J-Q Model¶
2x2 unit cell¶
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class
models.jq.JQ(j1=0.0, q=1.0, global_args=<config.GLOBALARGS object>)[source]¶ - Parameters
j1 (float) – nearest-neighbour interaction
q (float) – ring-exchange interaction
global_args (GLOBALARGS) – global configuration
Build Spin-1/2 \(J-Q\) Hamiltonian
\[H = J_1\sum_{<i,j>} h2_{ij} - Q\sum_p h4_p \]on the square lattice. Where the first sum runs over the pairs of sites i,j which are nearest-neighbours (denoted as <.,.>), and the second sum runs over all plaquettes p:
y\x _:__:__:__:_ ..._|__|__|__|_... ..._|__|__|__|_... ..._|__|__|__|_... ..._|__|__|__|_... ..._|__|__|__|_... : : : :
where
\(h2_{ij} = \mathbf{S}_i.\mathbf{S}_j\) with indices of h2 corresponding to \(s_i s_j;s'_i s'_j\)
\(h4_p = (\mathbf{S}_i.\mathbf{S}_j-1/4)(\mathbf{S}_k.\mathbf{S}_l-1/4) + (\mathbf{S}_i.\mathbf{S}_k-1/4)(\mathbf{S}_j.\mathbf{S}_l-1/4)\) where i,j,k,l labels the sites of a plaquette. Hence the Q term in the Hamiltian correspond to the following action over plaquette:
{ij,kl} and {ik,jl} (double lines denote the (S.S-1/4) terms) i===j i---j | | || || k===l + k---l
and the indices of h4 correspond to \(s_is_js_ks_l;s'_is'_js'_ks'_l\)
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energy_2x2_4site(state, env)[source]¶ - Parameters
- Returns
energy per site
- Return type
float
We assume iPEPS with 2x2 unit cell containing four tensors A, B, C, and D with simple PBC tiling:
A B A B C D C D A B A B C D C D
Taking the reduced density matrix \(\rho_{2x2}\) of 2x2 cluster given by
ctm.generic.rdm.rdm2x2()with indexing of sites as follows \(\rho_{2x2}(s_0,s_1,s_2,s_3;s'_0,s'_1,s'_2,s'_3)\):s0--s1 | | s2--s3
and without assuming any symmetry on the indices of individual tensors a set of four \(\rho_{2x2}\)’s are needed over which \(h2\) and \(h4\) operators are evaluated:
A3--1B B3--1A C3--1D D3--1C 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 C3--1D & D3--1C & A3--1B & B3--1A
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eval_corrf_DD_V(coord, direction, state, env, dist, verbosity=0)[source]¶ Evaluates correlation functions of two vertical dimers DD_v(r)= <(S(0).S(y))(S(r*x).S(y+r*x))>
or= <(S(0).S(x))(S(r*y).S(x+r*y))>
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eval_obs(state, env)[source]¶ - Parameters
- Returns
expectation values of observables, labels of observables
- Return type
list[float], list[str]
Computes the following observables in order
average magnetization over the unit cell,
magnetization for each site in the unit cell
\(\langle S^z \rangle,\ \langle S^+ \rangle,\ \langle S^- \rangle\) for each site in the unit cell
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
\[\begin{split}\begin{align*} S^+ &=S^x+iS^y & S^x &= 1/2(S^+ + S^-)\\ S^- &=S^x-iS^y\ \Rightarrow\ & S^y &=-i/2(S^+ - S^-) \end{align*}\end{split}\]
1x1 C4v¶
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class
models.jq.JQ_C4V(j1=0.0, q=1.0, global_args=<config.GLOBALARGS object>)[source]¶ - Parameters
j1 (float) – nearest-neighbour interaction
q (float) – ring-exchange interaction
global_args (GLOBALARGS) – global configuration
Build Spin-1/2 \(J-Q\) Hamiltonian
\[H = J_1\sum_{<i,j>} h2_{ij} - Q\sum_p h4_p \]on the square lattice. Where the first sum runs over the pairs of sites i,j which are nearest-neighbours (denoted as <.,.>), and the second sum runs over all plaquettes p:
y\x _:__:__:__:_ ..._|__|__|__|_... ..._|__|__|__|_... ..._|__|__|__|_... ..._|__|__|__|_... ..._|__|__|__|_... : : : :
where
\(h2_{ij} = \mathbf{S}_i.\mathbf{S}_j\) with indices of h2 corresponding to \(s_i s_j;s'_i s'_j\)
\(h4_p = (\mathbf{S}_i.\mathbf{S}_j-1/4)(\mathbf{S}_k.\mathbf{S}_l-1/4) + (\mathbf{S}_i.\mathbf{S}_k-1/4)(\mathbf{S}_j.\mathbf{S}_l-1/4)\) where i,j,k,l labels the sites of a plaquette. Hence the Q term in the Hamiltian correspond to the following action over plaquette:
{ij,kl} and {ik,jl} (double lines denote the (S.S-1/4) terms) i===j i---j | | || || k===l + k---l
and the indices of h4 correspond to \(s_is_js_ks_l;s'_is'_js'_ks'_l\)
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energy_1x1(state, env_c4v)[source]¶ - Parameters
- Returns
energy per site
- Return type
float
We assume 1x1 C4v iPEPS which tiles the lattice with tensor A on every site:
1x1 C4v A A A A A A A A A A A A A A A A
Due to C4v symmetry it is enough to construct a single reduced density matrix
ctm.one_site_c4v.rdm_c4v.rdm2x2()of a 2x2 plaquette. Afterwards, the energy per site e is computed by evaluating a single plaquette term \(h_p\) containing two nearest-neighbour terms \(\bf{S}.\bf{S}\) and h4_p as:\[e = \langle \mathcal{h_p} \rangle = Tr(\rho_{2x2} \mathcal{h_p})\]
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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
\[\begin{split}\begin{align*} S^+ &=S^x+iS^y & S^x &= 1/2(S^+ + S^-)\\ S^- &=S^x-iS^y\ \Rightarrow\ & S^y &=-i/2(S^+ - S^-) \end{align*}\end{split}\]