The Worldline Formulation

We can find an exact rewriting of the constrained Villain model by first ‘integrating’ by parts,

\[\begin{align} S_J[\phi, n, v] &= \frac{\kappa}{2} \sum_{\ell} (d\phi - 2\pi n)_\ell^2 + 2\pi i \sum_\ell \delta(v/W + J/2\pi)_\ell n_\ell \end{align}\]

and applying link-by-link the Poisson summation formula

\[\sum_n \exp\left\{- \frac{\kappa}{2} (\theta - 2\pi n)^2 + i n \tilde{\theta}\right\} = \frac{1}{\sqrt{2\pi\kappa}} \sum_m \exp\left\{ - \frac{1}{2\kappa} \left(m - \frac{\tilde{\theta}}{2\pi}\right)^2 - i \left(m - \frac{\tilde{\theta}}{2\pi}\right) \theta\right\}\]

with \(\theta \rightarrow\; d\phi\) and \(\tilde{\theta} \rightarrow\; \delta (2\pi v / W + J)\) to find

\[\begin{split}\begin{align} Z[J] &= (2\pi\kappa)^{-|\ell|/2}\sum\hspace{-1.33em}\int D\phi\; Dm\; Dv\; e^{-S_J[\phi, m, v]} \\ S_J[\phi, m, v] &= \frac{1}{2\kappa} \sum_\ell \left(m - \delta\left(\frac{v}{W} + \frac{J}{2\pi}\right)\right)_\ell^2 - i \sum_\ell \left(m - \delta\left(\frac{v}{W} + \frac{J}{2\pi}\right)\right)_\ell (d\phi)_\ell. \end{align}\end{split}\]

‘Integrating’ by parts again transforms the action to

\[S_J[\phi, m, v] = \frac{1}{2\kappa} \sum_\ell \left(m - \delta\left(\frac{v}{W} + \frac{J}{2\pi} \right)\right)_\ell^2 - i \sum_x \left(\delta m\right)_x \phi_x\]

and we dropped the \(\delta^2\) term because \(\delta^2=0\). That leaves us with

\[\begin{split}\begin{align} Z[J] &= (2\pi\kappa)^{-|\ell|/2} \sum\hspace{-1.33em}\int D\phi\; Dm\; Dv\; e^{-S_J[\phi, m, v]} \\ S_J[\phi, m, v] &= \frac{1}{2\kappa} \sum_\ell \left(m - \delta\left(\frac{v}{W} + \frac{J}{2\pi}\right)\right)_\ell^2 - i \sum_x (\delta m)_x \phi_x \end{align}\end{split}\]

However, we can now execute the integral over \(\phi\), which just sets \(\delta m=0\) everywhere,

\[\begin{split}\begin{align} Z[J] &= (2\pi)^{|x|}(2\pi\kappa)^{-|\ell|/2} \sum Dm\; Dv\; e^{-S_J[m, v]} \left[\delta m = 0\right] \\ S_J[m, v] &= \frac{1}{2\kappa} \sum_\ell \left(m - \delta\left(\frac{v}{W} + \frac{J}{2\pi} \right)\right)_\ell^2 \end{align}\end{split}\]

where \([\delta m = 0]\) is the Iverson bracket and we picked up a \(2\pi\) for every site since \(\int d\phi\; e^{i o \phi} = 2\pi \delta(o)\). We can cast the dimensionless constants up into the action

\[\begin{split}\begin{align} Z[J] &= \sum Dm\; Dv\; e^{-S_J[m, v]} \left[\delta m = 0\right] \\ S_J[m, v] &= \frac{1}{2\kappa} \sum_\ell \left(m - \delta\left(\frac{v}{W} + \frac{J}{2\pi} \right)\right)_\ell^2 + \frac{|\ell|}{2} \ln (2\pi \kappa) - |x| \ln 2\pi \end{align}\end{split}\]

which will make functional differentiation more straightforward.

We can repeat the whole calculation with W=∞.

