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Time-Step Targeting Method (Local Krylov)
$ \newcommand{\braket}[1]{ \langle #1 \rangle } $
Deriving a Lie-Trotter integration scheme while working in the local reduced spaces is in fact equivalent to translating the time-step targeting DMRG[1][2][3][4][5] into the MPS framework. Crucially, the MPS framework makes it possible to precisely analyze the errors made, something which would be very difficult – and to our knowledge has not been done – in the standard environment-system DMRG picture. To integrate the local time-dependent Schr"odinger equations resulting from the Lie-Trotter decomposition, we will use a Krylov-based approach[3][4][5]. This approach has the advantage of a very precise solution of the local equations and a large degree of similarity with both ground-state search DMRG and the time-dependent variational principle. Alternatively, Runge-Kutta integrators have also been used extensively with only minor changes to the overall method[1][2]. In particular, the error analysis presented here is also valid for the Runge-Kutta integrator, though one of course also has to include the additional time-step error of this integrator.
We begin by looking at the Lie-Trotter decomposition of the time-dependent Schrödinger equation \begin{align} -i\frac{d}{dt}\ket{\psi} &= \hat{H}\ket{\psi} \equiv \sum_{\nu}\hat{H}_{\nu}\ket{\psi}\; . \end{align} The goal is to find a decomposition scheme $\hat H = \sum_{\nu} \hat H_{\nu}$ such that we can integrate each summand separately by taking advantage of the MPS representation of the state vector $\ket{\psi}$. Therefore we define orthogonal projectors $\hat{P}^{L,\ket{\psi}}_{j}$ and $\hat{P}^{R,\ket{\psi}}_{j}$ acting on the physical degrees of freedom in a partition of the Hilbert space. For that purpose we introduce bipartitions $\mathcal{H}=\mathcal{H}^{L}_{j}\otimes\mathcal{H}^{R}_{j+1}$ where $\mathcal{H}^{L}_{j} = \mathcal{H}_{1}\otimes\cdots\otimes\mathcal{H}_{j}$ and $\mathcal{H}^{R}_{j+1} = \mathcal{H}_{j+1}\otimes\cdots\otimes\mathcal{H}_{L}$ and declare
\begin{align} \hat{P}^{L,\ket{\psi}}_{j} &: \mathcal{H}^{L}_{j} \otimes \mathcal{H}^{R}_{j+1} \longrightarrow \mathcal{H}^{L}_{j} \otimes \mathcal{H}^{R}_{j+1} \notag \\ \hat{P}^{L,\ket{\psi}}_{j} &= \sum_{ \substack{ \sigma_{1}, \ldots, \sigma_{j}, \\ \bar{\sigma}_{1}, \ldots, \bar{\sigma}_{j}, \\ m_{j} } } \underbrace{ A_{1} \cdots A_{j} }_{ \equiv \psi^{L}_{j;m_j} } \underbrace{ \bar{A}_{j} \cdots \bar{A}_{1} }_{ \equiv \bar{\psi}^{L}_{j;m_{j}} } \ket{\sigma_{1} \cdots \sigma_{j}} \bra{\bar\sigma_{1} \cdots \bar\sigma_{j}} \otimes \mathbf{\hat{1}}^{R}_{j+1} \\ \hat{P}^{R,\ket{\psi}}_{j} &: \mathcal{H}^{L}_{j-1} \otimes \mathcal{H}^{R}_{j} \longrightarrow \mathcal{H}^{L}_{j-1} \otimes \mathcal{H}^{R}_{j} \notag \\ \hat{P}^{R,\ket{\psi}}_{j} &= \mathbf{\hat{1}}^{L}_{j-1} \otimes \sum_{ \substack{ \sigma_{j}, \ldots, \sigma_{L}, \\ \bar{\sigma}_{j}, \ldots, \bar{\sigma}_{L}, \\ m_{j-1} } } \underbrace{ \bar{B}_{L} \cdots \bar{B}_{j} }_{ \equiv \bar{\psi}^{R}_{j;m_{j-1}} } \underbrace{ B_{j} \cdots B_{L} }_{ \equiv \psi^{R}_{j;m_{j-1}} } \ket{\sigma_{j} \cdots \sigma_{L}} \bra{\bar\sigma_{j} \cdots \bar\sigma_{L}} \end{align} with mappings $\psi^{L}_{j;m_{j}}$ from a part of the physical Hilbert space into the bond space $m_j$.
