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Condensed Matter Physics Feng Duan Overview

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