Emergent geometry of fractional quantum Hall fluids

F. D. M. Haldane, Princeton University

The complex coordinate “z” in Laughlin’s wavefunction, and in other modelwavefunctions inspired by conformal field theory, is commonly taken to bedefined by the shape of the lowest-Landau level cyclotron orbit. This is afundamental misinterpretation: it is defined by the shape of the “flux attach-ment” that binds a correlation hole to the partlcles, which is an emergentlocal geometric property. The ⌫ = p/q FQH fluids can be understood as con-densates of “composite bosons” formed by “attachment” of q “flux quanta”to p particles. The mean shape of these elementary incompressible dropletsof FQH fluid defines a spatial metric field gab(x, t) that can fluctuate in spaceand time, and adjust to the local environment. More precisely, the expecta-tion value of a quasi-local operator defines a quantity ⌘abH (x) = hs✏ac✏bdgcd(x),which is the “guiding center” contribution to the “Hall viscosity tensor”,where s is a topologically-quantized (signed) “geometric spin” (related tothe so-called “shift” of FQHE states), and ✏ab is the antisymmetric symbol.The composite particles behave like neutral bosons that carry a geometricspin s (distinct from the more-familiar topological spin 1

2pq of the attachedflux) and their density is incompressibly-pinned to B(x)/q�0 + sKG(x)/2⇡,where �0 = h/e is the magnetic flux quantum for charge-e particles, B(x) isthe magnetic flux density, and KG(x) is the Gaussian curvature of the emer-gent spatial metric. The zero-point fluctuations of the metric (with fixeddet g) lead to the O(q4) long-wavelength behavior of the FQH guiding-centerstructure factor found by Girvin, MacDonald and Platzman.

1

Simons Center for Geometry and Physics, SUNY, Stony Brook, New York, USA

• Non-commutative geometry

• Origin of FQHE incompressibility

• geometry of flux attachment

• Topological (chiral) Virasoro anomaly,

Workshop on : Quantum Anomalies and Hydrodynamics: Applications to Nuclear and Condensed Matter Physics

February 17, 2014

• 2D electron gas in a magnetic field

Belectrons

filled

empty

1/3 partially-filled

E

gap

gap

Landau levels quantized kinetic energy of cyclotron

orbits

“flat bands” withoutkinetic energy!

xe-

~!c

~!cincompressible!

• original form of the Laughlin FQHE state

/Y

i<j

(zi � zj)qY

i

e�14 z

⇤i zi/`

2B

• proposed as a “lowest Landau level” state

• no apparent variational parameter!

• For q > 1, it describes a topologically-ordered highly-correlated incompressible fluid state

• The usual interpretation of the holomorpic character of the Laughlin state explains it as a “lowest Landau level property”

• However, the state also occurs in the second Landau level, and in lattice models!

• something is wrong with the usual interpretation!

• Let’s reexamine the Laughlin state (and drop what turns out to be an inappropraite interpretation of it as a “wavefunction”)

• Its rich internal geometry will then get exposed!

• 2D Landau levels are formed when electrons are bound to a 2D surface through which a magnetic flux passes

• The FQHE problem is often thought as being in the “high magnetic field limit”, but this is also incorrect: To have flat Landau levels, the field must be low enough so that the 2D area per flux quantum is much larger than the atomic-scale unit cell of the 2D surface

• It will be important NOT to impose 2D rotational symmetry: this allows two distinct sources of geometry to be distingished

⇡i = pi � eA(ri)H =X

i

"(⇡i) +X

i<j

V (ri � rj)

Peierls substitution2D kinetic energy

Coulombrepulsion

x e− e−

Landau orbits around “guiding center” Coulomb equipotentials around electron determined by effective mass tensor determined by dielectric tensor

• Macroscopically-degenerate Landau levels (extensive degeneracy) when λ = 0

H =X

i

"(⇡i) + �X

i<j

V (ri � rj)

uniform Bsmall λ

E1

E2

E3

E4

r = R+ R ⇡a = eB✏abRb

[⇡a,⇡b] = i~eB✏ab

[Ra, Rb] = i✏ab`2B

[Ra, Rb] = �i✏ab`2B

[Ra, Rb] = 0guiding center

• non-commutative guiding-center components obey an uncertainty principle

2⇡`2Bis area per

flux quantum

Landau levels

• electrons in a partially-filled Landau level have residual dynamics given by

H =X

i<j

Vn(Ri �Rj)

[Rai , R

bj ] = �i�ij✏

ab`2B

!a!⇤b = 1

2 (gab + i✏ab)

det g = 1

[a, a†] = 1

a† = !aRa a = !⇤

aRa

• Vn depends on Landau level

| L(g)i /Y

i<j

⇣a†i � a†j

⌘q| 0(g)i

ai| 0(g)i = 0• Laughlin states are parametrized by a metric gab !

(a unimodular Euclidean metric)

• Q. If the Laughlin state is a variational state, what determines its metric?

