Kagome¶
full name: tenpy.models.lattice.Kagome
parent module:
tenpy.models.lattice
type: class
Inheritance Diagram
Methods

Initialize self. 

Count e.g. 
Calculate correct shape of the strengths for a coupling. 


Get the distance for a given coupling between two sites in the lattice. 

Repeat the unit cell for infinite MPS boundary conditions; in place. 

Automatically find coupling pairs grouped by distances. 

Load instance from a HDF5 file. 

Translate lattice indices 
Translate MPS index i to lattice indices 


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

Reshape/reorder an array 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 

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. 

Plot arrows indicating the basis vectors of the lattice. 

Mark two sites indified by periodic boundary conditions. 

Plot lines connecting nearest neighbors of the lattice. 

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

Plot the sites of the lattice with markers. 

return ‘space’ position of one or multiple sites. 

Find possible MPS indices for twosite couplings. 

Generalization of 

Export self into a HDF5 file. 

return 
Sanity check. 
Class Attributes and Properties
the (expected) number of sites in the unit cell, 

Humanreadable list of boundary conditions from 

the dimension of the lattice 







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

class
tenpy.models.lattice.
Kagome
(Lx, Ly, sites, **kwargs)[source]¶ Bases:
tenpy.models.lattice.Lattice
A Kagome lattice.
(Source code, png, hires.png, pdf)
 Parameters
Lx (int) – The length in each direction.
Ly (int) – The length in each direction.
sites ((list of)
Site
) – The two local lattice sites making the unit_cell of theLattice
. If only a singleSite
is given, it is used for both sites.**kwargs – Additional keyword arguments given to the
Lattice
. basis, pos and pairs are set accordingly.

property
boundary_conditions
¶ Humanreadable list of boundary conditions from
bc
andbc_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
 Returns
number – Number of nearest neighbors (or whatever key specified) for the uth 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

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. Corresponds to dx argument of
tenpy.models.model.CouplingModel.add_multi_coupling()
. 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 origin to the lower left corner of box spanned by dx.

distance
(u1, u2, dx)[source]¶ Get the distance for a given coupling between two sites in the lattice.
The u1, u2, dx parameters are defined in analogy with
add_coupling()
, i.e., this function calculates the distance between a pair of operators added with add_coupling (using thebasis
andunit_cell_positions
of the lattice).Warning
This function ignores “wrapping” arround the cylinder in the case of periodic boundary conditions.
 Parameters
u1 (int) – Indices within the unit cell; the u1 and u2 of
add_coupling()
u2 (int) – Indices within the unit cell; the u1 and u2 of
add_coupling()
dx (array) – Length
dim
. The translation in terms of basis vectors for the coupling.
 Returns
distance – The distance between site at lattice indices
[x, y, u1]
and[x + dx[0], y + dx[1], u2]
, ignoring any boundary effects. Return type

enlarge_mps_unit_cell
(factor=2)[source]¶ Repeat the unit cell for infinite MPS boundary conditions; in place.
 Parameters
factor (int) – The new number of sites in the MPS unit cell will be increased from N_sites to
factor*N_sites_per_ring
. Since MPS unit cells are repeated in the xdirection in our convetion, the lattice shape goes from(Lx, Ly, ..., Lu)
to(Lx*factor, Ly, ..., Lu)
.

find_coupling_pairs
(max_dx=3, cutoff=None, eps=1e10)[source]¶ Automatically find coupling pairs grouped by distances.
Given the
unit_cell_positions
andbasis
, the couplingpairs
of nearest_neighbors, next_nearest_neighbors etc at a given distance are basically fixed (although not uniquely, since we take out half of them to avoid doublecounting couplings in both directionsA_i B_j + B_i A_i
). This function iterates through all possible couplings up to a given cutoff distance and then determines the possiblepairs
at fixed distances (up to roundoff errors). Parameters
max_dx (int) – Maximal index for each index of dx to iterate over. You need large enough values to include every possible coupling up to the desired distance, but choosing it too large might make this function run for a long time.
cutoff (float) – Maximal distance (in the units in which
basis
andunit_cell_positions
is given).eps (float) – Tolerance up to which to distances are considered the same.
 Returns
coupling_pairs – Keys are distances of nearestneighbors, nextnearestneighbors etc. Values are
[(u1, u2, dx), ...]
as inpairs
. Return type

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_{D1}, u)
to MPS index i. Parameters
lat_idx (array_like [.., dim+1]) – The last dimension corresponds to lattice indices
(x_0, ..., x_{D1}, u)
. All lattice indices should be positive and smaller than the corresponding entry inself.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_{dim1}, 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_{dim1}, 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 isNone
, orA.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 toself.unit_cell[u]
, as returned bymps_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 MPS indices along the specified axes are replaced by lattice indices, i.e., if MPS index j maps to lattice site (x0, x1, x2), then
res_A[..., x0, x1, x2, ...] = A[..., j, ...]
. 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[tuple(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

mps2lat_values_masked
(A, axes= 1, mps_inds=None, include_u=None)[source]¶ Reshape/reorder an array A to replace an MPS index by lattice indices.
This is a generalization of
mps2lat_values()
allowing for the case of an arbitrary set of MPS indices present in each axis of A. Parameters
A (ndarray) – Some values.
axes ((iterable of) int) – Chooses the axis of A which should be replaced. If multiple axes are given, you also need to give multiple index arrays as mps_inds.
mps_inds ((list of) 1D ndarray) – Specifies for each axis in axes, for which MPS indices we have values in the corresponding axis of A. Defaults to
[np.arange(A.shape[ax]) for ax in axes]
. For indices accross the MPS unit cell and “infinite” bc_MPS, we shift x_0 accordingly.include_u ((list of) bool) – Specifies for each axis in axes, whether the u index of the lattice should be included into the output array res_A. Defaults to
len(self.unit_cell) > 1
.
 Returns
res_A – Reshaped and reordered copy of A. Such that MPS indices along the specified axes are replaced by lattice indices, i.e., if MPS index j maps to lattice site (x0, x1, x2), then
res_A[..., x0, x1, x2, ...] = A[..., mps_inds[j], ...]
. Return type
np.ma.MaskedArray

