Epsilon expansion for multicritical fixed points and exact renormalisation group equations

The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the nonlinear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order ε and also for the field anomalous dimension to order ε². An exact marginal operator for the full RG equations is also constructed.     

Dynamical and spatial aspects of sandpile cellular automata

The Bak, Tang, and Wiesenfeld cellular automaton is simulated in 1, 2, 3, 4, and 5 dimensions. We define a (new) set of scaling exponents by introducing the concept of conditional expectation values. Scaling relations are derived and checked numerically and the critical dimension is discussed. We address the problem of the mass dimension of the avalanches and find that the avalanches are noncompact for dimensions larger than 2. The scaling of the power spectrum derives from the assumption that the instantaneous dissipation rate of the individual avalanches obeys a simple scaling relation. Primarily, the results of our work show that the flow of sand down the slope does not have a 1/f power spectrum in any dimension, although the model does show clear critical behavior with scaling exponents depending on the dimension. The power spectrum behaves as 1/f ² in all the dimensions considered.     

Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice

This is a general and exact study of multiple Hamiltonian walks (HAW) filling the two-dimensional (2D) Manhattan lattice. We generalize the original exact solution for a single HAW by Kasteleyn to a system ofmultiple closed walks, aimed at modeling a polymer melt. In 2D, two basic nonequivalent topological situations are distinguished. (1) the Hamiltonian loops are allrooted andcontractible to a point:adjacent one to another, and, on a torus,homotopic to zero. (2) the loops can encircle one another and, on a torus, canwind around it. Forcase 1, the grand canonical partition function and multiple correlation functions are calculated exactly as those of multiple rooted spanningtrees or of a massive 2Dfree field, critical at zero mass (zero fugacity). The conformally invariant continuum limit on a Manhattantorus is studied in detail. The melt entropy is calculated exactly. We also consider the relevant effect of free boundary conditions. The number of single HAWs on Manhattan lattices with other perimeter shapes (rectangular, Kagomé, triangular, and arbitrary) is studied and related to the spectral theory of the Dirichlet Laplacian. This allows the calculation of exact shape-dependent configuration exponents y. An exact surface critical exponent is obtained. Forcase 2, nested and winding Hamiltonian circuits are allowed. An exact equivalence to thecritical Q-state Potts model exists, whereQ ^1/2 is the walk fugacity. The Hamiltonian system is then always critical (forQ<-4). The exact critical exponents, in infinite numbers, are universal and identical to those of theO(n=Q ^1/2) model in its low-temperature phase, i.e. are those of dense polymers. The exact critical partition functions on the torus are given from conformai invariance theory. These models 1 and 2 yield the two first exactly solved models of polymer melts.     

Critical exponents of random Ising-like systems in general dimensions

Critical exponents of weakly dilute Ising-like systems are computed for non-integer space dimensionalities in the range 2</d4. The calculations are performed in the framework of the Callan-Symanzik field-theoretic approach. Two-loop renormalization group functions are obtained as renormalized perturbation theory series expansions directly in noninteger dimensions. The values of the critical exponents are estimated with the use of the two-variable Borel resummation method.     

Conformal invariance and surface critical behavior

Conformal invariance constrains the form of correlation functions near a free surface. In two dimensions, for a wide class of models, it completely determines the correlation functions at the critical point, and yields the exact values of the surface critical exponents. They are related to the bulk exponents in a non-trivial way. For the Q-state Potts model (0 Q 4) we find η_<|; = 2/(3v − 1), and for the O(N) model (−2 N 2), η_<|; = (2v − 1)/(4v − 1).     

Critical Exponents from the Resummed Effective Potential of the ({\frac{g}{4}}\phi^{4}-J\phi)_{1+1} Scalar Field Theory

We establish a unified way for the calculation of the critical exponents, without the use of epsilon expansion, through the improvement of the perturbative effective potential of the 1+1 dimensional $({\frac{g}{4}}\phi^{4}-J\phi)$ scalar field theory. First, we obtain the perturbation series for the effective potential up to g ³. We improved the perturbative effective potential by establishing a parameter-free resummation algorithm, originally due to Kleinert, Thoms and Janke, which has the privilege of using the strong coupling as well as the large coupling behaviors rather than the conventional resummation techniques which use only the large order behavior. Accordingly, although the perturbation series available is up to g ³ order, we found a complete agreement between our resummed effective potential and the well known features from constructive field theory. We prove that the 1-PI correlation functions and the effective potential ought to have the same large order as well as strong coupling behaviors. We computed the critical exponents and our results show a good agreement with the exact Ising model values.     

Monte Carlo renormalization group study of φ4 field theory

A previously proposed general method for evaluating block renormalized coupling constants within the framework of the Monte Carlo renormalization group (MCRG) is applied to φ⁴ field theory. The flow diagrams, fixed points, and critical exponents are determined in two, three and four dimensions. Results in four dimensions are consistent with the idea that φ⁴ field theory is trivial (non-interacting) in the continuum limit. The possibility of using MCRG techniques to ascertain whether a general non-asymptotically free theory is trivial or not is also discussed.     

