Critical exponents for random knots

The size of a zero-thickness (no excluded volume) nonphantom polymer ring is shown to scale with chain length N in the same way as the size of the excluded-volume (self-avoiding) linear polymer, that is, as Nnu, where nu approximately 0.588. The consequences of this fact are examined, including the sizes of trivial and nontrivial knots.     

Critical exponents for long-range interactions

Long-range components of the interaction in statistical mechanical systems may affect the critical behavior, raising the system's effective dimension. Presented here are explicit implications to this effect of a collection of rigorous results on the critical exponents in ferromagnetic models with one-component Ising (and more genrally Griffiths=Simon class) spin variables. In particular, it is established that even in dimensions d<4 if a ferromagnetic Ising spin model has a reflection-positive pair interaction with a sufficiently slow decay, e.g. as J _x=1/|x| ^d+ with 0<d/2, then the exponents , , and ₄ exist and take their mean-field values. This proves rigorously an early renormalization-group prediction of Fisher, Ma and Nickel. In the converse direction: when the decay is by a similar power law with >-2, then the long-range part of the interaction has no effect on the existent critical exponent bounds, which coincide then with those obtained for short-range models.Also in the Physics Department. Research supported in part by the National Science Foundation Grant PHY 86-05164.     

Critical exponents for one-dimensional percolation clusters

For d=1, percolation clusters follow a scaling law with critical exponents σ=1 and τ=2. For the limit d→1, critical exponents can differ from their d=1 values, a complication which can already be studied in the simple Bethe lattice solution for cluster numbers.     

Density-matrix spectra for integrable models

The spectra which occur in numerical density-matrix renormalization group (DMRG) calculations for quantum chains can be obtained analytically for integrable models via corner transfer matrices. This is shown in detail for the transverse Ising chain and the uniaxial XXZ Heisenberg model and explains in particular their exponential character in these cases.     

Neumann-like integrable models

A countable class of integrable dynamical systems, with four-dimensional phase space and conserved quantities in involution (Hn,In)$(H_n,I_n)$ are exhibited. For n=1$n=1$ we recover Neumann system on T^∗S²$T^{\ast }{S}^2$. All these systems are also integrable at the quantum level.     

Dyonic integrable models

A class of non-abelian affine Toda models arising from the axial gauged two-loop WZW model is presented. Their zero curvature representation is constructed in terms of a graded Kac–Moody algebra. It is shown that the discrete multivacua structure of the potential together with non-abelian nature of the zero grade subalgebra allows soliton solutions with non-trivial electric and topological charges. The dressing transformation is employed to explicitly construct one and two soliton solutions and their bound states in terms of the tau functions. A discussion of the classical spectra of such solutions and the time delays are given in detail.     

Critical exponents and elementary particles

Particles are shown to exist for a.e. value of the mass in single phase ⁴ lattice and continuum field theories and nearest neighbor Ising models. The particles occur in the form of poles at imaginary (Minkowski) momenta of the Fourier transformed two point function. The new inequalitydm ²/dZ, where =m ₀ ² is a bare mass² andZ is the strength of the particle pole, is basic to our method. This inequality implies inequalities for critical exponents.Supported in part by the National Science Foundation under grant PHY 76-17191Supported in part by the National Science Foundation under grant MPS 75-21212     

Critical exponents for alternation in Heisenberg antiferromagnetic chains

The ground-state energy per site, λ(δ), and singlet-triplet gap, ΔE(δ), of regular (δ=0) and alternating (δ ≤ 0.01) Heisenberg antiferromagnetic chains are obtained by extrapolating exact numerical solutions of even and odd rings and chains of N ≤ 20 spins. The critical exponents of 1.72 ± 0.02 for λ(δ) and 0.9 ± 0.1 for ΔE(δ) are compared with recent theoretical results for spin-Peierls transitions. Interactions among spinless fermions alter the instability to alternation.     

Critical exponents of uranium telluride

Neutron and magnetization experiments have been used to determine the critical exponents β, γ and δ of the ferromagnet UTe, T_c = 100.7(3) K, near the Curie temperature. The values are β = 0.291(4), γ = 1.326(9) and δ = 5.23(7). Comparisons are made with values from other ferromagnetic systems.     

