This course targets to learn about the theory for quantum matter, in particular the recent progresses of it.

Graduate-level quantum mechanics, statistical mechanics, quantum field theory.

Final term paper 100%

Fradkin, Field Theories of Condensed Matter Physics
Mahan, Many-Particle Physics
Sachdev, Quantum Phase Transitions
Yellow Book (CFT)

Topic 1: Symmetry breakings in classical statistical mechanics
1-1: Universality & Critical Exponents
1-2: Landau-Ginzburg theory & Order parameter
1-2-1: Critical dimensions & Mermin-Wagner theorem
1-3: Momentum-shell renormalization group
1-3-1: Wilson-Fisher fixed point

Topic 2: Classical-to-quantum mapping
2-1: Breakdown of classical-to-quantum mapping
2-2: Quantum Monte-Carlo Simulation - (학생 발표)

Topic 3: Transverse-Field Ising Model
3-1: Kramer-Wannier duality
3-2: Jordan-Wigner transformation (Fermionization)
3-2-1: Kitaev chain - topological superconductor
3-3: D=2+1 Quantum Ising model as the KW dual of Z2 gauge theory
3-3-1: Deconfinement & Confinement
3-3-2: Wilson loops
3-4: Z2 gauge theory with matter fields
3-4-1: Toric code
3-4-2: Honeycomb Kitaev model
3-4-2-1: Thermal Hall experiment

Topic 4: Bose-Hubbard model
4-1: Insulator & superfluid phases
4-1-1: Ultra-cold atom experiments
4-2: Large-N description of the transition between insulator & superfluid
4-3: (D=1) KT transition
4-4: (D=2) Duality to QED3

Topic 5: Quantum Hall Effects
5-1: Integer quantum Hall phases
5-1-1: Chern insulator & Berry phase (topological insulator)
5-2: Fractional quantum Hall phases
5-2-0: A few important states: Laughlin, (n,l,m) states, pfaffian, etc
5-2-1: Fractional excitations & Chern-Simons theory
5-2-1-1: Edge theory
5-2-2: Composite Boson theory - “Bosonization in 2+1D”
5-2-3: Composite Fermion theory
5-2-3-1: Composite Fermi liquid
5-2-4: Parton description
5-2-5: General Abelian Chern-Simons theory
5-2-5-1: Edge theory
5-2-6: Coupled wire construction for Quantum Hall effects
5-3: More 2+1D dualities
5-3-1: “derivation” of fermion-fermion duality from QH transition
5-3-2: rederivation of QED3 & SF-Insulator transition
5-4: 1+1D chiral edges of quantum Hall states
5-4-1: Chiral CFT
5-4-2: Bulk-Boundary correspondence
5-4-3: Anomaly

(Interim) Parton Description of Spin Liquids
(Interim-0): A few natural ground states in spin models
(Interim-1): Resonant Valence Bond Picture
(Interim-2): U(1) spin liquid
(Interim 2-1): Polyakov’s confinement
(Interim 2-2): Paths for deconfinement
(Interim-3): Z2 spin liquids, revisited
(Interim 3-1): PSGs & Classifications
(Interim 3-2): fermionic-bosonic spinon duality
(Interim 3-3): Nearby proximate phases of Z2 spin liquids
(Interim 3-4): Field theory for Z2 spin liquids & Higgs mechanism
(Interim 3-5): Kitaev honeycomb model, revisited
(Interim-4): Deconfined quantum criticality
(Interim 4-1): Physical picture
(Interim 4-2): Field theory

Topic 6: Fermions
6-1: Metals
6-1-1: Universality of Fermi liquid (Shankar’s RG)
6-1-1-1: Phenomenologies
6-1-1-2: Transport
6-1-1-2-1: Drude model & Wiedemann-Franz law
6-1-1-2-2: Kubo formula
6-1-1-2-2-1: Linear response, in general.
6-1-2: Anderson localization
6-2: Instabilities of metals
6-2-1: Stoner’s instability
6-2-2: Importance of density of states & density waves (nesting)
6-2-2-1: Exotic density wave - pair density wave
6-2-2-1-2: Topological Defects
6-2-2-1-1: Thermal phase diagram - 4e superconductor
6-2-2-2: van Hove singularity
6-2-2-3: Experiments on Kagome materials
6-2-3: Superconductivity
6-2-3-1: p+ip superconductor
6-2-3-2: Majorana zero mode & Non-abelian braiding
6-2-3-3: Doping Z2 spin liquid as a path to superconductivity
6-2-3-3-1: holon-spinon parton ansatz


