Dynamical spin-spin susceptibility \(\chi_{S_z S_z}(\tau)\)

CTHYB can measure dynamical susceptibilities \(\chi_{\hat{O}_1 \hat{O}_2}(\tau)\) of operators \(\hat{O}_i\) that commute with the local Hamiltonian \([H_{loc}, \hat{O}_i] = 0\). This is performed using “sampling by insertion”.

Here we reuse the Anderson model in a Wilson bath from the first tutorail and sample its spin-spin susceptibility

\[\chi_{S_z S_z} = \langle S_z(\tau) S_z(0) \rangle\]

using the measure_O_tau parameter, by passing the pair of operators to sample. In this case the spin-z operator \(S_z = (n_\uparrow - n_\downarrow)/2\).

The one-orbital Anderson impurity embedded in a flat (Wilson) conduction 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 \(H_\mathrm{int} = U n_\uparrow n_\downarrow\), with Hubbard \(U\), local energy \(\epsilon_f\), bandwidth \(D\) and hybridization \(V\) at the inverse temperature of \(\beta\).

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 = 5., -2.5, 1., 1., 10.
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=10000,
    length_cycle=20,
    n_warmup_cycles=100,
    measure_O_tau=(Sz, Sz),
    measure_O_tau_min_ins=100,
    )

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The local Hamiltonian of the problem:
(-2.5,0)*c_dag('down',0)*c('down',0) + (-2.5,0)*c_dag('up',0)*c('up',0) + (5,0)*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 ...
17:19:09   1% ETA 00:00:07 cycle 140 of 10000
17:19:11  35% ETA 00:00:03 cycle 3587 of 10000
17:19:14  78% ETA 00:00:01 cycle 7869 of 10000


[Rank 0] Collect results: Waiting for all mpi-threads to finish accumulating...
[Rank 0] Timings for all measures:
Measure                 | seconds
Average sign            | 0.00148106
G_tau measure           | 0.0117463
O_tau insertion measure | 4.88952
Total measure time      | 4.90274
[Rank 0] Acceptance rate for all moves:
Move set Insert two operators: 0.0881532
  Move  Insert Delta_up: 0.0870787
  Move  Insert Delta_down: 0.0892352
Move set Remove two operators: 0.0876721
  Move  Remove Delta_up: 0.0871474
  Move  Remove Delta_down: 0.0881937
Move  Shift one operator: 0.441899
[Rank 0] Warmup lasted: 0.0381063 seconds [00:00:00]
[Rank 0] Simulation lasted: 5.81935 seconds [00:00:05]
[Rank 0] Number of measures: 10000
Total number of measures: 10000
Average sign: (1,0)

Starting on 1 Nodes at : 2019-06-05 17:19:09.074932

The measured susceptibility is stored in the member property S.O_tau and can be plotted using the TRIQS oplot command.

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

oplotr(S.O_tau)
plt.ylabel(r'$\chi_{S_z S_z}(\tau)$');
plt.ylim([0, 0.25])

(0, 0.25)

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