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
(10 28 eV)
2. QCD  Nuclear physics scale
(10 8 -10 9 eV) e d u d u u u d d d u u e
4 He + 2e

+
-
+
+ 3. Condensed matter physics scale
(10 0 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…”

 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?

 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
J s
  c
4

2
L
A , d J dt s
 c
2
4

2
L
E ,

2
L
 m * c
2
4
 n s e *
2
Wave function “rigidity” ansatz (London brothers)
J
 ne
(
 m
0 | P | 0
  e c
A )

Superconductivity
1950 Ginzburg-Landau equation ， introducing macroscopic wave function    e i

1
4 m
(
 i   
2 e c
A )
2
  a ( T

T c
)
  
|

|
2
 
0
J s
( r )
  ie 
2 m
(

*
     
*)

2 e
2 mc
|

|
2
A
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 ：
   k
( u k
 v k a k

 a

 k

) | 0

; u k
2  v k
2 
1
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”: “high temperature － high symmetry”,
“low temperature － low symmetry”

Broken continuous symmetry
Ferromagnet--broken rotational symmetry
Antiferromagnetic order – staggered magnetization
(Landau & Néel) ， －－ not conserved quantity
Macroscopic superconducting wave function
   e i

－ order parameter (Landau)
 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


J
0
J
0 sin( sin(

1
 
 
2
);
0

2 e

V
0 t ),
 
 t

2 eV
0

Using two Josephson junctions-- SQUID
I max

2 I c cos( 2

/

0
),

0
 hc / 2 e
Most profound exhibition of emergence!

Josephson Effect
S
2
S
1
 e i

2
 e i

1
 e

1

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
3 He Superfluidity
Atom traps, BEC, Superfluidity
10 -9 10 -6 10 -3 1 10 3 10 6 10 9 10 12
Nano-K micro-K milli-K K kilo-K mega-K giga-K tera-K

 Collective excitations—quasi-particles
 Symmetry breaking
 Renormalization
 ……

！！
MFT
Experiment a
0 (jump ） 
0
 1/2 
1/3 !
g 1 
4/3 !
d 3 
5 !
n 1/2 
2/3 !

0

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 =-0.011
 0.004
Space experiment
(7 decades) a =-0.0127
 0.0003
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


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

e 2 / h =1/(25 812.807 572 Ω) accuracy 10
－ 9
N=n Chern number

Qi & Zhang

Bulk-insulator, surface-metallic, no timereversal symmetry breaking, no backscattering, guaranteed by topological
Chern parity!!

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

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

600
500
400
300
200
100
0
0
0.97
0.92
0.87
0.80
0.76
0.73
0.67
0.57
0.48
0.43
50 100
(c)
150
T(K)
200
0.38
0.29
0.16
250 300
Missing of entropy at low energies
Concept of quasi-
Particle not applicable

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

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