# DualSquare¶

Inheritance Diagram

Methods

 DualSquare.__init__(Lx, Ly, sites, **kwargs) Initialize self. DualSquare.count_neighbors([u, key]) Count e.g. Calculate correct shape of the strengths for a coupling. DualSquare.from_hdf5(hdf5_loader, h5gr, subpath) Load instance from a HDF5 file. DualSquare.lat2mps_idx(lat_idx) Translate lattice indices (x_0, ..., x_{D-1}, u) to MPS index i. Translate MPS index i to lattice indices (x_0, ..., x_{dim-1}, u). DualSquare.mps2lat_values(A[, axes, u]) Reshape/reorder A to replace an MPS index by lattice indices. return an index array of MPS indices for which the site within the unit cell is u. Similar as mps_idx_fix_u(), but return also the corresponding lattice indices. Return a list of sites for all MPS indices. Calculate correct shape of the strengths for a multi_coupling. Deprecated. Deprecated. Provide possible orderings of the N lattice sites. DualSquare.plot_basis(ax, **kwargs) Plot arrows indicating the basis vectors of the lattice. Mark two sites indified by periodic boundary conditions. DualSquare.plot_coupling(ax[, coupling]) Plot lines connecting nearest neighbors of the lattice. DualSquare.plot_order(ax[, order, textkwargs]) Plot a line connecting sites in the specified “order” and text labels enumerating them. DualSquare.plot_sites(ax[, markers]) Plot the sites of the lattice with markers. DualSquare.position(lat_idx) return ‘space’ position of one or multiple sites. DualSquare.possible_couplings(u1, u2, dx) Find possible MPS indices for two-site couplings. Generalization of possible_couplings() to couplings with more than 2 sites. DualSquare.save_hdf5(hdf5_saver, h5gr, subpath) Export self into a HDF5 file. return Site instance corresponding to an MPS index i Sanity check.

Class Attributes and Properties

 DualSquare.boundary_conditions Human-readable list of boundary conditions from bc and bc_shift. DualSquare.dim The dimension of the lattice. DualSquare.nearest_neighbors DualSquare.next_nearest_neighbors DualSquare.next_next_nearest_neighbors DualSquare.order Defines an ordering of the lattice sites, thus mapping the lattice to a 1D chain.
class tenpy.models.toric_code.DualSquare(Lx, Ly, sites, **kwargs)[source]

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 Square lattice with the same dimensions Lx, Ly.

Parameters
• Ly (Lx,) – Dimensions of the original lattice. This lattice has 2*Lx*Ly sites.

• sites (Site) – The sites for the horizontal (first entry) and vertical (second entry) bonds.

• **kwargs – Additional keyword arguments given to the Lattice. basis, pos and pairs are set accordingly.

property boundary_conditions

Human-readable list of boundary conditions from bc and bc_shift.

Returns

boundary_conditions – List of "open" or "periodic", one entry for each direction of the lattice.

Return type

list of str

count_neighbors(u=0, key='nearest_neighbors')[source]

Count e.g. the number of nearest neighbors for a site in the bulk.

Parameters
• u (int) – Specifies the site in the unit cell, for which we should count the number of neighbors (or whatever key specifies).

• key (str) – Key of pairs to select what to count.

Returns

number – Number of nearest neighbors (or whatever key specified) for the u-th site in the unit cell, somewhere in the bulk of the lattice. Note that it might not be the correct value at the edges of a lattice with open boundary conditions.

Return type

int

coupling_shape(dx)[source]

Calculate correct shape of the strengths for a coupling.

Parameters

dx (tuple of int) – Translation vector in the lattice for a coupling of two operators.

Returns

• coupling_shape (tuple of int) – Len dim. The correct shape for an array specifying the coupling strength. lat_indices has only rows within this shape.

• shift_lat_indices (array) – Translation vector from lower left corner of box spanned by dx to the origin.

property dim

The dimension of the lattice.

classmethod from_hdf5(hdf5_loader, h5gr, subpath)[source]

Load instance from a HDF5 file.

