Fractionally charged skyrmions in fractional quantum Hall effect
Identifieur interne : 000693 ( Ncbi/Merge ); précédent : 000692; suivant : 000694Fractionally charged skyrmions in fractional quantum Hall effect
Auteurs : Ajit C. Balram [États-Unis] ; U. Wurstbauer [Allemagne] ; A. W Js [Pologne] ; A. Pinczuk [États-Unis] ; J. K. Jain [États-Unis]Source :
- Nature Communications [ 2041-1723 ] ; 2015.
Abstract
The fractional quantum Hall effect has inspired searches for exotic emergent topological particles, such as fractionally charged excitations, composite fermions, abelian and nonabelian anyons and Majorana fermions. Fractionally charged skyrmions, which support both topological charge and topological vortex-like spin structure, have also been predicted to occur in the vicinity of 1/3 filling of the lowest Landau level. The fractional skyrmions, however, are anticipated to be exceedingly fragile, suppressed by very small Zeeman energies. Here we show that, slightly away from 1/3 filling, the smallest manifestations of the fractional skyrmion exist in the excitation spectrum for a broad range of Zeeman energies, and appear in resonant inelastic light scattering experiments as well-defined resonances slightly below the long wavelength spin wave mode. The spectroscopy of these exotic bound states serves as a sensitive tool for investigating the residual interaction between composite fermions, responsible for delicate new fractional quantum Hall states in this filling factor region.
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DOI: 10.1038/ncomms9981
PubMed: 26608906
PubMed Central: 4674824
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Wurstbauer</name><affiliation wicri:level="1"><nlm:aff id="a2"><institution>Walter Schottky Institut and Physik-Department, Am Coulombwall 4a, Technische Universität München</institution>, D-85748 Garching,<country>Germany</country></nlm:aff><country xml:lang="fr">Allemagne</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation><affiliation wicri:level="1"><nlm:aff id="a3"><institution>Nanosystems Initiative Munich (NIM)</institution>, Schellingstraße 4, 80799 München,<country>Germany</country></nlm:aff><country xml:lang="fr">Allemagne</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author><author><name sortKey="W Js, A" sort="W Js, A" uniqKey="W Js A" first="A." last="W Js">A. W Js</name><affiliation wicri:level="1"><nlm:aff id="a4"><institution>Department of Theoretical Physics, Wrocław University of Technology</institution>, 50-370 Wrocław,<country>Poland</country></nlm:aff><country xml:lang="fr">Pologne</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author><author><name sortKey="Pinczuk, A" sort="Pinczuk, A" uniqKey="Pinczuk A" first="A." last="Pinczuk">A. Pinczuk</name><affiliation wicri:level="1"><nlm:aff id="a5"><institution>Department of Applied Physics and Applied Mathematics and Department of Physics, Columbia University</institution>, New York 10027,<country>USA</country></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author><author><name sortKey="Jain, J K" sort="Jain, J K" uniqKey="Jain J" first="J. K." last="Jain">J. K. Jain</name><affiliation wicri:level="1"><nlm:aff id="a1"><institution>Department of Physics, 104 Davey Lab, Pennsylvania State University</institution>, University Park, Pennsylvania 16802,<country>USA</country></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">26608906</idno><idno type="pmc">4674824</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4674824</idno><idno type="RBID">PMC:4674824</idno><idno type="doi">10.1038/ncomms9981</idno><date when="2015">2015</date><idno type="wicri:Area/Pmc/Corpus">000143</idno><idno type="wicri:Area/Pmc/Curation">000143</idno><idno type="wicri:Area/Pmc/Checkpoint">000105</idno><idno type="wicri:Area/Ncbi/Merge">000693</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Fractionally charged