# SU(3) model on Kagome lattice¶

## Dense¶

This implementation of SU(3)-symmetric model on Kagome lattice does not assume any symmetry and hence works with dense tensors.

class models.su3_kagome.KAGOME_SU3(phys_dim=3, j=0.0, k=1.0, h=0.0, global_args=<config.GLOBALARGS object>)[source]
Parameters
• j (int) – nearest-neighbour pairing

• k (float) – real part of triangle exchange

• h (float) – imaginary part of triangle exchange

• global_args (GLOBALARGS) – global configuration

The SU(3) Hamiltonian on Kagome lattice

$H = J \sum_{\langle ij \rangle} P_{ij} + K \sum_{t_{up},t_{down}} (P_{ijk} + P^{-1}_{ijk}) + ih \sum_{t_{up},t_{down}} (P_{ijk} - P^{-1}_{ijk})$

or in parametrization through angles

$H = cos \phi \sum_{\langle ij \rangle} P_{ij} + sin \phi \sum_{t_{up},t_{down}} exp(i\theta) P_{ijk} + exp(-i\theta)P^{-1}_{ijk}$

where $$J = cos \phi,\ K = sin \phi cos \theta,\ h= sin \phi sin \theta$$. The $$\phi= 0.5\pi$$ and $$\theta=0$$ corresponds to the AKLT point.

energy_1site(state, env, **kwargs)[source]
Parameters
• state (IPEPS_KAGOME) – wavefunction

• env (ENV) – CTM environment

• force_cpu (bool) – perform computation on CPU

Returns

energy per site

Return type

float

Evaluate energy per site from contributions from down triangle within 1x1 subsystem embedded in environment, and from upper triangle embedded in 2x2 environment. See ctm.pess_kagome.rdm_kagome.trace1x1_dn_kagome() and ctm.pess_kagome.rdm_kagome.rdm2x2_kagome() respectively.

energy_down_t_1x1subsystem(state, env, force_cpu=False, fail_on_check=False, warn_on_check=True)[source]
Parameters
• state (IPEPS_KAGOME) – wavefunction

• env (ENV) – CTM environment

• force_cpu (bool) – perform computation on CPU

Returns

energy per site

Return type

float

Evaluate energy contribution from down triangle within 1x1 subsystem embedded in environment, see ctm.pess_kagome.rdm_kagome.trace1x1_dn_kagome().

energy_per_site_2x2subsystem(state, env, force_cpu=False)[source]
Parameters
• state (IPEPS_KAGOME) – wavefunction

• env (ENV) – CTM environment

• force_cpu (bool) – perform computation on CPU

Returns

energy per site

Return type

float

Evaluate energy per site from contributions from up and down triangle. See energy_triangles_2x2subsystem().

energy_triangles_2x2subsystem(state, env, force_cpu=False)[source]
Parameters
• state (IPEPS_KAGOME) – wavefunction

• env (ENV) – CTM environment

• force_cpu (bool) – perform computation on CPU

Returns

energy contributions from down and up triangle

Return type

float, float

Evaluate energy contributions from down triangle within 2x2 subsystem embedded in environment, and from upper triangle embedded in 2x2 environment. See ctm.pess_kagome.rdm_kagome.rdm2x2_dn_triangle_with_operator() and ctm.pess_kagome.rdm_kagome.rdm2x2_kagome() respectively.

eval_obs(state, env, force_cpu=False, **kwargs)[source]
Parameters
• state (IPEPS_KAGOME) – wavefunction

• env (ENV) – CTM environment

• force_cpu (bool) – perform computation on CPU

Returns

expectation values of observables, labels of observables

Return type

list[float], list[str]

Evaluate observables for IPESS_KAGOME wavefunction. In particular

• average nearest-neighbour pairing on up and down triangles

• chiralities on up and down triangles

• vector of spontaneous magnetization $$\langle \vec{S} \rangle$$ for each site and its length $$m=|\langle \vec{S} \rangle|$$

The observables on down triangle are evaluated on 1x1 subsystem, while observables on up triangle are evaluated on 2x2 subsystem.

eval_obs_2x2subsystem(state, env, force_cpu=False, **kwargs)[source]
Parameters
• state (IPEPS_KAGOME) – wavefunction

• env (ENV) – CTM environment

• force_cpu (bool) – perform computation on CPU

Returns

expectation values of observables, labels of observables

Return type

list[float], list[str]

Evaluate observables for IPESS_KAGOME wavefunction. In particular

• energy contributions from up and down triangle

• average nearest-neighbour pairing on up and down triangles

• chiralities on up and down triangles

• vector of spontaneous magnetization $$\langle \vec{S} \rangle$$ for each site and its length $$m=|\langle \vec{S} \rangle|$$

The observables on both down triangle and up triangle are evaluated on 2x2 subsystem.

## With explict U(1)xU(1) symmetry¶

This implementation of SU(3)-symmetric model on Kagome lattice assumes explicit U(1)xU(1) abelian symmetry (subgroup) of iPEPS tensors.