University of Illinois at Urbana-Champaign · Department of Physics

Physics 598

Topological Phases in Quantum Condensed Matter

Academic Year 2022/2023

Fall Semester 2022

Instructor: Professor Eduardo Fradkin

Department of Physics
University of Illinois at Urbana-Champaign
Room 2119 ESB
1110 West Green Street, Urbana, IL 61801-3080
Phone: 217-333-4409
Fax: 217-333-9819

Time: 2:00 pm - 3:20 pm, Tuesdays and Thursdays
Place: Rm 276 Loomis Laboratory
Call Number: 60312
Credit: 1 unit.
Office Hours: Wednesdays 2:30 pm to 3:30 pm, Rm 2119 ESB

TA: Marcus Rosales
Office: TBA
Phone: TBA
Office Hours: TBA
e-mail: TBA

This course will cover current developments on topological phases in condensed matter physics, including: the theory of the fractional quantum Hall states, spin liquids, topological insulators and superconductors, effective field theories of topological phases, quasiparticles, fractional statistics (Abelian and non-Abelian), experimental detection of quasiparticles, quantum interferometers and the manipulation of quasiparticles, topological phases and topological quantum computing, quantum entanglement at quantum criticality and in topological phases. The course material will consist primarily of a review of recent literature in leading journals. I will also rely heavily on chapters of the second edition of my book "Field Theories of Condensed Matter Systems" (Cambridge University Press, 2013).

Concerning the required background I will assume that the students have taken at least the equivalent of our second graduate course in Quantum Mechanics (Physics 581: Advanced Quantum Mechanics) and at least the first course in Quantum Field Theory (Physics 582). Although no real expertise field theory will be assumed, students that have taken the Advanced Quantum Field Theory class (Physics 583) and/or the Advanced Condensed Matter Physics class (Physics 561) will profit much more from this class.

There will not be homeworks in this class but all students will be required to write a term paper and make an oral (powerpoint) presentation. To encourage participation, from time to time I may ask students to go over some material in front of the class on some previously agreed material.


Updated on Thursday December 8, 2022

Course Plan

Ordered Phases in Condensed Matter

The concept of order in condensed matter. Characterization of a phase of matter in terms of order parameters and its correlations. Effective field theory description. Review of classical and quantum phase transitions to an ordered state. Topology and order: topological excitations. Examples: superconductors and superfluids; antiferromagnets.

Characterization of Topological Phases

What is a topological phase. Absence of an order parameter description. Universal properties of topological phases and topological invariants. Chern numbers. Ground state degeneracies and the quantum numbers of the spectrum of excitations. Examples: quantum Hall phases, quantum dimers, topological superconductors and topological insulators.

Frustration, Quantum Disorder, and Gauge Theory

Frustration in classical and quantum spin systems. Quantum disordered states and quantum spin liquids. Dimer models and their phases. Topological phases in dimer models. Gauge theory description and deconfinement. Kitaev' Toric Code. Differences and similarities between one and two dimensional systems. Kitaev materials.

The Integer Quantum Hall effect

Electrons in two dimensions in large magnetic fields. Single electron picture: Landau quantization and degeneracy of Landau levels. Lattice effects. Quantization of the Hall conductance and plateaus. Chern numbers. Role of disorder and localization. Topological quantization. The plateau transition.

Topological Insulators

Topological classification of band structures. Band crossings and Dirac fermions. Berry's phases and Chern numbers. Spin orbit. The quantized anomalous Hall effect and the spin quantum Hall effect. Weak and strong topological insulators. 3D topological insulators, the Callan-Harvey effect and the parity anomaly in condensed matter. The role of time reversal breaking. Relation with topological superconductors. Examples: graphene (single, bilayer and others), HgTe heterostructures, three-dimensional \( \mathbb{Z}_2\) topological insulators, topological crystalline insulators, and Weyl semimetals. Classification schemes.