This method reconstructs a class instance from the data saved with save_hdf5().

Parameters
• hdf5_loader (Hdf5Loader) – Instance of the loading engine.

• h5gr (Group) – HDF5 group which is represent the object to be constructed.

• subpath (str) – The name of h5gr with a '/' in the end.

Returns

obj – Newly generated class instance containing the required data.

Return type

cls

lat2mps_idx(lat_idx)[source]

Translate lattice indices (x_0, ..., x_{D-1}, u) to MPS index i.

Parameters

lat_idx (array_like [.., dim+1]) – The last dimension corresponds to lattice indices (x_0, ..., x_{D-1}, u). All lattice indices should be positive and smaller than the corresponding entry in self.shape. Exception: for “infinite” bc_MPS, an x_0 outside indicates shifts accross the boundary.

Returns

i – MPS index/indices corresponding to lat_idx. Has the same shape as lat_idx without the last dimension.

Return type

array_like

mps2lat_idx(i)[source]

Translate MPS index i to lattice indices (x_0, ..., x_{dim-1}, u).

Parameters

i (int | array_like of int) – MPS index/indices.

Returns

lat_idx – First dimensions like i, last dimension has len dim+1 and contains the lattice indices (x_0, …, x_{dim-1}, u) corresponding to i. For i accross the MPS unit cell and “infinite” bc_MPS, we shift x_0 accordingly.

Return type

array

mps2lat_values(A, axes=0, u=None)[source]

Reshape/reorder A to replace an MPS index by lattice indices.

Parameters
• A (ndarray) – Some values. Must have A.shape[axes] = self.N_sites if u is None, or A.shape[axes] = self.N_cells if u is an int.

• axes ((iterable of) int) – chooses the axis which should be replaced.

• u (None | int) – Optionally choose a subset of MPS indices present in the axes of A, namely the indices corresponding to self.unit_cell[u], as returned by mps_idx_fix_u(). The resulting array will not have the additional dimension(s) of u.

Returns

res_A – Reshaped and reordered verions of A. Such that an MPS index j is replaced by res_A[..., self.order, ...] = A[..., np.arange(self.N_sites), ...]

Return type

ndarray

Examples

Say you measure expection values of an onsite term for an MPS, which gives you an 1D array A, where A[i] is the expectation value of the site given by self.mps2lat_idx(i). Then this function gives you the expectation values ordered by the lattice:

>>> print(lat.shape, A.shape)
(10, 3, 2) (60,)
>>> A_res = lat.mps2lat_values(A)
>>> A_res.shape
(10, 3, 2)
>>> A_res[lat.mps2lat_idx(5)] == A[5]
True


If you have a correlation function C[i, j], it gets just slightly more complicated:

>>> print(lat.shape, C.shape)
(10, 3, 2) (60, 60)
>>> lat.mps2lat_values(C, axes=[0, 1]).shape
(10, 3, 2, 10, 3, 2)


If the unit cell consists of different physical sites, an onsite operator might be defined only on one of the sites in the unit cell. Then you can use mps_idx_fix_u() to get the indices of sites it is defined on, measure the operator on these sites, and use the argument u of this function.

>>> u = 0
>>> idx_subset = lat.mps_idx_fix_u(u)
>>> A_u = A[idx_subset]
>>> A_u_res = lat.mps2lat_values(A_u, u=u)
>>> A_u_res.shape
(10, 3)
>>> np.all(A_res[:, :, u] == A_u_res[:, :])
True


Todo

make sure this function is used for expectation values…

mps_idx_fix_u(u=None)[source]

return an index array of MPS indices for which the site within the unit cell is u.

If you have multiple sites in your unit-cell, an onsite operator is in general not defined for all sites. This functions returns an index array of the mps indices which belong to sites given by self.unit_cell[u].