By construction, these operators fulfill $\left(\hat{P}^{L/R,\ket{\psi}}_{j}\right)^2 = \hat{P}^{L/R,\ket{\psi}}_{j}$ and $\left(\hat{P}^{L/R,\ket{\psi}}_{j}\right)^{\dagger} = \hat{P}^{L/R,\ket{\psi}}_{j}$, i.e., they are projectors. They are explicitly constructed from left-/right orthogonalized MPS site tensors and each term can be visualised as follows:
The action of such projectors onto an MPS representation of $\ket{\psi}$ in canonical form with orthogonality center at site $j+1$ $(j-1)$ is then given via \begin{align} \hat{P}^{L,\ket{\psi}}_{j}\ket{\psi} &= \sum_{ \substack{ \sigma_{1}, \ldots, \sigma_{j},\\ \sigma^{\prime}_{1}, \ldots, \sigma^{\prime}_{j},\\ \bar{\sigma}^{\prime}_{1}, \ldots, \bar{\sigma}^{\prime}_{j},\\ m_{1}, \ldots, m_{j}, \bar{m}^{\prime}_{j} } } \psi^{L; \sigma^{\prime}_{1}, \ldots, \sigma^{\prime}_{j}}_{j;m^{\prime}_{j}} \underbrace{ \bar{\psi}^{L;\bar{\sigma}^{\prime}_{1}, \ldots, \bar{\sigma}^{\prime}_{j}}_{j;\bar{m}^{\prime}_j} A^{\sigma_{1}}_{1;m_{1}} \cdots A^{\sigma_{j}}_{j;m_{j-1}, m_{j}} }_{ = \delta_{\bar{m}^{\prime}_{j}, m_j} \delta_{\bar{\sigma}^{\prime}_{1}, \sigma_1} \cdots\, \delta_{\bar{\sigma}^{\prime}_{j}, \sigma_j} } \sum_{ \sigma_{j+1}, \ldots, \sigma_L} M_{j+1} B_{j+2} \cdots B_{L} \ket{\sigma_{1} \cdots \sigma_{L}} \notag \\ &= \sum_{\sigma_{1}, \ldots, \sigma_{L}} A_{1} \cdots A_{j} M_{j+1} B_{j+2} \cdots B_L \ket{\sigma_{1} \cdots \sigma_{L}} = \ket{\psi} \\ \hat{P}^{R,\ket{\psi}}_{j}\ket{\psi} &= \sum_{\sigma_1, \ldots, \sigma_{j-1}} A_{1} \cdots A_{j-2} M_{j-1} \sum_{ \substack{ \sigma_{j}, \ldots, \sigma_{L},\\ \sigma^{\prime}_{j}, \ldots, \sigma^{\prime}_{L},\\ \bar{\sigma}^{\prime}_{j}, \ldots, \bar{\sigma}^{\prime}_{L},\\ m_{j}, \ldots, m_{L}, \bar{m}^{\prime}_{j} } } \underbrace{ B^{\sigma_{j}}_{j;m_{j-1}, m_{j}} \cdots B^{\sigma_{L}}_{L;m_{L-1}} \bar{\psi}^{R;\bar{\sigma}^{\prime}_{j}, \ldots, \bar{\sigma}^{\prime}_{L}}_{j;\bar{m}^{\prime}_{j-1}} }_{ \delta_{m_{j},\bar{m}^{\prime}_{j}} \delta_{\bar{\sigma}^{\prime}_{j}, \sigma_j} \cdots\, \delta_{\bar{\sigma}^{\prime}_{L}, \sigma_L} } \psi^{R;\sigma^{\prime}_{j}, \ldots, \sigma^{\prime}_{L}}_{j;m^{\prime}_{j-1}} \ket{\sigma_{1} \cdots \sigma_{L}} \notag \\ &= \sum_{\sigma_{1}\ldots\sigma_{L}} A_{1} \cdots A_{j-2} M_{j-1} B_{j} \cdots B_{L} \ket{\sigma_{1} \cdots \sigma_{L}} = \ket{\psi} \;. \end{align} That is, if the state $\ket{\psi}$ has orthogonality center to the right (left) of the target index $j$, the projectors constructed from it act as an identity on their Hilbert space partition.
(Above) Shifting the center of orthogonality from $3\rightarrow 5$ under the action of $\hat \Pi^{\ket{\psi}}_{5} \ket{\psi} = \hat{P}^{L,\ket{\psi}}_{4}\otimes \mathbf{\hat{1}}_{5} \otimes \hat{P}^{R,\ket{\psi}}_{6} \ket{ \psi }$. The orthogonality center is implicitly shifted to site 5 (without changing the state content, c.f. gray boxes) which gives identities on sites 1 through 4 and sites 6 and 7 and a new orthogonality center tensor $M_5$. The completely contracted upper two rows then define the new tensor $M_5$.
