• hidden geometry of the Laughlin state

• geometrodynamics of the FQHE

• geometry and entanglement

Geometry Topology and Entanglement in the FQHE

F. Duncan M. HaldanePrinceton University

ICTP conference on Geometrical Aspects of Quantum States in Condensed Matter, Trieste, July 5 , 2013

Laughlin state• originally introduced as a “lowest Landau level wavefunction”

(I will explain why this is a misleading characterization)

qL({ri}) =

Y

i<j

(zi � zj)qY

i

e�12 z

⇤i zi

• usual interpretation of z is

z =x+ iyp2`2B

2⇡`2Bmagnetic area:

(contains one flux quantum h/e)

The most striking feature for theorists

is that this is holomorphic!

• Laughlin explained that his wavefunction had a holomorphic factor because it was a lowest-Landau level wavefunction.

• I will explain why the holomorphic character has a quite different origin!

• This will explain why the Laughlin state can be found in systems unrelated to lowest Landau level systems

• It will also reveal the fundamental geometric degree of freedom of the FQHE state.

• non-relativistic Galileian-invariant Landau levels

H =|~p� e ~A(r)|2

2m= 1

2~!c(a†a+ aa†)

• Landau level ladder operators (in the “symmetric gauge”):

a = 12z +

@

@z⇤a† = 1

2z⇤ � @

@z [a, a†] = 1

a (r) = 0

lowest Landau level wavefunctions

holomorphicfunction

Gaussian⇥

(Note isotropiceffective mass)

(r) = f(z)e�12 z

⇤z

standard derivation

• The q = 3 Laughlin state was confirmed (by numerical exact diagonalization studies) to be the essential description of the 1/3 FQHE

• The holomorphic factor is incidentally noticed to be a cft correlator (conformal block) of the free boson cft with boson radius R= √(2/q) . (why?)

qL({ri}) =

Y

i<j

(zi � zj)qY

i

e�12 z

⇤i zi

• 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

problems with this• The “explanation” of why the Laughlin state is correct are vague

rationalizations, without quantitative content.

• The relation to cft is an empirical observation, and remains unexplained

• The 1/3 FQHE state also occurs in the second Landau level and is described by the same Laughlin state (but not the same “wavefunction”)

• It is recently also found on Chern-insulator lattice systems (by numerical diagonalization)

The physics of the FQHE in Landau levels is the physics of non-commuting “guiding centers” (quantum geometry) which cannot be described in terms of Schrodinger wavefunctions

• q =1 case is Slater determinant of filled lowest Landau level, uncorrelated, no topological order

• q >1 case is highly correlated, topologically-ordered, related to a Jack polynomial

/Y

i<j

(zi � zj)qY

i

e�12 z

⇤i zi

Y

i<j

(zi � zj)q =

Y

i<j

(zi � zj)q�1J↵

� (z1, . . . , zN )

↵ = �2 Jack parameter� = {q(N � 1), q(N � 2), . . . , q, 0}padded partition of N parts

• Jack symmetric polynomials are:

• homogeneous and symmetric

• eigenfunctions of a Laplace-Beltrami operator

↵X

i

(zi@zi)2J↵

� (z) +X

i<j

(zi � zj)�1(zi@zi � zj@zj )J

↵� (z) = E�(↵)J

↵� (z)

↵ = �2

↵ = �3

↵ = �k, k � 1 = 3, 4, 5 . . .

Laughlin (and quasi holes)

Moore-Read

Read-Rezayi Zk-1

parafermion states

• single particle states

• radially ordered

m / zme�12 z

⇤z

• “root partition” of Jack:nm(�) =

X

i

�m,�i

10010010010010|000000 . . . , q = 3

11111111111111|000000 . . . , q = 1

“not more than one particle in any group of q consecutive orbitals”

“Fermi point”

exclusion statistics ( q > 1)

(fuzzy) boundary ofcircular droplet

⌫ = 1/q ⌫ = 0

• “squeezing property” of Jacks

(z1 � z2)3 = (z31z

02 � z01z

32)� 3(z21z

12 � z11z

22)

10010|000 . . .

