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Physical review. B./Physical review. B

Effect of fractional spectra on conductor-insulator transitions in systems of interacting fermions

Andreas Sinner, V. A. Stephanovich

Published Oct 9, 2024• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.110.165125

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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 ddd and the degree of disorder. The disorder is modeled by a fractional Laplacian with L'evy index \ensuremathα\ensuremath{\alpha}\ensuremathα 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 \ensuremathγ=(d\ensuremath−\ensuremathα)/\ensuremathα\ensuremath{\gamma}=(d\ensuremath{-}\ensuremath{\alpha})/\ensuremath{\alpha}\ensuremathγ=(d\ensuremath−\ensuremathα)/\ensuremathα 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 \ensuremathγ\ensuremath{\gamma}\ensuremathγ varies. Additionally, by including the extended Coulomb interaction with a general power law r\ensuremath−\ensuremathβ{r}^{\ensuremath{-}\ensuremath{\beta}}r\ensuremath−\ensuremathβ (where \ensuremathβ\ensuremath≥0\ensuremath{\beta}\ensuremath{\ge}0\ensuremathβ\ensuremath≥0) into the model, we observe the effect of dimensional transmutation, in which the Coulomb exponent \ensuremathβ\ensuremath{\beta}\ensuremathβ becomes the effective spatial dimension. As our combined parameter \ensuremathγ\ensuremath{\gamma}\ensuremathγ 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 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 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)N({E}_{F})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.
The low--energy excited states of a system of interacting one--dimensional fermions in a conducting state are collective charge and spin density oscillations. The unusual physical properties of such a system (called ``Luttinger liquid'') are characterized by the velocities uρu_ρuρ​ and uσu_σuσ​ of the charge and spin excitations, as well as by a parameter KρK_ρKρ​ that determines the power law behavior of correlation functions. Umklapp scattering occuring at half--filling or other commensurate bandfilling can lead to a transition into an insulating state, characterized in particular by a gap in the charge excitations (the Mott--Hubbard gap). The properties in the vicinity of the transition are shown to depend on both the way the transition is approached (constant bandfilling and varying interaction, or constant interaction and varying bandfilling) and on the ``order'' of the commensurability. In particular, even and odd fractional fillings show quite different behavior. This behavior is illustrated in detail using lattice models like the Hubbard model and its extensions.
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=3d=3d=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 ϵϵϵ expansion in d=2+ϵd=2+ϵd=2+ϵ dimensions. We find that in d=2d=2d=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>2d>2d>2 dimensions (ϵ>0)(ϵ>0)(ϵ>0) occurs for disorder strengths of order ϵϵϵ, and is therefore perturbatively accessible for ϵ⪡1ϵ⪡1ϵ⪡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 one-dimensional disordered fermions that either undergo metal-insulator transitions or topological phase transitions to become trivial Anderson insulators. We focus on using entanglement to elucidate how the spatial, momentum, and internal degrees of freedom of fermions are affected by the presence of disorder in such cases. We develop entanglement tools that reveal the existence of metallic states in the presence of disorder and further show clear signatures of the corresponding localization transition even in the presence of interactions. In systems where the internal degrees of freedom are coupled with the motion of the electrons, topological phases develop. We subject a topological insulator model to different types of disorder and discuss how the topological aspects of the system can be captured through entanglement, even at strong disorder.
We study a three-dimensional Anderson-Hubbard model under the coexistence of short-range interaction and diagonal disorder within the Hartree-Fock approximation. We show that the density of states at the Fermi energy is suppressed in the metallic phases near the metal-insulator transition as a proximity effect of the soft Hubbard gap in the insulating phases. The transition to the insulator is characterized by a vanishing DOS in contrast to formation of a quasiparticle peak at the Fermi energy obtained by the dynamical mean field theory in pure systems. Furthermore, we show that there exist frozen spin moments in the paramagnetic metal.