Scientific Series

The spectral gap problem consist in deciding, given a local interaction, whether the corresponding translationally invariant Hamiltonian on a lattice has a spectral gap independent of the system size or not. In the simplest case of nearest-neighbour frustration-free qubit interactions, there is a complete classification. On the other extreme, for two (or higher) dimensional models with nearest-neighbour interactions this problem can be reduced to the Halting Problem, and it is therefore undecidable.

There are a lot of indications that one dimensional spin chain are relatively simpler than their counterparts in higher dimensions. Nonetheless, I will present a construction of a family of nearest-neighbour, translationally invariant Hamiltonians on a spin chain, for which the spectral gap problem is undecidable.

The basic geometry of the Solar System - the shapes, spacings, and

orientations of the planetary orbits - has long been a subject of

fascination as well as inspiration for planet-formation theories. For

exoplanetary systems, those same properties have only recently come

into focus. I will review our current knowledge of the occurrence of

planets around other stars, their orbital distances and

eccentricities, the orbital spacings and mutual inclinations in

multiplanet systems, the orientation of the host star's rotation axis,

and the properties of planets in binary-star systems. I will also

discuss opportunities to improve our understanding, with data from the

recently launched Transiting Exoplanet Survey Satellite.

Hyper Suprime-Cam (HSC) is an imaging camera mounted at the Prime Focus of the Subaru 8.2-m telescope operated by the National Astronomical Observatory of Japan on the summit of Maunakea in Hawaii. A consortium of astronomers from Japan, Taiwan and Princeton University is carrying out a three-layer, 300-night, multiband survey from 2014-2019 with this instrument. In this talk, I will focus on the HSC survey Wide Layer, which will cover 1400 square degrees in five broad bands (grizy), to a 5 sigma point-source depth of r~26. We have covered 240 square degrees of the Wide Layer in all five bands, and the median seeing in the i band is 0.60 arcseconds. This powerful combination of depth and image quality makes the HSC survey unique compared to other ongoing imaging surveys. In this talk I will describe the HSC survey dataset and the completed and ongoing science analyses with the survey Wide layer, including galaxy studies, strong and weak gravitational lensing, but with an emphasis on weak lensing. I will demonstrate the level of systematics control, the latest cosmology results, and describe some lessons learned that will be of use for other ongoing and future lensing surveys.

In gauge theories, there is an inherent tension between locality and gauge invariance.   This is precisely expressed by the failure to factorize the physical Hilbert space into local tensor products.   The extended Hilbert space offers a resolution to this problem by embedding physical states into a larger Hilbert space containing gauge variant degrees of freedom.  For two dimensional TQFT's, we show how this extension fits inside the framework of extended quantum field theory, as described by the Moore-Segal axioms.  These are sewing axioms that impose a sort of spacetime covariance for the extended Hilbert space.   For the case of two-dimensional Yang Mills and it's string theory dual, we show how this framework leads to exact calculations of multi-interval modular flow,  ,negativity, and entanglement entropy.  Moreover, we given an axiomatic formulation of the "entanglement brane", which nicely compliments the worldsheet description given in previous work.

We perform a full (3+1)-dimensional numerical simulation of the axion field around the QCD epoch. Our aim is to fully resolve large dynamical non-linear effects in the inhomogenous axion field. These effects are important as they lead to large overdensities in the field at late times. Those overdensities will eventually evolve into axion minicluster, which have various phenomenological implications like microlensing events. It is therefore important to have a reliable estimate of the number of overdensities and their mass relation.

The 8-vertex model and the XYZ spin chain have been found to emerge from gauge theories in various ways, such as 4d and 2d Nekrasov-Shatashvili correspondences, the action of surface operators on the supersymmetric indices of class-Sk theories, and correlators of line operators in 4d Chern-Simons theory.  I will explain how string theory unifies these phenomena.  This is based on my work with Kevin Costello [arXiv:1810.01970].

Positive geometries are real semialgebraic spaces that are
equipped with a meromorphic ``canonical form" whose residues reflect
the boundary structure of the space.  Familiar examples include
polytopes and the positive parts of toric varieties.  A central, but
conjectural, example is the amplituhedron of Arkani-Hamed and Trnka.
In this case, the canonical form should essentially be the tree
amplitude of N=4 super Yang-Mills.

I will talk about the definition and examples of positive geometries,
and discuss what is known about the geometry and combinatorics of the
amplituhedron.  The talk will be based on various joint works with
Arkani-Hamed, Bai, Galashin, and Karp.

