Recently, a novel real-space renormalization group (RG) algorithm was introduced. By maximizing an information-theoretic quantity, the real-space mutual information, the algorithm identifies the relevant low-energy degrees of freedom. Motivated by this insight, we investigate the information-theoretic properties of coarse-graining procedures for both translationally invariant and disordered systems. We prove that a perfect real-space mutual information coarse graining does not increase the range of interactions in the renormalized Hamiltonian, and, for disordered systems, it suppresses the generation of correlations in the renormalized disorder distribution, being in this sense optimal. We empirically verify decay of those measures of complexity as a function of information retained by the RG, on the examples of arbitrary coarse grainings of the clean and random Ising chain. The results establish a direct and quantifiable connection between properties of RG viewed as a compression scheme and those of physical objects, i.e., Hamiltonians and disorder distributions. We also study the effect of constraints on the number and type of coarse-grained degrees of freedom on a generic RG procedure.

中文翻译：

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基于信息论的最优重整化群转换

最近，引入了一种新颖的实空间重归一化组（RG）算法。通过最大化信息理论量（实际空间互信息），该算法可以识别相关的低能量自由度。基于这种见识，我们研究了平移不变和无序系统的粗粒度过程的信息理论特性。我们证明，理想的实空间互信息粗粒度不会增加重新规范化的哈密顿量中的相互作用范围，并且，对于无序系统，它抑制了重新规范化的无序分布中的相关性生成，因此在这种意义上是最优的。我们以干净和随机的伊辛链的任意粗粒度为例，根据RG保留的信息经验性地验证了这些复杂性度量的衰减。结果在被视为压缩方案的RG属性与物理对象（即哈密顿量和无序分布）的属性之间建立了直接且可量化的联系。我们还研究了通用RG程序上约束对粗粒度自由度的数量和类型的影响。