10010|000 . . .

a “squeeze”

�

µ � �

root partition

01100|000 . . . partitiondominated by root

01100|000 . . .

• actual occupation of orbitals differs from root.

A Luttinger-type sum rule

• in thermodynamic limit (map to cylinder, with circumference L, then infinite plane)

N =X

m

n0m =

X

m

nm

root occupationpattern

true occupationpattern after squeezing

k =2⇡m

LZ 1

�1dk (n(k)� n0(k)) = 0 n0(k) = 1/q k < qkF

n0(k) = 0 k > qkF{

• q = 1

⇢ ! ⌫

2⇡`2B particle density

kn = 0

n = 1“guiding center”occupation function

• q = 3

“3kF”

n → 1/3

n = 0n / (3kF � k)2

particle density⇢ ! ⌫

2⇡`2B

“guiding center”occupation function

“Luttinger”

6

because the other half can be obtained by mirror symme-try. The mean occupations hnmi

0

for each k = (2⇡/L)mare plotted against k rather than m for several values ofL.

Each occupation plot seems to follow a smooth profilethat might appear in the limit L ! 1. This observa-tion allows us to observe how well these wavefunctionsof finite numbers of particles agree with the behavior ofn(k) near k = 0 described by the chiral boson theory. Ineach occupation profile plot, we calculate the linear fit oflog n(k) versus log k with those momenta k = (2⇡/L)m

0

,the first non-vanishing occupation numbers. We observethat they are quite linear, and their linear fit coe�cientis the exponent r in n(k) / kr as k ! 0. For each Laugh-lin 1/2, 1/3, 1/4 state, the exponent is calculated to be0.963, 1.853, 2.722 respectively, while the expected expo-nents are 1, 2 and 3. For each Moore-Read 2/2 and 2/4state, the exponent is calculated to be 1.076 and 1.879while the expected exponents are 1 and 2.

Moreover, if we assume the exponents from the chiralboson theory and accept the form of occupation numbern(k) = Akq�1+. . . near k = 0, then we can calculate thenumerical factor A which is inaccessible in the field the-ory. Defining derivatives of n(k) by finite di↵erence, weobtain n0(k) for Laughlin 1/2 state and n00(k) for Laugh-lin 1/3 state. These are plotted in Fig. 8 and 9, respec-tively. We find that near k = 0, n(k) = 1.013k + O(k2)for Laughlin 1/2 state, and n(k) = 0.870k2 + O(k3) forLaughlin 1/3 state. We notice that these numerical fac-tors A might become rational numbers such as 1 and 7/8respectively in the thermodynamical limit.

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

k = 2⇡mL

n(k)

1/2 Laughlin state occupation numbers

FIG. 3: ⌫ = 1/2 Laughlin state density profile : ⇥ markersfor N = 14 data, and • markers for N = 15 data. It plotsdata obtained with di↵erent L = 15 to 24 with increments by0.5 (in units of `B). The horizontal line is 1/2. The linear fitof log(k = 2⇡/L) versus log n(k) gives log n(k) = 0.963 log k�0.010 with the norm of residues 0.003

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

k = 2⇡mL

n(k)

1/3 Laughlin state occupation numbers

FIG. 4: ⌫ = 1/3 Laughlin state density profile : same as Fig. 3 with L = 12.5 to 22. The horizontal line is 1/3. Thelinear fit of log(k = 3⇡/L) versus log n(k) gives log n(k) =1.853 log k � 0.609 with the norm of residues 0.017.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

k = 2⇡mL

n(k)

1/4 Laughlin state occupation numbers

FIG. 5: ⌫ = 1/4 Laughlin state density profile : • markersfor N = 11 data, L = 12.5 to 22 . The horizontal line is1/4. The linear fit of log(k = 4⇡/L) versus log n(k) giveslog n(k) = 2.722 log k� 0.530 with the norm of residues 0.028