The Fractional Quantum Hall Effect

Laughlin, Haldane/Halperin and Jain wavefunctions. Plasma analogy, collective modes and vortices. Fractional charge and statistics. Hierarchical states and the classification of Abelian Fractional quantum Hall states. Non-Abelian fractional quantum Hall phases: Moore-Read and Read-Rezayi states. Classification of non-Abelian states. Pairing and clustering. Topological Superconductors. The concept of non-Abelian fractional statistics. Topological protection and Topological Quantum Computing.

Gauge Theory and the FQHE

The Landau-Ginzburg picture and Chern-Simons gauge theory. Hydrodynamic picture of the FQH fluid. Composite bosons and composite fermions. Collective modes and Raman Scattering experiments. Experimental evidence for composite fermions: focusing experiments, gaps and Shubnikov-de haas experiments. Bilayer systems and partially polarized states. An integer as a fraction: Skyrmions in the integer FQH state and NMR experiments. What happens when the fractional quantum Hall state fails: compressible states, stripe and nematic phases.


Duality in classical models of statistical mechanics: Ising, \(\mathbb{Z}_N \) , and XY models in different dimensions. Abelian gauge theories and duality. Bosonization and duality. Particle-vortex duality and its generalizations. Duality in quantum theories in 2+1 dimensions. Applications to the FQH states.

Edge States

Luttinger liquid of chiral fermions and bosons. Universal effective theories of edge states and relation with conformal field theory. Tunneling signatures and experimental evidence for Luttinger liquids. Noise measurements at point contacts and the measurement of fractional charge. Quantum interferometers and the detection and measurement of fractional statistics, Abelian and non-Abelian.

Generalizations: SPTs, SETs and all that

Generalizations of the concept of topological phases: Symmetry Protected and Symmetry Enriched Topological Phases. Role of anomalies ('t Hooft, gauge, and mixed). Topological superconductors and Majorana zero-modes. Parafermions.

Quantum Entanglement in Condensed Matter

Scaling of quantum entanglement in condensed matter as a characterization of its phases. Quantum entanglement in a generic phase with finite correlation length: the "area" law. Scaling of quantum entanglement in 1D critical systems. Quantum entanglement and dynamics: quenches. Scaling of the quantum entanglement entropy in topological phases and at quantum phase transitions. Quantum entanglement and the AdS/CFT correspondence.

Scanned Lecture Notes

Lecture 1 : The Concept of order in Condensed Matter Physics. Symmetries and Phases of Matter. Global symmetries and spontaneous symmetry breaking. Vortices and Topological Excitations.

Lecture 2 :Quantum Phase Transitions. The Ising model in a transverse field and its global \(\mathbb{Z}_2 \) symmetry. Josephson junction array and \( U(1) \) global symmetry.

Lecture 3 : The Classical-Quantum Connection. Classical Statistical Mechanics and Quantum Field Theory in imaginary time. Quantum Systems at finite temperature. Path integral on a cylinder.

Lecture 4 : Quantum Antiferromagnets. The non-linear sigma model. One dimension and the role of the topological invariant. Integer and half-integer spin chains. Haldane's gap.

Lecture 5 : Theories without a local order parameter. Brief discussion of the quantization of Maxwell's electrodynamics. \(\mathbb{Z}_2 \) lattice gauge theory. Wilson-Wegner's picture and Hamiltonian Picture. Elitzur's Theorem and Observables. Wilson loops: area law and confinement, perimeter law and deconfinement. The deconfined phase of the \(\mathbb{Z}_2 \) gauge theory as a Topological Phase. Topology and ground state degeneracy.

Lecture 6 : Ising dualities: Kramers-Wannier-Wegner picture and Hamiltonian picture. Sums over paths and domain wall configurations. Self-duality in 2D. \(\mathbb{Z}_2 \) lattice gauge theory as the dual of the 3D Ising model. Matter fields and extended self-duality. Hamiltonian duality. Disorder operators. Gauge invariance and duality. Self-duality of the 3+1 dimensional \(\mathbb{Z}_2 \) gauge theory.

