Carpentier D, Le Doussal P
Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Feb;63(2 Pt 2):026110. doi: 10.1103/PhysRevE.63.026110. Epub 2001 Jan 24.
We study via renormalization group (RG), numerics, exact bounds, and qualitative arguments the equilibrium Gibbs measure of a particle in a d-dimensional Gaussian random potential with translationally invariant logarithmic spatial correlations. We show that for any d>/=1 it exhibits a transition at T=T(c)>0. The low-temperature glass phase has a nontrivial structure, being dominated by a few distant states (with replica symmetry breaking phenomenology). In finite dimension this transition exists only in this "marginal glass" case (energy fluctuation exponent straight theta=0) and disappears if correlations grow faster (single ground-state dominance straight theta>0) or slower (high-temperature phase). The associated extremal statistics problem for correlated energy landscapes exhibits universal features which we describe using a nonlinear Kolmogorov (KPP) RG equation. These include the tails of the distribution of the minimal energy (or free energy) and the finite-size corrections, which are universal. The glass transition is closely related to Derrida's random energy models. In d=2, the connection between this problem and Liouville and sinh-Gordon models is discussed. The glass transition of the particle exhibits interesting similarities with the weak- to strong-coupling transition in Liouville (c=1 barrier) and with a transition that we conjecture for the sinh-Gordon model, with correspondence in some exact results and RG analysis. Glassy freezing of the particle is associated with the generation under RG of new local operators and of nonsmooth configurations in Liouville. Applications to Dirac fermions in random magnetic fields at criticality reveal a peculiar "quasilocalized" regime (corresponding to the glass phase for the particle), where eigenfunctions are concentrated over a finite number of distant regions, and allow us to recover the multifractal spectrum in the delocalized regime.
我们通过重整化群（RG）、数值计算、精确界和定性论证，研究了在具有平移不变对数空间相关性的d维高斯随机势中粒子的平衡吉布斯测度。我们表明，对于任何d≥1，它在T = T(c)>0时表现出转变。低温玻璃相具有非平凡的结构，由少数远距离状态主导（具有副本对称性破缺现象）。在有限维中，这种转变仅存在于这种“边缘玻璃”情况（能量涨落指数θ = 0），如果相关性增长更快（单一基态主导θ>0）或更慢（高温相），则转变消失。相关的相关能量景观的极值统计问题展现出我们使用非线性柯尔莫哥洛夫（KPP）重整化群方程描述的普遍特征。这些特征包括最小能量（或自由能）分布的尾部以及有限尺寸修正，它们都是普遍的。玻璃转变与德里达的随机能量模型密切相关。在d = 2时，讨论了这个问题与刘维尔模型和 sinh - 戈登模型之间的联系。粒子的玻璃转变与刘维尔模型中从弱耦合到强耦合的转变（c = 1势垒）以及我们推测的 sinh - 戈登模型的转变表现出有趣的相似性，在一些精确结果和重整化群分析中存在对应关系。粒子的玻璃态冻结与重整化群下刘维尔模型中新的局部算符和非光滑构型的产生有关。对处于临界状态的随机磁场中的狄拉克费米子的应用揭示了一种特殊的“准局域化” regime（对应于粒子的玻璃相），其中本征函数集中在有限数量的远距离区域，并使我们能够在非局域化 regime 中恢复多重分形谱。
