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The single-channel Kondo impurity problem is a classic example of strongly coupled physics. In the Kondo problem, a single magnetic impurity is placed in a metal — the resulting system exhibits interesting properties such as a resistance minimum as a function of temperature. The problem was solved by Wilson’s numerical renormalization group and later by the Bethe ansatz technique. The Bethe ansatz exactly diagonalizes the Kondo hamiltonian for arbitrary impurity spin $s$ and numerically computes the impurity free energy for all temperatures. In this talk, I’ll present an alternate analytic solution for the Kondo problem at large $s$ that builds on recent results in boundary conformal field theory. This solution allows us to access analytically intermediate scales of the Kondo problem at large $s$; our results in this regime agree with the numeric results of the Bethe ansatz.

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Spatial symmetries can enrich the topological classification of interacting quantum matter and endow additional ``weak” topological indices upon systems with non-trivial strong topological invariants (protected by internal symmetries). In this talk, I will discuss the boundary physics of charge conserving systems with a non-zero shift invariant, which is protected by either a continuous U(1) or discrete CN rotation symmetry. In particular, I will discuss an interface between two systems with the same Chern number but distinct shift invariants and show that the interface hosts protected gapless edge modes. For general Abelian topological orders in 2D, I will prove sufficient conditions for gapless edge states protected by continuous rotation symmetry. For the case of discrete rotation symmetries, I will show that the Chern-Simons field theory for systems with gappable edges predicts fractional corner charges. These can also be computed when the system is placed on the two-dimensional surface of a Platonic solid, which relates to the fractional charge bound at disclination defects. Time permitting, I will discuss recent results regarding the relation between the shift and many-body real-space invariants for 2D systems with crystalline symmetry.

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Weak measurements can be viewed as a soft projection that interpolates between an identity operator and a projection operator, and can induce an effective central charge distinct from the unmeasured CFT. In the first part, I will discuss the effect of measurement and postselection on the critical ground state of a Luttinger liquid theory. Depending on the Luttinger parameter K, the effect of measurement is irrelevant, marginal, or relevant, respectively. When the measurement is marginal, and we find a critical state whose entanglement entropy exhibits a logarithmic behavior with a continuous effective central charge as a function of measurement strength. Inspired by this result, in the second part, I will discuss a holographic description of the weak measurement. The weak measurement is modeled by an interface brane, separating different geometries dual to the post-measurement state and the unmeasured CFT. In an infinite system, the weak measurement is related to ICFT via a spacetime rotation. We find that the holographic entanglement entropy with twist operators located on the defect is consistent in both calculations for ICFT and weak measurements. In a finite system, the weak measurement can lead to a rich phase diagram, in which the post-measurement geometry can realize a Python’s lunch.

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