Lattice Green’s functions

triqs_tprf.lattice.lattice_dyson_g0_wk()

Signature : (float mu, triqs_tprf::e_k_cvt e_k, triqs::mesh::imfreq mesh) -> triqs_tprf::g_wk_t Construct a non-interacting Matsubara frequency lattice Green’s function \(G^{(0)}_{a\bar{b}}(i\omega_n, \mathbf{k})\)

Computes

\[G^{(0)}_{a\bar{b}}(i\omega_n, \mathbf{k}) = \left[ (i\omega_n + \mu ) \cdot \mathbf{1} - \epsilon(\mathbf{k}) \right]^{-1}_{a\bar{b}},\]

using a discretized dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\), chemical potential \(\mu\), and a Matsubara frequency Green’s function mesh.

Parameters:

mu: chemical potential \(\mu\)
e_k: discretized lattice dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\)
mesh: imaginary frequency mesh

Returns:

out: Matsubara frequency lattice Green’s function \(G^{(0)}_{a\bar{b}}(i\omega_n, \mathbf{k})\)

triqs_tprf.lattice.lattice_dyson_g0_fk()

Signature : (float mu, triqs_tprf::e_k_cvt e_k, triqs::mesh::refreq mesh, float delta) -> triqs_tprf::g_fk_t Construct a non-interacting real frequency lattice Green’s function \(G^{(0)}_{a\bar{b}}(\omega, \mathbf{k})\)

Computes

\[G^{(0)}_{a\bar{b}}(\omega, \mathbf{k}) = \left[ (\omega + i\delta + \mu ) \cdot \mathbf{1} - \epsilon(\mathbf{k}) \right]^{-1}_{a\bar{b}},\]

using a discretized dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\), chemical potential \(\mu\), broadening \(\delta\), and a real frequency Green’s function mesh.

Parameters:

mu: chemical potential \(\mu\)
e_k: discretized lattice dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\)
mesh: real frequency mesh
delta: broadening \(\delta\)

Returns:

out: Matsubara frequency lattice Green’s function \(G^{(0)}_{a\bar{b}}(\omega, \mathbf{k})\)

triqs_tprf.lattice.lattice_dyson_g_wk()

Signature : (float mu, triqs_tprf::e_k_cvt e_k, triqs_tprf::g_w_cvt sigma_w) -> triqs_tprf::g_wk_t Construct an interacting Matsubara frequency lattice Green’s function \(G_{a\bar{b}}(i\omega_n, \mathbf{k})\)

Computes

\[G_{a\bar{b}}(i\omega_n, \mathbf{k}) = \left[ (i\omega_n + \mu ) \cdot \mathbf{1} - \epsilon(\mathbf{k}) - \Sigma(i\omega_n) \right]^{-1}_{a\bar{b}},\]

using a discretized dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\), chemical potential \(\mu\), and a momentum independent Matsubara frequency self energy \(\Sigma_{\bar{a}b}(i\omega_n)\).

Parameters:

mu: chemical potential \(\mu\)
e_k: discretized lattice dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\)
sigma_w: imaginary frequency self-energy \(\Sigma_{\bar{a}b}(i\omega_n)\)

Returns:

out: Matsubara frequency lattice Green’s function \(G_{a\bar{b}}(i\omega_n, \mathbf{k})\)

triqs_tprf.lattice.lattice_dyson_g_fk()

Signature : (float mu, triqs_tprf::e_k_cvt e_k, triqs_tprf::g_fk_cvt sigma_fk, float delta) -> triqs_tprf::g_fk_t Construct an interacting real frequency lattice Green’s function \(G_{a\bar{b}}(\omega, \mathbf{k})\)

Computes

\[G_{a\bar{b}}(\omega, \mathbf{k}) = \left[ (\omega + i\delta + \mu ) \cdot \mathbf{1} - \epsilon(\mathbf{k}) - \Sigma(\omega, \mathbf{k}) \right]^{-1}_{a\bar{b}},\]

using a discretized dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\), chemical potential \(\mu\), broadening \(\delta\), and a real frequency self energy \(\Sigma_{\bar{a}b}(\omega, \mathbf{k})\).

Parameters:

mu: chemical potential \(\mu\)
e_k: discretized lattice dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\)
sigma_fk: real frequency self-energy \(\Sigma_{\bar{a}b}(\omega, \mathbf{k})\)
delta: broadening \(\delta\)

Returns:

out: real frequency lattice Green’s function \(G_{a\bar{b}}(\omega, \mathbf{k})\)

triqs_tprf.lattice.lattice_dyson_g_f()

Signature : (float mu, triqs_tprf::e_k_cvt e_k, triqs_tprf::g_f_cvt sigma_f, float delta) -> triqs_tprf::g_f_t Construct an interacting real frequency local (\(\mathbf{r}=\mathbf{0}\)) lattice Green’s function \(G_{a\bar{b}}(\omega)\)

Computes

\[G_{a\bar{b}}(\omega) = \frac{1}{N_k} \sum_\mathbf{k} \left[ (i\omega_n + i\delta + \mu ) \cdot \mathbf{1} - \epsilon(\mathbf{k}) - \Sigma(\omega) \right]^{-1}_{a\bar{b}},\]

using a discretized dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\), chemical potential \(\mu\), and a momentum independent real frequency self energy \(\Sigma_{\bar{a}b}(\omega)\).

Parameters:

mu: chemical potential \(\mu\)
e_k: discretized lattice dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\)
sigma_f: real frequency self-energy \(\Sigma_{\bar{a}b}(\omega)\)
delta: broadening \(\delta\)

Returns:

out: Real frequency lattice Green’s function \(G_{a\bar{b}}(\omega, \mathbf{k})\)

triqs_tprf.lattice.lattice_dyson_g_w()

Signature : (float mu, triqs_tprf::e_k_cvt e_k, triqs_tprf::g_w_cvt sigma_w) -> triqs_tprf::g_w_t Construct an interacting Matsubara frequency local (\(\mathbf{r}=\mathbf{0}\)) lattice Green’s function \(G_{a\bar{b}}(i\omega_n)\)

Computes

\[G_{a\bar{b}}(i\omega_n) = \frac{1}{N_k} \sum_\mathbf{k} \left[ (i\omega_n + \mu ) \cdot \mathbf{1} - \epsilon(\mathbf{k}) - \Sigma(i\omega_n) \right]^{-1}_{a\bar{b}},\]

using a discretized dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\), chemical potential \(\mu\), and a momentum independent Matsubara frequency self energy \(\Sigma_{\bar{a}b}(i\omega_n)\).

Parameters:

mu: chemical potential \(\mu\)
e_k: discretized lattice dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\)
sigma_w: imaginary frequency self-energy \(\Sigma_{\bar{a}b}(i\omega_n)\)

Returns:

out: Matsubara frequency lattice Green’s function \(G_{a\bar{b}}(i\omega_n, \mathbf{k})\)

Non-interacting generalized susceptibility

triqs_tprf.lattice_utils.imtime_bubble_chi0_wk(g_wk, nw=1, save_memory=False, verbose=True)[source]

triqs_tprf.lattice.lindhard_chi00()

Signature : (triqs_tprf::e_k_cvt e_k, triqs::mesh::imfreq mesh, float mu) -> triqs_tprf::chi_wk_t Generalized Lindhard susceptibility in the particle-hole channel \(\chi^{(00)}_{\bar{a}b\bar{c}d}(i\omega_n, \mathbf{q})\).

