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Numerical method for disordered quantum phase transitions in the large$-N$ limit

We develop an efficient numerical method to study the quantum critical behavior of disordered systems with $\mathcal{O}(N)$ order-parameter symmetry in the large$-N$ limit. It is based on the iterative solution of the large$-N$ saddle-point equations combined with a fast algorithm for inverting the arising large sparse random matrices. As an example, we consider the superconductor-metal quantum phase transition in disordered nanowires. We study the behavior of various observables near the quantum phase transition. Our results agree with recent renormalization group predictions, i.e., the transition is governed by an infinite-randomness critical point, accompanied by quantum Griffiths singularities. Our method is highly efficient because the numerical effort for each iteration scales linearly with the system size. This allows us to study larger systems, with up to 1024 sites, than previous methods. We also discuss generalizations to higher dimensions and other systems including the itinerant antiferomagnetic transitions in disordered metals.