Exact results for the Kardar-Parisi-Zhang equation with spatially correlated noise

We investigate the Kardar-Parisi-Zhang (KPZ) equation in d spatial dimensions with Gaussian spatially long-range correlated noise -- characterized by its second moment -- by means of dynamic field theory and the renormalization group. Using a stochastic Cole-Hopf transformation we derive exact exponents and scaling functions for the roughening transition and the smooth phase above the lower critical dimension . Below the lower critical dimension, there is a line marking the stability boundary between the short-range and long-range noise fixed points. For , the general structure of the renormalization-group equations fixes the values of the dynamic and roughness exponents exactly, whereas above , one has to rely on some perturbational techniques. We discuss the location of this stability boundary in light of the exact results derived in this paper, and from results known in the literature. In particular, we conjecture that there might be two qualitatively different strong-coupling phases above and below the lower critical dimension, respectively. Received 5 August 1998     

Monte Carlo renormalization group study of φ field theory

A previously proposed general method for evaluating block renormalized coupling constants within the framework of the Monte Carlo renormalization group (MCRG) is applied to φ⁴ field theory. The flow diagrams, fixed points, and critical exponents are determined in two, three and four dimensions. Results in four dimensions are consistent with the idea that φ⁴ field theory is trivial (non-interacting) in the continuum limit. The possibility of using MCRG techniques to ascertain whether a general non-asymptotically free theory is trivial or not is also discussed.     

Renormalization point invariance,twisted fans,and critical exponents at finite ϵ in the reggeon calculus.

A procedure for calculating critical exponents directly at finite ? is proposed. It relies on the invariance of the critical exponents at the critical coupling g^c of the full theory with respect to finite changes in the renormalization point. This is expressed as the coincidence of curves at the point β = 0 in the plane of β versus a critical exponent parametrically described by the renormalized coupling for various values of the renormalization point (the “twisted fan”). If more than one critical exponent is present the fan is a set of curves in a multidimensional space with the twist at β = 0 and the exact values of the critical exponents. In perturbative approximations, an approximate invariance may result whether or not a zero of β exists to that order. We show that in the one and two loop approximations to the Reggeon calculus this approximate invariance does occur. The values of the critical exponents at the approximate twists show remarkable stability properties. We obtain σ_tot ≈ (lns)^?γ where ?γ ≈ 0.11 and 0.17 for one and two loops respectively.     

Random quantum magnets with broad disorder distribution

We study the critical behavior of Ising quantum magnets with broadly distributed random couplings (J), such that P(ln J) ∼ | ln J|^{-1 - α}, α > 1, for large | ln J| (Lévy flight statistics). For sufficiently broad distributions, α < , the critical behavior is controlled by a line of fixed points, where the critical exponents vary with the Lévy index, α. In one dimension, with = 2, we obtained several exact results through a mapping to surviving Riemann walks. In two dimensions the varying critical exponents have been calculated by a numerical implementation of the Ma-Dasgupta-Hu renormalization group method leading to ≈ 4.5. Thus in the region 2 < α < , where the central limit theorem holds for | ln J| the broadness of the distribution is relevant for the 2d quantum Ising model. Received 6 December 2000 and Received in final form 22 January 2001     

Universal critical properties of the Eulerian bond-cubic model

We investigate the Eulerian bond-cubic model on the square lattice by means of Monte Carlo simulations,using an efficient cluster algorithm and a finite-size scaling analysis.The critical points and four critical exponents of the model are determined for several values of n.Two of the exponents are fractal dimensions,which are obtained numerically for the first time.Our results are consistent with the Coulomb gas predictions for the critical O(n) branch for n < 2 and the results obtained by previous transfer matrix calculations.For n=2,we find that the thermal exponent,the magnetic exponent and the fractal dimension of the largest critical Eulerian bond component are different from those of the critical O(2) loop model.These results confirm that the cubic anisotropy is marginal at n=2 but irrelevant for n < 2.     

Scaling and infrared divergences in the replica field theory of the Ising spin glass

Replica field theory for the Ising spin glass in zero magnetic field is studied around the upper critical dimension d=6. A scaling theory of the spin glass phase, based on Parisi's ultrametrically organised order parameter, is proposed. We argue that this infinite step replica symmetry broken (RSB) phase is nonperturbative in the sense that amplitudes of scaling forms cannot be expanded in term of the coupling constant w². Infrared divergent integrals inevitably appear when we try to compute amplitudes perturbatively, nevertheless the -expansion of critical exponents seems to be well-behaved. The origin of these problems can be traced back to the unusual behaviour of the free propagator having two mass scales, the smaller one being proportional to the perturbation parameter w² and providing a natural infrared cutoff. Keeping the free propagator unexpanded makes it possible to avoid producing infrared divergent integrals. The role of Ward-identities and the problem of the lower critical dimension are also discussed. Received 23 December 1998 and Received in final form 23 March 1999     

Non-perturbative effects in supersymmetric sigma models

A scheme for taking account of crucial non-perturbative effects in a quantum field theory ahead of developing perturbation series for it is extended here from bosonic to supersymmetric sigma models in two dimensions. The scheme writes field products in the lagrangian in terms of suitably defined normal ordered products and VEVs of field products. The exact values of the latter can be inferred directly from the symmetry and supersymmetry Ward identities of the theory, so that a lagrangian with scale breaking effects explicitly treated, is available for use in perturbation theory. The supersymmetric sigma model on the manifold S^N is used to illustrate many aspects of the scheme.     

