Studying Nanophysics Using Methods from High Energy Theory

1. 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”

2. Renormalization group Bosonization Conformal field theory Sasha Polyakov Ken Wilson Sidney Coleman

3. 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

4. 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

5. 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

6. Antonio Castro Neto Ed Witten Lenny Susskind Eduardo Fradkin Xiaogang Wen Claudio Chamon

8. D-branes in string theory Boundary conformal field theory Quantum dots interacting with leads in nanostructures

9. 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

10. Continuum formulation: • 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?)

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

12. 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 )

13. 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 )

14. 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

15. 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

16. 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

17. bulk exponent  r exponent, ’ depends on universality class of boundary Boundary layer – non-universal Boundary - dynamics

18. 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

19. 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

20. -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 J x  0 Jc

21. 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

22. the recent advent of precision experimental techniques have lead to a quest for experimental realizations of this novel physics in nanoscale systems

23. 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)

24. STM tip

25. Semi-conductor Quantum Dots gates AlGaAs 2DEG GaAs

26. controllable gates dot lead .1 microns dots have S=1/2 for some gate voltages dot  impurity spin in Kondo model

27. 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

28. -the highly controllable interactions between semi-conductor quantum dots makes them an attractive candidate for qubits in a future quantum computer

29. 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)

30. Semi-conductor Quantum Dots gates AlGaAs 2DEG lead GaAs dot dots have s=1/2 for some gate voltages

31. 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

32. only essential relevant Kondo interaction: (pseudo-spin label) • we have found exact conformally • invariant boundary condition by: • 1. conformal embedding • 2. fusion

33. 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

34. We go from the free fermion boundary • condition to the fixed point b.c. by • a sequence of fusion operations: • Fuse with: • s=3/2 operator in SU(2)3 (spin) sector • s=1/2 operator in SU(2)8 (pseudospin) • 02 parafermion operator

36. 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