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A bstract We use the numerical conformal bootstrap in two dimensions to search for finite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity. We find the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism. All the solutions we find that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks. Our numerical analysis is a special case of the “Gliozzi” bootstrap method, and provides a simpler setting in which to study technical challenges with the method. In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and OPE coefficients for degenerate operators using the formulae of Dotsenko and Fateev.

Scattering experiments have revolutionized our understanding of nature. Examples include the discovery of the nucleus [R. G. Newton, Scattering Theory of Waves and Particles (1982)], crystallography [U. Pietsch, V. Holý, T. Baumback, High-Resolution X-Ray Scattering (2004)], and the discovery of the double-helix structure of DNA [J. D. Watson, F. H. C. Crick, Nature 171, 737–738]. Scattering techniques differ by the type of particles used, the interaction these particles have with target materials, and the range of wavelengths used. Here, we demonstrate a two-dimensional table-top scattering platform for exploring magnetic properties of materials on mesoscopic length scales. Long-lived, coherent magnonic excitations are generated in a thin film of yttrium iron garnet and scattered off a magnetic target deposited on its surface. The scattered waves are then recorded using a scanning nitrogen vacancy center magnetometer that allows subwavelength imaging and operation under conditions ranging from cryogenic to ambient environment. While most scattering platforms measure only the intensity of the scattered waves, our imaging method allows for spatial determination of both amplitude and phase of the scattered waves, thereby allowing for a systematic reconstruction of the target scattering potential. Our experimental results are consistent with theoretical predictions for such a geometry and reveal several unusual features of the magnetic response of the target, including suppression near the target edges and a gradient in the direction perpendicular to the direction of surface wave propagation. Our results establish magnon scattering experiments as a platform for studying correlated many-body systems.

When the Coulomb repulsion between electrons dominates over their kinetic energy, electrons in two-dimensional systems are predicted to spontaneously break continuous-translation symmetry and form a quantum crystal 1 . Efforts to observe 2 – 12 this elusive state of matter, termed a Wigner crystal, in two-dimensional extended systems have primarily focused on conductivity measurements on electrons confined to a single Landau level at high magnetic fields. Here we use optical spectroscopy to demonstrate that electrons in a monolayer semiconductor with density lower than 3 × 10 11  per centimetre squared form a Wigner crystal. The combination of a high electron effective mass and reduced dielectric screening enables us to observe electronic charge order even in the absence of a moiré potential or an external magnetic field. The interactions between a resonantly injected exciton and electrons arranged in a periodic lattice modify the exciton bandstructure so that an umklapp resonance arises in the optical reflection spectrum, heralding the presence of charge order 13 . Our findings demonstrate that charge-tunable transition metal dichalcogenide monolayers 14 enable the investigation of previously uncharted territory for many-body physics where interaction energy dominates over kinetic energy. The signature of a Wigner crystal—the analogue of a solid phase for electrons—is observed via the optical reflection spectrum in a monolayer transition metal dichalcogenide.

One of the first theoretically predicted manifestations of strong interactions in many-electron systems was the Wigner crystal 1 – 3 , in which electrons crystallize into a regular lattice. The crystal can melt via either thermal or quantum fluctuations 4 . Quantum melting of the Wigner crystal is predicted to produce exotic intermediate phases 5 , 6 and quantum magnetism 7 , 8 because of the intricate interplay of Coulomb interactions and kinetic energy. However, studying two-dimensional Wigner crystals in the quantum regime has often required a strong magnetic field 9 – 11 or a moiré superlattice potential 12 – 15 , thus limiting access to the full phase diagram of the interacting electron liquid. Here we report the observation of bilayer Wigner crystals without magnetic fields or moiré potentials in an atomically thin transition metal dichalcogenide heterostructure, which consists of two MoSe 2 monolayers separated by hexagonal boron nitride. We observe optical signatures of robust correlated insulating states at symmetric (1:1) and asymmetric (3:1, 4:1 and 7:1) electron doping of the two MoSe 2 layers at cryogenic temperatures. We attribute these features to bilayer Wigner crystals composed of two interlocked commensurate triangular electron lattices, stabilized by inter-layer interaction 16 . The Wigner crystal phases are remarkably stable, and undergo quantum and thermal melting transitions at electron densities of up to 6 × 10 12 per square centimetre and at temperatures of up to about 40 kelvin. Our results demonstrate that an atomically thin heterostructure is a highly tunable platform for realizing many-body electronic states and probing their liquid–solid and magnetic quantum phase transitions 4 – 8 , 17 . Optical signatures reveal correlated insulating Wigner crystals—electron solids—in a bilayer of a two-dimensional transition metal dichalcogenide, MoSe 2 , with hexagonal boron nitride between the layers.