Analytic calculation of the generalized (non-interacting) Lindhard susceptibility in the particle-hole channel. The analytic expression is obtained using residue calculus to explicitly evaluate the matsubara sum of the fourier transformed imaginary time bubble product of two non-interacting single-particle Green’s functions.

\[G^{(0)}_{a\bar{b}}(\mathbf{k}, i\omega_n) = \left[ i\omega_n \cdot \mathbf{1} - \epsilon(\mathbf{k}) \right]^{-1} .\]

The analytic evaluation of the bubble diagram gives

\[\begin{split}\chi^{(00)}_{\bar{a}b\bar{c}d}(i\omega_n, \mathbf{q}) \equiv \mathcal{F} \left\{ - G^{(0)}_{d\bar{a}}(\tau, \mathbf{r}) G^{(0)}_{b\bar{c}}(-\tau, -\mathbf{r}) \right\} = - \frac{1}{N_k} \sum_{\nu} \sum_{\mathbf{k}} G^{(0)}_{d\bar{a}}(\nu, \mathbf{k}) G^{(0)}_{b\bar{c}}(\nu + \omega, \mathbf{k} + \mathbf{q}) \\ = - \frac{1}{N_k} \sum_{\nu} \sum_{\mathbf{k}} \left( \sum_{i} U^\dagger_{di}(\mathbf{k}) \frac{1}{i\nu - \epsilon_{\mathbf{k}, i}} U_{i\bar{a}}(\mathbf{k}) \right) \left( \sum_j U^\dagger_{bj}(\mathbf{k} + \mathbf{q}) \frac{1}{i\nu + i\omega - \epsilon_{\mathbf{k} + \mathbf{q}, j}} U_{j\bar{c}}(\mathbf{k} + \mathbf{q}) \right) \\ = \frac{1}{N_k} \sum_{\mathbf{k}} \sum_{ij} \left( [1 - \delta_{0, \omega_n} \delta_{\epsilon_{\mathbf{k},i},\epsilon_{\mathbf{k}+\mathbf{q}, j}})] \frac{ f(\epsilon_{\mathbf{k}, i}) - f(\epsilon_{\mathbf{k}+\mathbf{q}, j}) } {i\omega_n + \epsilon_{\mathbf{k} + \mathbf{q}, j} - \epsilon_{\mathbf{k}, i}} + \delta_{0, \omega_n} \delta_{\epsilon_{\mathbf{k},i},\epsilon_{\mathbf{k}+\mathbf{q}, j}} \frac{\beta}{4 \cosh^2 (\beta \epsilon_{\mathbf{k}, i} / 2) } \right) \\ \times U_{\bar{a}i}(\mathbf{k}) U^\dagger_{id}(\mathbf{k}) U_{\bar{c}j}(\mathbf{k} + \mathbf{q}) U^\dagger_{jb}(\mathbf{k} + \mathbf{q})\end{split}\]

where the \(U(\mathbf{k})\) matrices are the diagonalizing unitary transform of the matrix valued dispersion relation \(\epsilon_{\bar{a}b}(\mathbf{k})\), i.e.

Parameters:

e_k: discretized lattice dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\)
mesh: bosonic Matsubara frequency mesh
mu: chemical potential \(\mu\)

Returns:

out

generalized Lindhard susceptibility in the particle-hole channel \(\chi^{(00)}_{\bar{a}b\bar{c}d}(i\omega_n, \mathbf{q})\)

\[\]

sum_{bar{a}b} U_{ibar{a}}(mathbf{k}) epsilon_{bar{a}b}(mathbf{k}) U^dagger_{bj} (mathbf{k}) = delta_{ij} epsilon_{mathbf{k}, i}

The analytic formula is sub-optimal in terms of performance for higher temperatures. The evaluation scales as \(\mathcal{O}(N_k^2)\) which is worse than computing the bubble explicitly in imaginary time, with scaling \(\mathcal{O}(N_k N_\tau \log(N_k N_\tau)\) for \(N_k \gg N_\tau\).

Care must be taken when evaluating the fermionic Matsubara frequency sum of the product of two simple poles. By extending the sum to an integral over the complex plane the standard expression for the Lindhard response is obtained when the poles are non-degenerate. The degenerate case produces an additional frequency independent contribution (the last term on the last row).

Random Phase Approximation

triqs_tprf.lattice.solve_rpa_PH()

Signature : (triqs_tprf::chi_wk_vt chi0, array_contiguous_view<std::complex<double>,4> U) -> triqs_tprf::chi_wk_t Random Phase Approximation (RPA) in the particle-hole channel

Computes the equation

\[\chi(\bar{a}b\bar{c}d) = \big( \mathbb{1} - \chi^{(0)}(\bar{a}b\bar{B}A) U(A\bar{B}D\bar{C}) \big)^{-1} \chi^{(0)}(\bar{C}D\bar{c}d)\,.\]

Parameters:

chi0: bare particle-hole bubble \(\chi^{(0)}_{\bar{a}b\bar{c}d}(\mathbf{k}, i\omega_n)\)
U: RPA static vertex as obtained from triqs_tprf.rpa_tensor.get_rpa_tensor \(U_{a\bar{b}c\bar{d}}\)

Returns:

out: RPA suceptibility \(\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, i\omega_n)\)

triqs_tprf.rpa_tensor.kanamori_quartic_tensor(norb, U, Up, J, Jp)[source]

Return Kanamori interaction as a quartic tensor

\[\begin{split}\hat{U}_{\text { Kanamori }} = U \sum_{i} \hat{n}_{i, \uparrow} \hat{n}_{i, \downarrow}+ \sum_{i>j, s, s^{\prime}}\left(U^{\prime}-J \delta_{\sigma, \sigma^{\prime}}\right) \hat{n}_{i, \sigma} \hat{n}_{j, \sigma^{\prime}} - \\ J \sum_{i \neq j}\left(\hat{c}_{i, \downarrow}^{\dagger} \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow} \hat{c}_{i, \uparrow}+\hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{i, \uparrow} \hat{c}_{i, \downarrow}+ \mathrm{h.c.}\right)\end{split}\]

Parameters:

norbint,: Number of orbitals excluding spin.
Ucomplex,: Strength of intra-orbital interaction.
Upcomplex,: Strength of inter-orbital interaction.
Jcomplex,: Strength of Hound’s coupling.
Jpcomplex,: Strength pair hopping and spin-flip.

Returns:

Unp.ndarray,: shape = (2*norb, 2*norb, 2*norb, 2*norb)

Impurity susceptibility and Bethe-Salpeter Equation

triqs_tprf.chi_from_gg2.chi0_from_gg2_PH()

Signature : (triqs_tprf::g_iw_vt g, triqs_tprf::g2_iw_vt g2) -> triqs_tprf::g2_iw_t Bubble susceptibility \(\chi^{(0)} = GG\) in the Particle-Hole channel

Computes

\[\chi^{(0)}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu') = - \beta \delta_{\nu, \nu'} G_{da}(\nu) \cdot G_{bc}(\omega + \nu)\]

Parameters:

g: single particle Green’s function \(G_{ab}(\nu)\)
g2: two-particle Green’s function with the mesh to use for \(\chi^{(0)}\)

Returns:

out: chi0 particle-hole bubble \(\chi^{(0)}_{\bar{a}b\bar{c}d}(\omega, \nu,\nu')\)

triqs_tprf.chi_from_gg2.chi0_from_gg2_PP()

Signature : (triqs_tprf::g_iw_vt g, triqs_tprf::g2_iw_vt g2) -> triqs_tprf::g2_iw_t Bubble susceptibility \(\chi^{(0)} = GG\) in the Particle-Particle

channel

Computes

\[\chi^{(0)}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu') = - \beta \delta_{\nu, \nu'} G_{da}(\nu) \cdot G_{bc}(\omega - \nu)\]

Parameters:

g: single particle Green’s function \(G_{ab}(\nu)\)
g2: two-particle Green’s function with the mesh to use for \(\chi^{(0)}\)

Returns:

out: chi0 particle-particle bubble \(\chi^{(0)}_{\bar{a}b\bar{c}d}(\omega, \nu,\nu')\)

triqs_tprf.chi_from_gg2.chi_from_gg2_PH()

Signature : (triqs_tprf::g_iw_vt g, triqs_tprf::g2_iw_vt g2) -> triqs_tprf::g2_iw_t Generalized susceptibility \(\chi^{(0)} = G^{(2)} - GG\) in the

Particle-Hole channel

Computes

\[\chi_{\bar{a}b\bar{c}d}(\omega, \nu, \nu') = G^{(2)}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu') - \beta \delta_{\omega} G_{ba}(\nu) \cdot G_{dc}(\nu')\]

