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UIUC-PI Joint Conference, 2014-10-16 Universal Geometric Response of Abelian and Non-Abelian FQH states Gil Young Cho Univ. of Illinois at Urbana-Champaign Main references! Based on the works/understanding from : [1] GYC, Y. You, and E. Fradkin, PRB, 90, 115139 (2014) [2] GYC, O. Parrikar, Y. You, R. Leigh, and T. Hughes, arxiv:1407.5637 [3] Y. You, GYC, and E. Fradkin, arxiv:1410.3390 [4] A. Gromov, GYC, Y. You, A. Abanov, and E. Fradkin, (in preparation) Contents 1. Geometric Responses of Abelian and Non-abelian FQHEs = Hall viscosity, Wen-Zee term, and Gravitational Chern-Simons term 2. Puzzles from (naΓ―ve) composite particle theories = Failure of naΓ―ve composite particle theories to capture the geometric responses 3. Revisiting Flux attachment and composite particle theories = correcting flux attachment in curved space 4. Projective Parton Approaches Goal of this Talk Composite Particle Theories (composite boson/fermion) are (1) successful in describing ground state properties and excitations of FQHEs (2) Generating Wen’s hydrodynamic theory with the correct (K, q) 1 1 𝐼𝐼 𝐽𝐽 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ (3) But fail to predict ``spin ” (necessary for geometric responses of FQH states) 0 𝑰𝑰𝑰𝑰 πœ‡πœ‡ 𝜈𝜈 πœ‡πœ‡ 𝑰𝑰 πœ‡πœ‡ 𝜈𝜈 πœ‡πœ‡ 𝐿𝐿 = πœŒπœŒΜ…π›Ώπ›Ώπ΄π΄ βˆ’ 4πœ‹πœ‹ 𝑲𝑲 𝑏𝑏 πœ•πœ• 𝑏𝑏 πœ–πœ– + 2πœ‹πœ‹ 𝒒𝒒 𝑏𝑏 πœ•πœ• 𝛿𝛿𝐴𝐴 πœ–πœ– + β‹― 𝐽𝐽 𝑆𝑆 More precisely, composite particle theories fail to capture : Wen-Zee term and Hall Viscosity Can we correct composite particle theories to derive the terms? Before jumping into the details.. 0. Review : Hydrodynamic Theory Ref. Wen’s textbook or Fradkin’s textbook 1 1 𝐼𝐼 𝐽𝐽 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ (1) , e0xternal prob𝑰𝑰𝑰𝑰e EMπœ‡πœ‡ g𝜈𝜈augπœ‡πœ‡e field 𝑰𝑰 πœ‡πœ‡ 𝜈𝜈 πœ‡πœ‡ 𝐿𝐿 = πœŒπœŒΜ…π›Ώπ›Ώπ΄π΄ βˆ’ 4πœ‹πœ‹ 𝑲𝑲 𝑏𝑏 πœ•πœ• 𝑏𝑏 πœ–πœ– + 2πœ‹πœ‹ 𝒒𝒒 𝑏𝑏 πœ•πœ• 𝛿𝛿𝐴𝐴 πœ–πœ– + β‹― πœ‡πœ‡ 𝛿𝛿𝐴𝐴 (2) , hydrodynamic gauge field, quasi-particle current: 1 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝑏𝑏 πœ‡πœ‡ 𝐽𝐽 πœ‡πœ‡ = 2πœ‹πœ‹ πœ–πœ–πœ‡πœ‡πœˆπœˆπœ‡πœ‡ πœ•πœ•πœˆπœˆπ‘π‘ πœ‡πœ‡ (3) , integer matrix, (encoding topological information) 𝐼𝐼𝐽𝐽 𝐾𝐾 ∈ 𝑍𝑍 ex, GS degeneracy on genus-g manifold 𝑔𝑔 ex, statistical angle of and is 𝐼𝐼𝐽𝐽 det 𝐾𝐾 𝑇𝑇 βˆ’1 1 2 12 1 2 (4) , integer vector, ch𝑙𝑙arge ve𝑙𝑙ctor πœƒπœƒ = 2πœ‹πœ‹ 𝑙𝑙 𝐾𝐾 𝑙𝑙 𝐼𝐼 π‘žπ‘ž ∈ 𝑍𝑍 = EM Hall response by integrating out 1 𝑇𝑇 βˆ’1 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 𝐿𝐿 = π‘žπ‘ž 𝐾𝐾 π‘žπ‘ž Γ— 4πœ‹πœ‹ πœ–πœ– π›Ώπ›Ώπ΄π΄πœ‡πœ‡πœ•πœ•πœˆπœˆπ›Ώπ›Ώπ΄π΄πœ‡πœ‡ 𝑏𝑏 πœ‡πœ‡ Contents 1. Geometric Responses of Abelian and Non-abelian FQHEs = Hall viscosity, Wen-Zee term, and Gravitational Chern-Simons term 2. Puzzles in naΓ―ve composite particle theories = Failure of naΓ―ve composite particle theories to capture the geometric responses 3. Revisiting Flux attachment and composite particle theories = correcting flux attachment in curved space 4. Projective Parton Approaches Part 1. Geometric Response of FQHE Imagine that the lattice is distorted & fluctuating and electrons are on top of the lattice… e e e e e e e e e Interesting Physics? Interesting Physics? 1. Wen-Zee term 2. Hall Viscosity Wen-Zee term : QH fluids see the curvature as ``magnetic flux’’ (1) QH fluids collects electric charge at the magnetic flux Wen-Zee 1992 Magnetic field Quantum Hall fluid Electric current Imagine : Induced field! ( ) Thus quantum Hall fluids accumulate the charge at the mag𝑑𝑑netic flux ! 𝑬𝑬 πœ•πœ• 𝐡𝐡 = 𝛻𝛻 Γ— 𝐸𝐸 (2) On the space with curvature… Collects or depletes charge when the curvature is present Wen-Zee term Wen-Zee 1992 Wen & Zee supplemented a phenomenological term into the hydrodynamic theory 1 1 𝑺𝑺𝑰𝑰 𝐼𝐼 𝐽𝐽 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝑰𝑰 𝝁𝝁𝝂𝝂𝝀𝝀 0 𝐼𝐼𝐽𝐽 πœ‡πœ‡ 𝜈𝜈 πœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡ 𝜈𝜈 πœ‡πœ‡ 𝝁𝝁 𝝂𝝂 𝝀𝝀 𝐿𝐿 = πœŒπœŒΜ…π›Ώπ›Ώπ΄π΄ βˆ’ 4πœ‹πœ‹ 𝐾𝐾 𝑏𝑏 πœ•πœ• 𝑏𝑏 πœ–πœ– + 2πœ‹πœ‹ π‘žπ‘ž 𝑏𝑏 πœ•πœ• 𝛿𝛿𝐴𝐴 πœ–πœ– + πŸπŸπŸπŸπ’ƒπ’ƒ 𝝏𝝏 𝝎𝝎 𝝐𝝐 β‹― with the spin vector (remember that ) 1 𝐼𝐼 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 𝑰𝑰 Here = spin connectioπœ‡πœ‡n describing 𝜈𝜈curπœ‡πœ‡vature of the background geo𝑺𝑺metry 𝐽𝐽 = 2πœ‹πœ‹ πœ–πœ– πœ•πœ• 𝑏𝑏 πœ‡πœ‡ πœ”πœ”i.e., Local curvature = (similar to local magnetic field ) βƒ— 𝛻𝛻 Γ— πœ”πœ” B = 𝛻𝛻 Γ— 𝐴𝐴 1 𝐼𝐼 𝐽𝐽 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡ 𝐿𝐿 = πœŒπœŒΜ…π›Ώπ›Ώπ΄π΄0 βˆ’ 4πœ‹πœ‹ 𝐾𝐾𝐼𝐼𝐽𝐽𝑏𝑏 πœ‡πœ‡πœ•πœ•πœˆπœˆπ‘π‘ πœ‡πœ‡πœ–πœ– + π‘žπ‘žπΌπΌπ½π½ πœ‡πœ‡π›Ώπ›Ώπ΄π΄ + 𝑠𝑠𝐼𝐼𝐽𝐽 πœ‡πœ‡πœ”πœ” β‹― the conserved particle current 1 𝐼𝐼 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡ 𝜈𝜈 πœ‡πœ‡ 𝐽𝐽 = 2πœ‹πœ‹ πœ–πœ– πœ•πœ• 𝑏𝑏 Gauge field and spin connection couples in the same way to As th𝛿𝛿e𝐴𝐴 mπœ‡πœ‡ agnetic field accumulaπœ”πœ”tπœ‡πœ‡es the electric charge, π½π½πœ‡πœ‡ the curvature accumulates the electric charge.

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(necessary for geometric responses of FQH states). More precisely, composite particle theories fail to capture : Wen-Zee term and Hall Viscosity.