B. Luttinger’s theorem

Given the occupation numbers, we can also verify thethat they satisfy the Luttinger sum rule[16]. For a onedimensional system with “Fermi surface” singularities inthe occupation numbers n(k) at “Fermi points” ki, thisstates that

N

L=

Zdk

2⇡n(k) =

Zdk

2⇡n0

(k), (30)

where in a Luttinger liquid (the 1D analog of a Fermiliquid), n

0

(k) is a integer topological index that is con-

(cft shows small k+3kF behavior is quadratic)

Finite-size Jack on cylinder(can’t get very close to -3kF)

1/3

-3kF

• The occupation functions are highly structured, with generalized Fermi point singularities, and directly reflect the properties of the Jack polynomials, which are deeply related to conformal field theory.

• In contrast, the lowest-Landau electron densities derived from the interpretation of the Laughlin state as a Schroedinger wavefunction are rather smooth and featureless ( WHY?)

In fact. all the non-trivial structure is present in the “guiding-center” degrees of freedom without

reference to Landau level structure.

• The FQH is a correlated state of the non-commuting GUIDING CENTERS of quantized Landau orbits, obeying the algebra

[Rx, Ry] = �i`2B

• The classical coordinate of the electrion combines this with the Landau orbit radius

r = R+ R

[Rx, Ry] = i`2B

[Rx, Ry] = 0

[rx, ry] = 0}• A Schroedinger wavefunction requires both

degrees of freedom

• The fundamental description of FQH states is a state in the many-guiding-center Hilbert space

• To make a wavefunction, with all particles in the same Landau level, we must “dress” it with a trivial state describing Landau orbits:

• We can recover by “undressing” Laughlin’s wavefunction.

Schrödinger vs Heisenberg

• resolution of conflict: the two formulations of QM are equivalent:

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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1 of 1 5/21/12 12:27 AM

(r) | i

wavefunctionin real space(classical geometry)

state inin Hilbert space

(r) = hr| i

iff ∃ s.t. |ri hr|r0i = 0, r 6= r0

requires an orthonormal basis in real space obeying classical locality

• classical locality (and Schrödinger-Heisenberg equivalence) fails after Landau quantization!

r = R+ R

⇥

O

r

R

Rclassicalcoordinate

guiding centercoordinate

Landau orbitradius vector

r = raea[ra, rb] = 0

`2B =~eB

> 0

e�

pa � eAa(r) ⌘ ✏ab~Ra/`2B

[Ra, Rb] = 0

[Ra, Rb] = �i`2B✏ab

[Ra, Rb] = i`2B✏ab

non-commutative algebra

• residual guiding center degrees of freedom are non-commutative

r = R+ Reliminatedby Landau

quantization

[Ra, Rb] = �i`2B✏ab

• isomorphic to phase space, obeys uncertainty principle

guiding centers cannot be localized within an area less than 2⇡`2B

• The Hamiltonian governing the residual guiding-center degrees of freedom:

H =

Zd2q`B2⇡

U(q)X

i<j

eiq·(Ri�Rj)

V (q) =

Zd2rij V (rij)e

iq·rij

Fourier transformed Coulomb interaction

Landau level form factor(n = landau level index)

u = 12 |q|

2`2B

fn(q) = h n|eiq·R| ni

(depends on Landau orbit)

= Ln(u)e12u

2

• in this limit, the state is an unentangled product of a non-trivial state of the guiding centers with a trivial state of the Landau orbits

| i = | Ri ⌦ | Rifull state(which has aSchrödingerrepresentation)

Y

⌦| n(Ri)i

Trivial stateof Landau orbits

FQHEis here !

characterized byform factor fn(q)

depends onlyon U(q)

• In what follows, I will regard the essential FQHE state as the purely-guiding center state defined by

H =

Zd2q`B2⇡

U(q)X

i<j

eiq·(Ri�Rj)

