MultiCouplingModel, it's time to add a simple model actually using it.I think the exactly solvable toric code is a good example, so here is my implementation:
Python: Select all
"""Kitaev's exactly solvable toric code model.
As we put the model on a cylinder, the name "toric code" is a bit misleading,
but it is established for this model...
"""
# Copyright 2018 TeNPy Developers
import numpy as np
from .lattice import Lattice
from ..networks.site import SpinHalfSite
from .model import MultiCouplingModel, MPOModel
from ..tools.params import get_parameter, unused_parameters
from ..tools.misc import any_nonzero
__all__ = ['DualSquare', 'ToricCode']
class DualSquare(Lattice):
"""The dual lattice of the square lattice (again square).
The sites in this lattice correspond to the vertical and horizontal (nearest neighbor) bonds
of a common :class:`~tenpy.models.lattice.Square` lattice with the same dimensions `Lx, Ly`.
Parameters
----------
Lx, Ly : int
Dimensions of the original lattice. This lattice has `2*Lx*Ly` sites.
site_hor, site_ver : :class:`~tenpy.networks.site.Site`
The sites for the horizontal and vertical bonds.
order : str | (priority, snake_winding)
A string or tuple specifying the order, given to :meth:`ordering`.
"""
def __init__(self, Lx, Ly, site_hor, site_ver, order='default', bc_MPS='finite'):
basis = np.eye(2)
pos = np.array([[0.5, 0.], [0., 0.5]])
self.nearest_neighbors = [(0, 1, np.array([0, 0])), (0, 1, np.array([1, 1])),
(1, 0, np.array([-1, 1])), (1, 0, np.array([0, 1]))]
self.next_nearest_neighbors = [(i, i, dx) for i in [0, 1]
for dx in [np.array([1,0]), np.array([0, 1])]]
super().__init__([Lx, Ly], [site_hor, site_ver], order, bc_MPS, basis, pos)
class ToricCode(MultiCouplingModel, MPOModel):
r"""Spin-S sites coupled by nearest neighbour interactions.
The Hamiltonian reads:
.. math ::
H = - \mathtt{Jv} \sum_{vertices v} \prod_{i \in v} \sigma^x_i
- \mathtt{Jp} \sum_{plaquettes p} \prod_{i \in p} \sigma^z_i
(Note that this are Pauli matrices, not spin-1/2 operators.)
All parameters are collected in a single dictionary `model_param` and read out with
:func:`~tenpy.tools.params.get_parameter`.
Parameters
----------
Lx, Ly : int
Dimension of the lattice, number of plaquettes around the cylinder.
conserve : 'parity' | None
What should be conserved. See :class:`~tenpy.networks.Site.SpinHalfSite`.
Jc, Jp: float | array
Couplings as defined for the Hamiltonian above.
bc_MPS : {'finite' | 'infinte'}
MPS boundary conditions. Coupling boundary conditions are chosen appropriately.
order : str
The order of the lattice sites in the lattice, see :class:`DualSquare`.
"""
def __init__(self, model_param):
# 0) read out/set default parameters
verbose = get_parameter(model_param, 'verbose', 1, self.__class__)
Lx = get_parameter(model_param, 'Lx', 2, self.__class__)
Ly = get_parameter(model_param, 'Ly', 2, self.__class__)
Jv = get_parameter(model_param, 'Jv', 1., self.__class__)
Jp = get_parameter(model_param, 'Jp', 1., self.__class__)
bc_MPS = get_parameter(model_param, 'bc_MPS', 'infinite', self.__class__)
order = get_parameter(model_param, 'order', 'default', self.__class__)
conserve = get_parameter(model_param, 'conserve', 'parity', self.__class__)
unused_parameters(model_param, self.__class__)
# 1) define Site and lattice
site = SpinHalfSite(conserve)
lat = DualSquare(Lx, Ly, site, site, order, bc_MPS)
# 2) initialize CouplingModel
bc_coupling = [None, 'periodic']
bc_coupling[0] = 'periodic' if bc_MPS == 'infinite' else 'open'
MultiCouplingModel.__init__(self, lat, bc_coupling)
# 3) add terms of the Hamiltonian
# (u is always 0 as we have only one site in the unit cell)
Jv = np.asarray(Jv)
Jp = np.asarray(Jp)
# vertex/star term
self.add_multi_coupling(Jv, 0, 'Sigmax', [(1, 'Sigmax', [0, 0]),
(0, 'Sigmax', [-1, 0]),
(1, 'Sigmax', [0, -1])])
# plaquette term
self.add_multi_coupling(Jp, 0, 'Sigmaz', [(1, 'Sigmaz', [0, 0]),
(0, 'Sigmaz', [0, 1]),
(1, 'Sigmaz', [1, 0])])
# 4) initialize MPO
MPOModel.__init__(self, lat, self.calc_H_MPO())
Is this multi-coupling intuitive enough regarding how to use it? Any suggestions how to make it more intuitive?
Writing the code, I felt that it might be more intuitive to have a single list
ops in add_multi_coupling, i.e. along the lines of
Python: Select all
self.add_multi_coupling(Jv, [(0, 'Sigmax', [0, 0]),
(1, 'Sigmax', [0, 0]),
(0, 'Sigmax', [-1, 0]),
(1, 'Sigmax', [0, -1])])
# instead of
self.add_multi_coupling(Jv, 0, 'Sigmax', [(1, 'Sigmax', [0, 0]),
(0, 'Sigmax', [-1, 0]),
(1, 'Sigmax', [0, -1])])
