Lecture 11: Z_2 gauge theory (continued)

Overview

Subir Sachdev completes the square-lattice Z₂ gauge-theory analysis: the even Gauss-law sector has a trivial large-coupling confined state, while the odd sector leads to dimer coverings and a constraint against a trivial symmetric gapped state. He then develops the square-lattice bosonic-parton description, deriving a CP¹ theory of spinons coupled to a U(1) photon and explaining why monopoles destabilize the apparent two-dimensional spin liquid.

Key takeaways

The local Gauss-law sector determines whether large-coupling confinement can yield a trivial product state: G_i = +1 permits one, while G_i = -1 imposes an odd background charge that obstructs a trivial symmetric gapped state.
In the odd Z₂ gauge theory, the large-coupling low-energy manifold is the set of hard-core dimer coverings, and projected plaquette dynamics generates the Rokhsar–Kivelson quantum dimer model.
Odd background gauge charge makes vison translations anticommute, TxTy = -TyTx, forcing symmetry-breaking, fractionalized, or gapless outcomes in the spirit of the Lieb–Schultz–Mattis constraint.
Square-lattice vison condensation can have emergent XY critical behavior even though microscopic lattice symmetry permits an eighth-order anisotropy that ultimately selects discrete valence-bond-solid orientations.
The square-lattice Schwinger-boson continuum limit yields a CP¹ theory of two complex spinons coupled to a U(1) photon; the gauge-invariant Néel vector is N = z†σz.
A two-dimensional compact U(1) spin liquid is only an intermediate or approximate description when monopoles are relevant, whereas three-dimensional pyrochlore systems can support stable U(1) spin liquids with emergent photons.

Chapters

0:00 Z₂ Gauge-Theory Recap and Course Roadmap

Sachdev frames the lecture as a wrap-up of Z₂ gauge theory before moving to gapless spin liquids.
The link variables are qubits with Pauli operators X and Z; plaquette-Z and link-X terms play roles analogous to magnetic and electric energy.
At small gauge coupling, the model has a deconfined phase with gapped visons; the vison band can have one or two minima depending on the Gauss-law sector.

4:30 Odd Gauss Law as a Background Z₂ Gauge Charge

The local Gauss-law operator G_i commutes with the Hamiltonian and fixes the allowed gauge-charge sector.
G_i = -1 corresponds to a background Z₂ gauge charge at every site, consistent with an odd number of bosonic partons per site.
That odd background charge changes the vison translation properties and distinguishes the odd gauge theory from the G_i = +1 case.

6:54 Large-Coupling Confinement and the Wilson–Wegner Loop

For G_i = +1 and large coupling, the ground state has all link X variables aligned, giving a unique, gapped product state.
A plaquette-Z operator flips four links and creates a simple local, gauge-neutral excitation, described as glueball-like.
The Wilson–Wegner loop, the product of Z operators around a loop, distinguishes phases through area-law behavior in the confined phase and perimeter-law behavior in the deconfined phase.

13:00 Vison Condensation and the Ising Critical Theory

Near the G_i = +1 confinement transition, the vison gap Δ closes and the low-energy mode is represented by a real scalar field φ.
The resulting long-wavelength theory is an Ising φ⁴ field theory, with the mass parameter tuning through the transition.
A single vison is not created by a local gauge-invariant operator, so φ is nonlocal while observables such as φ² are local; this affects boundary conditions and torus sectors.

16:31 Odd Gauss Law Produces Degenerate Dimer Coverings

For G_i = -1 at infinite coupling, the constraint requires an odd number of flipped links around every site.
The allowed lowest-energy configurations correspond to hard-core dimer coverings, with each dimer identified with a spin singlet.
Unlike the even-charge sector, the large-coupling ground-state manifold is extensively degenerate, so perturbation theory must act within that manifold.

19:45 Rokhsar–Kivelson Quantum Dimer Dynamics

Projecting the plaquette-Z term into the dimer manifold produces local resonance moves that flip two parallel dimers around a square.
The resulting effective Hamiltonian is the Rokhsar–Kivelson quantum dimer model, with a tunneling amplitude and potential terms that count flippable configurations.
The effective dimer model retains the constraint of one dimer touching each site and is generally analyzed with lattice-specific numerical methods.

