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. Haldane Princeton University Supported by DOE DE-SC0002140 and the W. M. Keck Foundation http://wwwphy.princeton.edu/~haldane/research.html Slides for this (and related) talks can be found at: APS March Meeting, Denver CO, March 4, 2014
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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:
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
• 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: