“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. Q: When is a “wavefunction” 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. Risultato della ricerca immagini di Google per http://upload.wikimedia.org/wikiped... http://www.google.it/imgres?imgurl=http://upload.wikimedia.org/wikipedia/comm... Werner Karl Heisenberg Sito web per questa immagine Werner Karl Heisenberg Da Wikipedia, l'enciclopedia libera. it.wikipedia.org Werner Karl Heisenberg (Würzburg, 5 Dimensione intera dicembre 1901 – Monaco di Baviera, 1º febbraio 220 × 349 (Stesse dimensioni), 13KB 1976) è stato un fisico tedesco. Ottenne il Premio Altre dimensioni Nobel per la Fisica nel 1932 ed è considerato uno Ricerca tramite immagine dei fondatori della meccanica quantistica. Immagini simili Indice Tipo: JPG 1 Meccanica quantistica Le immagini potrebbero essere soggette a 2 Il lavoro durante la guerra copyright. 3 Bibliografia 3.1 Autobiografie 3.2 Opere in italiano 3.3 Articoli di stampa Erwin_schrodinger1.jpg (JPEG Image, 485 × 560 pixels) - Scaled (71%) http://www.camminandoscalzi.it/wordpress/wp-content/uploads/2010/09/Erwin_sc... 4 Curiosità 5 Voci correlate 6 Altri progetti 7 Collegamenti esterni Meccanica quantistica Werner Karl Heisenberg Schrödinger vs HeisenbergQuando era studente, incontrò per la fisica 1932 Gottinga nel 1922. Ciò permise lo sviluppo di una fruttuosa collaborazione tra i due. Heisenberg ebbe l'idea della , la prima formalizzazione della meccanica quantistica, nel principio di indeterminazione, introdotto nel 1927, afferma che la misura simultanea di due variabili coniugate, come posizione e quantità di moto oppure energia e tempo, non può essere compiuta senza un'incertezza ineliminabile. Assieme a Bohr, formulò l' della meccanica quantistica. Ricevette il Premio Nobel per la fisica "per la creazione della meccanica • Schrödinger’s picture describes thequantistica, system la cui applicazione, by tra le altre cose, ha portato alla scoperta delle forme a wavefunction ( ) in real spaceallotrope dell'idrogeno". � r Il lavoro durante la guerra Heisenberg’s picture describes the Lasystem fissione nucleare venneby scoperta a in Germania nel 1939. Heisenberg rimase in • Germania durante la seconda guerra mondiale, lavorando sotto il regime nazista. Guidò il state ｜ in Hilbert space programma nucleare tedesco, ma i limiti della sua collaborazione sono controversi. �⟩ Rivelò l'esistenza del programma a Bohr durante un colloquio a Copenaghen nel settembre 1941. Dopo l'incontro, la lunga amicizia tra Bohr e Heisenberg terminò They are only equivalent if the basisbruscamente. r Bohr of si unì in seguito al progetto Manhattan. • Si è speculato sul fatto che Heisenberg avesse degli scrupoli morali e cercò di rallentare il progetto.| Heisenbergi stesso tentò di sostenere questa tesi. Il libro Heisenberg's War di states in real-space are orthogonal:Thomas Power e l'opera teatrale "Copenhagen" di Michael Frayn adottarono questa interpretazione. Nel febbraio 2002, emerse una letterathis scritta fails da Bohr ad Heisenberg nel 1957 (ma mai r r0 spedita)=0: vi si legge che Heisenberg, nella conversazione con Bohr del 1941, non espresse (r)= r requires alcun problema morale riguardo inal progetto a quantum di costruzione della bomba; si deduce inoltre h | iche Heisenberg aveva speso i precedenti due anni lavorandovi quasi esclusivamente, h | i (r =convintor0 )che la bomba avrebbe decisogeometry l'esito della guerra. 