Amorphous solid state: a locally stable thermodynamic phase of randomly constrained systems

Physical Review B - Condensed Matter and Materials Physics

Volumn 60, Issue 21, 1999, Pages 14702-14718

  Zippelius, Annette ^c  

^a LABORATOIRE DE PHYSIQUE STATISTIQUE   (France)

^b United States of America   (United States)

^c UNIVERSITY OF GÖTTINGEN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0003160646     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.60.14702     Document Type: Article

References (35)

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19. A similar phenomenon has been observed in the context of supervised learning in neural networks; see H. S. Seung, H. Sompolinsky, and N. Tishby, Phys. Rev. A 45, 6056 (1992).
20. A constant is ignored here that has to do with the symmetry under permutations of the macromolecules. See Ref. 4, Secs. 2.4 and 2.6 for a full discussion.
21. We use the notations (Formula presented) for the equilibrium value of the order parameter and (Formula presented) for the fluctuating variable (which features in the Landau free energy and of which (Formula presented) is the expectation value).
22. We consider here only the set of “small fluctuations” defined as those for which (Formula presented) is small for each individual wave vector (Formula presented). As, by Eq. (2.5), the stationary point order parameter is only nonzero in a lower dimensional subspace (defined by the condition (Formula presented), a more comprehensive definition of small fluctuations would additionally include those in which the support of the order parameter (i.e., the manifold on which it is nonzero) is slightly deformed. In those additional fluctuations, which one might call “capillary” or “interface” waves, the changes of the order-parameter field would not be small for individual values of the wave vector (Formula presented) (a full value of the order parameter would be replaced by zero for some values of (Formula presented), while for other values of (Formula presented) zero would be replaced by a full value of the order parameter). According to an enlarged definition of this sort, the set of small fluctuations would contain, among others, those fluctuations associated with “almost rigid” rotations of the system [notice that, by Eq. (2.10), a set of replica index dependent rigid rotations already produces a change of support for the order parameter].
23. See, e.g., Ref. 20, Vol. II, Chap. XI, Sec. 11.3.
24. Bateman Manuscript Project, Higher Transcendental Functions (McGraw-Hill, New York, 1953-55).
25. We have assumed here that the product (Formula presented) goes to zero continuously at the origin of (Formula presented) space. This assumption will hold true for the rest of the discussion, except for the problem of the spurious eigenvector in the case of MTI (i.e., (Formula presented) fluctuations.
26. The greatest lower bound of a nonempty set (Formula presented) of real numbers is the real number c such that for any lower bound b of the set (Formula presented) it is true that (Formula presented). Such a number always exists for any set that has at least a lower bound. See, e.g., H. L. Royden, Real Analysis (Macmillan, New York, 1968), Chap. 2.
27. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965).
28. The numerical errors quoted are estimated by using different numerical methods to compute the function (Formula presented) and comparing the highest and lowest estimates for the eigenvalues thus obtained. Other sources of error, not considered in the quoted estimate, include the discretization of the integral equation and the numerical error of the diagonalization algorithm itself.
29. One could attempt to solve Eq. (3.22a) by assuming that (Formula presented), but this leads to the condition (Formula presented)which can, at most, be satisfied for isolated values of n. In particular, by using Eq. (3.10) we see that for (Formula presented) the left-hand side is not an integer, and the equation is not satisfied.
30. To examine this assertion, let us consider a Hilbert space slightly different from the one defined in Sec. II C, in which the functions (Formula presented) are defined also for (Formula presented). This enlargement of the Hilbert space makes room for spurious fluctuations of the order parameter at the origin to appear. The scalar product in this enlarged space is defined by (Formula presented)The scalar product of the eigenvector (Formula presented) associated with a radial eigenfunction singular at the origin and the normalized function (Formula presented) concentrated at the zero-replica sector is given by (Formula presented)and thus (Formula presented) lies entirely in the zero-replica sector for (Formula presented).
31. A search for a replica symmetry-breaking stationary point of the free energy functional (2.4) has not yielded any such solution [W. Peng, H. E. Castillo, and P. M. Goldbart, (unpublished); see also
32. P. M. Goldbart and A. Zippelius, J. Phys. A 27, 6375 (1994)].
33. S. J. Barsky, Ph.D. thesis, Simon Fraser University, Canada, 1996;S. J. Barsky and M. Plischke (unpublished);
34. M. Plischke and B. Joós, Phys. Canada 53, 184 (1997), and references therein.See also the discussion in Ref. 9.
35. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York, 1965).