from supervillain.observable import Observable
from supervillain.lattice import d, delta
import numpy as np
[docs]class Links(Observable):
r'''
Both the :class:`~.Villain` and :class:`~.Worldline` formulations have the notion of invariant links.
The main purpose of ``Links`` is to ensure that we always write functions of the correct combination of raw fields.
.. warning ::
Unlike most observables these will NOT match${}^*$ between the different formulations.
${}^*$They both have 0 expectation value by the discrete rotational lattice symmetries, but they don't have related physical motivation; they are dual to one another under Poisson resummation.
'''
[docs] @staticmethod
def Villain(S, phi, n):
r'''
The only gauge-invariant link variables in the :class:`~.Villain` formulation are
.. math ::
\texttt{Links}_{\ell} = (d\phi - 2\pi n)_\ell
and these are what show up in any observables that talk to links.
You can see that this combination arises in the action itself and in the :class:`~.ActionTwoPoint`, for example.
Some observables, like the :class:`~.Spin_Spin` correlation function have 'bare' 0-form $\phi$s but they come in exponentials and are therefore gauge invariant.
Others, like :class:`~.TorusWrapping` live on loops where the $d\phi$ integrates away but are gauge invariant.
'''
return d(phi) - 2*np.pi*n
[docs] @staticmethod
def Worldline(S, m, v):
r'''
In the :class:`~.action.Worldline` action we always have the combination
.. math ::
\texttt{Links}_{\ell} = (m - \delta v / W)_\ell
and observables are generically functions of this combination, though there are some exceptional observables like :class:`~.TorusWrapping` where the $\delta v$ pieces cancel exactly.
'''
return m - delta(v) / S._W