When is a “wavefunction” not a wavefunction?haldane/talks/aps2013.pdfWhen is a “wavefunction” not a wavefunction? A quantum-geometric interpretation of the Laughlin state F

When is a “wavefunction” not a wavefunction?A quantum-geometric interpretation of the Laughlin state

F. D.M. Haldane, Princeton University

APS March Meeting, Baltimore March 20, 2013 N21.1

• The FQHE is a fundamentally problem of interacting “guiding centers” with a non commutative geometry: what is the true meaning of Laughlin’s “wavefunction”?

• An interpretation of “z” in the Laughlin “wavefunction” as the intrisic geometry of flux attachment.

Supported by DOE-SC0002140 and the W. M. Keck FoundationWednesday, March 20, 13

• In 1983 Bob Laughlin’s numerical diagonalization study of a system of 3 interacting electrons (!) in the lowest Landau level led him to his famous wavefunction.

Quantized motion of three two-dimensional electrons in a strong magnetic fieldR. B. LaughlinPhys. Rev. B 27, 3383 – Published 15 March 1983

=Y

i<j

(zi � zj)qY

i

e�12 z

⇤i zi

• It quickly became clear that this was the correct solution to the puzzle of the 1/3 FQHE and was the prototype model for the other FQHE states

z =x+ iyp2`B

Wednesday, March 20, 13

http://link.aps.org/doi/10.1103/PhysRevB.27.3383

http://link.aps.org/doi/10.1103/PhysRevB.27.3383

• The Laughlin state was convincing because it explained why ν = 1/3 was seen, but not ν =1/2.

• Its validity was subsequently established by numerical exact diagonalization studies of the adiabatic evolution between a model system with a “short range pseudopotential” for which the Laughlin state was the exact ground state, and a realistic Coulomb interaction.

FDMH and E. H. Rezayi, 1985


• So it is known to work, but why? (In my opinion, this question was never satisfactorily answered)

a common rationalization:

“Laughlin’s wavefunction cleverly lowers the

Coulomb correlation energy by placing its zeroes

at the locations of the particles”

we will see that this is an empty statementWednesday, March 20, 13

• A key aspect of the Laughlin wavefunction is that is a holomorphic function (times a non-holomorphic Gaussian)

• This is the lowest-Landau level property:

a (z, z⇤) = 0 a = 12z +

@@z⇤

a† = 12z

⇤ � @@z

[a, a†] = 1

(annihilated by Landau-level lowering operator)

(z, z⇤) = f(z)e�12 z

⇤z

holomorphicfunction

GaussianSchrödinger form of Landau level (harmonic oscillator) ladder operators


• Laughlin presented his state as a lowest-Landau-level wavefunction, and this interpretation seems to have been uncritically accepted. But I will argue that its holomorphic character has NOTHING to do with the lowest Landau level (LLL)!

• The Laughlin state also occurs in the second Landau level

• The Laughlin state occurs in models of Chern insulators with flat bands (no Landau levels)

some evidence against the LLL interpretation

(In fact, the Laughlin state should not even be interpreted as a Schrödinger wavefunction ....)


• When there is Landau quantization, the classical (with commuting components) displacement of an electron splits up into a non-classical guiding center displacement plus the cyclotron-orbit radial vector:

[Ra, Rb] = i`2B✏ab

antisymmetric symbol

O

~R~r x

~R

[ra, rb] = 0

area per London flux quantum = 2⇡`2B

[Ra, Rb] = 0independent[Ra, Rb] = �i`2B✏

ab

ra = Ra + Ra

O

guiding center of orbit


• The remaining degrees of freedom after Landau quantization are the non-classical guiding centers. They define a quantum geometry isomorphic to phase space, and obey an uncertainty principle

• They cannot be described by a Schrödinger wavefunction! (only by a Heisenberg state)

[Ra, Rb] = �i`2B✏ab

Erwin_schrodinger1.jpg (JPEG Image, 485 × 560 pixels) - Scaled (71%) http://www.camminandoscalzi.it/wordpress/wp-content/uploads/2010/09/Erwin_sc...

1 of 1 5/21/12 12:25 AM

Werner Karl Heisenberg

per la fisica 1932

Werner Karl HeisenbergDa Wikipedia, l'enciclopedia libera.

Werner Karl Heisenberg (Würzburg, 5dicembre 1901 – Monaco di Baviera, 1º febbraio1976) è stato un fisico tedesco. Ottenne il PremioNobel per la Fisica nel 1932 ed è considerato unodei fondatori della meccanica quantistica.