[Ra, Rb] = �i`2B✏ab

“quantum geometry*” ⇢(q) =X

i

eiq·R

[⇢(q), ⇢(q0)] = 2isin�12✏

abqaqb`2B

�⇢(q + q0) GMP 1985

* “triple” {algebra,representation,Hamiltonian} satisfies Connes’ definition

2

• given a complex structure (Kähler form) one can define ladder operators

[Ra, Rb] = �i`2B✏ab

!⇤a!b =

12 (gab � i✏ab) a = (!aR

a)/`B

[a, a†] = 1a Euclidean metric

det g = 12D antisymmetric

(Levi-Civita) symbol

• guiding-center “spin”:

[L, a†] = a†L(g) = gab⇤

ab

⇤ab ={Ra, Rb}

4`2Bgenerators of area-preservinglinear deformations of theguiding centers

• New insight: the choice of the Euclidean metric gab is (so far) arbitrary (previous work always chose it as diag(1,1) to be congruent to the shape of the Landau orbits)

• The metric is a (hidden) variational parameter of the Laughlin state, and is the fundamental physical degree of freedom of FQHE states.

(the metric is fixed as diag(1,1) in the “Laughlin wavefunction”)

• “symmetric gauge” basis Landau level states

• basis of Landau level states with general metric

wavefunctions in lowest Landau level:

hr| 0i / e�12 z

⇤z

| m(g)i = (a†)mpm!

| 0(g)i

a| 0(g)i = 0

hr| 0i / e12�z

2

e�12 z

⇤z

|�| < 1

(central coherent state)

• in the original Schroedinger/lowest Landau level language

• general relation

Landau levels Guiding centers

z = ↵z⇤ + �z (|↵|2 � |�|2 = 1)

• original (“Laughlin wavefunction”) relation

z = z⇤

ae�12 zz

⇤= ae�

12 z

⇤z = 0

a = 12z +

@

@z⇤a = 1

2 z +@

@z⇤

a†f(z)e�12 z

⇤z = zf(z)e�12 z

⇤z

• one can now write the Heisenberg form of the Laughlin state, liberated from any dependence on the Landau orbit geometry

• It is the exact zero-energy ground state of the “pseudopoential” model with

• coherent state basis

a|zi = z|zi |zi = eza†�z⇤a|0i

• non-null eigenstates of the overlap define an orthonormal basisZ

dz0dz0⇤

2⇡S(z, z⇤; z0, z0⇤) (z0, z0⇤) = � (z, z⇤)

• non-null eigenstates are degenerate with λ = 1

holomorphic!

(z, z⇤) = f(z⇤)e�12 z

⇤z “accidentally” coincidewith lowest-Landau levelwavefunctions if !!!z = z⇤

S(z, z⇤; z0, z0⇤) = hz|z0i = ez⇤z0� 1

2 (z0⇤z0+z⇤z)

• This is the true origin of holomorphic functions in the theory of the FQHE

• NOTHING to do with lowest Landau level states, derives from overlaps between states in a non-orthogonal overcomplete basis!

• Has obvious parallels in theory of flat-band Chern insulators, where the projected lattice-site basis is non-orthogonal and overcomplete

many-particlecoherent state

ai|z1, z2, . . . , zN i = zi|z1, z2, l . . . , zN i

“Laughlin wavefunction”

| qLi =

Y

i

Zdz⇤i dzi2⇡

Y

i<j

�z⇤i � z⇤j

�q Y

i

e�12 z

⇤i zi |z1, z2, . . . , zN i

• The metric is a physical degree of freedom that characterizes the shape of the correlation hole surrounding a particle in the Laughlin state

• The 1/q Laughlin state can be characterized as describing a “condensate” of “composite bosons” formed by “attaching” q “flux quanta” (orbitals) to the particles.

• more generally, the composite boson is formed by attaching q “flux quanta” to p particles.

