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Petrova EV, Tiunov ES, Bañuls MC, Fedorov AK. Fractal States of the Schwinger Model. PHYSICAL REVIEW LETTERS 2024; 132:050401. [PMID: 38364163 DOI: 10.1103/physrevlett.132.050401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2023] [Revised: 10/09/2023] [Accepted: 12/18/2023] [Indexed: 02/18/2024]
Abstract
The lattice Schwinger model, the discrete version of QED in 1+1 dimensions, is a well-studied test bench for lattice gauge theories. Here, we study the fractal properties of this model. We reveal the self-similarity of the ground state, which allows us to develop a recurrent procedure for finding the ground-state wave functions and predicting ground-state energies. We present the results of recurrently calculating ground-state wave functions using the fractal Ansatz and automized software package for fractal image processing. In certain parameter regimes, just a few terms are enough for our recurrent procedure to predict ground-state energies close to the exact ones for several hundreds of sites. Our findings pave the way to understanding the complexity of calculating many-body wave functions in terms of their fractal properties as well as finding new links between condensed matter and high-energy lattice models.
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Affiliation(s)
- Elena V Petrova
- Russian Quantum Center, Skolkovo, Moscow 121205, Russia
- Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
| | - Egor S Tiunov
- Russian Quantum Center, Skolkovo, Moscow 121205, Russia
- Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE
| | - Mari Carmen Bañuls
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
- Munich Center for Quantum Science and Technology (MCQST), Schllingstrasse 4, D-80799 München, Germany
| | - Aleksey K Fedorov
- Russian Quantum Center, Skolkovo, Moscow 121205, Russia
- National University of Science and Technology "MISIS," Moscow 119049, Russia
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2
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Chulaevsky V. An optimal result on localization in random displacements models. RANDOM OPERATORS AND STOCHASTIC EQUATIONS 2022. [DOI: 10.1515/rose-2022-2091] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
Abstract
Abstract
We study random displacements models with a long-range particle-media interaction potential
𝔲
(
r
,
θ
)
=
𝔣
(
θ
)
r
-
A
{\mathfrak{u}(r,\theta)=\mathfrak{f}(\theta)r^{-A}}
in polar coordinates, with a smooth function
𝔣
{\mathfrak{f}}
which can be sign-indefinite.
Spectral and dynamical localization, with an asymptotically exponential decay of eigenfunction correlators,
is proved under the optimal condition
A
>
d
{A>d}
.
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Affiliation(s)
- Victor Chulaevsky
- Département de Mathématiques , Université de Reims , Moulin de la Housse, B.P.1039, 51687 Reims Cedex , France
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3
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Alt J, Ducatez R, Knowles A. Delocalization Transition for Critical Erdős-Rényi Graphs. COMMUNICATIONS IN MATHEMATICAL PHYSICS 2021; 388:507-579. [PMID: 34720130 PMCID: PMC8550299 DOI: 10.1007/s00220-021-04167-y] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/26/2021] [Accepted: 07/03/2021] [Indexed: 06/13/2023]
Abstract
We analyse the eigenvectors of the adjacency matrix of a critical Erdős-Rényi graph G ( N , d / N ) , where d is of order log N . We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent γ ( w ) of an eigenvector w , defined through ‖ w ‖ ∞ / ‖ w ‖ 2 = N - γ ( w ) . Our results remain valid throughout the optimal regime log N ≪ d ⩽ O ( log N ) .
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Affiliation(s)
- Johannes Alt
- Section of Mathematics, University of Geneva, Rue du Conseil-Général 7-9, 1205 Geneva, Switzerland
| | - Raphael Ducatez
- Section of Mathematics, University of Geneva, Rue du Conseil-Général 7-9, 1205 Geneva, Switzerland
| | - Antti Knowles
- Section of Mathematics, University of Geneva, Rue du Conseil-Général 7-9, 1205 Geneva, Switzerland
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4
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Chulaevsky V. Fast decay of eigenfunction correlators in long-range continuous random alloys. RANDOM OPERATORS AND STOCHASTIC EQUATIONS 2019. [DOI: 10.1515/rose-2019-2004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
Abstract
We study random Anderson Hamiltonians in Euclidean spaces with a long-range particle-media interaction potential
{\mathfrak{u}(r)=r^{-A}}
. Improving earlier results, for any
{A>2d}
, we establish spectral and strong dynamical localization with sub-exponential decay of eigenfunction correlators, both in the strong disorder regime and at low energies.
