The importance of reductionism: recognizing microstructures to understand macroscopic manifestations
Complexity reduces to simplicity, simplicity constructs
complexity
coupling and decoupling between complex levels
decoupling: critical phenomena, renormalization group theory ->
ignore the influence of various microscopic effects (energy is much
greater than a certain cutoff value)
Emergent phenomena
Gauge
symmetry breaking (arbitrariness of wave function phase)
Particles in thermal equilibrium:\(v\sim\sqrt T\),thermal de Broglie
wavelength inversely proportional to the square root of
temperature
wavelength equal to distance between particles —>
quantum effect obvious
\(\frac{3}{2}kT=\frac{1}{2}mv^2\), \(\lambda=h/p\). estimate the quantum
degeneracy temperature
electron degeneracy temperature ~ \(10^5 K\), manifests wave properties
Ions
or atoms,\(\frac{50}{A} K\),manifests
at low temperatures
nature of condensation phenomena is \(compartmentalization\)
configuration
space: manifest in the emergence of free surface
momentum space:
condensation in momentum space
\(Boson\): BEC: when cooled to critical
temperature,macroscopic numbers of particles occupy ground state
\(Fermion\): Fermi sphere generalized
condensation in momentum space, exact critical temperature does not
exist
If exists sort of attractive interaction between fermions, will
pair to boson then condensate to macroscopic quantum phase
(metal
insulator and He3 superfluid)
different phase, result of competition between internal energy and
entropy
position order, correlation between different atom
positions
T<T0 what is important is order in momentum space
(momentum order, wave order))
root usually from interaction. while
non-interactive like ideal gas, could also be ordered. interaction
reduces wave order
e.g. interaction between bosons results in lower
BEC temperature
introduce interaction adiabatically, fermi gas
becomes fermi liquid, slight reduction in wave order, but still retains
sharp fermi surface
and meta-excitation corresponds to Fermi gas.
further increase interaction, transit to strong correlated Fermi system,
reveal abnormal phenomena
increase density might increase interaction strength in classical
systems, while it might be different in quantum systems
e.g. in
electron system, reduction in electron density instead results in
increase in interaction, strong correlated electron system usually be
low density electron system
due to Heisenberg uncertainty principle, uncertainty of position and
momentum are connected
thus could imagine, position order and
momentum order could be incompatible in quantum mechanics
e.g. BEC
does not emerge in crystalline solid, the formation of Wigner crystal is
at sacrificial of Fermi surface
The correlation of velocity fields in
fluid dynamics —> analogous to ordered states in momentum
space
The ordering phenomenon of spin particle systems
If spins
are confined to lattice sites, they manifest as disordered spin states
(paramagnets) transitioning into ordered spin states (ferromagnets)
through exchange interactions
Quantum description of the spin and
momentum ordering of itinerant electrons
Spin ordered phase, spin
polarization, interacting spin fermions may pair up
Schafroth pairs
in configuration space, Cooper pairs in momentum space
Condensed matter is a condensate in phase space, including
configuration space and momentum space, with rich implications
Although the interactions causing phase transitions are inherently
quantum in nature, the emergence of phase transitions is due to
classical fluctuations
Another type, quantum phase transition. At 0
K, thermal fluctuations vanish, and phase transitions occur through
quantum fluctuations required by the Heisenberg uncertainty
principle
Fundamentally different from thermodynamic phase
transitions. Can be confirmed by extrapolating experimental results
Paradigm
Wave propagation in periodic
structures
Simplifications introduced by translational
symmetry
Extensions and modifications
Aperiodicity, strong
disorder, and localization, Mott's physical interpretation of
these
Quasiperiodic structures. Another form of disorder:
inhomogeneous structures. Percolation theory. Fractal structures with
scale invariance
Surface physics. Low-dimensional
physics
Fluctuations destroy long-range periodic structures in one-
and two-dimensional structures, but due to coupling and finite size,
periodic structures can stably exist
Disorder effects are
particularly significant in low dimensions
Scaling theory of
localization suggests that electrons are localized in one- and
two-dimensional structures
Research on aperiodic and
non-three-dimensional structures extends the original paradigm
significantly, becoming an active field
Another active field relates
to the coherence effects of de Broglie waves
Anderson localization
and weak localization of electrons are fundamentally interference
effects of waves —> mesoscopic physics
Incorporating spin into
paradigms. Spin-polarized electrons can undergo spin-dependent
scattering with lattice waves and defects
Simplified models, energy band theory in the single-electron
approximation
Energy band theory emerged alongside the quantum
mechanical treatment of the hydrogen molecule
Although similar, they
differ. Both use single-electron approximations
Earlier, Heitler and
London, in their valence bond theory study of hydrogen molecules, had
roughly considered electron localization correlations