skyrmions in fractional quantum Hall effect</title><author><name sortKey="Balram, Ajit C" sort="Balram, Ajit C" uniqKey="Balram A" first="Ajit C." last="Balram">Ajit C. Balram</name><affiliation wicri:level="1"><nlm:aff id="a1"><institution>Department of Physics, 104 Davey Lab, Pennsylvania State University</institution>, University Park, Pennsylvania 16802,<country>USA</country></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author><author><name sortKey="Wurstbauer, U" sort="Wurstbauer, U" uniqKey="Wurstbauer U" first="U." last="Wurstbauer">U. Wurstbauer</name><affiliation wicri:level="1"><nlm:aff id="a2"><institution>Walter Schottky Institut and Physik-Department, Am Coulombwall 4a, Technische Universität München</institution>, D-85748 Garching,<country>Germany</country></nlm:aff><country xml:lang="fr">Allemagne</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation><affiliation wicri:level="1"><nlm:aff id="a3"><institution>Nanosystems Initiative Munich (NIM)</institution>, Schellingstraße 4, 80799 München,<country>Germany</country></nlm:aff><country xml:lang="fr">Allemagne</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author><author><name sortKey="W Js, A" sort="W Js, A" uniqKey="W Js A" first="A." last="W Js">A. W Js</name><affiliation wicri:level="1"><nlm:aff id="a4"><institution>Department of Theoretical Physics, Wrocław University of Technology</institution>, 50-370 Wrocław,<country>Poland</country></nlm:aff><country xml:lang="fr">Pologne</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author><author><name sortKey="Pinczuk, A" sort="Pinczuk, A" uniqKey="Pinczuk A" first="A." last="Pinczuk">A. Pinczuk</name><affiliation wicri:level="1"><nlm:aff id="a5"><institution>Department of Applied Physics and Applied Mathematics and Department of Physics, Columbia University</institution>, New York 10027,<country>USA</country></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author><author><name sortKey="Jain, J K" sort="Jain, J K" uniqKey="Jain J" first="J. K." last="Jain">J. K. Jain</name><affiliation wicri:level="1"><nlm:aff id="a1"><institution>Department of Physics, 104 Davey Lab, Pennsylvania State University</institution>, University Park, Pennsylvania 16802,<country>USA</country></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author></analytic><series><title level="j">Nature Communications</title><idno type="eISSN">2041-1723</idno><imprint><date when="2015">2015</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>The fractional quantum Hall effect has inspired searches for exotic emergent topological particles, such as fractionally charged excitations, composite fermions, abelian and nonabelian anyons and Majorana fermions. Fractionally charged skyrmions, which support both topological charge and topological vortex-like spin structure, have also been predicted to occur in the vicinity of 1/3 filling of the lowest Landau level. The fractional skyrmions, however, are anticipated to be exceedingly fragile, suppressed by very small Zeeman energies. Here we show that, slightly away from 1/3 filling, the smallest manifestations of the fractional skyrmion exist in the excitation spectrum for a broad range of Zeeman energies, and appear in resonant inelastic light scattering experiments as well-defined resonances slightly below the long wavelength spin wave mode. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Nat Commun</journal-id><journal-id journal-id-type="iso-abbrev">Nat Commun</journal-id><journal-title-group><journal-title>Nature Communications</journal-title></journal-title-group><issn pub-type="epub">2041-1723</issn><publisher><publisher-name>Nature Pub. Group</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">26608906</article-id><article-id pub-id-type="pmc">4674824</article-id><article-id pub-id-type="pii">ncomms9981</article-id><article-id pub-id-type="doi">10.1038/ncomms9981</article-id><article-categories><subj-group subj-group-type="heading"><subject>Article</subject></subj-group></article-categories><title-group><article-title>Fractionally charged skyrmions in fractional quantum Hall effect</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Balram</surname><given-names>Ajit C.