# The phase transition in the one-dimensional Ising Model with 1/r2 interaction energy

@article{Frhlich1982ThePT, title={The phase transition in the one-dimensional Ising Model with 1/r2 interaction energy}, author={J{\"u}rg Fr{\"o}hlich and Thomas J. Spencer}, journal={Communications in Mathematical Physics}, year={1982}, volume={84}, pages={87-101} }

We prove the existence of a spontaneous magnetization at low temperature for the one-dimensional Ising Model with 1/r2 interaction energy.

#### 203 Citations

The one-dimensional long-range ferromagnetic Ising model with a periodic external field

- Physics
- 2012

We consider the one-dimensional ferromagnetic Ising model with very long-range interaction under a periodic, biased and weak external field and prove that at sufficiently low temperatures the model… Expand

GROUND STATES OF ONE-DIMENSIONAL LONG-RANGE FERROMAGNETIC ISING MODEL WITH EXTERNAL FIELD

- Physics
- 2012

A zero-temperature phase-diagram of the one-dimensional ferromagnetic Ising model is investigated. It is shown that at zero temperature spins of any compact collection of lattice points with… Expand

Phase transition for a one-dimensional lattice gas with hard core

- Physics
- 1988

Existence of a phase transition is proved for a one-dimensional lattice gas with long-range interaction and nearest neighbor exclusion.

Restoring long-range order in one-dimensional superconductivity by power-law hopping

- Physics
- 2014

1. We demonstrate that long-range superconducting order is restored for 1=2 < < c, with 5=4 < c < 3=2. This conclusion is supported by a DMRG analysis of the model.

A one-dimensional model exhibiting phase transition

- Physics
- 1999

A one-dimensional model with two spin variables having a unique ground state and at least two extreme limit Gibbs states is constructed.

The phase transition in the discrete Gaussian chain with 1/r2 interaction energy

- Mathematics
- 1991

We exhibit a phase transition from a rough high-temperature phase to a rigid (localized) low-temperature phase in the discrete Gaussian chain with 1/r2 interaction energy. This transition is related… Expand

Study of a Long Range Perturbation of a One-Dimensional Kac Model

- Mathematics, Physics
- 2011

We consider a one dimensional ferromagnetic Ising spin system with interactions that correspond to a 1/r2 long range perturbation of the usual Kac model. We apply a coarse graining procedure widely… Expand

Phase diagram in the quantum XY model with long-range interactions

- Physics
- 2005

We study the d-dimensional quantum XY model with ferromagnetic long-range interaction decaying as r-p in terms of boson operators, by employing the coherent state path integral approach. We have… Expand

Bethe lattice approximation of the one-dimensional Potts model with long-range interaction

- Physics
- 1994

Abstract The mean-field approximation of the one-dimensional ferromagnetic p -state Potts model with interactions decaying as a power law is developed using a modified Bethe lattice recently… Expand

The relationship between phase transitions and percolation in models with ground-state degeneracy

- Mathematics
- 2002

A criterion for the uniqueness of limiting Gibbs states in classical models with unique ground states is formulated. Various applications of this criterion formulated in the terminology of… Expand

#### References

SHOWING 1-10 OF 12 REFERENCES

Rigorous entropy-energy arguments

- Physics
- 1981

We present a method for making rigorous various arguments which predict that certain situations are unstable because of a balance of energy vs. entropy. As applications, we give yet another proof… Expand

Correlations in Ising Ferromagnets. I

- Physics
- 1967

An inequality relating binary correlation functions for an Ising model with purely ferromagnetic interactions is derived by elementary arguments and used to show that such a ferromagnet cannot… Expand

Statistical mechanics of a one-dimensional lattice gas

- Mathematics
- 1968

We study the statistical mechanics of an infinite one-dimensional classical lattice gas. Extending a result ofvan Hove we show that, for a large class of interactions, such a system has no phase… Expand

An Ising ferromagnet with discontinuous long-range order

- Physics
- 1971

An infinite one-dimensional Ising ferromagnetM with long-range interactions is constructed and proved to have the following properties. (1)M has an order-disorder phase transition at a finite… Expand

Non-existence of spontaneous magnetization in a one-dimensional Ising ferromagnet

- Physics
- 1969

AbstractIt is proved that an infinite linear chain of spins μj=±1, with an interaction energy
$$H = - \Sigma J(i - j)\mu _i \mu _j $$
has zero spontaneous magnetization at all finite temperatures,… Expand

The random walk representation of classical spin systems and correlation inequalities

- Mathematics
- 1982

Ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion. By using a mixed spinrandom walk representation we present a unified approach to… Expand

Correlation inequalities and the decay of correlations in ferromagnets

- Mathematics
- 1980

AbstractWe prove a variety of new correlation inequalities which bound intermediate distance correlations from below by long distance correlations. Typical is the following which holds for spin 1/2… Expand