Parameters

u (None | int) – Selects a site of the unit cell. None (default) means all sites.

Returns

mps_idx – MPS indices for which self.site(i) is self.unit_cell[u]. Ordered ascending.

Return type

array

mps_lat_idx_fix_u(u=None)[source]

Similar as mps_idx_fix_u(), but return also the corresponding lattice indices.

Parameters

u (None | int) – Selects a site of the unit cell. None (default) means all sites.

Returns

• mps_idx (array) – MPS indices i for which self.site(i) is self.unit_cell[u].

• lat_idx (2D array) – The row j contains the lattice index (without u) corresponding to mps_idx[j].

mps_sites()[source]

Return a list of sites for all MPS indices.

Equivalent to [self.site(i) for i in range(self.N_sites)].

This should be used for sites of 1D tensor networks (MPS, MPO,…).

multi_coupling_shape(dx)[source]

Calculate correct shape of the strengths for a multi_coupling.

Parameters

dx (tuple of int) – Translation vector in the lattice for a coupling of two operators.

Returns

• coupling_shape (tuple of int) – Len dim. The correct shape for an array specifying the coupling strength. lat_indices has only rows within this shape.

• shift_lat_indices (array) – Translation vector from lower left corner of box spanned by dx to the origin.

number_nearest_neighbors(u=0)[source]

Deprecated.

number_next_nearest_neighbors(u=0)[source]

Deprecated.

property order

Defines an ordering of the lattice sites, thus mapping the lattice to a 1D chain.

This order defines how an MPS/MPO winds through the lattice.

ordering(order)[source]

Provide possible orderings of the N lattice sites.

This function can be overwritten by derived lattices to define additional orderings. The following orders are defined in this method:

order

equivalent priority

equivalent snake_winding

'Cstyle'

(0, 1, …, dim-1, dim)

(False, …, False, False)

'default'

'snake'

(0, 1, …, dim-1, dim)

(True, …, True, True)

'snakeCstyle'

'Fstyle'

(dim-1, …, 1, 0, dim)

(False, …, False, False)

'snakeFstyle'

(dim-1, …, 1, 0, dim)

(False, …, False, False)

Parameters

order (str | ('standard', snake_winding, priority) | ('grouped', groups)) – Specifies the desired ordering using one of the strings of the above tables. Alternatively, an ordering is specified by a tuple with first entry specifying a function, 'standard' for get_order() and 'grouped' for get_order_grouped(), and other arguments in the tuple as specified in the documentation of these functions.

Returns

order – the order to be used for order.

Return type

array, shape (N, D+1), dtype np.intp

See also

get_order()

generates the order from equivalent priority and snake_winding.

get_order_grouped()

variant of get_order.

plot_order()

visualizes the resulting order.

plot_basis(ax, **kwargs)[source]

Plot arrows indicating the basis vectors of the lattice.

Parameters
• ax (matplotlib.axes.Axes) – The axes on which we should plot.

• **kwargs – Keyword arguments specifying the “arrowprops” of ax.annotate.

plot_bc_identified(ax, direction=-1, shift=None, **kwargs)[source]

Mark two sites indified by periodic boundary conditions.

Works only for lattice with a 2-dimensional basis.

Parameters
• ax (matplotlib.axes.Axes) – The axes on which we should plot.

• direction (int) – The direction of the lattice along which we should mark the idenitified sites. If None, mark it along all directions with periodic boundary conditions.

• shift (None | np.ndarray) – The origin starting from where we mark the identified sites. Defaults to the first entry of unit_cell_positions.

• **kwargs – Keyword arguments for the used ax.plot.

plot_coupling(ax, coupling=None, **kwargs)[source]

Plot lines connecting nearest neighbors of the lattice.

Parameters
• ax (matplotlib.axes.Axes) – The axes on which we should plot.

