Disorder-Driven Quantum Transition in Relativistic Semimetals: Functional Renormalization via the Porous Medium Equation

Abstract

In the presence of randomness, a relativistic semimetal undergoes a quantum transition towards a diffusive phase. A standard approach relates this transition to the U (N ) Gross-Neveu model in the limit of N → 0. We show that the corresponding fixed point is infinitely unstable, demonstrating the necessity to include fluctuations beyond the usual Gaussian approximation. We develop a functional renormalization group method amenable to include these effects and show that the disorder distribution renormalizes following the so-called porous medium equation. We find that the transition is controlled by a nonanalytic fixed point drastically different from that of the U (N ) Gross-Neveu model. Our approach provides a unique mechanism of spontaneous generation of a finite density of states and also characterizes the scaling behavior of the broad distribution of fluctuations close to the transition. It can be applied to other problems where nonanalytic effects may play a role, such as the Anderson localization transition.Introduction. -The interplay between disorder and quantum fluctuations leads to unique phenomena, the most remarkable being the Anderson localization. After more than half a century of intensive efforts, it remains a topical subject of research with applications to various domains of physics ranging from condensed matter to cold atoms and light propagation [1]. Remarkably, a different type of disorder-driven quantum phase transition was discovered recently when considering waves with a quantum relativistic dispersion relation [2]. This transition happens between a pseudoballistic phase and a diffusive metal as a function of the disorder strength (or the energy). It is predicted to occur in particular in the recently discovered three-dimensional (3D) Weyl [3,4] and Dirac semimetals [5][6][7] in which, respectively, two and four electronic bands cross linearly at isolated points. However we expect these phenomena to be relevant to other relativistic waves beyond condensed matter, such as ultracold atoms [8].In spite of numerous efforts, the understanding of this transition remains elusive. In this Letter we show that the fluctuations of the randomness beyond the standard Gaussian approximation invalidate previous fieldtheoretic descriptions of this transition. A very similar mechanism occurs in the context of the Anderson transition: there discrepancies between the results obtained using renormalization group (RG) and numerical simulations grow with the number of loops [9]. One may attribute them to the existence of infinitely many relevant operators of the associated field theory [10] which destabilize the fixed point (FP) usually considered to describe the transition [11][12][13]. We find that the same problem appears at the new semimetal-diffusive metal transition. We demonstrate how to overcome this obstacle by deriving and solving a functional renormalization group (FRG) for the whole (non-Gaussian) disorder distribution. To our knowledge this solution consti-tutes the only example of an analytical de...

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“…We, therefore, believe that higher-order perturbation theory within the framework of an ϵ m expansion should be more controlled. Explicit higher-order calculation in ϵ m expansion and its corroboration with a newly proposed nonperturbative approach combined with the functional renormalization-group analysis [98] is, however, left as a challenging interesting problem for future investigation. Nonetheless, we note that leading-order ϵ d and ϵ m expansions, as well as the functional renormalization-group approach from Ref.…”

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