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UIUC-PI Joint Conference, 2014-10-16 Universal Geometric Response of Abelian and Non-Abelian FQH states Gil Young Cho Univ. of Illinois at Urbana-Champaign Main references! Based on the works/understanding from : [1] GYC, Y. You, and E. Fradkin, PRB, 90, 115139 (2014) [2] GYC, O. Parrikar, Y. You, R. Leigh, and T. Hughes, arxiv:1407.5637 [3] Y. You, GYC, and E. Fradkin, arxiv:1410.3390 [4] A. Gromov, GYC, Y. You, A. Abanov, and E. Fradkin, (in preparation) Contents 1. Geometric Responses of Abelian and Non-abelian FQHEs = Hall viscosity, Wen-Zee term, and Gravitational Chern-Simons term 2. Puzzles from (naΓ―ve) composite particle theories = Failure of naΓ―ve composite particle theories to capture the geometric responses 3. Revisiting Flux attachment and composite particle theories = correcting flux attachment in curved space 4. Projective Parton Approaches Goal of this Talk Composite Particle Theories (composite boson/fermion) are (1) successful in describing ground state properties and excitations of FQHEs (2) Generating Wen’s hydrodynamic theory with the correct (K, q) 1 1 𝐼𝐼 𝐽𝐽 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ (3) But fail to predict ``spin ” (necessary for geometric responses of FQH states) 0 𝑰𝑰𝑰𝑰 πœ‡πœ‡ 𝜈𝜈 πœ‡πœ‡ 𝑰𝑰 πœ‡πœ‡ 𝜈𝜈 πœ‡πœ‡ 𝐿𝐿 = πœŒπœŒΜ…π›Ώπ›Ώπ΄π΄ βˆ’ 4πœ‹πœ‹ 𝑲𝑲 𝑏𝑏 πœ•πœ• 𝑏𝑏 πœ–πœ– + 2πœ‹πœ‹ 𝒒𝒒 𝑏𝑏 πœ•πœ• 𝛿𝛿𝐴𝐴 πœ–πœ– + β‹― 𝐽𝐽 𝑆𝑆 More precisely, composite particle theories fail to capture : Wen-Zee term and Hall Viscosity Can we correct composite particle theories to derive the terms? Before jumping into the details.. 0. Review : Hydrodynamic Theory Ref. Wen’s textbook or Fradkin’s textbook 1 1 𝐼𝐼 𝐽𝐽 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ (1) , e0xternal prob𝑰𝑰𝑰𝑰e EMπœ‡πœ‡ g𝜈𝜈augπœ‡πœ‡e field 𝑰𝑰 πœ‡πœ‡ 𝜈𝜈 πœ‡πœ‡ 𝐿𝐿 = πœŒπœŒΜ…π›Ώπ›Ώπ΄π΄ βˆ’ 4πœ‹πœ‹ 𝑲𝑲 𝑏𝑏 πœ•πœ• 𝑏𝑏 πœ–πœ– + 2πœ‹πœ‹ 𝒒𝒒 𝑏𝑏 πœ•πœ• 𝛿𝛿𝐴𝐴 πœ–πœ– + β‹― πœ‡πœ‡ 𝛿𝛿𝐴𝐴 (2) , hydrodynamic gauge field, quasi-particle current: 1 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝑏𝑏 πœ‡πœ‡ 𝐽𝐽 πœ‡πœ‡ = 2πœ‹πœ‹ πœ–πœ–πœ‡πœ‡πœˆπœˆπœ‡πœ‡ πœ•πœ•πœˆπœˆπ‘π‘ πœ‡πœ‡ (3) , integer matrix, (encoding topological information) 𝐼𝐼𝐽𝐽 𝐾𝐾 ∈ 𝑍𝑍 ex, GS degeneracy on genus-g manifold 𝑔𝑔 ex, statistical angle of and is 𝐼𝐼𝐽𝐽 det 𝐾𝐾 𝑇𝑇 βˆ’1 1 2 12 1 2 (4) , integer vector, ch𝑙𝑙arge ve𝑙𝑙ctor πœƒπœƒ = 2πœ‹πœ‹ 𝑙𝑙 𝐾𝐾 𝑙𝑙 𝐼𝐼 π‘žπ‘ž ∈ 𝑍𝑍 = EM Hall response by integrating out 1 𝑇𝑇 βˆ’1 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 𝐿𝐿 = π‘žπ‘ž 𝐾𝐾 π‘žπ‘ž Γ— 4πœ‹πœ‹ πœ–πœ– π›Ώπ›Ώπ΄π΄πœ‡πœ‡πœ•πœ•πœˆπœˆπ›Ώπ›Ώπ΄π΄πœ‡πœ‡ 𝑏𝑏 πœ‡πœ‡ Contents 1. Geometric Responses of Abelian and Non-abelian FQHEs = Hall viscosity, Wen-Zee term, and Gravitational Chern-Simons term 2. Puzzles in naΓ―ve composite particle theories = Failure of naΓ―ve composite particle theories to capture the geometric responses 3. Revisiting Flux attachment and composite particle theories = correcting flux attachment in curved space 4. Projective Parton Approaches Part 1. Geometric Response of FQHE Imagine that the lattice is distorted & fluctuating and electrons are on top of the lattice… e e e e e e e e e Interesting Physics? Interesting Physics? 1. Wen-Zee term 2. Hall Viscosity Wen-Zee term : QH fluids see the curvature as ``magnetic flux’’ (1) QH fluids collects electric charge at the magnetic flux Wen-Zee 1992 Magnetic field Quantum Hall fluid Electric current Imagine : Induced field! ( ) Thus quantum Hall fluids accumulate the charge at the mag𝑑𝑑netic flux ! 𝑬𝑬 πœ•πœ• 𝐡𝐡 = 𝛻𝛻 Γ— 𝐸𝐸 (2) On the space with curvature… Collects or depletes charge when the curvature is present Wen-Zee term Wen-Zee 1992 Wen & Zee supplemented a phenomenological term into the hydrodynamic theory 1 1 𝑺𝑺𝑰𝑰 𝐼𝐼 𝐽𝐽 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝑰𝑰 𝝁𝝁𝝂𝝂𝝀𝝀 0 𝐼𝐼𝐽𝐽 πœ‡πœ‡ 𝜈𝜈 πœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡ 𝜈𝜈 πœ‡πœ‡ 𝝁𝝁 𝝂𝝂 𝝀𝝀 𝐿𝐿 = πœŒπœŒΜ…π›Ώπ›Ώπ΄π΄ βˆ’ 4πœ‹πœ‹ 𝐾𝐾 𝑏𝑏 πœ•πœ• 𝑏𝑏 πœ–πœ– + 2πœ‹πœ‹ π‘žπ‘ž 𝑏𝑏 πœ•πœ• 𝛿𝛿𝐴𝐴 πœ–πœ– + πŸπŸπŸπŸπ’ƒπ’ƒ 𝝏𝝏 𝝎𝝎 𝝐𝝐 β‹― with the spin vector (remember that ) 1 𝐼𝐼 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 𝑰𝑰 Here = spin connectioπœ‡πœ‡n describing 𝜈𝜈curπœ‡πœ‡vature of the background geo𝑺𝑺metry 𝐽𝐽 = 2πœ‹πœ‹ πœ–πœ– πœ•πœ• 𝑏𝑏 πœ‡πœ‡ πœ”πœ”i.e., Local curvature = (similar to local magnetic field ) βƒ— 𝛻𝛻 Γ— πœ”πœ” B = 𝛻𝛻 Γ— 𝐴𝐴 1 𝐼𝐼 𝐽𝐽 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡ 𝐿𝐿 = πœŒπœŒΜ…π›Ώπ›Ώπ΄π΄0 βˆ’ 4πœ‹πœ‹ 𝐾𝐾𝐼𝐼𝐽𝐽𝑏𝑏 πœ‡πœ‡πœ•πœ•πœˆπœˆπ‘π‘ πœ‡πœ‡πœ–πœ– + π‘žπ‘žπΌπΌπ½π½ πœ‡πœ‡π›Ώπ›Ώπ΄π΄ + 𝑠𝑠𝐼𝐼𝐽𝐽 πœ‡πœ‡πœ”πœ” β‹― the conserved particle current 1 𝐼𝐼 πœ‡πœ‡πœˆπœˆπœ‡πœ‡ 𝐼𝐼 πœ‡πœ‡ 𝜈𝜈 πœ‡πœ‡ 𝐽𝐽 = 2πœ‹πœ‹ πœ–πœ– πœ•πœ• 𝑏𝑏 Gauge field and spin connection couples in the same way to As th𝛿𝛿e𝐴𝐴 mπœ‡πœ‡ agnetic field accumulaπœ”πœ”tπœ‡πœ‡es the electric charge, π½π½πœ‡πœ‡ the curvature accumulates the electric charge.

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(necessary for geometric responses of FQH states). More precisely, composite particle theories fail to capture : Wen-Zee term and Hall Viscosity.
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