Towards the fractional quantum Hall eect: a noncommutative geometry perspective

8/3/2019 Towards the fractional quantum Hall eect: a noncommutative geometry perspective

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Based on

M.Marcolli and V.Mathai, Twisted index

theory on good orbifolds, II: fractional quan-

tum numbers, Communications in Mathe-

matical Physics, Vol.217, no.1 (2001) 55-

87.

M.Marcolli and V.Mathai, Twisted index

theory on good orbifolds, I: noncommuta-tive Bloch theory, Communications in Con-

temporary Mathematics, Vol.1 (1999) 553-

587.

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Electrons in solids Bloch theory

Crystals Bravais lattice Rd (d = 2,3)

Periodic potential (electronions interaction)

U(x) =

u(x ) TU = U,

Mutual interaction of electrons

Ni=1

(xi + U(xi)) +1

2

i=j

W(xi xj)

Independent electron approximation

Ni=1

(xi + V(xi))

modification V of single electron potential

(x1, . . . , xN) = det(i(xj)) for ( + V)i = Eii so(xi + V(xi)) = E with E =

Ei (reduction to

single electron)

Inverse problem: determine V

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Bloch electrons

T = unitary operator on H = L2(Rd) transla-

tion by

TH T1 = H

for H= + V

Simultaneously diagonalize: characters of

Pontrjagin dual

T = c(), with c : U(1) since T12 = T1T2

c() = e

ik,

, k

compact group isom to Td (dual of Rd mod

reciprocal lattice

= {k Rd : k, 2Z, }

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Brillouin zones

Fundamental domains of : n-th zone, points

such that line to the origin crosses exactly (n 1) Bragg

hyperplanes of the crystal

Zone 3

Zone 3

Zone 2

Zone 2

Zone 1

Zone 1

FCC

BCC

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Band structure

Self-adjoint elliptic boundary value problem

Dk = ( + V) = E

(x + ) = eik,

(x)

Eigenvalues {E1(k), E2(k), E3(k), . . .}

E(k) = E(k + u) u

Plot En(k) over the n-the Brillouin zone

k E(k) k Rd

energycrystal momentum dispersion relation

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Fermi surface

Surface F in space of crystal momenta k de-

termines electric and optical properties of the

solid

F(R) = {k Rd : E(k) = }

Chromium Vanadium Yttrium

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Complex Bloch variety

B(V) =(k, ) Cd+1 :

nontriv sol of

( + V) = (x + ) = eik,(x)

F(C) =

1() B(V)

complex hyperfurface in Cd singular projective

F Rd = F(R) cycle in Hd1(F(C),Z)

Integrated density of states

() = lim

1

#{eigenv of H }

H= + V on L2(Rd/)

Period

d

d=F(R)

= holom differential on F(C)

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Discretization on 2(Zd) Random Walk

R(n1

, . . . , nd

) = +di=1

(n1

, . . . , ni

+ 1, . . . , nd

)

+di=1 (n1, . . . , ni 1, . . . , nd)

Discretized Laplacian

(n1, . . . , nd) = (2d R) (n1, . . . , nd)

Bloch variety (polynomial equation in zi, z1

i)

B(V) =

(z1 . . . , zd, ) :

2() nontriv sol of

(R + V) = ( + 2d)

Ri = zi

Ri(n1, . . . , nd) = (n1, . . . , ni + ai, . . . , nd)

Random walk for discrete groups on H = 2()

Symmetric set of generators i of

R () =r

i=1

Ri () =r

i=1

(i)

Discretized Laplacian = r R

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The breakdown of classical Bloch theory

Magnetic field

Aperiodicity

In both cases TH = HT fails

noncommutativity

J.Bellissard, A.van Elst, H.Schulz-Baldes, The noncom-

mutative geometry of the quantum Hall effect,

J.Math.Phys. 35 (1994) 53735451

Noncommutative Bloch theory

Integer Quantum Hall Effect

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Hall Effect (Classical 1879)

thin metal, orthogonal strong magnetic field B, current

j electric field E, transverse Hall current

B

Ej

equilibrium of forces linear relation

N eE +j B = 0

Hall conductance

H

=Ne

Bintensity of Hall current/intensity of magnetic field

H =

RHfilling factor=density of charge carriers h

eB

Hall resistance RH = h/e2

universal constant(fine structure constant e2/c)

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Integer Quantum Hall Effect

(von Klitzing 1980)

low temperature ( 1K), strong magnetic field

H as function of has plateaux at integer

multiples of e2/h

At values of corresponding to plateaux,

conductivity along current density axis van-

ishes

(precision 108, independent of different samples, ge-

ometries, impurities)

Laughlin (1981): geometric origin

Thouless et al. (1982); Avron, Seiler, Simon

(1983): topological (GaussBonnet)

Bellissard et al. (1994): noncommutative ge-ometry (explains also vanishing of direct conductivity)

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Magnetic field2-form = d (field and potential B = curlA)

Magnetic Schrodinger operator

( i)2 + V

magnetic Laplacian := (d i) (d i)

Periodicity: = (e.g. constant field) 0 =

= d( )

=

d

Magnetic Laplacian no longer commutes with action

What symmetries?