Next we define the projector on the reduced site-space at site $j$ via \begin{align} \hat{\Pi}^{\ket{\psi}}_{j} \equiv \hat{P}^{L,\ket{\psi}}_{j-1}\otimes \mathbf{\hat 1}_{j} \otimes \hat{P}^{R,\ket{\psi}}_{j+1} \;. \label{eq:local-krylov:projector} \end{align}
The action of such a projector $\hat{\Pi}^{\ket{\psi}}_{j}$ on a state $\ket{\psi}$ is to shift the orthogonality center of $\ket{\psi}$ to the site $j$ which can be shown by applying the manipulation depicted in the figure above on to $\ket{\psi}$ repeatedly. Therein gauge invariance is employed so that the action of the projector is trivial on the site tensors $A_{k}/B_{k}, k \neq j$ redering $j$ the center of orthogonality. Thus, the quantum state $\ket{\psi}$ remains unchanged under the action of $\hat{\Pi}^{\ket{\psi}}_{j}$ and therefore
\begin{align} \braket{\phi|\hat{\Pi}^{\ket{\psi}}_{j}|\psi} = \braket{\phi | \psi} \label{eq:local-krylov:projector-action}\\ \Rightarrow \hat{\Pi}^{\ket{\psi}}_{j} \ket{\phi} = \left(\frac{\ket{\psi^{\perp}}\bra{\psi^{\perp}}}{\braket{\psi^{\perp}|\psi^{\perp}}} + \frac{\ket{\psi}\bra{\psi}}{\braket{\psi|\psi}} \right)\hat{\Pi}^{\ket{\psi}}_{j}\ket{\phi} = \frac{\braket{\psi^{\perp}|\hat{\Pi}^{\ket{\psi}}_{j}|\phi} }{\braket{\psi^{\perp}|\psi^{\perp}}}\ket{\psi^{\perp}} + \frac{\braket{\psi|\hat{\Pi}^{\ket{\psi}}_{j}|\phi} }{\braket{\psi|\psi}}\ket{\psi} = \frac{\braket{\psi|\phi}}{\braket{\psi|\psi}}\ket{\psi} \end{align}
It is instructive to think of $\hat{\Pi}^{\ket{\psi}}_{j}$ as an operator acting on both the physical and gauge degrees of freedom of the MPS representation. In the physical system it acts as a projector on the physical indices $\sigma_{j}$ of the source state $\ket{\psi}$. In the gauge degrees of freedom of the MPS representation, $\hat{\Pi}^{\ket{\psi}}_{j}\ket{\phi}$ fixes the orthogonality center to site $j$. As the physical content of the state is independent of the location of its orthogonality center, we must have \begin{align} \braket{\psi|\psi} &= \left(\bra{\psi} \hat \Pi_j^{\ket{\psi}}\right) \left( \hat \Pi_i^{\ket{\psi}} \ket{\psi} \right) \end{align} as also immediately follows. Now, we can reformulate the action of the Hamiltonian by decomposing it into representations $\hat{H}^{\hat \Pi^{\ket{\psi}}_{j}}$ acting only onto reduced site-spaces: \begin{align} \hat{H} &\approx \frac{1}{L\Vert \ket{\psi}\Vert}\sum_j \hat \Pi^{\ket{\psi}}_j \hat H \hat \Pi^{\ket{\psi}}_j \equiv \frac{1}{L\Vert \ket{\psi}\Vert}\sum_{j} \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}} \label{eq:local-krylov:approx-lie-decomp}\\ \hat{H}\ket{\psi} &\approx \frac{1}{L\Vert \ket{\psi}\Vert}\sum_{j}\hat{\Pi}^{\ket{\psi}}_{j}\hat{H} \hat \Pi^{\ket{\psi}}_j\ket{\psi} \\ & = \frac{1}{L\Vert \ket{\psi}\Vert}\sum_{j}\hat{H}^{\hat \Pi^{\ket{\psi}}_{j}}\ket{\psi} \end{align} which indeed yields a Lie-Trotter decomposition of the time-dependent Schrödinger equation \begin{align} -\mathrm i\frac{d}{dt}\ket{\psi(t)} &= \frac{1}{L\Vert \ket{\psi}\Vert}\sum_{j}\hat{H}^{\hat \Pi^{\ket{\psi}}_{j}}\ket{\psi(t)}\;.\label{eq:local-krylov:decomposition} \end{align} Typically, this decomposition is not exact, it depends for instance on the size of the chosen basis, but given a sufficiently large MPS bond dimension, the error made will be small.
Having obtained the Lie-Trotter decomposition, we proceed by formulating a recursive integration scheme which is suitable for the particular structure of the MPS representation. For this purpose note that a first-order approximation to the evolved state can be obtained by solving each problem $-\mathrm i\frac{d}{dt}\ket{\psi(t)} = \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}}\ket{\psi(t)}$ independently. The prefactors $\frac{1}{L\Vert \ket{\psi} \Vert}$ are absorbed into the normalization of the site tensor that is currently evolved. Let $\ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)}$ be the solution of the $j$-th problem. An approximation to the overall time-evolved state from $t\rightarrow t+\delta$ can then be obtained by sequentially solving the initial value problems (setting $\ket{\psi^{\hat \Pi^{\ket{\psi}}_{0}}(t)} \equiv \ket{\psi(t)}$)
\begin{align} -\mathrm i \frac{d}{dt} \ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} &= \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}}\ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} \nonumber \\ \textrm{and} \quad \ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} & = \ket{\psi^{\hat \Pi^{\ket{\psi}}_{j-1}}(t+\delta)} \label{eq:lie_trotter_approximation} \end{align}
and identifying $\ket{\psi(t+\delta)} \equiv \ket{\psi^{\hat \Pi^{\ket{\psi}}_{L}}(t+\delta)}$ with the approximated time evolved state. Comparing the formal Taylor expansions of the exactly integrated state $\ket{\psi(t+\delta)}_{\mathrm{exact}}$ with the approximation one readily finds
