“Reduced Density Matrices in Quantum Physics and Role of Fermionic Exchange Symmetry” : workshop Pauli2016 on Oxford University, April 12-15 2016 Quantum Geometry, Exclusion Statistics, and the Geometry of “Flux Attachment” in 2D Landau levels F. Duncan M. Haldane Princeton University The degenerate partially-filled 2D Landau level is a remarkable environment in which kinetic energy is replaced by "quantum geometry” (or an uncertainty principle) that quantizes the space occupied by the electrons quite differently from the atomic-scale quantization by a periodic arrangement of atoms. In this arena, when the short-range part of the Coulomb interaction dominates, it can lead to “flux attachment”, where a particle (or cluster of particles) exclusively occupies a quantized region of space. This principle underlies both the incompressible fractional quantum Hall fluids and the composite-fermion Fermi liquid states that occur in such systems.
When is a “wavefunction” Q: NOT a wavefunction? A: When it describes a “quantum geometry” • In this case space is “fuzzy”(non-commuting components of the coordinates), and the Schrödinger description in real space (i.e., in “classical geometry”) fails, though the Heisenberg description in Hilbert space survives • The closest description to the classical-geometry Schrödinger description is in a non-orthogonal overcomplete coherent - state basis of the quantum geometry.
antistica Schrödinger vs Heisenberg Werne ncontrò ò permise lo sviluppo di una tra i due. della , la prima , nel principio di indeterm imultanea di due variabili coniugate empo, non può essere compiuta senz • Schrödinger’s picture describes the system by a wavefunction 𝝎 ( r ) in real space • Heisenberg’s picture describes the system by a state ｜ 𝝎⟩ in Hilbert space • They are only equivalent if the basis of | r i states in real-space are orthogonal: this fails h r | r 0 i = 0 ψ ( r ) = h r | ψ i requires in a quantum ( r 6 = r 0 ) geometry
Schrödinger vs Heisenberg and quantum geometry • Schrödinger’s real-space form of quantum mechanics postulates a local basis of simultaneous eigenstates |x ⟩ of a commuting set of projection operators P(x), where P(x)P(x ′ ) = 0 for x ≠ x ′ . | Ψ i Ψ ( x ) = h x | Ψ i ? = Heisenberg Schrödinger only equivalent if this fails in a h x | x 0 i = 0 for x 6 = x 0 quantum geometry
• In “classical geometry” particles move from x to x’ because they have kinetic energy • In “quantum geometry”, they move because the states |x ⟩ and |x’ ⟩ are not only non- orthogonal, but overcomplete: In this case the positive Hermitian operator S ( x , x 0 ) = h x | x 0 i has null eigenstates X S ( x , x 0 ) ψ ( x 0 ) = 0 x 0 (so the basis cannot be reorthogonalized)
• If the Schrödinger basis is on a lattice, so |x ⟩ is normalizable x = x 0 d ( x , x 0 ) 2 = 1 − | S ( x , x 0 ) | 2 = 0 x 6 = x 0 = 1 Hilbert-Schmidt distance (trivial distance measure) In this case kinetic energy (Hamiltonian hopping matrix elements) sews the lattice together • In a quantum geometry there is a non-trivial Hilbert-Schmidt distance between (coherent) states on different lattice sites, and the Hamiltonian appears “local” X h x | x 0 i 6 = δ ( x , x 0 ) H = V ( x ) | x ih x | x
• Fractional quantum Hall effect in 2D electron gas in high magnetic field (filled Landau levels) Ψ 1 / 3 e − | z i | 2 / 4 ` 2 Y ( z i − z j ) 3 Y = B L i<j i ν = 1 3 • Laughlin (1983) found the wavefunction that correctly describes the 1/3 FQHE , and got Nobel prize, • Its known that it works, (tested by finite-size numerical diagonalization) but WHY it works has never really been satisfactorily explained!
�� 0.9 laughlin 10/30 0.8 goes into continuum 0.7 (2 quasiparticle 0.6 0.5 bosonic “roton” + 2 quasiholes) E 0.5 0.4 “roton” fermionic “roton” 0.3 ν = 2 “ k F ” 0.2 Moore-Read 4 0.1 0 1 2 0 k l B 0 0 0.5 1 1.5 2 2.5 3 3.5 k � B gap incompressibility Collective mode with short-range three-body Collective mode with short-range V 1 pseudopotential, 1/2 filling (Moore-Read state is pseudopotential, 1/3 filling (Laughlin state is exact ground state in that case) exact ground state in that case) • momentum ħ k of a quasiparticle-quasihole pair is proportional to its electric dipole moment p e ~ k a = � ab Bp b e gap for electric dipole excitations is a MUCH stronger condition than charge gap: doesn’t transmit pressure!
