We present a hypothesis for the universal properties of operators evolving under Hamiltonian dynamics in many-body systems. The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green's functions grow linearly with rate α in generic systems, with an extra logarithmic correction in 1d. The rate α -an experimental observable -governs the exponential growth of operator complexity in a sense we make precise. This exponential growth prevails beyond semiclassical or large-N limits. Moreover, α upper bounds a large class of operator complexity measures, including the out-of-time-order correlator. As a result, we obtain a sharp bound on Lyapunov exponents λL ≤ 2α, which complements and improves the known universal low-temperature bound λL ≤ 2πT . We illustrate our results in paradigmatic examples such as non-integrable spin chains, the Sachdev-Ye-Kitaev model, and classical models. Finally we use the hypothesis in conjunction with the recursion method to develop a technique for computing diffusion constants. CONTENTS

Sr & RuO ' is an unconventional superconductor that has attracted widespread study because of its high purity and the possibility that its superconducting order parameter has odd parity. We study the dependence of its superconductivity on anisotropic strain. Applying uniaxial pressures of up to ~1 GPa along a 〈100〉 direction ( -axis) of the crystal lattice results in . increasing from 1.5 K in the unstrained material to 3.4 K at compression by ≈0.6%, and then falling steeply. Calculations give evidence that the observed maximum . occurs at or near a Lifshitz transition when the Fermi level passes through a Van Hove singularity, and open the possibility that the highly strained, . =3.4 K Sr & RuO ' has an even-rather than an odd-parity order parameter.The formation of superconductivity by the condensation of electron pairs into a coherent

Hydrodynamics is a general description for the flow of a fluid, and is expected to hold even for fundamental particles such as electrons when inter-particle interactions dominate. While various aspects of electron hydrodynamics were revealed in recent experiments, the fundamental spatial structure of hydrodynamic electrons, the Poiseuille flow profile, has remained elusive. In this work we provide the first real-space imaging of Poiseuille flow of an electronic fluid, as well as visualization of its evolution from ballistic flow. Utilizing a scanning nanotube single electron transistor, we image the Hall voltage of electronic flow through channels of high-mobility graphene. We find that the profile of the Hall field across the channel is a key physical quantity for distinguishing ballistic from hydrodynamic flow. We image the transition from flat, ballistic field profiles at low temperature into parabolic field profiles at elevated temperatures, which is the hallmark of Poiseuille flow. The curvature of the imaged profiles is qualitatively reproduced by Boltzmann calculations, which allow us to create a 'phase diagram' that characterizes the electron flow regimes. Our results provide long-sought, direct confirmation of Poiseuille flow in the solid state, and enable a new approach for exploring the rich physics of interacting electrons in real space.

In metallic samples of small enough size and sufficiently strong momentum-conserving scattering, the viscosity of the electron gas can become the dominant process governing transport. In this regime, momentum is a long-lived quantity whose evolution is described by an emergent hydrodynamical theory. Furthermore, breaking time-reversal symmetry leads to the appearance of an odd component to the viscosity called the Hall viscosity, which has attracted considerable attention recently due to its quantized nature in gapped systems but still eludes experimental confirmation. Based on microscopic calculations, we discuss how to measure the effects of both the even and odd components of the viscosity using hydrodynamic electronic transport in mesoscopic samples under applied magnetic fields.The semiclassical theory of electronic conduction, based on relaxation of total momentum by impurities, phonons and umklapp scattering, occupies a central place in condensed matter physics. It is therefore of particular interest to study the cases for which it fails. One case that has attracted much interest is the possibility of a hydrodynamic regime, where transport is dominated by viscous effects . One needs a large separation of scales between momentum-relaxing and momentum-conserving scattering in order to see these effects. This was recently achieved in graphene [23,24] and PdCoO 2 [25,26].Interest in such a hydrodynamic regime also emanated from a conjectured bound on diffusion constants for the hydrodynamics of strongly interacting quantum systems [27,28]. Even though the physics described in this work is semiclassical and probably still quite far from these quantum-mechanical bounds, the observations that we hope to stimulate would constitute an important first step towards the understanding of emergent hydrodynamical regimes in electronic systems.A further motivation for the work is that reaching a viscous regime for a charged fluid enables one to break time-reversal symmetry by adding a magnetic field and hence to study a non-dissipative component to the viscosity tensor called the Hall viscosity. The recent interest in this Hall viscosity emanates from the fact that it is topologically quantized in gapped systems [29]. In order to study this effect experimentally, in analogy with the Hall conductivity, the first step would obviously be to measure the classical Hall viscosity. We show in this letter how this measurement could be done by describing specific size effects from Hall viscosity in transport in restricted 2D channels under transverse magnetic fields. This paper is organized as follows. We start by assuming a perfect hydrodynamic regime and calculate ρ xx and ρ xy . We show that the 1/W 2 component of ρ xy is proportional to the Hall viscosity, thereby providing a way of measuring it. In order to have realistic predictions to compare with experiments, one should also take into account other, non-viscous effects that can lead to a sizedependent resistivity. We thus perform a kinetic Boltzmann calculation in which t...

