Lecture 10: Z_2 gauge theory

Overview

Subir Sachdev derives the low-energy Z₂ gauge theory for two-dimensional quantum spin liquids, with link variables whose plaquette products encode vison flux and link-flip operators give visons dynamics. He relates the gauge theory to the transverse-field Ising model by duality, then shows how the microscopic spin parity fixes the gauge-charge sector: half-integer spins give visons a π Berry phase, projective translations, and a twofold-degenerate dispersion. The large-coupling limit connects this constraint to dimer coverings and the Lieb-Schultz-Mattis obstruction to a featureless symmetric state.

Key takeaways

The Z₂ gauge Hamiltonian has plaquette products of link Z operators for flux energy and link X operators for vison dynamics; its duality to a transverse-field Ising model establishes the stability of the weak-coupling deconfined phase.
The physical gauge-charge constraint is fixed by microscopic spin: Gᵢ = (−1)²ˢ, giving +1 for integer spins and −1 for half-integer spins.
A vison encircling one spin acquires Berry phase (−1)²ˢ; half-integer spin therefore produces π flux for vison motion on the dual lattice.
In the half-integer-spin case, vison translations satisfy TₓTᵧ = −TᵧTₓ, forcing at least twofold degeneracy and producing two symmetry-related band minima.
At large gauge coupling, integer-spin systems can admit a simple gauge-invariant product state, whereas the half-integer constraint produces dimer coverings and obstructs a featureless symmetric state.
On a torus, noncontractible X and Z loop operators label global sectors and supply the logical operators underlying toric-code quantum information.

Chapters

0:00 Partons, Spinons, and Vison Saddle Points

Sachdev reviews representing spins with bosonic partons subject to the local constraint that the parton number equals 2S.
Mean-field theory yields gapped spinon excitations and, for suitable frustrated lattices, a Z₂ spin-liquid saddle point without a gapless photon.
A vison saddle point can be represented by a branch cut across which the sign of the link variable Q changes; moving the cut is a gauge choice.

6:00 The Wegner Z₂ Gauge-Theory Hamiltonian

Each lattice link carries Pauli operators X and Z, and gauge-invariant flux is measured by the product of Z variables around a plaquette.
The plaquette-flux term favors zero flux, while the link X term flips Z and allows visons to move.
This is Wegner’s Z₂ lattice gauge theory, used here to test whether the spin-liquid saddle point remains stable beyond mean-field theory.

9:33 Duality Between Z₂ Gauge Theory and the Ising Model

Sachdev places dual Ising variables on the dual lattice and maps each gauge-theory plaquette product of Z operators to a dual-site X operator.
An Ising bond operator maps to the gauge-theory link Z crossed by that bond, exchanging dual sites and plaquettes under a 90-degree rotation.
Matching the operators’ squares and commutation relations establishes the duality, with boundary and global-sector subtleties on a torus.

13:30 Operator Mapping and Gauge–Ising Correlations

The local operators have simple duals: gauge plaquette flux maps to dual X, while dual Ising bonds map to gauge-link Z.
Their matching commutation and anticommutation relations preserve the operator algebra and therefore the spectra and corresponding correlation functions.
A single Z operator maps nonlocally, involving a string; the simple local mapping is sufficient to establish the duality.

20:30 Confinement and Deconfinement from Ising Duality

Under the duality, weak gauge coupling corresponds to the disordered, strong-coupling Ising regime, where the Z₂ spin liquid is stable.
The small-coupling gauge phase is deconfined: spinons remain distinct excitations and visons are gapped.
At the other end, vison proliferation corresponds to the dual Ising-ordered phase and confines spinons, destroying the fractionalized regime.

24:30 Why the Small-Coupling Gauge Phase Is Stable

The Ising duality supplies a stability argument for the small-gauge-coupling phase rather than relying only on the mean-field saddle point.
Sachdev notes that related duality constructions extend to discrete Z_N gauge groups, although the mapping is less straightforward for non-Abelian groups.
The rest of the analysis focuses on vison and spin-liquid properties in the deconfined regime.

