Percolation and number of phases in the 2D Ising model

We reconsider the percolation approach of Russo, Aizenman and Higuchi for showing that there exist only two phases in the Ising model on the square lattice. We give a fairly short alternative proof which is only based on FKG monotonicity and avoids the use of GKS-type inequalities originally needed for some background results. Our proof extends to the Ising model on other planar lattices such as the triangular and honeycomb lattice. We can also treat the Ising antiferromagnet in an external field and the hard-core lattice gas model on $Z^2$.Comment: 22 pages. Further details on extensions. To appear in J.Math.Phys., special issue on `Probabilistic Methods in Statistical Physics', March 200

A Finite-Volume Version of Aizenman-Higuchi Theorem for the 2d Ising Model

In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model are convex combinations of the two pure phases. We present here a new approach to this result, with a number of advantages: (i) We obtain an optimal finite-volume, quantitative analogue (implying the classical claim); (ii) the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon; (iii) this new approach might be applicable to systems for which the classical method fails.Comment: A couple of typos corrected. To appear in Probab. Theory Relat. Field

A model with simultaneous first and second order phase transitions

We introduce a two dimensional nonlinear XY model with a second order phase transition driven by spin waves, together with a first order phase transition in the bond variables between two bond ordered phases, one with local ferromagnetic order and another with local antiferromagnetic order. We also prove that at the transition temperature the bond-ordered phases coexist with a disordered phase as predicted by Domany, Schick and Swendsen. This last result generalizes the result of Shlosman and van Enter (cond-mat/0205455). We argue that these phenomena are quite general and should occur for a large class of potentials.Comment: 7 pages, 7 figures using pstricks and pst-coi

Mott transition in lattice boson models

We use mathematically rigorous perturbation theory to study the transition between the Mott insulator and the conjectured Bose-Einstein condensate in a hard-core Bose-Hubbard model. The critical line is established to lowest order in the tunneling amplitude.Comment: 20 page

Correlation inequalities for classical and quantum XY models

We review correlation inequalities of truncated functions for the classical and quantum XY models. A consequence is that the critical temperature of the XY model is necessarily smaller than that of the Ising model, in both the classical and quantum cases. We also discuss an explicit lower bound on the critical temperature of the quantum XY model.Comment: 13 pages. Submitted to the volume "Advances in Quantum Mechanics: contemporary trends and open problems" of the INdAM-Springer series, proceedings of the INdAM meeting "Contemporary Trends in the Mathematics of Quantum Mechanics" (4-8 July 2016) organised by G. Dell'Antonio and A. Michelangel

Identification of Melatonin-Regulated Genes in the Ovine Pituitary Pars Tuberalis, a Target Site for Seasonal Hormone Control

The pars tuberalis (PT) of the pituitary gland expresses a high density of melatonin (MEL) receptors and is believed to regulate seasonal physiology by decoding changes in nocturnal melatonin secretion. Circadian clock genes are known to be expressed in the PT in response to the decline (Per1) and onset (Cry1) of MEL secretion, but to date little is known of other molecular changes in this key MEL target site. To identify transcriptional pathways that may be involved in the diurnal and photoperiod-transduction mechanism, we performed a whole genome transcriptome analysis using PT RNA isolated from sheep culled at three time points over the 24-h cycle under either long or short photoperiods. Our results reveal 153 transcripts where expression differs between photoperiods at the light-dark transition and 54 transcripts where expression level was more globally altered by photoperiod (all time points combined). Cry1 induction at night was associated with up-regulation of genes coding for NeuroD1 (neurogenic differentiation factor 1), Pbef / Nampt (nicotinamide phosphoribosyltransferase) , Hif1α (hypoxia-inducible factor-1α), and Kcnq5 (K channel) and down-regulation of Rorβ, a key clock gene regulator. Using in situ hybridization, we confirmed day-night differences in expression for Pbef / Nampt, NeuroD1, and Rorβ in the PT. Treatment of sheep with MEL increased PT expression for Cry1, Pbef / Nampt, NeuroD1, and Hif1α, but not Kcnq5. Our data thus reveal a cluster of Cry1-associated genes that are acutely responsive to MEL and novel transcriptional pathways involved in MEL action in the PT

Surface tension in the dilute Ising model. The Wulff construction

We study the surface tension and the phenomenon of phase coexistence for the Ising model on \mathbbm{Z}^d ($d \geqslant 2$) with ferromagnetic but random couplings. We prove the convergence in probability (with respect to random couplings) of surface tension and analyze its large deviations : upper deviations occur at volume order while lower deviations occur at surface order. We study the asymptotics of surface tension at low temperatures and relate the quenched value $\tau^q$ of surface tension to maximal flows (first passage times if $d = 2$). For a broad class of distributions of the couplings we show that the inequality $\tau^a \leqslant \tau^q$ -- where $\tau^a$ is the surface tension under the averaged Gibbs measure -- is strict at low temperatures. We also describe the phenomenon of phase coexistence in the dilute Ising model and discuss some of the consequences of the media randomness. All of our results hold as well for the dilute Potts and random cluster models

Recurrent Variational Approach to the Two-Leg Hubbard Ladder

We applied the Recurrent Variational Approach to the two-leg Hubbard ladder. At half-filling, our variational Ansatz was a generalization of the resonating valence bond state. At finite doping, hole pairs were allowed to move in the resonating valence bond background. The results obtained by the Recurrent Variational Approach were compared with results from Density Matrix Renormalization Group.Comment: 10 pages, 14 Postscript figure

Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.