Oscillations of longitudina resistivity =Shubnikov-deHaas, minima close to 0.Plateaux in Hall resistivity r=h/(ne2) with integer n correspond to the minima

1(From Datta page 25)

discovery: 1980Nobel prize: 1985

K. v. Klitzing

Origin of the Oscillations of longitudina resistivity =Shubnikov-deHaas

0 02

2 1( ,0) ( ) ( , ) ( ),2 c c

m eB eBN E E E N E B E E nh m

2

r() without H

E

r() with H

E

2( ) ( )x yL L mE E

h

r

1( ) ( ) ( ( ) )2L cE N E E nr

full for ; decreasing H onestarts filling LL with 2. Number

The LL number

of fil

ν is

l

partially filled between and .1

ed : t

t t

tH

LLL H H LL

HH H

HH

resistivity minima close to 0

Rectangular conductor very thin in z direction uniform in x direction confined in y direction with B in z direction.

Assuming for the sake of argument that H is separable, the transverse dimensions yield infinite solutions that are called subbands: let us assume that only one z subband is occupied and the confinement along y is described by U(y).

22( )[ ( )] ( , ) ( , )2 2

yxpp eBy U y x y E x y

m m

( , ) ( )ikxx y e y

22( )[ ( )] ( , ) ( , )2 2

yxpk eBy U y x y E x y

m m

22 2

Expanding,

1[ ( ) ( ) ( )] ( , ) ( , ), 2 2

y xk k

p keBm y y U y x y E x y ym m eB

4

This is a paradox: one would expect minimum resistance when LL is at Fermi energy, but it is maximum; how does the sample carry the current when EF is between LL and there are no states at EF?

Reply: there are 1d states at the eges of the sample (hedge states ) that carry the current! The minimum resistivity is very low because of the suppression of relaxation. Carriers that go to opposite directions are far away and never meet.

r() with H

E

r() with H

E

Due to impurities, the DOS is not like this

but more like this

Due to disorder and impurities, it is possible to find the Fermi level away from the LL (otherwise it is unlikely to find it where the DOS is small and a small charge can move EF). The conduction takes place through the M hedge states that have very small resistance.

For =integer,..filled or empty LL gap, no scattering xx resistivity =0 For =0.5,1.5,2.5,.. Half filled LL maximum xx resistivity

5

Origin of zero resistance (see also Datta page 175)

Including confining potential in first order, states with k in x direction in the LL n have energy:

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

subband bottom, ( ) confining potential

, ( ) , ( ),

s c

s

xk k

E n k E n n k U y n k

E U ykn k U y n k U y yeB

In the middle of the sample, bulk-like eigenvalues and eigenvectors prevail, but near the edges the levels are shifted by U yielding a quasi-continuum of levels, also at fermi energy. Current in edge state can be evaluated by the group velocity:( ) ( )1 ( , ) 1 1 1 ( )v( , ) ,

edge states on opposite edges carry currents to opposite directions( ) since is opposite and the x component are assumed opposite.

No head-

k k kU y U y yE n k U yn kk k y k eB y

U yy

on collisions are possible.

We try to include confining potential U(y) along y as a perturbation, which is nearly constant over the extent of the LL wavefunction.

y

kx

6

No backscattering takes place.This situation when EF is between two LL, otherwise the LL at Fermi level carries current within the sample with scattering and maximum resistance (not an explanation but a description)

Could be measured i n balistic conductors within a few % in QHE with better than ppb accuracy !

2The Quantum of conductance2he

in QHE mm-sized electron mean free paths because curent carrying states in opposite directions are localized on opposite sides no backscattering

7

8

0

0

Degeneracy of each LL: D , number per spin 2

Condition for exactly filled LLL (for each spin direction): D .2

threshold field for having all electrons in LLL:

elL

elL

el

NN N

NN N

N

t 0H (LLL energy depends on spin)2elN

S

t

,1In termsof the surface density , Surface density per spin, .2

threshold field for having all electrons in L 1H2

LL .

elels s

x y x y

l se

NNn

hc

nL L L L

N ne

0 1Recall Hall resistance in Drude theory

for , ( )1 0| |

0 1since resistivity goes to 0 ( ( )

| |

Step hight

xx xy

yx yy s

BHe n

BHe nc

r r r

r r

r

�

�

t

2

in cgs units). Hall resistance 1 0

2For H , number of occupied LL, all others empty,

2 12 and theseare thequantized values if integer.

xys

xys

shcne

Hen c

H

hen c e

r

r

Quantized resistance

9Elaborated from a seminar by Michael Adler

Convincing explanation ? NO!Drude theory ia rough, the result is extremely precise, and why the plateaux?

2

2 ( ),

25.

Landauer formula along x direction

1

:

8 2802 2

L R

Hall L R

Longitudinal Hall L RLongitudinal Hall

eI Mh

eVV V h KR R

I I I e M M

Due to the applied bias, a current flows in tha sample; this produces the Hall field in the y direction. So the upper edge states, where electrons go to the left , have the chemicel potential R of the right electrode, the lower ones have the chemical potential L of the left electrode. The potential drop VLongitudina along x for both is zero. Since the potential drop along y is VH, R-L=VH.