Lecture 7 : Ising models and Majorana fermions. Jordan-Wigner transformation. Majorana and Dirac fermions. Fermion representation of the 1+1 dimensional Ising model in a transverse field. Boundary conditions and Hilbert space. Fermion parity. Relation to a the BCS theory of a p wave superconductor in one space dimension. Spectrum and quantum phase transition. Topology and Majorana fermions. Open chains and Majorana zero modes. Fermionization of the \(\mathbb{Z}_2 \) lattice gauge theory in 2+1 dimensions. Spectrum of the topological phase.

Lecture 8 : Frustration, Quantum Disorder and Gauge Theory. Valence bond states: resonating valence bond states (long and short). Spinons, holons, and valence bond states. Slave fermions and Schwinger bosons. Constraints and gauge symmetry. SU(2) and SU(N) quantum antiferromagnets. Large-N limits. Emergent gauge fields

Lecture 9 : Quantum antiferromagnets with SU(N) symmetry in the large N limit. Time-reversal invariant states. BZA state and spinon Fermi surface. Flux phases, Dirac fermions, and the Dirac spin liquid. Dimer states.

Lecture 10 : Quantum Dimer Models. Relation with frustrated Ising antiferromagnets in a transverse field. Extensive classical degeneracy and effective hamiltonian in the degenerate subspace. Flippable plaquettes. Space of dimer states and the short-range RVB state. Rokhsar-Kivelson point. Bipartite and non-bipartite lattices. Dimer configurations and loop configurations. Classical correlations. Resonon, and dynamic critical exponent. Ordered, critical and liquid (disordered) phases. Classical dimers at criticality. Theory of the compactified scalar (phase field). Compactification condition. Classical ground states and columnar states. Classical criticality. Electric and magnetic vertex operators, dimer density operators, columnar and orientational (nematic) operators

Lecture 11 : Critical behavior of classical dimers. Electric and magnetic vertex operators, dimer density operators, columnar, orientational (nematic), and hole ("monomer') operators. Correlation functions and scaling dimensions. Quantum field theory of the Quantum Dimer Model: the Quantum Lifshitz model (QLM). Phases of the QLM. Quantum critical point. Hamiltonian and path integral picture. Relation to Lifshitz transitions in nematic liquid crystals. The role of compactness (periodicity). Gauge theory dual. Ground state wave functional for the QLM (and the dual gauge theory). Relation with classical dimer model partition functions and observables. Vertex and dual (vortex) operators. Equal-time and unequal time correlators. Scaling analysis of the QLM. Relevant and irrelevant operators and quantum phase transitions.

Lecture 12 : Ground state wave functional for the QLM (and the dual gauge theory). Relation with classical dimer model partition functions and observables. Vertex and dual (vortex) operators. Equal-time and unequal time correlators. Scaling analysis of the QLM. Relevant and irrelevant operators and quantum phase transitions. Beginning of the treatment of Topology and the Quantum Hall Effect: the classical Hall effect, Landau levels in 2D, discovery of th integer quantum Hall effect.

Lecture 13 : Quantum states of a charged particle in a magnetic field in 2D. Landau levels. Disk, rectangular and toroidal geometries. Magnetic translations, magnetic algebra and the spectrum. The case of the torus and large gauge transformations. Landau levels on a lattice.

Lecture 14 : Landau levels on the square lattice. Lattice magnetic translations. Harper's equation. Symmetries. The case of fluxes with even denominators and the zero energy modes. Linear response theory and the computation of the conductivities from correlators. Gauge invariance, current conservation and Ward identities.

Lecture 15 : The polarization tensor, current conservation and transversality condition. Relation with the current-current correlator. Equal-time commutators, the Schwinger term and non-conservation of the current correlator. The polarization tensor for non-interacting fermions in 2D. Effective electromagnetic action for a gapped 2D system in an uniform magnetic field and the Hall conductivity. The Kubo formula for the Hall conductivity \( \sigma_{xy}\; \) in the Born-Oppenheimer approximation for a gapped system. Generalized toroidal BC's and the computation of \( \sigma_{xy}\; \) on a torus.

