Here we introduce the variational groundstate search for matrix product states. The goal is to find the MPS representation of the groundstate of the onedimensional lattice with a given hamiltonian H. Furthermore we limit the matrix dimension m of the matrices in the MPS to a maximal value. Therefore we try to find the best MPS representation of the groundstate for a given matrix dimension m.
The main idea is to use the variational principle. So we want to find the MPS $\lvert\psi\rangle$ that minimizes: \begin{align} E = min\frac{\langle\psi\lvert\hat H\lvert\psi\rangle}{\langle\psi\lvert\psi\rangle} \end{align}
This problem can be expressed by using the Lagrange-formalism. So we want to minimize $\langle\psi\lvert\hat H\lvert\psi\rangle$ under the constraint that the norm $\langle\psi\lvert\psi\rangle$ is constant. Introducing the groundstate energy E as the Lagrange-multiplier, we get the Langrange function: \begin{align} L=\langle\psi\lvert\hat H\lvert\psi\rangle - E(\langle\psi\lvert\psi\rangle-1) \end{align}
The usual way to continue is to differentiate the Lagrange function for all $p$ parameters of the trial wavefunction and set the differentiations to zero. This will lead to $p+1$ coupled equations that have to be solved consistently. Within MPS there are $p=d\cdot n$ matrices with $m^2$ coefficients. This makes $p=d\cdot n\cdot m^2$ parameters in total and therefore the global optimization will just work for very small dimensions m.
Therefore we follow the route of the DMRG: We just optimize one matrix of the MPS at a time and sweep through the system optimizing the matrices one after each other.
As stated beforehand the idea is not to optimize all matrices at the same time, but just optimize one matrix at site l in the MPS and then shift to the next matrix and continue.
In order to use the local DMRG optimization, we have to represent the expectation value $\langle\psi\lvert\hat H\lvert\psi\rangle$ in a compact form for the matrix at site l.
\begin{align} \langle\psi\lvert\hat H\lvert\psi\rangle = & \sum_{\boldsymbol{\sigma, \sigma^\prime}}\langle\boldsymbol\sigma\lvert \underbrace{B^{\sigma_L*}\cdots B^{\sigma_{l+1}*}M^{\sigma_l*}A^{\sigma_{l-1}*}\cdots A^{\sigma_1*}}_{B^{\sigma_L*}\cdots A^{\sigma_1*}} \hat H \underbrace{A^{\sigma_1^{\prime}}\cdots A^{\sigma_{l-1}^{\prime}}M^{\sigma_l^{\prime}}B^{\sigma_{l+1}^{\prime}}\cdots B^{\sigma_L^{\prime}}}_{A^{\sigma_1^{\prime}}\cdots B^{\sigma_L^{\prime}}}\lvert\boldsymbol{\sigma^\prime}\rangle\newline = & \sum_{\boldsymbol{\sigma, \sigma^\prime}}\sum_{\boldsymbol{\tilde\sigma, \tilde\sigma^\prime}}\langle\boldsymbol\sigma\lvert B^{\sigma_L*}\cdots A^{\sigma_1*} \lvert\boldsymbol{\tilde\sigma}\rangle W^{\tilde\sigma_1,\tilde\sigma_1^\prime}\cdots W^{\tilde\sigma_L,\tilde\sigma_L^\prime}\langle\boldsymbol{\tilde\sigma^\prime}\lvert A^{\sigma_1^{\prime}}\cdots B^{\sigma_L^{\prime}}\lvert\boldsymbol{\sigma^\prime}\rangle\newline = & \sum_{\boldsymbol{\sigma, \sigma^\prime}}\sum_{\boldsymbol{\tilde\sigma, \tilde\sigma^\prime}}\underbrace{\langle\boldsymbol\sigma\lvert\boldsymbol{\tilde\sigma}\rangle}_{\delta_{\sigma, \tilde\sigma}} \sum_{a_i, a_i^\prime, b_i} B^{\sigma_L*}_{b_{L-1}, 1}\cdots A^{\sigma_1*}_{1, a_1} W^{\tilde\sigma_1,\tilde\sigma_1^\prime}_{1, b_1}\cdots W^{\tilde\sigma_L,\tilde\sigma_L^\prime}_{b_{L-1}, 1} A^{\sigma_1^{\prime}}_{1, a_1^\prime}\cdots B^{\sigma_L^{\prime}}_{a_{L-1}^\prime, 1}\underbrace{\langle\boldsymbol{\tilde\sigma^\prime}\lvert\boldsymbol{\sigma^\prime}\rangle}_{\delta_{\sigma^\prime, \tilde\sigma^\prime}}\newline = & \sum_{a_{l-1},a_l}\sum_{a_{l-1}^{\prime}, a_{l}^\prime}\sum_{b_{l-1}, b_l}L^{a_{l-1}}_{a_{l-1}^{\prime}, b_{l-1}} \left(\sum_{\sigma_{l}, \sigma^\prime_{l}}M^{\sigma_{l}*}_{a_{l-1}, a_{l}}W^{\sigma_{l}, \sigma_{l}^\prime}_{b_{l-1}, b_{l}}M^{\sigma^{\prime}_{l}*}_{a_{l-1}^\prime, a_{l}^\prime}\right)R^{a_{l}}_{a_{l}^{\prime}, b_{l}} \end{align}
Whereby, $L^{a_{l-1}}_{a_{l-1}^{\prime}, b_{l-1}}$ represents the left side of the matrix at site l and $R^{a_{l}}_{a_{l}^{\prime}, b_{l}}$ the right side. They are explicitly given by:
\begin{align} L^{a_{l-1}}_{a_{l-1}^{\prime}, b_{l-1}}=\sum_{\{a_i, a_i^\prime, b_i; i<l-1\}}\left(\sum_{\sigma_1, \sigma^\prime_1}A^{\sigma_1*}_{1, a_1}W^{\sigma_1, \sigma_1^\prime}_{1, b_1}A^{\sigma^{\prime}_1}_{1, a_1^\prime}\right)\cdots\left(\sum_{\sigma_{l-1}, \sigma^\prime_{l-1}}A^{\sigma_{l-1}*}_{a_{l-2}, a_{l-1}}W^{\sigma_{l-1}, \sigma_{l-1}^\prime}_{b_{l-2}, b_{l-1}}A^{\sigma^{\prime}_{l-1}}_{a_{l-2}^\prime, a_{l-1}^\prime}\right) \end{align}
\begin{align} R^{a_{l}}_{a_{l}^{\prime}, b_{l}}=\sum_{\{a_i, a_i^\prime, b_i; i>l\}}\left(\sum_{\sigma_{l+1}, \sigma^\prime_{l+1}}B^{\sigma_{l+1}*}_{a_l, a_{l+1}}W^{\sigma_{l+1}, \sigma_{l+1}^\prime}_{b_l, b_{l+1}}B^{\sigma^{\prime}_{l+1}}_{a_l^{\prime}, a_{l+1}^\prime}\right)\cdots\left(\sum_{\sigma_{L}, \sigma^\prime_{L}}A^{\sigma_{L}*}_{a_{L-1}, 1}W^{\sigma_{L}, \sigma_{L}^\prime}_{b_{L-1}, 1}A^{\sigma^{\prime}_{L}}_{a_{L-1}^\prime, 1}\right) \end{align}
Also the scalar product $\langle\psi\lvert\psi\rangle$ can be given in local form:
\begin{align} \langle\psi\lvert\psi\rangle = &\sum_{\boldsymbol{\sigma, \sigma^\prime}}\langle\boldsymbol\sigma\lvert B^{\sigma_L*}\cdots B^{\sigma_{l+1}*}M^{\sigma_l*}\underbrace{A^{\sigma_{l-1}*}\cdots A^{\sigma_1*}A^{\sigma_1^{\prime}}\cdots A^{\sigma_{l-1}^{\prime}}}_{1}M^{\sigma_l^{\prime}}B^{\sigma_{l+1}^{\prime}}\cdots B^{\sigma_L^{\prime}}\lvert\boldsymbol{\sigma^\prime}\rangle\newline = &\sum_{\sigma_{l+1}, \ldots, \sigma_L}\sum_{a_l}\left(M^{\sigma_l*}M^{\sigma_l}\underbrace{B^{\sigma_{l+1}^{\prime}}\cdots B^{\sigma_L^{\prime}}B^{\sigma_L*}\cdots B^{\sigma_{l+1}*}}_{1}\right)_{a_l, a_l}\newline = &\sum_{\sigma_l}\text{Tr}\left(M^{\sigma_l*}M^{\sigma_l}\right) \end{align}
