### Abstract

The spectral correlation of a chaotic system with spin 1/2 is universally described by the GSE (Gaussian symplectic ensemble) of random matrices in the semiclassical limit. In semiclassical theory, the spectral form factor is expressed in terms of the periodic orbits and the spin state is simulated by the uniform distribution on a sphere. In this paper, instead of the uniform distribution, we introduce Brownian motion on a sphere to yield the parametric motion of the energy levels. As a result, the small time expansion of the form factor is obtained and found to be in agreement with the prediction of parametric random matrices in the transition within the GSE universality class. Moreover, by starting the Brownian motion from a point distribution on the sphere, we gradually increase the effect of the spin and calculate the form factor describing the transition from the Gaussian orthogonal ensemble class to the GSE class.

Original language | English |
---|---|

Article number | 004 |

Pages (from-to) | 12055-12070 |

Number of pages | 16 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 40 |

Issue number | 40 |

DOIs | |

Publication status | Published - 2007 Oct 5 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Modelling and Simulation
- Statistics and Probability

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*40*(40), 12055-12070. [004]. https://doi.org/10.1088/1751-8113/40/40/004

**Semiclassical approach to parametric spectral correlation with spin 1/2.** / Nagao, Taro; Saitou, Keiji.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 40, no. 40, 004, pp. 12055-12070. https://doi.org/10.1088/1751-8113/40/40/004

}

TY - JOUR

T1 - Semiclassical approach to parametric spectral correlation with spin 1/2

AU - Nagao, Taro

AU - Saitou, Keiji

PY - 2007/10/5

Y1 - 2007/10/5

N2 - The spectral correlation of a chaotic system with spin 1/2 is universally described by the GSE (Gaussian symplectic ensemble) of random matrices in the semiclassical limit. In semiclassical theory, the spectral form factor is expressed in terms of the periodic orbits and the spin state is simulated by the uniform distribution on a sphere. In this paper, instead of the uniform distribution, we introduce Brownian motion on a sphere to yield the parametric motion of the energy levels. As a result, the small time expansion of the form factor is obtained and found to be in agreement with the prediction of parametric random matrices in the transition within the GSE universality class. Moreover, by starting the Brownian motion from a point distribution on the sphere, we gradually increase the effect of the spin and calculate the form factor describing the transition from the Gaussian orthogonal ensemble class to the GSE class.

AB - The spectral correlation of a chaotic system with spin 1/2 is universally described by the GSE (Gaussian symplectic ensemble) of random matrices in the semiclassical limit. In semiclassical theory, the spectral form factor is expressed in terms of the periodic orbits and the spin state is simulated by the uniform distribution on a sphere. In this paper, instead of the uniform distribution, we introduce Brownian motion on a sphere to yield the parametric motion of the energy levels. As a result, the small time expansion of the form factor is obtained and found to be in agreement with the prediction of parametric random matrices in the transition within the GSE universality class. Moreover, by starting the Brownian motion from a point distribution on the sphere, we gradually increase the effect of the spin and calculate the form factor describing the transition from the Gaussian orthogonal ensemble class to the GSE class.

UR - http://www.scopus.com/inward/record.url?scp=34748872173&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34748872173&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/40/40/004

DO - 10.1088/1751-8113/40/40/004

M3 - Article

AN - SCOPUS:34748872173

VL - 40

SP - 12055

EP - 12070

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 40

M1 - 004

ER -