Dimensional Reduction for Directed Branched Polymers

Dimensional reduction occurs when the critical behavior of one system can be related to that of another system in a lower dimension. We show that this occurs for directed branched polymers (DBP) by giving an exact relationship between DBP models in D+1 dimensions and repulsive gases at negative activity in D dimensions. This implies relations between exponents of the two models: $\gamma(D+1)=\alpha(D)$ (the exponent describing the singularity of the pressure), and $\nu_{\perp}(D+1)=\nu(D)$ (the correlation length exponent of the repulsive gas). It also leads to the relation $\theta(D+1)=1+\sigma(D)$, where $\sigma(D)$ is the Yang-Lee edge exponent. We derive exact expressions for the number of DBP of size N in two dimensions.Comment: 7 pages, 1 eps figure, ref 24 correcte

Numerical study of the transition of the four dimensional Random Field Ising Model

We study numerically the region above the critical temperature of the four dimensional Random Field Ising Model. Using a cluster dynamic we measure the connected and disconnected magnetic susceptibility and the connected and disconnected overlap susceptibility. We use a bimodal distribution of the field with $h_R=0.35T$ for all temperatures and a lattice size L=16. Through a least-square fit we determine the critical exponents $\gamma$ and $\bar{\gamma}$. We find the magnetic susceptibility and the overlap susceptibility diverge at two different temperatures. This is coherent with the existence of a glassy phase above $T_c$. Accordingly with other simulations we find $\bar{\gamma}=2\gamma$. In this case we have a scaling theory with two indipendet critical exponentsComment: 10 pages, 2 figures, Late

Critical temperature and density of spin-flips in the anisotropic random field Ising model

We present analytical results for the strongly anisotropic random field Ising model, consisting of weakly interacting spin chains. We combine the mean-field treatment of interchain interactions with an analytical calculation of the average chain free energy (``chain mean-field'' approach). The free energy is found using a mapping on a Brownian motion model. We calculate the order parameter and give expressions for the critical random magnetic field strength below which the ground state exhibits long range order and for the critical temperature as a function of the random magnetic field strength. In the limit of vanishing interchain interactions, we obtain corrections to the zero-temperature estimate by Imry and Ma [Phys. Rev. Lett. 35, 1399 (1975)] of the ground state density of domain walls (spin-flips) in the one-dimensional random field Ising model. One of the problems to which our model has direct relevance is the lattice dimerization in disordered quasi-one-dimensional Peierls materials, such as the conjugated polymer trans-polyacetylene.Comment: 28 pages, revtex, 4 postscript figures, to appear in Phys. Rev.

Weighted Mean Field Theory for the Random Field Ising Model

We consider the mean field theory of the Random Field Ising Model obtained by weighing the many solutions of the mean field equations with Boltzmann-like factors. These solutions are found numerically in three dimensions and we observe critical behavior arising from the weighted sum. The resulting exponents are calculated.Comment: 15 pages of tex using harvmac. 8 postscript figures (fig3.ps is large) in a separate .uu fil

Criticality in one dimension with inverse square-law potentials

It is demonstrated that the scaled order parameter for ferromagnetic Ising and three-state Potts chains with inverse-square interactions exhibits a universal critical jump, in analogy with the superfluid density in helium films. Renormalization-group arguments are combined with numerical simulations of systems containing up to one million lattice sites to accurately determine the critical properties of these models. In strong contrast with earlier work, compelling quantitative evidence for the Kosterlitz--Thouless-like character of the phase transition is provided.Comment: To appear in Phys. Rev. Let

Real-space renormalization group for the random-field Ising model

We present real--space renormalization group (RG) calculations of the critical properties of the random--field Ising model on a cubic lattice in three dimensions. We calculate the RG flows in a two--parameter truncation of the Hamiltonian space. As predicted, the transition at finite randomness is controlled by a zero temperature, disordered critical fixed point, and we exhibit the universal crossover trajectory from the pure Ising critical point. We extract scaling fields and critical exponents, and study the distribution of barrier heights between states as a function of length scale.Comment: 12 pages, CU-MSC-757

