Asymptotically Universal Crossover in Perturbation Theory with a Field Cutoff

We discuss the crossover between the small and large field cutoff (denoted x_{max}) limits of the perturbative coefficients for a simple integral and the anharmonic oscillator. We show that in the limit where the order k of the perturbative coefficient a_k(x_{max}) becomes large and for x_{max} in the crossover region, a_k(x_{max}) is proportional to the integral from -infinity to x_{max} of e^{-A(x-x_0(k))^2}dx. The constant A and the function x_0(k) are determined empirically and compared with exact (for the integral) and approximate (for the anharmonic oscillator) calculations. We discuss how this approach could be relevant for the question of interpolation between renormalization group fixed points.Comment: 15 pages, 11 figs., improved and expanded version of hep-th/050304

Dyson instability for 2D nonlinear O(N) sigma models

For lattice models with compact field integration (nonlinear sigma models over compact manifolds and gauge theories with compact groups) and satisfying some discrete symmetry, the change of sign of the bare coupling g_0^2 at zero results in a mere discontinuity in the average energy rather than the catastrophic instability occurring in theories with integration over arbitrarily large fields. This indicates that the large order of perturbative series and the non-perturbative contributions should have unexpected features. Using the large-N limit of 2-dimensional nonlinear O(N) sigma model, we discuss the complex singularities of the average energy for complex 't Hooft coupling lambda= g_0^2N. A striking difference with the usual situation is the absence of cut along the negative real axis. We show that the zeros of the partition function can only be inside a clover shape region of the complex lambda plane. We calculate the density of states and use the result to verify numerically the statement about the zeros. We propose dispersive representations of the derivatives of the average energy for an approximate expression of the discontinuity. The discontinuity is purely non-perturbative and contributions at small negative coupling in one dispersive representation are essential to guarantee that the derivatives become exponentially small when lambda -> 0^+ We discuss the implications for gauge theories.Comment: 10 pages, 10 figures uses revte

The non-perturbative part of the plaquette in quenched QCD

We define the non-perturbative part of a quantity as the difference between its numerical value and the perturbative series truncated by dropping the order of minimal contribution and the higher orders. For the anharmonic oscillator, the double-well potential and the single plaquette gauge theory, the non-perturbative part can be parametrized as A (lambda)^B exp{-C/lambda} and the coefficients can be calculated analytically. For lattice QCD in the quenched approximation, the perturbative series for the average plaquette is dominated at low order by a singularity in the complex coupling plane and the asymptotic behavior can only be reached by using extrapolations of the existing series. We discuss two extrapolations that provide a consistent description of the series up to order 20-25. These extrapolations favor the idea that the non-perturbative part scales like (a/r_0)^4 with a/r_0 defined with the force method. We discuss the large uncertainties associated with this statement. We propose a parametrization of ln((a/r_0)) as the two-loop universal terms plus a constant and exponential corrections. These corrections are consistent with a_{1-loop}^2 and play an important role when beta<6. We briefly discuss the possibility of calculating them semi-classically at large beta.Comment: 13 pages, 16 figures, uses revtex, contains a new section with the uncertainties on the extrapolations, refs. adde

Fisher's zeros as boundary of renormalization group flows in complex coupling spaces

We propose new methods to extend the renormalization group transformation to complex coupling spaces. We argue that the Fisher's zeros are located at the boundary of the complex basin of attraction of infra-red fixed points. We support this picture with numerical calculations at finite volume for two-dimensional O(N) models in the large-N limit and the hierarchical Ising model. We present numerical evidence that, as the volume increases, the Fisher's zeros of 4-dimensional pure gauge SU(2) lattice gauge theory with a Wilson action, stabilize at a distance larger than 0.15 from the real axis in the complex beta=4/g^2 plane. We discuss the implications for proofs of confinement and searches for nontrivial infra-red fixed points in models beyond the standard model.Comment: 4 pages, 3 fig

Resummation of the Divergent Perturbation Series for a Hydrogen Atom in an Electric Field

We consider the resummation of the perturbation series describing the energy displacement of a hydrogenic bound state in an electric field (known as the Stark effect or the LoSurdo-Stark effect), which constitutes a divergent formal power series in the electric field strength. The perturbation series exhibits a rich singularity structure in the Borel plane. Resummation methods are presented which appear to lead to consistent results even in problematic cases where isolated singularities or branch cuts are present on the positive and negative real axis in the Borel plane. Two resummation prescriptions are compared: (i) a variant of the Borel-Pade resummation method, with an additional improvement due to utilization of the leading renormalon poles (for a comprehensive discussion of renormalons see [M. Beneke, Phys. Rep. vol. 317, p. 1 (1999)]), and (ii) a contour-improved combination of the Borel method with an analytic continuation by conformal mapping, and Pade approximations in the conformal variable. The singularity structure in the case of the LoSurdo-Stark effect in the complex Borel plane is shown to be similar to (divergent) perturbative expansions in quantum chromodynamics.Comment: 14 pages, RevTeX, 3 tables, 1 figure; numerical accuracy of results enhanced; one section and one appendix added and some minor changes and additions; to appear in phys. rev.

