- Initialize H-S field for all space-time point by choosing randomly from
$[-1,1]$ . - Calculate propagators
$B_{\phi}(\tau,0)$ for all time slice between$[\beta, 0]$ , use UDV stablization technique and store all U,D,V matrices. - Sweep backward then forward
- Perform measurements after each sweep
- Iterate 3-4 until measurement operator converge
Define direction of propagating: forward
For each
-
For each time-slice,
- flip each field at space according to acceptance ratio
- Update H-S field, Green function if accept proposal
- Replace the elements of
$U$ ,$D$ ,$V$ by the new one after Green function is computed
-
Advance equal time Green function by $$ G(\tau_2,\tau_2) = B(\tau_2,\tau_1) G(\tau_1,\tau_1) B(\tau_2,\tau_1)^{-1}. $$ PS. propagator of
$B(\tau_2,\tau_1)$ is calculated using the new configuration of H-S field. -
Every
$N_{stable}$ time slice, recompute equal time Green function from propagator using SVD decomposition
Starting from
Then replace the SVD decompostion of
For both backward sweep and forward sweep, we just need to do the replacement along the direction defined above.
Total number of electrons to determine the doping of electron. $$ \hat{N} = \sum_i c_i^\dagger c_i. $$
It is equal to average of $ N - \text{Tr}(G(\tau,\tau))$, where
If we sample the distribution
Sign of Monte Carlo weight $$ sign[\det[I-B(\beta,0)]] $$
In pratical, we collect quantity
Only flip one site each time
$$\Delta(\tau){ij} = \delta{i,n}\delta_{j,n} \Delta(\tau)_{nn}$$
Furthermore, we have
with
$$R = \det[I + \Delta(i,\tau) (I - G(\tau,\tau))] \ = 1+ \Delta(\tau){ii}(1 - G(\tau,\tau){ii})$$
$$ G_{jk} \rightarrow G_{jk} - \frac{1}{R} G_{ji} \Delta(\tau){ii} (I - G){ik} $$
For spin up or spin down
$R$ should be$R^{\sigma}$
$$ B_{\phi}(\tau,0) = U_R D_R V_R $$ and
Parts for spin up and spin down are decoupled.
The acceptance ratio should be product of contributions of spin up and spin down as $$ R = R^{\uparrow} R^{\downarrow} $$
Chapter 7 in Quantum Monte Carlo Methods
A brief introduction of DQMC study in itinerant quantum critical point