“Flux Attachment” in 2D Landau Levels

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“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 = (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

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

statisticsθ time = π/m e.g., m=3 2i ei 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

dz¯0dz¯0⇤ S(¯z,z¯0) (¯z0, z¯0⇤)= (¯z,z¯⇤) 2⇡ Z 1 z¯⇤z¯ (¯z,z¯⇤)=f(¯z⇤)e 2 The “puriﬁed” Laughlin state

1/q q = (¯a† a¯†) ¯ a¯i ¯ 0 =0 | L i i j | 0i | i i

• What deﬁnes a¯ i† ? It is the raising [L(g), a¯†(g)] =a ¯†(g) operator for the i i “guiding-center spin” g L(g)= ab RaRb L(g) 2`2 i i of particle i B i X gab 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 gab!

1/q q (g) = (¯a†(g) a¯†(g)) ¯ (g) | L i i j | 0 i i

H = U(R R ) • H has translation and i j i

[(Rx Rx), (Ry Ry)] = 2i`2 gap 1 2 1 2 B • relative coordinate of a pair of particles behaves like a single particle two-particle energy levels Solvable model! (“short-range pseudopotential”) E2 symmetric 1 • 2 (A + B) 1 (r )2 antisymmetric B 2 12 2 (r12) 2`2 U(r12)= A + B 2 e B rest all 0 `B 0 ⇣ ⇣ ⌘⌘

Laughlin state • • m=2: (bosons): all pairs m avoid the symmetric state m E2 = ½(A+B) = a† a† 0 | L i i j | i i

• repulsion of other particles make an attractive potential well strong enough to bind particle solid melts if well is not strong enough to contain zero-point motion (Helium liquids) but no broken symmetry • similar story in FQHE: • “flux attachment” creates correlation hole • defines an emergent e- geometry • potential well must be strong enough to bind electron • continuum model, but similar physics to Hubbard • new physics: Hall viscosity, model geometry...... • elementary unit of the FQHE fluid with ν= p/q is a “composite boson” of p electrons that exclude other electrons from a region with q London (h/e) flux quanta

central orbital central two orbitals occupied, occupied p=1, q=3 next three empty next two ⅓ Laughlin empty ν= ⅓ p=2, q=5 “exclusion ⅖ Hierarchy/Jain statistics” ν= ⅖ ⅓ Laughlin (with different shape) the rule formerly known as ν= ⅓ “odd-denominator”, (but Moore-Read has p=2, q=4)

Statistical selection rule composites ( 1)p ( 1)pq =+1 exchange as ⇥ bosons exchange of Berry phase composite p fermions (exchange of is a boson “exclusion zones”) • The metric (shape of the composite boson) has a preferred shape that minimizes the correlation energy, but ﬂuctuates around that shape • The zero-point ﬂuctuations of the metric are seen as the O(q4) behavior of the “guiding-center structure factor” (Girvin et al, (GMP), 1985) E (distortion)2 /

• The metric has a companion “guiding center spin” that is topologically quantized in incompressible states. total L conﬁguration of ⅓ Laughlin 1 3 5 “elementary droplet” L= 2 2 2 (composite boson) 1 1 0 0 2 3 ⅓ ⅓ ⅓ 2 subtract total L (=Lref) of 1 3 reference conﬁguration s =(2 2 )= 1 (uniform occupation p/q) • Origin of FQHE incompressibility is analogous to origin of Mott-Hubbard gap in lattice systems. • There is an energy gap for putting an extra particle in a quantized region that is already occupied

• On the lattice the “quantized region” is an atomic orbital with a - ﬁxed shape e • In the FQHE only the area of the “quantized region” is ﬁxed. energy gap prevents additional electrons The shape must adjust to from entering the minimize the correlation energy. region covered by the composite boson • The usual “lowest Landau level wavefunction” formalism has 1 2 z⇤z/` (x)=f(z)e 4 B holomorphic function • With a (quasi) periodic boundary condition, this becomes

N 1 z⇤z 4 `2 wi =0 (z,z⇤) (z wi) e B / i i=1 ! Y X Weierstrass (one for each flux quantum passing sigma function* N zeroes through the primitive *(slightly modified from Weierstrass’ original definition region of the pbc) when the pbc lattice is not square or hexagonal) • In the Heisenberg-algebra reinterpretation N = (a† w ) 0 wi =0 one particle | i i i | i i=1 i N =1 Y X • The filled Landau level is filled Level = (ai† aj†)( iai†) 0 | i 0 1 | i N = N i

correlation holes in two states with different metrics

(filled Landau level is a Slater-determinant state with no correlation hole) flux attachment creates a correlation hole that can bind one or more particles into a composite object p particles displacement of charge + q “flux” relative to center of δR (orbitals) x flux attachment gives an electric dipole