</given-names></name><xref ref-type="aff" rid="a1">1</xref></contrib><contrib contrib-type="author"><name><surname>Wurstbauer</surname><given-names>U.</given-names></name><xref ref-type="aff" rid="a2">2</xref><xref ref-type="aff" rid="a3">3</xref></contrib><contrib contrib-type="author"><name><surname>Wójs</surname><given-names>A.</given-names></name><xref ref-type="aff" rid="a4">4</xref></contrib><contrib contrib-type="author"><name><surname>Pinczuk</surname><given-names>A.</given-names></name><xref ref-type="aff" rid="a5">5</xref></contrib><contrib contrib-type="author"><name><surname>Jain</surname><given-names>J. K.</given-names></name><xref ref-type="corresp" rid="c1">a</xref><xref ref-type="aff" rid="a1">1</xref></contrib><aff id="a1"><label>1</label><institution>Department of Physics, 104 Davey Lab, Pennsylvania State University</institution>, University Park, Pennsylvania 16802,<country>USA</country></aff><aff id="a2"><label>2</label><institution>Walter Schottky Institut and Physik-Department, Am Coulombwall 4a, Technische Universität München</institution>, D-85748 Garching,<country>Germany</country></aff><aff id="a3"><label>3</label><institution>Nanosystems Initiative Munich (NIM)</institution>, Schellingstraße 4, 80799 München,<country>Germany</country></aff><aff id="a4"><label>4</label><institution>Department of Theoretical Physics, Wrocław University of Technology</institution>, 50-370 Wrocław,<country>Poland</country></aff><aff id="a5"><label>5</label><institution>Department of Applied Physics and Applied Mathematics and Department of Physics, Columbia University</institution>, New York 10027,<country>USA</country></aff></contrib-group><author-notes><corresp id="c1"><label>a</label><email>jkj2@psu.edu</email></corresp></author-notes><pub-date pub-type="epub"><day>26</day><month>11</month><year>2015</year></pub-date><pub-date pub-type="collection"><year>2015</year></pub-date><volume>6</volume><elocation-id>8981</elocation-id><history><date date-type="received"><day>22</day><month>03</month><year>2015</year></date><date date-type="accepted"><day>22</day><month>10</month><year>2015</year></date></history><permissions><copyright-statement>Copyright © 2015, Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved.</copyright-statement><copyright-year>2015</copyright-year><copyright-holder>Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved.</copyright-holder><license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by/4.0/"><pmc-comment>author-paid</pmc-comment>
<license-p>This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/4.0/">http://creativecommons.org/licenses/by/4.0/</ext-link></license-p></license></permissions><abstract><p>The fractional quantum Hall effect has inspired searches for exotic emergent topological particles, such as fractionally charged excitations, composite fermions, abelian and nonabelian anyons and Majorana fermions. Fractionally charged skyrmions, which support both topological charge and topological vortex-like spin structure, have also been predicted to occur in the vicinity of 1/3 filling of the lowest Landau level. The fractional skyrmions, however, are anticipated to be exceedingly fragile, suppressed by very small Zeeman energies. Here we show that, slightly away from 1/3 filling, the smallest manifestations of the fractional skyrmion exist in the excitation spectrum for a broad range of Zeeman energies, and appear in resonant inelastic light scattering experiments as well-defined resonances slightly below the long wavelength spin wave mode. The spectroscopy of these exotic bound states serves as a sensitive tool for investigating the residual interaction between composite fermions, responsible for delicate new fractional quantum Hall states in this filling factor region.