Lecture 12: Gapless and chiral spin liquids on the square lattice

Overview

Subir Sachdev connects the bosonic CP1 description of square-lattice antiferromagnets to fermionic partons, showing how compact U(1) gauge dynamics and monopoles affect candidate spin liquids. An Affleck–Marston π-flux ansatz gives four massless Dirac spinons and QED3; imaginary next-neighbor hopping gaps them into Chern bands whose induced level-2 Chern–Simons term produces a chiral spin liquid with semionic excitations.

Key takeaways

In compact 2+1-dimensional U(1) gauge theory, Polyakov monopoles proliferate and confine gauge charges; in square-lattice antiferromagnets, monopole Berry phases select valence-bond-solid order.
The square-lattice Affleck–Marston π-flux ansatz produces two Dirac valleys for each of two spin states, giving four massless spinon flavors coupled to U(1) gauge fields.
Imaginary next-neighbor hopping T2 opens a spinon mass gap and breaks time-reversal symmetry by selecting one of the ±π/2 internal-flux states.
When the gapped spinon bands have the appropriate valley mass signs, their occupied bands carry total Chern number 2, inducing a level-2 Chern–Simons term.
In the resulting chiral spin liquid, gauge dressing changes the fermionic partons’ exchange statistics: the spinon excitations are semions with exchange phases ±i.

Chapters

0:00 Bosonic CP1 Theory: Spinons, Gauge Flux, and Néel Order

The bosonic parton construction leads at low energies to a relativistic complex spinon field zα coupled to a U(1) gauge field.
Gauge-invariant combinations of zα describe the Néel vector, while gauge flux tracks collective spin-texture curvature and skyrmion number.
For negative mass-squared, spinon condensation produces Néel order through a Higgs mechanism.

5:30 The Positive-Mass Regime and the Apparent Gapless Photon

For positive spinon mass, the zα fields are gapped, but the noncompact U(1) description retains a massless photon.
That photon would represent a gapless excitation, unlike a fully gapped Z2 spin liquid.
Sachdev contrasts this 2+1-dimensional possibility with three-dimensional quantum spin ice, where emergent photons have experimental support.

10:30 Polyakov Monopoles Confine Compact U(1) Gauge Theory

In 2+1 dimensions, a monopole is a spacetime tunneling event that changes gauge flux and can change skyrmion number.
Polyakov showed that monopoles proliferate in compact pure U(1) gauge theory, eliminating the photon and confining gauge charges.
The mechanism parallels vortex proliferation in the high-temperature phase of the Kosterlitz–Thouless transition.

16:58 Berry Phases Turn Monopole Confinement into VBS Order

For square-lattice antiferromagnets, monopoles acquire Berry phases from the underlying spin system.
Those phases change the confining outcome: rather than a featureless spin gap, the resulting state has valence-bond-solid order.
Sachdev uses this bosonic route as motivation for switching to fermionic spinons.

21:00 Fermionic Partons and the One-Fermion-per-Site Constraint

For spin-1/2, the fermionic representation uses spin-up and spin-down partons to reproduce the local spin algebra.
The physical Hilbert space is enforced by the constraint f†ᵢαfᵢα = 1 on every lattice site.
The constraint introduces a gauge redundancy, while a particle-hole mean-field decoupling supplies candidate spin-liquid states.

25:50 Affleck–Marston π-Flux Mean-Field Ansatz

The square-lattice ansatz assigns real nearest-neighbor hopping and a staggered sign pattern whose product around each square gives π flux.
For a J1–J2 antiferromagnet, Sachdev also considers imaginary next-neighbor hopping, denoted T2.
The T2 = 0 state preserves time reversal; nonzero imaginary T2 will generate the chiral state.

32:30 Parton Gauge Transformations and the Two-Sublattice Bloch Hamiltonian

Under the local transformation fᵢ → eⁱθᵢfᵢ, a bond field transforms with the phase difference θᵢ − θⱼ.
Writing the bond field as its mean value times eⁱAᵢⱼ identifies the emergent U(1) gauge field.
The π-flux pattern requires an A/B sublattice Bloch Hamiltonian with two sites per unit cell.

38:00 π-Flux Bands Produce Dirac Nodes at Half Filling

With T2 = 0, the two-band spectrum has Dirac nodes where the relevant cosine and sine terms vanish.
The single-occupancy constraint fixes the parton chemical potential at zero, placing the Fermi level at the nodes.
The low-energy excitations carry spin but not electric charge, unlike graphene electrons with a similar Dirac dispersion.

41:20 Four Flavors of Massless Spinons Form QED3

The two Dirac valleys multiplied by physical spin up and down give four two-component massless Dirac fermions.
At low energy, these spinons couple to a dynamical U(1) gauge field, yielding a QED3 description.
The continuum theory has an enlarged SU(4) flavor symmetry that is not an exact symmetry of the microscopic spin model.

48:55 Imaginary T2 Hopping Opens a Time-Reversal-Breaking Mass

Nonzero imaginary next-neighbor hopping adds a mass term at the Dirac points and gaps the spinons.
The two choices of internal flux, +π/2 and −π/2 per oriented triangle, are related by time reversal.
Selecting one flux pattern breaks time-reversal and reflection symmetry and distinguishes the chiral state from the gapless π-flux state.

53:20 Gapped Spinon Bands Carry a Nonzero Chern Number

The T2-induced masses give the occupied spinon band a nonzero Chern number when the two valleys have the appropriate relative mass signs.
Reversing the flux reverses the chirality and the sign of the Chern response.
The Chern number is the Brillouin-zone integral of Berry curvature and characterizes the topology of the occupied band.

59:25 Integrating Out Spinons Generates a Chern–Simons Term

Integrating out the massive Dirac spinons produces both a Maxwell term and a parity-odd Chern–Simons term for the emergent gauge field.
The Chern–Simons contribution is allowed because the spinon mass breaks time reversal.
Its sign follows the common mass sign of the four Dirac flavors and flips when the chiral flux choice is reversed.

1:04:20 Band Topology Quantizes the Emergent Hall Response

The Chern–Simons coefficient is tied to the Hall response of the corresponding ungauged spinon bands.
Each occupied spin sector contributes a Chern number of one, giving a total dimensionless Hall coefficient σxy = 2.
The resulting quantized coefficient fixes the level-2 Chern–Simons term rather than leaving it as an arbitrary coupling.

1:08:45 A Chiral Spin Liquid Is a Gauged Chern Insulator

The state combines gapped spinons in occupied Chern bands with a fluctuating, dynamical U(1) gauge field.
Unlike an ordinary electronic Chern insulator, gauge fluctuations are important because the emergent gauge coupling is not weak.
The level-2 Chern–Simons term changes the gauge-field dynamics and prevents the simple Polyakov-confinement conclusion from applying unchanged.

1:11:25 Level-2 Topological Order and Semionic Spinons

The level-2 Chern–Simons theory gives the chiral spin liquid its characteristic topological order.
Although the partons began as fermions, the gauge dynamics make their physical excitations semions, acquiring a phase of +i or −i on exchange.
Sachdev relates this state to the Kalmeyer–Laughlin chiral spin liquid and to bosonic fractional quantum Hall intuition.

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.