LCDQM-15 Low Dimensional Quantum Condensed Matter workshop, University of Amsterdam, July 2, 2015 Geometrodynamics of the Fractional Quantum Hall Effect F. Duncan M. Haldane Princeton University • An effective field theory for the incompressible FQH fluid that describes its gap and long-wavelength collective excitations • The essential local fields are a flow- velocity, a polarization, and an emergent metric.
• in case I run out of time, I will quickly show the effective action that is finally obtained.
• three dynamical ingredients g ab , v a , P a : • a “dynamic emergent 2D spatial metric” g ab ( x , t ) with g ≡ det g , and Gaussian curvature current g = ✏ µ νλ @ ν ! λ ( x , t ) J µ • a flow velocity field v a ( x , t ) • an electric polarization field P a ( x , t ) • a composite boson current J µ b = √ g ( x , t ) ( δ µ 0 + v a ( x , t ) δ µ a ) here a is a 2D spatial index, and µ is a (2+1D) space-time index. The fluid motion is non-relativistic relative to the preferred inertial rest frame of the crystal background
σ H = ( pe ) 2 • effective bulk action: 2 π ~ K U (1) Chern-Simons field Z U (1) condensate field d 2 xdt L 0 − H S = “spin connection” L 0 = ~ 4 ⇡ ✏ µ νλ ( Kb µ @ ν b λ + �! µ @ ν ! λ ) of the metric + J µ b ( ~ ( ∂ µ ϕ − b µ − S ω µ ) + peA µ ) H = √ g " ( v , B ) − U ( g, B, P ) − ( E a + ✏ ab v b B ) P a � � kinetic energy metric-dependent of flow correlation energy
In 1983, Laughlin’s wavefunction was a unexpected gift to physics that seemed to emerge fully-formed from the void ..... (actually, this seems to be its genesis..) PRB 256 citations Ψ = Quantized motion of three two-dimensional electrons in a strong Y ( i < z magnetic field (/prb/abstract/10.1103/PhysRevB.27.3383) j i × − Y z ) j q e − 1 R. B. Laughlin i 2 z ∗ I i z i Phys. Rev. B 27 , 3383 (1983) - Published 15 March 1983 w i ! n We have found a simple, exact solution of the Schrödinger equation for three two-dimensional electrons in a strong magnetic field, given the assumption that they lie in a single Landau level. We find that the interelectronic spacing has characteristic values, not dependent on the form of the interaction, which change discontinuously as pressure is applied, and that the system has characteristic excitation energies 0.03 e 2 of approximately , where is the magnetic length. a 0 a 0 7/2/15, 3:03 A • It was quickly confirmed to be the solution to the FQHE mystery • but WHY it works has never really been explained
• Essentially all the key ideas in the FQHE have emerged as interpretions and generalizations of Laughlin’s wavefunction: topological order flux attachment braiding statistics shift composite particles fractional charge chiral cft edge states conformal block wavefunctions • I will describe a new insight from a feature of Laughlin’s wavefunction that remained FDMH undetected for 25 years: arXiv:0906-1854 emergent dynamical geometry PRL:107 116801 (2011) arXiv:1112-0990
2D Galilean-invariant Landau levels, uniform B a † = 1 2 z ∗ − ∂ ∂ z | p − e A | 2 a † a + aa † � = 1 � ∂ a = 1 2 ~ ω c 2 z + ∂z ∗ 2 m [ a, a † ] = 1 • mapping to the complex plane z = ( x + iy ) √ 2 ` B • Lowest Landau level wavefunctions holomorphic Ψ = f ( z ) e − 1 2 z ∗ z a Ψ = 0
• filled Lowest Landau level (vandermonde determinant) uncorrelated e − 1 Y Y 2 z ∗ i z i Ψ = ( z i − z j ) E n i<j i } empty • Laughlin state ( m > 1 ) partial filling ν highly correlated e − 1 Y ( z i − z j ) m Y 2 z ∗ Ψ m i z i L = i<j i ν = 1 bosons for even m, fermions for odd m m
some incorrect “conventional wisdom” about the Laughlin state e − 1 Y ( z i − z j ) m Y 2 z ∗ Ψ m i z i L = i<j i • “it is holomorphic (times a Gaussian) because it is a ✘ wrong lowest Landau level state” • “It is 2D rotationally invariant because the Landau orbits ✘ wrong are circular”. (angular momentum: ) L = 1 2 mN ( N − 1) ✘ wrong • “It has no continuously-variable variational parameters”. The probability that Laughlin himself may have believed these things does not make them true!
