Stability of solutions of the Sherrington-Kirkpatrick model with respect to replications of the phase space

We use real replicas within the Thouless, Anderson and Palmer construction to investigate stability of solutions with respect to uniform scalings in the phase space of the Sherrington-Kirkpatrick model. We show that the demand of homogeneity of thermodynamic potentials leads in a natural way to a thermodynamically dependent ultrametric hierarchy of order parameters. The derived hierarchical mean-field equations appear equivalent to the discrete Parisi RSB scheme. The number of hierarchical levels in the construction is fixed by the global thermodynamic homogeneity expressed as generalized de Almeida Thouless conditions. A physical interpretation of a hierarchical structure of the order parameters is gained.Comment: REVTeX4, 22 pages, second extended version to be published in Phys. Rev.

Replica Symmetry Breaking and the Renormalization Group Theory of the Weakly Disordered Ferromagnet

We study the critical properties of the weakly disordered $p$-component ferromagnet in terms of the renormalization group (RG) theory generalized to take into account the replica symmetry breaking (RSB) effects coming from the multiple local minima solutions of the mean-field equations. It is shown that for $p < 4$ the traditional RG flows at dimensions $D=4-\epsilon$, which are usually considered as describing the disorder-induced universal critical behavior, are unstable with respect to the RSB potentials as found in spin glasses. It is demonstrated that for a general type of the Parisi RSB structures there exists no stable fixed points, and the RG flows lead to the {\it strong coupling regime} at the finite scale $R_{*} \sim \exp(1/u)$, where $u$ is the small parameter describing the disorder. The physical concequences of the obtained RG solutions are discussed. In particular, we argue, that discovered RSB strong coupling phenomena indicate on the onset of a new spin glass type critical behaviour in the temperature interval $\tau < \tau_{*} \sim \exp(-1/u)$ near $T_{c}$. Possible relevance of the considered RSB effects for the Griffith phase is also discussed.Comment: 32 pages, Late

Mean-field glass transition in a model liquid

We investigate the liquid-glass phase transition in a system of point-like particles interacting via a finite-range attractive potential in D-dimensional space. The phase transition is driven by an `entropy crisis' where the available phase space volume collapses dramatically at the transition. We describe the general strategy underlying the first-principles replica calculation for this type of transition; its application to our model system then allows for an analytic description of the liquid-glass phase transition within a mean-field approximation, provided the parameters are chosen suitably. We find a transition exhibiting all the features associated with an `entropy crisis', including the characteristic finite jump of the order parameter at the transition while the free energy and its first derivative remain continuous.Comment: 12 pages, 6 figure

Genus Zero Correlation Functions in c<1 String Theory

We compute N-point correlation functions of pure vertex operator states(DK states) for minimal models coupled to gravity. We obtain agreement with the matrix model results on analytically continuing in the numbers of cosmological constant operators and matter screening operators. We illustrate this for the cases of the $(2k-1,2)$ and $(p+1,p)$ models.Comment: 11 pages, LaTeX, IMSc--92/35. (revised) minor changes plus one reference adde

An analogue of the Magnus problem for associative algebras

We prove an analogue of the Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by n+k generators and k relations and has an n-element system of generators, then this algebra is a free algebra of rank n

Coherence properties and quantum state transportation in an optical conveyor belt

We have prepared and detected quantum coherences with long dephasing times at the level of single trapped cesium atoms. Controlled transport by an "optical conveyor belt" over macroscopic distances preserves the atomic coherence with slight reduction of coherence time. The limiting dephasing effects are experimentally identified and are of technical rather than fundamental nature. We present an analytical model of the reversible and irreversible dephasing mechanisms. Coherent quantum bit operations along with quantum state transport open the route towards a "quantum shift register" of individual neutral atoms.Comment: 4 pages, 3 figure

Monte Carlo Simulation of a Random-Field Ising Antiferromagnet

Phase transitions in the three-dimensional diluted Ising antiferromagnet in an applied magnetic field are analyzed numerically. It is found that random magnetic field in a system with spin concentration below a certain threshold induces a crossover from second-order phase transition to first-order transition to a new phase characterized by a spin-glass ground state and metastable energy states at finite temperatures.Comment: 10 pages, 11 figure

Free-energy distribution functions for the randomly forced directed polymer

We study the $1+1$-dimensional random directed polymer problem, i.e., an elastic string $\phi(x)$ subject to a Gaussian random potential $V(\phi,x)$ and confined within a plane. We mainly concentrate on the short-scale and finite-temperature behavior of this problem described by a short- but finite-ranged disorder correlator $U(\phi)$ and introduce two types of approximations amenable to exact solutions. Expanding the disorder potential $V(\phi,x) \approx V_0(x) + f(x) \phi(x)$ at short distances, we study the random force (or Larkin) problem with $V_0(x) = 0$ as well as the shifted random force problem including the random offset $V_0(x)$; as such, these models remain well defined at all scales. Alternatively, we analyze the harmonic approximation to the correlator $U(\phi)$ in a consistent manner. Using direct averaging as well as the replica technique, we derive the distribution functions ${\cal P}_{L,y}(F)$ and ${\cal P}_L(F)$ of free energies $F$ of a polymer of length $L$ for both fixed ($\phi(L) = y$) and free boundary conditions on the displacement field $\phi(x)$ and determine the mean displacement correlators on the distance $L$. The inconsistencies encountered in the analysis of the harmonic approximation to the correlator are traced back to its non-spectral correlator; we discuss how to implement this approximation in a proper way and present a general criterion for physically admissible disorder correlators $U(\phi)$.Comment: 16 pages, 5 figure

Ising exponents in the two-dimensional site-diluted Ising model

We study the site-diluted Ising model in two dimensions with Monte Carlo simulations. Using finite-size scaling techniques we compute the critical exponents observing deviations from the pure Ising ones. The differences can be explained as the effects of logarithmic corrections, without requiring to change the Universality Class.Comment: 7 pages, 2 postscript figures. Reference correcte

On touching random surfaces, two-dimensional quantum gravity and non-critical string theory

A set of physical operators which are responsible for touching interactions in the framework of c<1 unitary conformal matter coupled to 2D quantum gravity is found. As a special case the non-critical bosonic strings are considered. Some analogies with four dimensional quantum gravity are also discussed, e.g. creation-annihilation operators for baby universes, Coleman mechanism for the cosmological constant.Comment: 22 pages, Latex2e, 3 figure