After optimization of a single matrix we want to continue with the next one. In order to setup the new L and R, we can use that the new L and R can be updated from the previous by:
\begin{align} L^{a_{l-1}}_{a_{l-1}^{\prime}, b_{l-1}}= &\sum_{a_{l-2}, a_{l-2}^\prime, b_{l-2}}L^{a_{l-2}}_{a_{l-2}^{\prime}, b_{l-2}}\left(\sum_{\sigma_{l-1}, \sigma^\prime_{l-1}}A^{\sigma_{l-1}*}_{a_{l-2}, a_{l-1}}W^{\sigma_{l-1}, \sigma_{l-1}^\prime}_{b_{l-2}, b_{l-1}}A^{\sigma^{\prime}_{l-1}}_{a_{l-2}^\prime, a_{l-1}^\prime}\right)\newline = &\sum_{\sigma_{l-1}, \sigma^\prime_{l-1}} \sum_{a_{l-2}, a_{l-2}^\prime}A^{\sigma_{l-1}*}_{a_{l-2}, a_{l-1}}\left(\sum_{b_{l-2}}L^{a_{l-2}}_{a_{l-2}^{\prime}, b_{l-2}}W^{\sigma_{l-1}, \sigma_{l-1}^\prime}_{b_{l-2}, b_{l-1}}\right)A^{\sigma^{\prime}_{l-1}}_{a_{l-2}^\prime, a_{l-1}^\prime}\newline = &\sum_{\sigma_{l-1}, \sigma^\prime_{l-1}} \sum_{a_{l-2}, a_{l-2}^\prime}A^{\sigma_{l-1}*}_{a_{l-2}, a_{l-1}}\left(L^{a_{l-2}}W^{\sigma_{l-1}, \sigma_{l-1}^\prime}\right)_{a_{l-2}^{\prime}, b_{l-1}}A^{\sigma^{\prime}_{l-1}}_{a_{l-2}^\prime, a_{l-1}^\prime}\newline = &\sum_{\sigma_{l-1}, \sigma^\prime_{l-1}} \sum_{a_{l-2}}A^{\sigma_{l-1}*}_{a_{l-2}, a_{l-1}}\left(\sum_{a_{l-2}^\prime}\left(\left(L^{a_{l-2}}W^{\sigma_{l-1}, \sigma_{l-1}^\prime}\right)^t\right)_{b_{l-1}, a_{l-2}^{\prime}}A^{\sigma^{\prime}_{l-1}}_{a_{l-2}^\prime, a_{l-1}^\prime}\right)\newline = &\sum_{\sigma_{l-1}, \sigma^\prime_{l-1}} \sum_{a_{l-2}}A^{\sigma_{l-1}*}_{a_{l-2}, a_{l-1}}\left(\left(L^{a_{l-2}}W^{\sigma_{l-1}, \sigma_{l-1}^\prime}\right)^tA^{\sigma^{\prime}_{l-1}}\right)_{b_{l-1}, a_{l-1}^\prime} \end{align}
Please note that $L^{a_{0}}$ is a scalar with value 1. There is an analog formulation for $R^{a_l}_{a_l^\prime, b_l}$:
\begin{align} R^{a_{l}}_{a_{l}^{\prime}, b_{l}}= \sum_{\sigma_{l+1},\sigma^\prime_{l+1}}\sum_{a_{l+1}}B^{\sigma_{l+1}*}_{a_{l}, a_{l+1}}\left(W^{\sigma_{l+1}, \sigma_{l+1}^\prime}\left(B^{\sigma^{\prime}_{l+1}}R^{a_{l+1}}\right)^t\right)_{b_{l}, a_{l}^\prime} \end{align}
We have shown how to setup local forms for the expectation value and the scalar product in respect to a single matrix at site l. Now we want to use the variational principle. Therefore we have to differentiate the Lagrange function: \begin{align} L=\langle\psi\lvert\hat H\lvert\psi\rangle - E(\langle\psi\lvert\psi\rangle-1) \end{align}
in respect to all coefficients $M^{\sigma_l}_{a_{l-1}, a_l}$ and set these differentiations to zero. By this we get: \begin{align} 0\stackrel{!}{=} & \frac{\partial L}{\partial M^{\sigma_L}_{a_{l-1}, a_l}} = \frac{\partial}{\partial M^{\sigma_L}_{a_{l-1}, a_l}}\left(\langle\psi\lvert\hat H\lvert\psi\rangle\right)-\lambda\frac{\partial}{\partial M^{\sigma_L}_{a_{l-1}, a_l}}\left(\langle\psi\lvert\lvert\psi\rangle\right)\newline = & \sum_{\sigma_l^\prime}\sum_{a_{l-1}^\prime, a_l^\prime}\sum_{b_{l-1}, b_l} L_{a_{l-1}^\prime, b_{l-1}}^{a_{l-1}} W_{b_{l-1}, b_l}^{\sigma_l, \sigma_l^\prime} R_{a_l^\prime, b_l}^{a_l}M_{a_{l-1}^\prime, a_l^\prime}^{\sigma_l^\prime} - \lambda\sum_{a_{l-1}^\prime, a_l^\prime} M^{\sigma_l^{\prime}}_{a_{l-1}^\prime, a_l^\prime} \end{align}
