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We present computations of the thermal Hall coefficient of phonons scattering off a defect with multiple energy levels. Using a microscopic formulation based on the Kubo formula, we find that the leading contribution perturbative in the phonon–defect coupling is proportional to the phonon lifetime and has a “side-jump” interpretation. Consequently, the thermal Hall angle is independent of the phonon lifetime. The contribution to the thermal Hall coefficient is at resonance when the phonon energy equals a defect-level spacing. Our results are obtained for three different defect models, which apply to different correlated electron materials. For the pseudogap regime of the cuprates, we propose a model of phonons coupled to an impurity quantum spin in the presence of quasistatic magnetic order with an isotropic Zeeman coupling to the applied field and without spin–orbit interaction.

We investigate a model of electrons with random and all-to-all hopping and spin exchange interactions, with a constraint of no double occupancy. The model is studied in a Sachdev–Ye–Kitaev-like large-Mlimit with SU(M) spin symmetry. The saddle-point equations of this model are similar to approximate dynamic mean-field equations of realistic, nonrandom,t-Jmodels. We use numerical studies on both real and imaginary frequency axes, along with asymptotic analyses, to establish the existence of a critical non–Fermi-liquid metallic ground state at large doping, with the spin correlation exponent varying with doping. This critical solution possesses a time-reparameterization symmetry, akin to Sachdev–Ye–Kitaev (SYK) models, which contributes a linear-in-temperature resistivity over the full range of doping where the solution is present. It is therefore an attractive mean-field description of the overdoped region of cuprates, where experiments have observed a linear-Tresistivity in a broad region. The critical metal also displays a strong particle–hole asymmetry, which is relevant to Seebeck coefficient measurements. We show that the critical metal has an instability to a low-doping spin-glass phase and compute a critical doping value. We also describe the properties of this metallic spin-glass phase.