H =X

i<j

Vn(Ri �Rj)

a = !⇤aR

a| L(g)i /Y

i<j

⇣a†i � a†j

⌘q| 0(g)i

• A. The metric must be chosen to minimize the correlation energy of the Hamiltonian

• The “naive” answer (that the metric is fixed by Lorentz or Galilean invariance) is not generally true in condensed matter, unless there is rotational invariance in the 2D plane in whuchthe electrons move.

• In fact, it is locally determined, if there is an inhomogeneous slowly-varying substrate potential

H =X

i

vn(Ri) +X

i<j

Vn(Ri �Rj)

vn(x)

deformationnear edgey

x

• Furthermore, the local electric charge density of the fluid with ν = p/q is determined by a combination of the magnetic flux density and the Gaussian curvature of the metric

J0e (x) =

e

2⇡q

✓peB

~ � sKg(x)

◆

Gaussian curvature of the metricTopologically quantized “guiding center spin”

• There are 4 fundamental properties of the FQHE state (in addition to the modular S, T matrices of its TQFT)

• the integers p,q where the chiral anomaly ν = p/q, and gcd(p,q) ≤ 2

• the geometric “guiding center spin” s

• the “gravitational anomaly” (Chiral energy anomaly) "

note that the TQFT modular matrices only determine c mod 8

c = (c� ⌫) + ⌫

(there is alsoa Landau orbit spin)

GC LO

• contributions from filled and partially-filled Landau levels to various topological quantities

1 1

11

c� 2p/q � 2

0

0

s

0

0 0

0

0

0

2⇥ 12

2⇥ 32

2⇥ ( sp + 52 )

0

0⌫ c

geometric spin “shift”

n = 0

n = 1

n = 2

(fully spin-polarized case)

standard picture• lowest-Landau level states: z =

x+ iyp2`B

`B =p(~/|eB|)

• filled Landau level (Slater determinant)

(x, y) = F (z)e�12 z

⇤z

=Y

i<j

(zi � zj)Y

i

e�12 z

⇤i zi

• 1/3 filled Landau level (Laughlin)

=Y

i<j

(zi � zj)3Y

i

e�12 z

⇤i zi

incompressible becauseof Pauli principle

incompressible becauseof electron repulsion

keeps electrons apart“flux” or “vortex” is

“attached” to electron

• elegant wavefunction, describes topologically-ordered fluid with fractional charge fractional statistics excitations

=Y

i<j

(zi � zj)3Y

i

e�12 z

⇤i zi

• exact ground state of modified model keeping only short range part of coulomb repulsion

• Validity confirmed by numerical exact diagonalization

Laughlin 1983

30 years later:unanswered question:we know it works, but why?

my answer:hidden geometry

• 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)

• 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

• shape of correlation hole (flux attachment) fluctuates, adapts to environment (electric field gradients)

e-e-

• polarizable, B x electric dipole = momentum,

e-x

origin of “inertial mass”

geometric distortion(preserving inversion symmetry)

electric polarizability

creates “curvature”of metric

shape=metric

new property:“spin” couplesto curvature

S

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 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 intocontinuum

gap incompressibility0 1 20

0.5

klB

Moore-Read ⌫ = 24

“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 = �abBpbe

k�B

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

(origin of Virasoro algebra in FQHE ?)

• electron in 2D Landau orbit (bound to 2D surface)

x

e-

guiding center R

magnetic fluxdensity B normal to 2D surface x=

Becomes a“fuzzy object”after kinetic

energy is quantized

`B =

✓~

|eB|

◆ 12

[Rx, Ry] = �i`2B

non-commutative geometry

cyclotron orbit

• Degenerate (flat) Landau levels

empty

filled

partially occupied

E

U(R1 �R2)

H =X

i<j

U(Ri �Rj)

effective Coulomb repulsion is analytic at

origin because of smoothing by Landau-

orbit form factor

[Rx, Ry] = �i`2B

quantum dynamics comesfrom non-commutativity

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`2B

[(Rx

1 �Rx

2), (Ry

1 �Ry

2)] = �2i`2B

• 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 avoidthis state

• Laughlin state

U(r12) =⇣A+B

⇣(r12)

2

`2B

⌘⌘e� (r12)2

2`2B 0

E2 symmetric

antisymmetric

• Solvable model! (“short-range pseudopotential”) 12 (A+B)

12B

rest all 0

| mL i =

Y

i<j

⇣a†i � a†j

⌘m|0i

ai|0i = 0 a†i

=Rx + iRy

p2`

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 ] = �ij

• “it describes particles in the lowest Landau level”

• “It is a Schrödinger wavefunction”

• “Its shape is determined by the shape of the Landau orbit”

• “It has no continuously-tunable variational parameter”

some widespread misconceptions about the Laughlin state

No Landau level was specified: all specifics of the Landau level are hidden in the form of U(r12)

Non-commutative geometry has no Schrödinger representation (this requires classical locality); it only has a Heisenberg representation.

The interaction potential U(r12) determines its geometry (shape)

Its geometry is a continuously-variable variational parameter

•Anatomy of Laughlin state

e-

e-

fractionally-chargede/3 quasiholes obeying

(Abelian) fractional statistics

electron with “flux attachment”to form a “composite boson”

Chiral edge mode with chiral anomalyand Virasoro anomaly

geometricedge dipole momentdetermined by Hall

viscosity

Topological and geometric bulk properties revealed by entanglement spectrum of cut

where is the geometry hiding?

