Spontaneous symmetry breaking in annealed and quenched gauge field models

The structure of annealed and quenched models with localU(1) gauge invariance is studied in terms of the Helmholtz free energy. The first non-trivial, or one-loop, account of fluctuations in the annealed model suggests that spontaneous symmetry breaking occurs in two and three dimensions, through a first-order phase transition. Within the same approximation scheme, the quenched model displays a continuous phase transition. A more complete account of the fluctuations in the annealed model changes the nature of the transition to a continuous one, whereas spontaneous symmetry breaking is then absent with quenched disorder.     

Phase diagram of randomly polymerized membrane

Using a replica formalism, a generalization of a recent mean field model corresponding to the observed wrinkling transition in randomly polymerized membranes is presented. In this model we study the effects of global fluctuations of the surface normals to the flat membrane, which can be introduced by a random local field. In absence of these global fluctuations, we show that, the model exhibits both continuous and discontinuous transitions between flat and wrinkled phases, contrary to what has been predicted by Bensimon et al. and Attal et al. Phase diagrams both in replica symmetry and in breaking of replica symmetry in sense of Almeida and Thouless are given. We have also investigated the effects of global fluctuations on the replica symmetry phase diagram. We show that, the wrinkled phase is favored and the flat phase is unstable. For large global fluctuations, the transition between wrinkled and flat phases becomes first order. Received: 3 December 1997 / Revised: 31 March 1998 / Accepted: 3 August 1998     

First-order phase transition and phase coexistence in a spin-glass model

We study the mean-field static solution of the Blume-Emery-Griffiths-Capel model with quenched disorder, an Ising-spin lattice gas with random magnetic interaction. The thermodynamics is worked out in the full replica symmetry breaking scheme. The model exhibits a high temperature/low density paramagnetic phase. As temperature decreases or density increases, a phase transition to a full replica symmetry breaking spin-glass phase occurs. The nature of the transition can be either of the second order or, at temperature below a given critical value, of the first order in the Ehrenfest sense, with a discontinuous jump of the order parameter, a latent heat, and coexistence of phases.     

Exothermic Transition to a Future Susy Universe

The Higgs sector of the MSSM may be extended to solve the μ problem by the addition of a gauge singlet scalar field. We consider an extended Higgs model. For simplicity we consider the case where all the fields in the scalar sector are real. We analyze the vacuum structure of the model. We address the question of an exothermic phase transition from a broken susy phase with electroweak symmetry breaking (our current universe) to an exact susy phase with electroweak symmetry breaking (future susy universe).     

Spontaneous breaking of four-fold rotational symmetry in two-dimensional electronic systems explained as a continuous topological transition

The Fermi liquid approach is applied to the problem of spontaneous violation of the C ₄ symmetry in strongly correlated two-dimensional electronic systems on a square lattice. The symmetry breaking is traced to the existence of a topological phase transition. This continuous transition is triggered when the Fermi line, driven by the quasiparticle interactions, reaches the van Hove saddle points, where the group velocity vanishes and the density of states becomes singular. An unconventional Fermi liquid emerges beyond the implicated quantum critical point.     

On the second-order phase transition initiated by a strong electron-phonon interaction

A model of a crystal with a strong electron-phonon interaction that initiates a second-order phase transition has been considered. The purpose of the study is to determine the temperature dependence of the thermodynamic potential of the symmetric phase in the temperature range in the immediate vicinity of the transition temperature T _C for this model. The problem has been solved using the quantum Matsubara Green’s function approach, which takes into account the influence of both thermal and quantum fluctuations. It has been demonstrated that fluctuation coherent deformations of the crystal lattice with the same symmetry as in the ordered phase appear to be energetically favorable at T > T _C due to the interaction with the electronic subsystem. The results obtained have made it possible to construct the model of the second-order phase transition near the Curie point T _C.     

Application of statistical mechanics to combinatorial optimization problems: The chromatic number problem andq-partitioning of a graph

Methods of statistical mechanics are applied to two important NP-complete combinatorial optimization problems. The first is the chromatic number problem, which seeks the minimal number of colors necessary to color a graph such that no two sites connected by an edge have the same color. The second is partitioning of a graph intoq equal subgraphs so as to minimize intersubgraph connections. Both models are mapped into a frustrated Potts model, which is related to theq- state Potts spin glass. For the first problem, we obtain very good agreement with numerical simulations and theoretical bounds using the annealed approximation. The quenched model is also discussed. For the second problem we obtain analytic and numerical results by evaluating the groundstate energy of theq=3 and 4 Potts spin glass using Parisi's replica symmetry breaking. We also perform some numerical simulations to test the theoretical result and obtain very good agreement.     

