Perturbation order histograms

To analyze the behaviour of the Markov chain in CTHYB the perturbation order histograms are a central tool. This is a simple example showing how to sample and plot these histograms.

As an example we solve the one-orbital Anderson impurity at inverse temperature \(\beta\), embedded in a flat (Wilson) bath with non-interacting Green’s function \(G^{-1}_{0,\sigma} (i\omega_n) = i \omega_n - \epsilon_f - V^2 \Gamma_\sigma(i \omega_n)\), and local interaction \(H_\mathrm{int} = U n_\uparrow n_\downarrow\), where \(U\) is the Hubbard interaction, \(\epsilon_f\) is the local energy. The bath \(\Gamma\), and the bath has bandwidth \(D\) and hybridization \(V\).

from triqs.operators import n
from h5 import HDFArchive
from triqs.gf import inverse, iOmega_n, Wilson

from triqs_cthyb import Solver

U, e_f, D, V, beta = 2., -1., 1., 1., 20.
Sz = 0.5 * ( n('up', 0) - n('down', 0) )

S = Solver(
    beta=beta, gf_struct=[('up',[0]), ('down',[0])],
    n_tau=400, n_iw=50,)

S.G0_iw << inverse(iOmega_n - e_f - V**2 * Wilson(D))

S.solve(
    h_int=U*n('up',0)*n('down',0),
    n_cycles=100000,
    length_cycle=20,
    n_warmup_cycles=100,
    measure_G_tau=False,
    measure_pert_order=True,
    )

Starting on 1 Nodes at : 2019-06-05 17:20:04.584678

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The local Hamiltonian of the problem:
(-1,0)*c_dag('down',0)*c('down',0) + (-1,0)*c_dag('up',0)*c('up',0) + (2,0)*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0)
Using autopartition algorithm to partition the local Hilbert space
Found 4 subspaces.

Warming up ...

Accumulating ...
17:20:04   0% ETA 00:00:14 cycle 667 of 100000
17:20:06  14% ETA 00:00:12 cycle 14816 of 100000
17:20:09  31% ETA 00:00:10 cycle 31517 of 100000
17:20:12  51% ETA 00:00:07 cycle 51752 of 100000
17:20:16  78% ETA 00:00:03 cycle 78933 of 100000


[Rank 0] Collect results: Waiting for all mpi-threads to finish accumulating...
[Rank 0] Timings for all measures:
Measure                   | seconds
Average sign              | 0.0121935
Perturbation order        | 0.017634
Perturbation order (down) | 0.0144553
Perturbation order (up)   | 0.011125
Total measure time        | 0.0554078
[Rank 0] Acceptance rate for all moves:
Move set Insert two operators: 0.056355
  Move  Insert Delta_up: 0.0565787
  Move  Insert Delta_down: 0.0561309
Move set Remove two operators: 0.0564614
  Move  Remove Delta_up: 0.0566373
  Move  Remove Delta_down: 0.0562848
Move  Shift one operator: 0.758509
[Rank 0] Warmup lasted: 0.0125826 seconds [00:00:00]
[Rank 0] Simulation lasted: 14.9179 seconds [00:00:14]
[Rank 0] Number of measures: 100000
Total number of measures: 100000
Average sign: (1,0)

Plotting perturbation order histograms

The measured histograms are available as the member properties S.perturbation_order and S.perturbation_order_total of the cthyb solver.

For an ergodic Markov chain with sufficient number of warmup sweeps, the perturbation order distribution should be approximately Gaussian (or Possonian for low average orders).

from triqs.plot.mpl_interface import oplot, oplotr, plt

oplot(S.perturbation_order_total, label='total')
for b in S.perturbation_order:
    oplot(S.perturbation_order[b], label='block {:s}'.format(b))

plt.semilogy([], [])
plt.xlim([0, 40]);

The total average perturbation order is important information to determine the length_cycle parameter that controls the number of attempted moves between measurements. To ensure low correlation between measurements the length_cycle should be high enough that the attempted moves can significanty change the configuration.

Autor: H. U.R. Strand (2019)