Parameters:

g: single particle Green’s function \(G_{ab}(\nu)\)
g2: two-particle Green’s function \(G^{(2)}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu')\)

Returns:

out: chi generalized particle-hole susceptibility \(\chi_{\bar{a}b\bar{c}d}(\omega, \nu,\nu')\)

triqs_tprf.chi_from_gg2.chi_from_gg2_PP()

Signature : (triqs_tprf::g_iw_vt g, triqs_tprf::g2_iw_vt g2) -> triqs_tprf::g2_iw_t Generalized susceptibility \(\chi^{(0)} = G^{(2)} - GG\) in the

Particle-Particle channel

Computes

\[\chi_{\bar{a}b\bar{c}d}(\omega, \nu, \nu') = G^{(2)}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu') - \beta \delta_{\nu + \nu' - \omega} G_{ba}(\nu) \cdot G_{dc}(\nu')\]

Parameters:

g: single particle Green’s function \(G_{ab}(\nu)\)
g2: two-particle Green’s function \(G^{(2)}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu')\)

Returns:

out: chi generalized particle-hole susceptibility \(\chi_{\bar{a}b\bar{c}d}(\omega, \nu,\nu')\)

triqs_tprf.bse.solve_local_bse(chi0_wnn, chi_wnn)[source]

Solve the Bethe-Salpeter equation for the local vertex function \(\Gamma_{abcd}(\omega, \nu, \nu')\).

Computes:

\[\Gamma_{abcd}(\omega, \nu, \nu') = [\chi^{(0)}]^{-1} - \chi^{-1}\]

where the inverses are taken in the particle-hole channel pairing of fermionic frequencies \(\nu\) and \(\nu'\) and orbital indices.

Parameters:

chi0_wnnGerealized local bubble susceptibility: \(\chi^{(0)}_{abcd}(\omega, \nu, \nu')\)
chi_wnnGeneralized local susceptibility: \(\chi_{abcd}(\omega, \nu, \nu')\)

Returns:

gamma_wnnParticle-hole vertex function: \(\Gamma_{abcd}(\omega, \nu, \nu')\)

Lattice Bethe-Salpeter Equation

triqs_tprf.bse.solve_lattice_bse(g_wk, gamma_wnn)[source]

Compute the generalized lattice susceptibility \(\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, \omega_n)\) using the Bethe-Salpeter equation (BSE).

Parameters:

g_wkGf,: Single-particle Green’s function \(G_{a\bar{b}}(i\nu_n, \mathbf{k})\).
gamma_wnnGf,: Local particle-hole vertex function \(\Gamma_{a\bar{b}c\bar{d}}(i\omega_n, i\nu_n, i\nu_n')\).

Returns:

chi_kwGf,: Generalized lattice susceptibility \(\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, i\omega_n)\).
chi0_kwGf,: Generalized bare lattice susceptibility \(\chi^0_{\bar{a}b\bar{c}d}(\mathbf{k}, i\omega_n)\).

triqs_tprf.bse.solve_lattice_bse_at_specific_w(g_wk, gamma_wnn, nw_index)[source]

Compute the generalized lattice susceptibility \(\chi_{\bar{a}b\bar{c}d}(i\omega_{n=\mathrm{nw\_index}}, \mathbf{k})\) using the Bethe-Salpeter equation (BSE) for a specific \(i\omega_{n=\mathrm{nw\_index}}\).

Parameters:

g_wkGf,: Single-particle Green’s function \(G_{a\bar{b}}(i\nu_n, \mathbf{k})\).
gamma_wnnGf,: Local particle-hole vertex function \(\Gamma_{a\bar{b}c\bar{d}}(i\omega_n, i\nu_n, i\nu_n')\).
nw_indexint,: The bosonic Matsubara frequency index \(i\omega_{n=\mathrm{nw\_index}}\) at which the BSE is solved.

Returns:

chi_kGf,: Generalized lattice susceptibility \(\chi_{\bar{a}b\bar{c}d}(i\omega_{n=\mathrm{nw\_index}}, \mathbf{k})\).
chi0_kGf,: Generalized bare lattice susceptibility \(\chi^0_{\bar{a}b\bar{c}d}(i\omega_{n=\mathrm{nw\_index}}, \mathbf{k})\).

triqs_tprf.bse.get_chi0_wnk(g_wk, nw=1, nwf=None)[source]

Compute the generalized bare lattice susceptibility \(\chi^{0}_{\bar{a}b\bar{c}d}(i\omega_n, i\nu_n, \mathbf{k})\) from the single-particle Green’s function \(G_{a\bar{b}}(i\nu_n, \mathbf{k})\).

Parameters:

g_wkGf,: Single-particle Green’s function \(G_{a\bar{b}}(i\nu_n, \mathbf{k})\).
nwint,: Number of bosonic frequencies in \(\chi^0\).
nwfint,: Number of fermionic frequencies in \(\chi^0\).

Returns:

chi0_wnkGf,: Generalized bare lattice susceptibility \(\chi^{0}_{\bar{a}b\bar{c}d}(i\omega_n, i\nu_n, \mathbf{k})\).

triqs_tprf.bse.get_chi0_nk_at_specific_w(g_wk, nw_index=1, nwf=None)[source]

Compute the generalized bare lattice susceptibility \(\chi^{0}_{\bar{a}b\bar{c}d}(i\omega_{n=\mathrm{nw\_index}}, i\nu_n, \mathbf{k})\) from the single-particle Green’s function \(G_{a\bar{b}}(i\nu_n, \mathbf{k})\) for a specific \(i\omega_{n=\mathrm{nw\_index}}\).

Parameters:

g_wkGf,: Single-particle Green’s function \(G_{a\bar{b}}(i\nu_n, \mathbf{k})\).
nw_indexint,: The bosonic Matsubara frequency index \(i\omega_{n=\mathrm{nw\_index}}\) at which \(\chi^0\) is calculated.
nwfint,: Number of fermionic frequencies in \(\chi^0\).

Returns:

chi0_nkGf,: Generalized bare lattice susceptibility \(\chi^{0}_{\bar{a}b\bar{c}d}(i\omega_{n=\mathrm{nw\_index}}, i\nu_n, \mathbf{k})\).

Dual Bethe-Salpeter Equation

triqs_tprf.dbse.solve_lattice_dbse(g_wk, F_wnn, L_wn, chi_imp_w)[source]

Compute the generalized lattice susceptibility \(\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, \omega_n)\) using the dual Bethe-Salpeter equation (DBSE).

Parameters:

g_wkGf,: Single-particle Green’s function \(G_{a\bar{b}}(i\nu_n, \mathbf{k})\).
F_wnnGf,: Local particle-hole reducible vertex function \(F_{a\bar{b}c\bar{d}}(i\omega_n, i\nu_n, i\nu_n')\).
L_wnGf,: Local particle-hole reducible triangle vertex function \(L_{a\bar{b}c\bar{d}}(i\omega_n, i\nu_n)\).
chi_imp_wGf,: Generalized DMFT impurity susceptibility \(\chi_{a\bar{b}c\bar{d}}(i\omega_n)\).

Returns:

chi_kwGf,: Generalized lattice susceptibility \(\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, i\omega_n)\).

triqs_tprf.dbse.impurity_reducible_vertex_F(g_w, g2_wnn)[source]

Compute the impurity reducible vertex function \(F_{abcd}(\omega, \nu, \nu')\).

Computes:

\[F_{abcd}(\omega, \nu, \nu') = [\chi^{(0)}]^{-1} (\chi - \chi^{(0)} ) [\chi^{(0)}]^{-1} \]

where the inverses are taken in the particle-hole channel pairing of fermionic frequencies \(\nu\) and \(\nu'\) and orbital indices.

