Quantum Dynamics in Rugged Energy Landscapes, and Additional Topics in Disordered Systems
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This thesis concerns the interplay of quantum mechanics with strong disorder, and the novel dynamical phases that are unique to disordered quantum systems. The results that we present apply to systems ranging from spin glasses to granular superconductors to quantum-computational problems. In the first part, we discuss the isolated quantum dynamics of mean-field spin glass models, using the random energy and p-spin models in a transverse field as tractable examples. We show that the low-energy configurations are organized into clusters separated by macroscopic Hamming distances, and that the tunneling amplitudes between clusters are exponentially suppressed. As a result, we find three distinct dynamical phases. At small transverse field, the system remains trapped within its starting cluster (trapped phase). At intermediate transverse field, the system tunnels between clusters (tunneling phase). At large transverse field, the system is excited out of clusters (excitation phase). We describe the similarities and differences between the trapped phase and a many-body localized phase. We also discuss at length the implications for quantum-computational approaches to ``matching'' problems, in which one solution to a computational problem is used as a starting point to find others. Only in the tunneling phase can quantum dynamics solve the matching problem. Although necessarily exponentially slow in system size, it may be exponentially faster than simple classical algorithms. In the second part, we discuss interfering directed paths in disordered media. Important physical realizations are hopping conduction in semiconductors, spin glasses at high temperature, and granular D-wave superconductors. Sign order, defined as the directed path sum having greater probability of being positive than negative at large distance, is a characterization of the role of interference with implications for the response of systems to a magnetic field. We show that path sums are necessarily sign-disordered in two dimensions but may be sign-ordered in three dimensions. Building on this result, we study the behavior of granular D-wave superconductors and show that the superconductivity is enhanced by a magnetic field, even beyond the directed-path regime.
- Physics