Good, but a serious doubt remains. We did several crude approximations. Why is the result so precise)?

10

11

Why the plateaux? Why so exact?

Consider a Metal ribbon long side along x, magnetic field along z.

11

x

y

2Hall resistivity quantized with , =sharp integerxyh

er

2

Hall conductivity quantized with , =sharp integerxyeh

22

Assume noninteracting electrons. The electronic obeys:

( ) [ ( )] ( , ) ( , )2 2

Setting: ( , ) ( )

one obtains

yx

ikx

pp eBy U y x y E x ym m

x y e y

22

22 2

( )[ ( )] ( , ) ( , ),2 2

1 [ ( ) ( ) ( )] ( , ) ( , ), 2 2

yx

y xk k

pk eBy U y x y E x ym m

p keBm y y U y x y E x y ym m eB

Laughlin does not even insert confining potential U(y) which plays no role in his argument. Now add an electric field along y.

22

0 0

2 22 2

0 0

2 2 2 2 2 20 0 0

( ) , ( )2 2

1 ( ) , . Next find y such that 2 2 2

1 1 1( ) ( )2 2 2

y ikxxkn n

y xc x c c

c x c c c

pp eByH eE y e y ym m

p k eBH m y k eE ym m mc

m y k eE y m y y m y

13

2 2 2 2 2 20 0 0

2 2 2 2 2 2 2 2 20 0 0 0

0 00 2

2 20 0

1 1 1( ) ( )2 2 21 1 1 1( )2 2 2 2

1 [ ]

1 just shifts oscillator : ( )2 2

c x c c c

c x c c c c c

x

c c c

n c c

m y k eE y m y y m y

m y k eE y m y m y m yy m y

k eE Eky cm m m B

mE n y

Consider closing it as a ring pierced by a flux, with opposite sides connected to charge reservoirs of infinite capacity , each as the same potential as the side to which it is connected. A current I flows around, the Hall potential VH exists between reservoirs.

15

Next replace E by a time-dependent flux inside; it produces a e.m.f that excites a current I around (along x)

, total energyUI c U

but the magnetic field along z then produces a Hall electric field VH, along the ribbon, that will transfer charge q from one reservoir to the other, contributing qVH to the energy.

0

Current due to flux piercing the ring:

for a cycle, using a macroscopic ring,(total energy) , since charge is transfered and system is restored

Hall co( )

H

H

U UI c c

U qVqVI c hce

nductivity .H

I eqV h

If the flux is one fluxon, the system is completely restored in previous state .

16

In order to have the ribbon in same state as before the fluxon is applied, q=ne with n integer. Hence,

2

Hall conductivity !!is exactly quantizedH

I enV h

Laughlin argues that the same holds true even in the presence of interactions.

Laughlin’s argument has been criticized on the grounds that different cycles of the pump may transport different amounts of charge, since q is not a conserved quantity. It is the mean transferred charge that must be quantized.

It appears to me that the criticism is rather sophistic, because if the average is an exact integer that does imply that every measurement gives an integer. One could envisage a situation where two exactly equally likely outcomes are 0.80000 and 1.20000 (sharp!) and so the average is 1,0000 without having integer outcomes each time. One should discover fractionally charged real electrons before accepting this explanation.

However some authors of the above criticisms have produced a remarkable alternative explanation.

17

According to the Authors, Laughlin’s argument is short in one important step, namely, the inclusion of topological quantum numbers.

( , ) flux in the ring, flux in the ammeter A that measures current along y.Parameter space: ( , ).

Current flowing through A:

H H

I c H

A

H i it t

18

The reading of the ammeter does not change energy

0

Inserting , setting c=1,

0 2Re

2Re 2Im .

H H H H

I c H

I H

I H i H

inserting and restoring

2 2Im

c

I i

H i it t

itc c t

19

2

, where 2 Im

curvature of the bundle of ground states

Hall conduct !ance B

I K Kt

c yK err

1 integer 2

where 2 Im

S= parameter spaceis a Chern number! What is it?

SKdA

K

20

Gauss and Charles Bonnet formula

1 2(1 )2

,S

KdA g

K curvature g number of handles

Chern formula

1 integer (Chern number)2

2 Im

SKdA

K

http://www.riken.go.jp/lab-www/theory/colloquium/furusaki.pdf

Bi2Se3

is a 3d topological insulator

Topological Insulators • (band) insulator with a nonzero gap to excitated states• topological number stable against any (weak) perturbation• gapless edge mode

• Low-energy effective theory of topological insulators = topological field theory (Chern-Simons)

The QHE is the prototype topological insulator

http://wwwphy.princeton.edu/~yazdaniweb/Research_TopoInsul.php

22

Fractional Quantum Hall effect

D.C. Tsui, H.L. Störmer and A.C. Gossard, prl (1982): quantization of Hall conductance at = 1/3 and 2/3 below 1 Kelvin

The fractional quantum Hall effect (FQHE) is a physical phenomenon in which a certain strongly correlated system at T under a very strong magnetic field behaves as if it were composed of particles with fractional charge (1998 Nobel Prize).