\begin{align} \ket{\psi(t+\delta)}_{\mathrm{exact}} &= \ket{\psi(t)} - \mathrm i \delta\hat{H}\ket{\psi(t)} - \frac{\delta^2}{2}\hat{H}^{2}\ket{\psi(t)} + \cdots \\ \ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t+\delta)} &= \ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} + \delta\frac{d}{dt}\ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} + \frac{\delta^2}{2}\frac{d^2}{dt^2}\ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} + \cdots \notag \\ &= \ket{\psi^{\hat \Pi^{\ket{\psi}}_{j-1}}(t+\delta)} - \mathrm i \delta \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}} \ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} - \frac{\delta^2}{2}\hat{H}^{\hat \Pi^{\ket{\psi}}_{j}} \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}}\ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} + \cdots \\ \Rightarrow \ket{\psi(t+\delta)} &= \ket{\psi(t)} - \mathrm i \delta\hat{H}\ket{\psi(t)} - \frac{\delta^2}{2}\sum_{i\leq j}\hat{H}^{\hat \Pi^{\ket{\psi}}_{j}}\hat{H}^{\hat \Pi^{\ket{\psi}}_{i}}\ket{\psi(t)} \cdots \label{eq:local-krylov:taylor-comparison} \end{align}
where we have already applied $\eqref{eq:lie_trotter_approximation}$ to the expansion of the partially time-evolved intermediate state $\ket{\psi^{\hat \Pi^{\ket{\psi}}_j}(t+\delta)}$ around $\delta = 0$. Replacing $\ket{\psi^{\hat \Pi^{\ket{\psi}}_{j-1}}(t+\delta)}$ by its Taylor expansion around $\delta=0$ generates a recursion for the partially time-evolved states. Applying this recursion $L$ times and using the initial condition $\ket{\psi^{\hat \Pi^{\ket{\psi}}_0}(t)} = \ket{\psi(t)}$ yields the last equation. As expected, $\ket{\psi^{\hat \Pi^{\ket{\psi}}_L} (t+\delta)}$ coincides with $\eqref{eq:local-krylov:decomposition}$ up to the first order of the expansion. The first error occurs at second-order terms in the expansion and is given by
\begin{align} \ket{\psi(t+\delta)}_{\mathrm{exact}} - \ket{\psi(t+\delta)} = \frac{\delta^2}{2}\sum_{i\lt j}\left[\hat{H}^{\hat \Pi^{\ket{\psi}}_{i}}, \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}} \right]\ket{\psi(t)} \end{align}
and further commutators at arbitrary high orders. Due to the dependence of the reduced problem currently solved onto the previous solution in $\eqref{eq:lie_trotter_approximation}$ the projectors $\hat{P}^{L,\ket{\psi}}_{j-1}$ in our integration scheme are in fact time-dependent because they need to be constructed from site tensors $A_{j^\prime\lt j}(t+\delta)$ that have already been evolved. In contrast, $\hat{P}^{R,\ket{\psi}}_{j+1}$ is constructed from unevolved site tensors $B_{j^\prime>j}(t)$. We hence have to actually solve the $L$ initial value problems
\begin{align} -\mathrm i \frac{d}{dt} \ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} &= \left[ \hat{\Pi}^{\ket{\psi}}_{j}(t,t+\delta)\hat{H}\hat{\Pi}^{\ket{\psi}}_{j}(t,t+\delta)\right]\ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} \nonumber \\ \textrm{and} \quad \ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} & = \ket{\psi^{\hat \Pi^{\ket{\psi}}_{j-1}}(t+\delta)} \end{align} with $\hat{\Pi}^{\ket{\psi}}_{j}(t,t+\delta) \equiv \hat{P}^{L,\ket{\psi^{\hat \Pi^{\ket{\psi}}_{j-1}}(t+\delta)}}_{j-1}\otimes\mathbf{\hat 1}_{j}\otimes\hat{P}^{R,\ket{\psi(t)}}_{j+1}$.
Note that while these problems appear to evolve over $L$ time steps with step size $\delta$, they only do so using different local Hamiltonians. Those local problems are then connected via their initial conditions (c.f. \eqref{eq:lie_trotter_approximation}) to generate a global time evolution by $\delta$ under (an approximation of) the global Hamiltonian.
Having decoupled the global Schrödinger equation into $L$ local problems, we can take advantage of the local representation of the state by multiplying each local differential equation at site $j$ with the bond maps $\bar{\psi}^{L}_{j-1}\otimes \mathbf{\hat 1}_{j} \otimes \bar{\psi}^{R}_{j+1}$ \begin{align} -\mathrm i \frac{d}{dt} \left[ \bar{\psi}^{L}_{j-1}\otimes \mathbf{\hat 1}_{j} \otimes \bar{\psi}^{R}_{j+1} \right] \ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}} &= \left[ \bar{\psi}^{L}_{j-1}\otimes \mathbf{\hat 1}_{j} \otimes \bar{\psi}^{R}_{j+1} \right] \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}} \ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}} \notag \\ \Rightarrow -\mathrm i \frac{d}{dt} M_{j} &= H^{\mathrm{eff}}_{j} M_{j} \label{eq:local-krylov:local-tdse} \end{align} where we defined effective reduced-site state and operator representations $M_{j}$ and $H^{\mathrm{eff}}_{j}$, respectively.
(Above) Visualisation of the local effective operator $H^{\mathrm{eff}}_{j}$ (top) and local effective state $M_j$.
It can be checked easily that the projection of the effective problems onto their reduced site spaces at site $j$ leaves the solution invariant. Carrying out the differentiation on the left-hand site explicitly yields a sum of partial differentiations. However, on the right hand site $\hat{H}^{\hat \Pi^{\ket{\psi}}_{j}}$ acts only on the reduced site spaces and hence all differentials must vanish except the one acting on the tensor in the reduced site space.