Parameters:

g_wSingle particle Green’s function: \(G_{ab}(\nu)\)
g2_wnnTwo-particle Green’s function: \(G^{(2)}_{abcd}(\omega, \nu, \nu')\)

Returns:

F_wnnParticle-hole reducible vertex function: \(F_{abcd}(\omega, \nu, \nu')\)

GW approximation

class triqs_tprf.gw_solver.GWSolver(e_k, V_k, wmesh, mu=None, g_wk=None, N_fix=False, N_tol=1e-05, mu_bracket=None)[source]

Methods

calc_real_freq
calc_real_space
calc_rho_k
calc_rho_loc
calc_rho_r
calc_total_density
dynamic_and_static_interaction
dyson_equation
fock_sigma
get_local_density_matrix
get_total_density
gw_dynamic_sigma
hartree_sigma
logo
polarization
screened_interaction
solve_iter
susceptibiltiy

triqs_tprf.lattice.dynamical_screened_interaction_W()

Signature : (triqs_tprf::chi_wk_cvt PI_wk, triqs_tprf::chi_k_cvt V_k) -> triqs_tprf::chi_wk_t Dynamical screened interaction \(W(i\omega_n, \mathbf{k})\) calculator for static momentum-dependent bare interactions \(V(\mathbf{k})\).

The full screened interaction \(W(i\omega_n, \mathbf{k})\) is given by

\[W^{(full)}_{abcd}(i\omega_n, \mathbf{k}) = V_{abcd}(\mathbf{k}) + \sum_{efgh} V_{abef}(\mathbf{k}) \cdot \Pi_{fegh}(i\omega_n, \mathbf{k}) \cdot W^{(full)}_{hgcd}(i\omega_n, \mathbf{k})\]

Parameters:

PI_wk: polarization bubble \(\Pi_{abcd}(i\omega_n, \mathbf{k})\)
V_k: static bare interaction \(V_{abcd}(\mathbf{k})\)

Returns:

out: dynamical screened interaction \(W_{abcd}(i\omega_n, \mathbf{k})\)

triqs_tprf.lattice.dynamical_screened_interaction_W_from_generalized_susceptibility()

Signature : (triqs_tprf::chi_wk_cvt chi_wk, triqs_tprf::chi_k_cvt V_k) -> triqs_tprf::chi_wk_t Dynamical screened interaction \(W(i\omega_n, \mathbf{k})\) calculator for static momentum-dependent bare interactions \(V(\mathbf{k})\) and known generalized susceptibility \(\chi(i\omega_n, \mathbf{k})\)

The full screened interaction \(W(i\omega_n, \mathbf{k})\) is given by

\[W^{(full)}_{abcd}(i\omega_n, \mathbf{k}) = V_{abcd}(\mathbf{k}) + \sum_{efgh} V_{abef}(\mathbf{k}) \cdot \chi_{fegh}(i\omega_n, \mathbf{k}) \cdot V_{hgcd}(\mathbf{k})\]

Parameters:

chi_wk: generalized susceptibility \(\chi_{abcd}(i\omega_n, \mathbf{k})\)
V_k: static bare interaction \(V_{abcd}(\mathbf{k})\)

Returns:

out: dynamical screened interaction \(W_{abcd}(i\omega_n, \mathbf{k})\)

triqs_tprf.gw.bubble_PI_wk(g_wk)[source]

Compute the particle-hole bubble from the single particle lattice Green’s function

\[\Pi_{abcd}(i\omega_n, k) = - \mathcal{F}_{\tau, \mathbf{r} \rightarrow i\omega_n, \mathbf{k}} \left\{ G_{d\bar{a}}(\tau, \mathbf{r}) G_{b\bar{c}}(-\tau, -\mathbf{r}) \right\}\]

Parameters:

g_wkTRIQS Green’s function (rank 2) on Matsubara and Brillouinzone meshes: Single particle lattice Green’s function.

Returns:

PI_wkTRIQS Green’s function (rank 4) on Matsubara and Brillouinzone meshes: Particle hole bubble.

triqs_tprf.gw.gw_sigma()

Signature : (triqs_tprf::chi_wk_cvt W_wk, triqs_tprf::g_wk_cvt g_wk) -> triqs_tprf::g_wk_t GW self energy \(\Sigma(i\omega_n, \mathbf{k})\) calculator for dynamic interactions

Splits the interaction into a dynamic and a static part

\[W_{abcd}(i\omega_n, \mathbf{k}) = W^{(dyn)}_{abcd}(i\omega_n, \mathbf{k}) + V_{abcd}(\mathbf{k})\]

by fitting the high-frequency tail.

Fourier transforms the dynamic part of the interaction and the single-particle Green’s function to imaginary time and real space.

\[G_{ab}(\tau, \mathbf{r}) = \mathcal{F}^{-1} \left\{ G_{ab}(i\omega_n, \mathbf{k}) \right\}\]

\[W^{(dyn)}_{abcd}(\tau, \mathbf{r}) = \mathcal{F}^{-1} \left\{ W^{(dyn)}_{abcd}(i\omega_n, \mathbf{k}) \right\}\]

computes the GW self-energy as the product

\[\Sigma^{(dyn)}_{ab}(\tau, \mathbf{r}) = - \sum_{cd} W^{(dyn)}_{abcd}(\tau, \mathbf{r}) G_{cd}(\tau, \mathbf{r})\]

and transforms back to frequency and momentum

\[\Sigma^{(dyn)}_{ab}(i\omega_n, \mathbf{k}) = \mathcal{F} \left\{ \Sigma^{(dyn)}_{ab}(\tau, \mathbf{r}) \right\}\]

The self-energy of the static part of the interaction is calculated as the sum

\[\Sigma^{(stat)}_{ab}(\mathbf{k}) = -\frac{1}{N_k} \sum_{\mathbf{q}} V_{abab}(\mathbf{k}) \rho(G(i\omega_n, \mathbf{k+q}))_{ab}\]

where \(\rho(G(i\omega_n, \mathbf{k+q}))\) is the density matrix of the single particle Green’s function.

The total GW self-energy is given by

\[\Sigma_{ab}(i\omega_n, \mathbf{k}) = \Sigma^{(dyn)}_{ab}(i\omega_n, \mathbf{k}) + \Sigma^{(stat)}_{ab}(\mathbf{k})\]

Parameters:

W_wk: interaction \(W_{abcd}(i\omega_n, \mathbf{k})\)
g_wk: single particle Green’s function \(G_{ab}(i\omega_n, \mathbf{k})\)

Returns:

out: GW self-energy \(\Sigma_{ab}(i\omega_n, \mathbf{k})\)

triqs_tprf.gw.g0w_sigma(): Signature : (float mu, float beta, triqs_tprf::e_k_cvt e_k, triqs_tprf::chi_k_cvt v_k, mesh::brzone::value_t kpoint) -> array<std::complex<double>, 2> Add some docs

Linearized Eliashberg equation

triqs_tprf.eliashberg.solve_eliashberg(Gamma_pp_wk, g_wk, initial_delta=None, Gamma_pp_const_k=None, tol=1e-10, product='FFT', solver='IRAM', symmetrize_fct=<function <lambda>>, k=6)[source]

Solve the linearized Eliashberg equation

Returns the biggest eigenvalues and corresponding eigenvectors of the linearized Eliashberg equation, for a particle-particle vertex in the random phase approximation, as described here Random phase approximation for the irreducible particle-particle vertex. The Eliashberg equation implementation is using fourier transformations for computational efficiency. The eigenvalues are found using an iterative algorithm from scipy.

Parameters:

Gamma_pp_wkGf,

Pairing vertex \(\Gamma(i\omega_n, \mathbf{k})\). The mesh attribute of the Gf must be a MeshProduct with the components (MeshImFreq, MeshBrZone).

g_wkGf,

Green’s function \(G(i\nu_n, \mathbf{k})\). The mesh attribute of the Gf must be a MeshProduct with the components (MeshImFreq, MeshBrZone).

initial_deltaGf, optional

An initial anomalous self-energy \(\Delta(i\nu_n, \mathbf{k})\) to start an iterative solver, given as a Gf with MeshProduct with the components (MeshImFreq, MeshBrZone). If not given semi_random_initial_delta() will be called.