• in the definition of ai

| mL i =

Y

i<j

⇣a†i � a†j

⌘m|0i

ai|0i = 0[ai, a†j ] = �ij

L(g) =gabRaRb

2`2B= 1

2

�a†a+ aa†

�

“guiding-center spin”

“unimodular”Euclidean metric

(det g = 1)

|0i = |0(g)i

| Li = | L(g)ia† = a†(g)

• composite boson: if the central orbital of a basis of eigenstates of L(g) is filled, the next two are empty

e-

e-

L(g)| mi = (m+ 12 )| mi

| 3Li =

Y

i<j

⇣a†i � a†j

⌘3|0i

• this correlation hole is equivalent to “attachment of three flux quanta” or vortices that travel with the particle, generating a Berry phase that cancels the Bohm-Aharonov phase and transmutes Fermi to Bose exchange statistics.

• this shape of the corelation hole - and hence its correlation energy - varies with the metric gab

differentmetrics

• If the Laughlin state is not a “wavefunction”, what are the holomorphic functions that are usually discussed?

• why are “wavefunctions” of model FQH states related to Euclidean 2D conformal field theory?

• What is the physical origin of incompressibility?

some questions

“wavefunction” coherent state amplitude

a(g)|r, gi = z⇤(r)|r, gi

• coherent states are parameterized by shape (gab) and location z(r)

• these are non-orthogonal and overcomplete

S(r, r0; g) = hr, g|r0, gi = ez⇤z0

e�12 z

⇤ze�12 z

0⇤z0

• eigenfunctions:Z

d2r0

2⇡`2BS(r, r0; g) (r0) = s (r)

solution:s = 1

(degenerate) (r) = F (z)e�

12 z

⇤z

holomorphic!

• coherent state representation

| (F, g)i = F (a†(g))|0(g)i

| (F, g)i =Z

d2r

2⇡`2Be�

12 z

⇤zF (z)|r(z), gi

• Schrodinger wavefunction in some Landau level

| n(r, g)i = |r(z), gi ⌦ | niLandauorbitstate

guiding-centercoherent statewith metric g

Landau-levelcoherent statewith metric g

nF (r0) =

Zd2r

2⇡`2Be�

12 z

⇤zF (z)hr0| n(r, g)i

• if “accidentally”, the Landau level is the n = 0 level of Galilean-invariant Landau levels with cyclotron effective mass tensor mgab

0F (r) = e�12 z

⇤zF (z)z =

x+ iyp2`B

!

• People who though they were working with lowest-Landau-level FQHE wavefunctions were in fact working with Landau-level-agnostic coherent state amplitudes!

• Laughlin (model state) coherent state amplitude

(r1, . . . , rN ) =Y

i<j

(zi � zj)mY

i

e�12 z

⇤i zi

This was recognized as a conformal block of a (free boson) 2D Euclidean conformal field theory

• Same thing for Moore-Read, Read-Rezayi states, etc.

• WHY? (there is no scale invariance in the gapped, incompressible FQH states!)

• usual suggestion: edge states of FQH states are described by 1+1d cft.

• But NO Lorentz invariance! so not cft.

• 1+1 cft requires a space-time metric

ds

2 = dx

2 � v

2dt

2

universal speedof massless excitations

There is no universal speed of FQH edge modes!They propagate with different speeds!

• suggested explanation (later)

Virasoro algebra

conformal field theory

QHE

playsa key role

playsa key role

no directconnection

(chiral “gravitational” orenergy anomaly)

seen inEntanglement

Spectrum

• The metric defines the shape of the coherent state at the center of the “symmetric gauge” basis of guiding-center states

centralcoherent

state

| m(g)i = (a†(g))mpm!

| 0(g)i

a(g)| 0(g)i = 0L(g) =

gab2`2B

RaRb

[L(g), a†(g)] = a†(g)• Guiding-center “spin” (rotation

operator) is defined by the metric

different choices of metric:(“squeezed” relative to each other)

| 0(g)i

• Model cft-based states such as the Laughlin state have a constant (flat, rigidly-fixed) metric

• In real FQH states of electrons contained in a non-uniform background potential, the metric varies locally and dynamically to allow the incompressible fluid to adjust to non-uniform flow induced by the background.

• The metric then becomes an emergent dynamical collective degree of freedom of the FQH state.

gab(r, t)

FDMH, Phys. Rev. Lett. 107, 116801 (2011)

• 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

• 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

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 bosonhas inversion symmetry

about its center

It has a “spin”

.....

.....−1 0 013

13

13

12

32

52

L = 12

L = 32−

s = �1

e

2/5 state

e.....

