New Foundation of a Low Temperature Theory for Spin Glasses and Frustrated Order with Replica Symmetry Breaking
Final Report Abstract
We achieved a ’New foundation of the (Parisi) spin glass theory’ in the following sense: The finite T = 0 theory of the frustrated Parisi spin glass phase was obtained, confirmed, and further developed in great numerical and analytical detail. The analytical theory was shaped in form of a T = 0 nonlinear variable partial differential equation (nv-PDE) for the exponential correction function, where one coefficient depends on the order parameter function q(a). Its initial condition was determined as the solution of an ordinary differential equation. The order parameter function q(a) was determined up to a 400-dimensional extremization of the free energy in the almost continuous regime of 200-fold replica symmetry breaking. We observed scaling with respect to physical parameters such as small external magnetic fields and small temperatures and determined the critical exponents. The variable a differs from the Parisi-variable x, since only a allows for a finite T = 0-theory. In addition it also allowed a finite temperature theory. In [9] we revealed two classes 0 ≤ a ≤ 1/kB T , and 1/(kB T ) ≤ a ≤ ∞ to be relevant for finite temperatures, the first condition suffices for T = 0. We explained this classification by means of the scaling function for the Parisi block size parameter m(T ). Temperature (T) and magnetic field (H-) dependence enter via the initial condition for the vn-PDE. T - and H-grids were refined and the RSB-order was extended up to κ = 64 to confirm scaling w.r.t. T and H near {κ = ∞, H = 0, T = 0} We found as well scaling with respect to the order of replica symmetry breaking, which refers to the flow of the discrete Parisi theory (for each order κ of replica symmetry breaking) towards the continuous theory κ = ∞. We determined as well a flow towards a discrete set of fixed point values qm (a∗ ), the number being limited by the order ∗ m κ of RSB-steps. This basis is important, since a divergence-free T = 0-theory is needed in order to incorporate many body phenomena like electronic transport for example occur in the presence of a spin glass phase, and mutual interplay with this type of magnetic order. We showed that scaling behavior exists in many ways in the Parisi construction of spin glass order. A (1+1)-dimensional partial differential equation, which is valid at T = 0, was derived. It belongs to the class of variable nonlinear PDEs, where a variable coefficient function contains the T = 0 order parameter function q(a). Thus, in principle, a solution must be found for many q(a), then used in expressing the free energy, which finally must be extremized in order to determine q(a). We found q(a) however already by our numerical procedure, and proposed a rather precise analytical model function. This has been used in the numerical solution of our T = 0 pde for the Parisi phase. Despite the fact that the SK-model is called a mean-field model due to the infinite-range Ising spin interactions, which suppresses spatial fluctuations, the Parisi scheme, proved to be the exact description of frustrated spin order, is a (1 + 1)-dimensional theory with two critical points at a = 0 and at a = ∞. We determined the corresponding critical power laws with 1D-type rational exponents in the almost continuous regime of large orders κ. The critical behavior, described in terms of physical and of nonphysical parameters, showed (up to high numerical precision) cubic root rational critical exponents, which are typical for 1D critical behavior. This was interpreted as a reducibility of the (1+1)-dimensional PDE into a one-dimensional ODE in the critical regime. The crossover between the scaling regimes at a = 0 and a = ∞ may prevent the PDE-reduction and be responsible for the difficulty in finding the exact solution for all a. In addition we observed and explained in detail, how the point a = ∞ contains also the break point at T = 0, in agreement with the finite Parisi parameters xbr . The most recent efforts are of mathematical nature. The variable nonlinear PDE for the exponential correction function was simplified by exact transformations and analyzed by Painleve tests, and also reshaped in order to achieve an effective potential method. This study was motivated by the work with Hannes Schenck in Op28/7, where we discovered and explained a pseudogap feature of the internal field distribution (or density of states in a fermionic spin glass analogy) by means of a quantum mechanical eigenstate of a special harmonic oscillator model with nonanalytic shift and tiny anharmonic corrections. We studied also hierarchical order in generalizations of the Galam voter model for close elections. The universal meaning of frustrated order, beyond its manifestation in physics, has been the motivation.
Publications
- Physical Review Letters 98, 127201 (2007)
R. Oppermann, M.J. Schmidt, D. Sherrington
- Physical Review E 77, 061104 (2008)
Manuel J. Schmidt, Reinhold Oppermann
- Physical Review E 78, 061124 (2008)
Reinhold Oppermann, Manuel J. Schmidt
- JETP Letters 91, 570 (2010)
E. Nakhmedov, R. Oppermann
- Phys. Rev. B 81, 134511 (2010)
E. Nakhmedov, R. Oppermann
- Philosophical Magazine, 1-5, iFirst (2011)
Reinhold Oppermann, Hannes Schenck
