Oppermann R, Schmidt M J
Institut für Theoretische Physik, Universität Würzburg, Am Hubland, 97074 Würzburg, Federal Republic of Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Dec;78(6 Pt 1):061124. doi: 10.1103/PhysRevE.78.061124. Epub 2008 Dec 23.
A scaling theory of replica symmetry breaking (RSB) in the Sherrington-Kirkpatrick (SK) model is presented in the framework of critical phenomena for the scaling regime of large RSB orders kappa , small temperatures T , and small (homogeneous) magnetic fields H . We employ the pseudodynamical picture [R. Oppermann, M. J. Schmidt, and D. Sherrington, Phys. Rev. Lett. 98, 127201 (2007)], where two critical points CP1 and CP2 are associated with the order function's pseudodynamical limits lim_{a-->infinity}q(a)=1 and lim_{a-->0}q(a)=0 at (T=0 , H=0 , 1kappa=0) . CP1 - and CP2 -dominated contributions to the free energy functional F[q(a)] require an unconventional scaling hypothesis. We determine the scaling contributions in accordance with detailed numerical self-consistent solutions for up to 200 orders of RSB. Power laws, scaling functions, and crossover lines are obtained. CP1 -dominated behavior is found for the nonequilibrium susceptibility, which decays like chi_{1}=kappa;{-53}f_{1}(Tkappa;{-53}) , for the entropy, which obeys S(T=0) approximately chi_{1};{2} , and for the subclass of diverging parameters a_{i}=kappa;{53}f_{a_{i}}(Tkappa;{-53}) [describing Parisi box sizes m_{i}(T) identical witha_{i}(T)T ], with f_{1}(zeta) approximately zeta and f_{a_{i}}(zeta) approximately 1zeta for zeta-->infinity , while f(0) is finite. CP2 -dominated behavior, controlled by the magnetic field H while temperature is irrelevant, is retrieved in the plateau height (or width) of the order function q(a) according to q_{pl}(H)=kappa;{-1}f_{pl}(H;{23}kappa;{-1}) with f_{pl}mid R:(zeta)mid R:{zeta-->infinity} approximately zeta and f{pl}(0) finite. Divergent characteristic RSB orders kappa_{CP1}(T) approximately T;{-35} and kappa_{CP2}(H) approximately H;{-23} , respectively, describe the crossover from mean field SK- to RSB-critical behavior with rational-valued exponents extracted with high precision from our RSB data. The order function q(a) is obtained as a fixed-point function q(a) of RSB flow, in agreement with integrated fixed-point energy and susceptibility distributions.
在大复制对称破缺（RSB）阶数κ、低温T和小（均匀）磁场H的标度区域的临界现象框架下，提出了Sherrington - Kirkpatrick（SK）模型中复制对称破缺（RSB）的标度理论。我们采用赝动力学图像[R. Oppermann, M. J. Schmidt, and D. Sherrington, Phys. Rev. Lett. 98, 127201 (2007)]，其中两个临界点CP1和CP2与序函数在(T = 0, H = 0, κ = 0)处的赝动力学极限lim_{a→∞}q(a) = 1和lim_{a→0}q(a) = 0相关联。对自由能泛函F[q(a)]的CP1和CP2主导贡献需要一个非常规的标度假设。我们根据高达200阶RSB的详细数值自洽解来确定标度贡献。得到了幂律、标度函数和交叉线。对于非平衡磁化率，发现其具有CP1主导行为，其衰减形式为χ₁ = κ^{-5/3}f₁(Tκ^{-5/3})；对于熵，其服从S(T = 0) ≈ χ₁²；对于发散参数子类aᵢ = κ^{5/3}f_{aᵢ}(Tκ^{-5/3}) [描述与aᵢ(T)T相同的帕里西盒尺寸mᵢ(T)]，当ζ→∞时，f₁(ζ) ≈ ζ且f_{aᵢ}(ζ) ≈ 1/ζ，而f(0)是有限的。在序函数q(a)的平台高度（或宽度）中恢复了由磁场H控制而温度无关的CP2主导行为，根据q_{pl}(H) = κ^{-1}f_{pl}(H^{2/3}κ^{-1})，当ζ→∞时，f_{pl}(ζ) ≈ ζ且f_{pl}(0)有限。分别为发散的特征RSB阶数κ_{CP1}(T) ≈ T^{-3/5}和κ_{CP2}(H) ≈ H^{-2/3}，描述了从平均场SK到RSB临界行为的交叉，从我们的RSB数据中高精度提取了有理值指数。序函数q(a)作为RSB流的不动点函数q(a)得到，与积分不动点能量和磁化率分布一致。