Gamma_pp_const_kfloat or np.ndarray or Gf, optional

Part of the pairing vertex that is constant in Matsubara frequency space \(\Gamma(\mathbf{k})\). If given as a Gf its mesh attribute needs to be a MeshBrZone. If not given, the constant part will be fitted.

tolfloat, optional

Relative accuracy for eigenvalues (stopping criterion).

productstr, [‘FFT’, ‘SUM’], optional

Which function of the Eliashberg product shall be used:

‘FFT’triqs_tprf.lattice.eliashberg_product_fft,
which uses Fourier transformation for optimal computational efficiency.

‘SUM’triqs_tprf.lattice.eliashberg_product, uses the explicit sum.
Restrictions : wmesh of Gamma_pp_wk must be atleast twice the size of the one of g_wk.

solverstr, [‘IRAM’, ‘PM’], optional

Which eigenvalue solver shall be used:

‘IRAM’ : Use the Implicitly Restarted Arnoldi Method implemented in implicitly_restarted_arnoldi_method().

‘PM’ : Use the Power Method implemented in power_method_LR().

symmetrize_fctfunction, optional

A function that takes one parameter: A Green’s function \(G(i\nu_n, \mathbf{k})\). The mesh attribute of the Gf must be a MeshProduct with the components (MeshImFreq, MeshBrZone). This function is applied after every iteration of the eigenvalue solver and can be used to enforce a specific symmetry. If no symmetries are enforced, caution is need, because unphysical symmetries can occur.

kint, optional

The number of leading superconducting gaps that shall be calculated. Does only have an effect, if ‘IRAM’ is used as a solver.

Returns:

Eslist of float,: Biggest eigenvalues of the linearized Eliashberg equation \(\lambda\).
eigen_modeslist of Gf,: Corresponding eigenvectors (anomalous self-energies) \(\Delta(i\nu_n, \mathbf{k})\) as Gf with MeshProduct with the components (MeshImFreq, MeshBrZone).

Hubbard atom analytic response functions

triqs_tprf.analytic_hubbard_atom.analytic_hubbard_atom(beta, U, nw, nwf, nwf_gf)[source]

Compute dynamical response functions for the Hubbard atom at half filling.

This function returns an object that contains the single-particle Greens function \(G(\omega)\), the magnetic two-particle generalized susceptibility \(\chi_m(\omega, \nu, \nu')\), and the corresponding bare bubble \(\chi^{(0)}_m(\omega, \nu, \nu')\), and the magnetic vertex function \(\Gamma_m(\omega, \nu, \nu')\).

This is implemented using analytical formulas from Thunstrom et al. [PRB 98, 235107 (2018)] please cite the paper if you use this function!

In particular this is one exact solution to the Bethe-Salpeter equation, that is the infinite matrix inverse problem:

\[\Gamma_m = [\chi^{(0)}_m]^{-1} - \chi_m^{-1}\]

Parameters:

betafloat: Inverse temperature.
Ufloat: Hubbard U interaction parameter.
nwint: Number of bosonic Matsubara frequencies in the computed two-particle response functions.
nwfint: Number of fermionic Matsubara frequencies in the computed two-particle response functions.
nwf_gfint: Number of fermionic Matsubara frequencies in the computed single-particle Greens function.

Returns:

pParameterCollection: Object containing all the response functions and some other observables, p.G_iw, p.chi_m, p.chi0_m, p.gamma_m, p.Z, p.m2, p.chi_m_static.

Two-particle response function linear-algebra operations

triqs_tprf.linalg.inverse_PH()

Signature : (triqs_tprf::g2_iw_vt g) -> triqs_tprf::g2_iw_t Two-particle response-function inversion \([g]^{-1}\) in the particle-hole channel (PH).

The two-particle response function \(g_{abcd}(\omega, \nu, \nu')\) is cast to matrix form and inverted

\[[g]^{-1} = [ g_{\{\nu\alpha\beta\}, \{\nu'\gamma\delta\}}(\omega) ]^{-1}\]

where the mapping of target-space indices \(\{a, b, c, d \}\) to \(\{\alpha, \beta\}, \{\gamma, \delta\}\) is channel dependent.

Storage is allocated and the inverse is returned by value.

Parameters:

g: two-particle response function to invert, \(g \equiv g_{abcd}(\omega, \nu, \nu')\)

Returns:

out: \([g]^{-1}\) in the given channel

triqs_tprf.linalg.inverse_PP()

Signature : (triqs_tprf::g2_iw_vt g) -> triqs_tprf::g2_iw_t Two-particle response-function inversion \([g]^{-1}\) in the particle-particle channel (PP).

The two-particle response function \(g_{abcd}(\omega, \nu, \nu')\) is cast to matrix form and inverted

\[[g]^{-1} = [ g_{\{\nu\alpha\beta\}, \{\nu'\gamma\delta\}}(\omega) ]^{-1}\]

where the mapping of target-space indices \(\{a, b, c, d \}\) to \(\{\alpha, \beta\}, \{\gamma, \delta\}\) is channel dependent.

Storage is allocated and the inverse is returned by value.

Parameters:

g: two-particle response function to invert, \(g \equiv g_{abcd}(\omega, \nu, \nu')\)

Returns:

out: \([g]^{-1}\) in the given channel

triqs_tprf.linalg.inverse_PH_bar()

Signature : (triqs_tprf::g2_iw_vt g) -> triqs_tprf::g2_iw_t Two-particle response-function inversion \([g]^{-1}\) in the particle-hole-bar channel (PH-bar).

The two-particle response function \(g_{abcd}(\omega, \nu, \nu')\) is cast to matrix form and inverted

\[[g]^{-1} = [ g_{\{\nu\alpha\beta\}, \{\nu'\gamma\delta\}}(\omega) ]^{-1}\]

where the mapping of target-space indices \(\{a, b, c, d \}\) to \(\{\alpha, \beta\}, \{\gamma, \delta\}\) is channel dependent.

Storage is allocated and the inverse is returned by value.

Parameters:

g: two-particle response function to invert, \(g \equiv g_{abcd}(\omega, \nu, \nu')\)

Returns:

out: \([g]^{-1}\) in the given channel

triqs_tprf.linalg.product_PH()

Signature : (triqs_tprf::g2_iw_vt A, triqs_tprf::g2_iw_vt B) -> triqs_tprf::g2_iw_t Two-particle response-function product \(A * B\) in the particle-hole channel (PH).

The two-particle response functions \(A \equiv A_{abcd}(\omega, \nu, \nu')\) and \(B \equiv B_{abcd}(\omega, \nu, \nu')\) are cast to matrix form and their product is computed

\[(A * B)_{\{\nu\alpha\beta\}, \{\nu'\gamma\delta\}}(\omega) = \sum_{\bar{\nu}ab} A_{\{\nu\alpha\beta\}, \{\bar{\nu}ab\}}(\omega) B_{\{\bar{\nu}ab\}, \{\nu'\gamma\delta\}}(\omega)\]

where the mapping of target-space indices \(\{a, b, c, d \}\) to \(\{\alpha, \beta\}, \{\gamma, \delta\}\) is channel dependent.

Storage is allocated and the product is returned by value.

Parameters:

A: two-particle response function \(A \equiv A_{abcd}(\omega, \nu, \nu')\)
B: two-particle response function \(B \equiv A_{abcd}(\omega, \nu, \nu')\)

Returns:

out: \((A * B)\) in the given channel

triqs_tprf.linalg.product_PP()

Signature : (triqs_tprf::g2_iw_vt A, triqs_tprf::g2_iw_vt B) -> triqs_tprf::g2_iw_t Two-particle response-function product \(A * B\) in the particle-particle channel (PP).

The two-particle response functions \(A \equiv A_{abcd}(\omega, \nu, \nu')\) and \(B \equiv B_{abcd}(\omega, \nu, \nu')\) are cast to matrix form and their product is computed

\[(A * B)_{\{\nu\alpha\beta\}, \{\nu'\gamma\delta\}}(\omega) = \sum_{\bar{\nu}ab} A_{\{\nu\alpha\beta\}, \{\bar{\nu}ab\}}(\omega) B_{\{\bar{\nu}ab\}, \{\nu'\gamma\delta\}}(\omega)\]

where the mapping of target-space indices \(\{a, b, c, d \}\) to \(\{\alpha, \beta\}, \{\gamma, \delta\}\) is channel dependent.