Unfortunately, a direct adoption of the recursive solution strategy proposed above is not possible because the current problem at site $j$ requires the projectors to be constructed from left-/right-canonical site-tensors $A_{1},\ldots, A_{j-1},B_{j+1},\ldots, B_{L}$. However, the solution $M_{j-1}$ of the previous reduced problem is not in general in a canonical representation so that in order to construct the next projector $\hat \Pi^{\ket{\psi}}_{j}$ we need to perform a basis transformation first. There is no prescription in the derived decomposition scheme which corresponds to such a basis transformation keeping the evolved site tensors in canonical form. On the other hand we can of course rewrite the updated site tensor as $M_{j-1}\rightarrow A_{j-1} R_{\underline{j-1}}$ in a left-canonical representation. But, as already pointed out, we have to discard the transformation matrix $R_{\underline{j-1}}$ to obey the decomposition scheme. To proceed with the next reduced problem, we need to resolve the situation of having $2$ different basis representations between the sites $(j-1,j)$. Hence, we introduce a transformation between the unevolved and evolved site tensors $A_{j-1},M_{j}$. The simplest guess is to consider a direct mapping between the unevolved and evolved site tensor basis sets. This transformation can be readily obtained from the iterative solution strategy and is performed by projecting the evolved onto the unevolved bond space. To see this we write the partially evolved state explicitely in the mixed canonical form
\begin{align} \ket{\psi(t,t+\delta)} &= \sum_{ \substack { \sigma_{1}, \ldots, \sigma_{L},\\ m_{j-1}, m_{j} } } \underbrace{ A^{\sigma_{1}}_{1}(t+\delta) \cdots A^{\sigma_{j-1}}_{j-1}(t+\delta) }_{ \equiv \psi^{L; \sigma_1, \ldots, \sigma_{j-1}}_{j-1; m_{j-1}}(t+\delta) } M^{\sigma_{j}}_{j;m_{j-1}, m_j}(t) \underbrace{ B^{\sigma_{j+1}}_{j+1}(t) \cdots B^{\sigma_{L}}_{L}(t) }_{ \equiv \psi^{R; \sigma_{j+1}, \ldots, \sigma_{L}}_{j+1; m_{j}}(t) } \ket{\sigma_{1} \cdots \sigma_{L}} \end{align}
and introduce a compact notation for the left/right partition’s basis states
\begin{align} \ket{\psi^L_{j}(t)}_{m_{j}} &= \sum_{\sigma_1, \ldots, \sigma_{j}} \psi^{L; \sigma_1, \ldots, \sigma_{j}}_{j; m_{j}}(t) \ket{\sigma_1 \cdots \sigma_{j}} & \ket{\psi^R_{j}(t)}_{m_{j-1}} &= \sum_{\sigma_j, \ldots, \sigma_L} \psi^{R; \sigma_{j}, \ldots, \sigma_{L}}_{j; m_{j-1}}(t) \ket{\sigma_{j} \cdots \sigma_{L}} \;. \end{align}
Without truncation $\ket{\psi^L_{j}(t)}_{m_{j}}$ and $\ket{\psi^R_{j}(t)}_{m_{j-1}}$ would constitute complete maps from the physical degrees of freedom in the left/right partition and we could perform an exact basis transformation mapping the evolved into the unevolved left basis
\begin{align} \ket{\psi(t)} &= \sum_{m_{j-1}} \ket{\psi^L_{j-1}(t)}_{m_{j-1}} {\vphantom{\ket{\psi^L_{j-1}(t)}}}_{m_{j-1}}\hspace{-0.4em}\braket{\psi^L_{j-1}(t) \vert \psi(t,t+\delta)} \\ &= \sum_{ \substack { m_{j-1}, m^{\prime}_{j-1}, \\ \sigma_{j}, m_{j} } } \ket{\psi^L_{j-1}(t)}_{m_{j-1}} \underbrace{ {\vphantom{\ket{\psi^L_{j-1}(t)}}}_{m_{j-1}}\hspace{-0.4em} \braket{\psi^L_{j-1}(t) \lvert \psi^{L}_{j-1}(t+\delta)}_{m^{\prime}_{j-1}} }_{ Q_{\underline{j-1};m_{j-1},m^{\prime}_{j-1}}(t,t+\delta) } M^{\sigma_{j}}_{j;m^{\prime}_{j-1}, m_j}(t)\ket{\sigma_{j}}\ket{\psi^{R}_{j}(t)}_{m_{j}} \\ &= \sum_{ \substack { \sigma_{1}, \ldots, \sigma_{L},\\ m_{j-1}, m_{j}\\ m^{\prime}_{j-1} } } \psi^{L; \sigma_1, \ldots, \sigma_{j-1}}_{j-1; m_{j-1}}(t) Q_{\underline{j-1};m_{j-1},m^{\prime}_{j-1}}(t,t+\delta) M^{\sigma_{j}}_{j;m^{\prime}_{j-1}, m_j}(t) \psi^{R}_{j+1; m_{j}}(t) \ket{\sigma_{1} \cdots \sigma_{L}} \; . \end{align}
and the matrix elements $Q_{\underline{j-1};m_{j-1},m^{\prime}_{j-1}}(t,t+\delta)$ of the basis transformation $Q_{\underline{j-1}}(t,t+\delta)$ are constructed from
\begin{align} Q_{\underline{j-1};m_{j-1},m^{\prime}_{j-1}}(t,t+\delta) &= \sum_{ \sigma_{1}, \ldots, \sigma_{j-1} } \bar{\psi}^{L;\sigma_{1},\ldots,\sigma_{j-1}}_{j-1;m_{j-1}}(t) \psi^{L;\sigma_{1},\ldots,\sigma_{j-1}}_{j-1;m^{\prime}_{j-1}}(t+\delta) \;. \end{align}