The metric describes the shape of 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 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

✄

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

of freedom

measured in diameters of composite bosons

• 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”

-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 Regnault

124 (c� ⌫) = 1

24 (1�13 ) =

136

136

Laughlin ν=1/3 state: topologically conserved “chiral central charge” is explicitly seen to be

MPS calculation: (“plevel” is Virasoro level at which auxilliary space is truncated, which causes errors at large L)

c = 1

• 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 = !ab.I will assume that translational symmetry is unbroken,

so "c†!c!!# = "#!!! , where " 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 "$(q)# = 2%"#2(q&B). Note that the fluctuation#$(q) = $(q) % "$(q)# also obeys the algebra (4). I willdefine a guiding-center structure factor s(q) = s(%q) by

12 "{#$(q), #$(q

!)}# = 2%s(q)#2(q&B + q!&B). (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) = "s(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†!c!!c†"c"!# % "c†!c!!#"c†"c"!#

$ s"#!"!#"!! , with s" = " + ("2, where ( = %1if the particles are spin-polarized fermions, and ( =+1 if they are bosons (which may be relevant forcold-atom systems). Note that for all temperatures,lim##" s()q) = s", while s(0) = lim##0 s()q) =kBT/

!

'2f(T, ")/'"2"

"

T

#

, where f(T, ") is the free en-ergy per flux quantum. s(0) vanishes at T = 0, and atall T if v()q) diverges as ) $ 0; the high temperatureexpansion at fixed ", for rq ! ea*abqb&2B, is

s(q)% s"(s")2

= %

$

v(q) + (v(rq)

kBT

%

+O

$

1

T 2

%

. (8)

The correlation energy per flux quantum is given by

+ =

&

d2q&2B4%

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" = (

&

d2q!&2B2%

eiq$q!$2B (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 " with the Hamiltonian(2). The free energy of this state is formally given byF [$eq], where $eq(T, ") is the equilibrium density-matrixZ%1 exp(%H/kBT ) and F [$] is the functional

F [$] = Tr ($(H + kBT log $)) , (11)

which, for fixed ", is minimized when $ = $eq. TheAPD corresponding to a shear is Ra $ Ra + *ab,bcRc,parametrized by a symmetric tensor ,ab = ,ba. Let $(,)= U(,)$eqU(,)%1, where U(,) is the unitary operatorthat implements the APD, and F (,) ! F [$(,)] = F [$eq]+ O(,2), which is is minimized when ,ab = 0. The freeenergy per flux quantum has the expansion

f(,) = f(T, ") + 12G

abcd(T, "),ab,cd +O(,3), (12)

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

bd= *be*dfGaecf , with Gac

bd = Gcadb, and Gac

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

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 onlya!ected term in the free energy is the correlation energy,which can be evaluated in terms of the deformed struc-ture factor s(q, ,), given by

s" + (

&

d2q!&2B2%

eiq$q!$2B (s(q!)% s")ei%abqaq

!

b$2

B , (13)

with ,ab ! *ac*bd,cd. This gives Gabcd(T, "),ab,cd as

,ab,cd*ae*cf

&

d2q&2B4%

v(q)qeqfsbd(q;T, "). (14)

Assuming only that the ground state |"0# of (2) hastranslational invariance, plus inversion symmetry (so ithas vanishing electric dipole moment parallel to the 2Dsurface), GMP[1] used the SMA variational state |"(q)#' $(q)|"0# to obtain an upper bound E(q) ( f(q)/s(q)to the energy of an excitation with momentum !q (orelectric dipole moment eea*abqb&2B), where

f(q) =

&

d2q!&2B4%

v(q!)!

2 sin 12q ) q!&2B

#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 o!er any further interpretation of f(q).It can now be seen to have the long-wavelength behavior

lim##0

f()q) $ 12)

4Gabcdqaqbqcqd&4B, (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

abcdqaqbqcqd&2B. (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)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

z=p*2/(norb−1)−1

A=0.05

Δ n

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 11

1.02

1.04

1.06

g xx

initial numerical study of Laughlin 1/3 on cylinder with edges

(with Zlatko Papic and Sonika Johri)

Variationof Gaussiancurvature of

pseudopotentials

predicted

edge effects?

• 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