Topic 7: Conformal Field Theory & 1+1D systems
7-1: 1+1D free fermion chain
7-2: 1+1D interacting fermion chain
7-2-1: Bosonization
7-2-2: Luttinger liquid - prototypical non-Fermi liquid
7-2-3: “Phases” of Luttinger liquid
7-2-4: Fisher-Kane
7-3: 1+1D spin chain & Abelian bosonization
7-4: Refermionization & Bosonization of bosonic systems
7-5: Compact boson CFT
7-6: Coulomb gas representation & Compactification
7-7: Torus partition function
7-7-1: Modular invariance
7-7-2: Gravitational anomaly of Quantum Hall edge states

Topic 8: Symmetry-Protected Topological Phases
8-1: Fermionic 10-fold classifications
8-1-1: 1D topological insulators & superconductors
8-1-1-1: Breakdown by interactions (Fidkowski-Kitaev)
8-1-2: 2D topological insulators & superconductors
8-1-2-1: Breakdown by interactions
8-1-2-2: Anomaly at the edge, orbifolding & orientifolding
8-1-2-3: Chern insulator, revisited
8-1-2-3-1: Spectral flow & Thouless pumping
8-1-2-3-2: Many-Body Invariant & Resta formula
8-1-3: 3D topological insulators & superconductors
8-1-3-1: Breakdown by interactions
8-1-3-2: Parity anomaly & half-level Chern-Simons theory
8-1-3-2-1: Axion electromagnetism
8-1-3-2-2: Witten effect
8-1-3-2-3: Many-Body invariant
8-1-3-3: Symmetry-preserving gapped boundary, T-pfaffian
8-1-3-4: 2+1D fermion-fermion duality, revisited
8-1-3-4-1: Coupled chain description
8-1-3-5: Half-filled Landau level, revisited
8-1-3-5-1: thermal Hall experiment
8-1-4: A few interesting heterostructures
8-1-4-1: Fractional charge at QSHE edge
8-1-4-2: Majorana at QSHE edge
8-1-4-2-1: Spin-orbit + magnetization + superconductor
8-1-4-2-2: Mapping to Kitaev chain models
8-1-4-3: Fu-Kane model on the surface of topological insulator
8-1-4-3-1: Spin-orbit + magnetization + superconductor
8-1-4-4: Parafermion at FQHE edge + superconductivity
8-1-4-4-1: Mapping to parafermion chain models
8-1-4-5: Spin liquid + superconductivity as a probe for spinon
8-2: Bosonic SPTs & Cohomology
8-2-1: 1D Haldane chain
8-2-1-1: instanton Berry phase
8-2-1-2: LSMOH perspective in instanton Berry phase
8-2-1-2: D2-symmetric model
8-2-1-3: MPS representation - (학생 발표)
8-2-2: 2D exactly-solvable models
8-2-2-1: Bosonic integer quantum Hall states
8-2-2-2: Gauging global symmetries
8-2-3: 3D bosonic SPTs
8-2-3-1: BF field theory
8-2-3-2: More 2+1D duality from surface of bosonic SPT
8-2-4: t’ Hooft anomaly
8-2-4-1: LSMOH theorem, rephrased
8-2-4-2: Deconfined quantum criticality, revisited
8-3: Miscellaneous recent topics
8-3-1: Topological semimetals (Basics)
8-3-1-1: RG on graphene + Coulomb interaction - logarithms
8-3-1-2: Axial anomaly in Weyl semimetal
8-3-2: Crystalline symmetries
8-3-2-1: A few no-go’s for symmetry fractionalizations
8-3-2-2: Variants of LSMOH theorem
8-3-2-3: Higher-order topology
8-3-3: Entanglement spectrum
8-3-3-1: Bulk-boundary correspondence in ES
8-3-3-2: Derivation of Modular Hamiltonian

Topic 9: Beyond (Only briefly touching)
9-1: 3+1D Coulomb spin liquids
9-1-1: Deconfinement & Confinement
9-1-2: Coulomb spin ice - pyrochlore lattice
9-2: 3+1D Z2 gauge theory
9-3: New possibility in 3+1D - fractonic states
9-4: Far-from-equilibrium quantum states
9-4-1: Floquet topological states
9-4-2: Volume-law states
9-4-3: Many-Body Localizations
9-4-4: Quantum scars
9-5: Random-unitary circuits
9-6: What we didn’t touch in this lecture.

Standard lecture style