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Ziegler K, Sinner A. Short Note on the Density of States in 3D Weyl Semimetals. PHYSICAL REVIEW LETTERS 2018; 121:166401. [PMID: 30387659 DOI: 10.1103/physrevlett.121.166401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2018] [Indexed: 06/08/2023]
Abstract
The average density of states in a disordered three-dimensional Weyl system is discussed in the case of a continuous distribution of random scattering. Our results clearly indicate that the average density of states does not vanish, reflecting the absence of a critical point for a metal-insulator transition. This calculation supports recent suggestions of an avoided quantum critical point in the disordered three-dimensional Weyl semimetal. However, the effective density of states can be very small such that the saddle-approximation with a vanishing density of states might be valid for practical cases.
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Affiliation(s)
- K Ziegler
- Institut für Physik, Universität Augsburg, D-86135 Augsburg, Germany
| | - A Sinner
- Institut für Physik, Universität Augsburg, D-86135 Augsburg, Germany
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6
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7
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Deng X, Kravtsov VE, Shlyapnikov GV, Santos L. Duality in Power-Law Localization in Disordered One-Dimensional Systems. PHYSICAL REVIEW LETTERS 2018; 120:110602. [PMID: 29601742 DOI: 10.1103/physrevlett.120.110602] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/14/2017] [Revised: 01/15/2018] [Indexed: 06/08/2023]
Abstract
The transport of excitations between pinned particles in many physical systems may be mapped to single-particle models with power-law hopping, 1/r^{a}. For randomly spaced particles, these models present an effective peculiar disorder that leads to surprising localization properties. We show that in one-dimensional systems almost all eigenstates (except for a few states close to the ground state) are power-law localized for any value of a>0. Moreover, we show that our model is an example of a new universality class of models with power-law hopping, characterized by a duality between systems with long-range hops (a<1) and short-range hops (a>1), in which the wave function amplitude falls off algebraically with the same power γ from the localization center.
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Affiliation(s)
- X Deng
- Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany
| | - V E Kravtsov
- Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
- L. D. Landau Institute for Theoretical Physics, Chernogolovka, 142432 Moscow Region, Russia
| | - G V Shlyapnikov
- LPTMS, CNRS, Universite Paris Sud, Universite Paris-Saclay, Orsay 91405, France
- SPEC, CEA, CNRS, Universite Paris-Saclay, CEA Saclay, Gif sur Yvette 91191, France
- Russian Quantum Center, Skolkovo, Moscow 143025, Russia
- Van der Waals-Zeeman Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
- Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, 430071 Wuhan, China
| | - L Santos
- Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany
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8
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Cook N, Hachem W, Najim J, Renfrew D. Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs. ELECTRON J PROBAB 2018. [DOI: 10.1214/18-ejp230] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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9
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Deng X, Altshuler BL, Shlyapnikov GV, Santos L. Quantum Levy Flights and Multifractality of Dipolar Excitations in a Random System. PHYSICAL REVIEW LETTERS 2016; 117:020401. [PMID: 27447492 DOI: 10.1103/physrevlett.117.020401] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/29/2016] [Indexed: 06/06/2023]
Abstract
We consider dipolar excitations propagating via dipole-induced exchange among immobile molecules randomly spaced in a lattice. The character of the propagation is determined by long-range hops (Levy flights). We analyze the eigenenergy spectra and the multifractal structure of the wave functions. In 1D and 2D, all states are localized, although in 2D the localization length can be extremely large leading to an effective localization-delocalization crossover in realistic systems. In 3D, all eigenstates are extended but not always ergodic, and we identify the energy intervals of ergodic and nonergodic states. The reduction of the lattice filling induces an ergodic to nonergodic transition, and the excitations are mostly nonergodic at low filling.