Lecture 16 : The Niu-Thouless-Wu formula for the Hall conductivity. The torus boundary conditions. Global obstructions to the definition of the phase of the wave function on the torus of boundary conditions. Topological invariance, Chern numbers and fiber bundles. Wave function of free fermions in a filled Landau level on a disk.

Lecture 17 : Wave function of free fermions in a filled Landau level on a torus and the computation of the Hall conductivity. Wave functions for the quantum Hall effect on a square lattice with commensurate flux. The Hall conductivity for filled Hofstadter bands and their Chern numbers.

Lecture 18 : Computation of the Hall conductivity for filled Hofstadter bands and their Chern numbers. Topological band structures and Topological Insulators. Chern insulators and their Berry connection. The anomalous quantum Hall effect. Two simple models: flux phases and Haldane's honeycomb model. Dirac fermions and the anomalous quantum Hall effect.

Lecture 19 : Dirac fermions and the anomalous quantum Hall effect. Graphene and Haldane's model. Effective low energy Dirac theory, masses, etc. Parity and time-reversal symmetries. Dirac theory in a background weak electromagnetic field. Field theoretic perturbative calculation of the effective action of the background field. Polarization tensor and gauge invariance. Parity even polarization and parity-odd amplitudes \( \Pi_0(p^2 \) and \(\Pi+A(p^2)\). IR behavior. Effective Maxwell coupling and Hall conductivity \( \sigma_{xy}\). The Chern-Simons term and the Hall conductivity. Effective masses of the Dirac fermions. Trivial insulator and the anomalous quantum Hall insulator. Fermion doubling and the quantization of the Hall conductance. The parity anomaly.

Lecture 20 : The Hall conductivity of the anomalous quantum Hall effect as a topological invariant. Homotopy group and the topological charge of the occupied states and the Pontryagin index. The Dirac approximation and meron charge. relation with the Nielsen-Ninomiya Theorem. The Quantum Spin hall effect. When it can be observed. Time-reversal invariance. Role oi spin-orbit coupling. The Kane-Mele model. The quantum spin hall effect in CdTe|HgTe|CdTe heterostructures. The Bernevig-Hughes-Zhang Model. Molenkamp's experiment and the observation of the quantum Spn Hall effect.

Lecture 21 : \(\mathbb{Z}_2 \; \) Topological Invariant for time-reversal invariant insulators; 2D and 3D \(\mathbb{Z}_2\; \) invariant insulators. The 3D case: Wilson fermion model. Edge states of topological insulators. The integer quantum Hall case. Kac-Moody algebra. The edge as a fluctuating chiral string. Polyacetylene: period 2 Peierls distortion and spontaneous breaking of translation symmetry. Effective low energy theory. Adiabatic and Gross-Neveu limits.

Lecture 22 : Soliton in 1D. Zero mode, spectral asymmetry and fractional charge. The Goldstone-Wilczek argument, adiabatic change and induced charge. Edge states in the quantum anomalous quantum Hall state: zero modes and their dispersion, chiral fermion and gauge anomaly. The quantum spin Hall case and its helical edge states. Three-dimensional( \mathbb{Z}_2\; \) topological insulators and their edge states. Construction of the edge states and Weyl fermions on the wall. Half-quantization of the surface Hall conductance and anomaly inflow. Bulk effective action and the boundary action. The Callan-Harvey effect.

Lecture 23 : The fractional quantum Hall effect. Materials and phenomenology. Fractional quantum hall plateaus, incompressibility and dissipationless transport. Laughlin's theory of the FQH state as an incompressible quantum fluid. The Laughlin wave function. Angular momentum eigenvalue. The plasma analogy. Laughlin's theory of the quasihole as the result of the adiabatic insertion of a flux quantum. Fractional charge and the Hall conductivity.

Lecture 24 : The Halperin wave function for two quasiholes as an argument for fractional statistics. The Arovas-Schrieffer-Wilczek Berry phase theory of fractional statistics. Heuristic hydrodynamic theory of the fractional quantum Hall effect: effective action and the Chern-Simons term. Quantization of the \( U(1)_k \) ChernSimons gauge theory. Canonical quantization and gauge invariance. Gauge anomaly and edge states.