The transformation $Q_{\underline{j}}(t,t+\delta)$ maps bond basis states $\ket{\psi^{L}_{j}(t+\delta)}_{m_j}$ which are optimized to represent the evolved state $\ket{\psi(t+\delta)}$ into bond basis states $\ket{\psi^{L}_{j}(t)}_{m_j}$ which are optimized to represent the unevolved state $\ket{\psi(t)}$. Now, if we would let act $Q_{\underline{j-1}}(t,t+\delta)$ to the already optimized canonical site tensor $A^{\sigma_{j-1}}_{j-1}(t+\delta)$ then the effect would be to undo the previous site optimization. Hence, the inverse transformation $\bar Q_{\underline{j-1}}(t,t+\delta)$ is employed to transform the unevolved bond basis labeled by $m_{j-1}$ of the current site tensor $M_{j}(t)$.
Observing that $\bar Q_{\underline{j-1}}$ is constructed from the above introduced bond maps $\bar{\psi}^{L}_{j-1}(t)$ and $\psi^{L}_{j-1}(t+\delta)$ which themselves should be build recursively, the transformation can be updated and applied before solving the $j$-th problem via
\begin{align} Q_{\underline{j-1}}(t, t+\delta) &= \bar A_{j-1}(t) Q_{\underline{j-2}}(t, t+\delta) A_{j-1}(t+\delta), \quad M_{j}(t) \longrightarrow \bar Q_{\underline{j-1}}(t, t+\delta) M_{j}(t) \; . \end{align}
If we allow for truncation the error incurred by this mapping depends on the overlap $\braket{\psi(t+\delta)|\psi(t)}$ as well as the discarded weight. This basis transformation is mostly motivated by its straightforward availability during the sweeping procedure. However, to the best of our knowledge there is no mathematical justification and we can only give the physical motivation that for small time steps $\delta$ the time-evolved state is expected to be relatively close to the unevolved state (deviation $\propto L\delta^2$ as follows from the consideration below.
Instead of mapping onto the space of a single site, in practice we map onto the space of two sites. The two-site local TDSE is solved using the time-dependent Lanczos approach to obtain $A_j(t+\delta)$. The original orthogonality center MPS tensor $M_{j+1}(t)$ is then projected onto the new left basis as described above. This allows for a flexible adaptation of not only the tensor $A_j$ itself but also of the bond basis and – if necessary – MPS bond dimension between sites $j$ and $j+1$.
Historically, only this two-site variant was used; but in analogy to the 2TDVP method presented later, it may well make sense to initially use the two-site local Krylov method until the desired maximal bond dimension has been obtained and then switch to the single-site integrator to save computational effort.
Errors
Four errors are present in the local Krylov method when used in its (standard) two-site variant. The smallest of those stems from the inexact solution of the local TDSE $\eqref{eq:local-krylov:local-tdse}$. This error can be made very small using a precise solver; in practice, a Krylov exponential as described on the global Krylov algorithm with very few (4-5) vectors is sufficient. The second error is the standard truncation error incurred during the SVD to split the merged two-site tensors again while truncating to the desired bond dimension. This error can be measured and observed throughout the calculation and is much the same as in the other methods.
The third error is due to the approximation in $\eqref{eq:local-krylov:approx-lie-decomp}$. This projection error is difficult to measure and strongly depends on the initial state. If the initial state has a reasonably large bond dimension and the Hamiltonian has reasonably short-range interactions, this error will be very small. The longer the interactions in the Hamiltonian, the larger the state has to be. In the two-site method, nearest-neighbor interactions can be handled at all bond dimensions, in the single-site variant, only on-site interactions are error-free at small bond dimensions. The projection error is in particular problematic when globally quenching from a product state with a very non-local Hamiltonian (e.g. resulting from a 2D $\to$ 1D map). When calculating equilibrium Green’s functions for short-range Hamiltonians, this error is quite negligible.
Finally, there is an error due to the sequential solution of the local TDSE as resulting from the Lie-Trotter decomposition. This error can be quantified, but doing so requires some additional work which will follow now: We continue from the Taylor expansion $\eqref{eq:local-krylov:taylor-comparison}$. We emphasize that the action of the commutators $\left[\hat{H}^{\hat \Pi^{\ket{\psi}}_{i}}, \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}} \right]\ket{\psi(t_{0})}$ need to be evaluated with respect to the iteration the commutators are generated from.