A finite size scaling analysis of the O(4)-invariant lattice δ4 theory

The critical behaviour of the three- and four-dimensional N=4 vector model is investigated by means of a Monte Carlo simulation on lattices with size between 4³ and 16³, and between 4⁴ and 12⁴, respectively. For obtaining information about some critical properties of the model, we use a method due to Binder which is based on the theory of finite size scaling. For the three-dimensional model we get estimates of the critical exponents ν and η which are compatible with estimates obtained from the ϵ-expansion. In four dimensions we study for two different values of the bare self-coupling λ (λ=1 in our normalization, and λ=∞) the scaling behaviour of some Green function ratios at the phase boundary. In both cases we find compatibility with the “predicted” scaling behaviour at the gaussian fixed point. This is another independent numerical hint that the continuum limit of the four-dimensional O(4)-invariant lattice δ⁴-model is a free field theory.     

Breakdown of universality in directed spiral percolation

Directed spiral percolation (DSP), percolation under both directional and rotational constraints, is studied on the triangular lattice in two dimensions (2D). The results are compared with that of the 2D square lattice. Clusters generated in this model are generally rarefied and have chiral dangling ends on both the square and triangular lattices. It is found that the clusters are more compact and less anisotropic on the triangular lattice than on the square lattice. The elongation of the clusters is in a different direction than the imposed directional constraint on both the lattices. The values of some of the critical exponents and fractal dimension are found considerably different on the two lattices. The DSP model then exhibits a breakdown of universality in 2D between the square and triangular lattices. The values of the critical exponents obtained for the triangular lattice are not only different from that of the square lattice but also different form other percolation models.Received: 12 March 2004, Published online: 23 July 2004PACS: 02.50.-r Probability theory, stochastic processes, and statistics - 64.60.-i General studies of phase transitions - 72.80.Tm Composite materials     

Multicritical behaviour at surfaces

The critical behaviour of a semi-infiniten-vector model with a surface term (c/2) ∫d Sφ² is studied in 4-ε dimensions near the special transition. It is shown that all critical surface exponents derive from bulk exponents and η_∥, the anomalous dimension of the order parameter at the surface. The surface exponents and the crossover exponent Φ for the variablec are calculated to second order in ε. It is found that Φ does not satisfy the relation Φ=1-ν predicted by Bray and Moore. The order-parameter profilem(z)=<ø> is calculated to first order in ε. In contrast to mean-field theory,m(z) is not flat nor does it satisfy a Neumann boundary condition. General aspects of the field-theoretic renormalization program for systems with surfaces are discussed with particular attention paid to the explanation of the unfamiliar new features caused by the presence of surfaces.     

A non-perturbative analysis of symmetry breaking in two-dimensional φ4 theory using periodic field methods

We describe the generalization of spherical field theory to other modal expansion methods. The main approach remains the same, to reduce a d-dimensional field theory into a set of coupled one-dimensional systems. The method we discuss here uses an expansion with respect to periodic-box modes. We apply the method to φ⁴ theory in two dimensions and compute the critical coupling and critical exponents. We compare with lattice results and predictions via universality and the two-dimensional Ising model.     

O(n) tricriticality in two dimensions

We present exact results for several universal parameters of the tricritical O(n) model in two dimensions. The results apply to the range −2⩽n⩽3/2, and include the central charge and three scaling dimensions, associated with temperature, magnetic field and the introduction of an interface. Since these results are based on an extrapolation of known relations between the O(n) and the Potts model, they cannot be considered as rigorous. For this reason, we perform an accurate numerical analysis of the central charge and the critical exponents. This analysis, which is based on transfer-matrix calculations on the honeycomb lattice, is in a full and precise agreement with the theoretical predictions.      

Critical wetting in the square Ising model with a boundary field

The Ising square lattice with nearest-neighbor exchangeJ>0 and a free surface at which a boundary magnetic fieldH ₁ acts has a second-order wetting transition. We study the surface excess magnetization and the susceptibility ofL×M lattices by Monte Carlo simulation and probe the critical behavior of this wetting transition, applying finite-size scaling methods. For the cases studied, the results are not consistent with the presumably exactly known values of the critical exponents, because the asymptotic critical region has not yet been reached. Implication of our results for critical wetting in three dimensions and for the application of the present model to adsorbed wetting layers at surface steps are briefly discussed.Alexander von Humboldt-Fellow