When is a “wavefunction” not a wavefunction? - Princeton University

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

When is a “wavefunction” not a wavefunction?

A quantum-geometric interpretation of the Laughlin state F. D.M. Haldane, Princeton University

• 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 Foundation

Wednesday, March 20, 13

Quantized motion of three two-dimensional electrons in a strong magnetic field R. B. Laughlin Phys. Rev. B 27, 3383 – Published 15 March 1983

• 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.

=

Y

i
(zi

zj )

q

Y

e

1 ⇤ 2 zi zi

i

• 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

Wednesday, March 20, 13

x + iy z= p 2`B

• 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

Wednesday, March 20, 13

• 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 statement Wednesday, 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=

a (z, z ) = 0 (annihilated by Landau-level lowering operator)



(z, z ) = f (z)e holomorphic function Wednesday, March 20, 13

1 ⇤ 2z z

Gaussian

a† =

@ 1 2 z + @z ⇤ @ 1 ⇤ z 2 @z †

[a, a ] = 1

Schrödinger 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)!

some evidence against the LLL interpretation

• •

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)

(In fact, the Laughlin state should not even be interpreted as a Schrödinger wavefunction ....) Wednesday, March 20, 13

• 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:

a

b

[r , r ] = 0

~ ˜ R

~r

x

~ R

O

antisymmetric symbol area per London flux quantum = 2⇡`2B

˜a, R ˜ b ] = i`2 ✏ab [R B

˜a + R ˜a ra = R

guiding center of orbit a

b

[R , R ] = Wednesday, March 20, 13

2 ab i`B ✏

˜ a , Rb ] = 0 [R

independent

a

b

[R , R ] =

2 ab i`B ✏

dicembre 1901 – Monaco di Baviera, 1º febbraio 1976) è stato un fisico tedesco. Ottenne il Premio Nobel per la Fisica nel 1932 ed è considerato uno dei fondatori della meccanica quantistica.

• The remaining degrees of freedom after

Indice

1 Meccanica quantistica 2 Il lavoro durante la guerra 3 Bibliografia 3.1 Autobiografie 3.2 Opere in italiano 3.3 Articoli di stampa 4 Curiosità 5 Voci correlate 6 Altri progetti 7 Collegamenti esterni

Landau quantization are the non-classical guiding centers. They define a quantum geometry isomorphic to phase space, and obey an uncertainty principle Meccanica quantistica

• They cannot be described by a

Quando era studente, incontrò Gottinga nel 1922. Ciò permise lo sviluppo di una fruttuosa collaborazione tra i due.

Werner Karl H

per la

Schrödinger wavefunction! (only by a Heisenberg state)

Heisenberg ebbe l'idea della , la prima formal meccanica quantistica, nel principio di indeterminazio afferma che la misura simultanea di due variabili coniugate, come moto oppure energia e tempo, non può essere compiuta senza un'in Assieme a Bohr, formulò l'

della me

Ricevette il Premio Nobel per la fisica "per la creazione d quantistica, la cui applicazione, tra le altre cose, ha portato alla sco allotrope dell'idrogeno".

The guiding centers are generic to any Landau level Il lavoro durante la guerra

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

1 of 1 La fissione nucleare venne scoperta in Germania nel 1939. Heisenb Germania durante la seconda guerra mondiale, lavorando sotto il re programma nucleare tedesco, ma i limiti della sua collaborazione s

Rivelò l'esistenza del programma a Bohr durante un colloquio a Co settembre 1941. Dopo l'incontro, la lunga amicizia tra Bohr e Heise bruscamente. Bohr si unì in seguito al progetto Manhattan.

Wednesday, March 20, 13

Si è speculato sul fatto che Heisenberg avesse degli scrupoli moral



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

a ¯



a ¯

1 = 2z 1 ⇤ = 2z



m

)/z e

@ @z ⇤ @ + @z

circular basis guiding center ladder operators

1 ⇤ 2z z

Landau orbit ladder operators

action on LLL states:

Wednesday, March 20, 13

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

|

q Li

0

[email protected]

Y⇣

i
† a ¯i

| ¯ 0 iA ⌦ |

a ¯i | ¯ 0 i = 0

at this point, we “purify” the Laughlin state by removing its LLL “baggage” Wednesday, March 20, 13

† a ¯j

⌘q

1

ai |

0i 0i

=0

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

† [L, a ¯i ]

=

and a ¯?

† a ¯i

x shape of Landau orbit

shape of basis of guiding center states

not necessarily congruent!

