Spin S=2 AKLT Model

2x1 and 2x2 unit cell

class models.akltS2.AKLTS2(global_args=<config.GLOBALARGS object>)[source]
energy_2x1_1x2(state, env)[source]
Parameters
  • state (IPEPS) – wavefunction

  • env (ENV) – CTM environment

Returns

energy per site

Return type

float

We assume iPEPS with 2x1 unit cell with tensors A, B and bipartite tiling or 2x2 unit cell containing four tensors A, B, C, and D with a simple PBC tiling:

A B A B  or  A B A B
B A B A      C D C D
A B A B      A B A B
B A B A      C D C D

Taking the reduced density matrix \(\rho_{2x1}\) (\(\rho_{1x2}\)) of 2x1 (1x2) cluster given by rdm.rdm2x1() (rdm.rdm1x2()) with indexing of sites as follows \(s_0,s_1;s'_0,s'_1\) for both types of density matrices:

rdm2x1   rdm1x2

s0--s1   s0
         |
         s1

and without assuming any symmetry on the indices of individual tensors a following set of terms has to be evaluated in order to compute energy-per-site for the case of 2x1 unit cell with bipartite tiling:

   0       0
1--A--3 1--B--3
   2       2                              A  B
   0       0                              2  2  
1--B--3 1--A--3           A--3 1--B,      0  0 
   2       2      , terms B--3 1--A, and  B, A

and for the case of 2x2 unit cell:

   0       0       0
1--A--3 1--B--3 1--A--3
   2       2       2
   0       0       0
1--C--3 1--D--3 1--C--3
   2       2       2             A--3 1--B,      A  B  C  D
   0       0                     B--3 1--A,      2  2  2  2
1--A--3 1--B--3                  C--3 1--D,      0  0  0  0
   2       2             , terms D--3 1--C, and  C, D, A, B

1x1 C4v

class models.akltS2.AKLTS2_C4V_BIPARTITE(global_args=<config.GLOBALARGS object>)[source]
eval_obs(state, env_c4v)[source]
Parameters
  • state (IPEPS_C4V) – wavefunction

  • env_c4v (ENV_C4V) – CTM c4v symmetric environment

Returns

expectation values of observables, labels of observables

Return type

list[float], list[str]

Computes the following observables in order

  1. magnetization

  2. \(\langle S^z \rangle,\ \langle S^+ \rangle,\ \langle S^- \rangle\)

where the on-site magnetization is defined as

\[\begin{split}\begin{align*} m &= \sqrt{ \langle S^z \rangle^2+\langle S^x \rangle^2+\langle S^y \rangle^2 } =\sqrt{\langle S^z \rangle^2+1/4(\langle S^+ \rangle+\langle S^- \rangle)^2 -1/4(\langle S^+\rangle-\langle S^-\rangle)^2} \\ &=\sqrt{\langle S^z \rangle^2 + 1/2\langle S^+ \rangle \langle S^- \rangle)} \end{align*}\end{split}\]

Usual spin components can be obtained through the following relations

\[\begin{split}\begin{align*} S^+ &=S^x+iS^y & S^x &= 1/2(S^+ + S^-)\\ S^- &=S^x-iS^y\ \Rightarrow\ & S^y &=-i/2(S^+ - S^-) \end{align*}\end{split}\]