### Abstract

A quantum mechanical lattice model of fermionic electrons interacting with infinitely massive nuclei is considered. (It can be viewed as a modified Hubbard model in which the spin-up electrons are not allowed to hop.) The electron-nucleus potential is "on-site" only. Neither this potential alone nor the kinetic energy alone can produce long range order. Thus, if long range order exists in this model it must come from an exchange mechanism. N, the electron plus nucleus number, is taken to be less than or equal to the number of lattice sites. We prove the following: (i) For all dimensions, d, the ground state has long range order; in fact it is a perfect crystal with spacing √2 times the lattice spacing. A gap in the ground state energy always exists at the half-filled band point (N = number of lattice sites). (ii) For small, positive temperature, T, the ordering persists when d ≥ 2. If T is large there is no long range order and there is exponential clustering of all correlation functions.

Original language | English (US) |
---|---|

Pages (from-to) | 320-358 |

Number of pages | 39 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 138 |

Issue number | 1-2 |

DOIs | |

State | Published - Sep 2 1986 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

*Physica A: Statistical Mechanics and its Applications*,

*138*(1-2), 320-358. https://doi.org/10.1016/0378-4371(86)90188-3

**An itinerant electron model with crystalline or magnetic long range order.** / Kennedy, Thomas G; Lieb, Elliott H.

Research output: Contribution to journal › Article

*Physica A: Statistical Mechanics and its Applications*, vol. 138, no. 1-2, pp. 320-358. https://doi.org/10.1016/0378-4371(86)90188-3

}

TY - JOUR

T1 - An itinerant electron model with crystalline or magnetic long range order

AU - Kennedy, Thomas G

AU - Lieb, Elliott H.

PY - 1986/9/2

Y1 - 1986/9/2

N2 - A quantum mechanical lattice model of fermionic electrons interacting with infinitely massive nuclei is considered. (It can be viewed as a modified Hubbard model in which the spin-up electrons are not allowed to hop.) The electron-nucleus potential is "on-site" only. Neither this potential alone nor the kinetic energy alone can produce long range order. Thus, if long range order exists in this model it must come from an exchange mechanism. N, the electron plus nucleus number, is taken to be less than or equal to the number of lattice sites. We prove the following: (i) For all dimensions, d, the ground state has long range order; in fact it is a perfect crystal with spacing √2 times the lattice spacing. A gap in the ground state energy always exists at the half-filled band point (N = number of lattice sites). (ii) For small, positive temperature, T, the ordering persists when d ≥ 2. If T is large there is no long range order and there is exponential clustering of all correlation functions.

AB - A quantum mechanical lattice model of fermionic electrons interacting with infinitely massive nuclei is considered. (It can be viewed as a modified Hubbard model in which the spin-up electrons are not allowed to hop.) The electron-nucleus potential is "on-site" only. Neither this potential alone nor the kinetic energy alone can produce long range order. Thus, if long range order exists in this model it must come from an exchange mechanism. N, the electron plus nucleus number, is taken to be less than or equal to the number of lattice sites. We prove the following: (i) For all dimensions, d, the ground state has long range order; in fact it is a perfect crystal with spacing √2 times the lattice spacing. A gap in the ground state energy always exists at the half-filled band point (N = number of lattice sites). (ii) For small, positive temperature, T, the ordering persists when d ≥ 2. If T is large there is no long range order and there is exponential clustering of all correlation functions.

UR - http://www.scopus.com/inward/record.url?scp=46149142828&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=46149142828&partnerID=8YFLogxK

U2 - 10.1016/0378-4371(86)90188-3

DO - 10.1016/0378-4371(86)90188-3

M3 - Article

AN - SCOPUS:46149142828

VL - 138

SP - 320

EP - 358

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

IS - 1-2

ER -