Critical exponents from power spectra

We have simulated the two- and three-dimensional Ising models at their respective critical points with a conventional Monte Carlo algorithm. From the power spectrum of the magnetization autocorrelations we have determined the dynamic critical exponents and obtained the valuesz = 2.16–2.19 andz = 2.05, in agreement with the results quoted in the literature. We have also studied the power spectrum for the Kardar-Parisi-Zhang and Sun-Guo-Grant equations describing interface dynamics. Arguments similar to what was recently used to conclude thatz = 4 - for model B in critical dynamics were applied to the Sun-Guo-Grant growth model and the known exact values for the roughening and dynamic exponents were obtained. From an analysis of the corresponding power spectrum in self-organized critical sand models one obtains a recently proposed hyperscaling relation.     

Critical exponents for the Reggeon quantum spin model

The Reggeon quantum spin (RQS) model on the transverse lattice in D dimensional impact parameter space has been conjectured to have the same critical behaviour as the Reggeon field theory (RFT). Thus from a high “temperature” series of ten (D = 2) and twenty (D = 1) terms for the RQS model we extrapolate to the critical temperature T = T_c by Padé approximants to obtain the exponents η=0.238±0.008, z=1.16±0.01, v=1.271±0.007 for D=2 and η=0.317±0.002, z=1.272±0.007, v=1.736±0.001, λ=0.57±0.03 for D=1. These exponents naturally interpolate between the D=0 and D=4?ε results for RFT as expected on the basis of the universality conjecture.     

Multicharged dyonic integrable models

We introduce and study new integrable models (IMs) of A_n⁽¹⁾-nonabelian Toda type which admit U(1)⊗U(1) charged topological solitons. They correspond to the symmetry breaking SU(n+1)→SU(2)⊗SU(2)⊗U(1)ⁿ⁻² and are conjectured to describe charged dyonic domain walls of N=1 SU(n+1) SUSY gauge theory in large n limit. It is shown that this family of relativistic IMs corresponds to the first negative grade q=−1 member of a dyonic hierarchy of generalized cKP type. The explicit relation between the 1-soliton solutions (and the conserved charges as well) of the IMs of grades q=−1 and q=2 is found. The properties of the IMs corresponding to more general symmetry breaking SU(n+1)→SU(2)^⊗p⊗U(1)^n−p as well as IM with global SU(2) symmetries are discussed.     

Multi-leg integrable ladder models

We construct integrable spin chains with inhomogeneous periodic disposition of the anisotropy parameter. The periodicity holds for both auxiliary (space) and quantum (time) directions. The integrability of the model is based on a set of coupled Yang–Baxter equations. This construction yields P-leg integrable ladder Hamiltonians. We analyse the corresponding quantum group symmetry and present algebraic Bethe ansatz (ABA) solution.     

Free field representation for massive integrable models

A new approach to massive integrable models is considered. It allows one to find symmetry algebras which define the spaces of local operators and to get general integral representations for form-factors in theSU(2) Thirring and Sine-Gordon models.     

Critical exponents for two-dimensional Heisenberg ferromagnet with long-range interaction

Summary Using renormalization group technique, the critical exponents are determined for two-dimensional Heisenberg ferromagnet in the presence of a magnetic field. Critical exponents are expressed as a function of potential range σ. For the long-range interaction potential between the spins, the spontaneous magnetization exponent β takes the poorer value of mean-field result. The other exponents agree with the known values of exponents. The susceptibility eigenvalue equation arising from renormalization group is also solved and eigenfunctions are determined.

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Riassunto Usando la tecnica del gruppo di rinormalizzazione si determinano gli esponenti critici per un ferromagnete di Heisenberg bidimensionale in presenza di un campo magnetico. Si esprimono gli esponenti critici in funzione di un intervallo di potenziale σ. Per un potenziale d’interazione a largo raggio tra gli spin, l’esponente di magnetizzazione spontanea β assume il valore più scarso del risultato del campo medio. Gli altri esponenti si accordano con i valori noti degli esponenti. Si risolve anche l’equazione degli autovalori di suscettibilità proveniente dal gruppo di rinormalizzazione e si determinano le autofunzioni.

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Резюме Используя технику грушы перенормировки, определяются критические экспоненты для двумерного ферромагнетика Гайзенберга в присутствии магнитного поля. Критические экспоненты выражаются через радиус действия потенциала. Для длиннодействующго потенциала взаимодействия между спинами зкспонента спонтанного нмагничивания β дает меньшую величину для среднего поля. Другие экспоненты согласуются с известными величинами для экспонент. Решается уравнение для собоственных значений восприимчивости, возникающее из группы перенормировки, и определяются собственные функции.