Results

Results#

Results are returned from the SCCE solver can be stored in a file using scce.save_result.

The result files contain information on:

  • unitcell: The specification of the unitcell in the calculation, corresponding to the unitcell input

  • system: The specification of the system in the calculations, corresponding to the system input

  • job parameters: Additional parameters used in the specific run of the SCCE solver

  • result: The result of the numerical calculation

The information is stored in a dictionary-like (or hash-map-like) data structure. The keys of the dictionary/hash-map are hierarchical text strings of the form unitcell_atom_0_... The stored values can be text strings, floating point values, integer values and one and two-dimensional lists of floating point or integer values.

Result Files#

The supported output file types are:

  • hdf5: A plain HDF5 file

  • pandas: A pandas DataSeries stored in a HDF5 file

  • yaml: A plain yaml file

The output format can be selected when creating a job.

After being downloaded, each file type can be imported into python with the following code snippets:

Plain HDF5#

After the import, the result data will be stored in a standard python dictionary. Values will be stored as numpy arrays. Requires the h5py python package.

import h5py
with h5py.File('$RESULT_FILE_LOCATION','r') as f:
    results = dict()
    for key in f.keys():
        results[key] = f[key][()]

Pandas#

After the import, the result data will be stored as a pandas DataSeries. Requires the pandas python package.

import pandas as pd
results = pd.hdf_read('$RESULT_FILE_LOCATION')

JSON#

deprecated: Fails for complex numbers.

After the import, the result data will be stored in a standard python dictionary. Matrices will not be stored as numpy arrays, but as lists of lists. No additional python packages are required.

import json
with open('$RESULT_FILE_LOCATION','r') as f:
    results = json.load(f)

YAML#

deprecated: Problems with complex numbers.

After the import, the result data will be stored in a standard python dictionary. Matrices will not be stored as numpy arrays, but as lists of lists. Requires the pyyaml python package.

import yaml
with open('$RESULT_FILE_LOCATION','r') as f:
    results = yaml.load(f)

Content#

Unitcell#

  • unitcell_number_atoms: Number of atoms in one unitcell.

  • dict_input_system: contains a yaml dump of the lattice builder input

System#

  • system_cluster_size: Size of one cluster in the system in unitcells.

  • system_size: Size of the system in units of system_cluster_size.

  • system_particle_number: Number of particles simulated in the system.

  • system_total_Sz: Total z-component of spin of the system. Only used when a system with spin is considered.

  • system_spinful: Simulated system with or without spin

  • system_boundary_conditions: Boundary conditions of the system.

  • system_cluster_number_basis_states: Number of basis states for one cluster in the system.

  • system_cluster_basis_state_$NUMBER_ID: Atom id in one unitcell for basis state number $NUMBER in the cluster.

  • system_cluster_basis_state_$NUMBER_position: Atom position for basis state number $NUMBER in the cluster.

  • system_cluster_basis_state_$NUMBER_unitcell_position_in_cluster: Position of the unitcell in the cluster for basis state number $NUMBER.

Run#

  • run_inverse_mean_field_method: Method used to calculate the inverse mean-field solution

  • run_number_iterations: Number of iterations of the self-consistency loop

  • run_MF_tolerance: Tolerance of the inverse mean-field cycle.

  • run_MF_Delta: Gap of the mean field. If it's close to zero, one must resort to finite temperature mean field to resolve the Fermi surface degeneracy. Should be implemented during the beta release cycle.

  • run_dmrg_quality: Maximum of the discarded entropiy as a measure of the quality of the DMRG simulation. Should be smaller than the energy scales of interest.

  • run_dmrg_quality_sum: A pessimistic measure of the quality of the DMRG simulation. It is the sum of the discarded entropies during the last sweeping.

  • run_dmrg_kept_max: Maximal number of states kept in a DMRG block.

  • run_scce_version: The version of the SCCE used in the simulation

Result#

  • result_number_states: Number of eigenstates returned by the SCCE solver.

  • result_system_cluster_energy: Array of the energies of the eigen states of the cluster-bath system. Only use as the true cluster eigenenergy when system_size = [1, 1, 1].

  • result_system_cluster_weights: The corresponding Boltzmann weight Z_n.

  • dark_energy: The mean-field energy of the disconnected single particle states that are not entering the SCCE solver and are therefore not accounted for.

  • result_cluster_mean_field_hamiltonian_real: The real part of the cluster mean-field Hamiltonian matrix.

  • result_cluster_mean_field_hamiltonian_imag: The imaginary part of the cluster mean-field Hamiltonian matrix.