Perturbative considerations account for the properties of conventional metals, including the range of temperatures where the transport scattering rate is 1/τtr = 2πλT, where λ is a dimensionless strength of the electron-phonon coupling. The fact that measured values satisfy λ ≲ 1 has been noted in the context of a possible \"Planckian\" bound on transport. However, since the electron-phonon scattering is quasielastic in this regime, no such Planckian considerations can be relevant. We present and analyze Monte Carlo results on the Holstein model which show that a different sort of bound is at play: a \"stability\" bound on λ consistent with metallic transport. We conjecture that a qualitatively similar bound on the strength of residual interactions, which is often stronger than Planckian, may apply to metals more generally.Perturbative considerations account for the properties of conventional metals, including the range of temperatures where the transport scattering rate is 1/τtr = 2πλT, where λ is a dimensionless strength of the electron-phonon coupling. The fact that measured values satisfy λ ≲ 1 has been noted in the context of a possible \"Planckian\" bound on transport. However, since the electron-phonon scattering is quasielastic in this regime, no such Planckian considerations can be relevant. We present and analyze Monte Carlo results on the Holstein model which show that a different sort of bound is at play: a \"stability\" bound on λ consistent with metallic transport. We conjecture that a qualitatively similar bound on the strength of residual interactions, which is often stronger than Planckian, may apply to metals more generally.

Perturbative considerations account for the properties of conventional metals, including the range of temperatures where the transport scattering rate is 1/τtr = 2πλT, where λ is a dimensionless strength of the electron—phonon coupling. The fact that measured values satisfy λ ≲ 1 has been noted in the context of a possible “Planckian” bound on transport. However, since the electron–phonon scattering is quasielastic in this regime, no such Planckian considerations can be relevant. We present and analyze Monte Carlo results on the Holstein model which show that a different sort of bound is at play: a “stability” bound on λ consistent with metallic transport. We conjecture that a qualitatively similar bound on the strength of residual interactions, which is often stronger than Planckian, may apply to metals more generally.

A bstract Loop-level scattering amplitudes for massless particles have singularities in regions where tree amplitudes are perfectly smooth. For example, a 2 → 4 gluon scattering process has a singularity in which each incoming gluon splits into a pair of gluons, followed by a pair of 2 → 2 collisions between the gluon pairs. This singularity mimics double parton scattering because it occurs when the transverse momentum of a pair of outgoing gluons vanishes. The singularity is logarithmic at fixed order in perturbation theory. We exploit the duality between scattering amplitudes and polygonal Wilson loops to study six-point amplitudes in this limit to high loop order in planar N = 4 super-Yang-Mills theory. The singular configuration corresponds to the limit in which a hexagonal Wilson loop develops a self-crossing. The singular terms are governed by an evolution equation, in which the hexagon mixes into a pair of boxes; the mixing back is suppressed in the planar (large N c ) limit. Because the kinematic dependence of the box Wilson loops is dictated by (dual) conformal invariance, the complete kinematic dependence of the singular terms for the self-crossing hexagon on the one nonsingular variable is determined to all loop orders. The complete logarithmic dependence on the singular variable can be obtained through nine loops, up to a couple of constants, using a correspondence with the multi-Regge limit. As a byproduct, we obtain a simple formula for the leading logs to all loop orders. We also show that, although the MHV six-gluon amplitude is singular, remarkably, the transcendental functions entering the non-MHV amplitude are finite in the same limit, at least through four loops.

Perturbative considerations account for the properties of conventional metals, including the range of temperatures where the transport scattering rate is 1/  = 2 , where is a dimensionless strength of the electron-phonon coupling. The fact that measured values satisfy  ≲ 1 has been noted in the context of a possible \"Planckian\" bound on transport. However, since the electron-phonon scattering is quasielastic in this regime, no such Planckian considerations can be relevant. We present and analyze Monte Carlo results on the Holstein model which show that a different sort of bound is at play: a \"stability\" bound on consistent with metallic transport. We conjecture that a qualitatively similar bound on the strength of residual interactions, which is often stronger than Planckian, may apply to metals more generally.

One of the most important interactions in a solid is that between electrons and the vibrational modes of the underlying lattice (phonons). This interaction determines many of the familiar features of conventional metals, e.g. resistivity at elevated temperatures, as well as their superconducting transition temperature, Tc. From a theoretical standpoint, the electron-phonon problem was considered to be solved 60 years ago, thanks to the pioneering work of Migdal and Eliashberg. We have revisited this problem using a combination of numerical and analytic techniques, and I will show that this \"solved\" problem harbors basic features that have not been properly understood. I will show that the Migdal-Eliashberg approach works remarkably well for sufficiently weak coupling but breaks down entirely as the coupling strength is increased (even while the nominal condition for its validity remains intact), giving way to a qualitatively new regime of strong coupling physics. I will discuss the implications of these results for superconductivity and argue they suggest an approximate, universal upper bound on Tc in conventional metals.