Storage is allocated and the product is returned by value.

Parameters:

A: two-particle response function \(A \equiv A_{abcd}(\omega, \nu, \nu')\)
B: two-particle response function \(B \equiv A_{abcd}(\omega, \nu, \nu')\)

Returns:

out: \((A * B)\) in the given channel

triqs_tprf.linalg.product_PH_bar()

Signature : (triqs_tprf::g2_iw_vt A, triqs_tprf::g2_iw_vt B) -> triqs_tprf::g2_iw_t Two-particle response-function product \(A * B\) in the particle-hole-bar channel (PH-bar).

The two-particle response functions \(A \equiv A_{abcd}(\omega, \nu, \nu')\) and \(B \equiv B_{abcd}(\omega, \nu, \nu')\) are cast to matrix form and their product is computed

\[(A * B)_{\{\nu\alpha\beta\}, \{\nu'\gamma\delta\}}(\omega) = \sum_{\bar{\nu}ab} A_{\{\nu\alpha\beta\}, \{\bar{\nu}ab\}}(\omega) B_{\{\bar{\nu}ab\}, \{\nu'\gamma\delta\}}(\omega)\]

where the mapping of target-space indices \(\{a, b, c, d \}\) to \(\{\alpha, \beta\}, \{\gamma, \delta\}\) is channel dependent.

Storage is allocated and the product is returned by value.

Parameters:

A: two-particle response function \(A \equiv A_{abcd}(\omega, \nu, \nu')\)
B: two-particle response function \(B \equiv A_{abcd}(\omega, \nu, \nu')\)

Returns:

out: \((A * B)\) in the given channel

triqs_tprf.linalg.identity_PH()

Signature : (triqs_tprf::g2_iw_vt g) -> triqs_tprf::g2_iw_t Two-particle response-function identity operator \(\mathbf{1}\) in the particle-hole channel (PH).

Constructs the unity-operator in the given channel

\[\mathbf{1}_{abcd}(\omega,\nu,\nu') = \mathbf{1}_{\{\nu\alpha\beta\}, \{\nu'\gamma\delta\}}(\omega) \equiv \delta_{\nu\nu'} \delta_{\alpha\gamma} \delta_{\beta\delta}\]

where the mapping of target-space indices \(\{a, b, c, d \}\) to \(\{\alpha, \beta\}, \{\gamma, \delta\}\) is channel dependent.

Storage is allocated and the result is returned by value.

Parameters:

A: two-particle response function \(A \equiv A_{abcd}(\omega, \nu, \nu')\) determinig the shape and size of the unity operator

Returns:

out: the unity operator \(\mathbf{1}\), in the given channel

triqs_tprf.linalg.identity_PP()

Signature : (triqs_tprf::g2_iw_vt g) -> triqs_tprf::g2_iw_t Two-particle response-function identity operator \(\mathbf{1}\) in the particle-particle channel (PP).

Constructs the unity-operator in the given channel

\[\mathbf{1}_{abcd}(\omega,\nu,\nu') = \mathbf{1}_{\{\nu\alpha\beta\}, \{\nu'\gamma\delta\}}(\omega) \equiv \delta_{\nu\nu'} \delta_{\alpha\gamma} \delta_{\beta\delta}\]

where the mapping of target-space indices \(\{a, b, c, d \}\) to \(\{\alpha, \beta\}, \{\gamma, \delta\}\) is channel dependent.

Storage is allocated and the result is returned by value.

Parameters:

A: two-particle response function \(A \equiv A_{abcd}(\omega, \nu, \nu')\) determinig the shape and size of the unity operator

Returns:

out: the unity operator \(\mathbf{1}\), in the given channel

triqs_tprf.linalg.identity_PH_bar()

Signature : (triqs_tprf::g2_iw_vt g) -> triqs_tprf::g2_iw_t Two-particle response-function identity operator \(\mathbf{1}\) in the particle-hole-bar channel (PH-bar).

Constructs the unity-operator in the given channel

\[\mathbf{1}_{abcd}(\omega,\nu,\nu') = \mathbf{1}_{\{\nu\alpha\beta\}, \{\nu'\gamma\delta\}}(\omega) \equiv \delta_{\nu\nu'} \delta_{\alpha\gamma} \delta_{\beta\delta}\]

where the mapping of target-space indices \(\{a, b, c, d \}\) to \(\{\alpha, \beta\}, \{\gamma, \delta\}\) is channel dependent.

Storage is allocated and the result is returned by value.

Parameters:

A: two-particle response function \(A \equiv A_{abcd}(\omega, \nu, \nu')\) determinig the shape and size of the unity operator

Returns:

out: the unity operator \(\mathbf{1}\), in the given channel

Wannier90 tight binding parsers

triqs_tprf.wannier90.parse_hopping_from_wannier90_hr_dat(filename)[source]

Wannier90 real space hopping parser of *_hr.dat files.

Returns a dictionary where the keys are the real-space hopping vectors, in terms of multiples of the lattice vectors, and the values are num_wann * num_wann numpy ndarrays with the hopping integrals.

Parameters:

filenamestr: Wannier90 *_hr.dat file to parse.

Returns:

hopp_dictdict: Dictionary of real space hoppings.
num_wannint: Total number of Wannier functions per unit-cell.

triqs_tprf.wannier90.parse_lattice_vectors_from_wannier90_wout(filename)[source]

Wannier90 real space lattice vectors parser of *.wout files.

Parameters:

filenamestr: Wannier90 *.wout file to parse.

Returns:

vectorslist of three three-tuples of floats: Lattice vectors.

triqs_tprf.wannier90.parse_reciprocal_lattice_vectors_from_wannier90_wout(filename)[source]

Wannier90 reciprocal space lattice vectors parser of *.wout files.

Parameters:

filenamestr: Wannier90 *.wout file to parse.

Returns:

vectorslist of three three-tuples of floats: Reciprocal lattice vectors.

triqs_tprf.wannier90.parse_band_structure_from_wannier90_band_dat(filename)[source]

Wannier90 band structure parser of *_band.dat` files.

Parameters:

filenamestr: Wannier90 *_band.dat file to parse.

Returns:

Endarray (2D): Band energies.
wndarray (1D): k-space path points.

Tight binding lattice model

class triqs_tprf.tight_binding.TBLattice(units, hoppings={}, orbital_positions=[(0, 0, 0)], orbital_names=None, hopping=None)[source]

Class describing a tight binding hamiltonian on top of a bravais lattice. Has objects of type BravaisLattice, BrillouinZone and TightBinding as attributes, and exposes part of their interfaces.

Attributes:

blBravaisLattice: The associated Bravais Lattice
bzBrillouinZone: The associated Brillouin Zone
tbTightBinding: The tight-binding Hamiltonian

Methods

`dispersion`(arg)	Signature : (k_cvt K) -> nda::array<double, 1> Evaluate the dispersion relation for a momentum vector k in units of the reciprocal lattice vectors
`fourier`(arg)	Signature : (k_cvt K) -> matrix<dcomplex> Evaluate the fourier transform for a momentum vector k in units of the reciprocal lattice vectors
`get_kmesh`(n_k)	Return a mesh on the Brillouin zone with a given discretization
`get_rmesh`(n_r)	Return a mesh on the Bravais lattice with a given periodicity
`lattice_to_real_coordinates`(x)	Signature : (r_t x) -> r_t Transform into real coordinates.

dispersion(arg)[source]: Signature : (k_cvt K) -> nda::array<double, 1> Evaluate the dispersion relation for a momentum vector k in units of the reciprocal lattice vectors

fourier(arg)[source]: Signature : (k_cvt K) -> matrix<dcomplex> Evaluate the fourier transform for a momentum vector k in units of the reciprocal lattice vectors

get_kmesh(n_k)[source]

Return a mesh on the Brillouin zone with a given discretization

Parameters:

n_kint or three-tuple of int: The linear dimension(s)