• coupling (list of (u1, u2, dx)) – By default (None), use self.pairs['nearest_neighbors']. Specifies the connections to be plotted; iteating over lattice indices (i0, i1, …), we plot a connection from the site (i0, i1, ..., u1) to the site (i0+dx[0], i1+dx[1], ..., u2), taking into account the boundary conditions.

• **kwargs – Further keyword arguments given to ax.plot().

plot_order(ax, order=None, textkwargs={}, **kwargs)[source]

Plot a line connecting sites in the specified “order” and text labels enumerating them.

Parameters
• ax (matplotlib.axes.Axes) – The axes on which we should plot.

• order (None | 2D array (self.N_sites, self.dim+1)) – The order as returned by ordering(); by default (None) use order.

• textkwargs (None | dict) – If not None, we add text labels enumerating the sites in the plot. The dictionary can contain keyword arguments for ax.text().

• **kwargs – Further keyword arguments given to ax.plot().

plot_sites(ax, markers=['o', '^', 's', 'p', 'h', 'D'], **kwargs)[source]

Plot the sites of the lattice with markers.

Parameters
• ax (matplotlib.axes.Axes) – The axes on which we should plot.

• markers (list) – List of values for the keywork marker of ax.plot() to distinguish the different sites in the unit cell, a site u in the unit cell is plotted with a marker markers[u % len(markers)].

• **kwargs – Further keyword arguments given to ax.plot().

position(lat_idx)[source]

return ‘space’ position of one or multiple sites.

Parameters

lat_idx (ndarray, (... , dim+1)) – Lattice indices.

Returns

pos – The position of the lattice sites specified by lat_idx in real-space.

Return type

ndarray, (..., dim)

possible_couplings(u1, u2, dx)[source]

Find possible MPS indices for two-site couplings.

For periodic boundary conditions (bc[a] == False) the index x_a is taken modulo Ls[a] and runs through range(Ls[a]). For open boundary conditions, x_a is limited to 0 <= x_a < Ls[a] and 0 <= x_a+dx[a] < lat.Ls[a].

Parameters
Returns

• mps1, mps2 (array) – For each possible two-site coupling the MPS indices for the u1 and u2.

• lat_indices (2D int array) – Rows of lat_indices correspond to rows of mps_ijkl and contain the lattice indices of the “lower left corner” of the box containing the coupling.

• coupling_shape (tuple of int) – Len dim. The correct shape for an array specifying the coupling strength. lat_indices has only rows within this shape.

possible_multi_couplings(u0, other_us, dx)[source]

Generalization of possible_couplings() to couplings with more than 2 sites.

Given the arguments of add_coupling() determine the necessary shape of strength.

Parameters
Returns

• mps_ijkl (2D int array) – Each row contains MPS indices i,j,k,l,… for each of the operators positions. The positions are defined by dx (j,k,l,… relative to i) and boundary coundary conditions of self (how much the box for given dx can be shifted around without hitting a boundary - these are the different rows).

• lat_indices (2D int array) – Rows of lat_indices correspond to rows of mps_ijkl and contain the lattice indices of the “lower left corner” of the box containing the coupling.

• coupling_shape (tuple of int) – Len dim. The correct shape for an array specifying the coupling strength. lat_indices has only rows within this shape.

save_hdf5(hdf5_saver, h5gr, subpath)[source]

Export self into a HDF5 file.

This method saves all the data it needs to reconstruct self with from_hdf5().

Specifically, it saves unit_cell, Ls, unit_cell_positions, basis, boundary_conditions, pairs under their name, bc_MPS as "boundary_conditions_MPS", and bc_MPS as "order_for_MPS". Moreover, it saves dim and N_sites as HDF5 attributes.

Parameters
• hdf5_saver (Hdf5Saver) – Instance of the saving engine.

• h5gr (:classGroup) – HDF5 group which is supposed to represent self.

• subpath (str) – The name of h5gr with a '/' in the end.

site(i)[source]

return Site instance corresponding to an MPS index i

test_sanity`()[source]

Sanity check.

Raises ValueErrors, if something is wrong.