Update theory.md

docs/src/theory.md

@@ -255,7 +255,7 @@ This equation must be true for any $|\phi\rangle \in \mathcal{H}$, Eq.~(\ref{eq:
     \left(\frac{\mathrm{d}}{\mathrm{d}t} - \frac{1}{\mathrm{i}\hbar}\hat{P}_{T_{|\varphi\rangle}\mathcal{M}}\hat{H}\right)|\varphi\rangle =0\ .
 ```
 
-In the context of MPS, the manifold $\mathcal{M}$ will correspond to the space of full-ranked MPS of a given bond dimension $D$, and the tangent space will be the space spanned by variations of single MPS tensors. The projector defines an effective Hamiltonian, $\hat{\mathcal{P}}_{|\psi(t)\rangle, \mathcal{M}}$, under which the dynamics are constrained on $\mathcal{M}$. Constraining the dynamics on a manifold, introduces a projection error: the time evolution will obey to an effective Hamiltonian different from the starting one. After the introduction of TDVP as a time evolution method for MPS, Haegeman \textit{et al.} pointed out [^haegeman_unifying_2016] that there exist an analytical decomposition for the projector operator $\hat{\mathcal{P}}$ that simplifies the resolution of the equation, turning the problem into one where each matrix $A_i$ can be updated with an effective \textit{on site} Hamiltonian $\hat H_\text{eff}$ via a Schroedinger like equation. The effective Hamiltonian $\hat H_\text{eff}$ is a contraction of the Hamiltonian MPO and the current state of the other matrices composing the MPS. This allows to do a sequential update.
+In the context of MPS, the manifold $\mathcal{M}$ will correspond to the space of full-ranked MPS of a given bond dimension $D$, and the tangent space will be the space spanned by variations of single MPS tensors. The projector defines an effective Hamiltonian under which the dynamics are constrained on $\mathcal{M}$. Constraining the dynamics on a manifold, introduces a projection error: the time evolution will obey to an effective Hamiltonian different from the starting one. After the introduction of TDVP as a time evolution method for MPS, Haegeman _et al._ pointed out [^haegeman_unifying_2016] that there exist an analytical decomposition for the projector operator $\hat{\mathcal{P}}$ that simplifies the resolution of the equation, turning the problem into one where each matrix $A_i$ can be updated with an effective _on site_ Hamiltonian $\hat H_\text{eff}$ via a Schroedinger like equation. The effective Hamiltonian $\hat H_\text{eff}$ is a contraction of the Hamiltonian MPO and the current state of the other matrices composing the MPS. This allows to do a sequential update.
 
 There exist different versions of the TDVP algorithm. In `MPSDynamics.jl` three methods have been so far implemented:
 - the one-site TDVP (1TDVP)