Evaluation of $\hat{H}^{\hat \Pi^{\ket{\psi}}_{2}} \hat{H}^{\hat \Pi^{\ket{\psi}}_{3}}\ket{\psi^{\hat \Pi^{\ket{\psi}}_{3}}(t)}$ at the example of a four-site system. The tensors $L_{i}/R_{i}$ correspond to partially contracted MPS-MPO-MPS-networks. The burgundy-shaded rectangular area at the top encloses the reduced site tensor $\psi^{C;\sigma_{i},\ldots,\sigma_{j}}_{i,\ldots,j;m_{i-1},m_{j}}$. The commutator $\left[\hat{H}^{\hat \Pi^{\ket{\psi}}_{2}} \hat{H}^{\hat \Pi^{\ket{\psi}}_{3}}\right]\ket{\psi^{\Pi}_{3}(t)}$ requires also the calculation of $\hat{H}^{\hat \Pi^{\ket{\psi}}_{3}} \hat{H}^{\hat \Pi^{\ket{\psi}}_{2}}\ket{\psi^{\hat \Pi^{\ket{\psi}}_{3}}(t)}$ which is obtained from vertically flipping the tensors covered by the central rectangular grey area between the sites $(i,j)$, i.e., $(2,3)$ in the presented example.
Consider, for instance, the action of $\hat{H}^{\hat \Pi^{\ket{\psi}}_{2}}\hat{H}^{\hat \Pi^{\ket{\psi}}_{3}}\ket{\psi(t)}$, which is generated from the first-order contribution $\hat{H}^{\hat \Pi^{\ket{\psi}}_{3}} \ket{\psi^{\hat \Pi^{\ket{\psi}}_{3}}}$ and subsequent application of $\hat{H}^{\hat \Pi^{\ket{\psi}}_{2}}$. Thus the commutator also needs to be evaluated considering the partial solution $\ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}}$, so that in general we have for $i \lt j$
\begin{align} \left[\hat{H}^{\hat \Pi^{\ket{\psi}}_{i}}, \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}} \right]\ket{\psi(t)} &= \left[\hat{H}^{\hat \Pi^{\ket{\psi}}_{i}}, \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}} \right]\ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)}\; . \end{align}
In the figure above, the action of $\hat{H}^{\hat \Pi^{\ket{\psi}}_{i}} \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}}\ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)}$ is demonstrated in case of a four-site system with $i=2$, $j=3$ by performing most of the contractions graphically.
In order to obtain the matrix element with an arbitrary state $\ket{\phi}$ we will introduce a compact notation for contractions of MPS and MPO site-tensors with the boundary tensors (partially contracted MPS-MPO-MPS-networks $L_j/R_j$)
\begin{align} L_{j-1 }E_{j} &\equiv L_{j-1} \bar{M}_{j} W_{j} M_{j} = \sum_{m_{j-1}, \sigma^{\prime}_{j}} \left( \sum_{w_{j-1}, \sigma_{j}} \left( \sum_{\bar{m}_{j-1}} L^{\bar{m}_{j-1},w_{j-1}}_{j-1; m_{j-1}} \bar{M}^{\sigma_{j}}_{j; \bar{m}_{j-1},\bar{m}_{j}} \right) W^{\sigma_{j},\sigma^{\prime}_{j}}_{j,w_{j-1},w_{j}} \right) M^{\sigma^{\prime}_{j}}_{j,m_{j-1},m_{j}} \\ E_{j} R_{j+1} &\equiv \bar{M}_{j} W_{j} M_{j} R_{j+1} = \sum_{\bar{m}_j, \sigma_j} \bar{M}^{\sigma_{j}}_{j; \bar{m}_{j-1},\bar{m}_{j}} \left( \sum_{w_j, \sigma^\prime_j} W^{\sigma_{j},\sigma^{\prime}_{j}}_{j; w_{j-1},w_{j}} \left( \sum_{m_j} M^{\sigma^{\prime}_{j}}_{j; m_{j-1},m_{j}} R^{\bar{m}_{j},w_{j}}_{j+1; m_{j}} \right) \right) \end{align}
with the transfer tensors $E_{j} = \bar{M}_{j} W_{j} M_{j}$. We also need transfer tensors with the target state which we define by $E^{\phi}_{j} = \bar{M}^{\phi}_{j} W_{j} M_{j} $. Finally, there will be open bonds at sites $i$ and $j$ that correspond to the contractions originating from the ‘brace’-contractions in the projectors $\hat{P}^{L, \ket{\psi}}_{i}$ and $\hat{P}^{R,\ket{\psi}}_{i}$ as well as $\hat{P}^{L,\ket{\psi}}_{j+1}$ and $\hat{P}^{R,\ket{\psi}}_{j+1}$, and we will label these bonds explicitly. Considering for instance the first summand of the commutator at $i=2,j=3$, that is $\braket{\phi| \hat{H}^{\hat \Pi^{\ket{\psi}}_{2}}\hat{H}^{\hat \Pi^{\ket{\psi}}_{3}} |\psi^{\hat \Pi^{\ket{\psi}}_{3}}(t)}$, the left part of the contractions can be written as
