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Register a new userCritical exponent for the magnetization of the weakly coupled $phi^4_4$ model
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Authors
Martin Lohmann
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We consider the weakly coupled $phi^4 $ theory on $mathbb Z^4 $, in a weak magnetic field $h$, and at the chemical potential $ u_c $ for which the theory is critical if $h=0$. We prove that, as $hto 0$, the magnetization of the model behaves as $(hlog h^{-1})^{frac 13} $, and so exhibits a logarithmic correction to mean field scaling behavior. This result is well known to physicists, but had never been proven rigorously. Our proof uses the classic construction of the critical theory by Gawedzki and Kupiainen, and a cluster expansion with large blocks.
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We introduce a transfer matrix formalism for the (annealed) Ising model coupled to two-dimensional causal dynamical triangulations. Using the Krein-Rutman theory of positivity preserving operators we study several properties of the emerging transfer matrix. In particular, we determine regions in the quadrant of parameters beta, mu >0 where the infinite-volume free energy converges, yielding results on the convergence and asymptotic properties of the partition function and the Gibbs measure.
We define a finite size renormalization scheme for $phi^4$ theory which in the thermodynamic limit reduces to the standard scheme used in the broken phase. We use it to re-investigate the question of triviality for the four dimensional infinite bare coupling (Ising) limit. The relevant observables all rely on two-point functions and are very suitable for a precise estimation with the worm algorithm. This contribution updates an earlier publication by analysing a much larger dataset.
The 1-arm exponent $rho$ for the ferromagnetic Ising model on $mathbb{Z}^d$ is the critical exponent that describes how fast the critical 1-spin expectation at the center of the ball of radius $r$ surrounded by plus spins decays in powers of $r$. Suppose that the spin-spin coupling $J$ is translation-invariant, $mathbb{Z}^d$-symmetric and finite-range. Using the random-current representation and assuming the anomalous dimension $eta=0$, we show that the optimal mean-field bound $rhole1$ holds for all dimensions $d>4$. This significantly improves a bound previously obtained by a hyperscaling inequality.
Four generalizations of the Phase Integral Approximation (PIA) to sets of N ordinary differential equations of the Schroedinger type: u_j(x) + Sum{k = 1 to N} R_{jk}(x) u_k(x) = 0, j = 1 to N, are described. The recurrence relations for higher order corrections are given in the form valid in arbitrary order and for the matrix R_{jk} either hermitian or non-hermitian. For hermitian and negative definite R matrices, the Wronskian conserving PIA theory is formulated which generalizes Fullings current conserving theory pertinent to positive definite R matrices. The idea of a modification of the PIA, well known for one equation: u(x) + R(x) u(x) = 0, is generalized to sets. A simplification of Wronskian or current conserving theories is proposed which in each order eliminates one integration from the formulas for higher order corrections. If the PIA is generated by a non-degenerate eigenvalue of the R matrix, the eliminated integration is the only one present. In that case, the simplified theory becomes fully algorithmic and is generalized to non-hermitian R matrices. General theory is illustrated by a few examples generated automatically by using authors program in Mathematica, published in arXiv:0710.5406.
We consider the $n$-component $|varphi|^4$ lattice spin model ($n ge 1$) and the weakly self-avoiding walk ($n=0$) on $mathbb{Z}^d$, in dimensions $d=1,2,3$. We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance $r$ as $r^{-(d+alpha)}$ with $alpha in (0,2)$. The upper critical dimension is $d_c=2alpha$. For $epsilon >0$, and $alpha = frac 12 (d+epsilon)$, the dimension $d=d_c-epsilon$ is below the upper critical dimension. For small $epsilon$, weak coupling, and all integers $n ge 0$, we prove that the two-point function at the critical point decays with distance as $r^{-(d-alpha)}$. This sticking of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion.
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