some beautiful theories can be carried over some beautiful theories can be carried over from one...
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Some beautiful theories can be carried over Some beautiful theories can be carried over
from one field of physics to anotherfrom one field of physics to another
-eg. High Energy to Condensed Matter-eg. High Energy to Condensed Matter ““The unreasonable effectiveness of The unreasonable effectiveness of
Mathematics in the Natural Sciences”Mathematics in the Natural Sciences”
Studying Nanophysics UsingStudying Nanophysics UsingMethods from High Energy TheoryMethods from High Energy Theory
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Bosonization
Sidney Coleman
Renormalizationgroup
Ken Wilson
Conformalfield theory
SashaPolyakov
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Renormalization GroupRenormalization Group
Low energy effective Hamiltonians sometimesLow energy effective Hamiltonians sometimes
have elegant, symmetric and universal form have elegant, symmetric and universal form
despite forbidding looking form of microscopicdespite forbidding looking form of microscopic
models models These effective Hamiltonians sometimes These effective Hamiltonians sometimes
contain “running” coupling constants that contain “running” coupling constants that
depend on characteristic energy/length scaledepend on characteristic energy/length scale
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Bosonization & Conformal Field TheoryBosonization & Conformal Field Theory
Interactions between nano-structures and Interactions between nano-structures and
macroscopic non-interacting electron gas can macroscopic non-interacting electron gas can
often be reduced to effective models in often be reduced to effective models in
(1+1) dimensions(1+1) dimensions
-eg. by projecting into s-wave channel-eg. by projecting into s-wave channel This can allow application of these powerful This can allow application of these powerful
methods of quantum field theory in (1+1) Dmethods of quantum field theory in (1+1) 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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ClaudioChamon
XiaogangWen
EdWitten
LennySusskind
EduardoFradkin
AntonioCastroNeto
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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 ProblemThe Kondo Problem
A famous model on which many ideas of RGA famous model on which many ideas of RG
were first developed, including perhaps were first developed, including perhaps
asymptotic freedomasymptotic freedom Describes a single quantum spin interactingDescribes a single quantum spin interacting
with conduction electrons in a metal with conduction electrons in a metal Since all interactions are at r=0 only we canSince all interactions are at r=0 only we can
normally reformulate model in (1+1) Dnormally reformulate model in (1+1) D
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)0(20
LLimpRRLL Sdx
d
dx
ddxiH
Continuum formulation:
•2 flavors of Dirac fermions on ½-lineinteracting with impurity spin (S=1/2) at origin(implicit sum over spin index)eff is small at high energies but gets largeat low energies•The “Kondo Problem” was how to understand low energy behaviour (like quark confinement?)
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1111
1 2)(
impjjj
jj SJtH
•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 spinsinglet with the impurity spin•electrons on sites 2, 3, … are freeexcept they cannot enter or leave site 1•In continuum model this corresponds to a simple change in boundary conditionL(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 spinsinglet with the impurity spin•electrons on sites 2, 3, … are freeexcept they cannot enter or leave site 1•In continuum model this corresponds to a simple change in boundary conditionL(0)=+R(0) (- sign at =0, + sign at )
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•A description of low energy behavior actually focuses on the other, approximatelyfree, electrons, not involved in the singlet formation•These electrons have induced self-interactions, localized near r=0, resulting from screeningof impurity spin•These interactions are “irrelevant” and corresponding corrections to free electronbehavior vanish as energy 0
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•a deep understanding of how this workscan be obtained using “bosonization”•i.e. replace free fermions by free bosons•this allows representation of the spin and charge degrees of freedom of electronsby 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
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Boundary Critical Phenomena & Boundary Critical Phenomena & Boundary CFT Boundary CFT
•Very generally, 1D Hamiltonians which are massless/critical in the bulk with interactions at the boundary renormalizeto conformally invariant boundary conditions at low energies•Basic Kondo model is a trivial examplewhere low energy boundary condition leaves fermions non-interacting•A “local Fermi liquid” fixed point
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Boundary layer – non-universal
rr
G1
'
1r
G
exponent, ’ depends on universality class of boundary
bulk exponent
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
1,1,11
1, 2)( a
aimpja
aj
jja
aj SJtH
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-if J 2 electrons get trapped at site #1 and “overscreen” S=1/2 impurity-this implies that stable low energy fixed pointof renormalization group is at intermediate coupling and is not a Fermi liquid
x
0 J
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 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
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the recent advent of precision experimentaltechniques have lead to a quest for experimental realizations of this novelphysics in nanoscale systems
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Cr Trimers on Au (111) Surface:Cr Trimers on Au (111) Surface:a non-Fermi liquid fixed pointa non-Fermi liquid fixed point
•Cr atoms can be manipulated and tunnelling current measured using a Scanning Tunnelling Microscope(M. Crommie)
Au
Cr (S=5/2)
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STM tip
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Semi-conductor Quantum Dots
GaAs2DEGAlGaAsgates
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.1 microns
controllable gates
lead dot
dots have S=1/2 for some gate voltagesdot 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:
2/1)0()( cVGVG
non-Fermi liquid exponent-many other low energy properties predicted
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-the highly controllable interactions between semi-conductor quantum dotsmakes 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-positionedto make important contributions to futuredevelopments in nano-science using methods from high energy theory (among other methods)
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Semi-conductor Quantum Dots
GaAsAlGaAs
2DEG
gates
lead
dot
dots have s=1/2 for some gate voltages
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)0()( 133221 JSSSSSSJH spins
• 2 doublet (s=1/2) groundstates with opposite helicity: |>exp[i2/3]|> under: SiSi+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
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only essential relevant Kondo interaction:
..)0)(( 1001 chSxJH impimpK
• we have found exact conformally invariant boundary condition by:1. conformal embedding2. fusion
(pseudo-spin label)
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We first represent the c=6 free fermion bulk theory in terms of Wess-Zumino-Witten non-linear modelsAnd 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 NLMC=9/5+9/5+12/5=6SU(2)8 = Z8 x U(1)C=7/5 + 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: 1. s=3/2 operator in SU(2)3 (spin) sector2. s=1/2 operator in SU(2)8 (pseudospin)3. 0
2 parafermion operator
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Conclusions about critical point:
• stable, even with broken particle-holesymmetry, (i.e. charge conjugation)and SU(2) symmetry as long as triangular symmetry is maintained• non-linear tunnelling conductancedI/dV A – B x V1/5