  • result_cluster_compensation_Zeeman: A local Zeeman field applied to the cluster for the spin Sz adaption.

  • NumTimeSteps: Number of time steps.

  • TimeSteps: Array of size \(NumTimeSteps + 1\) of the time steps performed including zero time. Only in td-simulations.

For fermions with spin:

  • result_state_{}_Sx: local \(S^x\) magnetization at site l: \(\left\langle \psi_{J} | \hat{S}^x_{l} | \psi_{J}\right\rangle\) with respect to eigenstate $J.

  • result_state_{}_Sy: local \(S^y\) magnetization at site l: \(\left\langle \psi_{J} | \hat{S}^y_{l} | \psi_{J}\right\rangle\) with respect to eigenstate $J. Note: always zero for real systems.

  • result_state_{}_Sz: local \(S^z\) magnetization at site l: \(\left\langle \psi_{J} | \hat{S}^z_{l} | \psi_{J}\right\rangle\) with respect to eigenstate $J.

  • result_state_$J_1rdm_up_up_real: The real part of the one particle reduced density matrix, with matrix element (l, k): \({\textrm{Re} \left( \left\langle \psi_{J} | c_{l, \uparrow}^{\dagger} c_{k, \uparrow} | \psi_{J} \right\rangle\right)}\) The real part of the expectation value of the correlator \({c_{l, \uparrow}^{\dagger} c_{k, \uparrow}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_up_up_imag: The imaginary part of the one particle reduced density matrix, with matrix element (l, k): \(\textrm{Im}\left(\left\langle \psi_{J} | c_{l, \uparrow}^{\dagger} c_{k, \uparrow} | \psi_{J}\right\rangle\right)\) The imaginary part of the expectation value of the correlator \({c_{l, \uparrow}^{\dagger} c_{k, \uparrow}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_down_down_real: The real part of the one particle reduced density matrix, with matrix element (l, k): \(\textrm{Re}\left(\left\langle \psi_{J} | c_{l, \downarrow}^{\dagger} c_{k, \downarrow} | \psi_{J}\right\rangle\right)\) The real part of the expectation value of the correlator \({c_{l, \downarrow}^{\dagger} c_{k, \downarrow}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_down_down_imag: The imaginary part of the one particle reduced density matrix, with matrix element (l, k): \(\textrm{Im}\left(\left\langle \psi_{J} | c_{l, \downarrow}^{\dagger} c_{k, \downarrow} | \psi_{J}\right\rangle\right)\) The imaginary part of the expectation value of the correlator \({c_{l, \downarrow}^{\dagger} c_{k, \downarrow}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_up_down_real: The real part of the one particle reduced density matrix, with matrix element (l, k): \({\textrm{Re} \left( \left\langle \psi_{J} | c_{l, \uparrow}^{\dagger} c_{k, \downarrow} | \psi_{J} \right\rangle\right)}\) The real part of the expectation value of the correlator \({c_{l, \uparrow}^{\dagger} c_{k, \uparrow}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_up_down_imag: The imaginary part of the one particle reduced density matrix, with matrix element (l, k): \(\textrm{Im}\left(\left\langle \psi_{J} | c_{l, \uparrow}^{\dagger} c_{k, \downarrow} | \psi_{J}\right\rangle\right)\) The imaginary part of the expectation value of the correlator \({c_{l, \uparrow}^{\dagger} c_{k, \uparrow}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_down_up_real: The real part of the one particle reduced density matrix, with matrix element (l, k): \(\textrm{Re}\left(\left\langle \psi_{J} | c_{l, \downarrow}^{\dagger} c_{k, \uparrow} | \psi_{J}\right\rangle\right)\) The real part of the expectation value of the correlator \({c_{l, \downarrow}^{\dagger} c_{k, \downarrow}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_down_up_imag: The imaginary part of the one particle reduced density matrix, with matrix element (l, k): \(\textrm{Im}\left(\left\langle \psi_{J} | c_{l, \downarrow}^{\dagger} c_{k, \uparrow} | \psi_{J}\right\rangle\right)\) The imaginary part of the expectation value of the correlator \({c_{l, \downarrow}^{\dagger} c_{k, \downarrow}}\) with respect to eigenstate $J.

  • result_state_$J_density_density_correlator: The density density correlation matrix, with matrix element (l, k): \(\left\langle \psi_{J} | (\hat n_{l, \downarrow} + \hat n_{l, \uparrow}) (\hat n_{k, \downarrow} + \hat n_{k, \uparrow}) | \psi_{J}\right\rangle\) The expectation value of the correlator \({(\hat n_{l, \downarrow} + \hat n_{l, \uparrow}) (\hat n_{k, \downarrow} + \hat n_{k, \uparrow})}\) with respect to eigenstate $J.