Experimental evidence of a phase transition in a closed turbulent flow

We experimentally study the susceptibility to symmetry breaking of a closed turbulent von Kármán swirling flow from Re=150 to Re?10?. We report a divergence of this susceptibility at an intermediate Reynolds number Re=Re(χ)?90,000 which gives experimental evidence that such a highly space and time fluctuating system can undergo a "phase transition." This transition is furthermore associated with a peak in the amplitude of fluctuations of the instantaneous flow symmetry corresponding to intermittencies between spontaneously symmetry breaking metastable states.     

Exact results on the random Potts model

We present a number of exact results on the random-bond,q-state Potts model. The quenched model on any finite planar graph or lattice is shown to obey a duality relation for general type of bond-randomness. In the annealed case, the solution of the model reduces to that of the regular (nonrandom) Potts model on the corresponding lattice. Explicit knowledge of the critical parameters of theq-state Potts model in two dimensions allows us to evaluate exactly the phase diagram of the annealed model on the square, triangular and honeycomb lattices. We discuss the behavior near the (random) critical point and comment on the relationship between the quenched and annealed systems. The exact phase diagram of the annealed system is obtained for the bond-diluted model and the spin-glass model with and without dilutions.Work supported in part by NSF grant No. DMR-78-18808     

Abelian duality, confinement, and chiral-symmetry breaking in a SU(2) QCD-like theory

We analyze the vacuum structure of SU(2) QCD with multiple massless adjoint representation fermions formulated on a small spatial S(1) x R(3). The absence of thermal fluctuations, and the fact that quantum fluctuations favor the vacuum with unbroken center symmetry in a weakly coupled regime, renders the interesting dynamics of these theories analytically calculable. Confinement and the generation of the mass gap in the gluonic sector are shown analytically. In this regime, theory exhibits confinement without continuous chiral-symmetry breaking. However, a flavor singlet chiral condensate (which breaks a discrete chiral symmetry) persists at arbitrarily small S(1). Under certain reasonable assumptions, we show that the theory exhibits a zero temperature chiral phase transition in the absence of any change in spatial center symmetry realizations.     

Freezing of Classical Fluids in a Quenched Random Potential

The phase diagram of a hard-sphere fluid in the presence of a random pinning potential is studied analytically and numerically. In the analytic work, replicas are introduced for averaging over the quenched disorder, and the hypernetted chain approximation is used to calculate density correlations in the replicated liquid. The freezing transition of the liquid into a nearly crystalline state is studied using a density-functional approach, and the liquid to glass transition is studied using a phenomenological replica symmetry breaking approach. In the numerical work, local minima of a discretized version of the Ramakrishnan-Yussouff free-energy functional are located and the phase diagram in the density-disorder plane is obtained from an analysis of the relative stability of these minima. Both approaches lead to similar results for the phase diagram. The first-order liquid to crystalline solid transition is found to change to a continuous liquid to glass transition as the strength of the disorder is increased above a threshold value.     

Analytical treatment of fluctuation spectra at the symmetry breaking chaos transition

The phase transition of characteristic exponents describing the fluctuations of temporal coarse grained quantities is investigated for the symmetry breaking bifurcation. The discussion of the eigenvalues of the transfer operator clarifies the origin of the phase transition and leads to scaling relations in the vicinity of the bifurcation point.     

Mean Field Analysis of Low–Dimensional Systems

For low–dimensional systems, (i.e. 2D and, to a certain extent, 1D) it is proved that mean–field theory can provide an asymptotic guideline to the phase structure of actual systems. In particular, for attractive pair interactions that are sufficiently “spead out” according to an exponential (Yukawa) potential it is shown that the energy, free energy and, in particular, the block magnetization (as defined on scales that are large compared with the lattice spacing but small compared to the range of the interaction) will only take on values near to those predicted by the associated mean–field theory. While this applies for systems in all dimensions, the significant applications are for d = 2 where it is shown: (a) If the mean–field theory has a discontinuous phase transition featuring the breaking of a discrete symmetry then this sort of transition will occur in the actual system. Prominent examples include the two–dimensional q = 3 state Potts model. (b) If the mean–field theory has a discontinuous transition accompanied by the breaking of a continuous symmetry, the thermodynamic discontinuity is preserved even if the symmetry breaking is forbidden in the actual system. E.g. the two–dimensional O(3) nematic liquid crystal. Further it is demonstrated that mean–field behavior in the vicinity of the magnetic transition for layered Ising and XY systems also occurs in actual layered systems (with spread–out interactions) even if genuine magnetic ordering is precluded.     

