The unrelated discoveries of quasicrystals and topological insulators have in turn challenged prevailing paradigms in condensed-matter physics. We find a surprising connection between quasicrystals and topological phases of matter: (i) quasicrystals exhibit nontrivial topological properties and (ii) these properties are attributed to dimensions higher than that of the quasicrystal. Specifically, we show, both theoretically and experimentally, that one-dimensional quasicrystals are assigned two-dimensional Chern numbers and, respectively, exhibit topologically protected boundary states equivalent to the edge states of a two-dimensional quantum Hall system. We harness the topological nature of these states to adiabatically pump light across the quasicrystal. We generalize our results to higher-dimensional systems and other topological indices. Hence, quasicrystals offer a new platform for the study of topological phases while their topology may better explain their surface properties.

One-dimensional (1D) quasicrystals exhibit physical phenomena associated with the 2D integer quantum Hall effect. Here, we transcend dimensions and show that a previously inaccessible phase of matter-the 4D integer quantum Hall effect-can be incorporated in a 2D quasicrystal. Correspondingly, our 2D model has a quantized charge-pump accommodated by an elaborate edge phenomena with protected level crossings. We propose experiments to observe these 4D phenomena, and generalize our results to a plethora of topologically equivalent quasicrystals. Thus, 2D quasicrystals may pave the way to the experimental study of 4D physics.

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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...