Ladder

class tenpy.models.lattice.Ladder(L, sites, **kwargs)[source]

Bases: tenpy.models.lattice.Lattice

A ladder coupling two chains.

../_images/Ladder.png
Parameters
Lint

The length of each chain, we have 2*L sites in total.

sites(list of) Site

The two local lattice sites making the unit_cell of the Lattice. If only a single Site is given, it is used for both chains.

**kwargs :

Additional keyword arguments given to the Lattice. basis, pos and [[next_]next_]nearest_neighbors are set accordingly.

Attributes
nearest_neighbors
next_nearest_neighbors
next_next_nearest_neighbors
order

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

Methods

count_neighbors(self[, u, key])

Count e.g.

coupling_shape(self, dx)

Calculate correct shape of the strengths for a coupling.

lat2mps_idx(self, lat_idx)

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

mps2lat_idx(self, i)

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

mps2lat_values(self, A[, axes, u])

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

mps_idx_fix_u(self[, u])

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

mps_lat_idx_fix_u(self[, u])

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

mps_sites(self)

Return a list of sites for all MPS indices.

multi_coupling_shape(self, dx)

Calculate correct shape of the strengths for a multi_coupling.

number_nearest_neighbors(self[, u])

Deprecated.

number_next_nearest_neighbors(self[, u])

Deprecated.

ordering(self, order)

Provide possible orderings of the N lattice sites.

plot_basis(self, ax, \*\*kwargs)

Plot arrows indicating the basis vectors of the lattice.

plot_bc_identified(self, ax[, direction, shift])

Mark two sites indified by periodic boundary conditions.

plot_coupling(self, ax[, coupling])

Plot lines connecting nearest neighbors of the lattice.

plot_order(self, ax[, order, textkwargs])

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

plot_sites(self, ax[, markers])

Plot the sites of the lattice with markers.

position(self, lat_idx)

return ‘space’ position of one or multiple sites.

possible_couplings(self, u1, u2, dx)

Find possible MPS indices for two-site couplings.

possible_multi_couplings(self, u0, other_us, dx)

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

site(self, i)

return Site instance corresponding to an MPS index i

test_sanity(self)

Sanity check.
count_neighbors(self, u=0, key='nearest_neighbors')

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

Parameters
uint

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

keystr

Key of pairs to select what to count.

Returns
numberint

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.

coupling_shape(self, dx)

Calculate correct shape of the strengths for a coupling.

Parameters
dxtuple of int

Translation vector in the lattice for a coupling of two operators.

Returns
coupling_shapetuple of int

Len dim. The correct shape for an array specifying the coupling strength. lat_indices has only rows within this shape.

shift_lat_indicesarray

Translation vector from lower left corner of box spanned by dx to the origin.

lat2mps_idx(self, lat_idx)

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

Parameters
lat_idxarray_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
iarray_like

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

mps2lat_idx(self, i)

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

Parameters
iint | array_like of int

MPS index/indices.

Returns
lat_idxarray

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.

mps2lat_values(self, A, axes=0, u=None)

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

Parameters
Andarray

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.

uNone | 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_Andarray

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), ...]

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(self, u=None)

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
uNone | int

Selects a site of the unit cell. None (default) means all sites.

Returns
mps_idxarray

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

mps_lat_idx_fix_u(self, u=None)

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

Parameters
uNone | int

Selects a site of the unit cell. None (default) means all sites.

Returns
mps_idxarray

MPS indices i for which self.site(i) is self.unit_cell[u].

lat_idx2D array

The row j contains the lattice index (without u) corresponding to mps_idx[j].

mps_sites(self)

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(self, dx)

Calculate correct shape of the strengths for a multi_coupling.

Parameters
dxtuple of int

Translation vector in the lattice for a coupling of two operators.

Returns
coupling_shapetuple of int

Len dim. The correct shape for an array specifying the coupling strength. lat_indices has only rows within this shape.

shift_lat_indicesarray

Translation vector from lower left corner of box spanned by dx to the origin.

number_nearest_neighbors(self, u=0)

Deprecated.

Use count_neighbors() instead.

number_next_nearest_neighbors(self, u=0)

Deprecated.

Use count_neighbors() instead.

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(self, order)

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
orderstr | ('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
orderarray, shape (N, D+1), dtype np.intp

the order to be used for order.

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(self, ax, **kwargs)

Plot arrows indicating the basis vectors of the lattice.

Parameters
axmatplotlib.axes.Axes

The axes on which we should plot.

**kwargs :

Keyword arguments specifying the “arrowprops” of ax.annotate.

plot_bc_identified(self, ax, direction=-1, shift=None, **kwargs)

Mark two sites indified by periodic boundary conditions.

Works only for lattice with a 2-dimensional basis.

Parameters
axmatplotlib.axes.Axes

The axes on which we should plot.

directionint

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

shiftNone | 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(self, ax, coupling=None, **kwargs)

Plot lines connecting nearest neighbors of the lattice.

Parameters
axmatplotlib.axes.Axes

The axes on which we should plot.

couplinglist 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(self, ax, order=None, textkwargs={}, **kwargs)

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

Parameters
axmatplotlib.axes.Axes

The axes on which we should plot.

orderNone | 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(self, ax, markers=['o', '^', 's', 'p', 'h', 'D'], **kwargs)

Plot the sites of the lattice with markers.

Parameters
axmatplotlib.axes.Axes

The axes on which we should plot.

markerslist

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(self, lat_idx)

return ‘space’ position of one or multiple sites.

Parameters
lat_idxndarray, (... , dim+1)

Lattice indices.

Returns
posndarray, (..., dim)

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

possible_couplings(self, u1, u2, dx)

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
u1, u2int

Indices within the unit cell; the u1 and u2 of add_coupling()

dxarray

Length dim. The translation in terms of basis vectors for the coupling.

Returns
mps1, mps2array

For each possible two-site coupling the MPS indices for the u1 and u2.

lat_indices2D 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_shapetuple 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(self, u0, other_us, dx)

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

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

Parameters
u0int

Argument u0 of add_multi_coupling().

other_uslist of int

The u of the other_ops in add_multi_coupling().

dxarray, shape (len(other_us), lat.dim+1)

The dx specifying relative operator positions of the other_ops in add_multi_coupling().

Returns
mps_ijkl2D 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_indices2D 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_shapetuple of int

Len dim. The correct shape for an array specifying the coupling strength. lat_indices has only rows within this shape.

site(self, i)

return Site instance corresponding to an MPS index i

test_sanity(self)

Sanity check.

Raises ValueErrors, if something is wrong.