“ﬂux attachment” b p¯a = B✏ab(eR ) Has a shape that deﬁnes a metric momentum

correlation energy dispersion "(P ,g) “kinetic energy” = electric polarization energy @" (velocitya)= @p¯a • The key idea is that (at the correct particle density) the Berry phase from motion of the attached vortex cancels the Bohm-Aharonov phase from motion of the charge • This means the Lorentz force is canceled by the Magnus force, and the composite object moves in straight lines like a neutral particle

Bosons Fermions can condense in the p = 0 can form a Fermi sea in (inversion-symmetric) state “momentum” (dipole)space with no electric dipole Berry curvature of the “Flux attachment” of a vortex-like correlation hole modiﬁes the statistics

p particles + q “ﬂux” (orbitals) • inversion symmetry of FQHE : gcd(p,q) = 1 or 2

• exchange phase composite object ( 1)pq⇠p = +1 is boson

-1 for electrons = −1 composite object is fermion e.g., one electron with p = 1, q = 2 1/3 Laughlin state If the central orbital is ﬁlled, (composite boson picture) the next two are empty The composite boson has inversion symmetry e about its center (that couples to It has a “spin” Gaussian curvature 1 3 5 of its metric) 2 2 2 1 1 0 0 ..... L = 2 1 1 1 L = 3 − 3 3 3 ..... − 2 s = 1 the electron excludes other particles from a region containing 3 ﬂux quanta, creating a potential well in which it is bound 2/5 hierarchy/Jain state (composite boson picture)

1 3 5 2 2 2 1 1 0 0 0 ..... L =2 e e 2 2 2 2 2 ..... − L =5 − 5 5 5 5 5 s = 3

g L = ab RaRb 2`2 i i B i X

ab 2 a b 2 ab Jain’s Q = d rr r ⇢(r)=s`Bg two ﬁlled Z “ -levels?” second moment of neutral composite boson charge distribution Model for 1/m CFL states • choose distinct “occupied orbitals” (allowed dipole moments, quantized by the pbc) d ,i=1,...N L { i } 2 { N } 1 2 1 ¯ 2 which minimize di dj = di d N | | 2 | | iquantum number that takes N2 distinct values. There is thus one such conﬁguration per sector of this many-body translational quantum number. • The model 1/m CFL states ( including the boson case m = 1) are ¯ ( zi,zi⇤ , di , w↵ ) det Mij( zk ,dj,dj⇤, d) { } { } { } / i,j { } m 2 ✓ ◆ m N 1 zi⇤zi 4 `2 (z z ) (( z ) w ) e B ⇥ 0 i j 1 i i ↵ i

F. D.M.H and E. H. Rezayi, unpublished; (m=2 case given in Shao et al, PRL 114, 206402 (2015)

mean value of dj • The matrix in the determinant is

d z 1 j⇤ i m 2`2 M ( z ,d ,d⇤, d¯)=e B (z z d + d¯) ij { k} j j i k j k=i Y6 m N ¯ also: w↵ = dj = Nd complex cf dipoles edj • L ↵=1 j=1 (dj is quantized in units ) X X N 0.028677091503 0 1110101000 0.0286770915235 0 0001010111 0.0171543754946 0 1110110000 0.0172785391733 0 0001001111 0.00272205658268 0 1111000001 0.00272205658096 0 0000111110 0.00741749061239 0 1111000010 0.00741749061624 0 0000111101 0.0131254758865 0 1111000100 0.0131430302064 0 0000111011 0.0172785391469 0 1111001000 0.017154375511 0 0000110111 0.0141743825022Now 0 we see 1111010000 that the 0.0141743825338 “Fermi sea”0 is invariant • 0000101111 0.00547651410185 0 1111100000 0.00547651412427under 0uniform translation0000011111 in “dipole space”

