Some beautiful theories can be carried over

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Studying Nanophysics UsingMethods 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

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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

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Bosonization Conformal Field Theory

• Interactions between nano-structures and
• macroscopic non-interacting electron gas can
• often be reduced to effective models in
• (11) dimensions
• -eg. by projecting into s-wave channel
• This can allow application of these powerful
• methods of quantum field theory in (11) D

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• 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

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Ed Witten
Lenny Susskind
Eduardo Fradkin
Xiaogang Wen
Claudio Chamon
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D-branes in string theory
Boundary conformal field theory
Quantum dots interacting with leads in
nanostructures
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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 r0 only we can
• normally reformulate model in (11) D

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Continuum formulation

• 2 flavors of Dirac fermions on ½-line
• interacting with impurity spin (S1/2) 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?)

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• A lattice version of model is useful for
• understanding strong coupling (as in Q.C.D.)

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• 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 ???)

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• 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 ???)

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• 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 r0, resulting from screening
• of impurity spin
• These interactions are irrelevant and
• corresponding corrections to free electron
• behavior vanish as energy ?0

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• 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 Wittens
• non-abelian bosonization which involves
• non-trivial conformal field theories

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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

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bulk exponent ?
r
exponent, ? depends on universality class of
boundary
Boundary layer non-universal
Boundary - dynamics
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• 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

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simplest example of a non-Fermi liquid
model -fermions have a channel index as well
as the spin index
(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
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-if J?? 2 electrons get trapped at site 1 and
overscreen S1/2 impurity -this implies that
stable low energy fixed point of renormalization
group is at intermediate coupling and is not a
Fermi liquid
J
x
?
0
Jc
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using field theory methods, this low energy
behavior is described by a Wess-Zumino-Witten
conformal field theory (with Kac-Moody central
charge k2) -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
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the recent advent of precision experimental techni
ques have lead to a quest for experimental
realizations of this novel physics in nanoscale
systems
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Cr Trimers on Au (111) Surfacea non-Fermi
liquid fixed point
Au
Cr (S5/2)

• Cr atoms can be manipulated
• and tunnelling current measured using
• a Scanning Tunnelling Microscope
• (M. Crommie)

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STM tip
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Semi-conductor Quantum Dots
gates
AlGaAs
2DEG
GaAs
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controllable gates
dot
lead
.1 microns
dots have S1/2 for some gate voltages dot ?
impurity spin in Kondo model
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These field theory techniques, predict, for
example, that the conductance through a
2-channel Kondo system scales with bias voltage
as
non-Fermi liquid exponent -many other low energy
properties predicted
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-the highly controllable interactions between
semi-conductor quantum dots makes them an
attractive candidate for qubits in a future
quantum computer
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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)
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Semi-conductor Quantum Dots
gates
AlGaAs
2DEG
lead
GaAs
dot
dots have s1/2 for some gate voltages
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• 2 doublet (s1/2) groundstates
• with opposite helicity
• ?gt?exp?i2?/3?gt under Si?Si1
• represent by s1/2 spin operators Saimp
• and p1/2 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

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only essential relevant Kondo interaction
(pseudo-spin label)

• we have found exact conformally
• invariant boundary condition by
• 1. conformal embedding
• 2. fusion

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We first represent the c6 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) C3k/(2k) for WZW
NL?M C9/59/512/56 SU(2)8 Z8 x U(1) C7/5
1 12/5
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• We go from the free fermion boundary
• condition to the fixed point b.c. by
• a sequence of fusion operations
• Fuse with
• s3/2 operator in SU(2)3 (spin) sector
• s1/2 operator in SU(2)8 (pseudospin)
• ?02 parafermion operator

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