Thank you for quick reply and explanation - this makes sense and is surely correct. I was trying to implement a more complicated Hamiltonian, which then made me generally confused...
I am looking at a Hamiltonian of the form:
\(
H = J \sum_{<i,j>}\sum_{a,b=1}^3 T^{ab}_i T^{ab}_j
\)
where \(\sigma^a\) and \(\tau^a\) are Pauli operators in spin and valley spaces, and \(T^{ab}=\sigma^a \otimes \tau^b \).
In other words, this is two spin sites grouped together, such that:
Python: Select all
ss = SpinSite(conserve='Sz')
gs = GroupedSite([ss, ss], labels=['spin', 'valley'], charges='same')
Expanding out this Hamiltonian yields:
\(
\begin{align}
\frac{H}{J}=\sum_{<i,j>}( &(\sigma^x_i \otimes \tau^x_i)(\sigma^x_j \otimes \tau^x_j)+(\sigma^x_i \otimes \tau^y_i)(\sigma^x_j \otimes \tau^y_j)+(\sigma^x_i \otimes \tau^z_i)(\sigma^x_j \otimes \tau^z_j) \\
+& (\sigma^y_i \otimes \tau^x_i)(\sigma^y_j \otimes \tau^x_j)+(\sigma^y_i \otimes \tau^y_i)(\sigma^y_j \otimes \tau^y_j)+(\sigma^y_i \otimes \tau^z_i)(\sigma^y_j \otimes \tau^z_j) \\
+& (\sigma^z_i \otimes \tau^x_i)(\sigma^z_j \otimes \tau^x_j)+(\sigma^z_i \otimes \tau^y_i)(\sigma^z_j \otimes \tau^y_j)+(\sigma^z_i \otimes \tau^z_i)(\sigma^z_j \otimes \tau^z_j))
\end{align}
\)
Focusing on the first summand, I can write:
\(\sigma^x_i \otimes \tau^x_i = \left( \frac{\sigma^+_i + \sigma^-_i}{2} \right)\left( \frac{\tau^+_i + \tau^-_i}{2} \right) = \frac{1}{4} (\sigma^+_i \tau ^+_i + \sigma^+_i \tau^-_i + \sigma^-_i \tau^+_i + \tau^-_i \sigma^-_i)\).
These operators do not commute here because they are acting on the same site. Expanding the Hamiltonian in this way yields:
\(
\frac{H}{J}=\sum_{<i,j>}\left( \frac{\sigma^+_i \tau^-_i \sigma^-_j \tau^+_j}{4}+\frac{\sigma^-_i \tau^+_i \sigma^+_j \tau^-_j}{4}+\frac{\sigma^-_i \tau^-_i \sigma^+_j \tau^+_j}{4} + \frac{\sigma^+_i \tau^+_i \sigma^-_j \tau^-_j}{4} + \frac{\sigma^+_i \tau^z_i \sigma^-_j \tau^z_j}{2}+\frac{\sigma^-_i \tau^z_i \sigma^+_j \tau^z_j}{2}+\frac{\sigma^z_i \tau^+_i \sigma^z_j \tau^-_j}{2} + \frac{\sigma^z_i \tau^-_i \sigma^z_j \tau^+_j}{2}\right)+\sigma^z_i \tau^z_i \sigma^z_j \tau^z_j
\)
...which I think can be written as...
\(
\frac{H}{J}=\sum_{<i,j>}\left( \frac{\sigma^+_i \tau^-_i \sigma^-_j \tau^+_j}{4}+\frac{\sigma^-_i \tau^-_i \sigma^+_j \tau^+_j}{4} + \frac{\sigma^+_i \tau^z_i \sigma^-_j \tau^z_j}{2}+\frac{\sigma^z_i \tau^+_i \sigma^z_j \tau^-_j}{2} +\text{H.c.}\right)+\sigma^z_i \tau^z_i \sigma^z_j \tau^z_j
\)
My attempt to implement this in TeNPy is then:
Python: Select all
for u1, u2, dx in self.lat.nearest_neighbors:
self.add_coupling(1 / 4 * J, u1, 'Spspin Smvalley', u2, 'Smspin Spvalley', dx)
self.add_coupling(np.conj(1 / 4 * J), u2, 'Spspin Smvalley', u1, 'Smspin Spvalley', -dx) # H.c.
self.add_coupling(1 / 4 * J, u1, 'Smspin Smvalley', u2, 'Spspin Spvalley', dx)
self.add_coupling(np.conj(1 / 4 * J), u2, 'Smspin Smvalley', u1, 'Spspin Spvalley', -dx) # H.c.
self.add_coupling(1 / 2 * J, u1, 'Spspin Szvalley', u2, 'Smspin Szvalley', dx)
self.add_coupling(np.conj(1 / 2 * J), u2, 'Spspin Szvalley', u1, 'Smspin Szvalley', -dx) # H.c.
self.add_coupling(1 / 2 * J, u1, 'Szspin Spvalley', u2, 'Szspin Smvalley', dx)
self.add_coupling(np.conj(1 / 2 * J), u2, 'Szspin Spvalley', u1, 'Szspin Smvalley', -dx) # H.c.
self.add_coupling(J, u1, 'Szspin Szvalley', u2, 'Szspin Szvalley', dx)
The code runs at least... Does this implementation look correct to you?
Also for the cases where one has compound operators on both sides, such as:
