Quantum Geometry, Exclusion Statistics, and the Geometry of Flux - - PowerPoint PPT Presentation

SLIDE 1

Quantum Geometry, Exclusion Statistics, and the Geometry of “Flux Attachment” in 2D Landau levels

“Reduced Density Matrices in Quantum Physics and Role of Fermionic Exchange Symmetry” : workshop Pauli2016 on Oxford University, April 12-15 2016

• 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.

SLIDE 2

When it describes a “quantum geometry” Q:

• 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.

A: When is a “wavefunction” NOT a wavefunction?

SLIDE 3

Schrödinger vs Heisenberg

• Schrödinger’s picture describes the system by

a wavefunction 𝝎(r) in real space

Werne

antistica

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

• Heisenberg’s picture describes the system by a

state ｜𝝎⟩ in Hilbert space

• They are only equivalent if the basis of

states in real-space are orthogonal:

|ri ψ(r) = hr|ψi hr|r0i = 0 (r 6= r0)

requires

this fails in a quantum geometry

SLIDE 4

• Schrödinger’s real-space form of quantum

mechanics postulates a local basis of simultaneous eigenstates |x⟩ of a commuting set of projection

• perators P(x), where P(x)P(x′) = 0 for x ≠ x′.

Schrödinger vs Heisenberg and quantum geometry

Ψ(x) = hx|Ψi

|Ψi

Heisenberg Schrödinger

?

=

• nly equivalent if

hx|x0i = 0 for x 6= x0

this fails in a quantum geometry

SLIDE 5

• 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-

• rthogonal, but overcomplete:

In this case the positive Hermitian operator

S(x, x0) = hx|x0i has null eigenstates

X

x0

S(x, x0)ψ(x0) = 0

(so the basis cannot be reorthogonalized)

SLIDE 6

• If the Schrödinger basis is on a lattice, so |x⟩

is normalizable

d(x, x0)2 = 1 − |S(x, x0)|2 = 0 = 1 x 6= x0 x = x0

(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”

Hilbert-Schmidt distance

H = X

x

V (x)|xihx| hx|x0i 6= δ(x, x0)

SLIDE 7

• Fractional quantum Hall effect in 2D

electron gas in high magnetic field (filled Landau levels)

Ψ1/3

L

= Y

i<j

(zi − zj)3 Y

i

e−|zi|2/4`2

B

• Laughlin (1983) found the

wavefunction that correctly describes the 1/3 FQHE , and got Nobel prize,

ν = 1

3

• Its known that it works, (tested by

finite-size numerical diagonalization) but WHY it works has never really been satisfactorily explained!

SLIDE 8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.5 1 1.5 2 2.5 3 3.5 laughlin 10/30

E

Collective mode with short-range V1 pseudopotential, 1/3 filling (Laughlin state is exact ground state in that case)

“roton”

(2 quasiparticle + 2 quasiholes)

goes into continuum

gap incompressibility

1 2 0.5 klB

• Moore-Read

ν = 2

4

“kF ”

fermionic “roton” bosonic “roton”

Collective mode with short-range three-body pseudopotential, 1/2 filling (Moore-Read state is exact ground state in that case)

• momentum ħk of a quasiparticle-quasihole pair is

proportional to its electric dipole moment pe ~ka = abBpb

e

kB

gap for electric dipole excitations is a MUCH stronger condition than charge gap: doesn’t transmit pressure!

SLIDE 9

fractional-charge, fractional statistics vortices

Chiral edge states at edge of finite droplet of fluid (Halperin, Wen) time

e2iθ eiθ

charge -e/m statisticsθ = π/m e.g., m=3

Ψ = Y

i,α

(zi − wα) Y

i<j

(zi − zj)m Y

i

e

−

1 4`2 B

z∗

i zi

SLIDE 10

• Non-commutative geometry of Landau-orbit

guiding centers

×

e−

O ~ r

~ R ~ Rc ~ R

~ r ~ Rc

displacement of electron from origin displacement of guiding center from origin displacement of electron relative to guiding center of Landau orbit

shape of orbit around guiding center is fixed by the cyclotron effective mass tensor

~ r = ~ R + ~ Rc [Rx, Ry] = −i`2

B

[Rx

c , Ry c] = +i`2 B

[rx, ry] = 0

classical geometry

Landau orbit (harmonic oscillator)

[Ra, Rb

c] = 0

(a, b ∈ {x, y})

guiding centers commute with Landau radii

quantum geometry guiding center

SLIDE 11

The one-particle Hilbert-space factorizes

• FQHE physics is *COMPLETELY*

defined in the many-particle generalization (coproduct) of

H = ¯ HGC ⊗ Hc [Rx

c , Ry c] = +i`2 B

[Rx, Ry] = −i`2

B

space isomorphic to phase space in which the Landau orbit radii act space isomorphic to phase space in which the guiding-centers act

HGC

Once is discarded, the Schrödinger picture is no longer valid!

Hc ~ r = ~ R + ~ Rc

classical electron coordinate

SLIDE 12

• Laughlin states also occur in the second

Landau level, and in graphene, and more recently in simulations of “flat-band” Chern insulators Previous hints that the Laughlin “wavefunction” should not be interpreted as a wavefunction: These don’t fit into the original paradigm of the Galileian-invariant Landau level

SLIDE 13

• First, translate Laughlin to the Heisenberg picture:

a = 1

2z + ∂z∗

a† = 1

2z∗ − ∂z

Landau-level ladder operators

ψ0(z, z∗) = e− 1

2 z∗z

aψ0(z, z∗) = 0

Guiding-center ladder operators

¯ a = 1

2 ¯

z + ∂¯

z∗

¯ a† = 1

2 ¯

z∗ − ∂¯

z

z ↔ ¯ z

¯ z = z∗

¯ aψ0(z, z∗) = 0 ¯ a†f(z)Ψ0(z, z∗) = zf(z)Ψ0(z, z∗) ¯ a† = 1

2z − ∂z∗

¯ a = 1

2z∗ + ∂z Gaussian lowest-weight state

action of guiding-center raising operators on LLL states

usual identification is

SLIDE 14

• Heisenberg form of Laughlin state (not

“wavefunction”)

|Ψ1/q

L i =

@Y

i<j

(¯ a†

i ¯

a†

j)q|¯

Ψ0i 1 A ⌦ (|Ψ0i) ¯ ai|¯ Ψ0i = 0

ai|Ψ0i = 0

∈ HGC ≡ ¯ H

∈ Hc ≡ H

Guiding-center factor (keep) Landau-orbit factor (discard)

• At this point we discard the Landau-orbit

Hilbert space.

• The only “memory” of the shape of the Landau
• rbits is “hidden” in the definition of ¯

a

SLIDE 15

• guiding-center Coherent states (single particle)

¯ a(g)|Ψg(0)i = 0

|Ψg(¯ z)i = e¯

za†(g)−¯ z∗a(g)|Ψg(0)i

• This is a non-orthogonal overcomplete basis

S(¯ z, ¯ z0) = hΨg(¯ z)|Ψg(¯ z0)i

• non-zero eigenvalues of the positive Hermitian
• verlap function are holomorphic!

Z d¯ z0d¯ z0⇤ 2π S(¯ z, ¯ z0)Ψ(¯ z0, ¯ z0⇤) = Ψ(¯ z, ¯ z⇤) Ψ(¯ z, ¯ z∗) = f(¯ z∗)e− 1

2 ¯

z∗¯ z

SLIDE 16

The “purified” Laughlin state

• This is now defined in the many-particle

guiding-center Hilbert space, without reference to any Landau-level structure

• What defines ?

|Ψ1/q

L i =

Y

i<j

(¯ a†

i ¯

a†

j)q|¯

Ψ0i ¯ ai|¯ Ψ0i = 0 L(g) = gab 2`2

B

X

i

Ra

i Rb i

[L(g), ¯ a†

i(g)] = ¯

a†

i(g)

It is the raising

• perator for the

“guiding-center spin” L(g)

• f particle i

¯ a†

i

gab is a 2x2 positive-definite unimodular (det = 1) 2D spatial metric tensor

SLIDE 17

• The Laughlin state has suddenly revealed its

well-kept secret- a hidden geometric degree of freedom! It is parameterized by a unimodular metric gab!

|Ψ1/q

L (g)i =

Y

i<j

(¯ a†

i(g) ¯

a†

j(g))q|¯

Ψ0(g)i

¯ ai(g)|¯ Ψ0(g)i = 0

• In the naive LLL wavefunction picture, the

unimodular metric gab is fixed to be proportional to the cyclotron effective mass tensor m*ab.

• In the reinterpretation it is a free parameter.

SLIDE 18

This is the entire problem: nothing other than this matters!

• generator of translations and

electric dipole moment!

H = X

i<j

U(Ri − Rj)

[Rx, Ry] = −i`2

B

[(Rx

1 − Rx 2), (Ry 1 − Ry 2)] = −2i`2 B

• relative coordinate of a pair of

particles behaves like a single particle

• H has translation and

inversion symmetry

[(Rx

1 + Rx 2), (Ry 1 − Ry 2)] = 0

[H, P

iRi] = 0

two-particle energy levels like phase-space, has Heisenberg uncertainty principle gap

want to avoid this state

SLIDE 19

• Laughlin state

U(r12) = ⇣ A + B ⇣

(r12)2 `2

B

⌘⌘ e

− (r12)2

2`2 B

E2

symmetric antisymmetric

• Solvable model! (“short-range pseudopotential”)

1 2(A + B)

1 2B

rest all 0

|Ψm

L i =

Y

i<j

⇣ a†

i a† j

⌘m |0i ai|0i = 0

a†

i = Rx + iRy

√2`B

EL = 0

maximum density null state

• m=2: (bosons): all pairs

avoid the symmetric state E2 = ½(A+B)

• m=3: (fermions): all pairs

avoid the antisymmetric state E2 = ½B

[ai, a†

j] = δij

SLIDE 20

• New feature is similar to FQH ferromagnet,

where electrons couple to a combination of magnetic flux and Berry curvature of the ferromagnetic order parameter(Skyrmions)

• The electron density is no longer rigidly tied

to the magnetic flux density, it can deviate from it at the expense of paying the correlation energy cost for geometric distortion.

• Old results of Girvin, Macdonald and Platzman

(O(q4) “guiding-center structure factor”) get a simple explanation as zero-point fluctuations

• f the geometry

SLIDE 21

• The key idea for understanding both the

Fractional Quantum Hall and Composite Fermi Liquid states is “Flux attachment”

SLIDE 22

• quantum solid
• repulsion of other particles make an attractive

potential well strong enough to bind particle

• unit cell is

correlation hole

• defines geometry

solid melts if well is not strong enough to contain zero-point motion (Helium liquids)

SLIDE 23

• similar story in FQHE:
• “flux attachment” creates

correlation hole

• potential well must be

strong enough to bind electron

• defines an emergent

geometry

• new physics: Hall viscosity,

geometry............ e-

• continuum model, but

similar physics to Hubbard model but no broken symmetry

SLIDE 24

(−1)p × (−1)pq = +1

exchange of p fermions

Berry phase (exchange of “exclusion zones”)

composite is a boson

Statistical selection rule

the rule formerly known as “odd-denominator”,

(but Moore-Read has p=2, q=4)

• 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

p=1, q=3 ⅓ Laughlin ⅓ Laughlin

(with different shape)

p=2, q=5 ⅖ Hierarchy/Jain

central orbital

• ccupied

next two empty central two orbitals occupied, next three empty

ν= ⅓ ν= ⅓ ν= ⅖

composites exchange as bosons

“exclusion statistics”

SLIDE 25

• 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)

• The metric has a companion “guiding center spin” that is topologically

quantized in incompressible states.

