Abstract
We calculate the effective action for disordered elastic manifolds in the ground state (equilibrium) up to 3-loop order. This yields the renormalization-group β-function to third order in ε=4−d, in an expansion in the dimension d around the upper critical dimension d=4. The calculations are performed using exact RG, and several other techniques, which allow us to resolve consistently the problems associated with the cusp of the renormalized disorder.
Highlights
Disordered systems are notoriously difficult to treat, since naive perturbation theory leads to absurd results, as exemplified by the phenomenon of dimensional reduction [1]
The latter goes back to the work by Wilson [3] and Wegner and Houghton [4]. For disordered systems these methods were first used by Daniel Fisher [5]. It took until 1992 that Narayan and Fisher [6, 7], shortly thereafter followed by Natterman, Stepanow, Tang and Leschhorn [8], recognized that the disorder correlator, which plays the role of the coupling constant in the functional renormalization group (FRG) treatment, has to assume a cuspy form
All methods, who are are genuinely different, give consistent results. This is strong evidence that the problem has a unique field theory, which we identify in this paper to 3-loop order
Summary
Disordered systems are notoriously difficult to treat, since naive perturbation theory leads to absurd results, as exemplified by the phenomenon of dimensional reduction [1]. Two main paths out of this dilemma have been pursued: Replica symmetry breaking [2], and the functional renormalization group The latter goes back to the work by Wilson [3] and Wegner and Houghton [4]. Once this question of principle solved, it remained the problems of feasibility and practicality: First, whether there is a controlled loop or ε-expansion, and second how to implement a method which makes sense of the cusp in this loop expansion, and more of the derivatives at the cusp The latter change sign, depending on whether the limit is taken for positive or negative argument, not to mention the additional problems arising for a higherdimensional field [16]. There we will extract the roughness exponent ζ, obtain the fixed-point functions R to 3-loop order, give the corrrection-to-scaling exponent ω, as well as the momentum dependent 2-point function
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