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Affiliation(s)
- X Deng
- Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany
| | - B L Altshuler
- Physics Department, Columbia University, 538 West 120th Street, New York, New York 10027, USA
- Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
| | - G V Shlyapnikov
- Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, Orsay 91405, France
- Van der Waals-Zeeman Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands
- Russian Quantum Center, Skolkovo, Moscow Region 143025, Russia
- Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, People's Republic of China
| | - L Santos
- Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany
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10
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Daley P, Wortis R. Persistence of energy-dependent localization in the Anderson-Hubbard model with increasing system size and doping. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:175601. [PMID: 27022884 DOI: 10.1088/0953-8984/28/17/175601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Non-interacting systems with bounded disorder have been shown to exhibit sharp density of state peaks at the band edge which coincide with an energy range of abruptly suppressed localization. Recent work has shown that these features also occur in the presence of on-site interactions in ensembles of two-site Anderson-Hubbard systems at half filling. Here we demonstrate that this effect in interacting systems persists away from half filling, and moreover that energy regions with suppressed localization continue to appear in ensembles of larger systems despite a loss of sharp features in the density of states.
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Affiliation(s)
- P Daley
- Department of Physics & Astronomy, Trent University, 1600 West Bank Dr., Peterborough ON, K9J 7B8, Canada
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11
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Vattay G, Salahub D, Csabai I, Nassimi A, Kaufmann SA. Quantum criticality at the origin of life. ACTA ACUST UNITED AC 2015. [DOI: 10.1088/1742-6596/626/1/012023] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/04/2023]
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12
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De Luca A, Altshuler BL, Kravtsov VE, Scardicchio A. Anderson localization on the Bethe lattice: nonergodicity of extended States. PHYSICAL REVIEW LETTERS 2014; 113:046806. [PMID: 25105646 DOI: 10.1103/physrevlett.113.046806] [Citation(s) in RCA: 44] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/30/2014] [Indexed: 06/03/2023]
Abstract
Statistical analysis of the eigenfunctions of the Anderson tight-binding model with on-site disorder on regular random graphs strongly suggests that the extended states are multifractal at any finite disorder. The spectrum of fractal dimensions f(α) defined in Eq. (3) remains positive for α noticeably far from 1 even when the disorder is several times weaker than the one which leads to the Anderson localization; i.e., the ergodicity can be reached only in the absence of disorder. The one-particle multifractality on the Bethe lattice signals on a possible inapplicability of the equipartition law to a generic many-body quantum system as long as it remains isolated.
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Affiliation(s)
- A De Luca
- Laboratoire de Physique Théorique de l'ENS and Institut de Physique Theorique Philippe Meyer 24, Rue Lhomond, 75005 Paris, France
| | - B L Altshuler
- Physics Department, Columbia University, 538 West 120th Street, New York, New York 10027, USA
| | - V E Kravtsov
- Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy and L. D. Landau Institute for Theoretical Physics, 2 Kosygina Street, 119334 Moscow, Russia
| | - A Scardicchio
- Physics Department, Columbia University, 538 West 120th Street, New York, New York 10027, USA and Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy and Physics Department, Princeton University, Princeton, New Jersey 08544, USA and INFN, Sezione di Trieste, Strada Costiera 11, 34151 Trieste, Italy
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13
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Johri S, Bhatt RN. Singular behavior of eigenstates in Anderson's model of localization. PHYSICAL REVIEW LETTERS 2012; 109:076402. [PMID: 23006388 DOI: 10.1103/physrevlett.109.076402] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2012] [Indexed: 06/01/2023]
Abstract
We observe an apparent singularity in the electronic properties of the Anderson model of localization with bounded diagonal disorder, which is clearly distinct from the well-established mobility edge (localization-delocalization transition) that occurs in dimensions d > 2. We present results of numerical calculations for Anderson's original uniform (box) distribution of on-site disorder in dimensions d = 1, 2, and 3. To establish this hitherto unreported behavior, and to understand its evolution with disorder, we contrast the behavior of two different measures of the localization length of the electronic wave functions-the averaged inverse participation ratio and the Lyapunov exponent. Our data suggest that Anderson's model exhibits richer behavior than has been established so far.
