Information Geometry and Phase Transitions

The introduction of a metric onto the space of parameters in models in Statistical Mechanics and beyond gives an alternative perspective on their phase structure. In such a geometrization, the scalar curvature, R, plays a central role. A non-interacting model has a flat geometry (R=0), while R diverges at the critical point of an interacting one. Here, the information geometry is studied for a number of solvable statistical-mechanical models.Comment: 6 pages with 1 figur

Fat and Thin Fisher Zeroes

We show that it is possible to determine the locus of Fisher zeroes in the thermodynamic limit for the Ising model on planar (``fat'') phi4 random graphs and their dual quadrangulations by matching up the real part of the high- and low-temperature branches of the expression for the free energy. Similar methods work for the mean-field model on generic, ``thin'' graphs. Series expansions are very easy to obtain for such random graph Ising models.Comment: 3 pages, LaTeX, Lattice2001(surfaces

Frustrating and Diluting Dynamical Lattice Ising Spins

We investigate what happens to the third order ferromagnetic phase transition displayed by the Ising model on various dynamical planar lattices (ie coupled to 2D quantum gravity) when we introduce annealed bond disorder in the form of either antiferromagnetic couplings or null couplings. We also look at the effect of such disordering for the Ising model on general $\phi^3$ and $\phi^4$ Feynman diagrams.Comment: 7pages, LaTex , LPTHE-ORSAY-94-5

Numerical studies of the two- and three-dimensional gauge glass at low temperature

We present results from Monte Carlo simulations of the two- and three-dimensional gauge glass at low temperature using the parallel tempering Monte Carlo method. Our results in two dimensions strongly support the transition being at T_c=0. A finite-size scaling analysis, which works well only for the larger sizes and lower temperatures, gives the stiffness exponent theta = -0.39 +/- 0.03. In three dimensions we find theta = 0.27 +/- 0.01, compatible with recent results from domain wall renormalization group studies.Comment: 7 pages, 10 figures, submitted to PR

Marginal Pinning of Quenched Random Polymers

An elastic string embedded in 3D space and subject to a short-range correlated random potential exhibits marginal pinning at high temperatures, with the pinning length $L_c(T)$ becoming exponentially sensitive to temperature. Using a functional renormalization group (FRG) approach we find $L_c(T) \propto \exp[(32/\pi)(T/T_{\rm dp})^3]$, with $T_{\rm dp}$ the depinning temperature. A slow decay of disorder correlations as it appears in the problem of flux line pinning in superconductors modifies this result, $\ln L_c(T)\propto T^{3/2}$.Comment: 4 pages, RevTeX, 1 figure inserte

A microscopic approach to critical phenomena at interfaces: an application to complete wetting in the Ising model

We study how the formalism of the Hierarchical Reference Theory (HRT) can be extended to inhomogeneous systems. HRT is a liquid state theory which implements the basic ideas of Wilson momentum shell renormalization group (RG) to microscopic Hamiltonians. In the case of homogeneous systems, HRT provides accurate results even in the critical region, where it reproduces scaling and non-classical critical exponents. We applied the HRT to study wetting critical phenomena in a planar geometry. Our formalism avoids the explicit definition of effective surface Hamiltonians but leads, close to the wetting transition, to the same renormalization group equation already studied by RG techiques. However, HRT also provides information on the non universal quantities because it does not require any preliminary coarse graining procedure. A simple approximation to the infinite HRT set of equations is discussed. The HRT evolution equation for the surface free energy is numerically integrated in a semi-infinite three-dimensional Ising model and the complete wetting phase transition is analyzed. A renormalization of the adsorption critical amplitude and of the wetting parameter is observed. Our results are compared to available Monte Carlo simulations.Comment: To be published in Phy. Rev.

The three-dimensional random field Ising magnet: interfaces, scaling, and the nature of states

The nature of the zero temperature ordering transition in the 3D Gaussian random field Ising magnet is studied numerically, aided by scaling analyses. In the ferromagnetic phase the scaling of the roughness of the domain walls, $w\sim L^\zeta$, is consistent with the theoretical prediction $\zeta = 2/3$. As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales as for a single second order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions. These dimensions are argued to be related to combinations of the energy scaling exponent, $\theta$, which determines the violation of hyperscaling, the correlation length exponent $\nu$, and the magnetization exponent $\beta$. The value $\beta = 0.017\pm 0.005$ is derived from the magnetization: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.Comment: 32 pp., 2 columns, 32 figure

Velocity-force characteristics of a driven interface in a disordered medium

Using a dynamic functional renormalization group treatment of driven elastic interfaces in a disordered medium, we investigate several aspects of the creep-type motion induced by external forces below the depinning threshold $f_c$: i) We show that in the experimentally important regime of forces slightly below $f_c$ the velocity obeys an Arrhenius-type law $v\sim\exp[-U(f)/T]$ with an effective energy barrier $U(f)\propto (f_{c}-f)$ vanishing linearly when f approaches the threshold $f_c$. ii) Thermal fluctuations soften the pinning landscape at high temperatures. Determining the corresponding velocity-force characteristics at low driving forces for internal dimensions d=1,2 (strings and interfaces) we find a particular non-Arrhenius type creep $v\sim \exp[-(f_c(T)/f)^{\mu}]$ involving the reduced threshold force $f_c(T)$ alone. For d=3 we obtain a similar v-f characteristic which is, however, non-universal and depends explicitly on the microscopic cutoff.Comment: 9 pages, RevTeX, 3 postscript figure

New Criticality of 1D Fermions

One-dimensional massive quantum particles (or 1+1-dimensional random walks) with short-ranged multi-particle interactions are studied by exact renormalization group methods. With repulsive pair forces, such particles are known to scale as free fermions. With finite $m$-body forces (m = 3,4,...), a critical instability is found, indicating the transition to a fermionic bound state. These unbinding transitions represent new universality classes of interacting fermions relevant to polymer and membrane systems. Implications for massless fermions, e.g. in the Hubbard model, are also noted. (to appear in Phys. Rev. Lett.)Comment: 10 pages (latex), with 2 figures (not included