TY - JOUR

T1 - A model for crystallization

T2 - A variation on the Hubbard model

AU - Lieb, Elliott H.

N1 - Funding Information: The support of the U.S. National Science Foundation, grant PHY-8515288, is gratefully acknowledged.

PY - 1986/12/1

Y1 - 1986/12/1

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 lattices 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 lattices 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.

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U2 - https://doi.org/10.1016/0378-4371(86)90228-1

DO - https://doi.org/10.1016/0378-4371(86)90228-1

M3 - Article

VL - 140

SP - 240

EP - 250

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

IS - 1-2

ER -