Einstein and quantum theory of

solids

Yu Lu

Institute of Theor. Phys. & InterdisciplinaryCenter of Theor. Studies, CAS

Einstein’s paper in March 1905

Planck proposed the radiation distribution ， while Einstein suggested that the radiation consists of a gas of “light energy quanta” (Lichtenergiequanten), or simply “light quanta” (Lichtquanten), each with energy proportional to frequency.

Among the 1905 papers Einstein only consideredthis paper “revolutionary”

Planck derived this relation

“Experiments are in unsolvable contradiction with classicalmechanics and classical electrodynamics”

Using Boltzmann’s entropy

Wien’s radiation law, for“high energy quanta”

Analogy of radiation with ideal gas--“gas of light quanta”

Einstein’s heuristic derivation

Not accepted by contemporaries ， strong objection by Planck himself ， Nobel prize only in 1922.

Proved in 1906 and 1907 papers ： Quantization of “light quanta” －－ Planck’law

1905 paper ： Quantization of interaction energy of radiation with matter

Stoke’s rule, photo-ionization, photo-effect

Dulong-Petit’s empirical law ： it should be a constant

Many solids, in particular insulators, SH much smaller, Strongly temperature dependent

Specific heat (SH) puzzle for solids

Boltzmann’a derivation in 1876 ： c=3Rn=5.94n cal/mole·grad

Boltzmann: motion of atoms constrained in solids, not as simple as he imagined

Lord Kelvin: doubt on Boltzmann’s derivation

Lord Rayleigh: Both theory and experiments are right ， genuine contradiction, new “insight” is needed!

Einstein’s quantum theory of specific heat for solids

What is the reason?

1907 paper: “Planck’s theory of radiation and theory of specific heat” Annalen der Physik 22 (1907) 180-190

1907 paper assumes quantization of energy of atom vibration,with the same frequency, the same average energy

Einstein founded the quantum theory of solids!!

derivation of typical frequency fromcompressibility and density

1910 Nenrst measured the temperature dependence for more solids

Comparison of theory with diamond’s SH in 1905’s paper

The earliest confirmation of quantum theorycame from solids

Millikan’s 1914 photoeffect experiments

Only after Compton scattering experiment in 1923 ， the concept of “light quanta” was accepted by physicists

1911 Debye model －－ continuum model

1911 Born- von Karman molecular chain －－ lattice

1924 Heisenberg quantum mechanical calculations

Born-Huang lattice dynamics theory

Drude-Lorentz free electron theory of metals

Electrons in metals are “free”, can conduct electricity, heat ，with some “mean free path”

22

3

e

k

T

Wiedemann-Franz law

Lorentz gave rigorous proof using Maxwell- Boltzmann distribution

Difficulties:1 ） Why some solids are metals ？ 2 ） Why SH of metals is not z times bigger ？

Pauli-Sommerfeld free electron theory of metals

Identity principle －－ Bose-Einstein and Fermi-Dirac statistics

Pauli with “great regret” gave up the Bose character assumption of electrons －－ derived Pauli paramagnetism

Sommerfeld systematically applied Fermi-Dirac statistics

For most metals F>>T, “low temperature phenomena”

Electron SH ~T, rediscovered Wiedemann-Franz law……

What is the difference between metals and insulators?

Extended Zone

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

-8 -6 -4 -2 0 2 4 6 8

k space

En

erg

y in

eV

Energy band theory of solids

Bloch theorem

Reduced Zone

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

-4 -3 -2 -1 0 1 2 3 4k space

En

erg

e in

eV

Metal—partially filled bands ；Insulator （ semiconductor ）－ fully filled bands

Two opposite views on the Nature

Reductionism: Everything is reduced to its constituents, governed by the most fundamental laws. “Ultimate Goal”—To establish THE THEORY OF EVERYTHING

Emergence: There are different levels of the real world, and there are fundamental laws at each level.Our mission is to start with the basic experimental facts, to unveil these fundamental laws and to understand “ How qualitatively new phenomena areEMERGING.”

