Critical properties of the random hierarchical model
Journal Article
·
· Phys. Rev., B: Condens. Matter; (United States)
The random hierarchical model is defined by introducing random couplings between spin blocks in Dyson's original hierarchical model for continuous spins. The spin-distribution functions are now site dependent and functions of random parameters that we take to be a quadratic and quartic potential, r/sup( m/)/sub l/ and U/sup( m/)/sub l/, respectively, for the mth spin at the l level of the hierarchy. Renormalization-group recursion relations for the cumulant averages are derived in an expansion in epsilon'= (2sigma-1) ln(2), where sigma is a parameter of the model and a measure of the interaction range. We find that only r/sub l/= < r/sup( m/)/sub l/ > , ..delta../sub l/= < (r/sup( m/)/sub l/-r/sub l/)/sup 2/ > , and U/sub l/= < U/sup( m/)/sub l/ > are relevant physical quantities within the epsilon' expansion. By solving the equations for U/sub l/ and ..delta../sub l/ to O (epsilon') we obtain in general four fixed points of the recursion relations. For sigma>1/2, there is no stable fixed point in the physical region ..delta.. > or = 0, U > or = 0; thus the behavior of the system is still open to study. In the particular case of uniform block coupling but random spin distributions, we find a fixed point with U* approx. = O ( vertical-bar epsilon' vertical-bar /sup 1/2/), ..delta..* approx. = O ( vertical-bar epsilon' vertical-bar /sup 1/2/) for epsilon'<0, but it is unstable.
- Research Organization:
- Department of Physics, Brookhaven National Laboratory, Upton, New York 11973
- DOE Contract Number:
- EY-76-C-02-0016
- OSTI ID:
- 5407301
- Journal Information:
- Phys. Rev., B: Condens. Matter; (United States), Journal Name: Phys. Rev., B: Condens. Matter; (United States) Vol. 21:7; ISSN PRBMD
- Country of Publication:
- United States
- Language:
- English
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