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Random quantum circuits yield minimally structured models for chaotic quantum dynamics, which are able to capture, for example, universal properties of entanglement growth. We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries. We study both 1+1D and higher dimensions and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems. In 1+1D, we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC and determines the butterfly speed v[subscript B]. We find that in 1+1D, the “front” of the OTOC broadens diffusively, with a width scaling in time as t[superscript 1/2]. We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front within a realization. Turning to higher dimensions, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a purely classical droplet growth problem. This implies that the width of the front of the averaged OTOC scales as t[superscript 1/3] in 2+1D and as t[superscript 0.240] in 3+1D (exponents of the Kardar-Parisi-Zhang universality class). We support our analytic argument with simulations in 2+1D. We point out that, in two or higher spatial dimensions, the shape of the spreading operator at late times is affected by underlying lattice symmetries and, in general, is not spherical. However, when full spatial rotational symmetry is present in 2+1D, our mapping implies an exact asymptotic form for the OTOC, in terms of the Tracy-Widom distribution. For an alternative perspective on the OTOC in 1+1D, we map it to the partition function of an Ising-like statistical mechanics model. As a result of special structure arising from unitarity, this partition function reduces to a random walk ca

Characterizing how entanglement grows with time in a many-body system, for example, after a quantum quench, is a key problem in nonequilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time-dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the “entanglement tsunami” in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated Kardar-Parisi-Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like (time)[superscript 1/3] and are spatially correlated over a distance ∝(time)[superscript 2/3]. We derive KPZ universal behavior in three complementary ways, by mapping random entanglement growth to (i) a stochastic model of a growing surface, (ii) a “minimal cut” picture, reminiscent of the Ryu-Takayanagi formula in holography, and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading-order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple coarse grained minimal cut picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the “velocity” of entanglement growth in the 1D entanglement tsunami. We show that in higher dimensions, noisy entanglement evolution maps to the well-studied problem of pinning of a membrane or domain wall by disorder. Subject Areas: Condensed Matter Physics, Quantum Information, Statistical Physics, Gordon and Betty Moore Foundation (Grant GBMF4303), Engineering and Physical Sciences Research Council (Grant EP/N028678/1), Kavli Institute for Theoretical Physics (Graduate Fellows Program), United States. Department of Energy. Division of Materials Sciences and Engineering (Award DE-SC0010526), Massachusetts Institute of Technology (MIT Pappalardo Fellowship in Physics)

We introduce a generalization of conventional lattice gauge theory to describe fracton topological phases, which are characterized by immobile, pointlike topological excitations, and subextensive topological degeneracy. We demonstrate a duality between fracton topological order and interacting spin systems with symmetries along extensive, lower-dimensional subsystems, which may be used to systematically search for and characterize fracton topological phases. Commutative algebra and elementary algebraic geometry provide an effective mathematical tool set for our results. Our work paves the way for identifying possible material realizations of fracton topological phases., David & Lucile Packard Foundation, MIT Department of Physics Pappalardo Program

The cubic code model is studied in the presence of arbitrary extensive perturbations. Below a critical perturbation strength, we show that most states with finite energy are localized; the overwhelming majority of such states have energy concentrated around a finite number of defects, and remain so for a time that is near exponential in the distance between the defects. This phenomenon is due to an emergent superselection rule and does not require any disorder. Local integrals of motion for these finite energy sectors are identified as well. Our analysis extends more generally to systems with immobile topological excitations., MIT Department of Physics Pappalardo Program (Fellowship)

We introduce exactly solvable models of interacting (Majorana) fermions in d≥3 spatial dimensions that realize a new kind of fermion topological quantum order, building on a model presented by S. Vijay, T. H. Hsieh, and L. Fu [Phys. Rev. X 5, 041038 (2015)10.1103/PhysRevX.5.041038]. These models have extensive topological ground-state degeneracy and a hierarchy of pointlike, topological excitations that are only free to move within submanifolds of the lattice. In particular, one of our models has fundamental excitations that are completely stationary. To demonstrate these results, we introduce a powerful polynomial representation of commuting Majorana Hamiltonians. Remarkably, the physical properties of the topologically ordered state are encoded in an algebraic variety, defined by the common zeros of a set of polynomials over a finite field. This provides a “geometric” framework for the emergence of topological order., David & Lucile Packard Foundation, United States. Dept. of Energy. Division of Materials Sciences and Engineering (Award DE-SC0010526), MIT Department of Physics Pappalardo Program

We study entanglement renormalization group transformations for the ground states of a spin model, called cubic code model H[subscript A] in three dimensions, in order to understand long-range entanglement structure. The cubic code model has degenerate and locally indistinguishable ground states under periodic boundary conditions. In the entanglement renormalization, one applies local unitary transformations on a state, called disentangling transformations, after which some of the spins are completely disentangled from the rest and then discarded. We find a disentangling unitary to establish equivalence of the ground state of H[subscript A] on a lattice of lattice spacing a to the tensor product of ground spaces of two independent Hamiltonians H[subscript A] and H[subscript B] on lattices of lattice spacing 2a. We further find a disentangling unitary for the ground space of H[subscript B] with the lattice spacing a to show that it decomposes into two copies of itself on the lattice of the lattice spacing 2a. The disentangling transformations yield a tensor network description for the ground state of the cubic code model. Using exact formulas for the degeneracy as a function of system size, we show that the two Hamiltonians H[subscript A] and H[subscript B] represent distinct phases of matter., Gordon and Betty Moore Foundation, MIT Department of Physics Pappalardo Program