Returns:

MeshBrZone: The mesh on the Brillouin zone

get_rmesh(n_r)[source]

Return a mesh on the Bravais lattice with a given periodicity

Parameters:

n_rint or three-tuple of int: The periodicity in each dimension

Returns:

MeshCycLat: The cyclic lattice mesh

lattice_to_real_coordinates(x)[source]: Signature : (r_t x) -> r_t Transform into real coordinates.

property n_orbitals: Number of orbitals in the unit cell

property ndim: Number of dimensions of the lattice

property orbital_names: The list of orbital names

property orbital_positions: The list of orbital positions

property units: Number of dimensions of the lattice

triqs_tprf.tight_binding.create_square_lattice(norb, t, tp=0.0, zeeman=0.0, spin=False, **kwargs)[source]

Retuns TBLattice that represents a model on a square lattice

The model is described by the Hamiltonian

\[H=-t \sum_{\langle j, l\rangle \sigma}\left(c_{j \sigma}^{\dagger} c_{l \sigma}+c_{l \sigma}^{\dagger} c_{j \sigma}\right) - t' \sum_{\langle\langle j, l\rangle\rangle \sigma}\left(c_{j \sigma}^{\dagger} c_{l \sigma}+c_{l \sigma}^{\dagger} c_{j \sigma}\right) + \xi \sum_j \left(n_{j \uparrow} - n_{j \downarrow} \right)\,,\]

where the angular bracket describes a sum over nearest neighbors and the double angular bracket over next-nearest neighbors.

Parameters:

norbint,: Number of orbitals excluding spin
tcomplex,: Kinetic energy of nearest neighbor hopping. Corresponds to \(t\).
tpcomplex, optional: Kinetic energy of next-nearest neighbor hopping. Corresponds to \(t'\).
zeemancomplex, optional: Strength of Zeeman term. Corresponds to \(\xi\).
spinbool,: True if spin index should be used explicitly, False otherwise. The Zeeman term can only be applied if spin=True.

Returns:

square_latticeTBLattice

class triqs_tprf.super_lattice.TBSuperLattice(tb_lattice, super_lattice_units, cluster_sites=None, remove_internal_hoppings=False)[source]

Builds a superlattice on top of a base TBLattice.

Parameters:

tb_latticeTBLattice instance: The base tight binding lattice.
super_lattice_unitsndarray (2D): The unit vectors of the superlattice in the tb_lattice (integer) coordinates.
cluster_sites: Coordinates of the cluster in tb_lattice coordinates. If None, an automatic computation of cluster positions is made as follows: it takes all points whose coordinates in the basis of the superlattice are in [0, 1[^dimension.
remove_internal_hoppingsbool: If true, the hopping terms are removed inside the cluster. Useful to add Hartree Fock terms at the boundary of a cluster, e.g.

Attributes:

hoppings
n_orbitals: Number of orbitals in the unit cell
ndim: Number of dimensions of the lattice
orbital_names: The list of orbital names
orbital_positions: The list of orbital positions
units: Number of dimensions of the lattice

Methods

`change_coordinates_L_to_SL`(x)	Given a point on the tb_lattice in lattice coordinates, returns its coordinates (R, alpha) in the Superlattice
`change_coordinates_SL_to_L`(R, alpha)	Given a point in the supercell R, site (number) alpha, it computes its position on the tb_lattice in lattice coordinates
`cluster_sites`()	Generate the position of the cluster site in the tb_lattice coordinates.
`dispersion`(arg)	Signature : (k_cvt K) -> nda::array<double, 1> Evaluate the dispersion relation for a momentum vector k in units of the reciprocal lattice vectors
`fold`(D1[, remove_internal, create_zero])	Input: a function r-> f(r) on the tb_lattice given as a dictionnary Output: the function R-> F(R) folded on the superlattice.
`fourier`(arg)	Signature : (k_cvt K) -> matrix<dcomplex> Evaluate the fourier transform for a momentum vector k in units of the reciprocal lattice vectors
`get_kmesh`(n_k)	Return a mesh on the Brillouin zone with a given discretization
`get_rmesh`(n_r)	Return a mesh on the Bravais lattice with a given periodicity
`lattice_to_real_coordinates`(x)	Signature : (r_t x) -> r_t Transform into real coordinates.
`pack_index_site_orbital`(n_site, n_orbital)	nsite and n_orbital must start at 0
`unpack_index_site_orbital`(index)	Inverse of pack_index_site_orbital

change_coordinates_L_to_SL(x)[source]: Given a point on the tb_lattice in lattice coordinates, returns its coordinates (R, alpha) in the Superlattice

change_coordinates_SL_to_L(R, alpha)[source]: Given a point in the supercell R, site (number) alpha, it computes its position on the tb_lattice in lattice coordinates

cluster_sites()[source]: Generate the position of the cluster site in the tb_lattice coordinates.

fold(D1, remove_internal=False, create_zero=None)[source]: Input: a function r-> f(r) on the tb_lattice given as a dictionnary Output: the function R-> F(R) folded on the superlattice. Only requirement is that f(r)[orbital1, orbital2] is properly defined. Hence f(r) can be a numpy, a GFBloc, etc…

pack_index_site_orbital(n_site, n_orbital)[source]: nsite and n_orbital must start at 0

unpack_index_site_orbital(index)[source]: Inverse of pack_index_site_orbital

Hartree-Fock and Hartree solvers

class triqs_tprf.hf_solver.HartreeFockSolver(e_k, beta, H_int=None, gf_struct=None, mu0=0.0, mu_max=None, mu_min=None)[source]

Hartree-Fock solver for local interactions.

Parameters:

e_kTRIQS Greens function on a Brillouin zone mesh: Single-particle dispersion.
betafloat: inverse temperature
H_intTRIQS Operator instance: Local interaction Hamiltonian
gf_structlist of pairs of block index and its size: gf_struct fixing orbital order between e_k and H_int
mu0float: chemical potential
mu_min, mu_maxfloat, optional: range for chemical potential search.

Methods

`chemical_potential`()
`density_matrix`()
`density_matrix_step`(rho_vec[, N_target])
`expectation_value`(op_ab)
`logo`()
`mat2vec`(mat)	Converts a unitary matrix to a vector representation with the order
`mean_field_matrix`()
`solve_iter`(N_target[, M0, mu0, nitermax, ...])	Solve the HF-equations using forward recursion at fixed density.
`solve_newton`(N_target[, M0, mu0])	Solve the HF-equations using a quasi Newton method at fixed density.
`solve_newton_mu`(mu[, M0])	Solve the HF-equations using a quasi Newton method at fixed chemical potential.
`solve_setup`(N_target[, M0, mu0])
`total_density`()
`update_chemical_potential`(N_target[, mu0])
`update_density_matrix`()
`update_interaction_energy`()
`update_kinetic_energy`()
`update_mean_field`(rho_ab)
`update_mean_field_dispersion`()
`update_momentum_density_matrix`()
`update_non_int_free_energy`()
`update_total_energy`()
`vec2mat`(vec)	Converts from vector representation to a unitary matrix see mat2vec(...) for details.

mat2vec(mat)[source]

Converts a unitary matrix to a vector representation with the order

the real diagonal entries
the real part of the upper triangular entries
the imaginary part of the upper triangular entries

solve_iter(N_target, M0=None, mu0=None, nitermax=100, mixing=0.5, tol=1e-09)[source]

Solve the HF-equations using forward recursion at fixed density.

Parameters:

N_targetfloat: Total density.
M0ndarray (2D), optional: Initial mean-field (0 if None).
mu0float, optional: Initial chemical potential.
nitermaxint, optional: Maximal number of self consistent iterations.
mixingfloat, optional: Linear mixing parameter.
tolfloat, optional: Convergence in relative change of the density matrix.

Returns:

rhondarray (2D): Local density matrix.

solve_newton(N_target, M0=None, mu0=None)[source]

Solve the HF-equations using a quasi Newton method at fixed density.

