"Tunneling spectroscopy near Anderson transitions with Coulomb interaction"
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We study [1,2] the tunneling density of states (TDOS) of a disordered electronic
system with Coulomb interaction on both the metallic and insulating sides of an
Anderson-localization metal-insulator transition (MIT). We discuss how the
zero-bias anomaly on the metal side transforms into the Coulomb-gap suppression
of the TDOS in the localized phase of the MIT. For tunneling into the insulating
phase, the average TDOS shows a critical behavior at high energies, with a
crossover to a soft Coulomb gap $\Delta$ at low energies. We demonstrate that
the single-particle excitations experience a localization transition (which
belongs to the noninteracting universality class) at energy $E=\pm E_c$. The
mobility edge $E_c$ scales with the distance $\mu_c-\mu$ from the interacting
critical point according to 
$E_c \propto (\mu_c-\mu)^{\nu z}$, where $\nu$ and $z$ are the interacting
localization-length and the dynamical critical exponents. Our theoretical
expectations and the "phase diagram" of the Anderson MIT in the presence of
Coulomb interaction are in an overall agreement with recent experimental results
[3] for the average TDOS in a device with a tunable doping level across the MIT. 
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We further explore mesoscopic fluctuations and correlations of the local density
of states (LDOS) near the MIT. It is shown that the LDOS multifractality
survives in the presence of Coulomb interaction. Specifically, the LDOS shows
strong fluctuations and long-range correlations which reflect the
multifractality associated with interacting and noninteracting fixed points as
well the localization of low-energy excitations. By using the onlinear
sigma-model approach, we calculate the spectrum of multifractal exponents of a
Coulomb-interacting system without time-reversal and spin symmetries in
$2+\epsilon$ spatial dimensions and show that it differs from that in the
absence of interaction. The multifractal character of fluctuations and
correlations of the LDOS can be studied experimentally by scanning tunneling
microscopy of two-dimensional and three-dimensional disordered structures. Our
results are in an overall agreement with the experimental data of Ref. [4]. In
addition to MIT and transitions between different phases of topological
insulators, we envision a possibility to extend our analysis also to
superconductor-insulator transitions. 
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[1] I.S. Burmistrov, I.V. Gornyi, and A.D. Mirlin, Phys. Rev. Lett. 111, 066601
(2013).
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[2] I.S. Burmistrov, I.V. Gornyi, and A.D. Mirlin, Phys. Rev. B 89, 035430
(2014).
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[3] A. Mottaghizadeh, Q. Yu, P. L. Lang, A. Zimmers, and H. Aubin, Phys. Rev.
Lett. 112, 066803 (2014).
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[4] A. Richardella, P. Roushan, S. Mack, B. Zhou, D.A. Huse, D.D. Awshalom, and
A. Yazdani, 
Science 327, 665 (2010).