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Magnetic translations

(x) =xx0

( )

T := exp(i)T

then (d i)T

= T

(d i)

Magnetic translations no longer commute

TT = (,

)T

projective representation of

(, ) = exp(i(x0))

(x) + (x) (x) indep of x

(except integer flux case, where commute)

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Pontrjagin dual vs. group C-algebra

discrete abelian compact abelian

Pontrjagin dual eik, characters

Algebra of functions

C() = Cr()

C

-algebra generated by regular representation on

2() = { : C :

|()|2 < }

Fourier transform

When non-abelian, Cr() still makes sense

(non-abelian)

exists as a noncommutative space

Magnetic field Brillouin zone becomes

noncommutative space

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Harper operator (discretized magnetic Laplacian)

H1,2(m, n) = ei1n (m + 1, n)

+ ei1n (m 1, n)

+ ei2m (m, n + 1)

+ ei2m (m, n 1)

Let ((m

, n

), (m,n)) = exp(i(1m

n + 2mn

))(2-cocycle : U(1))

Magnetic translations U= R(0,1)

, V = R(1,0)

U (m, n) = (m, n + 1)ei2m

V (m, n) = (m + 1, n)ei1n

Noncommutative torus ( = 2 1)

U V = eiV U

Harper operator H = U + U + V + V

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Harper operators for discrete groups

multiplier : U(1) (2-cocycle)

(1, 2)(12, 3) = (1, 23)(2, 3)

(, 1) = (1, ) = 1

Hilbert space: 2()

Left/right -regular representations

L() = (1)(, 1)

R() = ()(, )

satisfy

LL = (,

)L RR

= (,

)R

Harper operator

(symmetric set of generators {i}ri=1 of )

R =

r

i=1 R

i

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Twisted group algebras

C(, ) algebra of observables (magnetic

translations), rep in B(2()) by right -regularrep R

weak closure U(, ) twisted group

von Neumann algebra

norm closure Cr(, ) twisted

(reduced) group C-algebra

Integer QHE case

= Z2 ( and as above)

Cr(, )= A

irrational rotation algebra, NC torus

Connes-Chern character integerquantization of Hall conductance

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Spectral theory

Magnetic Schrodinger R + V

potential V C(, )

Selfadjoint bounded Spec(R + V) R

complement = collection of intervals

Finitely many (Band structure, gaps)

Infinitely many (Cantor-like spectrum)

For =Z2

, depends on rationality/irrationality of flux

= [], []

Hofstadter butterfly

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Fractional Quantum Hall Effect

(Stormer and Tsui, 1983)

(high quality semi-conductor interface, low carrier con-

centration, and extremely low temperatures 10mK,

strong magnetic field)

Plateaux at certain rational multiples ofe2/h

Strongly interacting electrons

Noncommutative geometry model?

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What is expected of such model?

Account for strong electron interactions

Exhibit observed fractions (+predictions)

Account for varying width of plateaux

Simulate interaction via curvature

single electron in curved geometry as if subject to an

average strong multi-electron interaction

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Hall effect in the hyperbolic plane

Cocompact Fuchsian group = (g, 1, . . . , n)

PSL(2,R) discrete cocompact, genus g, elliptic ele-

ments order 1, . . . , n

Presentation (i = 1, . . . , g, j = 1, . . . , n)

= a1, bi, cj, g

i=1

[ai, bi

]c1 cn

= 1, c

j

j= 1

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Good orbifolds = B = \E

(orbifold-covered by a smooth manifold)

gG

(g; 1, . . . , n) = \H

2

g = smooth compact Riemann surface genus

g = 1 +#G

2(2(g 1 ) + (n ))

=

n

j=1 1

j

Exception: teardrop orbifold

2 /p

Orbifold Euler characteristic orb() Q

Multiplicative over orbifold covers, usual for smooth,inclusionexclusion

orb(1 r) =

i orb(i)

i,j orb(i j)

+(1)r+1orb(1 r)

orb((g; 1, . . . , n)) = 2 2g + n

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Spectral theory

H C(, ) Cr (, ) U(, )

Spectral projections PE = (,E](H) U(, )

E / Spec(H) PE Cr(, )

F holom function on neighb of Spec(H)

PE = (,E](H) = F(H) =

C

d

H

C= contour around Spec(H) left ofE

Counting gaps in the spectrum counting

projections in Cr(, )

Trace (T) = Te1, e12()

tr = Tr : Proj(Cr (, ) K) R

[tr] on K0(Cr (, ))

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Range of the Trace

Morita equiv (A C0(G)) C0(\G, E)

E = A G \G

K(Cr())

= KSO(2)(P(g; 1, . . . , n))

= Korb((g; 1, . . . , n))=

Z2n+

j = even

Z2g = odd

Twisted case C0(\G, E) C0(\G, E) provided

() = 0

: H2(, U(1)) H3(,Z) surjection from

long exact sequence of 1 Z Rexp(2i)

U(1) 1

[tr](K0(Cr(, ))) = Z+ Z+

j

1j Z

= [], [] rational finitely many gaps

Fundamental class [] =[g]#G

= irrational ?