</p></abstract><abstract abstract-type="web-summary"><p><inline-graphic id="i1" xlink:href="ncomms9981-i1.jpg"></inline-graphic>It is predicted that fractionally charged skyrmions, topologically protected vortex-like spin configurations, may exist in systems exhibiting fractional quantum Hall states. Here, the authors demonstrate the existence of such objects in GaAs single quantum wells.</p></abstract></article-meta></front><floats-group><fig id="f1"><label>Figure 1</label><caption><title>Comparison between theory and experiment.</title><p>(<bold>a</bold>) shows the ground state for the fully polarized state for <italic>v</italic>≲1/3 with a single composite fermion (CF) hole (empty red circle) in the spin-up lowest ΛL (0↑); (<bold>b</bold>) shows an additional spin-flip exciton (SFE) that binds with the hole to produce a minimal positively charged fractional skyrmion (FS). (<bold>c</bold>) shows the state for <italic>v</italic>≳1/3 with a single CF particle in the spin-down lowest ΛL (0↓), and (<bold>d</bold>) has an additional SFE. (<bold>e</bold>) has a CF particle in the spin-up second ΛL (1↑), and (<bold>f</bold>) has an additional SFE. The composite fermions are shown as particles with two arrows, representing bound vortices, and their up and down spin ΛLs are shown as shaded blue and red rectangles, respectively. In <bold>g</bold>–<bold>i</bold> the red dashes (dots) show the exact (CF) energies of the ground states containing a single CF particle or hole (as shown in <bold>a</bold>,<bold>c</bold>,<bold>e</bold>) and the black symbols show the spectrum obtained when an additional SFE is created (as shown in <bold>b</bold>,<bold>d</bold>,<bold>f</bold>). The spherical geometry is used for calculations; panel (<bold>g</bold>) is for eight particles subjected to 22 flux quanta (a flux quantum is defined as <italic>φ</italic><sub>0</sub>=<italic>hc</italic>/<italic>e</italic>), and (<bold>h</bold>,<bold>i</bold>) correspond to 10 particles in 26 flux quanta. (<bold>j</bold>–<bold>l</bold>) show the experimentally measured energies of modes below the Zeeman energy. The theoretical energy of the FSs in the dilute limit of <italic>ν</italic>→1/3 including finite width correction is also shown by blue square. Panels (<bold>j</bold>,<bold>l</bold>) are for 50° tilt, whereas (<bold>k</bold>) is for 30° tilt. All energies in <bold>j</bold>–<bold>l</bold> are shown relative to the Zeeman energy, in units of <inline-formula id="d33e1073"><inline-graphic id="d33e1074" xlink:href="ncomms9981-m21.jpg"></inline-graphic></inline-formula>, where <inline-formula id="d33e1076"><inline-graphic id="d33e1077" xlink:href="ncomms9981-m22.jpg"></inline-graphic></inline-formula> is the dielectric constant of the material and <inline-formula id="d33e1079"><inline-graphic id="d33e1080" xlink:href="ncomms9981-m23.jpg"></inline-graphic></inline-formula> is the magnetic length. The modes depicted by red symbols are assigned to fractional skyrmions, green stars in panel <bold>k</bold> to the excitation shown in <xref ref-type="fig" rid="f8">Fig. 8d</xref>, and the black diamonds and purple stars in panel <bold>l</bold> to the excitation shown in <xref ref-type="fig" rid="f8">Fig. 8i</xref>. The theoretical error bars arise from the uncertainty in the Monte Carlo calculations and thermodynamic extrapolations, and the experimental error bar reflects the uncertainty in the Lorentzian fits.</p></caption><graphic xlink:href="ncomms9981-f1"></graphic></fig><fig id="f2"><label>Figure 2</label><caption><title>Contrasting negatively charged skyrmion with composite fermion (CF) particle.</title><p>(<bold>a</bold>–<bold>c</bold>) show charge density profiles of a spin-conserving CF particle, a spin-reversed CF particle, and a negatively charged fractional skyrmion. Their spin polarization, defined by <inline-formula id="d33e1107"><inline-graphic id="d33e1108" xlink:href="ncomms9981-m24.jpg"></inline-graphic></inline-formula> where <inline-formula id="d33e1110"><inline-graphic id="d33e1111" xlink:href="ncomms9981-m25.jpg"></inline-graphic></inline-formula> and <inline-formula id="d33e1113"><inline-graphic id="d33e1114" xlink:href="ncomms9981-m26.jpg"></inline-graphic></inline-formula> are the spatial densities of spin-up and spin-down composite fermions, is shown in <bold>d</bold>–<bold>f</bold>, respectively. The minimum/maximum values of the colour bars in each panel are: (<bold>a</bold>) 0.303/0.453, (<bold>b</bold>) 0.333/0.456, (<bold>c</bold>) 0.333/0.391, (<bold>d</bold>) 1.000/1.000, (<bold>e</bold>) −0.352/1.000, (<bold>f</bold>) −0.512/1.000. The disk has a radius of 12.5 <inline-formula id="d33e1142"><inline-graphic id="d33e1143" xlink:href="ncomms9981-m27.jpg"></inline-graphic></inline-formula>.</p></caption><graphic xlink:href="ncomms9981-f2"></graphic></fig><fig id="f3"><label>Figure 3</label><caption><title>Contrasting the positively charged skyrmion with the composite fermion (CF) hole.</title><p>(<bold>a</bold>,<bold>b</bold>) show charge density profiles of a CF hole and a positively charged fractional skyrmion. Their spin polarization, defined by <inline-formula id="d33e1157"><inline-graphic id="d33e1158" xlink:href="ncomms9981-m28.jpg"></inline-graphic></inline-formula> where <inline-formula id="d33e1160"><inline-graphic id="d33e1161" xlink:href="ncomms9981-m29.jpg"></inline-graphic></inline-formula> and <inline-formula id="d33e1163"><inline-graphic id="d33e1164" xlink:href="ncomms9981-m30.jpg"></inline-graphic></inline-formula> are the spatial densities of spin-up and spin-down composite fermions, is shown in <bold>c</bold>,<bold>d</bold>, respectively. The minimum/maximum values of the colour bars in each panel are: (<bold>a</bold>) 0.006/0.357, (<bold>b</bold>) 0.266/0.333, (<bold>c</bold>) 1.000/1.000, (<bold>d</bold>) −0.695/1.000. The disk shown has a radius of 12.5 <inline-formula id="d33e1186"><inline-graphic id="d33e1187" xlink:href="ncomms9981-m31.jpg"></inline-graphic></inline-formula>.</p></caption><graphic xlink:href="ncomms9981-f3"></graphic></fig><fig id="f4"><label>Figure 4</label><caption><title>Thermodynamic extrapolation of the binding energies of the fractional skyrmions.</title><p>The blue (red) symbols show the energies of negative (positive) fractional skyrmions for a system of <italic>N</italic> particles with zero transverse width, obtained from exact diagonalization. The inset shows the amount by which finite-width corrections lower the energy of the fractional skyrmion (FS) for a sample of width 33 nm.</p></caption><graphic xlink:href="ncomms9981-f4"></graphic></fig><fig id="f5"><label>Figure 5</label><caption><title>Comparison of exact and composite fermion (CF) density profiles for fractional skyrmions.</title><p>This figure shows the total density (<italic>ρ</italic>) and the density of spin-up particles <inline-formula id="d33e1207"><inline-graphic id="d33e1208" xlink:href="ncomms9981-m32.jpg"></inline-graphic></inline-formula> for fractional skyrmion (FS<sup>−</sup>) (blue) and FS<sup>+</sup> (red) obtained from exact (dotted and dashed lines) and CF diagonalization (filled and empty symbols). A near perfect overlay of the CF and exact curves shows that the wave function of the FS<inline-formula id="d33e1216"><inline-graphic id="d33e1217" xlink:href="ncomms9981-m36.jpg"></inline-graphic></inline-formula> obtained from CF diagonalization is almost identical to the exact one. The results are for 12 particles, and the density is quoted in units of the density of the uniform 1/3 state, denoted by <italic>ρ</italic><sub>0</sub>.</p></caption><graphic xlink:href="ncomms9981-f5"></graphic></fig><fig id="f6"><label>Figure 6</label><caption><title>Excitations in resonant inelastic light scattering spectra.</title><p>(<bold>a</bold>,<bold>b</bold>) show typical spectra at <italic>ν</italic>=0.36 for <italic>θ</italic>=30° and <italic>θ</italic>=50°, respectively, as a function of the incident laser energy <italic>E</italic><sub>laser</sub>.</p></caption><graphic xlink:href="ncomms9981-f6"></graphic></fig><fig id="f7"><label>Figure 7</label><caption><title>Lorentzian fits to the RILS spectra.</title><p>Resonant inelastic light scattering (RILS) spectra obtained at 30° tilt for <italic>ν</italic>=0.38 (<bold>a</bold>,<bold>c</bold>) and <italic>ν</italic>=0.37 (<bold>b</bold>,<bold>d</bold>). The raw RILS data are displayed as black bullets and the Lorentzian fits to the data as red solid line. The blue lines in each panel show the individual Lorentzians used to obtain the fit to the data. At 30° tilt the data do not fit well to two Lorentzians as seen in (<bold>a</bold>,<bold>b</bold>) but fit well to three Lorentzians (<bold>c</bold>,<bold>d</bold>).</p></caption><graphic xlink:href="ncomms9981-f7"></graphic></fig><fig id="f8"><label>Figure 8</label><caption><title>Elementary excitations in the vicinity of filling factor <italic>ν</italic>=1/3.</title><p>(<bold>b</bold>,<bold>g</bold>) indicate the partially spin-polarized and the fully spin-polarized ground states at <italic>ν</italic>≳1/3, and (<bold>m</bold>) indicates the fully spin-polarized ground state for <italic>ν</italic>≲1/3. The other panels (<bold>a</bold>, <bold>c</bold>–<bold>f</bold>, <bold>h</bold>–<bold>l</bold>, <bold>n</bold>) show the excitations obtained by either promoting (or demoting) a single composite fermion to a higher (lower) Λ level or flipping its spin or doing both. (<bold>c</bold>) shows the negatively charged fractional skyrmion (FS<sup>−</sup>) excitation and (<bold>n</bold>) shows the FS<sup>+</sup> excitation.</p></caption><graphic xlink:href="ncomms9981-f8"></graphic></fig><table-wrap position="float" id="t1"><label>Table 1</label><caption><title>Energy of the elementary excitations in the vicinity of <italic>ν</italic>=1/3 shown in <xref ref-type="fig" rid="f8">Fig. 8</xref>.</title></caption><table frame="hsides" rules="groups" border="1"><colgroup><col align="left"></col><col align="char" char="("></col><col align="char" char="("></col><col align="char" char="("></col></colgroup><thead valign="bottom"><tr><th align="left" valign="top" charoff="50"><bold>Mode</bold></th><th colspan="2" align="center" valign="top" char="(" charoff="50"><bold>Width,</bold><italic><bold>w</bold></italic><bold>=0</bold><hr></hr></th><th align="center" valign="top" char="(" charoff="50"><italic><bold>E</bold></italic><sub><italic><bold>w</bold></italic><bold>=33 nm</bold></sub><bold>−</bold><italic><bold>E</bold></italic><sub><italic><bold>w</bold></italic><bold>=0</bold></sub></th><th align="center" valign="top" charoff="50"> </th></tr><tr><th align="left" valign="top" charoff="50"> </th><th align="center" valign="top" char="(" charoff="50"><bold>Exact</bold></th><th align="center" valign="top" char="(" charoff="50"><bold>CF theory</bold></th><th align="center" valign="top" char="(" charoff="50"><bold>CF theory</bold></th><td> </td></tr></thead><tbody valign="top"><tr><td align="left" valign="top" charoff="50">(a)+(e),(d)</td><td align="char" valign="top" char="(" charoff="50">0.0284 (3)</td><td align="char" valign="top" char="(" charoff="50">0.0258 (0)</td><td align="char" valign="top" char="(" charoff="50">−0.0073 (0)</td></tr><tr><td align="left" valign="top" charoff="50">(c)</td><td align="char" valign="top" char="(" charoff="50">−0.0052 (2)</td><td align="char" valign="top" char="(" charoff="50">∼−0.0059 (7)</td><td align="char" valign="top" char="(" charoff="50">0.0010 (0)</td></tr><tr><td align="left" valign="top" charoff="50">(f)+(j)</td><td