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Physical review. B, Solid state

Crystalline Stability and Order in One and Two Dimensions

Julio F. Fernández

Published Feb 15, 1970• 9 citations
Publisher

Abstract

It is shown that no quantum system of particles, regardless of their interactions, can form a stable solid in one or two dimensions, if both the temperature and the compressibility are different from zero. By stable solid we mean that $〈{u}^{2}〉$ be finite. It is also proven that a neutral system of electrons and nuclei cannot exhibit crystalline order, of the usual kind, in one or two dimensions. This last proof is based on Bogolyubov's inequality.

DOI
10.1103/physrevb.1.1345

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It is shown that no quantum system of particles, regardless of their interactions, can form a stable solid in one or two dimensions, if both the temperature and the compressibility are different from zero. By stable solid we mean that 〈u2〉〈{u}^{2}〉〈u2〉 be finite. It is also proven that a neutral system of electrons and nuclei cannot exhibit crystalline order, of the usual kind, in one or two dimensions. This last proof is based on Bogolyubov's inequality.
We revisit the seminal 1968 proof of the absence of crystalline order in two dimensions and analyze the importance played in the quantum theorem by the assumption of local pair potentials. We relax the assumption of local potentials and consider instead nonlocal pair potentials. We show that the 1/k2 -singularity that occurs in the Bogoliubov inequality, which leads to no crystalline order in two dimensions for local potentials and nonzero temperatures, does not occur for nonlocal potentials. Accordingly, crystalline order in one and two dimensions cannot be ruled out for nonlocal pair potentials at finite temperatures.
We present recent works [15,16] on the thermodynamic limit of quantum Coulomb systems. We provide a general method which allows to show the existence of the limit for many different systems. Ordinary matter is composed of electrons (negatively charged) and nuclei (positively charged) interacting via Coulomb forces. The potential between two particles of charges z and z ′ located at x and x ′ in R 3 is zz ′ |x − x ′ |. There are two difficulties which occur when trying to describe systems composed of electrons and nuclei. Both have to do with the physical problem Talk given by M.L. at QMath10, 10th Quantum Mathematics International Conference, Moeciu, Romania, September 10–15, 2007. c○2008 by the authors. This work may be reproduced, in its entirety, for non-commercial purposes. 2 of stability of quantum systems. The first is due to the singularity of 1/|x | at 0: it is necessary to explain why a particle will not rush to a particle of the opposite charge. One of the first major triumphs of the theory of quantum mechanics is the explanation it gives of the stability of the hydrogen atom (and the complete description of its spectrum) and of other microscopic quantum Coulomb systems, via the uncertainty principle. Stability means that the total energy of the considered system cannot be arbitrarily negative. If there was no such lower bound to the energy it would be possible in principle to extract an infinite amount of energy. One often refers to this kind of stability as stability of the first kind [19,20]. If we denote by E(N) the ground state energy of the system under consideration, for N particles stability of the first kind can be written E(N)> −∞. (1) In proving (1) for Coulomb systems, a major role is played by the uncertainty principle which for nonrelativistic systems is mathematically expressed by the critical Sobolev embedding H 1 (R 3) ֒ → L 6 (R 3). The latter allows to prove Kato’s inequality
The proof of Jasnow and Fisher that no spontaneous ordering can occur in one- and two-dimensional systems with continuous symmetry is shown to fail for nonlocal potentials of sufficiently long range. The occurrence of Bose-Einstein condensation for finite temperature (in the infinite-volume limit) is, therefore, not forbidden for such potentials. This result also follows from the postulate of the completeness of the noninteracting equilibrium states. Thus, the completeness postulate represents a weaker assumption on the nature of an interacting equilibrium state (or, equivalently, on the basic interactions between particles) than is usually found in the literature.
The Bogoliubov inequality is used to prove the nonexistence of the excitonic insulating phase in one and two dimensions for systems of electrons described by three different many-band models with interactions. The proof is carried through (1) for arbitrary two-particle interactions with an interaction potential which goes to zero faster than [Formula: see text] (n being the number of dimensions) at large [Formula: see text], (2) for two-particle interactions of arbitrary range, provided only the so-called Coulomb and exchange parts of the interaction are retained and the remainder is neglected, and (3) for the simplified isotropic two-band model suggested for the excitonic insulator by Des Cloizeaux.