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

Initialize self. 

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


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. 
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
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.
Lattice
(Ls, unit_cell, order='default', bc='open', bc_MPS='finite', basis=None, positions=None, nearest_neighbors=None, next_nearest_neighbors=None, next_next_nearest_neighbors=None, pairs=None)[source]¶ Bases:
object
A general, regular lattice.
The lattice consists of a unit cell which is repeated in dim different directions. A site of the lattice is thus identified by lattice indices
(x_0, ..., x_{dim1}, u)
, where0 <= x_l < Ls[l]
pick the position of the unit cell in the lattice and0 <= u < len(unit_cell)
picks the site within the unit cell. The site is located in ‘space’ atsum_l x_l*basis[l] + unit_cell_positions[u]
(seeposition()
). (Note that the position in space is only used for plotting, not for defining the couplings.)In addition to the pure geometry, this class also defines an order of all sites. This order maps the lattice to a finite 1D chain and defines the geometry of MPSs and MPOs. The MPS index i corresponds thus to the lattice sites given by
(x_0, ..., x_{dim1}, u) = tuple(self.order[i])
. Infinite boundary conditions of the MPS repeat in the first spatial direction of the lattice, i.e., if the site at (x_0, x_1, …, x_{dim1},u)`` has MPS index i, the site at at(x_0 + a*Ls[0], x_1 ..., x_{dim1}, u)
corresponds to MPS indexi + N_sites
. Usemps2lat_idx()
andlat2mps_idx()
for conversion of indices. The functionmps2lat_values()
performs the necessary reshaping and reordering from arrays indexed in MPS form to arrays indexed in lattice form. Parameters
Ls (list of int) – the length in each direction
unit_cell (list of
Site
) – The sites making up a unit cell of the lattice. If you want to specify it only after initialization, useNone
entries in the list.order (str 
('standard', snake_winding, priority)
('grouped', groups)
) – A string or tuple specifying the order, given toordering()
.bc ((iterable of) {'open'  'periodic'  int}) – Boundary conditions in each direction of the lattice. A single string holds for all directions. An integer shift means that we have periodic boundary conditions along this direction, but shift/tilt by
shift*lattice.basis[0]
(~cylinder axis forbc_MPS='infinite'
) when going around the boundary along this direction.bc_MPS ('finite'  'segment'  'infinite') – Boundary conditions for an MPS/MPO living on the ordered lattice. If the system is
'infinite'
, the infinite direction is always along the first basis vector (justifying the definition of N_rings and N_sites_per_ring).basis (iterable of 1D arrays) – For each direction one translation vectors shifting the unit cell. Defaults to the standard ONB
np.eye(dim)
.positions (iterable of 1D arrays) – For each site of the unit cell the position within the unit cell. Defaults to
np.zeros((len(unit_cell), dim))
.nearest_neighbors (
None
 list of(u1, u2, dx)
) – Deprecated. Specify aspairs['nearest_neighbors']
instead.next_nearest_neighbors (
None
 list of(u1, u2, dx)
) – Deprecated. Specify aspairs['next_nearest_neighbors']
instead.next_next_nearest_neighbors (
None
 list of(u1, u2, dx)
) – Deprecated. Specify aspairs['next_next_nearest_neighbors']
instead.pairs (dict) – Of the form
{'nearest_neighbors': [(u1, u2, dx), ...], ...}
. Typical keys are'nearest_neighbors', 'next_nearest_neighbors'
. For each of them, it specifies a list of tuples(u1, u2, dx)
which can be used as parameters foradd_coupling()
to generate couplings over each pair of ,e.g.,'nearest_neighbors'
. Note that this adds couplings for each pair only in one direction!

order
¶  Type
ndarray (N_sites, dim+1)

boundary_conditions
¶

Ls
¶ the length in each direction.
 Type
tuple of int

shape
¶ the ‘shape’ of the lattice, same as
Ls + (len(unit_cell), )
 Type
tuple of int

N_sites_per_ring
¶ Defined as
N_sites / Ls[0]
, for an infinite system the number of cites per “ring”. Type

bc
¶ Boundary conditions of the couplings in each direction of the lattice, translated into a bool array with the global bc_choices.
 Type
bool ndarray

bc_shift
¶ The shift in xdirection when going around periodic boundaries in other directions.
 Type
None  ndarray(int)

bc_MPS
¶ Boundary conditions for an MPS/MPO living on the ordered lattice. If the system is
'infinite'
, the infinite direction is always along the first basis vector (justifying the definition of N_rings and N_sites_per_ring). Type
‘finite’  ‘segment’  ‘infinite’

basis
¶ translation vectors shifting the unit cell. The row i gives the vector shifting in direction i.
 Type
ndarray (dim, Dim)

unit_cell_positions
¶ for each site in the unit cell a vector giving its position within the unit cell.
 Type
ndarray, shape (len(unit_cell), Dim)

_strides
¶ necessary for
mps2lat_idx()
 Type
ndarray (dim, )

_perm
¶ permutation needed to make order lexsorted.
 Type
ndarray (N, )

_mps2lat_vals_idx
¶ index array for reshape/reordering in
mps2lat_vals()
 Type
ndarray shape

_mps_fix_u
¶ for each site of the unit cell an index array selecting the mps indices of that site.
 Type
tuple of ndarray (N_cells, ) np.intp

_mps2lat_vals_idx_fix_u
¶ similar as _mps2lat_vals_idx, but for a fixed u picking a site from the unit cell.
 Type
tuple of ndarray of shape Ls

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"
, andbc_MPS
as"order_for_MPS"
. Moreover, it savesdim
andN_sites
as HDF5 attributes.

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

property
dim
The dimension of the lattice.

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, …, 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)
 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.

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

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)

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,…).

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

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

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]
.

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 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…

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

possible_couplings
(u1, u2, dx)[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
u2 (u1,) – 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
mps1, mps2 (array) – For each possible twosite 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
u0 (int) – Argument u0 of
add_multi_coupling()
.other_us (list of int) – The u of the other_ops in
add_multi_coupling()
.dx (array, shape (len(other_us), lat.dim+1)) – The dx specifying relative operator positions of the other_ops in
add_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).
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.

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.

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.

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()
.

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

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 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.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
.