# Transverse Field Ising Model¶

## 1x1 unit cell¶

class models.ising.ISING(hx=0.0, q=0.0, global_args=<config.GLOBALARGS object>)[source]
Parameters
• hx (float) – transverse field

• q (float) – plaquette interaction

• global_args (GLOBALARGS) – global configuration

Build Ising Hamiltonian in transverse field with plaquette interaction

$H = - \sum_{<i,j>} h2_{<i,j>} + q\sum_{p} h4_p - h_x\sum_i h1_i$

on the square lattice. Where the first sum runs over the pairs of sites i,j which are nearest-neighbours (denoted as <.,.>), the second sum runs over all plaquettes p, and the last sum runs over all sites:

y\x
_:__:__:__:_
..._|__|__|__|_...
..._|__|__|__|_...
..._|__|__|__|_...
..._|__|__|__|_...
..._|__|__|__|_...
:  :  :  :


where

• $$h2_{ij} = 4S^z_i S^z_j$$ with indices of h2 corresponding to $$s_i s_j;s'_i s'_j$$

• $$h4_p = 16S^z_i S^z_j S^z_k S^z_l$$ where i,j,k,l labels the sites of a plaquette:

p= i---j
|   |
k---l


and the indices of h4 correspond to $$s_is_js_ks_l;s'_is'_js'_ks'_l$$

• $$h1_i = 2S^x_i$$

energy_1x1(state, env)[source]
Parameters
• state (IPEPS) – wavefunction

• env (ENV) – CTM environment

Returns

energy per site

Return type

float

For 1-site invariant iPEPS it’s enough to construct a single reduced density matrix of a 2x2 plaquette. Afterwards, the energy per site e is computed by evaluating individual terms in the Hamiltonian through $$\langle \mathcal{O} \rangle = Tr(\rho_{2x2} \mathcal{O})$$

$e = -(\langle h2_{<\bf{0},\bf{x}>} \rangle + \langle h2_{<\bf{0},\bf{y}>} \rangle) + q\langle h4_{\bf{0}} \rangle - h_x \langle h4_{\bf{0}} \rangle$
eval_obs(state, env)[source]
Parameters
• state (IPEPS) – wavefunction

• env (ENV) – CTM environment

Returns

expectation values of observables, labels of observables

Return type

list[float], list[str]

Computes the following observables in order

1. $$\langle 2S^z \rangle,\ \langle 2S^x \rangle$$ for each site in the unit cell

## 1x1 C4v¶

class models.ising.ISING_C4V(hx=0.0, q=0, global_args=<config.GLOBALARGS object>)[source]
Parameters
• hx (float) – transverse field

• q (float) – plaquette interaction

• global_args (GLOBALARGS) – global configuration

Build Ising Hamiltonian in transverse field with plaquette interaction

$H = - \sum_{<i,j>} h2_{<i,j>} + q\sum_{p} h4_p - h_x\sum_i h1_i$

on the square lattice. Where the first sum runs over the pairs of sites i,j which are nearest-neighbours (denoted as <.,.>), the second sum runs over all plaquettes p, and the last sum runs over all sites:

y\x
_:__:__:__:_
..._|__|__|__|_...
..._|__|__|__|_...
..._|__|__|__|_...
..._|__|__|__|_...
..._|__|__|__|_...
:  :  :  :


where

• $$h2_{ij} = 4S^z_i S^z_j$$ with indices of h2 corresponding to $$s_i s_j;s'_i s'_j$$

• $$h4_p = 16S^z_i S^z_j S^z_k S^z_l$$ where i,j,k,l labels the sites of a plaquette:

p= i---j
|   |
k---l


and the indices of h4 correspond to $$s_is_js_ks_l;s'_is'_js'_ks'_l$$

• $$h1_i = 2S^x_i$$

energy_1x1_nn(state, env_c4v)[source]
Parameters
• state (IPEPS) – wavefunction

• env_c4v (ENV_C4V) – CTM c4v symmetric environment

Returns

energy per site

Return type

float

For 1-site invariant c4v iPEPS with no 4-site term present in Hamiltonian it is enough to construct a single reduced density matrix of a 2x1 nearest-neighbour sites. Afterwards, the energy per site e is computed by evaluating individual terms in the Hamiltonian through $$\langle \mathcal{O} \rangle = Tr(\rho_{2x1} \mathcal{O})$$

$e = -\langle h2_{<\bf{0},\bf{x}>} \rangle - h_x \langle h1_{\bf{0}} \rangle$
energy_1x1_plaqette(state, env_c4v)[source]
Parameters
• state (IPEPS) – wavefunction

• env_c4v (ENV_C4V) – CTM c4v symmetric environment

Returns

energy per site

Return type

float

For 1-site invariant c4v iPEPS it’s enough to construct a single reduced density matrix of a 2x2 plaquette. Afterwards, the energy per site e is computed by evaluating individual terms in the Hamiltonian through $$\langle \mathcal{O} \rangle = Tr(\rho_{2x2} \mathcal{O})$$

$e = -(\langle h2_{<\bf{0},\bf{x}>} \rangle + \langle h2_{<\bf{0},\bf{y}>} \rangle) + q\langle h4_{\bf{0}} \rangle - h_x \langle h4_{\bf{0}} \rangle$
eval_obs(state, env_c4v)[source]
Parameters
• state (IPEPS) – 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. $$\langle 2S^z \rangle,\ \langle 2S^x \rangle$$ for each site in the unit cell

TODO 2site observable SzSz