Lattice Geometry#

Geometric information about 2D lattice is captured by a class yastn.tn.fpeps.SquareLattice or its special subclasses yastn.tn.fpeps.RectangularUnitcell, yastn.tn.fpeps.CheckerboardLattice, and yastn.tn.fpeps.TriangularLattice. They all operate on a square lattice, where yastn.tn.fpeps.SquareLattice can be used for a finite system or an infinite system with a rectangular unit cell. yastn.tn.fpeps.RectangularUnitcell works for an infinite system with a pattern of repeating tensors inside the unit cell. yastn.tn.fpeps.CheckerboardLattice is a special case of the latter, with a \(2 \times 2\) unit cell and checkerboard pattern, and yastn.tn.fpeps.TriangularLattice is a special case with a \(3 \times 3\) unit cell.

The geometry classes provide information on lattice sites (in particular, unique sites in the unit cell), unique bonds, and a way to navigate the lattice through the information on the neighborhood of each site.

Basic Elements#

Auxiliary objects in lattice definition are yastn.tn.fpeps.Bond representing a pair of nearest-neighbor lattice sites, and yastn.tn.fpeps.Site for sites.

class yastn.tn.fpeps.Bond(site0: Site = None, site1: Site = None)[source]#

A bond between two lattice sites.

Create new instance of Bond(site0, site1)

class yastn.tn.fpeps.Site(nx: int = 0, ny: int = 0)[source]#

Site coordinates (nx, ny) are consistent with matrix indexing with (row, column):

┌───── y(cols) ──────ᐳ
│
x(rows)  (0,0) (0,1) ...
│        (1,0) (1,1) ...
│         ...
ᐯ

Create new instance of Site(nx, ny)

Core Classes#

class yastn.tn.fpeps.SquareLattice(dims=(2, 2), boundary='infinite')[source]#

Geometric information about 2D square lattice.

Parameters:

  • dims (tuple[int, int]) – Size of the unit cell in a form of dims=(rows, columns). Site(0, 0) corresponds to top-left corner of the unit cell.

  • boundary (str) –

    Type of boundary conditions:

    • ‘infinite’ (the default) for an infinite lattice,

    • ‘obc’ for a finite lattice, or

    • ‘cylinder’ for a finite cylinder periodic along rows, i.e.:

      ┌───── y(cols) ──────ᐳ
│
│        (0, 0) (0, 1) ... (0, Ny-1)
x(rows)  (1, 0) (1, 1) ... (1, Ny-1)
│         ...
│        (Nx-1, 0)   ...   (Nx-1, Ny-1)
ᐯ        (0, 0)      ...   (0, Ny-1)
bonds(dirn=None, reverse=False) → Sequence[Bond][source]#

Sequence of unique nearest neighbor bonds between lattice sites.

Parameters:

  • dirn (None | str) – return horizontal followed by vertical bonds if None; ‘v’ and ‘h’ are, respectively, for vertical and horizontal bonds only.

  • reverse (bool) – whether to reverse the order of bonds.

nn_site(site, d) → Site | None[source]#

Index of the lattice site neighboring the site in the direction d.

For infinite lattices, this function simply shifts the site by provided vector d. For finite lattices with open/periodic boundary it handles corner cases where d is too large and the resulting site either doesn’t exist or it wraps around periodic boundary.

Return None if there is no neighboring site in a given direction.

Parameters:

d (str | tuple[int, int]) – Take values in: ‘t’, ‘b’, ‘l’, ‘r’, ‘tl’, ‘bl’, ‘tr’, ‘br’, or a tuple of shifts (dx, dy).

sites(reverse=False) → Sequence[Site][source]#

Sequence of unique lattice sites.

class yastn.tn.fpeps.CheckerboardLattice[source]#

Geometric information about infinite checkerboard lattice, which is an infinite lattice with \(2{\times}2\) unit cell and two unique tensors.

class yastn.tn.fpeps.RectangularUnitcell(pattern, **kwargs)[source]#

Rectangular unit cells supporting patterns characterized by a single momentum Q=(q_x, q_y).

Inspired by b1592/ad-peps by B. Ponsioen.

Parameters:

pattern (Sequence[Sequence[int]] | dict[tuple[int, int], int]) – Definition of a rectangular unit cell that tiles the square lattice. Integers are labels of unique tensors populating the sites within the unit cell.

Examples of such patterns can be:

  • [[0,],] : 1x1 unit cell, Q=0

  • {(0, 0): 0} : 1x1 unit cell, Q=0

  • [[0, 1],] : 1x2 unit cell, Q=(0, pi)

  • {(0, 0): 0, (0, 1): 1} : 1x2 unit cell, Q=(0, pi)

  • [[0, 1], [1, 0]] : 2x2 unit cell with bipartite pattern, Q=(pi, pi). Equivalent to yastn.tn.fpeps.CheckerboardLattice.

  • [[0, 1, 2], [1, 2, 0], [2, 0, 1]] : 3x3 unit cell with diagonal stripe order, Q=(2pi/3, 2pi/3)

Warning

It is assumed that the neighborhood of each unique tensor is identical. This excludes cases such as [[0, 1], [1, 1]].

class yastn.tn.fpeps.TriangularLattice(**kwargs)[source]#

Geometric information about infinite triangular lattice, which is an infinite lattice with \(3{\times}3\) unit cell and three unique tensors.