.....−

1 0

12

32

52

−

0 0125

25

25

25

25

L = 2

L = 5

s = �3

L =gab2`2B

X

i

RaiR

bi

Qab =

Zd2r rarb�⇢(r) = s`2Bg

ab

second moment of neutral composite boson

charge distribution

• The composite boson behaves as a neutral particle because the Berry phase (from the disturbance of the the other particles as its “exclusion zone” moves with it) cancels the Bohm-Aharonov phase

• It behaves as a boson provided its statistical spin cancels the particle exchange factor when two composite bosons are exchanged

(�1)pq = (�1)p

(�1)pq = 1

fermionsbosons

p particlesq orbitals

• 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)

• long-wavelength limit of GMP collective mode is fluctuations of (spatial) metric (analog of “graviton”)

�E / (distortion)

2

FDMH, Phys. Rev. Lett. 107, 116801 (2011)

• static guiding-center structure factor

⇢(q) =X

j

eiq·Rj

[⇢(q), ⇢(q0)] = 2i sin( 12q ⇥ q0`2B)⇢(q + q0)

• generators of area preserving diffeomorphisms

s(q) =⌫

N(h⇢(q)⇢(�q)i � h⇢(q)ih⇢(�q)i)

characterizes zero-point quantum fluctuationsvanishes as |q|4 at long wavelengths,

this is quantum fluctuations of the metric

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-or<:

L c.)

l&(do{i>

()P(gt

oqe0)t+io>

) )'d

o (di

4|oE(dtrMeQ

l.+ q

o9- F'i.1

f-O-

@d'-c )r

u! +, ij[r.

qAL

o-q)w

O

4-.O

rttd

59 5zi

x-

boi:9or3d f,'^

\oc!

^'l'.ll-

.= > ;1

-!-(U

i6^)

:JDTS

+Y

?".'C

. = rD

1u*"^-J

$Tl+

trrboo

o'*, ;"".

=Y

c6€t-

..3ai

Y :8&

Pt

rn o\

oOil tl

cD=-J

\O.i

\ouJ

o o

9€rFA

6-:O

t *

,r O

rXttffvb0+{ti

C)

hooPq>d(0()p

// a

$cr)

-'(U=>u (!,

':- r

L-iiO

J;+=€a:o=v

\)a

ri1 0\

ciollll=N

= t,-

\OT

\.OU

JS0(q) =1

N

X

ij

heiq·Rie�iq·Rj i � heiq·Riihe�iq·Rj i

s(q) = �S0(q)

(per flux , instead of per particle)

0 < �E(k) <O(k4)

s(k)

variational upper bound tocollective excitation energy

� =N

Norb

number of orbitals = number of flux quanta through

2D surface

must be O(k4) if gapped

Text

• crucial new physics:

composite bosons couple to the combination

peB(r)� ~sK(r)

charge of compositeboson guiding-center

“spin” of boson

Gaussian curvatureof metric

* gauge field is peAµ(r)� ~s⌦µ(r)

analog of spin-connectionrelated toWen and Zee 1992

+ Chern-Simons

• The shape of the composite boson is determined by minimizing the sum of the correlation energy and the background potential energy.

• If there is no background potential, the metric is flat and the charge density is uniform

• If there is a background potential gab(r) varies with position to give a charge density fluctuation

�⇢(r) = esK(r)K(r) = 1

2@a@bgab + 1

8gab✏cd✏ef@eg

ac@fgbd

Gaussian curvature of metric {from Berry phase associated with shape change

{from variation of

second moment of charge distribution“spin”

• metric deforms (preserving det g =1)in presence of non-uniform electric field

potential near edge

fluid compressedby Gaussian curvature!

produces a dipole momemt

• Hall viscosity

(plus a similar term from the Landau orbit degrees of freedom (Avron et al))

dv

y

dx

current of px

in x-direction(stress force)

⌘xxxy

�ab = ✏be⌘

aecfH ✏cf@cv

d

⌘abcd =eBs

4⇡q12

�gac✏bd + gbd✏ac + a $ b

�

Hall viscosity determines a dipole moment per unit length at the edge of the fluid

• Total guiding center angular momentum of a fluid disk of N elementary droplets

statistical(conformal) spin

geometric(guiding-center)

spin(dipole at edge)

momentum

electric dipole

dPa = ~e`2B

✏abdpb

momentum dP

The dipole at a segment of theedge has a momentum

momentum dipole

HdPa = 0

doesn’t contributeto total momentum:

�Lz(g) = ~H

✏abgbcrcdPa 6= 0

it does contribute an extra term to total angular momentum:

circular droplet

3

between the two Fermi momenta there are qN orbitalswhere N = pN . For example, for ⌫ = 1/2 and ⌫ = 1/3Laughlin states, we have the following root momentumoccupations (i.e. the occupations of the state correspond-ing to the monomial m

�0) for two circumferences L and2L, See FIG. 1. We have occupation numbers for sets ofdiscrete momenta {k : k = ⇡m/L, m even and m > 0}for ⌫ = 1/2 and {k : k = ⇡m/L, m odd and m > 0} for⌫ = 1/3.

In order to have the two edges not interact with eachother, we need N ! 1. Practically, we can have onlyfinite N , and this restricts the largest available L for thefixed N . If L increased furthur than this value, then theJack polynomial becomes a wavefunction of Calogero-Sutherland model with its two edges interacting strongly[3]. If L is too small, then the Jack becomes a charge-density-wave state. For fixed L, the occupation numbersconverge to some limits as the number of particles in-crease.

