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!
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