27:39 Gauge-Equivalent Ground States at Zero Coupling

At zero gauge coupling, setting every link Z to +1 minimizes the plaquette terms and gives a simple ground-state configuration.
Flipping all links incident on one site leaves every plaquette flux unchanged, so it produces a gauge-equivalent configuration rather than a distinct physical state.
Only gauge-invariant measurements distinguish physical states; the apparent ground-state multiplicity must be treated through gauge symmetry.

30:00 Local Gauge Charges and Gauss’s Law

Sachdev defines a local gauge charge Gᵢ as the product of X operators on all links incident on site i.
Each Gᵢ commutes with the Hamiltonian: its X factors commute with the link-flip terms and produce an even number of sign changes through each plaquette term.
The charges obey Gᵢ² = 1, so their eigenvalues are ±1; choosing their values specifies the physical gauge-theory sector, analogous to imposing Gauss’s law.

36:00 Spin Parity Fixes the Gauge-Charge Sector

The microscopic spin model selects a definite gauge-charge sector: Gᵢ = (−1)²ˢ at every site.
Half-integer spins require Gᵢ = −1, while integer spins require Gᵢ = +1; a site with a different spin acts like a gauge-charge defect.
This dependence of the low-energy gauge theory on microscopic spin parity is tied to translation symmetry and is understood through Lieb-Schultz-Mattis-type anomaly constraints.

42:30 Vison Berry Phase from Winding Around a Spin

Applying Gᵢ moves a vison around a closed path surrounding site i, so the phase acquired by that motion reveals the local gauge-charge value.
Returning the parton configuration to its original gauge requires a transformation whose sign depends on the number of bosonic partons at the enclosed site.
The resulting vison Berry phase is +1 for integer spin and −1 for half-integer spin, equivalently (−1)²ˢ.

51:00 Half-Integer Spins Put Visons in π Flux

At small gauge coupling, a vison hops between dual-lattice sites with amplitudes set by the link-flip term.
For half-integer spin, the Berry phase of −1 around each original site means the vison moves in a background of π flux per dual plaquette.
For integer spin there is no such required π flux, so the half-integer case has distinct band and symmetry properties.

55:03 Two-Sublattice Tight-Binding Model for a π-Flux Vison

A gauge choice represents the π flux with alternating hopping signs and doubles the vison hopping unit cell, without physically breaking translation symmetry.
The resulting tight-binding problem has two sublattices, A and B, and a reduced Brillouin zone.
The lowest vison band has two distinct minima, at momenta (0, 0) and (0, π) in the reduced-zone convention.

1:04:00 Projective Translations Enforce Vison Degeneracy

For a vison moving in π flux, translations in the x and y directions obey TₓTᵧ = −TᵧTₓ, even though each translation separately preserves the Hamiltonian.
This projective translation algebra has no one-dimensional representation, so vison states must occur in at least twofold multiplets.
The two dispersion minima are therefore symmetry-related; Sachdev identifies this as a physical consequence for half-integer-spin Z₂ spin liquids, unlike the integer-spin case.

1:08:58 Torus Loops and the Toric-Code Connection

On a torus, additional conserved operators are noncontractible loops of X or Z that cannot be written as products of local Gᵢ charges.
These topological loop operators distinguish global sectors and connect the gauge theory to the toric code.
In the toric-code setting, long loop operators provide the logical degrees of freedom used to store quantum information.

1:11:30 Large Coupling, Dimer Coverings, and the LSM Obstruction

At large coupling and integer spin, the local constraint allows X = +1 on every link, yielding a simple gauge-invariant state; Sachdev also cites AKLT-type singlet constructions as examples.
For half-integer spin, Gᵢ = −1 prevents every incident-link X configuration from being +1, so the strong-coupling states correspond to arrangements of dimers or valence bonds.
The many dimer configurations signal that a featureless symmetric product ground state is unavailable; depending on lattice and interactions, the system may form a valence-bond solid or another nontrivial phase, as in Lieb-Schultz-Mattis constraints.

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.