Anomalous density fluctuations in a random t−J model

A previous work [Joshi et al., Phys. Rev. X 10, 021033 (2020)] found a deconfined critical point at nonzero doping in a $t\text{\ensuremath{-}}J$ model with all-to-all and random hopping and spin exchange and argued for its relevance to the phenomenology of the cuprates. We extend this model to include all-to-all and random density-density interactions of mean-square strength $K$. In a fixed realization of the disorder, and for specific values of the hopping, exchange, and density interactions, the model is supersymmetric, but we find no supersymmetry after independent averages over the interactions. Using the previously developed renormalization group analysis, we find a new fixed point at nonzero $K$. However, this fixed point is unstable toward the previously found fixed point at $K=0$ in our perturbative analysis. We compute the exponent characterizing local density fluctuations at both fixed points: This exponent determines the spectrum of electron energy-loss spectroscopy.