δE ∝ (distortion)2

L=

1 2

3 2

5 2

⅓ ⅓ ⅓ 1 0 0

3 2 1 2

configuration of “elementary droplet” (composite boson) subtract total L (=Lref) of reference configuration (uniform occupation p/q)

total L ⅓ Laughlin

s = ( 1

2 − 3 2) = −1

SLIDE 26

• Origin of FQHE incompressibility is analogous to origin
• f 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

• In the FQHE only the area of

the “quantized region” is fixed. The shape must adjust to minimize the correlation energy. e-

energy gap prevents additional electrons from entering the region covered by the composite boson

SLIDE 27

• The usual “lowest Landau level wavefunction” formalism has

Ψ(x) = f(z)e− 1

4 z∗z/`2 B

holomorphic function

• With a (quasi) periodic boundary condition, this becomes

ψ(z, z∗) ∝ NΦ Y

i=1

σ(z − wi) ! e

− 1

4 z∗z `2 B

X

i

wi = 0

Weierstrass sigma function* zeroes

NΦ

(one for each flux quantum passing through the primitive region of the pbc)

*(slightly modified from Weierstrass’ original definition when the pbc lattice is not square or hexagonal)

SLIDE 28

• In the Heisenberg-algebra reinterpretation

X

i

wi = 0

• The filled Landau level is

|Ψi = @Y

i<j

σ(a†

i a† j)σ(P ia† i)

1 A |0i

• The Laughlin states are

|Ψi = @Y

i<j

σ(a†

i a† j)m

1 A

m

Y

k=1

σ(P

ia† i wk)|0i m

X

k=1

wk = 0.

|Ψi =

NΦ

Y

i=1

σ(a†

i wi)|0i

• ne particle

filled Level

N = NΦ

N = 1 NΦ = mN

Laughlin state

ν = 1

m

SLIDE 29

• Unlike the filled Landau level state, in which the
• nly metric-dependence is the normalization, the

Laughlin states depend on the metric choice which fixes the shape of the vortex-like correlation hole around each particle (“attached flux”)

|Ψi = @Y

i<j

σ(a†

i a† j)m

1 A

m

Y

k=1

σ(P

ia† i wk)|0i

ν = 1

m

correlation holes in two states with different metrics

(filled Landau level is a Slater-determinant state with no correlation hole)

SLIDE 30

x

“flux attachment” Has a shape that defines a metric δR displacement of charge relative to center of flux attachment gives an electric dipole momentum flux attachment creates a correlation hole that can bind one or more particles into a composite object

ε(P , g)

correlation energy dispersion p particles + q “flux” (orbitals) “kinetic energy” = electric polarization energy

(velocitya) = ∂ε ∂¯ pa ¯ pa = B✏ab(eRb)

SLIDE 31

• 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 can condense in the p = 0 (inversion-symmetric) state with no electric dipole Fermions can form a Fermi sea in “momentum” (dipole)space

SLIDE 32

• exchange phase

p particles + q “flux” (orbitals)

(−1)pqξp

• 1 for electrons

= +1 composite object is boson = −1 composite object is fermion

e.g., one electron with p = 1, q = 2

• inversion symmetry of FQHE : gcd(p,q) = 1 or 2

Berry curvature of the “Flux attachment” of a vortex-like correlation hole modifies the statistics

SLIDE 33

e the electron excludes other particles from a region containing 3 flux quanta, creating a potential well in which it is bound 1/3 Laughlin state If the central orbital is filled, the next two are empty The composite boson has inversion symmetry about its center It has a “spin” ..... ..... − 1 0 0

