# Triangular¶

Inheritance Diagram Methods

 `Triangular.__init__`(Lx, Ly, site, **kwargs) Shallow copy of self. `Triangular.count_neighbors`([u, key]) Count e.g. Calculate correct shape of the strengths for a coupling. `Triangular.distance`(u1, u2, dx) Get the distance for a given coupling between two sites in the lattice. Repeat the unit cell for infinite MPS boundary conditions; in place. `Triangular.extract_segment`([first, last, ...]) Extract a finite segment from an infinite/large system. `Triangular.find_coupling_pairs`([max_dx, ...]) Automatically find coupling pairs grouped by distances. `Triangular.from_hdf5`(hdf5_loader, h5gr, subpath) Load instance from a HDF5 file. `Triangular.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)`. `Triangular.mps2lat_values`(A[, axes, u]) same as `Lattice.mps2lat_values()`, but ignore `u`, setting it to `0`. `Triangular.mps2lat_values_masked`(A[, axes, ...]) 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 `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. `Triangular.plot_basis`(ax[, origin, shade]) Plot arrows indicating the basis vectors of the lattice. `Triangular.plot_bc_identified`(ax[, ...]) Mark two sites indified by periodic boundary conditions. `Triangular.plot_coupling`(ax[, coupling, wrap]) Plot lines connecting nearest neighbors of the lattice. `Triangular.plot_order`(ax[, order, textkwargs]) Plot a line connecting sites in the specified "order" and text labels enumerating them. `Triangular.plot_sites`(ax[, markers]) Plot the sites of the lattice with markers. `Triangular.position`(lat_idx) return 'space' position of one or multiple sites. `Triangular.possible_couplings`(u1, u2, dx[, ...]) Find possible MPS indices for two-site couplings. `Triangular.possible_multi_couplings`(ops[, ...]) Generalization of `possible_couplings()` to couplings with more than 2 sites. `Triangular.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

 `Triangular.Lu` the (expected) number of sites in the unit cell, `len(unit_cell)`. `Triangular.boundary_conditions` Human-readable list of boundary conditions from `bc` and `bc_shift`. `Triangular.dim` the dimension of the lattice `Triangular.nearest_neighbors` `Triangular.next_nearest_neighbors` `Triangular.next_next_nearest_neighbors` `Triangular.order` Defines an ordering of the lattice sites, thus mapping the lattice to a 1D chain.
class tenpy.models.lattice.Triangular(Lx, Ly, site, **kwargs)[source]

A triangular lattice.

Parameters
• Lx (int) – The length in each direction.

• Ly (int) – The length in each direction.

• site (`Site`) – The local lattice site. The unit_cell of the `Lattice` is just `[site]`.

• **kwargs – Additional keyword arguments given to the `Lattice`. pairs are set accordingly. If order is specified in the form `('standard', snake_windingi, priority)`, the snake_winding and priority should only be specified for the spatial directions. Similarly, positions can be specified as a single vector.

dim = 2

the dimension of the lattice

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

copy()[source]

Shallow copy of self.

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. 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 the `basis` and `unit_cell_positions` of the lattice).

Warning

This function ignores “wrapping” arround the cylinder in the case of periodic boundary conditions.

Parameters
Returns

distance – The distance between site at lattice indices `[x, y, u1]` and `[x + dx, y + dx, u2]`, ignoring any boundary effects.

Return type

float

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 x-direction in our convetion, the lattice shape goes from `(Lx, Ly, ..., Lu)` to `(Lx*factor, Ly, ..., Lu)`.

extract_segment(first=0, last=None, enlarge=None)[source]

Extract a finite segment from an infinite/large system.

Parameters
• first (int) – The first and last site to include into the segment. last defaults to `L` - 1, i.e., the MPS unit cell for infinite MPS.

• last (int) – The first and last site to include into the segment. last defaults to `L` - 1, i.e., the MPS unit cell for infinite MPS.

• enlarge (int) – Instead of specifying the first and last site, you can specify this factor by how much the MPS unit cell should be enlarged.

Returns

copy – A copy of self with “segment” `bc_MPS` and `segment_first_last` set.

Return type

`Lattice`

find_coupling_pairs(max_dx=3, cutoff=None, eps=1e-10)[source]

Automatically find coupling pairs grouped by distances.

Given the `unit_cell_positions` and `basis`, the coupling `pairs` 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 double-counting couplings in both directions `A_i B_j + B_i A_i`). This function iterates through all possible couplings up to a given cutoff distance and then determines the possible `pairs` at fixed distances (up to round-off 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` and `unit_cell_positions` is given).

• eps (float) – Tolerance up to which to distances are considered the same.

Returns

coupling_pairs – Keys are distances of nearest-neighbors, next-nearest-neighbors etc. Values are `[(u1, u2, dx), ...]` as in `pairs`.

Return type

dict

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” or “segment” 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” or “segment” bc_MPS, we shift x_0 accordingly.

Return type

array

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

same as `Lattice.mps2lat_values()`, but ignore `u`, setting it to `0`.

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

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 (2D array, shape (N_ops, `dim`)) – `dx[i, :]` is the translation vector in the lattice for the i-th operator. Corresponds to the dx of each operator given in the argument ops 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. (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, …, 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)

Note

For lattices with a non-trivial 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'` 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

`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 arrows indicating the basis vectors of the lattice.

Parameters
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 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.

• 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`), 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, i1+dx, ..., 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. If `False`, 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`) 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, strength=None)[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
• 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 two-site 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 where `strength_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 of `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).

• 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 where `strength_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"`, and `order` as `"order_for_MPS"`. Moreover, it saves `dim` and `N_sites` as HDF5 attributes.

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

• h5gr (:class`Group`) – 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.