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

Bosonization

Sidney Coleman

Renormalizationgroup

Ken Wilson

Conformalfield theory

SashaPolyakov

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

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

•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

ClaudioChamon

XiaogangWen

EdWitten

LennySusskind

EduardoFradkin

AntonioCastroNeto

D-branes in string theory

Boundary conformal field theory

Quantum dots interacting with leads in nanostructures

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

)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?)

1111

1 2)(

impjjj

jj SJtH

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

•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 )

•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 )

•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

•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

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

Boundary layer – non-universal

rr

G1

'

1r

G

exponent, ’ depends on universality class of boundary

bulk exponent

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

(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

-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

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 experimentaltechniques have lead to a quest for experimental realizations of this novelphysics in nanoscale systems

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)

STM tip

Semi-conductor Quantum Dots

GaAs2DEGAlGaAsgates

.1 microns

controllable gates

lead dot

dots have S=1/2 for some gate voltagesdot 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:

2/1)0()( cVGVG

non-Fermi liquid exponent-many other low energy properties predicted

-the highly controllable interactions between semi-conductor quantum dotsmakes 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-positionedto make important contributions to futuredevelopments in nano-science using methods from high energy theory (among other methods)

Semi-conductor Quantum Dots

GaAsAlGaAs

2DEG

gates

lead

dot

dots have s=1/2 for some gate voltages

)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

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)

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

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

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