Wednesday, March 20, 13

L=

1 2

X⇣ i

a ¯†i a ¯i + a ¯i a ¯†i



1 X a b = 2 g¯ab Ri Ri 2`B i 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) =

Z

d2 q`2B X Vm Lm (qg2 `2B )e 2⇡ m

1 2 2 2 qg `B

• It is the shape of the

composite boson formed by “flux attachment”

X

e

~i R ~j) i~ q ·(R

i
2 qg

ab

⌘ g qa qb

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

area-preserving zero-point fluctuations of shape give q4 structure factor Wednesday, March 20, 13

• In general a= a† = x

1 @ a ¯ = 2 z¯ + @ z¯⇤ † 1 ⇤ @ a ¯ = 2 z¯ @ z¯

@ 1 2 z + @z ⇤ @ 1 ⇤ z 2 @z

guiding-center flux attachment shape congruent to

Landau orbits congruent to

|¯ z | = constant

|z| = constant ⇤

z¯ = ↵z + z |↵|

2

2

| | =1

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

Wednesday, March 20, 13

• So where do the holomorphic functions come from?

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

a ¯|¯ z i = z¯|¯ zi 0

0

~) h¯ z |¯ z i ⌘ Sg¯ (~ r, r

Wednesday, March 20, 13

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

• The overlap defines a “quantum geometry” • a “quantum distance” measure is defined by 0 2

0

~ )| |S(~ r, r

~) =1 d(~ r, r

• The non-null eigenfunctions of the overlap define an orthonormal basis:

Z

2 0

d r 0) ~ S (~ r , r g ¯ 2 2⇡`B |

Wednesday, March 20, 13

1 i= p s

Z

0

(~ r)=s d2 r 2⇡`2B

(~ r)

(~ r )|¯ z (~ r )i

• For the guiding-center coherent states, the non-null eigenstates of S are degenerate, ⇤ 1 ⇤ z ¯ z ¯ with the form f (¯ z )e 2

• Coherent-state 0

holomorphic! representation of Laughlin state:

Y Z d2 ri Y ⇤ @ | i/ (¯ zi 2 2⇡` B i i
⇤ q z¯j )

Y

e

⇤ 1 z ¯ z ¯ i i 2

i

Laughlin “wavefunction”, with replacement

unchanged when composite boson has same shape as Landau orbit! Wednesday, March 20, 13

1

A|{¯ zi }i



z ! z¯

(¯ z = z⇤)

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

Wednesday, March 20, 13

relation to Hall viscosity through edge currents produced by entanglement cuts

• V(x)

fluid is compressed at edges ⇤ e s 0 by creating Gaussian 0 near edges: J = J e g curvature 2⇡ fluid density fixed by flux density

x✓

g=

Jg0 =

↵(x) 0

0 1 ↵(x)

1 d2 1 2 dx2 ↵(x)

For larger s, fluid becomes more compressible (less distortion needed for a given density change) Wednesday, March 20, 13



|R, ii = p “flat | bands”, i s.t. get h a similar | i=1 • On ChernPˆ Insulator ei

(R1 , R2 , R3 )



= ei'↵

(R1 , R2 ) i'

e

R,i

(R2 , R3 ) i'

e

R, i

↵ (R3 , R1 )

R, i

R, i

structure by projecting lattice-site orbitals !  We renormalize each band, projected orbital to unit weight: back to into a single and renormalizing X † X X † i p chosen s.t. h | i = 1 0 ,j + h.c.] [c e cR0 ,j + h.c.] + M H = tunit [c c + t R,i R,i R,i 1 2 R weight: R,i R,i

R,

hNNNi

hNNi

B

A

Hf /

X

R,i,R0 ,j

Vij (R

R0 ) · c¯†R,i c¯†R0 ,j c¯R0 ,j c¯R,i

!  Resulting orbitals are non-orthogonal†and overcomplete. 0

{¯ cR,i , c¯R0 ,j } = Sij R, R Analogous to normalized, single-particle states in the LLL The overlap matrix encodes coherent the quantum (natural basis forindescribing FQHtoo, states) geometry this case difference a shows a !a (R r ) | g (r)i = 0



between topological and non-topological ⇤ ⇤ ! + ! g ⌘ ! ab a b b !a bands det(g) = 1

Wednesday, March 20, 13

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

Wednesday, March 20, 13

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When is a “wavefunction” not a wavefunction? - Princeton University

APS March Meeting, Baltimore March 20, 2013 N21.1 When is a “wavefunction” not a wavefunction? A quantum-geometric interpretation of the Laughlin st...

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