Einstein and quantum theory of solids Yu Lu Institute of Theor. Phys. & Interdisciplinary Center 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 considered this paper “revolutionary” Planck derived this relation “Experiments are in unsolvable contradiction with classical mechanics 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 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)

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 from compressibility 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” 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?

Energy band theory of solids Bloch theorem 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 are EMERGING.”

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<kj<k 

Achievements: atoms, molecules, solids ······N k 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 simplest constituents and understand the behavior of these constituents, as ancient Greeks dreamed, but we know nothing 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

1950 Ginzburg-Landau equation ， introducing macroscopic wave function Bardeen realized ： gap in spectrum leads to “rigidity” Superconductivity Cooper pairing ： arbitrarily weak attraction gives rise to bound states at the Fermi surface—pairing energy is the gap

Is SC Bose-Einstein condensation of Cooper pairs?-- a bit more complicated! BCS wave function ： 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 Most profound demonstration of emergence! Bardeen’s objection Using two Josephson junctions-- SQUID

Discovery of the integer quantum Hall effect － 1985 Nobel prize in physics T ～ 1 K B ～ 8 T

QHE as an emergent phenomenon Precision ： Self-organization ： － /cm 2 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 Physics Nobel Daniel C. Tsui Horst L. Störmer Robert Laughlin

Comparison of fractional and integer QHE Common features ： exact Hall plateau －－ constant×e 2 /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 transitions John Hertz (1976), Andrews Millis, Subir Sachdev Singularity of ground state properties as function of g T Quantum-critical g gcgc

1D Ising chain in transverse field （ qubits ） （ g=0) （ g=  ) Quantum critical point Relaxation of qubits depends only on 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!