6 1 of 1 5/21/12 12:25 AM 1 of 1 5/21/12 12:27 AM 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′. ? (x)= x | i h | i Heisenberg = Schrödinger only equivalent if this fails in a x x0 =0for x = x0 quantum geometry h | i 6 • 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, x0)= x x0 has null eigenstates h | i S(x, x0) (x0)=0 Xx0 (so the basis cannot be reorthogonalized) • If the Schrödinger basis is on a lattice, so |x⟩ is normalizable 2 2 =0 x = x0 d(x, x0) =1 S(x, x0) x = x0 − | | =1 6 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” H = V (x) x x x x0 = δ(x, x0) | ih | h | i6 x X • Fractional quantum Hall effect in 2D electron gas in high magnetic field (filled Landau levels) 1/3 3 z 2/4`2 = (z z ) e−| i| B L i − j i<j i Y Y1 ⌫ = 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 ތࠓ goes into continuum 0.7 0.6 (2 quasiparticle 0.5 bosonic “roton” E 0.5 + 2 quasiholes) 0.4 fermionic 0.3 “roton” “roton” 0.2 2 Moore-Read “kF ” ⌫ = 0.1 4 0 0 0 0.5 1 1.5 2 2.5 3 3.5 0 1 2k k lB gap incompressibility B Collective mode with short-range V1 Collective mode with short-range three-body pseudopotential, 1/3 filling (Laughlin state is pseudopotential, 1/2 filling (Moore-Read state is exact ground state in that case) exact ground state in that case) momentum ħk of a quasiparticle-quasihole pair is • b proportional to its electric dipole moment pe ~ka = abBpe gap for electric dipole excitations is a MUCH stronger condition than charge gap: doesn’t transmit pressure! fractional-charge, fractional statistics vortices 1 z⇤zi m 4`2 i = (z w ) (z z ) e− B i − ↵ i − j i,↵ i<j i Y Y Y Chiral edge states at edge of finite droplet of fluid (Halperin, Wen) charge -e/m statisticsθ time = π/m e.g., m=3 2iθ eiθ e • Non-commutative geometry of Landau-orbit guiding centers ~r displacement of electron from origin shape of orbit around guiding ~ displacement of guiding center from origin center is fixed by the cyclotron R effective mass tensor ~ displacement of electron relative to Rc guiding center of Landau orbit e− ~ Rc r~ = R~ + R~ c ⇥ ~r guiding [rx,ry]=0 ~ R center classical geometry x y 2 x y 2 O [R ,R ]= i` [Rc ,Rc ]=+i`B − B Landau orbit quantum geometry (harmonic oscillator) guiding centers commute with Landau radii [Ra,Rb]=0 (a, b x, y ) c 2{ } classical electron coordinate r~ = R~ + R~ c The one-particle Hilbert-space factorizes H = H¯ H GC ⌦ c space isomorphic space isomorphic to phase space in which to phase space in which the guiding-centers act the Landau orbit radii act [Rx,Ry]= i`2 [Rx,Ry]=+i`2 − B c c B • FQHE physics is *COMPLETELY* defined in the many-particle generalization (coproduct) of HGC Once H c 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: 1 1 a† = z⇤ @ a¯† = z¯⇤ @z¯ 2 − z 2 − 1 z z¯ a = z + @z $ 1 2 ⇤ a¯ = 2 z¯ + @z¯⇤ Landau-level Guiding-center ladder operators ladder operators Gaussian lowest-weight state usual identification is 1 z⇤z 0(z,z⇤)=e− 2 z¯ = z⇤ a¯ 0(z,z⇤)=0 a 0(z,z⇤)=0 action of guiding-center raising operators on LLL states 1 1 a¯† = z @ a¯ = z + @ 2 − z⇤ 2 ⇤ z a¯†f(z) 0(z,z⇤)=zf(z) 0(z,z⇤) • Heisenberg form of Laughlin state (not “wavefunction”) a¯ ¯ =0 ai 0 =0 i| 0i | i 1/q q = (¯a† a¯†) ¯ ( ) | L i 0 i − j | 0i1 ⌦ | 0i i<j H H Y 2 c ⌘ H H¯ @ 2 GC ⌘ A 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) (0) =0 | g i za¯ †(g) z¯⇤a(g) (¯z) = e − (0) | g i | g i • This is a non-orthogonal overcomplete basis S(¯z,z¯0)= (¯z) (¯z0) h g | g i • non-zero