Parameters:

N_totfloat: Total density.
M0ndarray (2D), optional: Initial mean-field (0 if None).
mu0float, optional: Initial chemical potential.

solve_newton_mu(mu, M0=None)[source]

Solve the HF-equations using a quasi Newton method at fixed chemical potential.

Parameters:

mu0float: Initial chemical potential.
M0ndarray (2D), optional: Initial mean-field (0 if None).

vec2mat(vec)[source]: Converts from vector representation to a unitary matrix see mat2vec(…) for details.

class triqs_tprf.hf_response.HartreeFockResponse(hartree_fock_solver, eps=1e-09)[source]

Hartree-Fock linear response calculator.

Parameters:

hartree_fock_solverHartreeFockSolver instance: Converged Hartree-Fock solver.
epsfloat: Step size in finite difference linear response calculation.

Methods

logo()

bare_response
compute_chi0_k
mode_decomposition
response

class triqs_tprf.hf_solver.HartreeSolver(e_k, beta, H_int=None, gf_struct=None, mu0=0.0, mu_max=None, mu_min=None)[source]

Hartree solver for local interactions.

Parameters:

e_kTRIQS Greens function on a Brillouin zone mesh: Single-particle dispersion.
betafloat: inverse temperature
H_intTRIQS Operator instance: Local interaction Hamiltonian
gf_structlist of lists of block index and list of internal orbital indices: gf_struct fixing orbital order between e_k and H_int
mu0float: chemical potential
mu_min, mu_maxfloat, optional: range for chemical potential search.

Methods

`chemical_potential`()
`density_matrix`()
`density_matrix_step`(rho_vec[, N_target])
`expectation_value`(op_ab)
`logo`()
`mat2vec`(mat)	Converts a unitary matrix to a vector representation with the order
`mean_field_matrix`()
`solve_iter`(N_target[, M0, mu0, nitermax, ...])	Solve the HF-equations using forward recursion at fixed density.
`solve_newton`(N_target[, M0, mu0])	Solve the HF-equations using a quasi Newton method at fixed density.
`solve_newton_mu`(mu[, M0])	Solve the HF-equations using a quasi Newton method at fixed chemical potential.
`solve_setup`(N_target[, M0, mu0])
`total_density`()
`update_chemical_potential`(N_target[, mu0])
`update_density_matrix`()
`update_interaction_energy`()
`update_kinetic_energy`()
`update_mean_field`(rho_ab)
`update_mean_field_dispersion`()
`update_momentum_density_matrix`()
`update_non_int_free_energy`()
`update_total_energy`()
`vec2mat`(vec)	Converts from vector representation to a unitary matrix see mat2vec(...) for details.

mat2vec(mat)[source]

Converts a unitary matrix to a vector representation with the order

the real diagonal entries
the real part of the upper triangular entries
the imaginary part of the upper triangular entries

vec2mat(vec)[source]: Converts from vector representation to a unitary matrix see mat2vec(…) for details.

class triqs_tprf.hf_response.HartreeResponse(hartree_solver, eps=1e-09)[source]

Hartree linear response calculator.

Parameters:

hartree_solverHartreeSolver instance: Converged Hartree solver.
epsfloat: Step size in finite difference linear response calculation.

Methods

logo()

bare_response
extract_dens_dens
response

Utility functions

triqs_tprf.lattice_utils.k_space_path(paths, num=100, bz=None, relative_coordinates=False)[source]

Generate an array of k-vectors along a path defined by a list of pairs of k-vectors

Parameters:

pathslist of pairs of three-vectors of floats: List of pairs of k-vectors in reciprocal units to create a path in-between.
numint, default=100: Number of k-vectors along each segment of the overall path
bzbrillouin_zone, optional: When a Brillouin Zone is passed, calculate distance in absolute units
relative_coordinatesbool, optional: Return k-vectors in reciprocal units. (Default False)

Returns:

kvecs: numpy.ndarray [shape=(len(paths)*num,3)]: Two-dimensional numpy array containing the path vectors (in reciprocal units) as rows
dist: numpy.ndarray [shape=(kvecs.shape[0])]: One-dimensional numpy array containing, for each element in kvecs, the distance travelled along the path. Useful for plotting. If bz is provided, calculate the distance in absolute units. The distances for the relevant k-points in paths can be obtained with dist[num::num].
ticksnumpy.ndarray [shape=(len(paths)+1)]: Array with tick points, i.e. distances at the interfaces of path segments. Only returned if return_ticks is True.

triqs_tprf.lattice_utils.chi_contraction(chi, op1, op2)[source]

Contract a susceptibility with two operators

Parameters:

chiGf,: Susceptibility \(\chi(i\omega_n, \mathbf{k})\). The mesh attribute of the Gf must be a MeshProduct with the components (MeshImFreq, MeshBrZone) and its target_rank 4.
op1, op2np.ndarray,: Operators in matrix representation.

Returns:

Gf,
Susceptibilitymath:chi(iomega_n, mathbf{k}). With a target_rank of 0.

Parameter collections

class triqs_tprf.ParameterCollection.ParameterCollection(**kwargs)[source]

Helper class for handing collections of parameters.

Parameters:

kwargsdict: Key-word argument list of parameters.

Examples

A ParameterCollection has any number of attributes, accessible with the dot operator.

>>> p = ParameterCollection(beta=10., U=1.0, t=1.0)
>>> print p
U = 1.0
beta = 10.0
t = 1.0
>>> print p.beta
10.0
>>> p.W = 1.2
>>> print p
U = 1.0
W = 1.2
beta = 10.0
t = 1.0

and can be stored and loaded to/from TRIQS hdf archives.

>>> from h5 import HDFArchive
>>> with HDFArchive('data.h5', 'w') as arch: arch['p'] = p
>>> with HDFArchive('data.h5', 'r') as arch: p_ref = arch['p']
>>> print p_ref
U = 1.0
beta = 10.0
t = 1.0

Methods

`alter`(**kwargs)	Change or add attributes
`convert_keys_from_string_to_python`(dict_key)	h5.HDFArchive incorrectly mangles tuple keys to string running this on the affected dict tries to revert this by running eval on the string representation.
`copy`()	Shallow copy

dict
get_my_name
grab_attribs
items
keys

alter(**kwargs)[source]

Change or add attributes

Returns:

pParameterCollection

convert_keys_from_string_to_python(dict_key)[source]: h5.HDFArchive incorrectly mangles tuple keys to string running this on the affected dict tries to revert this by running eval on the string representation. UGLY FIX…

copy()[source]: Shallow copy

class triqs_tprf.ParameterCollection.ParameterCollections(objects=[])[source]

Helper class for handing a series of collections of parameters.

Parameters:

objectslist: List of ParameterCollection instances.

Examples

The ParameterCollections class makes it easy to get vectors of parameters from a list of ParameterCollection objects.

>>> p1 = ParameterCollection(beta=10., U=1.0, t=1.0)
>>> p2 = ParameterCollection(beta=5., U=2.0, t=1.337)
>>> ps = ParameterCollections(objects=[p1, p2])
>>> print ps.beta
[10.  5.]
>>> print ps.U
[1. 2.]

Methods

append
getattr_from_objects
set_sorted_order
sort_on

triqs_tprf.ParameterCollection.parameter_scan(p, **kwargs)[source]

Return ParameterCollections with copies of ParameterCollection for different parameters

Uses a given ParameterCollection as a template to create copies of it with one or more parameters changing. Stores all of these copies in a ParameterCollections for easy access.

Parameters:

pParameterCollection,: The ParameterCollection that shall be used as a template for all the others
**kwargsSequence,: The keyword gives the parameter name and the Sequence the values that shall be scanned through.

Returns:

ParameterCollections

Examples

>>> p = ParameterCollection(beta=10., U=1.0, t=1.0)
>>> ps = parameter_scan(p, U=[1.0, 1.5, 2.0])
>>> print ps[0]
U = 1.0
beta = 10.0
t = 1.0
>>> print ps[1]
U = 1.5
beta = 10.0
t = 1.0
>>> print ps[2]
U = 2.0
beta = 10.0
t = 1.0