We consider a disordered system of interacting fermions, for which we demonstrate, that depending on the relationship between the degree of disorder and real physical dimensionality d , this system undergoes a delocalization (e.g. a metal-insulator) phase transition. The effect of the disorder is accounted for by replacing the ordinary single-particle kinetic energy operator by that with a fractional spectrum. These fractional spectra represent the microscopic equivalent of the well-known non-trivial power-law decay of the low-energy density of states of non-interacting disordered systems. We develop a powerful approach to analyze the density of states in interacting disordered system, combining both the saddle-point and renormalization group methods. These two methods complement each other effectively. Through this approach, we identify a fixed point in the renormalization group flow of the model’s parameters, which dictates the behavior of the system in the conducting phase. Additionally, we extend our analysis to include static conductivity. Our examination of the system’s conductivity allows us to establish several strict conditions regarding the stability of the metallic state across different spatial dimensions d . Notably, we emphasize the unique significance of the two-dimensional case d = 2 . We also compare our findings with salient experimental features observed in paradigmatic physical systems.
We demonstrate that a disordered Fermi system with density-density interaction can undergo a metal-dielectric phase transition, which depends on the relationship between the real physical dimensionality
d and the degree of disorder. The disorder is modeled by a fractional Laplacian with L\'evy index $\ensuremath{\alpha}$ in place of the ordinary one in the single-particle kinetic energy operator. Our studies of various benchmark regimes marked by an effective system dimension $\ensuremath{\gamma}=(d\ensuremath{-}\ensuremath{\alpha})/\ensuremath{\alpha}$ show a pattern of dimensional flow of the system's density of states (DOS) at the Fermi surface. This pattern describes the continuous change of the DOS as $\ensuremath{\gamma}$ varies. Additionally, by including the extended Coulomb interaction with a general power law ${r}^{\ensuremath{-}\ensuremath{\beta}}$ (where $\ensuremath{\beta}\ensuremath{\ge}0$) into the model, we observe the effect of dimensional transmutation, in which the Coulomb exponent $\ensuremath{\beta}$ becomes the effective spatial dimension. As our combined parameter $\ensuremath{\gamma}$ permits to manipulate the properties of above metal-dielectric phase transition, our findings can be used to improve functionality of the transition metal dichalcogenide channel transistors and other nanoelectronic devices.
We study the effects of electron-electron interactions in disordered metals in and close to two dimensions (2D). We consider physical situations in which localization effects are suppressed. The field-theoretical renormalization-group (RG) calculation performed recently by Finkelstein is interpreted and rederived in terms of perturbative results. Surprisingly, except for the density of states, the scaling behavior is independent of the interaction range. We further extend the model to several new universality classes. In the presence of a strong magnetic field the metal is unstable in 2D and undergoes a metal-insulator transition in $d=2+\ensuremath{\epsilon}$. The conductivity exponent, defined by $\ensuremath{\sigma}\ensuremath{\sim}{(n\ensuremath{-}{n}_{c})}^{\ensuremath{\mu}}$, is universal with $\ensuremath{\mu}=1+O(\ensuremath{\epsilon})$ but
N(EF) depends not only on the range of the interaction but also on its strength for short-ranged interactions. In 2D the conductivity has a universal temperature dependence [$\ensuremath{\delta}\ensuremath{\sigma}(T)={\ensuremath{\sigma}}_{N}(2\ensuremath{-}2\mathrm{ln}2)\mathrm{ln}(T\ensuremath{\tau})$, ${\ensuremath{\sigma}}_{N}=\frac{{e}^{2}}{2{\ensuremath{\pi}}^{2}\ensuremath{\hbar}}$] if the interaction is Coulombic. If magnetic impurities (or strong spin-orbit scattering with a weak magnetic field) are present instead, the noninteracting fixed point is stable for short-ranged interactions ($\ensuremath{\mu}=\frac{1}{2}$). For the Coulomb interaction the interaction is relevant and drives a metal-insulator transition in $d=2+\ensuremath{\epsilon}$ with universal critical properties ($\ensuremath{\mu}=1$). In 2D the conductivity also has a universal temperature dependence [$\ensuremath{\delta}\ensuremath{\sigma}(T)={\ensuremath{\sigma}}_{N}\mathrm{ln}(T\ensuremath{\tau})$]. We also discuss the behavior of the dielectric constant on the insulator side and the frequency (temperature) dependence of the conductivity at criticality. Remarks are made on the relationship of the above to experiments.