Lecture 25 : Vacuum degeneracy of the U(1) Chern-Simons theory on a 2-torus and on surfaces of genus \( g \). Link invariants and fractional abelian statistics. Braids and half-braids. Chern-Simons duality and statistical transmutation. Comments on the BF topological theory and its relation with \( \mathbb{Z}_k \; \) toric code.

Lecture 26 : Field theories of the fractional quantum Hell effect. Flux attachment, Chern-Simons and statistical transmutation. Landau-Ginzburg theory and the composite boson picture. Average field approximation and superfluid picture. Laughlin fractions. Quantum fluctuations and effective low energy theory. Computation of the Hall conductivity. Vortices as Laughlin quasiholes. Fractional charge. Relation wit the hydrodynamic theory. Fractional statistics. Vacuum degeneracy and the number of distinct anyon types. Fusion rules.

Lecture 27 : Composite fermion picture. Partial screening of the magnetic field and effective integer quantum Hall state. Jian fractions. Compressible states. Effects of quantum fluctuations. Computation of the effective action of the chern-Simons gauge fields. Composite fermion polarization tensor. The full polarization tensor and the effective action of a background electromagnetic gauge field. Sum rules, gauge invariance and Galilean invariance. Low energy effective action and the hall conductivity. Fractional charge and fractional statistics.

Lecture 28 : Quantum Hall wavefunctions and Conformal Field Theory. Euclidean compactified boson. Relation between the compactification radius and the filling fraction. Quasiholes and vertex operators. Multi-component 2DEGs and Halperin wavefunctions, and CFT correlators. Spin-singlet states, spin-charge separation: FQH state of semions and SU(1)$_1$ correlators. U(1)$_2$ correlators. Moore-read non abelian states. Pfaffians. Ising CFT, majorana correlators and allowed operators. Fusion rules and nonabelian statistics. TQFT hydrodynamic theory and geometrical response. Shift, topological spin and the Wen-Zee term. Effective hydrodynamic theory of the multi-component states and the K-matrix.

Lecture 29 : Edge states of quantum Hall fluids. Wen's hydrodynamic theory of edge states. Relation between the total charge at the edge and boundary conditions. Classical picture of the edge states and quantization. Chiral U(1) Kac-Moody algebra. Compactified chiral boson and chiral Luttinger liquid. Lagrangian and Hamiltonian. Euclidean compactified boson. Relation between the compactification radius and the filling fraction. Vertex operator representation of the electron. Quasiparticle/quasihole vertex operators. Spectrum of allowed quasiholes and extended symmetry algebra. Interactions at the edge. Electron and quasihole propagators. Scaling dimensions. Edge states of l Jain states 2/5 and 2/3. Co-propagating and counter-propagating edge states. Tunneling Hamiltonians. Constrictions and point contacts. Computation of the tunneling current at a point contact. Scaling of the differential conductance at a constriction and at a point contact from a Fermi liquid.

Term Paper

There will be a Term Paper that will play the role of the final exam. A list of topics for the Term Paper can be found here. The paper must be at least 10 pages long (references not included), it must be a single column, double space, 12 pt font document formatted using LaTEX 2e. The Term paper will be due at 9:00 pm on Wednesday December 14, 2022 (the day before the Final Exam). You must sent me the pdf files of your paper and of your slide presentation by email . The Final exam for Physics 598, will be on Thursday December 15, 2022. in room 3110 ESB at 9:00 am . Each student will give a 15-20 minute long PowerPoint style presentation. If needed we will use a zoom format.

Recommended Bibliography

Eduardo Fradkin, "Field Theories of Condensed Matter Physics, Second Edition" (Cambridge University Press, 2013).

B. Andrei Bernevig and Taylor L. Hughes, "Topological Insulators and Superconductors" (Princeton University Press, 2013).

Roderich Moessner and Joel E. Moore, "Topological Phases of Matter" (Cambridge University Press, 2021).

Last updated 12/8/2022