On infrared divergences in spin glasses

By studying the structure of infrared divergences in a toy propagator in the replica approach to the Ising spin glass below $T_c$, we suggest a possible cancellation mechanism which could decrease the degree of singularity in the loop expansion.Comment: 13 pages, Latex , revised versio

Symmetry breaking via fermion 4-point functions

We construct the effective action and gap equations for nonperturbative fermion 4-point functions. Our results apply to situations in which fermion masses can be ignored, which is the case for theories of strong flavor interactions involving standard quarks and leptons above the electroweak scale. The structure of the gap equations is different from what a naive generalization of the 2-point case would suggest, and we find for example that gauge exchanges are insufficient to generate nonperturbative 4-point functions when the number of colors is large.Comment: 36 pages, uses Revtex and eps files for figure

Recursive Graphical Construction of Feynman Diagrams in phi^4 Theory: Asymmetric Case and Effective Energy

The free energy of a multi-component scalar field theory is considered as a functional W[G,J] of the free correlation function G and an external current J. It obeys non-linear functional differential equations which are turned into recursion relations for the connected Greens functions in a loop expansion. These relations amount to a simple proof that W[G,J] generates only connected graphs and can be used to find all such graphs with their combinatoric weights. A Legendre transformation with respect to the external current converts the functional differential equations for the free energy into those for the effective energy Gamma[G,Phi], which is considered as a functional of the free correlation function G and the field expectation Phi. These equations are turned into recursion relations for the one-particle irreducible Greens functions. These relations amount to a simple proof that Gamma[G,J] generates only one-particle irreducible graphs and can be used to find all such graphs with their combinatoric weights. The techniques used also allow for a systematic investigation into resummations of classes of graphs. Examples are given for resumming one-loop and multi-loop tadpoles, both through all orders of perturbation theory. Since the functional differential equations derived are non-perturbative, they constitute also a convenient starting point for other expansions than those in numbers of loops or powers of coupling constants. We work with general interactions through four powers in the field.Comment: 34 pages; abstract expanded; section IV.E about absorption of tadpoles and one related reference added; eqs. (20) and (23) corrected; further references added; some minor beautifications; to be published by Phys.Rev.

Local excitations in mean field spin glasses

We address the question of geometrical as well as energetic properties of local excitations in mean field Ising spin glasses. We study analytically the Random Energy Model and numerically a dilute mean field model, first on tree-like graphs, equivalent to a replica symmetric computation, and then directly on finite connectivity random lattices. In the first model, characterized by a discontinuous replica symmetry breaking, we found that the energy of finite volume excitation is infinite whereas in the dilute mean field model, described by a continuous replica symmetry breaking, it slowly decreases with sizes and saturates at a finite value, in contrast with what would be naively expected. The geometrical properties of these excitations are similar to those of lattice animals or branched polymers. We discuss the meaning of these results in terms of replica symmetry breaking and also possible relevance in finite dimensional systems.Comment: 7 pages, 4 figures, accepted for publicatio

Statistical mechanics of the random K-SAT model

The Random K-Satisfiability Problem, consisting in verifying the existence of an assignment of N Boolean variables that satisfy a set of M=alpha N random logical clauses containing K variables each, is studied using the replica symmetric framework of diluted disordered systems. We present an exact iterative scheme for the replica symmetric functional order parameter together for the different cases of interest K=2, K>= 3 and K>>1. The calculation of the number of solutions, which allowed us [Phys. Rev. Lett. 76, 3881 (1996)] to predict a first order jump at the threshold where the Boolean expressions become unsatisfiable with probability one, is thoroughly displayed. In the case K=2, the (rigorously known) critical value (alpha=1) of the number of clauses per Boolean variable is recovered while for K>=3 we show that the system exhibits a replica symmetry breaking transition. The annealed approximation is proven to be exact for large K.Comment: 34 pages + 1 table + 8 fig., submitted to Phys. Rev. E, new section added and references update

Is the droplet theory for the Ising spin glass inconsistent with replica field theory?