We consider the quantum critical properties of a disordered two dimensional Bose gas based on the renormalization group approach. To account for disorder, we consider that the Bose gas interacts with an external random field. The system’s behavior in the critical region of the quantum phase transition is obtained in the universality class z =2 and includes contributions beyond the Gaussian approximation by considering interactions between bosonic degrees of freedom. The evaluation of the system’s conductivity close to the transition point proves the occurrence of a nonzero conductivity (metallic-like) state in the system in the ω =0 and T →0 K limit.
We uncover a disorder-driven instability in the diffusive Fermi liquid phase of a class of many-fermion systems, indicative of a metal-insulator transition of first-order type, which arises solely from the competition between quenched disorder and interparticle interactions. Our result is expected to be relevant for sufficiently strong disorder in
d=3 spatial dimensions. Specifically, we study a class of half-filled, Hubbard-like models for spinless fermions with (complex) random hopping and short-ranged interactions on bipartite lattices in $d\ensuremath{\geqslant}2$. In a given realization, the hopping disorder breaks time-reversal invariance but preserves the special ``nesting'' symmetry responsible for the charge density wave instability of the ballistic Fermi liquid. This disorder may arise, e.g., from the application of a random magnetic field to the otherwise clean model. We derive a low-energy effective field theory description for this class of disordered, interacting fermion systems, which takes the form of a Finkel'stein nonlinear sigma model $(\mathrm{FNL}\ensuremath{\sigma}\mathrm{M})$ (A. M. Finkel'stein, Zh. Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)]). We analyze the $\mathrm{FNL}\ensuremath{\sigma}\mathrm{M}$ using a perturbative, one-loop renormalization group analysis controlled via an
ϵ expansion in
d=2+ϵ dimensions. We find that in
d=2 dimensions, the interactions destabilize the conducting phase known to exist in the disordered, noninteracting system. The metal-insulator transition that we identify in
d>2 dimensions
(ϵ>0) occurs for disorder strengths of order
ϵ, and is therefore perturbatively accessible for
ϵ⪡1. We emphasize that the disordered system has no localized phase in the absence of interactions, so that a localized phase and the transition into it can only appear due to the presence of the interactions.
We study the effects of generic short-ranged interactions on a system of two-dimensional (2D) Dirac fermions subject to a special kind of static disorder, often referred to as ``chiral.'' The noninteracting system is a member of the disorder class
BDI [M. R. Zirnbauer, J. Math. Phys. 37, 4986 (1996)]. It emerges, for example, as a low-energy description of a time-reversal invariant tight-binding model of spinless fermions on a honeycomb lattice, subject to random hopping, and possessing particle-hole symmetry. It is known that, in the absence of interactions, this disordered system is special in that it does not localize in 2D, but possesses extended states and a finite conductivity at zero energy, as well as a strongly divergent low-energy density of states. In the context of the hopping model, the short-range interactions that we consider are particle-hole symmetric density-density interactions. Using a perturbative one-loop renormalization group analysis, we show that the same mechanism responsible for the divergence of the density of states in the noninteracting system leads to an instability, in which the interactions are driven strongly relevant by the disorder. This result should be contrasted with the limit of clean Dirac fermions in 2D, which is stable against the inclusion of weak short-ranged interactions. Our work suggests a mechanism wherein a clean system, initially insensitive to interaction effects, can be made unstable to interactions upon the inclusion of weak static disorder. We dub this mechanism a ``disorder-driven Mott transition.'' Our result for 2D fermions also contrasts sharply with known results in one dimension, where a similar delocalized phase has been shown to be robust against the inclusion of weak interaction effects.
In this thesis, we study the effects of both disorder and low carrier density in low-dimensional systems, employing a variety of analytical and numerical methods. The work is structured in two main parts, each addressing a distinct research direction. The first part focuses on the impact of disorder in one-dimensional and quasi-one dimensional systems. Using tools such as bosonization, renormalization group techniques, and exact diagonalization, we investigate two key aspects. First, we explore how specific spatial correlations in the disorder affect the properties of the metallic-insulating transition in a one-dimensional system, uncovering significant deviations from the behaviour typically associated with uncorrelated disorder. Second, we examine the effects of different types of disorder on a quasi-one-dimensional singlet superconductor, where we recover Anderson’s theorem under certain conditions and observe a competition between disorder and interactions. The second part of the thesis examines magneto-transport in two-dimensional, locally correlated materials. Using a microscopic quantum theory, we demonstrate pronounced deviations of the Hall constant and the conductivities of the system from their semiclassical predictions, in regimes where the Fermi energy is no longer the dominant energy scale.