fractional-charge, fractional statistics vortices 1 z ∗ i z i Y Y ( z i − z j ) m Y − 4 ` 2 Ψ = ( z i − w α ) B e i, α i<j i Chiral edge states at edge of finite droplet of fluid (Halperin, Wen) charge -e/m statistics θ time = π /m e.g., m=3 e 2 i θ e i θ
• Non-commutative geometry of Landau-orbit guiding centers ~ displacement of electron from origin r ~ shape of orbit around guiding displacement of guiding center from origin R center is fixed by the cyclotron displacement of electron relative to ~ effective mass tensor R c guiding center of Landau orbit e − ~ r = ~ R + ~ R c ~ R c × ~ guiding [ r x , r y ] = 0 r ~ center R classical geometry [ R x c , R y c ] = + i ` 2 O [ R x , R y ] = − i ` 2 B B Landau orbit quantum geometry (harmonic oscillator) guiding centers commute with Landau radii [ R a , R b c ] = 0 ( a, b ∈ { x, y } )
r = ~ R + ~ classical electron coordinate ~ R c The one-particle Hilbert-space factorizes ¯ H = H GC ⊗ H c space isomorphic space isomorphic to phase space in which to phase space in which the guiding-centers act the Landau orbit radii act [ R x , R y ] = − i ` 2 [ R x c , R y c ] = + i ` 2 B B • FQHE physics is *COMPLETELY* defined in the many-particle generalization (coproduct) of H GC H c Once is discarded, the Schrödinger picture is no longer valid!
Previous hints that the Laughlin “wavefunction” should not be interpreted as a wavefunction: • Laughlin states also occur in the second Landau level, and in graphene, and more recently in simulations of “flat-band” Chern insulators These don’t fit into the original paradigm of the Galileian-invariant Landau level
• First, translate Laughlin to the Heisenberg picture: a † = 1 a † = 1 z ∗ − ∂ ¯ 2 z ∗ − ∂ z ¯ 2 ¯ z z ↔ ¯ z a = 1 2 z + ∂ z ∗ a = 1 z + ∂ ¯ ¯ 2 ¯ z ∗ Landau-level Guiding-center ladder operators ladder operators usual identification is Gaussian lowest-weight state z = z ∗ ¯ ψ 0 ( z, z ∗ ) = e − 1 2 z ∗ z a ψ 0 ( z, z ∗ ) = 0 a ψ 0 ( z, z ∗ ) = 0 ¯ action of guiding-center raising operators on LLL states a † = 1 2 z ∗ + ∂ z a = 1 2 z − ∂ z ∗ ¯ ¯ a † f ( z ) Ψ 0 ( z, z ∗ ) = zf ( z ) Ψ 0 ( z, z ∗ ) ¯
• Heisenberg form of Laughlin state (not “wavefunction”) a i | ¯ Ψ 0 i = 0 a i | Ψ 0 i = 0 ¯ 0 1 | Ψ 1 /q @Y j ) q | ¯ a † a † A ⌦ ( | Ψ 0 i ) L i = (¯ i � ¯ Ψ 0 i i<j ∈ H c ≡ H ¯ ∈ H GC ≡ H Guiding-center Landau-orbit factor (keep) factor (discard) • At this point we discard the Landau-orbit Hilbert space. • The only “memory” of the shape of the Landau orbits is “hidden” in the definition of ¯ a
• guiding-center Coherent states (single particle) a ( g ) | Ψ g (0) i = 0 ¯ za † ( g ) − ¯ z ) i = e ¯ z ∗ a ( g ) | Ψ g (0) i | Ψ g (¯ • This is a non-orthogonal overcomplete basis z 0 ) = h Ψ g (¯ z 0 ) i S (¯ z, ¯ z ) | Ψ g (¯ • non-zero eigenvalues of the positive Hermitian overlap function are holomorphic! Z d ¯ z 0 d ¯ z 0⇤ z 0 ) Ψ (¯ z 0 , ¯ z 0⇤ ) = Ψ (¯ z ⇤ ) S (¯ z, ¯ z, ¯ 2 π z ∗ ) e − 1 z ∗ ¯ 2 ¯ Ψ (¯ z, ¯ z ∗ ) = f (¯ z
The “purified” Laughlin state | Ψ 1 /q j ) q | ¯ a i | ¯ a † a † Y Ψ 0 i = 0 L i = (¯ i � ¯ Ψ 0 i ¯ i<j • This is now defined in the many-particle guiding-center Hilbert space, without reference to any Landau-level structure • What defines ? a † It is the raising ¯ i operator for the a † a † [ L ( g ) , ¯ i ( g )] = ¯ i ( g ) “guiding-center spin” L ( g ) = g ab L ( g ) X R a i R b i 2 ` 2 of particle i B i g ab is a 2x2 positive-definite unimodular (det = 1 ) 2D spatial metric tensor
• The Laughlin state has suddenly revealed its well-kept secret- a hidden geometric degree of freedom! It is parameterized by a unimodular metric g ab ! | Ψ 1 /q j ( g )) q | ¯ a † a † Y L ( g ) i = (¯ i ( g ) � ¯ Ψ 0 ( g ) i i<j a i ( g ) | ¯ ¯ Ψ 0 ( g ) i = 0 • In the naive LLL wavefunction picture, the unimodular metric g ab is fixed to be proportional to the cyclotron effective mass tensor m * ab . • In the reinterpretation it is a free parameter .
This is the entire problem: nothing other than this matters! • H has translation and X H = U ( R i − R j ) inversion symmetry i<j [ R x , R y ] = − i ` 2 B 2 ) , ( R y 1 − R y [( R x 1 + R x 2 )] = 0 like phase-space, [ H, P i R i ] = 0 has Heisenberg uncertainty principle • generator of translations and want to avoid electric dipole moment! this state gap 2 ) , ( R y 1 − R y [( R x 1 − R x 2 )] = − 2 i ` 2 B • relative coordinate of a pair of particles behaves like a single two-particle energy levels particle
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