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arXiv:1411.4635 · Physical Review B

Unconventional localization transition in high dimensions

Sergey Syzranov, Victor Gurarie, Leo Radzihovsky

Published Jan 28, 2015• 79 citations
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Abstract

We study noninteracting systems with a power-law quasiparticle dispersion ${\ensuremath{\xi}}_{\mathbf{k}}\ensuremath{\propto}{k}^{\ensuremath{\alpha}}$ and a random short-range-correlated potential. We show that, unlike the case of lower dimensions, for $d>2\ensuremath{\alpha}$, there exists a critical disorder strength (set by the bandwidth), at which the system exhibits a disorder-driven quantum phase transition at the bottom of the band that lies in a universality class distinct from the Anderson transition. In contrast to the conventional wisdom, it manifests itself in, e.g., the disorder-averaged density of states. For systems in symmetry classes that permit localization, the striking signature of this transition is a nonanalytic behavior of the mobility edge, which is pinned to the bottom of the band for subcritical disorder and grows for disorder exceeding a critical strength. Focusing on the density of states, we calculate the critical behavior (exponents and scaling functions) at this novel transition, using a renormalization group, controlled by an $\ensuremath{\varepsilon}=d\ensuremath{-}2\ensuremath{\alpha}$ expansion. We also apply our analysis to Dirac materials, e.g., Weyl semimetals, where this transition takes place in physically interesting three dimensions.

