### Abstract

We study the entropy of pure shift-invariant states on a quantum spin chain. Unlike the classical case, the local restrictions to intervals of length N are typically mixed and have therefore a nonzero entropy S_{N} which is, moreover, monotonically increasing in N. We are interested in the asymptotics of the total entropy. We investigate in detail a class of states derived from quasi-free states on a CAR algebra. These are characterized by a measurable subset of the unit interval. As the entropy density is known to vanish, S_{N} is sublinear in N. For states corresponding to unions of finitely many intervals, S_{N} is shown to grow slower than log ^{2} N Numerical calculations suggest a log N behavior. For the case with infinitely many intervals, we present a class of states for which the entropy S_{N} increases as N^{α} where α can take any value in (0,1).

Original language | English |
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Pages (from-to) | 6005-6019 |

Number of pages | 15 |

Journal | Journal of Mathematical Physics |

Volume | 44 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 2003 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*44*(12), 6005-6019. https://doi.org/10.1063/1.1623616

**Entropy growth of shift-invariant states on a quantum spin chain.** / Fannes, M.; Haegeman, B.; Mosonyi, M.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 44, no. 12, pp. 6005-6019. https://doi.org/10.1063/1.1623616

}

TY - JOUR

T1 - Entropy growth of shift-invariant states on a quantum spin chain

AU - Fannes, M.

AU - Haegeman, B.

AU - Mosonyi, M.

PY - 2003/12

Y1 - 2003/12

N2 - We study the entropy of pure shift-invariant states on a quantum spin chain. Unlike the classical case, the local restrictions to intervals of length N are typically mixed and have therefore a nonzero entropy SN which is, moreover, monotonically increasing in N. We are interested in the asymptotics of the total entropy. We investigate in detail a class of states derived from quasi-free states on a CAR algebra. These are characterized by a measurable subset of the unit interval. As the entropy density is known to vanish, SN is sublinear in N. For states corresponding to unions of finitely many intervals, SN is shown to grow slower than log 2 N Numerical calculations suggest a log N behavior. For the case with infinitely many intervals, we present a class of states for which the entropy SN increases as Nα where α can take any value in (0,1).

AB - We study the entropy of pure shift-invariant states on a quantum spin chain. Unlike the classical case, the local restrictions to intervals of length N are typically mixed and have therefore a nonzero entropy SN which is, moreover, monotonically increasing in N. We are interested in the asymptotics of the total entropy. We investigate in detail a class of states derived from quasi-free states on a CAR algebra. These are characterized by a measurable subset of the unit interval. As the entropy density is known to vanish, SN is sublinear in N. For states corresponding to unions of finitely many intervals, SN is shown to grow slower than log 2 N Numerical calculations suggest a log N behavior. For the case with infinitely many intervals, we present a class of states for which the entropy SN increases as Nα where α can take any value in (0,1).

UR - http://www.scopus.com/inward/record.url?scp=0345149787&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0345149787&partnerID=8YFLogxK

U2 - 10.1063/1.1623616

DO - 10.1063/1.1623616

M3 - Article

AN - SCOPUS:0345149787

VL - 44

SP - 6005

EP - 6019

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 12

ER -