# Z_{COM} overlap with PH-conjugate in opposite charge sector 1- overlap 0 0.999998870263 1.1297367517e-06 1 0.999999369175 6.3082507884e-07 Computing ph symmetry 2 0.99999860296 1.39704033186e-06 3 0.99999860296 1.3970403312e-06 (with Scott Geraedts) 4 0.999999369175 6.30825078063e-07 5 0.999998870263 1.12973675237e-06 model state is numerically very 6 0.999999369175 6.30825079173e-07 7 0.99999860296 1.39704032942e-06 close to p-h symmetry 8 0.99999860296 1.39704032909e-06 9 0.999999369175 6.30825078507e-07

py pbc 2⇡~ empty L ﬁlled

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ﬂuctuations of

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!t .U: r--j sa -'- -i-.til ] , ) ⟩ ) a 2 b .) ). c ) ac bd q γ , σ eq q (or ( q ( R ) ac bd ρ G ( ( (11) (13) (15) (12) (16) (17) (14) ρ [ˆ q =0, bc f G q as /s limit, , Ψ F ) γ ′ | , ! .The T q q q cd 2 B ab , ( ( eq ℓ ) ϵ γ ′ b s . f ρ of (2) has .Letˆ 3 . q )] = ) 2 ab γ a + , ⟩ )) ba # γ =0.(Note ν q ( γ ≤ ′ =ˆ 0 ( 4 B γ ) a (the gradient . 2 B , q ) =0.Thefree ab ℓ O ρ ν Ψ ℓ ( T, [ˆ R γ ρ d 2 B | d ac bc q ′ = i s ; )) ( q ℓ + u q F ab e 2 T, G c ρ q d c ) ( γ E ( q → q ab ∂ cd × b ≡ c − γ ∞ bd γ guiding-center shear q a q log ˆ ) s q ) s b a abcd ab 1 2 R q ), where γ q f q with the Hamiltonian T ( − γ q a G 2 B ,and ) B e q − ) ℓ F ν ′ q ν k ′ b ) is the unitary operator ca db abcd ) q q q γ 2 sin ( .The“ G q + ( G ( T, ! abcd ( s ] is the functional ab ( 4 s ( ) ϵ ˜ v ρ U G ′ λ [ˆ H = a 2 B 2 1 cdab ( q 2 1 2 B ℓ e F ′ ( ρ ℓ abcd )+ G e π q ˜ ac bd v q ≤ q 4 G → 2 ) is the equilibrium density-matrix × G 2 B ) = 1 2 q d ν . This gives ) + ℓ i er any further interpretation of q ′ π ′ e q ( ff , is minimized when ˆ cd q )and 4 ,where T, s q & λ ν γ 2 2 B ( 1 )+ ), given by ) ( ( bacd ]=Tr(ˆ T ℓ d s ν γ − q ′ ρ bd f cf π eq G B ( [ˆ ) ( , ϵ ϵ q 2 0 ρ , with γ 1 2 q 2 T, F & E ac = ( and filling factor ( ae ( → d to obtain an upper bound ϵ lim ϵ s λ f U H/k ⟩ aecf ≡ T cd ≡ 0 & − eq γ G ” (per flux quantum) of the state is given by abcd ρ )= ) ), which is is minimized when ξ Ψ )= )ˆ | 2 ab ab df G q q ) γ ϵ γ ( ], where ˆ γ , + γ γ ( ′ ( q exp( ( f be ( eq f q 1 ∞ ϵ U O ( ρ ected term in the free energy is the correlation energy, ρ Assuming only that the ground state Consider the equilibrium state of a system with tem- s [ˆ − s ff of the displacementare field) linearly are related mixed-index byThe tensors the entropy that is elastic left modulusa invariant tensor by thewhich APD, can and be the evaluatedture only in factor terms of the deformed struc- (the momentum current) and strain with Other than noting it was quartic in the small- that in a spatially-covariant formalism, both stress APD corresponding to a shear is energy per flux quantum has the expansion translational invariance, plus inversionhas symmetry vanishing electric (so dipole it surface), moment GMP[1] parallel used the to SMA the variational 2D state to the energyelectric of dipole an moment excitation with momentum perature + parametrized by a symmetric tensor modulus = ∝ and is controlledThen by the the SMA result guiding-center is, shear-modulus. at long wavelengths, at which, for fixed that implements the APD, and where (2). TheF free energy ofZ this state is formally given by = GMP did not not o It can now be seen to have the long-wavelength behavior ) ⟩ 1 ′ T = q 0). β (6) (9) (7) (8) − ( B c )by (10) a ξ )= k † β s q q → c = , spin, λ . − . ≪ . ⟩⟨ ( ) ( T ′ ξ s ) s | % B α e.g. ) 0 ) of Ref.[1] ) ℓ c 2 ∞ ′ j r 1 q s † α =0,andat → ( . T q ( R c λ )= v . s | ab − $ , T i q ) + η b ( ) , is R ′ q O s ∞ ( ) is the free en- B etc. · 2 B a q s = ℓ ,where q ν ⟩−⟨ ℓ , ( q i + 2 ′ q b b e s − ( ). I also define β q q ab ab ( T, % 2 c ξν ) q ( g g ∂ ij δ ) ab 2 B ( † β q a X f ) ℓ ϵ ¯ q c s ( ′ q + ≡ ′ (0) = lim a ˜ q 0; the high temperature r s ν q ∂ ( α s 1 ( e 2 g ν × c s / N v )( ; this will only occur if the q ) is the “filling factor” of the T → † α π ξ i ≡ q q )= c ab e ( B = ( ⟨ ). Note that the fluctuation ν λ q g q )= ˜ ,q v k (0) vanishes at s ( 2 B ˜ ) B r =2 q )+ s 2 ℓ ,where s ℓ 2 B g ( ′ ∞ π q # ⟩ ∂ ℓ 0 q also obeys the algebra (4). I will q π s ( q 2 } ( ( T S q , while 2 ˜ 4 v ) " " ⟩ v 2 2 ≡ ′ d ) 2 ,for ∞ remains much smaller than the gap d $ q ) q ν s ,where ( ( ) q ′ ∂ν & − πνδ T )=˜ ⌫ ( ρ & / δρ ξ q B ) αα ± , ab ( = , with k ) = ν s ′ ˜ v −⟨ = (1 =2 FDMH arXiv:1112.0990 νδ q )= ¯ ) diverges as ε , ⌫ ) ( ∞ T, βα q a q ⟩ s ∞ q ( 2 δ = q ) λ is the inverse of a positive-definite unimodular λ ( ) ′ s δρ O) f ( )= ( ∂ q ,! ⟩ dac ρ ; { − .vv