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Dense -subalgebra

For set De = ()e with () = word length

Unbounded closed derivation = [D, ] on Cr (, )

R :=kN

Dom(k)

C(, ) R, closed under holo functional calculus

Polynomial growth group cocycles on define cyclic

cocycles continuous on R (Haagerup type inequality)

PE = (,E](H + V) R

Cyclic cocycles t : R R C

t(a0, a1, . . . , an) = t(an, a0, a1, . . . , an1). . . = t(a1, . . . , an, a0)

and

t(aa0, a1, . . . , an) t(a, a0a1, . . . , an)

(1)n+1t(ana, a0, . . . , an1) = 0

Cyclic cocycles pair with K-theory

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Conductance

In lattice = Z2, current density in e1 direction func-tional derivative 1 ofH by A1 (component of magneticpotential)

Expected value of current tr(P1H) for state P

Using tP = i[P,H] and t =A2

t

2 (for e2 e1) get

itr(P[tP, 1P]) = iE2tr(P[2P, 1P])

(electrostatic potential gauged away: E = At

)

In zero temperature limit charge carriers

occupy all levels below Fermi level, so P = PF

Conductance

H = tr(PF[1PF, 2PF])

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In curved geometry

On Riemann surfaces changes of potential by real and

imaginary parts of holomorphic 1-forms

H1(,Z) = Z2g {ai, bi}i=1,...,g symplectic basis

By effect of electron-electron interaction, to a moving

elector the directions {e1, e2} appear split into {ei, ei+g}i=1,...,g

corresponding to ai, bi

Kubo formula

1-cocycle a on derivation

a(f)() = a()f() on f C(, )

Conductance cocycle:

gi=1

tr

f0

ai(f1)bi(f

2) bi(f1)ai(f

2)

defines cyclic 2-cocycle trK(f0, f1, f2) on R

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Area cocycle On G = PSL(2,R) area cocycle

C(1, 2) = Area((x0, 11 x0, 2x0)

hyperbolic area of geodesic triangle with given vertices

Restriction of PSL(2,R) area cocycle on

Area cocycle:012=1

f0(0)f1(1)f

2(2)C(1, 2)(1, 2)

defines cyclic 2-cocycle trC(f0, f1, f2) on R

Coboundaries t1, t2 cyclic 2-cocycles

t1(a0, a1, a2) t2(a0, a1, a2) = (a0a1, a2) (a0, a1a2)

+ (a2a0, a1),

where is a cyclic 1-cocycle

Conductance and area differ by coboundary

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Comparison Difference between the hyperbolic area

of a geodesic triangle and the Euclidean area of its image

under the Abel-Jacobi map (curve and Jacobian)

U(1, 2) = h(12 , 1) h(

11 , 2) + h(1, 1)

each term difference of line integrals, one along a geodesic

segment in H2 and one along a straight line in the uni-

versal cover of the Jacobian

trK(f0, f1, f2) trC(f0, f1, f2) =

012=1

f0(0)f1(1)f2(2)U(1, 2)(1, 2)

This can be written as (f0f1, f2)(f0, f1f2)+(f2f0, f1)

(f0, f1) =

01=1

f0(0)f1(1)h(1, 1)(0, 1)

[trK] = [trC] same values on K-theory

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Values of the conductance

(ConnesMoscovici higher index theorem, twisted)

Indc,,(+E ) =

12#G

g

A tr(eRE)euc

= d 2-form of magnetic field 2 = i

cocycle c and lift uc to 2-form on g

By dimension

Indc,,(+E ) =

rankE

2#G

g

uc

Dependence on magnetic field only through E = orbifold

bundle representing class of PE spectral projection in

K0(Cr (, ) (using BaumConnes)

For area cocycle c 2-form uc is hyperbolic volume formg

uc = 2(2g 2)

Orbifold Euler characteristic

(2g 2)

#G = orb() Q

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Rational values of the onductance

H = trK(PF, PF, PF)

= trC(PF, PF, PF) Zorb()

experimental g = 0 n = 3

1/3 (0; 3, 6, 6)2/5 (0; 5, 5, 5)

2/3 (0; 9, 9, 9)3/5 (0; 5, 10, 10)4/9 (0; 3, 9, 9)5/9 (0; 6, 6, 9)4/5 (0; 15, 15, 15)3/7 (0; 4, 4, 14)4/7 (0; 7, 7, 7)5/7 (0; 7, 14, 14)

predicted g = 0 n = 3

8/15 (0; 5, 6, 10)11/15 (0; 10, 10, 15)

7/9 (0; 12, 12, 18)11/21 (0; 6, 6, 7)16/21 (0; 12, 12, 14)

Problem: does not discriminate against even denomina-tors (too many fractions)

Relation to ChernSimons approach?

NC versions of Bloch varieties and periods?

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Towards the fractional quantum Hall eect: a noncommutative geometry perspective