We study the superconductivity pairing symmetry in Sr2RuO4 in the limit of small interaction by extending a renormalization group calculation developed by Raghu et al. [Phys. Rev. B 81, 224505 (2010)] to include spin-orbit coupling and multiband effects. We show these effects to be crucial to discriminate between the possible order parameters. In contrast to previous results and without the necessity of fine-tuning, we obtain pseudospin-triplet gaps of the same order of magnitude on the two-dimensional γ band and the quasi-one-dimensional α and β bands. The ratio of the gap amplitude on the different bands varies continuously with the interaction parameter. The favored pairing symmetry is shown to be chiral when γ is slightly dominant and helical when α and β are slightly dominant.

In this short review, we aim to provide a topical update on the status of efforts to understand the superconductivity of Sr2RuO4. We concentrate on the quest to identify a superconducting order parameter symmetry that is compatible with all the major pieces of experimental knowledge of the material, and highlight some major discrepancies that have become even clearer in recent years. As the pun in the title suggests, we have tried to start the discussion from scratch, making no assumptions even about fundamental issues such as the parity of the superconducting state. We conclude that no consensus is currently achievable in Sr2RuO4, and that the reasons for this go to the heart of how well some of the key probes of unconventional superconductivity are really understood. This is therefore a puzzle that merits continued in-depth study.normal state is a well-understood Fermi liquid. Secondly, the extremely high purity of the best available samples means that disorder is not nearly as big a complicating factor in experiments as it is in most other materials. Thirdly, the disorder sensitivity of the superconductivity comes because as well as the order parameter being unconventional, the coherence length in the superconducting state is rather long: approximately 750 Å. This means that the thermodynamic features expected of a mean-field, BCS-like transition are seen, and that the superconducting state averages over microscopic detail in a way that is seldom the case for materials with unconventional order parameters whose coherence volumes contain only a few electrons.The fact that full understanding of the superconducting state of Sr2RuO4 has not yet been achieved shows the level of challenge that still exists at the interface between theory and experiment in quantum materials, and strongly motivates a new generation of research on this fascinating material. Our goal is to frame that research by highlighting the main problems with finding a fully self-consistent description of the key experimentally determined features of the superconducting state, and to speculate about how the current mysteries might, in future, be resolved. Because a number of lengthy and detailed reviews of the properties of Sr2RuO4 already exist 3,6,[19][20][21][22] , we will not attempt to be comprehensive. Instead, we will select the issues that we believe to be the most important, and highlight those. We believe that the discussion we will give has

We introduce exactly solvable gapless quantum systems in d dimensions that support symmetry protected topological (SPT) edge modes. Our construction leads to long-range entangled, critical points or phases that can be interpreted as critical condensates of domain walls "decorated" with dimension (d − 1) SPT systems. Using a combination of field theory and exact lattice results, we argue that such gapless SPT systems have symmetry-protected topological edge modes that can be either gapless or symmetry-broken, leading to unusual surface critical properties. Despite the absence of a bulk gap, these edge modes are robust against arbitrary symmetry-preserving local perturbations near the edges. In two dimensions, we construct wavefunctions that can also be interpreted as unusual quantum critical points with diffusive scaling in the bulk but ballistic edge dynamics.

Focusing on semiclassical systems, we show that the parametrically long exponential growth of outof-time order correlators (OTOCs), also known as scrambling, does not necessitate chaos. Indeed, scrambling can simply result from the presence of unstable fixed points in phase space, even in an integrable model. We derive a lower bound on the OTOC Lyapunov exponent which depends only on local properties of such fixed points. We present several models for which this bound is tight, i.e. for which scrambling is dominated by the local dynamics around the fixed points. We propose that the notion of scrambling be distinguished from that of chaos.