  • result_state_$J_parity_parity_correlator: The parity parity correlation matrix mith matrix element (l, k): \(\left\langle \psi_{J} | (-1)^{\hat n_{l, \downarrow} + \hat n_{l, \uparrow}} (-1)^{\hat n_{k, \downarrow} + \hat n_{k, \uparrow}} | \psi_{J}\right\rangle\) The expectation value of the correlator \({(-1)^{\hat n_{l, \downarrow} + \hat n_{l, \uparrow}} (-1)^{\hat n_{k, \downarrow} + \hat n_{k, \uparrow}}}\) with respect to eigenstate $J.

  • result_state_$J_Sz_Sz_correlator: \(\left\langle \psi_{J} | \hat S^z_{l} \hat S^z_{k} | \psi_{J}\right\rangle\) The real part of the spin \({S^z}\) correlation matrix with respect to eigenstate $J.

  • result_state_$J_Sp_Sm_correlator: \(\left\langle \psi_{J} | \hat S^+_{l} \hat S^-_{k} + \hat S^-_{l} \hat S^+_{k}| \psi_{J}\right\rangle\) The real part of the spin \({S^+ S^- + S^- S^+}\) correlation matrix with respect to eigenstate $J.

  • result_state_$J_double_occupancy_correlator: The correlation matrix of double occupancy of sites, with matrix element (l, k): \(\left\langle \psi_{J} | \hat n_{l, \downarrow} \hat n_{l, \uparrow} \hat n_{k, \downarrow} \hat n_{k, \uparrow} | \psi_{J}\right\rangle\) The expectation value of the correlator \({\hat n_{l, \downarrow} \hat n_{l, \uparrow} \hat n_{k, \downarrow} \hat n_{k, \uparrow}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_BCS_up_down_real: The real part of the one particle reduced anomalous pairing matrix, with matrix element (l, k): \({\textrm{Re} \left( \left\langle \psi_{J} | c_{l, \uparrow}^{} c_{k, \downarrow} | \psi_{J}\right\rangle \right)}\) The expectation value of the correlator \({c_{l, \uparrow}^{} c_{k, \downarrow}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_BCS_up_down_imag: The imaginary part of the one particle reduced anomalous pairing matrix, with matrix element (l, k): \(\textrm{Im}\left(\left\langle \psi_{J} | c_{l, \uparrow }^{} c_{k, \downarrow} | \psi_{J}\right\rangle\right)\) The imaginary part of the expectation value of the correlator \({c_{l,\uparrow}^{} c_{k, \downarrow}}\) with respect to eigenstate $J.

Note: The anomalous up-down pairing is only available if mod_N > 0.

  • result_state_$J_1rdm_BCS_up_up_real: The real part of the one particle reduced anomalous pairing matrix, with matrix element (l, k): \({\textrm{Re} \left( \left\langle \psi_{J} | c_{l, \uparrow}^{} c_{k, \uparrow} | \psi_{J}\right\rangle \right)}\) The expectation value of the correlator \({c_{l, \uparrow}^{} c_{k, \uparrow}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_BCS_up_up_imag: The imaginary part of the one particle reduced anomalous pairing matrix, with matrix element (l, k): \(\textrm{Im}\left(\left\langle \psi_{J} | c_{l, \uparrow }^{} c_{k, \uparrow} | \psi_{J}\right\rangle\right)\) The imaginary part of the expectation value of the correlator \({c_{l,\uparrow}^{} c_{k, \uparrow}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_BCS_down_down_real: The real part of the one particle reduced anomalous pairing matrix, with matrix element (l, k): \({\textrm{Re} \left( \left\langle \psi_{J} | c_{l, \downarrow}^{} c_{k, \downarrow} | \psi_{J}\right\rangle \right)}\) The expectation value of the correlator \({c_{l, \downarrow}^{} c_{k, \downarrow}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_BCS_down_down_imag: The imaginary part of the one particle reduced anomalous pairing matrix, with matrix element (l, k): \(\textrm{Im}\left(\left\langle \psi_{J} | c_{l, \downarrow }^{} c_{k, \downarrow} | \psi_{J}\right\rangle\right)\) The imaginary part of the expectation value of the correlator \({c_{l,\downarrow}^{} c_{k, \downarrow}}\) with respect to eigenstate $J.

