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Exploring New Paradigm in Physics Yu Lu Institute of Physics Chinese Academy of Sciences “…The underlying physical laws necessary for the mathematical 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 equations much too complicated to be soluble.” P.A.M. Dirac, Proc. Roy. Soc. A123, 713 (1929) How do you do to get the Theory of Everything? 1. Planck/unification scale (1028 eV) 2. QCD Nuclear physics scale (108-109 eV) d u u du d e d u u + 2e 4 He u d d e 3. Condensed matter physics scale (100 eV) + + + + + + + + + + + + - Na metal The Theory of Everyday Everything! 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 Is this truly The theory of Everything? Can one derive ALL exotic properties, from the Schrödinger equation?? “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. 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…” Emergent features of condensed matter systems Collective excitations—quasi-particles Symmetry breaking Renormalization …… 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 ! 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 matrix (metal) from the Their properties not sensitive to microscopic interactions,which cannot be derived from these ‘coarse grained’ properties Basic assumption: Adiabaticity Question: How to justify it, if no gaps? Emergent features of condensed matter systems Collective excitations—quasi-particles Symmetry breaking Renormalization …… Superconductivity 1911 Kamerlingh Onnes discovered zero resistance Early 30s Meissner effect discovered, complete diamagnetism more fundamental London equations 2 d Js c2 m * c 2 Js A , E , L 2 2 4L dt 4L 4ns e *2 c Wave function “rigidity” ansatz (London brothers) ne e J ( 0 | P | 0 A) m c Superconductivity 1950 Ginzburg-Landau equation,introducing macroscopic wave function e i 1 2e 2 (i A) a(T Tc ) | |2 0 4m c ie 2e 2 J s (r ) ( * *) | |2 A 2m mc Bardeen realized: gap in spectrum leads to “rigidity” Cooper pairing:arbitrarily weak attraction gives rise to bound states at the Fermi surface —pairing energy is the gap Is SC a Bose-Einstein condensation of Cooper pairs?--a bit more complicated! BCS wave function: (uk vk ak ak ) | 0; uk2 vk2 1 k 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 Symmetry Breaking Discrete symmetry--from up or down to definite up(down) Broken symmetry-reduction of symmetry elements Displacive phase transition “Usually”: symmetry”, “high temperature-high “low temperature-low symmetry” Broken continuous symmetry Ferromagnet--broken rotational symmetry Antiferromagnetic order – staggered magnetization (Landau & Néel), --not conserved quantity Macroscopic superconducting wave function - order parameter (Landau) i e breaking of U(1) gauge symmetry Anderson-Higgs mechanism Goldstone mode: collective excitations, recovering the symmetry – like spin waves When external (gauge) field coupled, becomes massive -- Meissner effect Unified weak-electromagnetic interactions- 1979 Nobel prize in physics Weinberg- Salam- Glashow Josephson effect: visualization of the phase J J 0 sin( 1 2 ); 2e 2eV0 J J 0 sin( 0 V0t ), t Using two Josephson junctions-- SQUID I max 2I c cos(2 / 0 ), 0 hc / 2e Most profound exhibition of emergence! Josephson Effect S2 S1 e1 e i 2 e i1 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!! Quark-Gluon Plasma Neutron Stars, Color Superconductivity High Tc Superconductivity Low Tc Superconductivity Heavy Electron Superconductivity 3He Superfluidity Atom traps, BEC, Superfluidity 10-9 10-6 10-3 Nano-K micro-K milli-K 1 K 103 kilo-K 106 109 1012 mega-K giga-K tera-K Emergent features of condensed matter systems Collective excitations—quasi-particles Symmetry breaking Renormalization …… Failure of Mean Field Theory!! MFT a g d n Experiment 0 (jump) 1/2 1 3 1/2 0 0 1/3 ! 4/3 ! 5 ! 2/3 ! 0 Theory valid in space dimensions beyond 4 ! Renormalization Group (RG) Theory of Critical Phenomena -- 1982 Physics Nobel Kenneth K. Wilson 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. Experimental verification of RG theory Newest results of RG a=-0.0110.004 Space experiment (7 decades) a=-0.01270.0003 Full agreement within accuracy Power of Theoretical Physics !! Justification of Landau Fermi -liquid theory —Weakly interacting fermion systems renormalize to its ‘fixed Point’—Free fermions 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?? Integer Quantum Hall Effect - 1985 Nobel in Physics No symmetry breaking Failure of Landau paradigm !! X.G. Wen Topological properties of QHE e2/h=1/(25 812.807 572 Ω) accuracy 10-9 N=n Chern number QHE and Quantum Spin Hall Effect Qi & Zhang Topological insulators Bulk-insulator, surface-metallic, no timereversal symmetry breaking, no backscattering, guaranteed by topological Chern parity!! Plausible exotic excitations Charge+monopole-‘Dyon’ Majorana fermion Axion? X.L. Qi et al. No answer yet to the challenge Posed by Müller-Bednorz!! LSCO –La2-xSrxCuO4+d YBCO -- YBa2Cu3O6+y Not so much the Tc so high, super-glue? Even more profound problem: the Fermi liquid theory fails! “Anomalous” normal state properties mysterious linear resistivity H. Takagi et al. PRL, 1992 Pseudogap of High-Tc (dark entropy) Missing of entropy at low energies 600 (c) 0.97 0.92 0.87 0.80 0.76 0.73 0.67 0.57 0.48 0.43 500 400 300 200 0.38 0.29 0.16 100 0 0 50 Concept of quasiParticle not applicable 100 150 T(K) 200 250 300 Attempts to explore new paradigm Topology + quantum geometry (D. Haldane) Topology + long range entanglements (X.G. Wen) Laughlin’s wave function for FQHE Fractional charge, fractional statistics, …… Is this a complete description?? 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’!! 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 Quantum Entanglement EPR paradox Classical orders (crystals, ferromagnets)-untangled Even the ‘quantum order’-superfluidity-untangled 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 Some key words Topology Geometry (non-commutative) Long-range entanglements Entanglement spectrum, instead of just a number (von Neumann entropy) …… Thank you all!