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Physical Review

Existence of Long-Range Order in One and Two Dimensions

Pierre C. Hohenberg

Published Jun 10, 1967• 2,415 citations
Publisher

Abstract

It is pointed out that a rigorous inequality first proved by Bogoliubov may be used to rule out the existence of quasi-averages (or long-range order) in Bose and Fermi systems for one and two dimensions and $T\ensuremath{\ne}0$.

DOI
10.1103/physrev.158.383

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It is pointed out that a rigorous inequality first proved by Bogoliubov may be used to rule out the existence of quasi-averages (or long-range order) in Bose and Fermi systems for one and two dimensions and T\ensuremath≠0T\ensuremath{\ne}0T\ensuremath=0.
The off-diagonal ordering of one- and two-dimensional Bose particle systems of finite thickness and cross-section is considered in the presence and absence of a symmetry-breaking field. It is shown rigorously, by applying Bogoliubov's inequality to a subdomain of the system, that no spontaneous ordering can occur; several definitions of long-range order are discussed. Explicit bounds on the order-order correlation function, integrated over a subdomain, are obtained which indicate how the short-range order decays with distance. Conditions under which information on the pointwise behavior of the correlation function can be inferred are also discussed. The inequalities are assessed numerically for real situations.
In this paper we explore the low-dimension long-range superfluid order in the trapped Bose-Einstein condensate driven by spin-orbit coupling at any finite temperature. With the help of the Bogoliubov inequality we give rigorous proofs in both one and two dimensions. For one dimension, no long-range superfluid order can be found, even if spin-orbit coupling exists. For two dimensions, we find that the long-range superfluid order cannot be ruled out in the presence of the unequal Rashba-Dresselhaus-type spin-orbit coupling. However, this long-range superfluid order disappears for the equal case, which has been achieved in recent experiments.
On the basis of Bogoliubov's 1/k2 theorem the relation between the long-range order in the ground state and the collective mode, namely, the Goldstone particle is investigated. It is pointed out that Bogoliubov's inequality rules out the long-range orders in the ground states of the isotropic Heisenberg model, the hafl-filled Hubbard model and the interacting Bose system for one dimension while it admits the long-range orders for two dimensions. The present proof is based on the fact that the lowest-excited state that can be regarded as the Goldstone particle has the energy E(k)∝|k| for small k. This energy spectrum is exactly given in the one-dimensional models and is shown to be proven in the ordered state on a reasonable assumption except for the ferromagnetic case.