2 +)d 6g;i ,but s ( translationally-invariant density-matrix, and as- ′ v ∞ / O rr'd OA +J

⟨ @ 1 ab αβ −

) U)*o ∂ ) ρ r ( α .g; s . q.i 1 2 δ

g o ⟨ tnc ) ( c q w- if ˜ ! S q " r! Y 'A9v.d ( ( q † α ∞ fr.oJt i5. )= tn-1 →∞ O O s ( s any c s (U Q+) T q λ Y'A a s ⟨ T/ E*,>F- ∂ (U q S) 4H) ( I will assume that translational symmetry is unbroken, In the “high-temperature limit” where o..l - (U The correlation energy per ﬂux quantum is givenH-.\(U by

B Ê \o d"9Y E 9: rD rI, s." ro I + d define a guiding-center structure factor δρ Then where so (determinant = 1) metric ≡ Landau level. (Inat a finite 2D temperatures, system, but this may will break always down be as true sumes that no additional degrees of freedom ( (defined per particle) by mal to the surface,relative and to no this “tilting” axis, of the in magnetic which field case shape of the Landauof orbits are the congruent Coulomb with equipotentials thethe around shape surface. a In point practice, chargean this on atomic-scale only three-fold happens or when four-fold there rotation is axis nor- The fundamental dualityready of apparent the in (8), structure and function derived (al- below) is valley, or layer indices) distinguish the particles. This ising valid for a structure function calculated us- This is a structure factor definedgiven in per terms flux quantum, of and the is GMP structure factor ¯ The interaction also has rotational invariance if between Landau levels, the guidingpletely centers uncorrelated, become with com- for all expansion at fixed if the particles are spin-polarized fermions, and → ergy per flux quantum. all k cold-atom systems).lim Note that for all temperatures, (nc +1 if they are bosonsa(i (which may be relevant for a a x.E nl o. (h rn o. €t&E a$: Y^.4 A '-i ^.Y O.O.g $ qt + 3T: H'EA llll t.\E il il- H H ,\ X H Ett B rv.lv! o -.E 9..1 6ql En(U

Jcroa dA9- ± livi F rJto n- oJF d fA-P )^\ .9-8 z r-dE a f, c o '1 c9 6 a f i'0. " = r^:H0)YVrYhn F

o d -., d3lEEH ⇠ h ?A 00ge; o 0)nE fr LLv g5.E'i;

ai3ar bosons ir LA, l'1 6oE because of Pauli, function is its own Fourier transform! 5 .r{ O-.i .H6-o U-5 ha +) F. > qJ\ tr fermions trOe tr o\\'d E pE d oo (ud ()vll pO '-,! O+)d

!t .U: r--j sa -'- -i-.til