Recent advances in the theory of disordered systems

Mogus Degoyae Mochena

Hossein Hosseinabadi

Recent advances in theoretical physics have increasingly focused on the dynamics of quantum systems with many interacting degrees of freedom, while ultracold atom platforms now enable the realization of diverse theoretical models with precise control over system parameters and advanced monitoring techniques. A prominent example is cavity quantum electrodynamics (CQED), where large atomic or molecular ensembles are coupled to high-finesse optical cavities, enabling strong interaction with confined electromagnetic modes and leading to phenomena such as the superradiant phase transition, in which atoms and photons form a coherent state. Most theoretical studies of CQED have concentrated on regimes of spatially uniform atom–cavity coupling, where collective behavior allows a description in terms of the classical dynamics of a few macroscopic variables. However, experimental capabilities now permit exploration beyond this limit, though theoretical progress is hindered by the exponential computational complexity of simulating large quantum systems, particularly when environmental coupling is included. This thesis addresses such challenges by developing efficient approaches, within the Keldysh formalism of non-equilibrium quantum field theory, to study systems with strongly disordered interactions coupled to external reservoirs. Two representative models are considered: a disordered spin–boson model describing light–matter interactions with quenched disorder, relevant to recent CQED experiments in confocal cavities, and a fermion–phonon system governed by the Yukawa–Sachdev–Ye–Kitaev (Yukawa-SYK) model. Both systems are quantum critical, exhibiting a continuum of low-energy modes above their stationary states, yet disorder produces qualitatively distinct dynamical behaviors: in the spin–boson system it induces slow, glassy relaxation, whereas in the Yukawa-SYK model it leads to rapid thermalization. These contrasting outcomes reveal the non-trivial role of disorder in quantum many-body systems and motivate further investigation into its effects on non-equilibrium quantum dynamics.

V. Dobrosavljević

The Mott and the Anderson routes to localization have long been recognized as the two basic processes that can drive the metal-insulator transition (MIT).Theories separately describing each of these mechanisms were discussed long ago, but an accepted approach that can include both has remained elusive.The lack of any obvious static symmetry distinguishing the metal from the insulator poses another fundamental problem, since an appropriate static order parameter cannot be easily found.More recent work, however, has revisited the original arguments of Anderson and Mott, which stressed that the key diference between the metal end the insulator lies in the dynamics of the electron.This physical picture has suggested that the "typical" (geometrically averaged) escape rate τ -1 typ = exp ln τ -1 esc from a given lattice site should be regarded as the proper dynamical order parameter for the MIT, one that can naturally describe both the Anderson and the Mott mechanism for localization.This article provides an overview of the recent results obtained from the corresponding Typical-Medium Theory, which provided new insight into the the two-fluid character of the Mott-Anderson transition. I. FROM METAL TO INSULATOR: A NEW PERSPECTIVETo understand many, if not most exotic new materials, one has to tackle the difficult problem of understanding the metal-insulator quantum phase transition, as driven by the combined effects of strong electronic correlations and disorder.Traditional approaches to the problem, which emerged in the early 1980s, have focused on examining the perturbative effects of disorder within the Fermi liquid framework.Despite their mathematical elegance, these theories, unfortunately, prove ill-suited to describe several key physical processes, such as tendency to local magnetic moment formation and the approach to the Mott insulating state.In addition, such weak-coupling theories cannot easily describe strongly inhomogeneous phases, with behavior often dominated by broad distributions and rare disorder configurations.This new insight, which is largely driven by experimental advances, seems to suggest that an alternative theoretical picture may provide a better starting point.In this article we describe recent advances based on a new theoretical method, which offers a complementary perspective to the conventional weak-coupling theories.By revisiting the original ides of Anderson and Mott, it examines the typical escape rate from a given site as the fundamental dynamical order parameter to distinguish between a metal and an insulator.This article describes the corresponding Typical-Medium Theory (TMT) and discusses some of its recent results and potential applications.We fist discuss discuss, in some detail, several experimental and theoretical clues suggesting that a new theoretical paradigm is needed.The formulation of TMT for Anderson localization of noninteracting electrons is then discussed, with emphasis on available analytical results.Finally, we review recent progress in applying TMT to the Mott-Anderson transition for disordered Hubbard models, and discuss resulting the two-fluid behavior at the critical point. II. THEORETICAL CHALLENGES: BEYOND CINDERELLA'S SLIPPER?The existence of a sharp metal-insulator transition at T = 0 has been appreciated for many years [1].Experiments on many systems indeed have demonstrated that a well defined critical carrier concentration can easily be identified.On the theoretical side, ambiguities on how to describe or even think of the metal-insulator transition have made it difficult to directly address the nature of the critical region.In practice, one often employs the theoretical tools that are available, even if possibly inappropriate.Even worse, one often focuses on those systems and phenomena that fit an available theoretical mold, ignoring and brushing aside precisely those features that seem difficult to understand.This "Cinderella's slipper" approach is exactly what one should not do; unfortunately it happens all too often.A cure is, of course, given by soberly confronting the experimental reality: what seems paradoxical at first sight often proves to be the first clue to the solution.A. Traditional approaches to disordered interacting electrons

Ferenc Pázmándi, R. T. Scalettar, Gergely T. Zimányi

Hsiang-Hsuan Hung, Aaron Barr, Emil V. Prodan +1 more

Using exact diagonalization and quantum Monte Carlo calculations we investigate the effects of disorder on the phase diagram of both noninteracting and interacting models of two-dimensional topological insulators. In the fermion sign problem-free interacting models we study, electron-electron interactions are described by an on-site repulsive Hubbard interaction and disorder is included via the one-body hopping operators. In both the noninteracting and interacting models we make use of recent advances in highly accurate real-space numerical evaluation of topological invariants to compute phase boundaries and in the noninteracting models determine critical exponents of the transitions. We find different models exhibit distinct stability conditions of the topological phase with respect to interactions and disorder. We provide a general analytical theory that accurately predicts these trends.

Uwe Krey