1 3 1 3 1 3

1 2 3 2 5 2

L = 1

2

L = 3

2

−

s = −1

(composite boson picture)

(that couples to Gaussian curvature

• f its metric)

SLIDE 34

e 2/5 hierarchy/Jain state e ..... ..... − 1

1 2 3 2 5 2

− 0 0 1

2 5 2 5 2 5 2 5 2 5

L = 2 L = 5 s = −3 L = gab 2`2

B

X

i

Ra

i Rb i

Qab = Z d2r rarb⇢(r) = s`2

Bgab

second moment of neutral composite boson charge distribution

(composite boson picture)

Jain’s two filled

• “ -levels?”

λ

SLIDE 35

• choose distinct “occupied orbitals” (allowed

dipole moments, quantized by the pbc)

{di, i = 1, . . . N} ∈ { L

N }

which minimize for fixed

¯ d = 1 N X

i

di 1 N X

i<j

|di − dj|2 = 1 2 X

i

|di − ¯ d|2

Model for 1/m CFL states

• is a many-body quantum number that

takes N2 distinct values. There is thus one such configuration per sector of this many-body translational quantum number.

¯ d mod { L N }

SLIDE 36

• The model 1/m CFL states ( including the boson case m = 1) are

Ψ({zi, z∗

i }, {di}, {wα}) ∝

✓ det

i,j Mij({zk}, dj, d∗ j, ¯

d) ◆

× @Y

i<j

σ(zi − zj) 1 A

m−2 m

Y

α=1

σ((P

izi) − wα) N

Y

i=1

e

− 1

4 z∗ i zi `2 B

Mij({zk}, dj, d⇤

j, ¯

d) = e

1 m d∗ j zi 2`2 B

Y

k6=i

σ(zi − zk − dj + ¯ d)

• The matrix in the determinant is
• also:

m

X

α=1

wα =

N

X

j=1

dj = N ¯ d

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

complex cf dipoles edj (dj is quantized in units )

L N

mean value of dj

SLIDE 37

• Now we see that the “Fermi sea” is invariant

under uniform translation in “dipole space”

py px

filled empty cluster of adjacent occupied states pbc

2π~ L

# Z_{COM} overlap with PH-conjugate in opposite charge sector 1-

• verlap

0 0.999998870263 1.1297367517e-06 1 0.999999369175 6.3082507884e-07 2 0.99999860296 1.39704033186e-06 3 0.99999860296 1.3970403312e-06 4 0.999999369175 6.30825078063e-07 5 0.999998870263 1.12973675237e-06 6 0.999999369175 6.30825079173e-07 7 0.99999860296 1.39704032942e-06 8 0.99999860296 1.39704032909e-06 9 0.999999369175 6.30825078507e-07

Computing ph symmetry (with Scott Geraedts)

model state is numerically very close to p-h symmetry

SLIDE 38 q ^E I d

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• A remarkable self-duality of the guiding-

center structure function: something related to “generalized Pauli constraints”

S0(q) = 1 NΦ X

ij

eiq·(Ri−Rj)

O(q4)

Imprint of shape- fluctuations of the metric

(GMP)

Pij = 1 NΦ X

q

eiq·(RiRj)

S(∞) = ν(1 ± ν)

FDMH arXiv:1112.0990

fermions

guiding center exchange operator

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S0(q) = 1 NΦ X

ij

eiq·(Ri−Rj)

S(∞) = ν(1 ± ν)

FDMH arXiv:1112.0990

ξ = ±1

bosons fermions because of Pauli, function is its own Fourier transform!

The fundamental duality of the structure function (al- ready apparent in (8), and derived below) is s(q) − s∞ = ξ d2q′ℓ2

B

2π eiq×q′ℓ2

B (s(q′) − s∞) .

(10) This is valid for a structure function calculated us- ing any translationally-invariant density-matrix, and as- sumes that no additional degrees of freedom (e.g., spin, valley, or layer indices) distinguish the particles.