23

experiments performed on gallium arsenide heterostructures 

0

2

Strong fields : all electrons in2

25.8Integer : quantized in units of , integer number2 2

interpretation: number of edgestates

el

xy

N LLL

hQHE k Me M M

M

r

0

Somewhat Stronger fields : allelectrons with spin in

(typically8Tesla) , but integer QHE

elN LLL

2

Still Stronger fields :FQHE :1 2 4quantized in units of , , , ,3 5 7xy

h ppe M

r

Denominators almost ever odd. The FQHE is a different phenomenon, requiring a different explanation.

It is believed that the effect is due to the Coulomb interaction.

All electrons in LLL treating interactions as a perturbation that tends to lower the symmetry.

25

Wave functions for the LLL in polar coordinates

Using the Curl in cylindrical coordinates1 1ˆˆ ˆ ( )z r z rrotA r A A A A z rA Ar z z r r r

1one finds a Gauge : 0, , 0 such that (0,0, )2r zA A rB A rotA B

2 2 2 2

2 2 2 2 2

1 1To write the SE we also need the Laplacian .x y r r r r

21 1 , that is, 2 2e iep p eA i i rB r B

r r

( ) ( )

( ) ( )2

2

22 2

4

2

| |

2 2

1 1 , , .2 2

Specialize for the LLL 0, set and define magnetic length:

, , with integer, M

iM

c

il lzl

ie r B r E rm r r r r

n z x iy re lm eB

r eN lr z e

26

Laughlin sought thee Many-Body wave function for LLL in the form

( ) ( )

( ) ( )

2

2| |4 2

1 2 1 21

, ,..., , ,..., , , ,

where , , ensures Pauli principle.

i

M

zNl

N N i i i Mi

i j j i

z z z f z z z e z x iy leB

f z z f z z

We want it to be an eigenfunction of total angular momentum i i

i

( ) ( )2

2| |4

1 2

Laughlin ansatz :

, ,..., ,

Pauli odd

i

Mi

zN q lN j k

j k

z

q

z z z z e

( )2

2| |4 2

setting =il, angular momentum l, the wavefunction of an harmonic oscillator at large r,

, ,

2where 2 . Minimum at r=r A circle of this radius en

M

zll

M

l M l

r Nz e z x iy leB

lr l leB

closes l fluxons.

27

2 2 21 2 3

1 2 3 1 2 1 3 2 3 2

For 3 electrons Laughlin Wave Function :

( , , ) ( ) ( ) ( ) exp[ ].4

q q q

M

z z zz z z z z z z z z

l

max0

jThis is the maximum power that any z can have.

l

Physical Picture: Outer electron encloses lmax fluxons

R. Laughlin

28

In terms of such wavefunctions one can estimate correlations and compute successfully the relevant quantities. Quasiparticles of fractional charge e/q obey anyon statistics (exchange of two quasiparticles brings a phase , which can be calculated as a Berry phase).

0max

0

number of quanta per electron in sample.

1Equivalently, the number of electrons per fluxon in sample=

elel

l N q qN

q

Let us evaluate q. Expanding the product in Laughlin ansatz the maximum power of z of any particle evidently turns out to be: lmax= Nelq

( ) ( )2

2| |4

1 2, ,..., dd, oi

Mi

zN q lN j k

j k

z z z z z e q

Very accurate wave function when compared with numerical estimates. q=1 yields integer QHE wavefunction

29From a seminar by Michael Adler

30

Some CONCLUSIONS

The dimensionality conditions the many-body behaviour already at the classical level- e.g. phase transitions

At the quantum level, the transport properties are very different when nanoscopic objects are considered, and this requires new intriguing concepts and funny mathematical methods, many of which involve the Berry phase too. Here I just recall some. The subject is in rapid evolution, and new applications are also under way. Ballistic conduction, nonlinear magnetic behaviour, possibility of various kinds of pumping. The role of correlation effects is much more critical in 2d (e.g. QHE, charge fractionalization, anyons , Kosterlitz-Thouless ) and above all in 1d (Peierls transition and charge fractionalization, ) and there was no time to introduce others, that would require the Luttinger Liquid formalism….

31

Charge-spin separation

1 d antiferromagnet

1 d antiferromagnet with hole

spins are assumed to jump freely to empty sites

hole has moved

hole has moved

spinonholon

32

nearby sites are assumed to exchange spins

spinon propagation

hole has moved

holon

Fermi particle becomes a pair of boson excitations that can propagate with different speeds (in 1d only)

Bosonization

Fermion operators can be expressed in terms of bosons:

spinons and holons and problem is solved

hole has moved

spinonholon

holon also travels

33

interaction-dependent exponent

( ) ( ) ( )

r

1 12 2 2 2

Typical power-law dependence of the DOS and many quantities in 1d

( , ) spinon holon holonq v v v

Separation Realized experimentally!