To calculate the expectation value of the occupationnumber operator n

m

for each momentum k = 2⇡m/L,we evaluate

hnm

i

↵

�0=

h ↵�0|n

m

| ↵�0i

h ↵�0| ↵�0i =

P��0

a

�0,�(↵)2h

�

|nm

| �

iP��0

a

�0,�(↵)2

(17)where h

�

|nm

| �

i is the multiplicity of m in the parti-tion �. Here, it is understood that n

0

= 0 for ⌫ = 1/2Laughlin state, n

1/2

= 0 for ⌫ = 1/3 Laughlin state, andso on. See FIG. 2, 3, 4, 5 and 6.

FIG. 2: ⌫ = 1/2 Laughlin state density profile : } markersfor N = 14 data, and + markers for N = 15 data. It plotsdata with di↵erent L = 15 to 24 with increments by 0.5 (inunits of `B). A color gradient from blue (L = 15) to red(L = 24) is used to distinguish data with di↵erent Ls. Forlog n(k) versus log k plot, we used data points with k = 2⇡

L fordi↵erent L. The linear fit of the log log plot gives log n(k) =0.963 log k � 0.010 with the norm of residues 0.003

Using the occupation numbers, we can first check if they

FIG. 3: ⌫ = 1/3 Laughlin state density profile : same asFIG. 2 with L = 12.5 to 22. For logn(k) versus log k plot,we used k = 3⇡

L . The linear fit of the log log plot giveslog n(k) = 1.853 log k�0.609 with the norm of residues 0.017.

FIG. 4: ⌫ = 1/4 Laughlin state density profile : X markersfor N = 11 data, L = 12.5 (blue) to 22 (red) , and for log n(k)versus log k plot, used k = 4⇡

L . The linear fit of the log log plotgives log n(k) = 2.722 log k � 0.530 with the norm of residues0.028

satisfy Luttinger’s sum rule :

�N(k) = �⌫m+mX

m

0=0 or 1/2

hnm

0i

↵

�0(18)

where k = 2⇡m/L. We expect �N(k) should vanish ask gets large. Because we are limited by the finite size, wecalculate �N(k) only up to the center of the fluid. SeeFIG. 7, 8, 9, 10 and 11.

The dipole moment p

y per length relative to theuniform background with density ⌫/2⇡`2

B

calculated from

n(k)edge of Laughlin 1/3

orbital occupation

chiral Virasoro algebra at the edge, and in entanglement spectra.

• multicomponent quantum Hall edge states do not have a universal speed, so are not Lorentz and conformally invariant.

• components of the cft energy momentum tensor:

Momentum density is independent of v:

Energy density and stress are proportional to v

Tracelessness (in flat space-time) is independent of vT 00 + T x

x

= 0Energy current density is proportional to v2

T x

0 = v2(T � T )

T 00 = �T x

x

= v(T + T )

T 0x

= T � T

The only speed-independent properties are

• The (signed) Virasoro algebra of the Fourier components of the momentum density (with the topologically-conserved chiral central charge

c = c� cThis is a fundamental quantity that has nothing to do with conformal invariance (and in fact must vanish in a “true” (modular-invariant) 1+1d cft )

• Tracelessness of the energy-momentum tensor (1d pressure = energy density), which is true for linearly-dispersing modes, independent of their speed.

It controls a “Casimir momentum”124~c/L

• Tracelessness of the 2D stress tensor in the Euclidean field theory is the absence of hydrostatic 2D pressure

• Incompressible 2D quantum fluids at T = 0 do not transmit pressure through their bulk because of their energy gap (no gapless sound modes) - they only transmit pressure around their edges via gapless edge excitations.

• The traceless stress tensor of Euclidean 2d conformal field theory (and its dependence on a metric) may explain its applicability to FQHE

P = � 12

�T x

x

+ T y

y

�= 0

One final result

• In the “trivial” non-topologically-ordered integer QHE (due to the Pauli principle)

c = ⌫ = Chern number

c� ⌫ = 0

• the (guiding-center) “orbital entanglement spectrum” of Li and Haldane is insensitive to filled (or empty) Landau levels or bands, and allows direct determination of non-zero c� ⌫

previous methods used the onerous calculation of the“real-space” entanglement spectrum to find c

• Hall viscosity gives “thermally excited” momentum density on entanglement cut, relative to “vacuum”, at von Neumann temperature T = 1

0

20

40

60

80

100

120

140

0 5 10 15 20 25 30 35 40

"laughlin3_n_13.cyl16.14" using 2:4

(NOT “real-space cut” which requiresthe Landau orbit degrees of freedom and theirform factor to be included

ORBITAL CUT

signed conformalanomaly (chiral stress-

energy anomaly)chiral

anomaly

virasoro levelof sector

“CASIMIR MOMENTUM” term

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 100 200 300 400 500 600 700 800 900

mea

n Vi

raso

ro le

vel -

exp

ecte

d

L^2

L_A = 148, plevel = 12, +1/6L_A = 149, plevel = 12

L_A = 150, plevel = 12, +1/6L_A = 149, plevel = 11

1/36

Yeje Park, Z Papic, N Regnault1

24(c� ⌫) =

1

24

�1� 1

3

�=

1

36

1

36

Matrix-product state calculation on cylinder with circumference L (“plevel” is Virasoro level at which the auxialliary space is truncated)