DOI
10.1103/physrevb.91.035133

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We study noninteracting systems with a power-law quasiparticle dispersion \ensuremathξk\ensuremath∝k\ensuremathα{\ensuremath{\xi}}_{\mathbf{k}}\ensuremath{\propto}{k}^{\ensuremath{\alpha}}\ensuremathξk​\ensuremath∝k\ensuremathα and a random short-range-correlated potential. We show that, unlike the case of lower dimensions, for d>2\ensuremathαd>2\ensuremath{\alpha}d>2\ensuremathα, there exists a critical disorder strength (set by the bandwidth), at which the system exhibits a disorder-driven quantum phase transition at the bottom of the band that lies in a universality class distinct from the Anderson transition. In contrast to the conventional wisdom, it manifests itself in, e.g., the disorder-averaged density of states. For systems in symmetry classes that permit localization, the striking signature of this transition is a nonanalytic behavior of the mobility edge, which is pinned to the bottom of the band for subcritical disorder and grows for disorder exceeding a critical strength. Focusing on the density of states, we calculate the critical behavior (exponents and scaling functions) at this novel transition, using a renormalization group, controlled by an \ensuremathε=d\ensuremath−2\ensuremathα\ensuremath{\varepsilon}=d\ensuremath{-}2\ensuremath{\alpha}\ensuremathε=d\ensuremath−2\ensuremathα expansion. We also apply our analysis to Dirac materials, e.g., Weyl semimetals, where this transition takes place in physically interesting three dimensions.
We study a one-dimensional (1D) system with a power-law quasiparticle dispersion ${\ensuremath{\propto}|k|}^{\ensuremath{\alpha}}sgnk$ in the presence of a short-range-correlated random potential, and demonstrate that for $\ensuremath{\alpha}<1/2$ it exhibits a disorder-driven quantum phase transition with critical properties similar to those of the localization transition near the edge of the band of a semiconductor in high dimensions, as studied recently [Phys. Rev. Lett. 114, 166601 (2015); Phys. Rev. B 91, 035133 (2015)]. Despite the absence of localization in the considered 1D system, the disorder-driven transition manifests itself, for example, in a critical form of the disorder-averaged density of states. We confirm the existence of the transition by numerical simulations and find the critical exponents and the critical disorder strength as a function of $\ensuremath{\alpha}$. The proposed system thus presents a convenient platform for numerical studies of the recently predicted unconventional high-dimensional localization effects and has the potential for experimental realizations in chains of ultracold atoms in optical traps.
Disordered noninteracting systems in sufficiently high dimensions have been predicted to display a non-Anderson disorder-driven transition that manifests itself in the critical behavior of the density of states and other physical observables. Recently, the critical properties of this transition have been extensively studied for the specific case of Weyl semimetals by means of numerical and renormalisation-group approaches. Despite this, the values of the critical exponents at such a transition in a Weyl semimetal are currently under debate. We present an independent calculation of the critical exponents using a two-loop renormalization-group approach for Weyl fermions in $2\ensuremath{-}\ensuremath{\varepsilon}$ dimensions and resolve controversies currently existing in the literature.
Weyl or Dirac fermions with the iconic linear energy-momentum relation and average density of states (ADOS) ρ(E)∼∣E∣d−1ρ(E) \sim |E|^{d-1}ρ(E)∼∣E∣d−1 at energy EEE in ddd spatial dimensions, constitute a unique setup to study the disorder-driven semimetal-to-metal quantum phase transition (QPT) between ballistic (realized for weak disorder) and diffusive (stabilized at stronger disorder) quasiparticles. Such a QPT takes place only for d>2d>2d>2 and falls beyond the realm of the Anderson metal-to-insulator transition. From numerically computed ADOS (using the kernel polynomial method) in dirty Weyl systems in d=2d=2d=2 to 666, here we show that d=2d=2d=2 and d=4d=4d=4 are the lower (dℓd_\elldℓ​) and upper (dud_udu​) critical dimensions for such an unconventional QPT, respectively, as suggested from the solution of quasiparticle lifetime within the self-consistent Born approximation. Consequently, for d≥4d \geq 4d≥4 the associated correlation length exponent is found to be ν≈0.5ν\approx 0.5ν≈0.5 (within numerical accuracy). However, the dynamic scaling exponent at the quantum critical point is pinned close to z≈d/2z \approx d/2z≈d/2 (numerically) for any d≥3d \geq 3d≥3, which is shown to be an exact result from a field-theoretic renormalization group calculation. Therefore, Weyl semimetal-to-metal QPTs can be studied field theoretically around both dℓd_\elldℓ​ and dud_udu​.
Exploiting the enabling power of the Lanczos method in momentum space, we determine accurately the quasiparticle and scaling properties of disordered three-dimensional Dirac semimetals surrounding the quantum critical point separating the semimetal and diffusive metal regimes. We unveil that the imaginary part of the quasiparticle self-energy obeys a common power law before, at, and after the quantum phase transition, but the power law is nonuniversal, whose exponent is dependent on the disorder strength. More intriguingly, whereas a common power law is also found for the real part of the self-energy before and after the phase transition, a distinctly different behavior is identified at the critical point, characterized by the existence of a nonanalytic logarithmic singularity. This nonanalytical correction serves as the very basis for the unusual power-law behaviors of the quasiparticles and many other physical properties surrounding the quantum critical point. Our approach also allows the ready and reliable determination of the scaling properties of the correlation length and dynamical exponents. We further show that the central findings are valid for both uncorrelated and correlated disorder distributions and should be directly comparable with future experimental observations.
We describe the critical behavior of the Anderson-like transition predicted to occur in the 2D tight-binding model with isotropic scale-free long-range correlated disorder characterized by a power-law spectral density. We explore the scale invariance of the participation function relative fluctuation at the critical point to locate the mobility edge as a function of the power-law spectral density exponent α 2D . The states near the band center, which exhibit power-law localization for uncorrelated disorder, become delocalized for α 2D >2. In addition, we consider the finite-size scaling hypothesis to estimate the correlation length critical exponent. We find that the critical exponent ν depends on α 2D , thus indicating that correlations in the disorder distribution are indeed relevant in this regime, in agreement with the extended Harris criterion.
Anderson localization is fundamentally controlled by dimensionality, yet the nature of the Anderson transition in continuously tunable noninteger dimensions remains largely unexplored. Here, we introduce a family of three-dimensional fractal lattices with continuously tunable spectral dimension ds∈[2,3]d_s\in[2,3]ds​∈[2,3], providing a controlled platform for studying localization physics beyond integer dimensions and across the lower critical dimension ds=2d_s=2ds​=2. Using large-scale finite-size scaling analysis, we systematically investigate the Anderson transition and identify mobility edges throughout the fractal family. The critical disorder strength evolves continuously from 000 to 16.616.616.6 as the spectral dimension increases from 222 to 333. We show that the spectral dimension predominantly governs the universality class of the transition, while the precise critical point is additionally influenced by microscopic geometric details of the underlying fractal lattice. The critical exponent exhibits an approximate inverse dependence on dsd_sds​, providing quantitative insight into scaling theory in noninteger dimensions. Our results establish tunable fractal lattices as a versatile framework for exploring localization and quantum critical phenomena beyond conventional integer-dimensional systems.