CornerTransferMethods.jl

requires TensorNetworkTensors.jl

Basics

This package implements Corner-Transfer-Algorithms (CTM) for infinite Projected Pair Entangled States (iPEPS).

The input of the algorithm is a rank-4 tensor A_ijkl that could represent a spin in a classical 2d partition function or the norm of a iPEPS. Graphically that would be represented as

    k
    |
l--[A]--j
    |
    i

The problem we're solving is this: Given a tensor A which is used to build an infinite grid, as in

   |    |    |    |    |    |    |    |    |
--[A]--[A]--[A]--[A]--[A]--[A]--[A]--[A]--[A]--
   |    |    |    |    |    |    |    |    |
--[A]--[A]--[A]--[A]--[A]--[A]--[A]--[A]--[A]--
   |    |    |    |    |    |    |    |    |
--[A]--[A]--[A]--[A]--[A]--[A]--[A]--[A]--[A]--
   |    |    |    |    |    |    |    |    |
--[A]--[A]--[A]--[A]--[A]--[A]--[A]--[A]--[A]--
   |    |    |    |    |    |    |    |    |
--[A]--[A]--[A]--[A]--[A]--[A]--[A]--[A]--[A]--
   |    |    |    |    |    |    |    |    |

How can we construct the effective environment of a particular tensor?

In CTM-methods, we choose an ansatz for the environment consisting of tensors C_ and T_ that are connected with legs of dimension χ (== or || below)

[C1]==[T1]==[C2]
 ||     |    ||
[T4]--[ A]--[T2]
 ||     |    ||
[C4]==[T3]==[C3]

and iteratively improve the tensors C_ and T_ which represent infinite corners and half-infinite columns/rows respectively as e.g.

   ||       |    |    |    |    |    |    |
--[T2] = --[A]--[A]--[A]--[A]--[A]--[A]--[A]-⋯
   ||       |    |    |    |    |    |    |

until they are represent the best approximation of the environment given the ansatz-bond dimension χ.

Notation

The bulk-tensors A_ijkl are always oriented as in

    k
    |
l--[A]--j
    |
    i

i.e. the first index points down and the subsequent indices are counter-clockwise around the tensor.

Corner tensors C1_ij, C2_ij, C3_ij and C4_ij are oriented as

[C1]==j   i==[C2]        i     j
 ||           ||        ||     ||
 i             j    j==[C3]   [C4]==i

i.e. moving clockwise in the environment, the first index encountered is the first index of the corner.

For the row- and column-tensors T1_ijk,T2_ijk,T3_ijk,T4_ijk, the indices are oriented as

i==[T1]==k       i         j         k
    ||          ||        ||         ||
    j       j==[T2]   k==[T3]==i    [T4]==j
                ||                   ||
                 k                   i

i.e. the again following the sequence when traversing the environment clockwise.

Algorithms

Depending on the symmetry of A, there are different algorithms implemented:

No Symmetry

The algorithm chosen for this situation is the directional variant of CTMRG, as described in Simulation of two-dimensional quantum…, page 2f.

Rotational Symmetry

If A_ijkl = A_jkli, i.e. A is rotationally symmetric, we can restrit the environment ansatz to

[ C]==[ T]==[C*]
 ||     |     ||
[ T]--[ A]--[T*]
 ||     |     ||
[C*]==[T*]==[ C]

Updating is then done using and enforcing this symmetry, including on the tensor-level where C_ij ≡ C_ji and T_ijk ≡ T_kji.

The algorithm chosen here is

Hermitian w.r.t left/right and up/down

If A_ijkl ≡ A*_kjil ≡ A*_ilkj, i.e. A is Hermitian w.r.t to both the left-right and up-down indices, we can restrict the environment ansatz to

[ C]==[T1 ]==[ C*]
 ||     |     ||
[T2]--[ A ]--[T2*]
 ||     |     ||
[C*]==[T1*]==[ C ]

The algorithm chosen for this situation is the simplified one-directional 1D method, as described in Exploring corner transfer matrices…, page 9ff.

This can be used for e.g. imaginary time evolution to find ground states of 1D systems if you can provide an MPO-Tensor A that satisfies the restrictions above.