24:30 Odd-Sector Outcomes: Topological Liquid or VBS Order

Depending on the lattice and effective dimer couplings, the large-coupling phase can remain a deconfined Z₂ spin liquid or become a confined valence-bond solid.
On the square lattice, visons have two symmetry-related band minima, represented by fields φ₁ and φ₂.
Condensing the visons breaks lattice symmetry in the odd sector, leading to a dimer crystal rather than a featureless confined state.

28:30 Torus Symmetries and Winding Operators

Putting the gauge theory on an Lx × Ly torus reveals conserved local Gauss-law operators, lattice translations, and noncontractible winding operators Vx and Vy.
Products of local Gauss-law operators generate closed-loop operators, while Vx and Vy wrap around the two independent cycles of the torus.
These global operators expose constraints on the ground-state structure that are hidden in an infinite-plane discussion.

34:20 Translation–Winding Algebra and the LSM Obstruction

In the odd Gauss-law sector, translating a winding operator can produce a sign depending on the system size; for example, Ty and Vx acquire a factor of (-1) raised to a transverse length.
The noncommuting symmetry algebra means states retain information about boundary conditions and cannot behave like a simple local product state.
This is a gauge-theory version of the Lieb–Schultz–Mattis constraint: the system must avoid a trivial symmetric gapped state, for example through symmetry breaking, fractionalization, or gaplessness.

40:30 Projective Visons and the Odd-Sector Critical Modes

Visons moving around one lattice cell pick up a π flux, so their translations obey TxTy = -TyTx.
The two low-energy vison modes transform into one another under lattice symmetries, forming a projective representation rather than ordinary scalar translation rules.
For square-lattice vison condensation, the symmetry action constrains the order parameter and links the confined phase to valence-bond-solid order.

44:30 Emergent XY Symmetry and Eighth-Order Lattice Anisotropy

Combining the two vison fields as s = φ₁ + iφ₂ gives an XY-like continuum theory near a possible critical point.
Vison number is conserved only modulo two, while square-lattice rotations impose additional restrictions on allowed anisotropies.
The leading lattice anisotropy appears at eighth order in s, so the critical behavior can exhibit emergent U(1) symmetry even though the microscopic lattice has discrete rotational symmetry.

56:20 From Z₂ Liquids to Square-Lattice Bosonic Partons

Sachdev turns to the square-lattice Schwinger-boson construction, writing each spin using bosonic partons subject to a local constraint.
A bond mean-field amplitude Q carries a phase that becomes the spatial component of an emergent U(1) gauge field; fluctuations of the constraint supply its temporal component.
The low-energy theory contains both charged spinons and an emergent photon, extending beyond the earlier gapped Z₂ gauge-theory description.

1:00:50 Long-Wavelength Spinons and the Emergent Gauge Field

Sachdev expands the lattice bosons around low-energy momentum minima and introduces slowly varying fields on the two square-lattice sublattices.
The two complex spinon fields are combined into continuum variables, while the bond-phase fluctuations become components of the U(1) gauge field.
Integrating out the canonical momentum converts the phase-space description into a configuration-space field theory with gauge-covariant spinon dynamics.

1:05:30 The CP¹ Gauge Theory and Néel Order

The continuum theory is the CP¹ model: two complex fields z₁ and z₂ couple to a U(1) gauge field, and a common phase rotation is gauge redundancy.
For positive spinon mass parameter r, the spinons are gapped and the mean-field state resembles a U(1) spin liquid with a gapless photon.
When the spinons condense, the gauge-invariant vector N = z†σz acquires an expectation value with Néel symmetry, producing antiferromagnetic order.

1:12:30 Monopoles Destabilize the 2D U(1) Spin Liquid

The initial CP¹ derivation omits monopole events, which are allowed in a compact two-dimensional U(1) gauge theory and ultimately destabilize the apparent spin liquid.
Monopoles can be sufficiently dilute that finite-scale experiments or simulations still observe behavior resembling a photon-bearing spin liquid.
In three-dimensional pyrochlore systems, monopole effects are less destabilizing, allowing genuine U(1) spin-liquid phases with emergent photon signatures.

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.