align="center" valign="top" char="(" charoff="50">0</td><td align="center" valign="top" char="(" charoff="50">0</td><td align="center" valign="top" char="(" charoff="50">0</td></tr><tr><td align="left" valign="top" charoff="50">(h)</td><td align="char" valign="top" char="(" charoff="50">0.0422 (33)</td><td align="char" valign="top" char="(" charoff="50">∼0.0436 (113)</td><td align="char" valign="top" char="(" charoff="50">−0.0095 (207)</td></tr><tr><td align="left" valign="top" charoff="50">(i)</td><td align="char" valign="top" char="(" charoff="50">−0.0284 (3)</td><td align="char" valign="top" char="(" charoff="50">−0.0258 (0)</td><td align="char" valign="top" char="(" charoff="50">0.0073 (0)</td></tr><tr><td align="left" valign="top" charoff="50">(k)</td><td align="center" valign="top" char="(" charoff="50">—</td><td align="char" valign="top" char="(" charoff="50">0.0867 (1)</td><td align="char" valign="top" char="(" charoff="50">−0.0224 (1)</td></tr><tr><td align="left" valign="top" charoff="50">(l)</td><td align="char" valign="top" char="(" charoff="50">0.0369 (17)</td><td align="char" valign="top" char="(" charoff="50">0.0366 (47)</td><td align="char" valign="top" char="(" charoff="50">−0.0117 (77)</td></tr><tr><td align="left" valign="top" charoff="50">(n)</td><td align="char" valign="top" char="(" charoff="50">−0.0096 (2)</td><td align="char" valign="top" char="(" charoff="50">∼−0.0108 (24)</td><td align="char" valign="top" char="(" charoff="50">0.0013 (7)</td></tr></tbody></table><table-wrap-foot><fn id="t1-fn1"><p>The Coulomb energy of the the elementary excitations near <italic>ν</italic>=1/3 determined by an extrapolation of the finite system results, obtained by exact diagonalization (second column) and the CF theory (third column), for quantum well width <italic>w</italic>=0. The last column gives the difference in the energies of each mode for quantum wells of widths <italic>w</italic>=33 nm and <italic>w</italic>=0, obtained by the CF theory. All energies are quoted in units of <inline-formula id="d33e1498"><inline-graphic id="d33e1499" xlink:href="ncomms9981-m33.jpg"></inline-graphic></inline-formula>. The cases where linear extrapolation in 1/<italic>N</italic> to the thermodynamic limit is not very accurate are marked by the symbol ∼ to indicate larger uncertainty. The total energy of <xref ref-type="fig" rid="f8">Fig. 8c, i and n</xref> (<xref ref-type="fig" rid="f8">Fig. 8d</xref>) is obtained by adding (subtracting) the Zeeman splitting <italic>E</italic><sub>Z</sub> as explained in the text. (Note: A combination of <xref ref-type="fig" rid="f8">Fig. 8a and e</xref> is needed to obtain <italic>S</italic><sup>2</sup> eigen states; the same is true of f and j. k is an eigen state of the Hamiltonian but it is in general an excited state.)</p></fn></table-wrap-foot></table-wrap></floats-group></pmc><affiliations><list><country><li>Allemagne</li><li>Pologne</li><li>États-Unis</li></country></list><tree><country name="États-Unis"><noRegion><name sortKey="Balram, Ajit C" sort="Balram, Ajit C" uniqKey="Balram A" first="Ajit C." last="Balram">Ajit C. Balram</name></noRegion><name sortKey="Jain, J K" sort="Jain, J K" uniqKey="Jain J" first="J. K." last="Jain">J. K. Jain</name><name sortKey="Pinczuk, A" sort="Pinczuk, A" uniqKey="Pinczuk A" first="A." last="Pinczuk">A. Pinczuk</name></country><country name="Allemagne"><noRegion><name sortKey="Wurstbauer, U" sort="Wurstbauer, U" uniqKey="Wurstbauer U" first="U." last="Wurstbauer">U. Wurstbauer</name></noRegion><name sortKey="Wurstbauer, U" sort="Wurstbauer, U" uniqKey="Wurstbauer U" first="U." last="Wurstbauer">U. Wurstbauer</name></country><country name="Pologne"><noRegion><name sortKey="W Js, A" sort="W Js, A" uniqKey="W Js A" first="A." last="W Js">A. W Js</name></noRegion></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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