1. Exploring New Paradigm in Physics Yu Lu Institute of Physics Chinese Academy of Sciences

2. P.A.M. Dirac, Proc. Roy. Soc. A123, 713 (1929) “…The underlying physical laws necessary for the mathema- tical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equat- ions much too complicated to be soluble.”

3. How do you do to get the Theory of Everything? 1. Planck/unification scale (10 28 eV) d d u d u u d d u du u e e 4 He + 2e  2.QCD  Nuclear physics scale (10 8 -10 9 eV) - + ++++ +++ ++++ - - - - - - - - - - - Na metal 3.Condensed matter physics scale (10 0 eV) The Theory of Everyday Everything!

4. Great achievements of quantum theory and relativity: Civilization of the information Age  Structure of matter: how chemistry ‘works’  Electronic theory: transistors, IC, memories  Lasing principle: lasers, optical fibers…  Fission and fusion: nuclear energy…  Nuclear Techniques: MRI, PET, CT… Observations and exploitations of these remarkable quantum phenomena

5. Is this truly The theory of Everything? Can one derive ALL exotic properties, from the Schrödinger equation??

6. “We often think that when we have completed our study of one we know all about two, because ‘two’ is ‘one and one.’ We forget that we have still to make a study of ‘and.’ ” --. --Sir Arthur Eddington.

7. Philip W. Anderson: More is different (1972) “The behavior of large and complex aggregations of elementary particles, … is not to be understood in terms of a simple extrapolation of the properties of a few particles. Instead, at each new level of complexity, entirely new properties appear, and the understanding of this behavior requires research as fundamental in its nature as any other…”

8. Emergent features of condensed matter systems  Collective excitations—quasi-particles  Symmetry breaking  Renormalization  ……

9. Lattice vibration and phonons  If ground state stable: low energy excitations —harmonic oscillations. Quantization of these oscillations — phonons  “Like” ordinary particles ， dispersion  (p)  No restrictions on generation: bosons  They cease to exist, while away from crystals: quasi-particles  Not sensitive to microscopic details ， those details cannot be recovered from the phonons This was initiated by Einstein ！

10. Landau Fermi Liquid Theory  Low energy excitations of interacting Fermi systems （ like electrons in metals ） can be mapped onto weakly interacting Fermi gas  These quasi-pariticles follow Fermi statistics ， with dispersion  (p) ， with the same Fermi volume as free fermions (Luttinger theorem).  They cease to exist if taken away from the matrix (metal)  Their properties not sensitive to microscopic interactions ， which cannot be derived from these ‘coarse grained’ properties

11. Basic assumption: Adiabaticity Question: How to justify it, if no gaps?

12. Emergent features of condensed matter systems  Collective excitations—quasi-particles  Symmetry breaking  Renormalization  ……

13. Superconductivity 1911 Kamerlingh Onnes discovered zero resistance Early 30s Meissner effect discovered, complete diamag- netism more fundamental Wave function “rigidity” ansatz (London brothers) London equations

14. 1950 Ginzburg-Landau equation ， introducing macroscopic wave function Bardeen realized: gap in spectrum leads to “rigidity” Superconductivity Cooper pairing ： arbitrarily weak attraction gives rise to bound states at the Fermi surface —pairing energy is the gap

15. Is SC a Bose-Einstein condensation of Cooper pairs?--a bit more complicated! BCS wave function ： Problem solved ！ Nobel prize was delayed by 15 years ！ ! Particle number not conserved ， change from one Hilbert space to another one — symmetry breaking—conceptual breakthrough

16. Symmetry Breaking Discrete symmetry －－ from up or down to definite up （ down ） Broken symmetry － reduction of symmetry elements “Usually”: “high temperature － high symmetry”, “low temperature － low symmetry” Displacive phase transition

17. Ferromagnet--broken rotational symmetry Broken continuous symmetry Antiferromagnetic order – staggered magnetization (Landau & Néel) ， －－ not conserved quantity Macroscopic superconducting wave function － order parameter (Landau)  breaking of U(1) gauge symmetry

18. Goldstone mode: collective excitations, recovering the symmetry – like spin waves Anderson-Higgs mechanism Unified weak-electromagnetic interactions － 1979 Nobel prize in physics Weinberg- Salam- Glashow When external (gauge) field coupled, becomes massive -- Meissner effect

19. Josephson effect ： visualization of the phase Most profound exhibition of emergence! Using two Josephson junctions-- SQUID

20. Josephson Effect S2S2 S1S1

21. Bardeen - Josephson dispute  Anderson’s lecture  Josephson’s calculation  Bardeen’s added note  Dispute at LT 8 BCS mentor against the most convincing proof of his theory!!

