A geometric generalization of field theory to manifolds of arbitrary dimension

We introduce a generalization of the O(N) field theory to N-colored membranes of arbitrary inner dimension D. The O(N) model is obtained for , while leads to self-avoiding tethered membranes (as the O(N) model reduces to self-avoiding polymers). The model is studied perturbatively by a 1-loop renormalization group analysis, and exactly as .Freedom to choose the expansion point D, leads to precise estimates of critical exponents of the O(N) model. Insights gained from this generalization include a conjecture on the nature of droplets dominating the 3d-Ising model at criticality; and the fixed point governing the random bond Ising model. Received: 15 October 1998 / Accepted: 4 November 1998     

Branching chains and critical clusters of Ising spins

Ising critical clusters are related to the excess of neighbor spins of similar, over anti-similar, orientation. The clusters are approximately described by self-avoiding branching chains. γ, ν andalso a “time” (of growth from the origin) and a “perimeter/bulk” exponent are measured with the help of Monte-Carlo sampling.     

The φ4 lattice field theory as an asymptotic expansion about the Ising limit

For a d-dimensional φ⁴ lattice field theory consisting of N spins, an asymptotic expansion of expectations about the Ising limit is established in inverse powers of the bare coupling constant λ. In the thermodynamic limit (N → ∞), the expansion is expected to be valid in the noncritical region of the Ising system.     

An improved theory of higher-order correlations for an Ising model with general spin

A recently developed theory, applying a correlation reduction idea in a completely new formulation, is improved. The stability of the results for the critical temperature with respect to various choices of the reduction factors is shown. A more detailed and accurate determination of the criticality condition, yielded by the self-avoiding representation of spin thermal averages, leads to highly accurate (within 0.56% of series results for arbitrary spin S and all cubic lattices) values of the transition temperature of the Ising ferromagnets. A comparable accuracy (≈1%) is attained for the FCC Blume-Capel model in the whole range of the anisotropy parameter where the system undergoes a second-order phase transition. The method also reproduces the general tendency of the quadrupole moment variable near the tricritical point exhibited by series expansions.     

New renormalization group results for scaling of self-avoiding tethered membranes

The scaling properties of self-avoiding polymerized two-dimensional membranes are studied via renormalization group methods based on a multilocal operator product expansion. The renormalization group functions are calculated to second order. This yields the scaling exponent ν to order ε² Our extrapolations for ν agree with the Gaussian variational estimate for large space dimension d and are close to the Flory estimate for d = 3. The interplay between self-avoidance and rigidity at small d is briefly discussed.     

Vertex models on Feynman diagrams

We discuss the statistical mechanics of vertex models on both generic (“thin”) and planar (“fat”) random graphs. Such models can be formulated as the N → 1 and N → ∞ limits of N × N complex matrix models, respectively. From the graph theoretic perspective one is using matrix model and field theory inspired methods to count various classes of directed graphs. For the thin random graphs we use saddle point methods to solve the models in the thermodynamic, large number of vertices limit and note that, as in the case of the eight-vertex model on the square lattice, various other models such as the Ising model appear as particular limits. The generic solution of the fat graph model is rather more elusive, but we show that for several choices of the couplings the models can be reduced to eigenvalue integrals and their critical behaviour deduced.     

Soluble renormalization groups and scaling fields for low-dimensional Ising systems

A variety of one-dimensional Ising spin systems, including staggered and parallel magnetic fields, alternating and second neighbor interactions, four-spin coupling, etc., are discussed in terms of renormalization group theory. A continuous range of distinct renormalization groups is constructed in exact closed form, analyzed in detail, and compared with exactly calculated thermodynamic properties. Fixed point linearization yields relevant, irrelevant, and marginal operators. All groups yield identical “critical” behavior (at T = 0) with η = 1, δ = ∞, γ = ν = 2 ? α, and with identical linear scaling fields. A generalization of Wegner's analysis to discrete groups yields explicit power series for the nonlinear scaling fields; these are seen to depend on the particular renormalization group and, hence, are physically nonunique. A planar, multiconnected “truncated tetrahedron” model of effective dimensionality log₂ 3 is analyzed via a dedecoration and star-triangle group revealing highly singular behavior as T → T_c = 0.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

Ground state of an impurity in a homogeneous system in the hypernetted chain approximation

The hypernetted chain theory of the ground state of a homogeneous N-particle medium NM with an impurity particle is presented. The N identical particles are fermions with spin-isospin degeneracy ν, or bosons (in the limit of ν → ∞). The ground-state wave-function of the system is assumed in the Jastrow form with central, state-independent correlation functions. Central, spin-isospin-dependent two-body interactions both in NM and between the impurity particle and the particles of NM are considered. Expressions for the ground-state energy of the system and for the separation energy of the impurity particle are derived. The simplified case of the chain approximation is also considered.     

