Origin of composite particle “mass” in the fractional quantum Hall effect

• geometry of flux attachment and the origin of pseudo-inertial mass through electric polarizability.

F 45.8

F. D. M. HaldanePrinceton University

Supported by DOE DE-SC0002140 and the W. M. Keck Foundation

http://wwwphy.princeton.edu/~haldane/research.htmlSlides for this (and related) talks can be found at:

APS March Meeting, Denver CO, March 4, 2014

• In the vacuum, the gravitational mass of particles is identical to their inertial mass:

• for a slowly-moving neutral particle ( |v| << c) in a non-inertial (e.g. rotating) frame

H = "(p�mAg(r))�mAg0(r))

"(p) =1

2mgabpapb

mab = mgab

gravitational vector potential(Coriolis vector potential)

gravitational scalar potential(-Newtonian gravitational potential)

Euclidean spatial metric

Kinetic energy Galileian mass tensor(as a function of momentum)

• For low-kinetic-energy electrons moving inside condensed matter (near the band-minimum of a conduction band) the link between “inertial” and “gravitational” mass is broken

unrenormalized gravitational (rest) mass of the electron(derived from coupling to the Higgs field)

Kinetic energy

H = "(p�meAg(r)� eAem(r))�meA

g0(r)� eAem

0 (r)coupling to electromagnetic and gravitational fields

"(p) = 12 (m

�1e↵ )abpapb me↵

ab(Pseudo-Galileian)

effective mass-tensor

a band-structure property

(strongly renormalized by the electromagnetic condensate that binds condensed matter together)

• Condensed matter (e.g. the 2D electron gas in AlxGa1-x As heterostructures) is usually described in an inertial frame with no gravitational potentials

• The electrons have pseudo-inertial properties governed by the pseudo-Galileian effective-mass tensor

⇡ = p� eA(r)

[⇡a,⇡b] = i~eB✏abantisymmetric symbol

...

12~!c

32~!c

52~!c

Landau levels(uniform B)H = 1

2 (m�1e↵ )ab⇡a⇡b

E

!c =|eB|p

det |me↵ |“cyclotron frequency”

(not really the cyclotron frequency, which involves the bare me)

• The residual dynamics of electrons in a partially-filled Landau level is governed by the projected two-body interaction

H =X

i<j

Vn(Ri �Rj) Vn(r) = PnVCoulombPn

analytic in rdepends on Landau index

[Rai , R

bi ] = �i✏ab`2B

Landau orbitradius vector

⇥

O

rR

R̃classicalcoordinate

guiding centercoordinate

e�

• Non-commutative geometry of the “guiding centers” gives rise to “composite particle” physics

• The entire partially-filled Landau-level problem is given in the form

H =X

i<j

Vn(Ri �Rj) [Rai , R

bi ] = �i✏ab`2B

• This depends only on the projected interaction Vn(r) and the area per h/e flux quantum2⇡`2B

• In particular, there is NO direct knowledge of (a) the effective mass tensor or (b) the Euclidean spatial metric of flat space-time.

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

• generator of translations and electric dipole moment!

[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

H =X

i<j

Vn(Ri �Rj)

• 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

V

• The solvable model is in fact a continuous family of solvable models defined by a metric gab where|r12|2 ⌘ gabr

a12r

b12 det g = 1

• This metric is the shape of flux attachment that forms the “composite boson” of the Laughlin state

L = 12`2B

gabRaRb

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

a g-dependent basis of 1-particle states

centralcoherent

state

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

| 0(g)i

e

If the central orbital is filled, the next two are empty

“flux attachment”

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

+−− +

++

“gaffnian”LLL hierarchy/Jain statehigher LL state

S = �3S = �2

• The metric adjusts locally to minimize the sum of correlation plus substrate potential energy of the guiding centers

• A key property is that the charge density is pinned not just to the magnetic flux density but is given locally by

⇢e =e⇤

2⇡

✓peB

~ � sKg

◆ pe = charge of composite bosonq = flux attachment

e* = e/q = elementary fractional charge

s = (topologically-quantized) guiding center spinKg = Gaussian curvature of metric

⌫ = pq (�1)pq = (�1)p gcd(p, q) 2

• A key property of the uniform FQHE state is its inversion symmetry: electrical polarization is proportional to momentum and is conserved: there is a gap for all excitations carrying an electric dipole moment

Pa = ~`2B

✏abX

i

Rbi = ✏abB

X

i

eRbi

!

momentum dipole moment

Energy as a function of momentum

Energy as a function of

dipole moment

pseudo-inertia defines “inertial mass” of composite particle

0 1 20

0.5

klB

Moore Read2

4

• two distinct degrees of freedom of the composite particle:

+−+

�E1�E2

• dipolar distortion (energy scale ΔE1)E

P(dipole)

dispersion!

e• fluctuations of metric preserve inversion

symmetry (energy scale ΔE2)

parity1

~2d2E

dP 2=

1

me↵

• the geometric “flux-attachment metric” degree of freedom is complemented by an independent dipolar distortion degree of freedom that gives “pseudo-inertia” to composite particles (bosons or fermions)

• This depends only on Coulomb interaction, through dependence of the correlation energy on the shape of the flux attachment.

conclusions

http://wwwphy.princeton.edu/~haldane/research.htmlSlides for this (and related) talks can be found at: