The Compact Boson

The compact boson in 1+1D

\[\mathcal{Z} = \int \mathcal{D}\varphi\; \exp\left\{ - \int d^2x\; \frac{1}{8\pi} (\partial\varphi)^2\right\}\]

is dual to the free fermion and enjoys two interesting global symmetries,

\[\begin{split}\begin{align} \text{shift} && U(1)_S && \varphi \rightarrow \varphi+\epsilon && J^S_\mu &= \frac{i}{4\pi} \partial_\mu \varphi \\ \text{winding} && U(1)_W && \text{topological, not Noetherian} && J^V_\mu &= \frac{1}{2\pi} \epsilon_{\mu\nu} \partial^\nu \varphi \end{align}\end{split}\]

which correspond to the vector and axial currents on the fermionic side. The first is always conserved by the equations of motion of \(\varphi\), but the second is only conserved so long as partial derivatives commute. In other words, the winding symmetry is conserved as long as there are no vortices: parallel transport around vortices can yield a net winding number.

The modified Villain [1] formulation of the compact boson is a lattice discretization which allows us to easily control the winding subgroup, allowing it to break completely (yielding the traditional XY model), forcing it to maintain a \(\mathbb{Z}_W\) subgroup, or to keep it in its entirety. This discretization and related physical models are implemented in supervillain.

The discretization is given by

\[\begin{align} Z[J] &= \sum\hspace{-1.33em}\int D\phi\; Dn\; Dv\; e^{-S[\phi, n, v]} & S[\phi, n, v] &= \frac{\kappa}{2} \sum_{\ell} (d\phi - 2\pi n)_\ell^2 + 2\pi i \sum_p v_p (dn)_p / W \end{align}\]

with \(\phi\in\mathbb{R}\) on sites, \(n\in\mathbb{Z}\) on links, and a Lagrange multiplier field \(v\in\mathbb{Z}\) on plaquettes, and a careful choice of finite differencing \(d\) that obeys \(d^2=0\). The path integral over \(v\) restricts the vorticity plaquette-by-plaquette, setting \((dn) \equiv 0\; (\text{mod }W)\).

Changing the coupling \(\kappa\) corresponds to dialing Thirring terms on the fermionic side; a particular value corresponds to the free fermion. In general we do not have a simple map of the lattice coupling \(\kappa\) to the radius of the compact boson; there may be special points where enhanced symmetries or self-duality protects \(\kappa\) from renormalization.