Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Crystalline Order in Restricted Geometries

Julio F. Fernández

Published Dec 1, 1970• 6 citations
Publisher

Abstract

It is proven that a system of nuclei and electrons, confined by walls in the shape of an infinite slab or a rectangular rod of infinite length, cannot have maximum long-range crystalline order. The proof is based on the Bogoliubov inequality.
DOI
10.1103/physreva.2.2555

Related papers

Existence of Long-Range Order in One and Two Dimensions

Pierre C. Hohenberg

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 $T\ensuremath{\ne}0$.

Physical Review•Jun 10, 1967•2,415 citations

Crystalline Stability and Order in One and Two Dimensions

Julio F. Fernández

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.

Physical review. B, Solid state•Feb 15, 1970•9 citations

Casimir forces in Bose–Einstein condensates: finite-size effects in three-dimensional rectangular cavities

Ariel Edery

The Casimir force due to thermal fluctuations (or the pseudo-Casimir force) was previously calculated for a perfect Bose gas in the slab geometry for various boundary conditions. The Casimir pressure due to quantum fluctuations in a weakly-interacting dilute Bose–Einstein condensate (BEC) confined to a parallel plate geometry was recently calculated for Dirichlet boundary conditions. In this paper we calculate the Casimir energy and pressure due to quantum fluctuations in a zero-temperature homogeneous weakly-interacting dilute BEC confined to a parallel plate geometry with periodic boundary conditions and include higher-order corrections to the leading-order result which we refer to as Bogoliubov corrections. The leading-order term is identified as the Casimir energy of a massless scalar field moving with a wave velocity equal to the speed of sound in the BEC. We then obtain the leading-order Casimir pressure in a general three-dimensional rectangular cavity of arbitrary length and obtain the finite-size correction to the parallel plate scenario.

arXiv:hep-th/0510238•Journal of Statistical Mechanics Theory and Experiment•Jun 8, 2006•27 citations

Peierls Argument and Long-Range Order Behavior of Quantum Lattice Systems with Unbounded Spins

Sergio A Albeverio, A. Yu. Kondratiev, A. L. Rebenko

Journal of Statistical Physics•Sep 1, 1998•13 citations

Destruction of Long-range Order in One-dimensional and Two-dimensional Systems Possessing a Continuous Symmetry Group. II. Quantum Systems.

V. L. Berezinskiǐ

Journal of Experimental and Theoretical Physics•Jan 1, 1972•288 citations

Long-Range Orders in Ground States and Collective Modes in One- and Two-Dimensional Models

Satoshi Takada

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.

Progress of Theoretical Physics•Oct 1, 1975•11 citations
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.
The Casimir force due to thermal fluctuations (or the pseudo-Casimir force) was previously calculated for a perfect Bose gas in the slab geometry for various boundary conditions. The Casimir pressure due to quantum fluctuations in a weakly-interacting dilute Bose–Einstein condensate (BEC) confined to a parallel plate geometry was recently calculated for Dirichlet boundary conditions. In this paper we calculate the Casimir energy and pressure due to quantum fluctuations in a zero-temperature homogeneous weakly-interacting dilute BEC confined to a parallel plate geometry with periodic boundary conditions and include higher-order corrections to the leading-order result which we refer to as Bogoliubov corrections. The leading-order term is identified as the Casimir energy of a massless scalar field moving with a wave velocity equal to the speed of sound in the BEC. We then obtain the leading-order Casimir pressure in a general three-dimensional rectangular cavity of arbitrary length and obtain the finite-size correction to the parallel plate scenario.
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.