• How universal is the Thermal Hall effect formula? It also depends on

• If Lorentz invariance is present, its essentially the same calculation as Casimir momentum

• When Lorentz invariance is broken by different speeds for different modes, but the remain independent, the result still stands

• How much information about the Hamiltonian (T00 ) is needed? Is there a clean “gravitational” derivation just based on the momentum T0x Virasoro anomaly?

c

JaE =

c

12

(2⇡kBT )2

2⇡~ ✏abgbc2

~g

c2= �

~rT

T

T

0x

=

1

L

X

m

T

m

exp(2⇡ix/L)

[Tm, Tn] = (m� n)Tm+n + 112 cm(m2 � 1)�m+n,0

chiral central charge

Can we obtain the Thermal Hall effect just from this plus

“gravity”?

Momentum density is universal:

• Hall viscosity

(plus a similar term from the Landau orbit degrees of freedom (Avron et al))

dv

y

dx

current of px

in x-direction(stress force)

⌘xxxy

�ab = ✏be⌘

aecfH ✏cf@cv

d

⌘abcd =eBs

4⇡q12

�gac✏bd + gbd✏ac + a $ b

�

Hall viscosity determines a dipole moment per unit length at the edge of the fluid

• Total guiding center angular momentum of a fluid disk of N elementary droplets

statistical(conformal) spin

geometric(guiding-center)

spin(dipole at edge)

momentum

electric dipole

dPa = ~e`2B

✏abdpb

momentum dP

The dipole at a segment of theedge has a momentum

momentum dipole

HdPa = 0

doesn’t contributeto total momentum:

�Lz(g) = ~H

✏abgbcrcdPa 6= 0

it does contribute an extra term to total angular momentum:

circular droplet

3

between the two Fermi momenta there are qN orbitalswhere N = pN . For example, for ⌫ = 1/2 and ⌫ = 1/3Laughlin states, we have the following root momentumoccupations (i.e. the occupations of the state correspond-ing to the monomial m

�0) for two circumferences L and2L, See FIG. 1. We have occupation numbers for sets ofdiscrete momenta {k : k = ⇡m/L, m even and m > 0}for ⌫ = 1/2 and {k : k = ⇡m/L, m odd and m > 0} for⌫ = 1/3.

In order to have the two edges not interact with eachother, we need N ! 1. Practically, we can have onlyfinite N , and this restricts the largest available L for thefixed N . If L increased furthur than this value, then theJack polynomial becomes a wavefunction of Calogero-Sutherland model with its two edges interacting strongly[3]. If L is too small, then the Jack becomes a charge-density-wave state. For fixed L, the occupation numbersconverge to some limits as the number of particles in-crease.

To calculate the expectation value of the occupationnumber operator n

m

for each momentum k = 2⇡m/L,we evaluate

hnm

i

↵

�0=

h ↵�0|n

m

| ↵�0i

h ↵�0| ↵�0i =

P��0

a

�0,�(↵)2h

�

|nm

| �

iP��0

a

�0,�(↵)2

(17)where h

�

|nm

| �

i is the multiplicity of m in the parti-tion �. Here, it is understood that n

0

= 0 for ⌫ = 1/2Laughlin state, n

1/2

= 0 for ⌫ = 1/3 Laughlin state, andso on. See FIG. 2, 3, 4, 5 and 6.

FIG. 2: ⌫ = 1/2 Laughlin state density profile : } markersfor N = 14 data, and + markers for N = 15 data. It plotsdata with di↵erent L = 15 to 24 with increments by 0.5 (inunits of `B). A color gradient from blue (L = 15) to red(L = 24) is used to distinguish data with di↵erent Ls. Forlog n(k) versus log k plot, we used data points with k = 2⇡

L fordi↵erent L. The linear fit of the log log plot gives log n(k) =0.963 log k � 0.010 with the norm of residues 0.003

Using the occupation numbers, we can first check if they

FIG. 3: ⌫ = 1/3 Laughlin state density profile : same asFIG. 2 with L = 12.5 to 22. For logn(k) versus log k plot,we used k = 3⇡

L . The linear fit of the log log plot giveslog n(k) = 1.853 log k�0.609 with the norm of residues 0.017.

FIG. 4: ⌫ = 1/4 Laughlin state density profile : X markersfor N = 11 data, L = 12.5 (blue) to 22 (red) , and for log n(k)versus log k plot, used k = 4⇡

L . The linear fit of the log log plotgives log n(k) = 2.722 log k � 0.530 with the norm of residues0.028

satisfy Luttinger’s sum rule :

�N(k) = �⌫m+mX

m

0=0 or 1/2

hnm

0i

↵

�0(18)

where k = 2⇡m/L. We expect �N(k) should vanish ask gets large. Because we are limited by the finite size, wecalculate �N(k) only up to the center of the fluid. SeeFIG. 7, 8, 9, 10 and 11.

