Physical systems differring in their microscopic details often display strikingly similar behaviour when probed at macroscopic scales. Those universal properties, largely determining their physical characteristics, are revealed by the powerful renormalization group (RG) procedure, which systematically retains "slow" degrees of freedom and integrates out the rest. However, the important degrees of freedom may be difficult to identify. Here we demonstrate a machine learning algorithm capable of identifying the relevant degrees of freedom and executing RG steps iteratively without any prior knowledge about the system. We introduce an artificial neural network based on a modelindependent, information-theoretic characterization of a real-space RG procedure, performing this task. We apply the algorithm to classical statistical physics problems in one and two dimensions. We demonstrate RG flow and extract the Ising critical exponent. Our results demonstrate that machine learning techniques can extract abstract physical concepts and consequently become an integral part of theory-and model-building.Machine learning has been captivating public attention lately due to groundbreaking advances in automated translation, image and speach recognition [1], gameplaying [2], and achieving super-human performance in tasks in which humans excelled while more traditional algorithmic approaches struggled [3]. The applications of those techniques in physics are very recent, initially leveraging the trademark prowess of machine learning in classification and pattern recognition and applying them to classify phases of matter [4][5][6][7][8], study amorphous materials [9, 10], or exploiting the neural networks' potential as efficient non-linear approximators of arbitrary functions [11,12] to introduce a new numerical simulation method for quantum systems [13,14]. However, the exciting possibility of employing machine learning not as a numerical simulator, or a hypothesis tester, but as an integral part of the physical reasoning process is still largely unexplored and, given the staggering pace of progress in the field of artificial intelligence, of fundamental importance and promise.The renormalization group (RG) approach has been one of the conceptually most profound tools of theoretical physics since its inception. It underlies the seminal work on critical phenomena [15], the discovery of asymptotic freedom in quantum chromodynamics [16], and of the Kosterlitz-Thouless phase transition [17,18]. The RG is not a monolith, but rather a conceptual framework comprising different techniques: real-space RG [19], functional RG [20], density matrix renormalization group (DMRG) [21], among others. While all those schemes differ quite substantially in details, style and applicability there is an underlying physical intuition which encompasses all of them -the essence of RG lies in identifying the "relevant" degrees of freedom and integrating out the "irrelevant" ones iteratively, thereby arriving at a universal, low-energy effective theory. However...

We construct exactly soluble lattice models for fractionalized, time reversal invariant electronic insulators in 2 and 3 dimensions. The low energy physics of these models is exactly equivalent to a non-interacting topological insulator built out of fractionally charged fermionic quasiparticles. We show that some of our models have protected edge modes (in 2D) and surface modes (in 3D), and are thus fractionalized analogues of topological insulators. We also find that some of the 2D models do not have protected edge modes -that is, the edge modes can be gapped out by appropriate time reversal invariant, charge conserving perturbations. (A similar state of affairs may also exist in 3D). We show that all of our models are topologically ordered, exhibiting fractional statistics as well as ground state degeneracy on a torus. In the 3D case, we find that the models exhibit a fractional magnetoelectric effect.

In this work, the library spinney is presented, which provides an implementation of helicity spinors and related algorithms for the symbolical manipulation program Form. The package is well suited for symbolic amplitude calculations both in traditional, Feynman diagram based approaches and unitarity-based techniques.

The connections between information theory, statistical physics and quantum field theory have been the focus of renewed attention. In particular, the renormalization group (RG) has been explored from this perspective. Recently, a variational algorithm employing machine learning tools to identify the relevant degrees of freedom of a statistical system by maximizing an information-theoretic quantity, the real-space mutual information (RSMI), was proposed for real-space RG. Here we investigate analytically the RG coarse-graining procedure and the renormalized Hamiltonian, which the RSMI algorithm defines. By a combination of general arguments, exact calculations and toy models we show that the RSMI coarse-graining is optimal in a sense we define. In particular, a perfect RSMI coarse-graining generically does not increase the range of a short-ranged Hamiltonian, in any dimension. For the case of the 1D Ising model we perturbatively derive the dependence of the coefficients of the renormalized Hamiltonian on the real-space mutual information retained by a generic coarse-graining procedure. We also study the dependence of the optimal coarse-graining on the prior constraints on the number and type of coarse-grained variables. We construct toy models illustrating our findings.

The analysis of complex physical systems hinges on the ability to extract the relevant degrees of freedom from among the many others. Though much hope is placed in machine learning, it also brings challenges, chief of which is interpretability. It is often unclear what relation, if any, the architectureand training-dependent learned "relevant" features bear to standard objects of physical theory. Here we report on theoretical results which may help to systematically address this issue: we establish equivalence between the field-theoretic relevance of the renormalization group, and an information-theoretic notion of relevance we define using the information bottleneck (IB) formalism of compression theory. We show analytically that for statistical physical systems described by a field theory the relevant degrees of freedom found using IB compression indeed correspond to operators with the lowest scaling dimensions. We confirm our field theoretic predictions numerically. We study dependence of the IB solutions on the physical symmetries of the data. Our findings provide a dictionary connecting two distinct theoretical toolboxes, and an example of constructively incorporating physical interpretability in applications of deep learning in physics.