# SU(2) model on Kagome lattice¶

## Dense¶

This implementation of SU(2)-symmetric model on Kagome lattice does not assume any symmetry and hence works with dense tensors.

class models.spin_half_kagome.S_HALF_KAGOME(j1=1.0, JD=0, j1sq=0, j2=0, j2sq=0, jtrip=0.0, jperm=0j, h=0, phys_dim=2, global_args=<config.GLOBALARGS object>)[source]
Parameters
• j1 (float) – nearest-neighbour spin-spin interaction

• JD (float) – Dzyaloshinskii-Moriya interaction

• jtrip (float) – scalar chirality

• jperm (complex) – triangle exchange

• global_args (GLOBALARGS) – global configuration

Build spin-1/2 Hamiltonian on Kagome lattice

$\begin{split}H &= J_1 \sum_{<ij>} S_i.S_j + J_2 \sum_{<<ij>>} S_i.S_j - J_{trip} \sum_t (S_{t_1} \times S_{t_2}).S_{t_3} \\ &+ J_{perm} \sum_t P_t + J^*_{perm} \sum_t P^{-1}_t\end{split}$

where the first sum runs over the pairs of sites i,j which are nearest-neighbours (denoted as <.,.>), the second sum runs over pairs of sites i,j which are next nearest-neighbours (denoted as <<.,.>>). The $$J_{trip}$$ and $$J_{perm}$$ terms represent scalar chirality and triangle exchange respectively. The $$\sum_t$$ runs over all triangles. The sites $$t_1,t_2,t_3$$ on the triangles are always ordered anti-clockwise.

energy_triangle_dn(state, env, force_cpu=False, **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 contribution from down triangle within 2x2 subsystem embedded in environment, see ctm.pess_kagome.rdm_kagome.rdm2x2_dn_triangle_with_operator().

energy_triangle_dn_1x1(state, env, force_cpu=False, **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 contribution from down triangle within 1x1 subsystem embedded in environment, see ctm.pess_kagome.rdm_kagome.rdm1x1_kagome().

energy_triangle_up(state, env, force_cpu=False, **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 contribution from up triangle within 2x2 subsystem embedded in environment, see ctm.pess_kagome.rdm_kagome.rdm2x2_up_triangle_open().

eval_corrf_SS(coord, direction, state, env, dist, site=0)[source]
Parameters
• coord (tuple(int,int)) – reference site

• direction (tuple(int,int)) –

• state (IPEPS_KAGOME) – wavefunction

• env (ENV) – CTM environment

• dist (int) – maximal distance of correlator

• site (int) – selects one of the non-equivalent physical degrees of freedom within the unit cell

Returns

dictionary with full and spin-resolved spin-spin correlation functions

Return type

dict(str: torch.Tensor)

Evaluate spin-spin correlation functions $$\langle\mathbf{S}(r).\mathbf{S}(0)\rangle$$ up to r = dist in given direction. See ctm.generic.corrf.corrf_1sO1sO().

The on-site tensor of the underlying IPEPS_KAGOME contains all three DoFs of the down triangle. Choosing site selects one of them to be used for evaluating correlation function:

   a
|
b--\
\
s0--s2--d
| /
|/   <- down triangle
s1
|
c

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

• env (ENV) – CTM environment

• force_cpu (bool) – perform computation on CPU

• cast_real (bool) – if False keep imaginary part of energy contributions from up and down triangles

• disp_corre_len (bool) – compute correlation lengths from transfer matrices

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 triangles

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

• nearest-neighbour spin-spin correlations for all bonds in the unit cell

• (optionally) correlation lengths

## With explict U(1) symmetry¶

This implementation of SU(2)-symmetric model on Kagome lattice assumes explicit U(1) abelian symmetry (subgroup) of iPEPS tensors.