Indice

1 Meccanica quantistica2 Il lavoro durante la guerra3 Bibliografia

3.1 Autobiografie3.2 Opere in italiano3.3 Articoli di stampa

4 Curiosità5 Voci correlate6 Altri progetti7 Collegamenti esterni

Meccanica quantisticaQuando era studente, incontrò Gottinga nel 1922. Ciò permise lo sviluppo di unafruttuosa collaborazione tra i due.

Heisenberg ebbe l'idea della , la prima formalizzazione dellameccanica quantistica, nel principio di indeterminazione, introdotto nel 1927,afferma che la misura simultanea di due variabili coniugate, come posizione e quantità dimoto 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 meccanicaquantistica, la cui applicazione, tra le altre cose, ha portato alla scoperta delle formeallotrope dell'idrogeno".

Il lavoro durante la guerraLa fissione nucleare venne scoperta in Germania nel 1939. Heisenberg rimase inGermania durante la seconda guerra mondiale, lavorando sotto il regime nazista. Guidò ilprogramma nucleare tedesco, ma i limiti della sua collaborazione sono controversi.

Rivelò l'esistenza del programma a Bohr durante un colloquio a Copenaghen nelsettembre 1941. Dopo l'incontro, la lunga amicizia tra Bohr e Heisenberg terminòbruscamente. Bohr si unì in seguito al progetto Manhattan.

Si è speculato sul fatto che Heisenberg avesse degli scrupoli morali e cercò di rallentare ilprogetto. Heisenberg stesso tentò di sostenere questa tesi. Il libro Heisenberg's War diThomas Power e l'opera teatrale "Copenhagen" di Michael Frayn adottarono questainterpretazione.

Nel febbraio 2002, emerse una lettera scritta da Bohr ad Heisenberg nel 1957 (ma maispedita): vi si legge che Heisenberg, nella conversazione con Bohr del 1941, non espressealcun problema morale riguardo al progetto di costruzione della bomba; si deduce inoltreche Heisenberg aveva speso i precedenti due anni lavorandovi quasi esclusivamente,convinto che la bomba avrebbe deciso l'esito della guerra.

Sito web per questa immagineWerner Karl Heisenbergit.wikipedia.org

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The guiding centers are generic to any Landau level

We must abandon the Schrödinger picture and and use a Heisenberg description of the guiding center


• In Laughlin’s “symmetric gauge” scheme, the guiding centers are described in a basis of circular states centered at an arbitrary origin

• This circular shape is also the shape of the Landau orbits

0m(z, z⇤) / zme�12 z

⇤z

a = 12z

⇤ + @@z

a† = 12z �

@@z⇤

circular basis guiding centerladder operators

Landau orbit ladder operators

action on LLL states:


• The Heisenberg form of Laughlin is therefore an unentangled product of a correlated guiding-center state with a (trivial) harmonic oscillator state of the Landau orbits:

ai| 0i = 0ai| 0i = 0

| qLi =

0

@Y

i<j

⇣a†i � a†j

⌘q| 0i

1

A⌦ | 0i

at this point, we “purify” the Laughlin state by removing its

LLL “baggage”


• The Landau orbits are now gone: what defines and ?a a†

L = 12

X

i

⇣a†i ai + aia

†i

⌘

[L, a†i ] = a†i

=1

2`2B

X

i

gabRaiR

bi

x

shape of Landau orbit

shape of basis of guiding center states

not necessarilycongruent!

The guiding center metric gab that defines the basis of guiding-center ladder operators is an INDEPENDENT variational geometric parameter of the Laughlin state, that must be chosen to minimize the correlation energy


• the guiding-center metric is a parameter of the pseudopotential Hamiltonian that can be used to define the Laughlin state:

H(g) =

Zd2q`2B2⇡

X

m

VmLm(q2g`2B)e

� 12 q

2g`

2B

X

i<j

ei~q·(~Ri�~Rj)

q2g ⌘ gabqaqb• It is the shape of the composite boson formed by “flux attachment”

“Elementary droplet” or“composite boson” of 1/3 Laughlin state: central orbital filled, next two empty

area-preserving zero-pointfluctuations of shape give q4 structure factor


• In general

a = 12 z +

@@z⇤a = 1

2z +@

@z⇤

a† = 12z

⇤ � @@z

a† = 12 z

⇤ � @@z

Landau orbits congruent to

guiding-center flux attachment shape congruent to

|z| = constant

|z| = constant

z = ↵z⇤ + �z|↵|2 � |�|2 = 1

• “zeroes of wavefunction” only coincide with locations of particles if β = 0.

x


• So where do the holomorphic functions come from?

