Download Boundary Conformal Field Theory & Nano

Studying Nanophysics Using
Methods from High Energy Theory
Some beautiful theories can be carried over
from one field of physics to another
-eg. High Energy to Condensed Matter
 “The unreasonable effectiveness of
Mathematics in the Natural Sciences”

Renormalization Bosonization
group
Conformal
field theory
Ken Wilson
Sasha
Polyakov
Sidney Coleman
Renormalization Group
Low energy effective Hamiltonians sometimes
have elegant, symmetric and universal form
despite forbidding looking form of microscopic
models
 These effective Hamiltonians sometimes
contain “running” coupling constants that
depend on characteristic energy/length scale

Bosonization & Conformal Field Theory
Interactions between nano-structures and
macroscopic non-interacting electron gas can
often be reduced to effective models in
(1+1) dimensions
-eg. by projecting into s-wave channel

This can allow application of these powerful
methods of quantum field theory in (1+1) D

•Another way of seeing the influence of
High Energy Physics on Condensed Matter
Physics is to look at some “academic
family trees”
-eg. Condensed Matter Theory group
At Boston University
Ed
Witten
Xiaogang
Wen
Claudio
Chamon
Lenny
Susskind
Eduardo
Fradkin
Antonio
Castro
Neto
D-branes in string theory
Boundary conformal field theory
Quantum dots interacting with leads
in nanostructures
The Kondo Problem
A famous model on which many ideas of RG
were first developed, including perhaps
asymptotic freedom

Describes a single quantum spin interacting
with conduction electrons in a metal

Since all interactions are at r=0 only we can
normally reformulate model in (1+1) D

Continuum formulation:


  d

 d
 
H  i  dx  L  L  R  R    Simp  L  L (0)
dx 
2
 dx
0

•2 flavors of Dirac fermions on ½-line
interacting with impurity spin (S=1/2) at origin
(implicit sum over spin index)
•eff is small at high energies but gets large
at low energies
•The “Kondo Problem” was how to understand
low energy behaviour (like quark confinement?)
•A lattice version of model is useful for
understanding strong coupling (as in Q.C.D.)





H  t  ( j  j 1  j 1 j )  JSimp  1  1
2
j 1

•at J fixed point, 1 electron is
“confined” at site 1 and forms a spin
singlet with the impurity spin
•electrons on sites 2, 3, … are free
except they cannot enter or leave site 1
•In continuum model this corresponds
to a simple change in boundary condition
L(0)=+R(0)
(- sign at =0, + sign at )
•at J fixed point, 1 electron is
“confined” at site 1 and forms a spin
singlet with the impurity spin
•electrons on sites 2, 3, … are free
except they cannot enter or leave site 1
•In continuum model this corresponds
to a simple change in boundary condition
L(0)=+R(0)
(- sign at =0, + sign at )
•A description of low energy behavior
actually focuses on the other, approximately
free, electrons, not involved in the singlet
formation
•These electrons have induced self-interactions,
localized near r=0, resulting from screening
of impurity spin
•These interactions are “irrelevant” and
corresponding corrections to free electron
behavior vanish as energy 0
•a deep understanding of how this works
can be obtained using “bosonization”
•i.e. replace free fermions by free bosons
•this allows representation of the spin
and charge degrees of freedom of electrons
by independent boson fields
•it can then be seen that the Kondo interaction
only involves the spin boson field
•an especially elegant version is Witten’s
“non-abelian bosonization” which involves
non-trivial conformal field theories
Boundary Critical Phenomena &
Boundary CFT
•Very generally, 1D Hamiltonians which
are massless/critical in the bulk with
interactions at the boundary renormalize
to conformally invariant boundary
conditions at low energies
•Basic Kondo model is a trivial example
where low energy boundary condition
leaves fermions non-interacting
•A “local Fermi liquid” fixed point
bulk exponent 
r
1
G  '
r
1
G 
r
exponent, ’ depends
on universality class
of boundary
Boundary layer – non-universal
Boundary - dynamics
• for non-Fermi liquid boundary conditions,
boundary exponents bulk exponents
• trivial free fermion bulk exponents
turn into non-trivial boundary exponents
due to impurity interactions
simplest example of a non-Fermi liquid model:
-fermions have a “channel” index as well as
the spin index