Critical exponents at the ferromagnetic transition in tetrakis(diethylamino)ethylene-C60_{60} (TDAE-C60_{60})

Critical exponents at the ferromagnetic transition were measured for the first time in an organic ferromagnetic material tetrakis(dimethylamino)ethylene fullerene[60] (TDAE-C$_{60}$). From a complete magnetization-temperature-field data set near $T_{c}=16.1\pm 0.05,$ we determine the susceptibility and magnetization critical exponents $\gamma =1.22\pm 0.02$ and $\beta =0.75 \pm 0.03$ respectively, and the field vs. magnetization exponent at $T_{c}$ of $\delta =2.28\pm 0.14$. Hyperscaling is found to be violated by $\Omega \equiv d^{\prime}-d \approx -1/4$, suggesting that the onset of ferromagnetism can be related to percolation of a particular contact configuration of C$_{60}$ molecular orientations.Comment: 5 pages, including 3 figures; to appear in Phys. Rev. Let

Disorder-Induced Critical Phenomena in Hysteresis: Numerical Scaling in Three and Higher Dimensions

We present numerical simulations of avalanches and critical phenomena associated with hysteresis loops, modeled using the zero-temperature random-field Ising model. We study the transition between smooth hysteresis loops and loops with a sharp jump in the magnetization, as the disorder in our model is decreased. In a large region near the critical point, we find scaling and critical phenomena, which are well described by the results of an epsilon expansion about six dimensions. We present the results of simulations in 3, 4, and 5 dimensions, with systems with up to a billion spins (1000^3).Comment: Condensed and updated version of cond-mat/9609072,``Disorder-Induced Critical Phenomena in Hysteresis: A Numerical Scaling Analysis'

BF models, Duality and Bosonization on higher genus surfaces

The generating functional of two dimensional $BF$ field theories coupled to fermionic fields and conserved currents is computed in the general case when the base manifold is a genus g compact Riemann surface. The lagrangian density $L=dB{\wedge}A$ is written in terms of a globally defined 1-form $A$ and a multi-valued scalar field $B$. Consistency conditions on the periods of $dB$ have to be imposed. It is shown that there exist a non-trivial dependence of the generating functional on the topological restrictions imposed to $B$. In particular if the periods of the $B$ field are constrained to take values $4\pi n$, with $n$ any integer, then the partition function is independent of the chosen spin structure and may be written as a sum over all the spin structures associated to the fermions even when one started with a fixed spin structure. These results are then applied to the functional bosonization of fermionic fields on higher genus surfaces. A bosonized form of the partition function which takes care of the chosen spin structure is obtainedComment: 17 page

Asymptotically Improved Convergence of Optimized Perturbation Theory in the Bose-Einstein Condensation Problem

We investigate the convergence properties of optimized perturbation theory, or linear $\delta$ expansion (LDE), within the context of finite temperature phase transitions. Our results prove the reliability of these methods, recently employed in the determination of the critical temperature T_c for a system of weakly interacting homogeneous dilute Bose gas. We carry out the explicit LDE optimized calculations and also the infrared analysis of the relevant quantities involved in the determination of $T_c$ in the large-N limit, when the relevant effective static action describing the system is extended to O(N) symmetry. Then, using an efficient resummation method, we show how the LDE can exactly reproduce the known large-N result for $T_c$ already at the first non-trivial order. Next, we consider the finite N=2 case where, using similar resummation techniques, we improve the analytical results for the nonperturbative terms involved in the expression for the critical temperature allowing comparison with recent Monte Carlo estimates of them. To illustrate the method we have considered a simple geometric series showing how the procedure as a whole works consistently in a general case.Comment: 38 pages, 3 eps figures, Revtex4. Final version in press Phys. Rev.

Higher Order Evaluation of the Critical Temperature for Interacting Homogeneous Dilute Bose Gases

We use the nonperturbative linear \delta expansion method to evaluate analytically the coefficients c_1 and c_2^{\prime \prime} which appear in the expansion for the transition temperature for a dilute, homogeneous, three dimensional Bose gas given by T_c= T_0 \{1 + c_1 a n^{1/3} + [ c_2^{\prime} \ln(a n^{1/3}) +c_2^{\prime \prime} ] a^2 n^{2/3} + {\cal O} (a^3 n)\}, where T_0 is the result for an ideal gas, a is the s-wave scattering length and n is the number density. In a previous work the same method has been used to evaluate c_1 to order-\delta^2 with the result c_1= 3.06. Here, we push the calculation to the next two orders obtaining c_1=2.45 at order-\delta^3 and c_1=1.48 at order-\delta^4. Analysing the topology of the graphs involved we discuss how our results relate to other nonperturbative analytical methods such as the self-consistent resummation and the 1/N approximations. At the same orders we obtain c_2^{\prime\prime}=101.4, c_2^{\prime \prime}=98.2 and c_2^{\prime \prime}=82.9. Our analytical results seem to support the recent Monte Carlo estimates c_1=1.32 \pm 0.02 and c_2^{\prime \prime}= 75.7 \pm 0.4.Comment: 29 pages, 3 eps figures. Minor changes, one reference added. Version in press Physical Review A (2002