coherent states of the guiding centers are non-orthogonal and overcomplete:

a|zi = z|zi

complex positive-indefinite Hermitian overlap matrix that depends implicitly on the metric

hz|z0i ⌘ Sg(~r, ~r0)


• The overlap defines a “quantum geometry”

• a “quantum distance” measure is defined by

d(~r, ~r0)2 = 1� |S(~r, ~r0)|• The non-null eigenfunctions of the overlap define

an orthonormal basis:Z

d2r0

2⇡`2BSg(~r, ~r0) �(~r

0) = s� �(~r)

| �i =1ps�

Zd2r

2⇡`2B �(~r)|z(~r)i


• For the guiding-center coherent states, the non-null eigenstates of S are degenerate, with the form f(z⇤)e�

12 zz

⇤

holomorphic!

• Coherent-state representation of Laughlin state:

| i /Y

i

Zd2ri2⇡`2B

0

@Y

i<j

(z⇤i � z⇤j )qY

i

e�12 ziz

⇤i

1

A|{zi}i

Laughlin “wavefunction”, with replacement z ! z⇤

unchanged when composite boson has same shape as Landau orbit! (z = z⇤)Wednesday, March 20, 13

• The metric is the true collective degree of freedom of the FQHE state: it adjusts to non-uniform electric fields. Its curvature from spatial nonuniformity provides an additional gauge field similar to Berry curvature in quantum Hall ferromagnets

cutrelation toHall viscosity through edgecurrents produced byentanglementcuts

gives “Area” (perimeter) term in “momentum polarization”


• near edges:

V(x)

g =

✓↵(x) 00 1

↵(x)

◆

J

0g = �1

2

d

2

dx

2

1

↵(x)

x

fluid densityfixed by flux density

fluid is compressed at edges by creating Gaussiancurvature �J0

e =e⇤s

2⇡J0g

For larger s, fluid becomes more compressible (less distortion needed for a given density change)


• On Chern Insulator “flat bands”, get a similar structure by projecting lattice-site orbitals into a single band, and renormalizing back to unit weight:

!  We project single-particle orbitals into the lowest band

!  We renormalize each projected orbital to unit weight:

!  Resulting orbitals are non-orthogonal and overcomplete.

Analogous to normalized, single-particle coherent states in the LLL (natural basis for describing FQH states)

S-Matrix Diagonalization Sagar Vijay

| 1/qL (g)i /

2

4Y

i<j

⇣a†i (g)� a†j(g)

⌘q|0ig

3

5

s.t. ai(g) |0ig = 0

[Ra, Rb] = �i`2B✏ab

ˆP |R, ii = pR,i | R, ii s.t. h R, i | R, ii = 1

H = t1X

hNNi

[c†R,icR0,j + h.c.] + t2X

hNNNi

[c†R,iei�cR0,j + h.c.] +M

X

R,i

✏R,ic†R,icR,i

Hf /X

R,i,R0,j

Vij(R�R0) · c†R,ic

†R0,j cR0,j cR,i

{cR,i, c†R0,j} = Sij

�R, R0�

!a(Ra � ra) | g(r)i = 0

gab ⌘ !⇤a!b + !⇤

b!a

det(g) = 1

d�(r � r0) = 1� | h g(r) | g(r0)i |�

S(r, r0) ⌘ h g(r) | g(r0)i

Zd2r

2⇡`2BS(r, r0) · '�(r

0) = s�'�(r) with s� = 0 or 1

'(r) = f(z⇤) · e� 12 z

⇤z

z⇤ = z

1

S-Matrix Diagonalization Sagar Vijay

lim

v!1| h R,B | R,Ai | =

p3

3

S↵�(R1, R2) = | h R1,↵ | R2, �i | · ei'↵�(R1,R2)

d↵�(R, R0) ⌘ 1� | h R,↵ | R0, �i |

ei�↵��(R1,R2,R3)= ei'↵�(R1,R2)ei'��(R2,R3)ei'�↵(R3,R1)

t2, �

pR,i chosen s.t. h R,i | R,ii = 1

2

A

B

The “Fuzzy Lattice” Hubbard Model

• The overlap matrix encodes the quantum geometry in this case too, shows difference between topological and non-topological bands


summary

• The variable “z” in the Laughlin “wavefunction” depends on guiding center geometry, not Landau orbit shape, which are chosen to be congruent in Laughlin’s treatment

• The Laughlin “wavefunction” is really a guiding-center coherent-state amplitude


Documents

When is a “wavefunction” not a wavefunction?haldane/talks/aps2013.pdfWhen is a “wavefunction” not a wavefunction? A quantum-geometric interpretation of the Laughlin state F