Hidetoshi Nishimori and Gerardo Ortiz
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577224
- eISBN:
- 9780191722943
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577224.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Real materials always contain randomness or disorder that cannot be expressed by idealized simple model systems. The present chapter studies the effects of randomness on phase transitions and ...
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Real materials always contain randomness or disorder that cannot be expressed by idealized simple model systems. The present chapter studies the effects of randomness on phase transitions and critical phenomena. Although randomness may seem to obscure singular behaviour such as divergence of physical quantities at the critical temperature, it is established that well-defined phase transitions exist as long as randomness is not too strong, but the critical behaviour may get modified with respect to the pure sample. After the introduction of basic concepts and methods such as self-averaging and replica method, it is elucidated what type of phase transitions exist in the random-field Ising model and the SK model of spin glasses. Also explained are the percolation transitions using the fractal structure and the Potts model.Less
Real materials always contain randomness or disorder that cannot be expressed by idealized simple model systems. The present chapter studies the effects of randomness on phase transitions and critical phenomena. Although randomness may seem to obscure singular behaviour such as divergence of physical quantities at the critical temperature, it is established that well-defined phase transitions exist as long as randomness is not too strong, but the critical behaviour may get modified with respect to the pure sample. After the introduction of basic concepts and methods such as self-averaging and replica method, it is elucidated what type of phase transitions exist in the random-field Ising model and the SK model of spin glasses. Also explained are the percolation transitions using the fractal structure and the Potts model.
Hidetoshi Nishimori
- Published in print:
- 2001
- Published Online:
- January 2010
- ISBN:
- 9780198509417
- eISBN:
- 9780191709081
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509417.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter continues the analysis of the Sherrington–Kirkpatrick (SK) model started in the preceding chapter. The free energy of the SK model derived under the ansatz of replica symmetry has the ...
More
This chapter continues the analysis of the Sherrington–Kirkpatrick (SK) model started in the preceding chapter. The free energy of the SK model derived under the ansatz of replica symmetry has the problem of negative entropy at low temperatures. It is therefore natural to investigate the possibility that the order parameter may assume various values depending upon the replica indices. The theory of replica symmetry breaking started in this way as a mathematical effort to avoid unphysical conclusions of the replica-symmetric solution. It turned out, however, that the scheme of replica symmetry breaking developed by Parisi has a very rich physical implication, namely the existence of a vast variety of stable states with ultrametric structure in the phase space. The chapter is devoted to the elucidation of this story.Less
This chapter continues the analysis of the Sherrington–Kirkpatrick (SK) model started in the preceding chapter. The free energy of the SK model derived under the ansatz of replica symmetry has the problem of negative entropy at low temperatures. It is therefore natural to investigate the possibility that the order parameter may assume various values depending upon the replica indices. The theory of replica symmetry breaking started in this way as a mathematical effort to avoid unphysical conclusions of the replica-symmetric solution. It turned out, however, that the scheme of replica symmetry breaking developed by Parisi has a very rich physical implication, namely the existence of a vast variety of stable states with ultrametric structure in the phase space. The chapter is devoted to the elucidation of this story.