Replacing the Lagrange multiplier in the Villain formulation with a real-valued multiplier as in the vortex-free model

\[\begin{align} 2\pi i \sum_p (v/W)_p (dn)_p &\rightarrow i \sum_p \tilde{v}_p (dn)_p & \end{align}\]

and path integrating over \(\tilde{v}\) rather than path summing over \(v\) replaces Kronecker-δs with Dirac-δs, killing all vortices rather than restricing them to \(0\;(\text{mod }W)\). The \(W=\infty\) result is

\[\begin{split}\begin{align} Z[J] &= \sum\hspace{-1.33em}\int Dm\; D\tilde{v}\; e^{-S_J[m, \tilde{v}]} \left[\delta m = 0\right] \\ S_J[m, \tilde{v}] &= \frac{1}{2\kappa} \sum_\ell \left(m - \delta\left(\frac{\tilde{v} + J}{2\pi} \right)\right)_\ell^2 + \frac{|\ell|}{2} \ln (2\pi \kappa) - |x| \ln 2\pi \end{align}\end{split}\]

Perhaps obviously, we could have just set \(v\rightarrow 0\) and integrated over \(J\rightarrow \tilde{v}\). We can make this dual look more like the original Villain formulation by multiplying by \(1 = (2\pi)^2 / (2\pi)^2\),

\[S_J[m, \tilde{v}] = \frac{1}{2\kappa (2\pi)^2} \sum_\ell \left(\delta\left(\tilde{v} + J \right) - 2\pi m \right)_\ell^2 + \frac{|\ell|}{2} \ln (2\pi \kappa) - |x| \ln 2\pi\]

and comparing with the original Villain formulation we can find the self-dual radius by setting

\[\begin{align} \frac{\kappa}{2} &= \frac{1}{2\kappa (2\pi)^2} & \rightarrow&& \kappa &= \frac{1}{2\pi} \end{align}\]

The rule is that we can go to \(W=\infty\) by replacing

\[\begin{align} 2\pi v/W &\rightarrow \tilde{v} \in \mathbb{R} & v/W & \rightarrow \tilde{v} / 2\pi \end{align}\]

and integrating \(\tilde{v}\) over the real numbers.

class supervillain.action.Worldline(lattice, kappa, W=1)[source]

The dual (worldline) action is

\[\begin{split}\begin{align} Z[J] &= \sum Dm\; Dv\; e^{-S_J[m, v]} \left[\delta m = 0\right] \\ S_J[m, v] &= \frac{1}{2\kappa} \sum_\ell \left(m - \delta\left(\frac{v}{W} + \frac{J}{2\pi} \right)\right)_\ell^2 + \frac{|\ell|}{2} \ln (2\pi \kappa) - |x| \ln 2\pi \end{align}\end{split}\]

In other words, it is a sum over all configurations where \(\delta m\) vanishes on every site.

This formulation has no obvious sign problem when \(W\neq 1\), but maintaining the constraint \(\delta m = 0\) requires a nontrivial algorithm.

Parameters

lattice (supervillain.Lattice2D) – The lattice on which \(m\) lives.
kappa (float) – The \(\kappa\) in the overall coefficient.
W (int) – The winding symmetry is \(\mathbb{Z}_W\). If \(W=1\) the vortices are completely unconstrained.

valid(configuration)[source]

Returns true if the constraint \([\delta m = 0]\) is satisfied everywhere and false otherwise.

Parameters: configuration (dict) – A dictionary that at least contains m.
Returns: Is the constraint satisfied everywhere?
Return type: bool

__call__(m, v, **kwargs)[source]

Parameters

m (np.ndarray) – An integer-valued 1-form.
v (np.ndarray) – An integer-valued 2-form, (real-valued when \(W=\infty\)).

Returns

\(S_0[m]\)

Return type

float

Raises

ValueError – If \(m\) does not satisfy the constraint.

configurations(count)[source]

Parameters: count (int) – How many configurations to return.
Returns: count configurations of a zeroed 1-form m a zeroed 2-form v.
Return type: Configurations

equivalence_class_v(configuration)[source]

Assuming \(W<\infty\) the constrained model has a gauge symmetry \(v \rightarrow v \pm W\) with the gauge-invariant combination \(m-\delta v / W\).

We can take any configuration and send

\[\begin{align} v &\rightarrow v + \lambda W & m &\rightarrow m - \delta \lambda \end{align}\]

for integer \(\lambda\). We fix \(\lambda\) on every plaquette so that after the transformation \(v\in[0,W)\).