Disordered Ground States of Ergodic Quantum Spin Systems

We study the problem of existence and properties of ground states for quantum spin systems on $\mathbb{Z}^d$ with disordered, bounded-range interactions. In the ergodic case, it is known that the ground state energy density exists and is non-random. Here, we prove existence of ground states and analyze their properties. Specifically, we show that for ergodic interactions, there exist translation-covariant sequences of finite-volume ground states converging to an infinite-volume ground state. Moreover, we demonstrate that this infinite-volume ground state satisfies a strong clustering property (exponential decay of correlations), generalizing existing results for ordered systems to the disordered ergodic setting. This work contributes to the fundamental understanding of quantum phases of matter and has implications for the design of robust quantum computing architectures.