Translation Invariant Equilibrium States Research Articles

Vanishing self-diffusivity in Ginibre interacting Brownian motions in two dimensions

AbstractWe prove that the tagged particles of infinitely many Brownian particles in $$ {\mathbb {R}} ^2$$ R 2 interacting via a logarithmic (two-dimensional Coulomb) potential with inverse temperature $$ \beta = 2 $$ β = 2 are sub-diffusive. The associated unlabeled diffusion is reversible with respect to the Ginibre random point field, and the dynamics are thus referred to as the Ginibre interacting Brownian motion. If the interacting Brownian particles have interaction potential $$ \Psi $$ Ψ of Ruelle class and the total system starts in a translation invariant equilibrium state, then the tagged particles are always diffusive if the dimension $$ d$$ d of the space $$ {\mathbb {R}}^{d} $$ R d is greater than or equal to two. That is, the tagged particles are always non-degenerate under diffusive scaling. Our result is, therefore, contrary to known results. The Ginibre random point field has various levels of geometric rigidity. Our results reveal that the geometric property of infinite particle systems affects the dynamical property of the associated stochastic dynamics.

Equilibrium States and Their Macroscopic Uniformity in Fermion Lattice Systems

We study equilibrium states and their macroscopic uniformity in the fermion lattice system satisfying the assumptions presented by Araki and Moriya [5]. We show that every translation-invariant KMS state of the even part of the fermion algebra extends uniquely to a translation-invariant KMS state of the whole fermion algebra. The macroscopic uniformity is established for an extremal translation-invariant equilibrium state of the fermion algebra.

Breaking of periodicity at positive temperatures

We discuss a classical lattice gas model without periodic or quasiperiodic ground states. The only ground state configurations of our model are nonperiodic Thue-Morse sequences. We show that low temperature phases of such models can be ordered. In fact, we prove the existence of an ordered (nonmixing) low temperature translation invariant equilibrium state which has nonperiodic Gibbs states in its extremal decomposition.

Generic triviality of phase diagrams in spaces of long-range interactions

We show that interactions with multiple translation-invariant equilibrium states form a very “thin” set in spaces of long-range interactions of classical or quantum lattice systems. For example, generic finite-dimensional subspaces do not intersect this set. This constitutes a severe violation of the Gibbs Phase Rule.

On some variational approximations in two-dimensional classical lattice systems

A new derivation is presented of some variational approximations for classical lattice systems that belong to the class of cluster-variation methods, among them the well-known Bethe-Peierls and Kramers-Wannier approximations. The limiting behavior of a hierarchical sequence of cluster-variation approximations, the so-calledC hierarchy, is discussed. It is shown that this hierarchy provides a monotonically decreasing sequence of upper boundsfn on the free energy per lattice sitef and thatfn → f asn → ∞. Our results are based on extension theorems for states given on subsets of the lattice, which might be of some independent interest, and on an application of transfer matrix concepts to the variational characterization of translation-invariant equilibrium states.

Translation symmetry breaking in four dimensional lattice gauge theories

We consider lattice gauge theories with finite abelian groupG in the weak coupling regime. It is shown that there is only one translation invariant equilibrium state for the infinite system. In four dimensions we construct a nontranslation invariant equilibrium state, describing an infinite system with localized magnetic flux tube, starting and ending at infinity.

Exact variational methods and cluster-variation approximations

It is shown that for classical,d-dimensional lattice models with finite-range interactions the translation-invariant equilibrium states have the property that their mean entropy is completely determined by their restriction to a subset of the lattice that is infinite in (d−1) dimensions and has a width equal to the range of the interaction in the dth dimension. This property is used to show proper convergence toward the exact result for a hierarchy of approximations of the cluster-variation method that uses one-dimensionally increasing basis clusters in a two-dimensional lattice.

Spin waves, vortices, and the structure of equilibrium states in the classicalXY model

We prove that, for spin systems with a continuous symmetry group on lattices of arbitrary dimension, the surface tension vanishes at all temperatures. For the classicalXY model in zero magnetic field, this result is shown to imply absence of interfaces in the thermodynamic limit, at arbitrary temperature. We show that, at values of the temperature at which the free energy of that model is continuously differentiable, i.e. at all except possibly countably many temperatures, there iseither aunique translation-invariant equilibrium state, or all such states are labelled by the elements of the symmetry group, SO(2). Moreover, there areno non-translation-invariant, but periodic equilibrium states. We also reconsider the representation of theXY model as a gas of spin waves and vortices and discuss the possibility that, in four or more dimensions, translation invariance may be broken by imposing boundary conditions which force an (open) vortex sheet through the system. Among our main tools are new correlation inequalities.

More surprises in the general theory of lattice systems

I use Israel's methods to prove new theorems of “ubiquitous pathology” for classical and quantum lattice systems. The main result is the following: Let Φ be any interaction and ϱ be any translation-invariant equilibrium state for Φ (extremal or not). Then there exists a sequence {Φ k } of interactions converging to Φ, having extremal (or even unique) translation-invariant equilibrium states ϱ k , such that {ϱ k } converges to ϱ. In certain situations the perturbations Φ k −Φ can be chosen to lie in a cone of “antiferromagnetic pair interactions.” I discuss the connection with results of Daniëls and van Enter, and point out an application to the one-dimensional ferromagnetic Ising model with 1/r 2 interaction (Thouless effect).

Translation invariant equilibrium states of ferromagnetic Abelian lattice systems

The structure of the set of all translation invariant equilibrium states is determined for all temperatures, for which the free energy is differentiable. Models with several phase transitions are discussed rigorously.

On the uniqueness of the equilibrium state for plane rotators

We study the classical statistical mechanics of the plane rotator, and show that there is a unique translation invariant equilibrium state in zero external field, if there is no spontaneous magnetization. Moreover, this state is then extremal in the equilibrium states. In particular there is a unique phase for the two dimensional rotator, and a unique phase for the three dimensional rotator above the critical temperature. It is also shown that in a sufficiently large external field the Lee-Yang theorem implies uniqueness of the equilibrium state.

Ferromagnetic spin systems at low temperatures

Finite-spin systems with ferromagnetic, finite range interactions are considered. Using Ruelle's theorem on zeros of polynomials contracted according to Asano, analyticity of pressure and correlation functions is proved. A description of all translation invariant equilibrium states at low temperatures for a large class of systems is given.

A family of equilibrium states relevant to low temperature behavior of spin 1/2 classical ferromagnets. Breaking of translation symmetry

An analysis of a family of equilibrium states is performed which, combined with our previous work, allows to describe all translation invariant equilibrium states of spin 1/2 classical ferromagnetic systems with finite range interactions at low temperatures. A model is described with continuously many equilibrium states for low temperatures.