Finitely coordinated models for low-temperature phases of amorphous systems

We introduce models of heterogeneous systems with finite connectivity defined on random graphs to capture finite-coordination effects on the low-temperature behaviour of finite-dimensional systems. Our models use a description in terms of small deviations of particle coordinates from a set of reference positions, particularly appropriate for the description of low-temperature phenomena. A Born–von Karman-type expansion with random coefficients is used to model effects of frozen heterogeneities. The key quantity appearing in the theoretical description is a full distribution of effective single-site potentials which needs to be determined self-consistently. If microscopic interactions are harmonic, the effective single-site potentials turn out to be harmonic as well, and the distribution of these single-site potentials is equivalent to a distribution of localization lengths used earlier in the description of chemical gels. For structural glasses characterized by frustration and anharmonicities in the microscopic interactions, the distribution of single-site potentials involves anharmonicities of all orders, and both single-well and double-well potentials are observed, the latter with a broad spectrum of barrier heights. The appearance of glassy phases at low temperatures is marked by the appearance of asymmetries in the distribution of single-site potentials, as previously observed for fully connected systems. Double-well potentials with a broad spectrum of barrier heights and asymmetries would give rise to the well-known universal glassy low-temperature anomalies when quantum effects are taken into account.

[1]  G. Parisi,et al.  Replica field theory for deterministic models: I. Binary sequences with low autocorrelation , 1994, hep-th/9405148.

[2]  S. Berezovsky,et al.  A possible origin of anomalous properties of proper uniaxial ferroelectrics near the lock-in transition , 2000 .

[3]  M. Mézard,et al.  Self induced quenched disorder: a model for the glass transition , 1994, cond-mat/9405075.

[4]  M. Mézard,et al.  Glass models on Bethe lattices , 2003, cond-mat/0307569.

[5]  Cavity approach to the random solid state. , 2005, Physical review letters.

[6]  J. Jäckle.  On the ultrasonic attenuation in glasses at low temperatures , 1972 .

[7]  Parshin.  Interactions of soft atomic potentials and universality of low-temperature properties of glasses. , 1994, Physical review. B, Condensed matter.

[8]  M. Weigt,et al.  A hard-sphere model on generalised Bethe lattices: Dynamics , 2005, cond-mat/0505202.

[9]  T. R. Kirkpatrick,et al.  p-spin-interaction spin-glass models: Connections with the structural glass problem. , 1987, Physical review. B, Condensed matter.

[10]  Towards finite-dimensional gelation , 2002, cond-mat/0204494.

[11]  T. R. Kirkpatrick,et al.  Connections between some kinetic and equilibrium theories of the glass transition. , 1987, Physical review. A, General physics.

[12]  D. Sherrington,et al.  Graph bipartitioning and spin glasses on a random network of fixed finite valence , 1987 .

[13]  Weber,et al.  Computer simulation of local order in condensed phases of silicon. , 1985, Physical review. B, Condensed matter.

[14]  A. Coolen,et al.  Dynamical replica analysis of disordered Ising spin systems on finitely connected random graphs. , 2005, Physical review letters.

[15]  T. R. Kirkpatrick,et al.  Dynamics of the structural glass transition and the p-spin-interaction spin-glass model. , 1987, Physical review letters.

[16]  E. Donth.  The glass transition : relaxation dynamics in liquids and disordered materials , 2001 .

[17]  A. Bray,et al.  Phase diagrams for dilute spin glasses , 1985 .

[18]  Glassy behavior induced by geometrical frustration in a hard-core lattice-gas model , 2002, cond-mat/0210054.

[19]  Parshin Da.  Interactions of soft atomic potentials and universality of low-temperature properties of glasses. , 1994 .

[20]  Cugliandolo,et al.  Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model. , 1993, Physical review letters.

[21]  Guilhem Semerjian,et al.  Dynamics of dilute disordered models: A solvable case , 2002, cond-mat/0204613.