Now we get introduce $H_{\text{Eff}}$ by: \begin{align} H_{\text{Eff}(\sigma_la_{l-1}a_l)(\sigma_l^{\prime}a_{l-1}^{\prime}a_l^{\prime})}=\sum_{b_{l-1}, b_l}L_{a_{l-1}^\prime, b_{l-1}}^{a_{l-1}} W_{b_{l-1}, b_l}^{\sigma_l, \sigma_l^\prime} R_{a_l^\prime, b_l}^{a_l} \end{align} and $\nu_{\sigma_la_{l-1}a_l}=M_{a_{l-1}, a_l}^{\sigma_l}$. So we get a local Eigenvalue problem: \begin{align} H_{\text{Eff}}\nu - \lambda\nu = 0 \end{align} This eigenvalue problem can be solved exactly (or approximately by a Lanczos) and the smallest eigenvalue gives the optimized value for the overall groundstate energy. The related eigenvector $\nu_0$ gives the new optimized matrix: $M^{\sigma_L}$
In order to optimize a given starting state \(\lvert \psi \rangle\) iteratively to represent the groundstate we will proceed as follows:
At first we will fill the matrices of the starting state with random values. Then we will right-normalize the state and by this create all the \(R\)-matrices The we will create the Hamilton-MPO. The sweep will start at site \(l=1\) and the MPS has the form : \begin{equation*} M^{\sigma_1}B^{\sigma_{2}}\cdots B^{\sigma_L} \end{equation*}
At first we will setup the \(H_{\text{Eff}}\) at site \(l\) and solve the related local Eigenvalue problem. Then we will update the matrix \(M^{\sigma_l}\) by the (reshaped) eigenvector of the smallest eigenvalue (1). In order to extend the left-normalized block we will apply a SVD on the new matrix \(\tilde M^{\sigma_l}\) and save the \(U\) as the new \(A^{\sigma_l}\) while the \(S V^{\dagger}\) is multiplied to the matrix \(B^{\sigma_{l+1}}\) which shrinks the right block (2) : \begin{align*} A^{\sigma_1}\cdots A^{\sigma_{l-1}} M^{\sigma_{l}} B^{\sigma_{l+1}}\cdots B^{\sigma_{L}} &\xrightarrow{(1)}A^{\sigma_1}\cdots A^{\sigma_{l-1}} \tilde{M}^{\sigma_{l}} B^{\sigma_{l+1}}\cdots B^{\sigma_{L}}\newline &\xrightarrow{(2)}A^{\sigma_1}\cdots A^{\sigma_{l}} M^{\sigma_{l+1}} B^{\sigma_{l+2}}\cdots B^{\sigma_{L}} \end{align*} From the singular values (diagonal elements of \(S\)) we can setup the density matrix \(\rho=S^2\) and calculate the “von Neumann”-Entanglement-Entropy \(S_{\text{vN}}(\lvert\psi\rangle)=-\text{Tr}\rho \log_2 \rho\). Then we will proceed with the next site (\(l\rightarrow l+1\)).
Repeating step 2 until we have reached \(l=L\) will give one complete right sweep. The MPS is now left-normalized: \begin{equation*} A^{\sigma_{1}}\cdots A^{\sigma_{L-1}}M^{\sigma_L} \end{equation*}
Analogously to step 3 we can sweep also to the left side. We will start with site \(l=L\) and end with site \(l=1\).
Repeating steps 3 and 4 is called sweeping. We will stop the sweeping untill a certain criteria is fullfilled (e.g. convergence, max. number of sweeps)
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