This was later
extended to Heisenberg’s treatment forming the theoretical foundation of
localized spin ferromagnetism
Subsequently, Pauli introduced hybrid
orbital theory, adopting a more intuitive and localized valence bond
orbital approach
Chemists emphasize localization (atomic orbital
shapes, bond formation, charge transfer)
Physicists emphasize
delocalized valence electrons and dispersion relations in wavevector
space
Each has its strengths and weaknesses. Energy band theory is
the most successful for handling transport problems, but it is less
effective in dealing with bonding problems compared to quantum
chemistry
Quantum chemistry's approach of summing atomic
contributions becomes difficult to extend to more atoms
Modifications to the single-electron approximation. Introducing
correlation terms characterizing electron interactions
Density
functional theory. Local density approximation. Standard methods of ab
initio calculations
Since the 1970s, nanostructures intermediate
between molecules and bulk solids, particularly quantum wells, wires,
dots, and superlattices, have garnered widespread interest
Many of
their properties are precisely intermediate between molecules and
crystals
Limitations of energy band theory have become
evident
Wigner theoretically considered: as free electron density
decreases, the kinetic energy of electrons becomes less significant
compared to Coulomb energy
If the electron density becomes
sufficiently low, Coulomb energy dominates, causing electrons to arrange
themselves in an ordered lattice configuration, forming a Wigner
crystal
For transition metal oxides such as NiO, CoO, and MnO, a unit
cell contains an odd number of valence electrons, leading energy band
theory to predict conductivity, whereas they are actually
insulators
Peierls and Mott attributed the issue to electron
correlation effects caused by interactions. Mott insulators
On-site
correlation energy can prevent the formation of mobile configurations.
Using on-site correlation energy to distinguish between conductors and
insulators is a more chemistry-oriented approach
Hubbard combined
this concept with energy band theory to form the Hubbard model,
addressing the Mott transition (metal-insulator transition due to
electron correlations)
In the 1950s, Anderson formulated the
superexchange interaction theory for antiferromagnetic and ferrimagnetic
oxides
On-site correlation energy also plays a crucial role
here
Meanwhile, Zener developed the double exchange mechanism to
explain the ferromagnetic metallic state of certain manganites
By the
1990s, it was clear that these oxides exhibited not only spin ordering
but also possible orbital and charge ordering
Various types of
ordering may couple together. Orbital physics in oxides developed.
Colossal magnetoresistance (CMR)
Studies of magnetic impurities in
normal nonmagnetic metals revealed intriguing effects, such as anomalies
in resistivity due to magnetism
This is clearly related to
hybridization and interaction between s & d (or f)
electrons
Anderson proposed a simplified model considering on-site
correlation energy, known as the Anderson model
The Anderson
Hamiltonian can be transformed into the Kondo Hamiltonian, explaining
the Kondo effect
The Anderson model can be extended to the periodic
Anderson model, just as the Kondo model extends to the Kondo lattice
model
Since 1986, a series of high-temperature superconductors have
been discovered. Their prototype phases are nearly Mott insulators,
while the superconducting phases are doped Mott insulators
Manganites
exhibiting colossal magnetoresistance share a similar
scenario
Therefore, strong electron correlations are expected to play
a crucial role in both the anomalous normal-state resistivity and the
superconducting pairing mechanism
Cooperative phenomena, such as phase transitions, are central to
many-body physics
Heisenberg’s ferromagnetism theory and BCS
superconductivity theory stand as peaks of the quantum theory of
cooperative phenomena
Antiferromagnetism, \(He_{3}\) and \(He_{4}\) superfluidity, ordered phases via
symmetry breaking
General difficulties in handling many-body problems
—> mean-field approximation
Universal Landau phenomenological
theory of second-order phase transitions
Studies on excitations in
ordered phases
Debye phonon theory introduced the elementary
excitation concept, Bloch introduced spin waves or magnons, Bohm and
Pines’ plasmon theory, and Landau’s Fermi liquid theory marked
significant advancements in many-body problem theory
Dislocation
theory explains why metals are easily plastically
deformed
Simultaneously, Landau and Lifshitz proposed the theory of
magnetic domains in ferromagnets
... Classification schemes of
topological defects, providing a unified theoretical treatment of
defects for the first time
Topological defects are considered
singular regions in ordered media, with their topological stability
depending only on the spatial dimensionality of the medium and the
number of components in the order parameter, reflecting
universality
Landau theory fails in the critical region, leading to
the emergence of modern critical phenomena theories handling strong and
long-range fluctuations
Since the 1970s, these cooperative phenomena
concepts have been extended into new fields by P.G. de Gennes, S.F.
Edwards, and others
Such as liquid crystal physics, polymer physics,
and other soft-matter physics