Journal of Mathematical Physics

Absence of Long-Range Order in Thin Films

J. C. Garrison, Jack Wong, Harry L. Morrison

Published Nov 1, 1972• 11 citations

Abstract

Thin films are described as idealized systems having finite extent in one direction but infinite extent in the other two. For systems of particles interacting through smooth potentials (e.g., no hard cores), it is shown that an equilibrium state for a homogeneous thin film is necessarily invariant under any continuous internal symmetry group generated by a conserved density. For short-range interactions it is also shown that equilibrium states are necessarily translation invariant. The absence of long-range order follows from its relation to broken symmetry. The only properties of the state required for the proof are local normality, spatial translation invariance, and the Kubo-Martin-Schwinger boundary condition. The argument employs the Bogoliubov inequality and the techniques of the algebraic approach to statistical mechanics. For inhomogeneous systems, the same argument shows that all order parameters defined by anomalous averages must vanish. Identical results can be obtained for systems with infinite extent in one direction only.

DOI: 10.1063/1.1665901

Related papers

GAUGE INVARIANCE AND BROKEN SYMMETRIES IN ANYON SUPERFLUIDS

D. Boyanovsky

We review aspects of broken symmetry and the nature of long range order in theories of anyons starting with bosons with a statistical interaction. We introduce a novel gauge invariant quantization scheme that allows the identification of local and gauge invariant order parameters. The connection between spin and statistics is reviewed and the consequences of broken symmetries in the anyon representation are discussed. An anyon gas is studied in the Bogoliubov approximation, it is determined that the ground state is a condensate of charge-flux composites with “quasi-long-range order” at zero temperature, a “weak” gap in the spectrum and finite helicity modulus. The system is disordered at nonzero temperatures. The disorder is not caused by Goldstone bosons but by the strong infrared behavior arising from the Coulomb interaction induced by the long-range statistical interaction. The properties of topological vortices in nonrelativistic and in relativistic Landau-Ginzburg theories are studied in detail. We study the physics of the mean-field ansatz and quasi-long range order in a simple exactly soluble relativistic model. This model exhibits a novel phenomenon of charge redistribution to the boundaries and restoration of translational invariance in the infinite volume limit. It also illuminates the physics of quasi-long-range order with a gap in the spectrum, statistical charge polarization by external magnetic fields and the role of “large” gauge transformations.

International Journal of Modern Physics ASep 30, 199212 citations

Non-existence of spontaneous symmetry breakdown of time-translation symmetry on general quantum systems: Any macroscopic order parameter moves not!

Hajime Moriya

The Kubo–Martin–Schwinger (KMS) condition is a well-founded general definition of equilibrium states on quantum systems. The time invariance property of equilibrium states is one of its basic consequences. From the time invariance of any equilibrium state it follows that the spontaneous breakdown of time-translation symmetry is impossible. Moreover, triviality of the temporal long-range order is derived from the KMS condition. Therefore, the manifestation of genuine quantum time crystals is impossible as long as the standard notion of spontaneous symmetry breakdown is considered.

arXiv:2201.06051Reviews in Mathematical PhysicsSep 30, 20232 citations

Proof of the Absence of Long-Range Temporal Orders in Gibbs States

Haruki Watanabe, Masaki Oshikawa, Tohru Koma

Abstract We address the question whether time translation symmetry can be spontaneously broken in a quantum many-body system. One way of detecting such a symmetry breaking is to examine the time-dependence of a correlation function. If the large-distance behavior of the correlation function exhibits a nontrivial time-dependence in the thermodynamic limit, the system would develop a temporal long-range order, realizing a time crystal. In an earlier publication, we sketched a proof for the absence of such time dependence in the thermal equilibrium described by the Gibbs state (Watanabe and Oshikawa in Phys Rev Lett 114:251603, 2015). Here we present a complete proof and extend the argument to a more general class of stationary states than the Gibbs states.

arXiv:1911.12939Journal of Statistical PhysicsJan 16, 202024 citations

Critical behaviour of thin films with quenched impurities

L. Craco, L. De Cesare, Ileana Rabuffo +2 more

The critical behaviour of thin films containing quenched random impurities and inhomogeneities is investigated by the renormalization-group method to the one-loop order within the framework of the n-component φ4-model. The finite-size crossover in impure films has been considered on the basis of the fundamental relationship between the effective dimensionality Deff and the characteristic lengths of the system. The fixed points, their stability properties and the critical exponents are obtained and analyzed, using an ε̃=(4−Deff)-expansion near the effective spatial dimensionality Deff of the fluctuation modes in D-dimensional hyperslabs with two types of quenched impurities: point-like impurities with short-range random correlations and extended (linear) impurities with infinite-range random correlations along the small-size spatial direction. The difference between the critical properties of infinite systems and films is demonstrated and investigated. A new critical exponent, describing the scaling properties of the thickness of films with extended impurities has been derived and calculated. A special attention is paid to the critical behaviour of real impure films (D=3).

Jan 1, 19997 citations

Long-range order in the spin van der Waals model

M. Howard Lee

Long-range order in the spin van der Waals model is considered when the number of spins is finite and also when infinite. We show explicitly that a finite system cannot support long-range order. An infinite system at high temperatures is found to be dominated by the entropy of degenerate states of the system and, as a result, the system behaves essentially like an ideal system. In an infinite system at low temperatures, long-range order exists, fully reflecting the spin symmetry of the Hamiltonian. For the XY-like regime (J≳Jz ), mx is finite but mz vanishes, where mx and mz are reduced order parameters for the transverse and longitudinal directions, respectively. For the Ising-like regime (J<Jz), mx vanishes but mz is finite. The isotropic interaction (Jz = J) behaves as a singularity and it must be considered separately. A physical interpretation of the behavior of long-range order is offered using the geometry of spin space.

Journal of Mathematical PhysicsMar 1, 198223 citations

Absence of local order in topologically frustrated spin chains

Vanja Marić, S. M. Giampaolo, Fabio Franchini

We show that a wide class of spin chains with topological frustration cannot develop any local order. In particular, we consider translational-invariant one-dimensional chains with frustrated boundary conditions, i.e. periodic boundary conditions and an odd number of sites, which possess a global SU(2) symmetry. This condition implies, even at a finite sizes, an exact degeneracy of the ground state and is quite general in absence of external fields. We directly evaluate the expectation value of operators with support over a finite range of lattice sites and show that, except for some precise conditions, they all decay algebraically, or faster, with the chain length and vanish in the thermodynamic limit. The exceptions that admit a finite order are cases with a higher ground state degeneracy in which the translational symmetry is broken by the ground state choice.

Jan 18, 20215 citations