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 unitcell, 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 (2D array, shape (N_ops,
dim
)) –dx[i, :]
is the translation vector in the lattice for the ith operator. Corresponds to the dx of each operator given in the argument ops oftenpy.models.model.CouplingModel.add_multi_coupling()
. 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 origin to the lower left corner of box spanned by dx. (Unlike for
coupling_shape()
it can also contain entries > 0)

number_nearest_neighbors
(u=0)[source]¶ Deprecated.
Deprecated since version 0.5.0: Use
count_neighbors()
instead.

number_next_nearest_neighbors
(u=0)[source]¶ Deprecated.
Deprecated since version 0.5.0: Use
count_neighbors()
instead.

property
order
¶ Defines an ordering of the lattice sites, thus mapping the lattice to a 1D chain.
Each row of the array contains the lattice indices for one site, the order of the rows thus specifies a path through the lattice, along which an MPS will wind through through the lattice.
You can visualize the order with
plot_order()
.

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, …, dim1, dim)
(False, …, False, False)
'default'
'snake'
(0, 1, …, dim1, dim)
(True, …, True, True)
'snakeCstyle'
'Fstyle'
(dim1, …, 1, 0, dim)
(False, …, False, False)
'snakeFstyle'
(dim1, …, 1, 0, dim)
(False, …, False, False)
(Source code, png, hires.png, pdf)
Note
For lattices with a nontrivial unit cell (e.g. Honeycomb, Kagome), the grouped order might be more appropriate, see
get_order_grouped()
 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'
forget_order()
and'grouped'
forget_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, origin=(0.0, 0.0), shade=None, **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 for
ax.arrow
.

plot_bc_identified
(ax, direction= 1, origin=None, cylinder_axis=False, **kwargs)[source]¶ Mark two sites indified by periodic boundary conditions.
Works only for lattice with a 2dimensional 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.cylinder_axis (bool) – Whether to plot the cylinder axis as well.
origin (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, wrap=False, **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
), useself.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.wrap (bool) – If
True
, plot couplings going around the boundary by directly connecting the sites it connects. This might be hard to see, as this puts lines from one end of the lattice to the other. IfFalse
, plot the couplings as dangling lines.**kwargs – Further keyword arguments given to
ax.plot()
.

plot_order
(ax, order=None, textkwargs={'color': 'r'}, **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
) useorder
.textkwargs (
None
 dict) – If notNone
, we add text labels enumerating the sites in the plot. The dictionary can contain keyword arguments forax.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 markermarkers[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 realspace.
 Return type
ndarray,
(..., dim)

possible_couplings
(u1, u2, dx, strength=None)[source]¶ Find possible MPS indices for twosite couplings.
For periodic boundary conditions (
bc[a] == False
) the indexx_a
is taken moduloLs[a]
and runs throughrange(Ls[a])
. For open boundary conditions,x_a
is limited to0 <= x_a < Ls[a]
and0 <= x_a+dx[a] < lat.Ls[a]
. Parameters
u1 (int) – Indices within the unit cell; the u1 and u2 of
add_coupling()
u2 (int) – Indices within the unit cell; the u1 and u2 of
add_coupling()
dx (array) – Length
dim
. The translation in terms of basis vectors for the coupling.strength (array_like  None) – If given, instead of returning lat_indices and coupling_shape directly return the correct strength_12.
 Returns
mps1, mps2 (1D array) – For each possible twosite coupling the MPS indices for the u1 and u2.
strength_vals (1D array) – (Only returend if strength is not None.) Such that
for (i, j, s) in zip(mps1, mps2, strength_vals):
iterates over all possible couplings with s being the strength of that coupling. Couplings wherestrength_vals == 0.
are filtered out.lat_indices (2D int array) – (Only returend if strength is None.) Rows of lat_indices correspond to entries of mps1 and mps2 and contain the lattice indices of the “lower left corner” of the box containing the coupling.
coupling_shape (tuple of int) – (Only returend if strength is None.) Len
dim
. The correct shape for an array specifying the coupling strength. lat_indices has only rows within this shape.

possible_multi_couplings
(ops, strength=None)[source]¶ Generalization of
possible_couplings()
to couplings with more than 2 sites. Parameters
ops (list of
(opname, dx, u)
) – Same as the argument ops ofadd_multi_coupling()
. 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).
strength_vals (1D array) – (Only returend if strength is not None.) Such that
for (ijkl, s) in zip(mps_ijkl, strength_vals):
iterates over all possible couplings with s being the strength of that coupling. Couplings wherestrength_vals == 0.
are filtered out.lat_indices (2D int array) – (Only returend if strength is None.) 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) – (Only returend if strength is None.) 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"
, andorder
as"order_for_MPS"
. Moreover, it savesdim
andN_sites
as HDF5 attributes.

dim
= 2¶ the dimension of the lattice

Lu
= 3¶ the (expected) number of sites in the unit cell,
len(unit_cell)
.