• Guiding centers of Landau levels obey the algebra [ R a , ( p − eA ) b ] = 0 [ R a , R b ] = − i ` 2 B ✏ ab Antisymmetric 2D Levi-Civita symbol b † = R x + iR y ∂ = 1 a † = 1 2 z ∗ − ∂ 2 z − ∂ z ∗ ∂z √ 2 ` B ∂ a = 1 b = R x − iR y 2 z + 2 z ∗ + ∂ = 1 ∂z ∗ ∂z √ 2 ` B 2 z ∗ z = zf ( z ) e − 1 action on LLL states b † f ( z ) e − 1 2 z ∗ z
• A new look at the Laughlin state: ( b † i � b † Y | Ψ m j ) m | Ψ 0 (˜ L (˜ g ) i = g ) i b i | Ψ 0 (˜ g ) i = 0 i<j ε ( p i � a A ( x i )) | Ψ 0 (˜ g ) i = E n | Ψ 0 (˜ g ) i unimodular g ab Λ ab = 1 metric ( b † i b i + b i b † X L (˜ g ) = ˜ i ) 2 det ˜ g = 1 i 2 mN 2 � 1 g ) | Ψ m 2 ( m � 1) N ) | Ψ m L i = ( 1 L ((˜ L i extensive superextensive “geometrical spin” “statistical spin” generator of linear 1 Λ ab = X ( R a i R b i + R b i R a i ) area-preserving 4 ` 2 B i distortions
• Changes from the original Laughlin formulation: no need for Galilean invariance (not a property of electrons in solids) δ ab p a p b → ε ( p ) = ε ( − p ) any Landau level will do, not just 2 m the “lowest” the Laughlin state is PARAMETRIZED δ ab → ˜ by a unimodular emergent 2D spatial g ab metric that SHOULD NOT be identified with the Euclidean metric of Galilean invariance g ab is also NOT related to an intrinsic Riemannian metric of the surface ˜ on which the particles move: (This would essentially just be a local form of Galilean invariance on curved generalizations of the 2D plane, where the kinetic energy is identified with the Laplace-Beltrami operator.).
• Lie Algebra (SL(2,R)) ✏ ac Λ bd + ✏ ad Λ bc + ✏ bc Λ ad + ✏ bd Λ ac � [ Λ ab , Λ cd ] = 1 � 2 i • quadratic Casimir: [ Λ ab , C 2 ] = 0 , 2 ✏ ac ✏ bd Λ ab Λ cd C 2 = det Λ = 1 • unitary deformation operator U ( β ) = exp i β ab Λ ab β ab = β ba , real det β > 0 : pseudo rotation (elliptic) det β = 0 : shear (parabolic) det β < 0 : squeeze (hyperbolic)
• Laughlin states with different intrinsic metrics are related by transformations | Ψ m g 0 ) i = U ( β ) | Ψ m L (˜ L (˜ g ) i g 0 = A ( β )˜ gA T ( β ) ˜ det A ( β ) = 1 SL (2 , R ) • In contrast (in the absence of a boundary), the filled Landau level is not parametrized by a metric, and is left invariant by the action of U ( β )
• The Laughlin state is parametrized by a unimodular metric: what is its physical meaning? correlation holes in two states with different metrics • In the ν = 1/3 Laughlin state, each electron sits in a correlation hole with an area containing 3 flux quanta. The metric controls the shape of the correlation hole. • In the ν = 1 filled LL Slater-determinant state, there is no correlation hole (just an exchange hole), and this state does not depend on a metric
• Q: If we use the Laughlin state as a variational approximation to a true ground state, what determines the choice of the metric parameter? • A: it should be chosen to minimize the correlation energy
• The generic Hamiltonian has 2D translation and inversion symmetry, but does not have any O(2) rotation symmetry X H = V n ( R i − R j ) R i 7! a ± R i i<j • The uniform incompressible quantum Hall states do not break inversion or translation symmetry, and have no electric polarization. Z d 2 q ` 2 d 2 r 0 Z V ( r 0 ) e i q · ( r � r 0 ) | f n ( q ) | 2 B V n ( r ) = 2 ⇡` 2 2 ⇡ B Coulomb interaction Landau level regular at potential, modified at short form factor short distance distance by finite layer width (property of 2D band structure) (property of 3D dielectric tensor)
• If the repulsive short-distance interaction has rotational symmetry with respect to a metric, then the unimodular metric parameter that minimizes the correlation energy will be proportional to that metric. • The 1/2 (boson) and 1/3 (fermion) Laughlin states with metric are exact zero- ˜ g ab energy ground states of a model interaction V n ( r ) = ( A + Bu ( r )) e − u ( r ) A, B > 0 g ab r a r b / ` 2 u ( r ) = 1 2 ˜ B
• Let be the exact FQH ground state of H | Ψ 0 i β → 0 2 Γ abcd β ab β cd > E 0 h Ψ 0 | U − 1 ( β ) HU ( β ) | Ψ 0 i ! E 0 + 1 • The rank-4 tensor Γ abcd is a kind of “shear modulus” of the FQH fluid. • Girvin MacDonald and Platzman found an inequality equivalent as wavevector k → 0 to S ( k ) ∆ E ( k ) < Γ abcd k a k b k c kd excitation guiding-center energy structure factor ∝ k 4 constant
Recommend
More recommend
