Spin S=2 AKLT Model¶
2x1 and 2x2 unit cell¶
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class
models.akltS2.
AKLTS2
(global_args=<config.GLOBALARGS object>)[source]¶ -
energy_2x1_1x2
(state, env)[source]¶ - Parameters
- 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
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1x1 C4v¶
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class
models.akltS2.
AKLTS2_C4V_BIPARTITE
(global_args=<config.GLOBALARGS object>)[source]¶ -
eval_obs
(state, env_c4v)[source]¶ - Parameters
- Returns
expectation values of observables, labels of observables
- Return type
list[float], list[str]
Computes the following observables in order
magnetization
\(\langle S^z \rangle,\ \langle S^+ \rangle,\ \langle S^- \rangle\)
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
\[\begin{split}\begin{align*} S^+ &=S^x+iS^y & S^x &= 1/2(S^+ + S^-)\\ S^- &=S^x-iS^y\ \Rightarrow\ & S^y &=-i/2(S^+ - S^-) \end{align*}\end{split}\]
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