</sub> to U(1) Symmetry Breaking
We investigate the coupled-top model, which describes two large spins interacting along both <i>x</i> and <i>y</i> directions. By tuning coupling strengths along distinct directions, the system exhibits different symmetries, ranging from a discrete <inline-formula><m...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
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| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/27/5/474 |
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| Summary: | We investigate the coupled-top model, which describes two large spins interacting along both <i>x</i> and <i>y</i> directions. By tuning coupling strengths along distinct directions, the system exhibits different symmetries, ranging from a discrete <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mn>2</mn></msub></semantics></math></inline-formula> to a continuous U(1) symmetry. The anisotropic coupled-top model displays a discrete <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mn>2</mn></msub></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mn>2</mn></msub></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mn>2</mn></msub></semantics></math></inline-formula> 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. |
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| ISSN: | 1099-4300 |