[22]  W. Kauzmann.  The Nature of the Glassy State and the Behavior of Liquids at Low Temperatures. , 1948 .

[23]  Translational invariance in models for low-temperature properties of glasses , 1999, cond-mat/9910195.

[24]  Andrea Montanari,et al.  From Large Scale Rearrangements to Mode Coupling Phenomenology in Model Glasses , 2005 .

[25]  S. Nagel,et al.  Supercooled Liquids and Glasses , 1996 .

[26]  Random Matrix Approach to Glassy Physics: Low Temperatures and Beyond , 1997, cond-mat/9704162.

[27]  P. Goldbart,et al.  Randomly crosslinked macromolecular systems : vulcanization transition to and properties of the amorphous solid state , 1996, cond-mat/9604062.

[28]  M. Mézard,et al.  A ferromagnet with a glass transition , 2001, cond-mat/0103026.

[29]  P. Anderson,et al.  Anomalous low-temperature thermal properties of glasses and spin glasses , 1972 .

[30]  R. Monasson.  Optimization problems and replica symmetry breaking in finite connectivity spin glasses , 1997, cond-mat/9707089.

[31]  G. Parisi,et al.  Recipes for metastable states in spin glasses , 1995 .

[32]  M. Weigt,et al.  Approximation schemes for the dynamics of diluted spin models: the Ising ferromagnet on a Bethe lattice , 2004, cond-mat/0402451.

[33]  R. Pohl,et al.  Thermal Conductivity and Specific Heat of Noncrystalline Solids , 1971 .

[34]  J. H. Gibbs,et al.  Nature of the Glass Transition and the Glassy State , 1958 .

[35]  M. Mézard,et al.  The Bethe lattice spin glass revisited , 2000, cond-mat/0009418.

[36]  A. Crisanti,et al.  The sphericalp-spin interaction spin glass model: the statics , 1992 .

[37]  M. Mézard,et al.  THERMODYNAMICS OF GLASSES : A FIRST PRINCIPLES COMPUTATION , 1998, cond-mat/9807420.

[38]  G. Parisi,et al.  Phase Diagram of Coupled Glassy Systems: A Mean-Field Study , 1997 .

[39]  Monasson.  Structural glass transition and the entropy of the metastable states. , 1995, Physical Review Letters.

[40]  W. Kob.  Supercooled Liquids and Glasses , 1999, cond-mat/9911023.

[41]  Y. Kagan,et al.  On the nature of the universal properties of amorphous solids , 1996 .

[42]  G. Adam,et al.  On the Temperature Dependence of Cooperative Relaxation Properties in Glass‐Forming Liquids , 1965 .

[43]  Marc Mézard,et al.  Lattice glass models. , 2002, Physical review letters.

[44]  Lattice glass model with no tendency to crystallize. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[45]  P. Goldbart,et al.  Elastic heterogeneity of soft random solids , 2006, cond-mat/0610407.

[46]  W. A. Phillips,et al.  Tunneling states in amorphous solids , 1972 .

[47]  G. Parisi,et al.  Replica field theory for deterministic models: II. A non-random spin glass with glassy behaviour , 1994, cond-mat/9406074.

[48]  Universality in glassy low-temperature physics , 2002, cond-mat/0202340.

[49]  Harmonic Vibrational Excitations in Disordered Solids and the “Boson Peak” , 1998, cond-mat/9801249.

[50]  J. Jäckle,et al.  Anomalous Sound Velocity in Vitreous Silica at Very Low Temperatures , 1974 .

[51]  D. Saad,et al.  Random graph coloring: statistical physics approach. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  G. Parisi,et al.  Phonon interpretation of the ‘boson peak’ in supercooled liquids , 2003, Nature.

[53]  M. Weigt,et al.  A hard-sphere model on generalized Bethe lattices: statics , 2005, cond-mat/0501571.