22. 10 -9 10 -6 10 -3 1 10 3 10 6 10 9 10 12 Atom traps, BEC, Superfluidity 3 He Superfluidity Heavy Electron Superconductivity Low Tc Superconductivity High Tc Superconductivity Neutron Stars, Color Superconductivity Quark-Gluon Plasma Nano-K micro-K milli-K K kilo-K mega-K giga-K tera-K

23. Emergent features of condensed matter systems  Collective excitations—quasi-particles  Symmetry breaking  Renormalization  ……

24. Failure of Mean Field Theory ！！ MFT Experiment  4/3 !  0 (jump ）  0  1/3 !  5 !  2/3 !  0  0 Theory valid in space dimensions beyond 4 ！

25. Kenneth K. Wilson Renormalization Group (RG) Theory of Critical Phenomena -- 1982 Physics Nobel Basic Ideas: First integrate out short range fluctuations to find out how coupling constant changes with scale. Using expansion around “ fixed ” point to calculate the critical exponents, in full agreement with experiments, without any adjustable parameters.

26. Experimental verification of RG theory Newest results of RG  =-0.011  0.004 Space experiment (7 decades)  =-0.0127  0.0003 Full agreement within accuracy Power of Theoretical Physics !!

27. Justification of Landau Fermi -liquid theory —Weakly interacting fermion systems renormalize to its ‘fixed Point’—Free fermions

28. Paradigm in studying Emergent phenomena  Low energy excitations: quasi particles  Landau Fermi liquid theory  Symmetry breaking  Renormalization  ……. Very successful, common features of phenomena at very different scales, but is it a universal recipe??

29. Integer Quantum Hall Effect － 1985 Nobel in Physics No symmetry breaking Failure of Landau paradigm !!

30. X.G. Wen

31. Topological properties of QHE e 2 /h=1/(25 812.807 572 Ω) accuracy 10 － 9 N=n Chern number

32. QHE and Quantum Spin Hall Effect Qi & Zhang

33. Bulk-insulator, surface-metallic, no time- reversal symmetry breaking, no back- scattering, guaranteed by topological Chern parity!! Topological insulators

34. Plausible exotic excitations Charge+monopole-‘Dyon’ Majorana fermion Axion? X.L. Qi et al.

35. YBCO -- YBa 2 Cu 3 O 6+y No answer yet to the challenge Posed by Müller-Bednorz!! LSCO –La 2-x Sr x CuO 4+ 

36.  Not so much the Tc so high, super-glue?  Even more profound problem: the Fermi liquid theory fails!

37. “Anomalous” normal state properties mysterious linear resistivity H. Takagi et al. PRL, 1992

38. Pseudogap of High-Tc (dark entropy) Missing of entropy at low energies Concept of quasi- Particle not applicable

39. Attempts to explore new paradigm  Topology + quantum geometry (D. Haldane)  Topology + long range entanglements (X.G. Wen)

40. Fractional charge, fractional statistics, …… Is this a complete description?? Laughlin’s wave function for FQHE

41. New question raised by Haldane Are these two ‘circles’ the same? Using geometrical approach they are not the same!! The latter is described by the “guiding centers” which obey ‘non-commutative geometry’!!

43. How to characterize topological order?  No symmetry breaking, nor local order parameter, different quantum Hall states have the same symmetry  Non-local topological order parameter  Ground state degeneracy-Berry phase  Abelian-Non-Abelian edge states (CFT)  Gapped spin-liquid states, protected by symmetry, chiral spin state, …… What is the most fundamental?? X.G. Wen

44. Quantum Entanglement Classical orders (crystals, ferromagnets)-untangled Even the ‘quantum order’-superfluidity-untangled EPR paradox

45. Classification of entanglements  Short range entanglement Landau symmetry breaking states No symmetry breaking- Symmetry protected  topological order  like topological insulators,  Haldane spin 1 chain……  Long range entanglement Symmetry breaking like P+iP superconductivity No symmetry breaking: FQHE, spin liquids Non-trivial topological order = long range entanglement in MB states

46. Some key words  Topology  Geometry (non-commutative)  Long-range entanglements  Entanglement spectrum, instead of just a number (von Neumann entropy)  ……

47. Thank you all!