The unexpected diversity of planetary systems has posed challenges to our classical understanding of planetary formation. For instance, Jupiter sized planets have been detected with short orbital periods of a few days in misaligned orbits with respect to the spin-axis of their host stars. I will first describe the statistical implication of detecting misaligned hot Jupiters and will suggest how dynamical interactions between an outer perturber and the inner planet, can naturally lead to the formation of such misaligned hot Jupiters. Next, I will discuss a similar dynamical process in the outer Solar System, far away from our Sun, which could cause the observed clustering of extreme trans-Neptunian objects. This can constrain properties of a possible outer planet, Planet Nine, in our own Solar System. 

Applying a chemical potential bias to a conductor drives the system out of equilibrium into a current carrying non-equilibrium state.  This current flow is associated with entropy production in the leads, but it remains poorly understood under what conditions the system is driven to local equilibrium by this process.  We investigate this problem using two toy models for coherent quantum transport of diffusive fermions:  Anderson models in the conducting phase and a class of random quantum circuits acting on a chain of qubits, which exactly maps to an interacting fermion problem. Under certain conditions, we find that the long-time states in both models exhibit volume-law mutual information and entanglement, in striking violation of local equilibrium.  Extending this analysis to Anderson metal-insulator transitions, we find that the volume-law entanglement scaling persists at the critical point up to mobility edge effects.  This work points towards a broad class of examples of physical systems where volume-law entanglement can be sustained, and potentially harnessed, despite strong coupling of the system to its surrounding environment. 

In the second lecture, I will extend the previous discussion to gravity, and show that the conformal trace anomaly must play a special role in the effective field theory of low energy gravity. Classical General Relativity receives an infrared relevant modification from the conformal trace anomaly of the energy-momentum tensor of massless, or nearly massless, quantum fields, which implies the existence of a new long range massless scalar degree of freedom, not present in the classical Einstein theory, that contributes to gravitational scattering processes and has long range gravitational effects. Similar to the axial anomaly, the local form of the effective action associated with the conformal anomaly is expressible in terms of a dynamical scalar field that couples to the conformal factor of the spacetime metric, allowing it to propagate over macroscopic distances. Among the significant implications of this effective field theory of gravity are the prediction of scalar gravitational wave solutions—a spin-0 breather mode— in addition to the transversely polarized tensor waves of the classical Einstein theory. Astrophysical sources for scalar gravitational waves are considered, with the excited gluonic condensates in the interiors of neutron stars in merger events with other compact objects likely to provide the strongest burst signals. The conformal anomaly also implies generically large quantum back reaction effects and conformal correlators in the vicinity of black hole horizons which are relevant to the formation of a non-singular interior, as well as an additional scalar degree of freedom in cosmology, providing a theoretical foundation for dynamical dark energy.

Ubiquitous in the behavior of physical systems is the competition between an energy term E and an entropy term S of their free energy F = E - beta S. These concepts are also relevant for error correction, where the `energy` E is the number of qubits afflicted by an error, the `entropy' S(E) is the logarithm of the number of energy-E failing errors, and beta relates to the probability of each qubit's error. Error-correction schemes with larger minimum free energy have better performance. Often distance d (which correct all errors with energy less than d/2) is used as a proxy for a code's performance since it increases the minimal E and therefore tends to increases the minimal F. However, a sufficiently large entropy S can counteract a large d and reduce the free energy (negatively impacting a code's performance). A great example of these principles is the surface code, which is at present the leading architecture for fault tolerant quantum computing. Rotating a square lattice geometry over the surface of the torus can increase the distance of the code by a factor of root two, but at the cost of increased entropy. We obtain exact expressions for this entropic effect in the low error regime, and introduce an analytical model that qualitatively describes the behavior for error rates all the way up to threshold. Our predictions are corroborated by numerical estimates of the low error failure rate, using the splitting method algorithm introduced by Bravyi et al. We find that although the rotated lattice outperforms the non-rotated lattice with the same number of qubits for low error rates, the two codes have very similar performance for error rates which are an appreciable fraction of the threshold error rate. Surprisingly, we also find some system sizes and error rates for which the non-rotated lattice has marginally better performance. 

I will discuss the status and future of numerical lattice Quantum Chromodynamics (QCD) calculations for nuclear physics. With advances in supercomputing, we are beginning to quantitatively understand nuclear structure and interactions directly from the fundamental quark and gluon degrees of freedom of the Standard Model. Recent studies provide insight into the neutrino-nucleus interactions relevant to long-baseline neutrino experiments, double beta decay, and nuclear sigma terms needed for theory predictions of dark matter cross-sections at underground detectors. I will also address new work constraining `exotic glue’ in nuclei, which will be measurable for the first time at a future electron-ion collider, and explain how machine learning tools are providing new possibilities in this field.