Lecture 14: Fractional quantum Hall and square lattice spin liquids

Overview

Subir Sachdev connects fractional quantum Hall states to parton mean-field theories, showing how integer-filled parton Landau levels produce the Laughlin 1/3 state, its exchange angle of π/3, and the Jain sequence. He then develops the square-lattice spin-liquid description from bosonic spinons and valence-bond order to fermionic π-flux spinons coupled to an emergent SU(2) gauge field, whose 2+1-dimensional theory resembles a reduced version of QCD and helps organize Néel and valence-bond-solid phases.

Key takeaways

For three fermionic partons with integer Landau-level fillings n₁, n₂, and n₃, the filling obeys 1/ν = 1/n₁ + 1/n₂ + 1/n₃; n₁ = n₂ = n₃ = 1 gives the Laughlin ν = 1/3 state.
Projecting three filled-parton Vandermonde wavefunctions onto coincident electron coordinates produces the Laughlin factor ∏ᵢ<ⱼ(zᵢ − zⱼ)³, while the induced Chern-Simons theory gives quasiparticle exchange angle π/3.
The choice n₁ = n₂ = 1 and n₃ = n generates the Jain sequence ν = n/(2n + 1), including the experimentally prominent 2/5 and 3/7 fractions.
In a square-lattice valence-bond solid, separating spin defects creates a string of disrupted singlet bonds whose energy grows with separation, providing an intuitive picture of spinon confinement.
The fermionic π-flux state has massless Dirac spinons, and its particle-hole redundancy enlarges the apparent U(1) gauge symmetry to local SU(2).
The proposed square-lattice continuum theory has two Dirac flavors coupled to SU(2) in 2+1 dimensions, and its symmetry-constrained confined phases connect the fermionic description to Néel and valence-bond-solid order.

Chapters

0:00 Course Setup and the Correct Chern-Simons Anyon Phase

Sachdev opens with presentation scheduling for December 1, 2, and 3, then corrects a 2π error from the prior discussion.
For a Chern-Simons term with level k, a particle sources gauge flux 2π/k.
The resulting exchange angle is π/k, so k = 2 gives semionic statistics with phase ±i.

10:12 Fractional Quantum Hall Filling and the Three-Parton Electron

The filling fraction ν is the electron count divided by the number of magnetic flux quanta through the sample.
Sachdev represents a spin-polarized electron as c = f₁f₂f₃, a product of three fermionic partons.
The physical electron states correspond to all three partons being empty or all three being occupied.

14:14 Two Internal Gauge Fields Enforce the Parton Constraints

Equal parton densities impose the constraints that identify physical electron states.
Two independent local phase redundancies follow from the three-parton representation, requiring two internal gauge fields, B₁ and B₂.
The external electromagnetic vector potential couples to the partons alongside these emergent fields.

17:38 Integer Parton Landau Levels Generate Allowed Hall Fractions

The three partons experience effective magnetic fields B₁, B₂, and B₃ whose sum equals the physical field B.
A useful mean-field state occurs when each parton fills an integer number nᵢ of Landau levels, producing a parton excitation gap.
The density and field constraints give 1/ν = 1/n₁ + 1/n₂ + 1/n₃; integer choices yield odd-denominator fractions.

21:16 Deriving the Laughlin 1/3 Wavefunction and Statistics

Taking n₁ = n₂ = n₃ = 1 gives ν = 1/3, with one effective flux quantum per parton.
Each filled parton Landau level contributes a Vandermonde factor; imposing coincident parton coordinates yields the electron factor ∏ᵢ<ⱼ(zᵢ − zⱼ)³.
Integrating out the gapped partons generates Chern-Simons terms and an exchange angle of π/3 for Laughlin quasiparticles.

27:51 Jain Fractions, Projected Wavefunctions, and Stability

Choosing n₁ = n₂ = 1 and n₃ = n gives ν = n/(2n + 1), including 1/3, 2/5, and 3/7.
Allowing negative integer parton fillings also produces the reverse Jain sequence, including n/(2n − 1).
Jain wavefunctions combine filled-Landau-level factors and are often projected into the electron lowest Landau level.
Sachdev notes that the Laughlin 1/3 state is stabilized by a pseudopotential Hamiltonian that excludes relative angular momentum one.

34:23 Bosonic Spinons and Néel Order on the Square Lattice

The bosonic parton theory represents the square-lattice antiferromagnet using spinons coupled to a U(1) gauge field.
A nonzero spinon condensate produces a nonzero Néel order parameter, the staggered magnetization.
When the bosonic spinons remain gapped, gauge fluctuations and monopole effects instead lead toward confinement and valence-bond order.

38:17 Why Confinement Produces Valence-Bond-Solid Order

In a valence-bond solid, spins form spatially ordered singlet bonds, so separating spin defects disrupts the bond pattern.
The disrupted region acts like a string with energy proportional to its length, confining the two spinons.
Sachdev compares this linear confinement to the energy cost of separating quarks in quantum chromodynamics.

41:50 Fermionic Spinons in the Square-Lattice π-Flux State

The fermionic parton construction includes a π flux through each square plaquette when the time-reversal-breaking second-neighbor hopping is set to zero.
The π-flux mean-field band has massless Dirac points at half filling.
A U(1) gauge theory of these Dirac spinons is complicated by strong gauge fluctuations and monopole effects.

45:38 Particle-Hole Redundancy Reveals an Emergent SU(2) Gauge Symmetry

At one fermion per site, the parton representation can be viewed using either particles relative to the empty state or holes relative to the filled state.
Sachdev packages spin-up and spin-down fermions and their particle-hole partners into a matrix with a reality constraint.
The redundant representation exposes a local SU(2) gauge freedom that is larger than the initially apparent U(1) phase symmetry.

54:38 Separating Global Spin Rotations from Local Gauge Rotations

Physical spin rotations act on the spinor index, while the emergent SU(2) gauge rotations act on the particle-hole or number index.
The two transformations commute because they act on different sides of the parton matrix.
Spin rotation is a global symmetry shared across lattice sites; the gauge transformation can vary independently from site to site.

59:22 The π-Flux Saddle Point and Lattice SU(2) Gauge Fields

A saddle point with pure-imaginary hopping and π flux preserves the full SU(2) gauge structure.
The hopping links are promoted to SU(2) matrices whose site-dependent transformations cancel those of the partons.
This lattice gauge construction provides a symmetry-consistent framework for studying fluctuations around the π-flux state.

1:07:00 Two Dirac Flavors and the Connection to QCD

The continuum square-lattice theory contains Nf = 2 massless Dirac fermions coupled to an SU(2) gauge field in 2+1 dimensions.
Sachdev compares this structure with QCD, which has three light quark flavors coupled to an SU(3) gauge field in 3+1 dimensions.
The comparison motivates questions about whether the 2+1-dimensional SU(2) theory confines and what phases its confined states represent.

1:11:06 Confinement, Néel and Valence-Bond Phases, and Superconductivity

Anomaly and symmetry arguments associated with work by Wang, Senthil, and collaborators constrain the confined phases of the SU(2) theory.
The resulting phase structure matches the Néel and valence-bond-solid states also obtained from the bosonic spinon approach.
Sachdev presents the agreement between bosonic and fermionic descriptions as a useful unification and previews applying the SU(2) gauge structure to high-temperature superconductivity.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Subir Sachdev.