| ... | ... | @@ -3,13 +3,41 @@ This code is based on the method described in the supplementary information of [ |
|
|
|
|
|
|
|
|
|
|
|
<h1>Section 1: Hamiltonian</h1>
|
|
|
|
We consider spinful Fermions on a lattice with L sites.
|
|
|
|
<a href="https://www.codecogs.com/eqnedit.php?latex=\hat{c}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\hat{c}" title="\hat{c}" /></a>
|
|
|
|
We consider spinful Fermions on a lattice with L sites. The Hamiltonian we consider is
|
|
|
|
|
|
|
|
<a href="https://www.codecogs.com/eqnedit.php?latex=H=&space;J\sum_{i=1,&space;\sigma=&space;\downarrow,&space;\uparrow}^L&space;\hat{c}_{i+1,&space;\sigma}^{\dagger}&space;\hat{c}_{i,&space;\sigma}&space;+&space;h.c&space;+&space;\sum_{i=1,&space;\sigma=&space;\downarrow,&space;\uparrow}^L&space;V_{i,&space;\sigma}&space;\hat{c}_{i,&space;\sigma}^{\dagger}&space;\hat{c}_{i,&space;\sigma}&space;+&space;U&space;\sum_{i=1}^L&space;\hat{c}_{i,&space;\uparrow}^{\dagger}&space;\hat{c}_{i,&space;\uparrow}\hat{c}_{i,&space;\downarrow}^{\dagger}&space;\hat{c}_{i,&space;\downarrow}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?H=&space;J\sum_{i=1,&space;\sigma=&space;\downarrow,&space;\uparrow}^L&space;\hat{c}_{i+1,&space;\sigma}^{\dagger}&space;\hat{c}_{i,&space;\sigma}&space;+&space;h.c&space;+&space;\sum_{i=1,&space;\sigma=&space;\downarrow,&space;\uparrow}^L&space;V_{i,&space;\sigma}&space;\hat{c}_{i,&space;\sigma}^{\dagger}&space;\hat{c}_{i,&space;\sigma}&space;+&space;U&space;\sum_{i=1}^L&space;\hat{c}_{i,&space;\uparrow}^{\dagger}&space;\hat{c}_{i,&space;\uparrow}\hat{c}_{i,&space;\downarrow}^{\dagger}&space;\hat{c}_{i,&space;\downarrow}" title="H= J\sum_{i=1, \sigma= \downarrow, \uparrow}^L \hat{c}_{i+1, \sigma}^{\dagger} \hat{c}_{i, \sigma} + h.c + \sum_{i=1, \sigma= \downarrow, \uparrow}^L V_{i, \sigma} \hat{c}_{i, \sigma}^{\dagger} \hat{c}_{i, \sigma} + U \sum_{i=1}^L \hat{c}_{i, \uparrow}^{\dagger} \hat{c}_{i, \uparrow}\hat{c}_{i, \downarrow}^{\dagger} \hat{c}_{i, \downarrow}" /></a>
|
|
|
|
|
|
|
|
Although the general code Exact_Diagonalization.py' is designed for any onsite potential, in the script 'run_this_script.py' we use an on-site potential of the form <a href="https://www.codecogs.com/eqnedit.php?latex=V_{i,&space;\sigma}=&space;\Delta_{\sigma}(i-L/2)&space;+&space;\alpha&space;(i-L/2)^2&space;+&space;\Delta_{r}&space;+&space;\Delta_{aa}\cos(2\pi&space;\beta&space;i&space;+&space;\phi_{aa}&space;)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?V_{i,&space;\sigma}=&space;\Delta_{\sigma}(i-L/2)&space;+&space;\alpha&space;(i-L/2)^2&space;+&space;\Delta_{r}&space;+&space;\Delta_{aa}\cos(2\pi&space;\beta&space;i&space;+&space;\phi_{aa}&space;)" title="V_{i, \sigma}= \Delta_{\sigma}(i-L/2) + \alpha (i-L/2)^2 + \Delta_{r} + \Delta_{aa}\cos(2\pi \beta i + \phi_{aa} )" /></a>.
|
|
|
|
|
|
|
|
Below is a decription of the variables that appear in secion 1 of the script 'run_this_script.py'
|
|
|
|
|
|
|
|
1. J is the hopping rate (as it appears in the above Hamiltonian)
|
|
|
|
2. Delta_random = <a href="https://www.codecogs.com/eqnedit.php?latex=\Delta_{r}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\Delta_{r}" title="\Delta_{r}" /></a>
|
|
|
|
3. Delta_aubry_andre = <a href="https://www.codecogs.com/eqnedit.php?latex=\Delta_{aa}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\Delta_{aa}" title="\Delta_{aa}" /></a>
|
|
|
|
4. phi_aubry_andre= <a href="https://www.codecogs.com/eqnedit.php?latex=\phi_{aa}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\phi_{aa}" title="\phi_{aa}" /></a>
|
|
|
|
5. Delta_dn = <a href="https://www.codecogs.com/eqnedit.php?latex=\Delta_{\downarrow}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\Delta_{\downarrow}" title="\Delta_{\downarrow}" /></a>
|
|
|
|
6. Delta_up = <a href="https://www.codecogs.com/eqnedit.php?latex=\Delta_{\downarrow}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\Delta_{\uparrow}" title="\Delta_{\uparrow}" /></a>
|
|
|
|
7. U is as in the Hamiltonian.
|
|
|
|
8. ti is the initial time in milliseconds.
|
|
|
|
9. tf is the final time in milliseconds
|
|
|
|
10. n is the number of samples desired per millisecond.
|
|
|
|
11. alpha = <a href="https://www.codecogs.com/eqnedit.php?latex=\alpha" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\alpha" title="\alpha" /></a> in the Hamiltonian above.
|
|
|
|
|
|
|
|
<h1>Section 2: System size and initial state</h1>
|
|
|
|
The varaibles that appear in this section are described below.
|
|
|
|
|
|
|
|
1. L = number of lattice sites. The sites are numbered as 0, 1, 2, ... , L-1.
|
|
|
|
2. up_atm = an ordered array of integers less than L indicating the sites occupied by a spin up atom in the initial state.
|
|
|
|
3. dn_atm = an ordered array of integers less than L indicating the sites occupied by a spin down atom in the initial state.
|
|
|
|
|
|
|
|
<h1>Example</h1>
|
|
|
|
|
|
|
|
We consider L=21 sites and an initial state with up_atm=[1, 5, 9, 13, 17] and dn_atm=[3, 7, 11, 15, 19] and compute its time evolution under the stark Hamiltonian with <img src="http://www.sciweavers.org/tex2img.php?eq=%5CDelta%3D3J&bc=White&fc=Black&im=jpg&fs=12&ff=arev&edit=0" align="center" border="0" alt="\Delta=3J" width="56" height="19" /> and U=5J.
|
|
|
|
We consider L=21 sites and an initial state with
|
|
|
|
|
|
|
|
up_atm=[1, 5, 9, 13, 17] and
|
|
|
|
|
|
|
|
dn_atm=[3, 7, 11, 15, 19]
|
|
|
|
|
|
|
|
We compute its time evolution under the stark Hamiltonian with <a href="https://www.codecogs.com/eqnedit.php?latex=\Delta&space;=&space;3J" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\Delta&space;=&space;3J" title="\Delta = 3J" /></a> and U=5J.
|
|
|
|
|
|
|
|
 |
|
|
\ No newline at end of file |
