Ensemble dependence in the Random transverse-field Ising chain

In a disordered system one can either consider a microcanonical ensemble, where there is a precise constraint on the random variables, or a canonical ensemble where the variables are chosen according to a distribution without constraints. We address the question as to whether critical exponents in these two cases can differ through a detailed study of the random transverse-field Ising chain. We find that the exponents are the same in both ensembles, though some critical amplitudes vanish in the microcanonical ensemble for correlations which span the whole system and are particularly sensitive to the constraint. This can \textit{appear} as a different exponent. We expect that this apparent dependence of exponents on ensemble is related to the integrability of the model, and would not occur in non-integrable models.Comment: 8 pages, 12 figure

Percolation in random environment

We consider bond percolation on the square lattice with perfectly correlated random probabilities. According to scaling considerations, mapping to a random walk problem and the results of Monte Carlo simulations the critical behavior of the system with varying degree of disorder is governed by new, random fixed points with anisotropic scaling properties. For weaker disorder both the magnetization and the anisotropy exponents are non-universal, whereas for strong enough disorder the system scales into an {\it infinite randomness fixed point} in which the critical exponents are exactly known.Comment: 8 pages, 7 figure

Correlated disordered interactions on Potts models

Using a weak-disorder scheme and real-space renormalization-group techniques, we obtain analytical results for the critical behavior of various q-state Potts models with correlated disordered exchange interactions along d1 of d spatial dimensions on hierarchical (Migdal-Kadanoff) lattices. Our results indicate qualitative differences between the cases d-d1=1 (for which we find nonphysical random fixed points, suggesting the existence of nonperturbative fixed distributions) and d-d1>1 (for which we do find acceptable perturbartive random fixed points), in agreement with previous numerical calculations by Andelman and Aharony. We also rederive a criterion for relevance of correlated disorder, which generalizes the usual Harris criterion.Comment: 8 pages, 4 figures, to be published in Physical Review

The Anderson-Mott Transition as a Random-Field Problem

The Anderson-Mott transition of disordered interacting electrons is shown to share many physical and technical features with classical random-field systems. A renormalization group study of an order parameter field theory for the Anderson-Mott transition shows that random-field terms appear at one-loop order. They lead to an upper critical dimension $d_{c}^{+}=6$ for this model. For $d>6$ the critical behavior is mean-field like. For $d<6$ an $\epsilon$-expansion yields exponents that coincide with those for the random-field Ising model. Implications of these results are discussed.Comment: 8pp, REVTeX, db/94/

Order Parameter Description of the Anderson-Mott Transition

An order parameter description of the Anderson-Mott transition (AMT) is given. We first derive an order parameter field theory for the AMT, and then present a mean-field solution. It is shown that the mean-field critical exponents are exact above the upper critical dimension. Renormalization group methods are then used to show that a random-field like term is generated under renormalization. This leads to similarities between the AMT and random-field magnets, and to an upper critical dimension $d_{c}^{+}=6$ for the AMT. For $d<6$, an $\epsilon = 6-d$ expansion is used to calculate the critical exponents. To first order in $\epsilon$ they are found to coincide with the exponents for the random-field Ising model. We then discuss a general scaling theory for the AMT. Some well established scaling relations, such as Wegner's scaling law, are found to be modified due to random-field effects. New experiments are proposed to test for random-field aspects of the AMT.Comment: 28pp., REVTeX, no figure

Spin glass transition in a magnetic field: a renormalization group study

We study the transition of short range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number n to 0, and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin glass transitions in zero and non-zero field are analyzed in a common framework. The mean field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in non-zero field, at the de Almeida-Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order epsilon = 6-d. In the general theory we find a stable fixed-point associated to the spin glass transition in zero field. This fixed-point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed-point to describe the AT transition, in agreement with previous results of other authors.Comment: 36 pages with 4 tables. To be published in Phys. Rev.

Finite-size scaling properties of random transverse-field Ising chains : Comparison between canonical and microcanonical ensembles for the disorder

The Random Transverse Field Ising Chain is the simplest disordered model presenting a quantum phase transition at T=0. We compare analytically its finite-size scaling properties in two different ensembles for the disorder (i) the canonical ensemble, where the disorder variables are independent (ii) the microcanonical ensemble, where there exists a global constraint on the disorder variables. The observables under study are the surface magnetization, the correlation of the two surface magnetizations, the gap and the end-to-end spin-spin correlation $C(L)$ for a chain of length $L$. At criticality, each observable decays typically as $e^{- w \sqrt{L}}$ in both ensembles, but the probability distributions of the rescaled variable $w$ are different in the two ensembles, in particular in their asymptotic behaviors. As a consequence, the dependence in $L$ of averaged observables differ in the two ensembles. For instance, the correlation $C(L)$ decays algebraically as 1/L in the canonical ensemble, but sub-exponentially as $e^{-c L^{1/3}}$ in the microcanonical ensemble. Off criticality, probability distributions of rescaled variables are governed by the critical exponent $\nu=2$ in both ensembles, but the following observables are governed by the exponent $\tilde \nu=1$ in the microcanonical ensemble, instead of the exponent $\nu=2$ in the canonical ensemble (a) in the disordered phase : the averaged surface magnetization, the averaged correlation of the two surface magnetizations and the averaged end-to-end spin-spin correlation (b) in the ordered phase : the averaged gap. In conclusion, the measure of the rare events that dominate various averaged observables can be very sensitive to the microcanonical constraint.Comment: 24 page

The Random-bond Potts model in the large-q limit

We study the critical behavior of the q-state Potts model with random ferromagnetic couplings. Working with the cluster representation the partition sum of the model in the large-q limit is dominated by a single graph, the fractal properties of which are related to the critical singularities of the random Potts model. The optimization problem of finding the dominant graph, is studied on the square lattice by simulated annealing and by a combinatorial algorithm. Critical exponents of the magnetization and the correlation length are estimated and conformal predictions are compared with numerical results.Comment: 7 pages, 6 figure

Smeared phase transition in a three-dimensional Ising model with planar defects: Monte-Carlo simulations

We present results of large-scale Monte Carlo simulations for a three-dimensional Ising model with short range interactions and planar defects, i.e., disorder perfectly correlated in two dimensions. We show that the phase transition in this system is smeared, i.e., there is no single critical temperature, but different parts of the system order at different temperatures. This is caused by effects similar to but stronger than Griffiths phenomena. In an infinite-size sample there is an exponentially small but finite probability to find an arbitrary large region devoid of impurities. Such a rare region can develop true long-range order while the bulk system is still in the disordered phase. We compute the thermodynamic magnetization and its finite-size effects, the local magnetization, and the probability distribution of the ordering temperatures for different samples. Our Monte-Carlo results are in good agreement with a recent theory based on extremal statistics.Comment: 9 pages, 6 eps figures, final version as publishe

The effect of rare regions on a disordered itinerant quantum antiferromagnet with cubic anisotropy

We study the quantum phase transition of an itinerant antiferromagnet with cubic anisotropy in the presence of quenched disorder, paying particular attention to the locally ordered spatial regions that form in the Griffiths region. We derive an effective action where these rare regions are described in terms of static annealed disorder. A one loop renormalization group analysis of the effective action shows that for order parameter dimensions $p<4$ the rare regions destroy the conventional critical behavior. For order parameter dimensions $p>4$ the critical behavior is not influenced by the rare regions, it is described by the conventional dirty cubic fixed point. We also discuss the influence of the rare regions on the fluctuation-driven first-order transition in this system.Comment: 6 pages RevTe