1. Breaking electrons apart in condensed matter physics T. Senthil (MIT) Group at MIT Predrag Nikolic Dinesh Raut O. Motrunich (now at KITP) A. Vishwanath Other main collaborators L. Balents (UCSB) Matthew P.A Fisher (KITP) Subir Sachdev (Yale) D. Ivanov (Zurich)

2. Conventional condensed matter physics: Landau’s 2 great ideas 1.Theory of fermi fluids (electrons in a metal, liquid He-3, nuclear matter, stellar structure,……..) 2. Notion of ``order parameter’’ to describe phases of matter -related notion of spontaneously broken symmetry -basis of phase transition theory

3. Fermi liquid theory Electrons in a metal: quantum fluid of fermions Inter-electron spacing ~ 1 A  Very strong Coulomb repulsion ~ 1-10 eV. But effects dramatically weakened due to Pauli exclusion. Important `quasiparticle’ states near Fermi surface scatter only weakly off each other. Describes conventional metals extremely well. kxkx kyky kzkz Fermi surface Filled states unavailable for scattering

4. Order parameter Example - ferromagnetism Spontaneous magnetization: `order parameter’. Ordered phase spontaneously breaks spin rotation symmetry. Ferromagnet: Spins aligned Paramagnet: Spins disordered Increase temperature

5. Notion of order parameter and symmetry breaking Powerful unifying framework for thinking generally about variety of ordered phases (eg: superfluids, antiferromagnets, crystals, etc). Determine many universal properties of phases - eg: rigidity of crystals, presence of spin waves in magnets, vortices in superfluids,…….

6. Phase transitions -Theoretical paradigm Critical singularities: long wavelength fluctuations of order parameter field. Landau-Ginzburg-Wilson: Landau ideas + renormalization group - sophisticated theoretical framework

7. Modern quantum many-electron physics Many complex materials studied in last two decades DEFY understanding within Landau thinking Examples: 1.One dimensional metals (Carbon nanotubes) 2.Quantum Hall effects 3.High temperature superconductors 4.Various magnetic ordering transitions in rare-earth alloys Need new ideas, paradigms!! Well-developed theory ??!!

8. High temperature superconductors Cu O La Parent insulator remove electrons Superconductor at relatively high temperatures

9. Complex phase diagram T x = number of doped holes Insulating antiferro magnet Superconductor Fermi liquid Strange non-Fermi Liquid metal Another strange metal

10. T = 0 phase transitions in rare earth alloys Examples: CePd 2 Si 2, CeCu 6-x Au x, YbRh 2 Si 2,…… (Quantum) critical point with striking non-fermi liquid physics unexpected in Landau paradigm. Magnetic metal Fermi liquid Pressure/B-field/etc

11. In search of new ideas and paradigms Most intriguing – electron breaks apart!! (Somewhat) more precise: Fractional quantum numbers Excitations of many body ground state have quantum numbers that are fractions of those of the underlying electrons.

12. Fractional quantum numbers Relatively new theme in condensed matter physics. Solidly established in two cases d = 1 systems (eg: polyacetylene, nanotubes, …..), d = 2 fractional quantum Hall effect in strong magnetic fields

13. Broken electrons in d = 1 Remove an electron from a d = 1 antiferromagnet Removed electron

14. Broken electrons in d = 1 Remove an electron from a d = 1 antiferromagnet Removed charge Spin domain wall = removed spin

15. Broken electrons in d = 1 Remove an electron from a d = 1 antiferromagnet Removed charge Spin domain wall = removed spin

16. Broken electrons in d = 1 Remove an electron from a d = 1 antiferromagnet Removed charge Spin domain wall = removed spin

17. Broken electrons in d = 1 The charge and spin of the removed electron move separately – the electron has broken! ``Spin-charge separation’’ the rule in d = 1 metals

18. Quantum Hall effect Confine electrons to two dimensions Turn on very strong magnetic fields Make the sample very clean Go to low temperature  Extremely rich and weird phenomena (eg: quantization of Hall conductance)

19. Fractional charge If flux density (in units of flux quantum) is commensurate with electron density, get novel incompressible electron fluid. Excitations with fractional charge (and statistics) appear! (Experiment: Klitzing, Tsui, Stormer, Gossard,…… Theory: Laughlin, Halperin, …………) Physics Nobel: 1985, 1998.

