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The non-perturbative renormalization group in the ordered phase | | Jean-Michel Caillol
; | | Date: |
19 Sep 2011 | | Abstract: | We study some analytical properties of the solutions of the non perturbative
renormalization group equations for the scalar $phi^4$ model in the ordered
phase, i.e. at temperatures below the critical temperature. The study is made
in the framework of the local potential approximation. We show that the
required physical discontinuity of the magnetic susceptibility $chi(M)$ at
$M=pm M_0$ ($M_0$ spontaneous magnetization) is reproduced only if the cut-off
function which separates high and low energy modes satisfies to some
restrictive explicit mathematical conditions; we stress that these conditions
are not satisfied by a sharp cut-off in dimensions of space $d<4$.
By generalizing a method proposed earlier by Bonanno and Lacagnina
( extit{Nucl. Phys. B}, extbf{789} (2004) 693) to any kind of cut-off we
propose to solve numerically the flow equations for the threshold functions
rather than for the local potential. It yields an algorithm sufficiently robust
and precise to extract universal as well as non universal quantities from
numerical experiments at any temperature, in particular at sub-critical
temperatures in the ordered phase. Numerical results obtained with three
different cut-off functions are reported and compared. The data confirm our
theoretical predictions concerning the analytical behavior of $chi(M)$ at
$M=pm M_0$.
Fixed point solutions of the adimensionned flow equations are also obtained
in the same vein, that is by solving the fixed points equations and the
associated eigenvalue problem for the threshold functions rather than for the
potential. We report high precision data for the odd and even spectra of
critical exponents for different cut-offs obtained in this way. | | Source: | arXiv, 1109.4024 | | Services: | Forum | Review | PDF | Favorites | |
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