Gaussian random field p-spin-interaction ising model in a transverse field

The Gaussian random field Ising model with p-spin interactions in the presence of a transverse field is studied by combining the Suzuki-Trotter approach and the thermodynamic perturbation theory. The first-order phase transitions are found in the limit p → ∞, in contrast to the cases with p=2.     

Factorizable Z(N) models

The factorizable S-matrix with Z(N) symmetry is constructed. It is speculated that the field theory belonging to this S-matrix matrix is related to the scaling limit of Z(N) generalizations of the Ising model.     

Large q expansions for q-state gauge-matter Potts models in lagrangian form

We consider the lagrangian form of a q-state generalization of Ising gauge theories with matter fields in d = 3 and 4 dimensions. The theory is exactly soluble in the limit q → ∞ and corrections are easily calculable in power series in $\text{1}\text{q}{}^{\mathrm{<mtext>1</mtext><mtext>d</mtext>}}$. Extrapolating the series for the free energies and latent heats by the method of Padé approximants, we have constructed the phase diagrams for all values of q. Our results agree well with known results for pure spin systems and, for the case q = 2, with Ising Monte Carlo data.     

On the susceptibility exponent of random anisotropy magnets

The temperature dependence of the non-linear susceptibility ≈₂(T) of random anisotropy magnets in the Ising limit (speromagnets) is calculated for temperatures above the freezing temperature T_f within the framework of the correlated molecular field theory. For the effective susceptibility exponent λ_s(T) = (T?T_f)≈₂d^-1_≈2/dT a non-monotonic temperature dependence is found as for the case of spin glasses. This must be taken into account in order to obtain reliable values for the critical susceptibility exponent from experimental data.     

The critical behavior of φ 1 4

The eigenvalues, eigenfunctions, and Schwinger functions of the ordinary differential operator $$H(\lambda ,m) = \tfrac{1}{2}\{ p^2 + \lambda q^4 + (m^2 - \lambda m^{ - 2} )q^2 \} $$ are studied as λ → ∞. It is shown that the scaling limit of the Schwinger functions equals the scaling limit of a one dimensional Ising model. Critical exponents ofH(λ,m) are shown to equal critical exponents of the Ising model, while critical exponents of the renormalized theory are shown to agree with those of a harmonic oscillator.     

Mean-Field Behavior for Long- and Finite Range Ising Model,Percolation and Self-Avoiding Walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if d>2(α ∧2) for self-avoiding walk and the Ising model, and d>3(α ∧2) for percolation, where d denotes the dimension and α the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (Ann. Probab. 33(5):1886–1944, 2005).      

Critical properties of the three-dimensional frustrated Ising model on a cubic lattice

The critical properties of the three-dimensional fully frustrated Ising model on a cubic lattice are investigated by the Monte Carlo method. The critical exponents α (heat capacity), γ (susceptibility), β (magnetization), and ν (correlation length), as well as the Fisher exponent η, are calculated in the framework of the finite-size scaling theory. It is demonstrated that the three-dimensional frustrated Ising model on a cubic lattice forms a new universality class of the critical behavior.     

有限分枝分形上的自迴避迹行走

本文考虑在Sierpinski gasket及分支Koch曲线上的自迴避迹行走,运用实空间重整化群技术求出了相应的关联长度临界指数ν。结果表明,在Sierpinski gasket上,自迴避迹行走与自迴避行走属同一普适类;而在较高分枝度(R_max>3)的Koch曲线上,两者属不同普适类。 关键词：     

Higher order invariants in extended supergravity

On-shell linearized extended supergravity is presented in superspace for all N. The formalism is then used in the construction of higher order invariants which may serve as counterterm lagrangians. It is shown that three-loop counterterms exist for N ? 3 and (N ? 1)-loop counterterms for N ? 4. In the full non-linear theory, the presence of a global non-compact symmetry group for N ? 4 does not allow a simple extension of the (N ? 1)-loop term, but N-loop counterterms may be constructed.     

Locally critical point in an anisotropic Kondo lattice

We report the first numerical identification of a locally quantum critical point at which the criticality of the local Kondo physics is embedded in that associated with a magnetic ordering. We are able to numerically access the quantum critical behavior by focusing on a Kondo-lattice model with Ising anisotropy. We also establish that the critical exponent for the q-dependent dynamical spin susceptibility is fractional and compares well with the experimental value for heavy fermions.