Exploring New Paradigm in Physics

Yu LuInstitute of Physics

Chinese Academy of Sciences

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.”

How do you do to get the Theory of Everything?

1. Planck/unification scale (1028 eV)

ddu

du

u

ddu

d uu

e

e 4 He + 2e2. QCD Nuclear physics scale (108-109 eV)

-

+

+ + + +

+ + +

+ + + +

- -

-

--

-

-

-

-

-

Na metal

3. Condensed matter physics scale (100 eV)

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 theseremarkable quantum phenomena

Is this truly The theory ofEverything?

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 ofcondensed 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 from the matrix (metal)

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 ofcondensed matter

systems

Collective excitations—quasi-particles

Symmetry breaking Renormalization……

Superconductivity1911 Kamerlingh Onnes

discovered zero resistanceEarly 30s Meissner effect

discovered, complete diamag-netism more fundamental

Wave function “rigidity” ansatz (London brothers)

London equations

2

22

2

2

2 *4

* ,

4 ,

4 en

cmE

c

dt

JdA

cJ

sL

L

s

L

s

)0||0( Ac

eP

m

neJ

1950 Ginzburg-Landau equation ， introducing macroscopic wave function

ie

Bardeen realized: gap in spectrum leads to “rigidity”

Superconductivity

0||)() 2

(4

1 22 cTTaAc

ei

m

Amc

e

m

ierJ s

22

||2

*)*(2

)(

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 ：

1 ;0|)( 22

kkkkk

kk vuaavu

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

“ Usually”: “high temperature － high symmetry”, “low temperature － low symmetry”

Displacive phase transition

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

ie

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

Josephson effect ： visualization of the phase

0

000

210

2 ),

2sin(

);sin(

eV

ttV

eJJ

JJ

Most profound exhibition of emergence!

Using two Josephson junctions-- SQUID

ehcII c 2/ ),/2cos(2 00max

1e

1e i2e i

Josephson Effect

S2 S1

Bardeen - Josephson dispute

Anderson’s lecture

Josephson’s calculation

Bardeen’s added note

Dispute at LT 8 BCS mentor againstthe most convincing proof of his theory!!

10-9 10-6 10-3 1 103 106 109 1012

Atom traps, BEC, Superfluidity

3He 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

Emergent features ofcondensed matter

systems

Collective excitations—quasi-particles

Symmetry breaking Renormalization……

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 ！

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.

Experimental verification of RG theory

Newest results of RG=-0.0110.004

Space experiment (7 decades)=-0.01270.0003Full agreement within

accuracy

Power of Theoretical Physics !!

Justification of Landau Fermi -liquid theory —Weakly interacting fermionsystems renormalize to its ‘fixedPoint’—Free fermions

Paradigm in studying Emergent phenomena

Low energy excitations: quasi particles

Landau Fermi liquid theory

Symmetry breaking

Renormalization

…….

Very successful, common features ofphenomena at very different scales,but is it a universal recipe??

Integer Quantum Hall Effect

－ 1985 Nobel in Physics

No symmetry breakingFailure 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

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

Topological insulators

Plausible exotic excitations

Charge+monopole-‘Dyon’Majorana fermionAxion?

X.L. Qi et al.

YBCO -- YBa2Cu3O6+y

No answer yet to the challenge Posed by Müller-Bednorz!!

LSCO –La2-xSrxCuO4+

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

0

100

200

300

400

500

600

0 50 100 150 200 250 300

T(K)

0.16

0.38

0.29

0.970.920.870.800.760.730.670.570.480.43

(c)

Concept of quasi-Particle not applicable

Attempts to explore new paradigm

Topology + quantum geometry

(D. Haldane) Topology + long range entanglements

(X.G. Wen)

Fractional charge, fractional statistics,……

Is this a complete description??

Laughlin’s wave function for FQHE

New question raised by Haldane

Are these two ‘circles’ the same?

Using geometrical approach they are notthe 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

Classical orders (crystals, ferromagnets)-untangled

Even the ‘quantum order’-superfluidity-untangled

EPR paradox

Classification of entanglements Short range entanglement

• Landau symmetry breaking states• No symmetry breaking- Symmetry protectedtopological orderlike 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!