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 (naïve) composite particle theories = Failure of naïve 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 𝜇𝜇𝜈𝜈𝜇𝜇 𝐼𝐼 𝜇𝜇𝜈𝜈𝜇𝜇 (3) But fail to predict ``spin ” (necessary for geometric responses of FQH states) 0 𝑰𝑰𝑰𝑰 𝜇𝜇 𝜈𝜈 𝜇𝜇 𝑰𝑰 𝜇𝜇 𝜈𝜈 𝜇𝜇 𝐿𝐿 = 𝜌𝜌̅𝛿𝛿𝐴𝐴 − 4𝜋𝜋 𝑲𝑲 𝑏𝑏 𝜕𝜕 𝑏𝑏 𝜖𝜖 + 2𝜋𝜋 𝒒𝒒 𝑏𝑏 𝜕𝜕 𝛿𝛿𝐴𝐴 𝜖𝜖 + ⋯ 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 𝜇𝜇𝜈𝜈𝜇𝜇 𝐼𝐼 𝜇𝜇𝜈𝜈𝜇𝜇 (1) , e0xternal prob𝑰𝑰𝑰𝑰e EM𝜇𝜇 g𝜈𝜈aug𝜇𝜇e field 𝑰𝑰 𝜇𝜇 𝜈𝜈 𝜇𝜇 𝐿𝐿 = 𝜌𝜌̅𝛿𝛿𝐴𝐴 − 4𝜋𝜋 𝑲𝑲 𝑏𝑏 𝜕𝜕 𝑏𝑏 𝜖𝜖 + 2𝜋𝜋 𝒒𝒒 𝑏𝑏 𝜕𝜕 𝛿𝛿𝐴𝐴 𝜖𝜖 + ⋯ 𝜇𝜇 𝛿𝛿𝐴𝐴 (2) , hydrodynamic gauge field, quasi-particle current: 1 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝑏𝑏 𝜇𝜇 𝐽𝐽 𝜇𝜇 = 2𝜋𝜋 𝜖𝜖𝜇𝜇𝜈𝜈𝜇𝜇 𝜕𝜕𝜈𝜈𝑏𝑏 𝜇𝜇 (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 −1 1 2 12 1 2 (4) , integer vector, ch𝑙𝑙arge ve𝑙𝑙ctor 𝜃𝜃 = 2𝜋𝜋 𝑙𝑙 𝐾𝐾 𝑙𝑙 𝐼𝐼 𝑞𝑞 ∈ 𝑍𝑍 = EM Hall response by integrating out 1 𝑇𝑇 −1 𝜇𝜇𝜈𝜈𝜇𝜇 𝐼𝐼 𝐿𝐿 = 𝑞𝑞 𝐾𝐾 𝑞𝑞 × 4𝜋𝜋 𝜖𝜖 𝛿𝛿𝐴𝐴𝜇𝜇𝜕𝜕𝜈𝜈𝛿𝛿𝐴𝐴𝜇𝜇 𝑏𝑏 𝜇𝜇 Contents 1. Geometric Responses of Abelian and Non-abelian FQHEs = Hall viscosity, Wen-Zee term, and Gravitational Chern-Simons term 2. Puzzles in naïve composite particle theories = Failure of naïve 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 mag𝑡𝑡netic flux ! 𝑬𝑬 𝜕𝜕 𝐵𝐵 = 𝛻𝛻 × 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 𝜇𝜇𝜈𝜈𝜇𝜇 𝐼𝐼 𝜇𝜇𝜈𝜈𝜇𝜇 𝑰𝑰 𝝁𝝁𝝂𝝂𝝀𝝀 0 𝐼𝐼𝐽𝐽 𝜇𝜇 𝜈𝜈 𝜇𝜇 𝐼𝐼 𝜇𝜇 𝜈𝜈 𝜇𝜇 𝝁𝝁 𝝂𝝂 𝝀𝝀 𝐿𝐿 = 𝜌𝜌̅𝛿𝛿𝐴𝐴 − 4𝜋𝜋 𝐾𝐾 𝑏𝑏 𝜕𝜕 𝑏𝑏 𝜖𝜖 + 2𝜋𝜋 𝑞𝑞 𝑏𝑏 𝜕𝜕 𝛿𝛿𝐴𝐴 𝜖𝜖 + 𝟐𝟐𝟐𝟐𝒃𝒃 𝝏𝝏 𝝎𝝎 𝝐𝝐 ⋯ with the spin vector (remember that ) 1 𝐼𝐼 𝜇𝜇𝜈𝜈𝜇𝜇 𝐼𝐼 𝑰𝑰 Here = spin connectio𝜇𝜇n describing 𝜈𝜈cur𝜇𝜇vature of the background geo𝑺𝑺metry 𝐽𝐽 = 2𝜋𝜋 𝜖𝜖 𝜕𝜕 𝑏𝑏 𝜇𝜇 𝜔𝜔i.e., Local curvature = (similar to local magnetic field ) ⃗ 𝛻𝛻 × 𝜔𝜔 B = 𝛻𝛻 × 𝐴𝐴 1 𝐼𝐼 𝐽𝐽 𝜇𝜇𝜈𝜈𝜇𝜇 𝐼𝐼 𝜇𝜇 𝐼𝐼 𝜇𝜇 𝐿𝐿 = 𝜌𝜌̅𝛿𝛿𝐴𝐴0 − 4𝜋𝜋 𝐾𝐾𝐼𝐼𝐽𝐽𝑏𝑏 𝜇𝜇𝜕𝜕𝜈𝜈𝑏𝑏 𝜇𝜇𝜖𝜖 + 𝑞𝑞𝐼𝐼𝐽𝐽 𝜇𝜇𝛿𝛿𝐴𝐴 + 𝑠𝑠𝐼𝐼𝐽𝐽 𝜇𝜇𝜔𝜔 ⋯ the conserved particle current 1 𝐼𝐼 𝜇𝜇𝜈𝜈𝜇𝜇 𝐼𝐼 𝜇𝜇 𝜈𝜈 𝜇𝜇 𝐽𝐽 = 2𝜋𝜋 𝜖𝜖 𝜕𝜕 𝑏𝑏 Gauge field and spin connection couples in the same way to As th𝛿𝛿e𝐴𝐴 m𝜇𝜇 agnetic field accumula𝜔𝜔t𝜇𝜇es the electric charge, 𝐽𝐽𝜇𝜇 the curvature accumulates the electric charge.