The flattening of the average potential in models with fermions

The average potential is a scale dependent scalar effective potential. In a phase with spontaneous symmetry breaking its inner region becomes flat as the averaging extends over infinite volume and the average potential approaches the convex effective potential. Fermion fluctuations affect the shape of the average potential in this region and its flattening with decreasing physical scale. They have to be taken into account to find the true minimum of the scalar potential which determines the scale of spontaneous symmetry breaking.     

Phase transition in the scalar noise model of collective motion in three dimensions

We consider disorder-order phase transitions in the three-dimensional version of the scalar noise model (SNM) of flocking. Our results are analogous to those found for the two-dimensional case [CITE]. For small velocity (v≤0.1) a continuous, second-order phase transition is observable, with the diffusion of nearby particles being isotropic. By increasing the particle velocities the phase transition changes to first order, and the diffusion becomes anisotropic. The first-order transition in the latter case is probably caused by the interplay between anisotropic diffusion and periodic boundary conditions, leading to a boundary condition dependent symmetry breaking of the solutions.     

Statistical mechanics and stability of random lattice field theory

The averaging procedure in the random lattice field theory is studied by viewing it as a statistical mechanics of a system of classical particles. The corresponding thermodynamic phase is shown to determine the random lattice configuration which contributes dominantly to the generating function. The non-abelian gauge theory in four (space plus time) dimensions in the annealed and quenched averaging versions is shown to exist as an ideal classical gas, implying that macroscopically homogeneous configurations dominate the configurational averaging. For the free massless scalar field theory with O(n) global symmetry, in the annealed average, the pressure becomes negative for dimensions greater than two when n exceeds a critical number. This implies that macroscopically inhomogeneous collapsed configurations contribute dominantly. In the quenched averaging, the collapse of the massless scalar field theory is prevented and the system becomes an ideal gas which is at infinite temperature. Our results are obtained using exact scaling analysis. We also show approximately that SU(N) gauge theory collapses for dimensions greater than four in the annealed average. Within the same approximation, the collapse is prevented in the quenched average. We also obtain exact scaling differential equations satisfied by the generating function and physical quantities.     

The chiral magnetic effect and chiral symmetry breaking in <Emphasis Type="Italic">SU</Emphasis>(3) quenched lattice gauge theory

We study some properties of the non-Abelian vacuum induced by strong external magnetic field. We perform calculations in the quenched SU(3) lattice gauge theory with tadpole-improved Lüscher-Weisz action and chirally invariant lattice Dirac operator. The following results are obtained: The chiral symmetry breaking is enhanced by the magnetic field. The chiral condensate depends on the strength of the applied field as a power function with exponent ν = 1.6 ± 0.2. There is a paramagnetic polarization of the vacuum. The corresponding susceptibility and other magnetic properties are calculated and compared with the theoretical estimations. There are nonzero local fluctuations of the chirality and electromagnetic current, which grow with the magnetic field strength. These fluctuations can be a manifestation of the Chiral Magnetic Effect.     

Critical Properties and Finite-Size Estimates for the Depinning Transition of Directed Random Polymers

We consider models of directed random polymers interacting with a defect line, which are known to undergo a pinning/depinning (or localization/delocalization) phase transition. We are interested in critical properties and we prove, in particular, finite-size upper bounds on the order parameter (the contact fraction) in a window around the critical point, shrinking with the system size. Moreover, we derive a new inequality relating the free energy F and an annealed exponent μ which describes extreme fluctuations of the polymer in the localized region. For the particular case of a (1+1)-dimensional interface wetting model, we show that this implies an inequality between the critical exponents which govern the divergence of the disorder-averaged correlation length and of the typical one. Our results are based on the recently proven smoothness property of the depinning transition in presence of quenched disorder and on concentration of measure ideas.     

On the nature of the low-temperature phase in discontinuous mean-field spin glasses

The low-temperature phase of discontinuous mean-field spin glasses is generally described by a one-step replica symmetry breaking (1RSB) ansatz. The Gardner transition, i.e. a very-low-temperature phase transition to a full replica symmetry breaking (FRSB) phase, is often regarded as an inessential, and somehow exotic phenomenon. In this paper we show that the metastable states which are relevant for the out-of-equilibrium dynamics of such systems are always in a FRSB phase. The only exceptions are (to the best of our knowledge) the p-spin spherical model and the random energy model (REM). We also discuss the consequences of our results for aging dynamics and for local search algorithms in hard combinatorial problems. Received 10 February 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: Federico.Ricci@roma1.infn.it RID="b" ID="b"UMR 8549, Unité Mixte de Recherche du Centre National de la Recherche Scientifique et de l' école Normale Supérieure