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Exploring New Paradigm in Physics
Yu LuInstitute of Physics
Chinese Academy of Sciences
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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.”
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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!
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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
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Is this truly The theory ofEverything?
Can one derive ALL exotic properties,from the Schrödinger equation??
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“ 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..
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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…”
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Emergent features ofcondensed matter
systems
Collective excitations—quasi-particles
Symmetry breaking Renormalization ……
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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 !
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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
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Basic assumption: Adiabaticity
Question: How to justify it, if no gaps?
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Emergent features ofcondensed matter
systems
Collective excitations—quasi-particles
Symmetry breaking Renormalization……
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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
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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
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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
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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
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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
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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
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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
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1e
1e i2e i
Josephson Effect
S2 S1
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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!!
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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
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Emergent features ofcondensed matter
systems
Collective excitations—quasi-particles
Symmetry breaking Renormalization……
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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 !
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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.
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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 !!
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Justification of Landau Fermi -liquid theory —Weakly interacting fermionsystems renormalize to its ‘fixedPoint’—Free fermions
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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??
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Integer Quantum Hall Effect
- 1985 Nobel in Physics
No symmetry breakingFailure of Landau paradigm !!
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X.G. Wen
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Topological properties of QHE
e2/h=1/(25 812.807 572 Ω) accuracy 10 - 9
N=n Chern number
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QHE and Quantum Spin Hall Effect
Qi & Zhang
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Bulk-insulator, surface-metallic, no time-reversal symmetry breaking, no back-scattering, guaranteed by topological Chern parity!!
Topological insulators
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Plausible exotic excitations
Charge+monopole-‘Dyon’Majorana fermionAxion?
X.L. Qi et al.
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YBCO -- YBa2Cu3O6+y
No answer yet to the challenge Posed by Müller-Bednorz!!
LSCO –La2-xSrxCuO4+
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Not so much the Tc so high, super-glue?
Even more profound problem: the Fermi liquid theory fails!
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“Anomalous” normal state properties mysterious linear resistivity
H. Takagi et al.PRL, 1992
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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
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Attempts to explore new paradigm
Topology + quantum geometry
(D. Haldane) Topology + long range entanglements
(X.G. Wen)
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Fractional charge, fractional statistics,……
Is this a complete description??
Laughlin’s wave function for FQHE
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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’!!
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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
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Quantum Entanglement
Classical orders (crystals, ferromagnets)-untangled
Even the ‘quantum order’-superfluidity-untangled
EPR paradox
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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
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Some key words
Topology Geometry (non-commutative) Long-range entanglements Entanglement spectrum, instead of just a number (von Neumann entropy)……
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Thank you all!