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 \(s_is_js_ks_l;s'_is'_js'_ks'_l\)
\(h1_i = 2S^x_i\)
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energy_1x1(state, env)[source]¶ - Parameters
- 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 \(\langle \mathcal{O} \rangle = Tr(\rho_{2x2} \mathcal{O})\)
\[e = -(\langle h2_{<\bf{0},\bf{x}>} \rangle + \langle h2_{<\bf{0},\bf{y}>} \rangle) + q\langle h4_{\bf{0}} \rangle - h_x \langle h4_{\bf{0}} \rangle\]
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 \(s_is_js_ks_l;s'_is'_js'_ks'_l\)
\(h1_i = 2S^x_i\)
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energy_1x1_nn(state, env_c4v)[source]¶ - Parameters
- 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 \(\langle \mathcal{O} \rangle = Tr(\rho_{2x1} \mathcal{O})\)
\[e = -\langle h2_{<\bf{0},\bf{x}>} \rangle - h_x \langle h1_{\bf{0}} \rangle\]
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energy_1x1_plaqette(state, env_c4v)[source]¶ - Parameters
- 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 \(\langle \mathcal{O} \rangle = Tr(\rho_{2x2} \mathcal{O})\)
\[e = -(\langle h2_{<\bf{0},\bf{x}>} \rangle + \langle h2_{<\bf{0},\bf{y}>} \rangle) + q\langle h4_{\bf{0}} \rangle - h_x \langle h4_{\bf{0}} \rangle\]