The dipole moment p

y per length relative to theuniform background with density ⌫/2⇡`2

B

calculated from

n(k)edge of Laughlin 1/3

orbital occupation

✄

area (perimeter) part of entanglement is naturally measuredin the boundary length per degree

of freedom

measured in diameters of composite bosons

• other consequences of geometry

• long-wavelength of GMP mode is “graviton”

• q4 behavior of guiding-center structure factor is due to zero-point fluctuations f metric (components do not commute, determinant is Casimir)

• Conformal field theory has a fixed metric; geometrodynamics is like extension of special relativity to GR!

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5 3 3.5

laughlin 10/30

ΔE

“roton”

(2 quasiparticle + 2 quasiholes)

goes intocontinuum

gap incompressibilityk�B

Laughlin

⌫ = 13

unfortunately, long-wavelength limit of “graviton” collective mode is hidden in “two-roton continuum”

numericalfinite-size

diagonalization

2

The interaction also has rotational invariance if

v(q) = v(qg), q2g ≡ gabqaqb, (6)

where gab is the inverse of a positive-definite unimodular(determinant = 1) metric gab; this will only occur if theshape of the Landau orbits are congruent with the shapeof the Coulomb equipotentials around a point charge onthe surface. In practice, this only happens when there isan atomic-scale three-fold or four-fold rotation axis nor-mal to the surface, and no “tilting” of the magnetic fieldrelative to this axis, in which case gab = ηab.I will assume that translational symmetry is unbroken,

so ⟨c†αcα′⟩ = νδαα′ , where ν is the “filling factor” of theLandau level. (In a 2D system, this will always be trueat finite temperatures, but may break down as T → 0).Then ⟨ρ(q)⟩ = 2πνδ2(qℓB). Note that the fluctuationδρ(q) = ρ(q) − ⟨ρ(q)⟩ also obeys the algebra (4). I willdefine a guiding-center structure factor s(q) = s(−q) by

12 ⟨{δρ(q), δρ(q

′)}⟩ = 2πs(q)δ2(qℓB + q′ℓB). (7)

This is a structure factor defined per flux quantum, and isgiven in terms of the GMP structure factor s(q) of Ref.[1](defined per particle) by s(q) = νs(q). I also define sa(q)≡ ∂s(q)/∂qa, sab(q) ≡ ∂2s(q)/∂qa∂qb, etc.In the “high-temperature limit” where |v(r)| ≪ kBT

for all r, but kBT remains much smaller than the gapbetween Landau levels, the guiding centers become com-pletely uncorrelated, with ⟨c†αcα′c†βcβ′⟩ − ⟨c†αcα′⟩⟨c†βcβ′⟩

→ s∞δαβ′δβα′ , with s∞ = ν + ξν2, where ξ = −1if the particles are spin-polarized fermions, and ξ =+1 if they are bosons (which may be relevant forcold-atom systems). Note that for all temperatures,limλ→∞ s(λq) = s∞, while s(0) = limλ→0 s(λq) =kBT/

!

∂2f(T, ν)/∂ν2"

"

T

#

, where f(T, ν) is the free en-ergy per flux quantum. s(0) vanishes at T = 0, and atall T if v(λq) diverges as λ → 0; the high temperatureexpansion at fixed ν, for rq ≡ eaϵabqbℓ2B, is

s(q)− s∞(s∞)2

= −

$

v(q) + ξv(rq)

kBT

%

+O

$

1

T 2

%

. (8)

The correlation energy per flux quantum is given by

ε =

&

d2qℓ2B4π

v(q) (s(q)− s∞) . (9)

The fundamental duality of the structure function (al-ready apparent in (8), and derived below) is

s(q)− s∞ = ξ

&

d2q′ℓ2B2π

eiq×q′ℓ2B (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.

Consider the equilibrium state of a system with tem-perature T and filling factor ν with the Hamiltonian(2). The free energy of this state is formally given byF [ρeq], where ρeq(T, ν) is the equilibrium density-matrixZ−1 exp(−H/kBT ) and F [ρ] is the functional

F [ρ] = Tr (ρ(H + kBT log ρ)) , (11)

which, for fixed ν, is minimized when ρ = ρeq. TheAPD corresponding to a shear is Ra → Ra + ϵabγbcRc,parametrized by a symmetric tensor γab = γba. Let ρ(γ)= U(γ)ρeqU(γ)−1, where U(γ) is the unitary operatorthat implements the APD, and F (γ) ≡ F [ρ(γ)] = F [ρeq]+ O(γ2), which is is minimized when γab = 0. The freeenergy per flux quantum has the expansion

f(γ) = f(T, ν) + 12G

abcd(T, ν)γabγcd +O(γ3), (12)

where Gabcd = Gbacd = Gcdab. The “guiding-center shearmodulus” (per flux quantum) of the state is given by Gac

bd= ϵbeϵdfGaecf , with Gac

bd = Gcadb, and Gac

bc = 0. (Notethat in a spatially-covariant formalism, both stress σa

b(the momentum current) and strain ∂cud (the gradientof the displacement field) are mixed-index tensors thatare linearly related by the elastic modulus tensor Gac

bd.)The entropy is left invariant by the APD, and the onlyaffected term in the free energy is the correlation energy,which can be evaluated in terms of the deformed struc-ture factor s(q, γ), given by

s∞ + ξ

&

d2q′ℓ2B2π

eiq×q′ℓ2B (s(q′)− s∞)eiγabqaq

′

bℓ2

B , (13)

with γab ≡ ϵacϵbdγcd. This gives Gabcd(T, ν)γabγcd as

γabγcdϵaeϵcf

&

d2qℓ2B4π

v(q)qeqfsbd(q;T, ν). (14)