a
a
a 
H  t  ( j  a, j 1  j 1 a, j )  JSimp  1  a,1
2
j 1
(assume 2 channels: a is summed from 1 to 2)
-again J(T) gets larger as we lower T
-but now J is not a stable fixed point
-if J 2 electrons get trapped at site #1 and
“overscreen” S=1/2 impurity
-this implies that stable low energy fixed point
of renormalization group is at intermediate
coupling and is not a Fermi liquid
0
x
Jc
J

using field theory methods,
this low energy behavior is described by
a Wess-Zumino-Witten conformal field
theory (with Kac-Moody central charge k=2)
-this field theory approach predicts exact
critical behavior
-various other nanostructures with several
quantum dots and several channels also
exhibit non-Fermi liquid behavior and can
be solved by Conformal Field Theory/
Renormalization Group methods
the recent advent of precision experimental
techniques have lead to a quest for
experimental realizations of this novel
physics in nanoscale systems
Cr Trimers on Au (111) Surface:
a non-Fermi liquid fixed point
Au
Cr (S=5/2)
•Cr atoms can be manipulated
and tunnelling current measured using
a Scanning Tunnelling Microscope
(M. Crommie)
STM tip
Semi-conductor Quantum Dots
gates
2DEG
GaAs
AlGaAs
controllable gates
lead
dot
.1 microns
dots have S=1/2 for some gate voltages
dot  impurity spin in Kondo model
These field theory techniques, predict,
for example, that the conductance
through a 2-channel Kondo system scales
with bias voltage as:
G(V )  G(0)  cV
1/ 2

non-Fermi liquid exponent
-many other low energy properties predicted
-the highly controllable interactions
between semi-conductor quantum dots
makes them an attractive candidate
for qubits in a future quantum computer
the Boston University condensed matter
group, which Larry Sulak played a vital
role in assembling, is well-positioned
to make important contributions to future
developments in nano-science using
methods from high energy theory
(among other methods)
Semi-conductor Quantum Dots
gates
lead
2DEG
AlGaAs
GaAs
dot
dots have s=1/2 for some gate voltages
     
H spins  J ( S1  S2  S2  S3  S3  S1 )
( J  0)
• 2 doublet (s=1/2) groundstates
with opposite helicity:
|>exp[i2/3]|> under: SiSi+1
• represent by s=1/2 spin operators Saimp
and p=1/2 pseudospin operators aimp
• 3 channels of conduction electrons
couple to the trimer
• these can be written in a basis of
pseudo-spin eigenstates, p=-1,0,1
only essential relevant Kondo interaction:
 
H K  J (  0   1 )( x  0)  Simp imp  h.c.

1


0

(pseudo-spin label)
• we have found exact conformally
invariant boundary condition by:
1. conformal embedding
2. fusion
We first represent the c=6 free fermion
bulk theory in terms of
Wess-Zumino-Witten non-linear  models
And a “parafermion” CFT:
O(12)1  SU(2)3 x SU(2)3 x SU(2)8
(spin) (isospin) (pseudospin)
C=3k/(2+k) for WZW NLM
C=9/5+9/5+12/5=6
SU(2)8 = Z8 x U(1)
C=7/5 + 1 = 12/5
We go from the free fermion boundary
condition to the fixed point b.c. by
a sequence of fusion operations:
Fuse with:
1. s=3/2 operator in SU(2)3 (spin) sector
2. s=1/2 operator in SU(2)8 (pseudospin)
3. 02 parafermion operator
Conclusions about critical point:
• stable, even with broken particle-hole
symmetry, (i.e. charge conjugation)
and SU(2) symmetry as long as
triangular symmetry is maintained
• non-linear tunnelling conductance
dI/dV  A – B x V1/5