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Affiliation(s)
- S Johri
- Department of Electrical Engineering, Princeton University, New Jersey 08544, USA
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14
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Gruber MJ, Veselić I. The modulus of continuity of Wegner estimates for random Schrödinger operators on metric graphs. RANDOM OPERATORS AND STOCHASTIC EQUATIONS 2008. [DOI: 10.1515/rose.2008.001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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15
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Affiliation(s)
- G. A. Thomas
- a AT&T Bell Laboratories , Murray Hill , New Jersey , 07974 , U.S.A
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16
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Byczuk K, Hofstetter W, Vollhardt D. Mott-Hubbard transition versus Anderson localization in correlated electron systems with disorder. PHYSICAL REVIEW LETTERS 2005; 94:056404. [PMID: 15783669 DOI: 10.1103/physrevlett.94.056404] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/31/2004] [Indexed: 05/24/2023]
Abstract
The phase diagram of correlated, disordered electron systems is calculated within dynamical mean-field theory using the geometrically averaged ("typical") local density of states. Correlated metal, Mott insulator, and Anderson insulator phases, as well as coexistence and crossover regimes, are identified. The Mott and Anderson insulators are found to be continuously connected.
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Affiliation(s)
- Krzysztof Byczuk
- Institute of Theoretical Physics, Warsaw University, ulica Hoza 69, PL-00-681 Warszawa, Poland
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17
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18
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Castellani C, Castro CD, Forgacs G, Tabet E. Gauge invariance and the multiplicative renormalisation group in the Anderson transition. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0022-3719/16/1/018] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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19
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Fisch R. Band tails, length scales, and localization. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:6862-6864. [PMID: 9982107 DOI: 10.1103/physrevb.53.6862] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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20
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Belitz D, Kirkpatrick TR. Order parameter description of the Anderson-Mott transition. ACTA ACUST UNITED AC 1995. [DOI: 10.1007/bf01320853] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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21
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Kirkpatrick TR, Belitz D. Landau theory for a metal-insulator transition. PHYSICAL REVIEW LETTERS 1994; 73:862-865. [PMID: 10057558 DOI: 10.1103/physrevlett.73.862] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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22
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Belitz D, Kirkpatrick TR. Critical behavior of the density of states at the metal-insulator transition. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:14072-14079. [PMID: 10007818 DOI: 10.1103/physrevb.48.14072] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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23
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Harigaya K, Wada Y, Fesser K. Impurity distribution and electronic states in doped conjugated polymers in the coherent-potential approximation. PHYSICAL REVIEW. B, CONDENSED MATTER 1990; 42:11303-11309. [PMID: 9995418 DOI: 10.1103/physrevb.42.11303] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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24
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Xu BC, Taylor PL. Comment on "Absence of impurity bands in conjugated polymers". PHYSICAL REVIEW LETTERS 1990; 65:805. [PMID: 10043024 DOI: 10.1103/physrevlett.65.805] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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25
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De Raedt H, de Vries P. Simulation of two and three-dimensional disordered systems: Lifshitz tails and localization properties. ACTA ACUST UNITED AC 1989. [DOI: 10.1007/bf01313668] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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26
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Gold A. Scattering time and single-particle relaxation time in a disordered two-dimensional electron gas. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 38:10798-10811. [PMID: 9945936 DOI: 10.1103/physrevb.38.10798] [Citation(s) in RCA: 69] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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27
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Gold A, Serre J, Ghazali A. Density of states in a two-dimensional electron gas: Impurity bands and band tails. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 37:4589-4603. [PMID: 9945118 DOI: 10.1103/physrevb.37.4589] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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28
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29
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March P, Sznitman AS. Some connections between excursion theory and the discrete Schrödinger equation with random potentials. Probab Theory Relat Fields 1987. [DOI: 10.1007/bf00320079] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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30
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Kim Y, Harris AB. Density of states of the random-hopping model on a Cayley tree. PHYSICAL REVIEW. B, CONDENSED MATTER 1985; 31:7393-7407. [PMID: 9935663 DOI: 10.1103/physrevb.31.7393] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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31
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Simon B, Taylor M, Wolff T. Some rigorous results for the Anderson model. PHYSICAL REVIEW LETTERS 1985; 54:1589-1592. [PMID: 10031078 DOI: 10.1103/physrevlett.54.1589] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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