Assuming only that the ground state |Ψ0⟩ of (2) hastranslational invariance, plus inversion symmetry (so ithas vanishing electric dipole moment parallel to the 2Dsurface), GMP[1] used the SMA variational state |Ψ(q)⟩∝ ρ(q)|Ψ0⟩ to obtain an upper bound E(q) ≤ f(q)/s(q)to the energy of an excitation with momentum !q (orelectric dipole moment eeaϵabqbℓ2B), where

f(q) =

&

d2q′ℓ2B4π

v(q′)!

2 sin 12q × q′ℓ2B

#2s(q′, q),

s(q′, q) ≡ 12 (s(q

′ + q) + s(q′ − q)− 2s(q′)) . (15)

Other than noting it was quartic in the small-q limit,GMP did not not offer any further interpretation of f(q).It can now be seen to have the long-wavelength behavior

limλ→0

f(λq) → 12λ

4Gabcdqaqbqcqdℓ4B, (16)

and is controlled by the guiding-center shear-modulus.Then the SMA result is, at long wavelengths, at T = 0,

E(q)s(q) ≤ 12G

abcdqaqbqcqdℓ2B. (17)

Momentum ∝dipole moment

(inversion and translation invariant)

Gap for tangential electric polarization (no dielectric screening) single-mode approx.

• Geometric action

S =

Zd

3xL0 �H0

(reduces to electromagnetic Chern-Simons action when s = 0 (integer QHE))

H0 = J0U(J0g) J0 =1

2⇡pq~�peB � ~sJ0

g

�

Gaussian curvature

L0 =1

4⇡pq~✏µ⌫�

�peAµ � s⌦g

µ

�@⌫ (peA� � ~s⌦g

�)

electromagneticgauge potentials

spin connectionof metric

Jµg = ✏µ⌫�@⌫⌦

g�

correlation energy density

composite-boson density

energyfunction

(after Chern-Simons fields are integrated over)

correlation energy density

H0 = (detG)1/2U(G) = J0U(J0g)

µg = U(G) +Gab@U

@Gab

geometric chemical potential (of composite bosons)

Geometric distortion energy

shear-stress tensor (traceless) �a

b = 2Gbc@U

@Gac� �abGcd

@U

@Gcd

�aa = 0 �

ac (x)✏

bc = �

bc(x)✏ac

�

ac (x)g

bc(x) = �

bc(x)gac(x)

both expressions are symmetric in a $ b

Stress tensor is traceless because the gapped quantum incompressible fluid does not transmit pressure

(unlike incompressible limit of classical incompressible fluid,which has speed of sound vs ! 1

�abcdH (g) = 1

2

�✏acgbd + ✏adgbc + ✏bcgad + ✏bdgac

�

Euler equation

• action is minimized by Hall viscosity condition

⌘acbd(G) = 12~s✏be✏dfJ

0�aecfH (g)

covariant spatial gradientof Ja = J0va

J0�ab (G) = ⌘acbd(G)rg

cJd

Tracelessstress-tensor

Hall viscosity ⌘acbd = �⌘cadbdissipationless

⌘abac = ⌘baca = 0 incompressible

fluid flow-velocity

compositeboson current

• composite boson current

Ja =1

2⇡pq~�✏ab(peEb � @bµg)� ~sJa

g

�

responds to gradient of geometric chemical potential as well as electric field

peEaJa 6= 0

pe(J0Ea + ✏abJaB) 6= 0

J0 =1

2⇡pq~�✏abpeB � ~sJ0

g

�

Energy flow from electromagnetic field to FQH fluid

tangential momentum flow from electromagnetic field to FQH fluid

Gaussian curvature density and current

• Action gives gapped spin-2 (graviton-like) collective mode that coincides at long wavelengths with the “single-mode approximation” of Girvin-MacDonald and Platzman.

• charge fluctuations relative to the back-ground charge density fixed by the magnetic flux are given by the Gaussian curvature

J0g = � 1

2@a@bgab + 1

8gac✏bd✏ef

�@eg

ab� �

@fgcd�

�J0e =

e⇤s

2⇡J0g

zero-point fluctuations of gaussian curvaturegive quantitatively correct O(q4) structure factor

second derivative of metric

• 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)

• New collective geometric degree of freedom leads to a description of the origin of incompressibility in FQHE in a continuum “geometric field theory”

• many new relations: guiding-center spin characterizes coupling to Gaussian curvature of intrinsic metric, stress in fluid, guiding- center structure-factors, etc.

http://wwwphy.princeton.edu/~haldane

also see arXiv (search for author=haldane)

Can be also be accessed through Princeton University Physics Dept home page (look for Research:condensed matter theory)

SUMMARY