Monte Carlo study of the random-field Ising model

Using a cluster-flipping Monte Carlo algorithm combined with a generalization of the histogram reweighting scheme of Ferrenberg and Swendsen, we have studied the equilibrium properties of the thermal random-field Ising model on a cubic lattice in three dimensions. We have equilibrated systems of LxLxL spins, with values of L up to 32, and for these systems the cluster-flipping method appears to a large extent to overcome the slow equilibration seen in single-spin-flip methods. From the results of our simulations we have extracted values for the critical exponents and the critical temperature and randomness of the model by finite size scaling. For the exponents we find nu = 1.02 +/- 0.06, beta = 0.06 +/- 0.07, gamma = 1.9 +/- 0.2, and gammabar = 2.9 +/- 0.2.Comment: 12 pages, 6 figures, self-expanding uuencoded compressed PostScript fil

Directed polymers and interfaces in random media : free-energy optimization via confinement in a wandering tube

We analyze, via Imry-Ma scaling arguments, the strong disorder phases that exist in low dimensions at all temperatures for directed polymers and interfaces in random media. For the uncorrelated Gaussian disorder, we obtain that the optimal strategy for the polymer in dimension $1+d$ with $0<d<2$ involves at the same time (i) a confinement in a favorable tube of radius $R_S \sim L^{\nu_S}$ with $\nu_S=1/(4-d)<1/2$ (ii) a superdiffusive behavior $R \sim L^{\nu}$ with $\nu=(3-d)/(4-d)>1/2$ for the wandering of the best favorable tube available. The corresponding free-energy then scales as $F \sim L^{\omega}$ with $\omega=2 \nu-1$ and the left tail of the probability distribution involves a stretched exponential of exponent $\eta= (4-d)/2$. These results generalize the well known exact exponents $\nu=2/3$, $\omega=1/3$ and $\eta=3/2$ in $d=1$, where the subleading transverse length $R_S \sim L^{1/3}$ is known as the typical distance between two replicas in the Bethe Ansatz wave function. We then extend our approach to correlated disorder in transverse directions with exponent $\alpha$ and/or to manifolds in dimension $D+d=d_{t}$ with $0<D<2$. The strategy of being both confined and superdiffusive is still optimal for decaying correlations ($\alpha<0$), whereas it is not for growing correlations ($\alpha>0$). In particular, for an interface of dimension $(d_t-1)$ in a space of total dimension $5/3<d_t<3$ with random-bond disorder, our approach yields the confinement exponent $\nu_S = (d_t-1)(3-d_t)/(5d_t-7)$. Finally, we study the exponents in the presence of an algebraic tail $1/V^{1+\mu}$ in the disorder distribution, and obtain various regimes in the $(\mu,d)$ plane.Comment: 19 page

Fisher Renormalization for Logarithmic Corrections

For continuous phase transitions characterized by power-law divergences, Fisher renormalization prescribes how to obtain the critical exponents for a system under constraint from their ideal counterparts. In statistical mechanics, such ideal behaviour at phase transitions is frequently modified by multiplicative logarithmic corrections. Here, Fisher renormalization for the exponents of these logarithms is developed in a general manner. As for the leading exponents, Fisher renormalization at the logarithmic level is seen to be involutory and the renormalized exponents obey the same scaling relations as their ideal analogs. The scheme is tested in lattice animals and the Yang-Lee problem at their upper critical dimensions, where predictions for logarithmic corrections are made.Comment: 10 pages, no figures. Version 2 has added reference

Kosterlitz-Thouless Transition and Short Range Spatial Correlations in an Extended Hubbard Model

We study the competition between intersite and local correlations in a spinless two-band extended Hubbard model by taking an alternative limit of infinite dimensions. We find that the intersite density fluctuations suppress the charge Kondo energy scale and lead to a Fermi liquid to non-Fermi liquid transition for repulsive on-site density-density interactions. In the absence of intersite interactions, this transition reduces to the known Kosterlitz-Thouless transition. We show that a new line of non-Fermi liquid fixed points replace those of the zero intersite interaction problem.Comment: 11 pages, 2 figure