Topological terms in condensed matter physics


# Date Read Topic
1. Tue, Jan 27 Introduction. Course structure, grading, prerequisites, etc. Topology in condensed matter physics as an emergent phenomena. Spontaneous symmetry breaking. Effective actions. Particle on a ring. Motivating example: particle on a ring. Lagrangian and equations of motion. Hamiltonian. Quantum mechanics of the particle on a ring. Spectrum. Aharonov-Bohm effect.
2. Thu, Jan 29 Path integral. Topological sectors. Wick rotation. Quantum interference due to the topological term. Properties of theta terms. General definition of topological terms. Metric independence. Topology and topological spaces. Topological properties.
3. Tue, Feb 3 Homework 1 is given. Topology and topological spaces. Homeomorphism. Open sets. Metric topology. Examples of topological spaces. Euclidean spaces, spheres and balls. Real and complex projective spaces. Grassmann manifolds.
4. Thu, Feb 5 Direct product of topological spaces. Examples of topological spaces. Compact classic groups. Action of groups in spaces. Classic surfaces. Topological invariants. Euler characteristic.
5. Tue, Feb 10 Space of mappings. Homotopy theory. Paths and loops. Product of paths. Constant and inverse paths. Homotopy classes of loops. Fundamental or first homotopy group.
6. Thu, Feb 12 Fundamental group: examples. Free homotopy classes of loops. Fractional statistics of quantum particles.
7. Tue, Feb 17 Homework 2 is given. Fractional statistics of quantum particles. Braid group. Higher homotopy groups. Abelian nature of higher homotopy groups. Homotopic invariance of homotopy group. Examples of topological spaces and their homotopy groups.
8. Thu, Feb 19 Examples of topological spaces and their homotopy groups. Homotopy groups of spheres: degree of mapping, Hopf invariant, stability of homotopy groups. Homotopy groups of Lie groups. Bott periodicity. Topological defects and textures in ordered media. Effective field theory approach in condensed matter physics. Low energy degrees of freedom from the spontaneous symmetry breaking and from the gauge symmetry. Nontrivial topology of the order parameter space. Topological defects. Point and line defects. Domain walls.
9. Tue, Feb 24 Topological textures. Examples: Classical Heisenberg FM in 2d, spiral phase in 3d. Phase transitions driven by topological defects. XY model: definition, continuous limit, superfluid density (stiffness).
10. Thu, Feb 26 XY model: gauge invariance and transverse susceptibility,
11. Tue, Mar 3 XY model: absence of the long range order in 2d, entropy argument for BKT transition, decoupling of vortices. Defects, textures, and theta terms.
12. Thu, Mar 5 Homework 3 is given. Theta terms. Homotopy classes of spacetime configurations. Quantum interference between topological sectors. Unitary, one-dimensional representations of corresponding homotopy groups. Complex weights of topological sectors in the path integral. Examples: topological theta-termd in 2d O(3) NLSM, 3d O(4) NLSM, Hopf term in 3d O(3) NLSM, 4d O(4) NLSM, Pruisken's theta term for integer Quantum Hall Effect. WZW term in quantum mechanics: single spin. Path integral for a single spin. Commutation relations, Poisson brackets, Lagrangian and action. Multivaluedness of the action.
13. Tue, Mar 10 Expression for WZW in 0+1 dimensions. Multivaluedness of the action and quantization of coupling constants (2S - integer). Spin coupled to a unit vector. Integrating fermions out. Berry's phase and induced action.
14. Thu, Mar 12 Gradient expansion of fermionic determinant. WZW term induced by fermions. Independence of WZW term on metric. First look at differential forms. Reduction to theta term. Properties of WZW terms.
15. Tue, Mar 17 Brief introduction to differential forms. Exterior k-forms. Exterior product. Differential form. Integration of differential forms.
16. Thu, Mar 19 Chains. Integrals of forms over chains. Exterior differentiation. Stokes' formula. Closed forms and cycles. Cohomology and homology. Gauge fields.
17. Tue, Mar 24 Homework 4 is given. Gauge fields. Aharonov-Bohm effect. Dirac monopole. Theta and WZW terms for sphere target spaces. Spin chains. Path integral for quantum Heisenberg type magnet.
18. Thu, Mar 26 Continuum description of quantum ferromagnet. Derivation of NLSM for quantum antiferromagnet in the limit S going to infinity. NLSM for spin chains with S>>1. Haldane's theta-term.
19. Tue, Mar 31 Yan Xu: BKT phase transition.
Continuous non-Abelian symmetry. O(N) nonlinear sigma model in 2d. The leading order in 1/N expansion. Mass generation. Exponential decay of correlation functions.
20. Thu, Apr 2 Homework 5 is given. Remarks on chiral anomaly (problems 4, 5 of hw4). 1/N expansion of O(N) NLSM. RG for O(N) NLSM. (d-2) expansion. Nontrivial fixed point in d>2. Perturbative nature of RG. Appearance of the scale \xi\sim e^{\pi S} in NLSM for spin chains.
Tue, Apr 7 NO CLASSES (Spring Recess).
Thu, Apr 9 NO CLASSES (Spring Recess).
21. Tue, Apr 14 Rafael Lopes de Sa: Topological versus "spin-wave" transitions in statistical mechanics.
Topological term. RG-type diagram for NLSM with topological term. AKLT models.
22. Thu, Apr 16 AKLT models. Spin chains with integer spin. Boundary states. WZW as boundary theory for NLSM with integer-spin top. term. Heisenberg model as large U limit of Hubbard model.
23. Tue, Apr 21 Homework 6 is given. Hubbard model at half filling for small U. Critical behavior for half-integer spins? WZW as critical theory of spin chains. Relations between NLSM with theta term and WZW model.
24. Thu, Apr 23 Boundary-bulk correspondence of topological terms. Quantum magnets in higher dimensions. Hopf term etc. NLSM in two and higher spatial dimensions and allowed topological terms. Hopf term induced by fermions. Hopf term and spin-statistics of skyrmions.
25. Tue, Apr 28 Sriram Ganeshan: Topological order. Boundary states of spin chains (experiment). Integer Quantum Hall Effect. Classical Hall effect. IQHE experiment. Landau levels. Localization.
26. Thu, Apr 30 Ozan Erdogan: Compact QED in 2+1. Topological defects and mass of photon.
Saul Lapidus: Quantum Spin Hall Effect.
Laughlin's topological argument. Quantization of Hall conductivity. Edge states.
27. Tue, May 5 Manas Kulkarni: Topological defects in spinor BEC condensates with S=1.
Marcos Crighigno: Compact QED in 2+1 with fermions. Zero modes on topological defects.
28. Thu, May 7 Savvas Zafiropoulos: Polyakov-t'Hooft monopole.
Linear response and Chern-Simons theory. Gauge non-invariance of Chern-Simons theory on the manifold with boundaries. Edge states. Ballistic 1d wires and quantization of Hall conductance. Fractional Quantum Hall Effect. Experiment. Landau levels in radial gauge. Slater determinant for nu=1 (vandermmonde). Laughlin's wave function and quasihole excitations. Fractional charge of quasiholes. Conclusion and open problems.