Philip W. Anderson: More is different (1972)

…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.

Theory of Everything

R B Laughlin & D Pines

j<k

Achievements: atoms, molecules, solids ······Nk Approximate methods ： crystal structure, phonon spectrum, even Tc under el-ph model of SC

DFT － 1998 Nobel Prize in Chemistry

Walter Kohn

Quantum Molecular Dynamics －Car-Parrinello method

Dynamic Quantum Mean Field Theory （ DMFT ）

LDA ＋ DMFT

Phonon spectra of plutonium ， theoretical predictions (red circles) （ X. Dai et al., Science 300, 953 (2003)); Neutron scattering results (black squares) (Science 301,1078 (2003))

Failures ： Superconductivity, superfluidity, QHE Josephson effect······ High Tc······ not talking about protein functions understanding of conscience…….

We can decompose complex systems into the simplestconstituents and understand the behavior of these constituents, as ancient Greeks dreamed, but we knownothing about the complex systems themselves!!

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 do not survive, while leaving crystals:quasiparticles

• Not sensitive to microscopic details ， which 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 quasipariticles 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 properties• From the RG point of view, interacting and free fermion systems are controlled by the same fixed point

Superconductivity

1911 Kamerlingh Onnes discovered zero resistance

Early 30s Meissner effect was discovered ，complete diamagnetism 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 boundstates at the Fermi surface—pairing energy is the gap

Is SC 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

Goldstone mode: collective excitations, recovering the symmetry – like spin waves

When external (gauge) field coupled, becomes massive Meissner effect

Anderson-Higgs mechnism

Unified weak-electromagnetic interactions －－ 1979 Nobel prize in physics

Josephson effect ： visualization of the phase

0

000

210

2 ),

2sin(

);sin(

eV

ttV

eJJ

JJ

Most profound demonstration of emergence!

Bardeen’s objection

Using two Josephson junctions-- SQUID

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

Discovery of the integer quantum Hall effect － 1985 Nobel prize in physics

T ～ 1 K

B ～ 8 T

QHE as an emergent phenomenon

• Precision ： 10-9

h

exy

2

整数 0 ,0

yxxyyyxx

xxxxxx

• Self-organization ： 1011 － 1012 /cm2 particles synchronized

• Universality －“ robustness” － not sensitive to impurities, details of microscopic interactions, etc.

What guarantees “exactness” of quantization? – Disorder caused localization

Electron interactions can be “adiabatically” switched off

“pumping” integer number of electrons--emergence

Fractional QHE-1998 Physics Nobel

Daniel C. Tsui Horst L. Störmer Robert Laughlin

Comparison of fractional and integer QHE

Common features ：

• exact Hall plateau －－ constant×e2/h

• Zero longitudinal conductivity and resistance

• Thermal activation ， gap ， described by Mott VRH

Differences ： constant --integer or fractional ？“adiabatically” derivable from noninteracting model?disorder dominating －－ integer, interaction dominating －－ fractional

Laughlin wave function －－ new quantum state

• gap, incompressible, like rotons in SF helium• quasiparticle charge e/3• fractional statistics ， gauge interactions

Unusual properties ：

Another pronounced example of emergence!

Quantum phase transitionsJohn Hertz (1976), Andrews Millis, Subir Sachdev

Singularity of ground state properties as function of g

T Quantum-critical

ggc

)( 1 n

xn

zn

zn gJH

1D Ising chain in transverse field （ qubits ）

（ g=0) 321321 |||| |||| nnnnnn

（ g=) |||| 321 nnn

gg c 1 Quantum critical point

TkB

R )16/tan(2

Relaxation of qubits depends onlyon temperature

Einstein in 1905 explored the wave-particle duality for the radiation field ， the same idea leads to quantum theory of solids 。

100 years of quantum theory of solids—Outcome of complementary interplay of reductionism and emergence!