Symmetry arguments are used to derive a set of exact identities between irreducible vertex functions for the replica symmetric field theory of the Ising spin glass in zero magnetic field. Their range of applicability spans from mean field to short ranged systems in physical dimensions. The replica symmetric theory is unstable for d>8, just like in mean field theory. For 6<d<8 and d<6 the resummation of an infinite number of terms is necessary to settle the problem. When d<8, these Ward-like identities must be used to distinguish an Almeida-Thouless line from the replica symmetric droplet phase.Comment: 4 pages. Accepted for publication in J.Phys.A. This is the accepted version with the following minor changes: one extra sentence in the abstract; footnote 2 slightly extended; last paragraph somewhat reformulate

Quenched Random Graphs

Spin models on quenched random graphs are related to many important optimization problems. We give a new derivation of their mean-field equations that elucidates the role of the natural order parameter in these models.Comment: 9 pages, report CPTH-A264.109

Stability of self-consistent solutions for the Hubbard model at intermediate and strong coupling

We present a general framework how to investigate stability of solutions within a single self-consistent renormalization scheme being a parquet-type extension of the Baym-Kadanoff construction of conserving approximations. To obtain a consistent description of one- and two-particle quantities, needed for the stability analysis, we impose equations of motion on the one- as well on the two-particle Green functions simultaneously and introduce approximations in their input, the completely irreducible two-particle vertex. Thereby we do not loose singularities caused by multiple two-particle scatterings. We find a complete set of stability criteria and show that each instability, singularity in a two-particle function, is connected with a symmetry-breaking order parameter, either of density type or anomalous. We explicitly study the Hubbard model at intermediate coupling and demonstrate that approximations with static vertices get unstable before a long-range order or a metal-insulator transition can be reached. We use the parquet approximation and turn it to a workable scheme with dynamical vertex corrections. We derive a qualitatively new theory with two-particle self-consistence, the complexity of which is comparable with FLEX-type approximations. We show that it is the simplest consistent and stable theory being able to describe qualitatively correctly quantum critical points and the transition from weak to strong coupling in correlated electron systems.Comment: REVTeX, 26 pages, 12 PS figure

The 3-SAT problem with large number of clauses in ∞\infty-replica symmetry breaking scheme

In this paper we analyze the structure of the UNSAT-phase of the overconstrained 3-SAT model by studying the low temperature phase of the associated disordered spin model. We derive the $\infty$ Replica Symmetry Broken equations for a general class of disordered spin models which includes the Sherrington - Kirkpatrick model, the Ising $p$-spin model as well as the overconstrained 3-SAT model as particular cases. We have numerically solved the $\infty$ Replica Symmetry Broken equations using a pseudo-spectral code down to and including zero temperature. We find that the UNSAT-phase of the overconstrained 3-SAT model is of the $\infty$-RSB kind: in order to get a stable solution the replica symmetry has to be broken in a continuous way, similarly to the SK model in external magnetic field.Comment: 19 pages, 7 figures; some section improved; iopart styl

The spherical 2+p spin glass model: an analytically solvable model with a glass-to-glass transition

We present the detailed analysis of the spherical s+p spin glass model with two competing interactions: among p spins and among s spins. The most interesting case is the 2+p model with p > 3 for which a very rich phase diagram occurs, including, next to the paramagnetic and the glassy phase represented by the one step replica symmetry breaking ansatz typical of the spherical p-spin model, other two amorphous phases. Transitions between two contiguous phases can also be of different kind. The model can thus serve as mean-field representation of amorphous-amorphous transitions (or transitions between undercooled liquids of different structure). The model is analytically solvable everywhere in the phase space, even in the limit where the infinite replica symmetry breaking ansatz is required to yield a thermodynamically stable phase.Comment: 21 pages, 18 figure