Note: The anomalous up-up and down-down pairing are only available if mod_N > 0 and mod_Sz > 0

For fermions without spin:

  • result_state_$J_1rdm_real: The real part of the one particle reduced density matrix, with matrix element (k, l): \({\textrm{Re} \left( \left\langle \psi_{J} | c_{l}^{\dagger} c_{k} | \psi_{J}\right\rangle \right)}\) The expectation value of the correlator \({c_{l}^{\dagger} c_{k}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_imag: The imaginary part of the one particle reduced density matrix, with matrix element (k, l): \(\textrm{Im}\left(\left\langle \psi_{J} | c_{l }^{\dagger} c_{k} | \psi_{J}\right\rangle\right)\) The imaginary part of the expectation value of the correlator \({c_{l}^{\dagger} c_{k}}\) with respect to eigenstate $J.

  • result_state_$J_density_density_correlator: The density density correlation matrix, with matrix element (k, l): \(\left\langle \psi_{J} | \hat n_{l} \hat n_{k} | \psi_{J}\right\rangle\) The expectation value of the correlator \({\hat n_{l} \hat n_{k}}\) with respect to eigenstate $J.

  • result_state_$J_parity_parity_correlator: The parity parity correlation matrix, with matrix element (k, l): \(\left\langle \psi_{J} | (-1)^{\hat n_{l}} (-1)^{\hat n_{k}} | \psi_{J}\right\rangle\) The expectation value of the correlator \({(-1)^{\hat n_{l} } (-1)^{\hat n_{k}}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_BCS_real: The real part of the one particle reduced anomalous pairing matrix, with matrix element (k, l): \({\textrm{Re} \left( \left\langle \psi_{J} | c_{l}^{} c_{k} | \psi_{J}\right\rangle \right)}\) The expectation value of the correlator \({c_{l}^{} c_{k}}\) with respect to eigenstate $J.

  • result_state_$J_1rdm_BCS_imag: The imaginary part of the one particle reduced anomalous pairing matrix, with matrix element (k, l): \(\textrm{Im}\left(\left\langle \psi_{J} | c_{l }^{} c_{k} | \psi_{J}\right\rangle\right)\) The imaginary part of the expectation value of the correlator \({c_{l}^{} c_{k}}\) with respect to eigenstate $J.

  • result_cluster_mean_field_hamiltonian_BCS_real: The real part of the cluster mean-field Hamiltonian matrix for the anomalous BCS type pairing.

  • result_cluster_mean_field_hamiltonian_BCS_imag: The imaginary part of the cluster mean-field Hamiltonian matrix for the anomalous BCS type pairing.

  • rdm1_t: time resolved single particle reduced density matrix. Only in td-simulations.

Note: The anomalous pairing and BCS mean field is only available if mod_N > 0.

For spin systems:

  • result_state_{}_Sx: local \(S^x\) magnetization at site l: \(\left\langle \psi_{J} | \hat{S}^x_{l} | \psi_{J}\right\rangle\) with respect to eigenstate $J.

  • result_state_{}_Sy: local \(S^y\) magnetization at site l: \(\left\langle \psi_{J} | \hat{S}^y_{l} | \psi_{J}\right\rangle\) with respect to eigenstate $J. Note: This is currently always zero, which will be fixed in later.

  • result_state_{}_Sz: local \(S^z\) magnetization at site l: \(\left\langle \psi_{J} | \hat{S}^z_{l} | \psi_{J}\right\rangle\) with respect to eigenstate $J.

  • result_state_{}_Sz2: local squared \(S^z\) magnetization at site l: \(\left\langle \psi_{J} | \left(\hat{S}^z_{l}\right)^2 | \psi_{J}\right\rangle\) with respect to eigenstate $J. For spin 1/2 systems this will always be 1/4.

  • result_state_$J_Sz_Sz_correlator: \(\left\langle \psi_{J} | \hat S^z_{l} \hat S^z_{k} | \psi_{J}\right\rangle\) The spin \({S^z}\) correlation matrix with respect to eigenstate $J.

  • result_state_$J_Sp_Sm_correlator: \(\left\langle \psi_{J} | \hat S^+_{l} \hat S^-_{k} + \hat S^-_{l} \hat S^+_{k}| \psi_{J}\right\rangle\) The real part of the spin \({S^+ S^- + S^- S^+}\) correlation matrix with respect to eigenstate $J.

  • result_state_$J_Sz2_Sz2: The correlation matrix of the \((\hat{S}^z)^2\) operator, with matrix element (l, k): \(\left\langle \psi_{J} | \left(\hat{S}^z_{l}\right)^2 \left(\hat{S}^z_{k}\right)^2 | \psi_{J}\right\rangle\).

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