20. All important question Are broken electrons restricted to such exotic situations (d = 1 or d = 2 in strong magnetic fields)? Inspiration: Very appealing ideas on cuprate superconductors based on 2d avatars of spin-charge separation (Anderson, Kivelson et al, P.A Lee et al, …..)

21. All important question Are broken electrons restricted to such exotic situations (d = 1 or d = 2 in strong magnetic fields)? NO!!!

22. Recent theoretical progress Electrons can break apart in regular solids with strong interactions in 2 or 3 dimensions and in zero B-fields 1.Novel quantum phases with fractional quantum numbers (spin- charge separation) (Many people: Anderson, Read, Sachdev, Wen, TS, Fisher, Moessner, Sondhi, Balents, Girvin, Misguich, Motrunich, Nayak, Freedman, Schtengel,……..) 2. Novel phase transitions described by fractionalized excitations separating two conventional phases. (TS, Vishwanath, Balents, Sachdev, Fisher,Science March 04) Complete demonstrable breakdown of Landau paradigms!!

23. Some highlights Theoretical description of fractionalized phases (eg: nature of excitation spectrum) Concrete (and simple) microscopic models showing fractionalization Prototype wavefunctions for fractionalized ground states Precise characterization of nature of ordering in the ground state: replace notion of broken symmetry.

24. Where might it occur? Always a hard question: hints from theory  Frustrated quantum magnets with paramagnetic ground states  ``Intermediate’’ correlation regime – neither potential nor kinetic energy overwhelmingly dominates the other. (i) Quantum solids near the melting transition (ii) Mott insulators that are not too deeply into the insulating regime  Possibly in various 3d transition metal oxides  Perhaps even very common but we just haven’t found out!!

25. One specific simple model – small superconducting islands on a regular lattice (quantum Josephson junction array) Competition between Josephson coupling and charging energy: H = H J + H ch Josephson : Cooper pairs hop between islands to delocalize Charging energy: prefer local charge neutrality, i.e localized Cooper pairs. Superconductivity if Josephson wins, insulator otherwise.... Motrunich, T.S, Phys Rev Lett 2002

26. Phase diagram in d = 2 Josephson Charging energy Fractionalized insulator sandwiched between superfluid and conventional insulator. Fractionalized phase: excitations with half of Cooper pair charge.

27. Broken symmetry versus fractionalization Goldstone modes (spin waves, phonons, etc) Stiffness (crystal rigidity, persistent superflow,…) Topological defects (vortices, dislocations, etc) Hartree-Fock mean field theory Coexistence of different broken symmetries (magnetic superconductors, supersolids,etc) Tools to detect (Bragg scattering, Josephson, etc)

28. Broken symmetry versus fractionalization Goldstone modes (spin waves, phonons, etc) Gauge excitations Stiffness (crystal rigidity, persistent superflow,…) Topological defects (vortices, dislocations, etc) Hartree-Fock mean field theory Coexistence of different broken symmetries (magnetic superconductors, supersolids,etc) Tools to detect (Bragg scattering, Josephson, etc)

29. Why gauge? Relic of glue that confines broken pieces together in conventional phases. -Analogous to quark confinement. Conventional phases: Broken pieces (like quarks) are bound together by a confining gauge field. Fractionalized phases: Gauge field is deconfined; liberates the fractional particles.

30. Broken symmetry versus fractionalization Goldstone modes (spin waves, phonons, etc) Gauge excitations Stiffness (crystal rigidity, persistent superflow,…) Robustness to all perturbations Topological defects (vortices, dislocations, etc) Hartree-Fock mean field theory Coexistence of different broken symmetries (magnetic superconductors, supersolids,etc) Tools to detect (Bragg scattering, Josephson, etc)

31. Robustness to all perturbations (gauge rigidity) Gauge excitations preserved for arbitrary local perturbations to the Hamiltonian (including ones that break symmetries) Stable to dirt, random noise, coupling to lattice vibrations, etc. (``Topological/quantum order’’ – Wen) Protected against decoherence by environment (Potential application to quantum computing – Kitaev)

