### Abstract

We present a proof of the central limit theorem for a pair of mutually non-commuting operators in mixing quantum spin chains. The operators are not necessarily strictly local but quasi-local. As a corollary we obtain a direct construction of the time evolution of the algebra of normal fluctuation for Gibbs states of finite range interactions on a one-dimensional lattice. We show that the state of the algebra of normal fluctuation satisfies the β-KMS condition if the microscopic state is a β-KMS state. We show that any mixing finitely correlated state satisfies our assumption for the central limit theorem.

Original language | English |
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Pages (from-to) | 63-83 |

Number of pages | 21 |

Journal | Annales Henri Poincare |

Volume | 4 |

Issue number | 1 |

DOIs | |

Publication status | Published - May 6 2003 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics

### Cite this

**On the algebra of fluctuation in quantum spin chains.** / Matsui, Taku.

Research output: Contribution to journal › Article

*Annales Henri Poincare*, vol. 4, no. 1, pp. 63-83. https://doi.org/10.1007/s00023-003-0122-z

}

TY - JOUR

T1 - On the algebra of fluctuation in quantum spin chains

AU - Matsui, Taku

PY - 2003/5/6

Y1 - 2003/5/6

N2 - We present a proof of the central limit theorem for a pair of mutually non-commuting operators in mixing quantum spin chains. The operators are not necessarily strictly local but quasi-local. As a corollary we obtain a direct construction of the time evolution of the algebra of normal fluctuation for Gibbs states of finite range interactions on a one-dimensional lattice. We show that the state of the algebra of normal fluctuation satisfies the β-KMS condition if the microscopic state is a β-KMS state. We show that any mixing finitely correlated state satisfies our assumption for the central limit theorem.

AB - We present a proof of the central limit theorem for a pair of mutually non-commuting operators in mixing quantum spin chains. The operators are not necessarily strictly local but quasi-local. As a corollary we obtain a direct construction of the time evolution of the algebra of normal fluctuation for Gibbs states of finite range interactions on a one-dimensional lattice. We show that the state of the algebra of normal fluctuation satisfies the β-KMS condition if the microscopic state is a β-KMS state. We show that any mixing finitely correlated state satisfies our assumption for the central limit theorem.

UR - http://www.scopus.com/inward/record.url?scp=0037272548&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037272548&partnerID=8YFLogxK

U2 - 10.1007/s00023-003-0122-z

DO - 10.1007/s00023-003-0122-z

M3 - Article

AN - SCOPUS:0037272548

VL - 4

SP - 63

EP - 83

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

IS - 1

ER -