home.md

 <h1>Introduction</h1>
-This code is based on the method described in the supplementary information of [this paper](https://arxiv.org/abs/2010.12965). It can be used to compute the time-evolution of a given inital state of spinful fermions in a 1D lattice under the Fermi-Hubbard Hamiltonian.  The script 'Exact_Diagonalization.py' consists of the actual time evolution function. The script 'run_this_script.py' consists of all the parameters of the Hamiltonian and the initial state. Typically, one needs to edit only the latter script. The parameters are divided into two sections.
+This code is based on the method described in the supplementary information of [this paper](https://arxiv.org/abs/2010.12965). It can be used to compute the time-evolution of a given initial state of spinful fermions in a 1D lattice under the Fermi-Hubbard Hamiltonian.  The script 'Exact_Diagonalization.py' consists of the actual time evolution function. The script 'run_this_script.py' consists of all the parameters of the Hamiltonian and the initial state. Typically, one needs to edit only the latter script. The parameters are divided into two sections.
 
 
 <h1>Section 1: Hamiltonian</h1>
@@ -20,9 +20,11 @@ Below is a decription of the variables that appear in secion 1 of the script 'ru
 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. 
+10. n is the number of samples desired per millisecond. Total number of samples is m= int(n*(tf-ti)) +1. 
 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.
 
@@ -30,6 +32,15 @@ The varaibles that appear in this section are described below.
 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>Section 3: Time evolution</h1>
+The time evolution produces three arrays.
+
+1. T = a 1D array of size m+1 consisting of points in time between ti and tf. 
+2. NEUP= a 1D array of size m+1 consisting of the total spin-up occupancy of even sites at each of the m+1 points in time. 
+3. NOUP= a 1D array of size m+1 consisting of the total spin-down occupancy of even sites at each of the m+1 points in time. 
+
+The spin resolved even-odd imbalance can be computed as shown in the script. 
+
 <h1>Example</h1>
 
 We consider L=21 sites and an initial state with 
@@ -38,6 +49,6 @@ 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. 
+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. We show the even-odd imbalance of the spin up atoms from this computation below:
 
 ![Figure_1](uploads/96d14f8fb29acd16f899c7aa7a98bd66/Figure_1.png)
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