32. Broken symmetry versus fractionalization Goldstone modes (spin waves, phonons, etc) Gauge excitations Stiffness (crystal rigidity, persistent superflow,…) Robustness to all local perturbations Topological defects (vortices, dislocations, etc) Fractional charge Hartree-Fock mean field theory Coexistence of different broken symmetries (magnetic superconductors, supersolids,etc) Tools to detect (Bragg scattering, Josephson, etc)

33. Fractional charge: defects in gauge field configuration Fractional charges carry the gauge charge that couples to the gauge field - hence defects in the gauge field (as in ordinary electromagnetism) Electric chargeElectric field lines

34. Broken symmetry versus fractionalization Goldstone modes (spin waves, phonons, etc) Gauge excitations Stiffness (crystal rigidity, persistent superflow,…) Robustness to all local perturbations Topological defects (vortices, dislocations, etc) Fractional charge Hartree-Fock mean field theorySlave particle mean field theory Coexistence of different broken symmetries (magnetic superconductors, supersolids,etc) Tools to detect (Bragg scattering, Josephson, etc)

35. Slave particle mean field theory (Coleman, Read, Kotliar, Lee,….) Write electron operator c α = b † f α Charged spinless boson (``holon’’) Neutral spinful fermion (``spinon’’) Replace microscopic Hamiltonian with equivalent non-interacting Hamiltonian for holons and spinons with self-consistently determined parameters.

36. Broken symmetry versus fractionalization Goldstone modes (spin waves, phonons, etc) Gauge excitations Stiffness (crystal rigidity, persistent superflow,…) Robustness to all local perturbations Topological defects (vortices, dislocations, etc) Fractional charge Hartree-Fock mean field theorySlave particle mean field theory Coexistence of different broken symmetries (magnetic superconductors, supersolids,etc) Coexistence with conventional broken symmetry Tools to detect (Bragg scattering, Josephson, etc)

37. Coexistence (Balents, Fisher, Nayak, TS) Fractionalization may coexist with conventional broken symmmetry (eg: fractionalized magnet, fractionalized superfluid,…) Important implication: Presence of conventional order may hide more subtle fractionalization physics. (Is Nickel Sulfide fractionalized?)

38. Broken symmetry versus fractionalization Goldstone modes (spin waves, phonons, etc) Gauge excitations Stiffness (crystal rigidity, persistent superflow,…) Robustness to all local perturbations Topological defects (vortices, dislocations, etc) Fractional charge Hartree-Fock mean field theorySlave particle mean field theory Coexistence of different broken symmetries (magnetic superconductors, supersolids,etc) Coexistence with conventional broken symmetry Tools to detect (Bragg scattering, Josephson, etc) Flux memory, noise, ??

39. Detecting the gauge field Largely an open problem in general !! In some cases can use proximate superconducting states to create and then detect the gauge flux (TS, Fisher PRL 2001; TS, Lee forthcoming) Cuprate experiments (Bonn, Moler) find no evidence for Z 2 gauge flux expected for one possible phase with spin-charge separation. Other possibilities exist and haven’t been checked for yet.

40. Outlook Theoretical progress dramatic (rapid important developments every year) But no unambiguous experimental identification yet (though many promising candidates exist) Theoretically important answer to 0 th order question posed by experiments: Can Landau paradigm be violated at phases and phase transitions of strongly interacting electrons?

41. Outlook (cont’d) Extreme pessimist: Why bother? Might not be seen in any material. Extreme optimist: Might be happening everywhere without us knowing (eg: in Nickel Sulfide,…..)

42. Outlook (cont’d) Strong need for probes to tell if fractionalized (completely new experimental toolbox). Ferromagnetism (relatively rare)– known for centuries Antiferromagnetism (much more common) – known only for < 70 years Had to await development of new probes like neutron scattering

43. Questions for the future Will these ideas ``solve’’ existing mysteries like the cuprates? Will they have deep implications for other branches of physics (much like ideas of broken symmetry did)? See X.-G. Wen, Origin of Light for some suggestions. Will they form the basis of quantum computing technology?

44. Quantum Hall effect Hall conductance

45. Fractional charge in FQHE More pictures

46. Outline Some basic ideas in condensed matter physics Complex new materials – crisis in quantum many body physics! New ideas needed! Why break the electron? What does it mean? How can you tell? Why should anyone care?