\begin{align} \sum_{m_{2},\bar{m}_{2}} \left(L_{1} E_{2} \right)_{\bar{m}_2} \otimes \left(L_{1} E^{\phi}_{2} \right)_{m_{2}} \delta_{\bar{m}_{2},m_{2}} &\equiv \sum_{m_{2},\bar{m}_{2}}\left(L_1 \otimes L_{1} \right) \left( E_{2} \otimes E^{\phi}_{2} \right)_{\bar{m}_{2},m_{2}} \delta_{\bar{m}_{2},m_{2}}\; . \end{align}
To obtain a compact notation for the right part we introduce the ‘dangling’ transfer tensors $D_{j} = W_{j}M_{j}$ if only the ket site-tensor of the state is included and $\bar{D}_{j} = \bar{M}_{j} W_{j}$ if only the bra site-tensor is considered. In the same manner as for the left part we can now write for the right contractions compactly
\begin{align} \sum_{m_{3},\bar{m}_{3}} \left(D_{3}R_{4} \right)_{\bar{m}_{3}} \otimes \left( \bar{D}^{\phi}_{3} R_{4} \right)_{m_{3}}\delta_{\bar{m}_{3},m_{3}} &\equiv \sum_{m_{3},\bar{m}_{3}} \left( D_{3} \otimes \bar{D}^{\phi}_{3} \right) \left( R_{4} \otimes R_{4} \right)_{\bar{m}_{3},m_{3}}\delta_{\bar{m}_{3},m_{3}} \;. \end{align}
In general we obtain for the matrix element of the commutator with the target state $\ket{\phi}$ (suppressing the Kronecker-$\delta$ for brevity)
\begin{align} &\braket{\phi|\left[\hat{H}^{\hat \Pi^{\ket{\psi}}_{i}}, \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}} \right]|\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} \notag \\ &= \sum_{\substack{m_{j-1},\bar{m}_{j-1} \\ m_{j},\bar{m}_{j} }} \left(L_{i-1} \otimes L_{i-1} \right) \left(E_{i}\cdots E_{j-1} \otimes E^{\phi}_{i}\cdots E^{\phi}_{j-1} \right)_{\bar{m}_{j-1},m_{j-1}}\left(D_{j}\otimes \bar{D}^{\phi}_{j} \right) \left( R_{j+1} \otimes R_{j+1} \right)_{\bar{m}_{j},m_{j}} + \cdots \notag \\ &- \sum_{\substack{m_{i-1},\bar{m}_{i-1} \\ m_{i},\bar{m}_{i} }} \left(L_{i-1} \otimes L_{i-1} \right)_{\bar{m}_{i-1},m_{i-1}} \left(D_{i}\otimes \bar{D}^{\phi}_{i} \right)\left(E_{i+1}\cdots E_{j}\otimes E^{\phi}_{i+1} \cdots E^{\phi}_{j} \right)_{\bar{m}_{i},m_{i}} \left(R_{j+1}\otimes R_{j+1} \right)\; . \end{align}
It becomes immediately clear that only those terms in the Hamiltonian contribute to the commutator which cross the bond $(j-1,j)$ or $(i,i+1)$. For the current purpose it suffices to find a general estimate for the incurred error $\propto \delta^2$ in the Taylor expansion. Thus, we will only consider the contributions from nearest-neighbor interactions and hence set $i=j-1$ so that
\begin{align} &\braket{\phi|\left[\hat{H}^{\hat \Pi^{\ket{\psi}}_{j-1}}, \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}} \right]|\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} \notag \\ &= \sum_{\substack{m_{j-1},\bar{m}_{j-1} \\ m_{j},\bar{m}_{j} }} \left(L_{j-2}\otimes L_{j-2} \right)\left(E_{j-1}\otimes E^{\phi}_{j-1} \right)_{\bar{m}_{j-1},m_{j-1}} \left(D_{j} \otimes \bar{D}^{\phi}_{j} \right)\left(R_{j+1} \otimes R_{j-1} \right)_{\bar{m}_{j},m_{j}} + \cdots \notag \\ &- \sum_{\substack{m_{j-2},\bar{m}_{j-2} \\ m_{j-1},\bar{m}_{j-1} }} \left(L_{j-2} \otimes L_{j-2} \right)_{\bar{m}_{j-2},m_{j-2}} \left(D_{j-1} \otimes \bar{D}^{\phi}_{j-1} \right) \left(E_{j} \otimes E^{\phi}_{j} \right)_{\bar{m}_{j-1},m_{j-1}} \left(R_{j+1} \otimes R_{j+1} \right) \;. \end{align}
The crucial observation here is that for each open bond index pair we can treat the combined MPS-MPS and MPO-MPO tensor-contractions over the bond $(j-1,j)$ as scalar product between the left and right part of the system. We can thus write for the first summand for each open index pair
\begin{align} &\underbrace{\left(L_{j-2}\otimes L_{j-2} \right)\left(E_{j-1}\otimes E^{\phi}_{j-1} \right)_{\bar{m}_{j-1},m_{j-1}}}_{\equiv \bra{e^{L}_{j-1}\otimes e^{L,\phi}_{j-1}}} \underbrace{\vphantom{\left(L_{j-2}\otimes L_{j-2} \right)\left(E_{j-1}\otimes E^{\phi}_{j-1} \right)_{\bar{m}_{j-1},m_{j-1}}}\left(D_{j} \otimes \bar{D}^{\phi}_{j} \right)\left(R_{j+1} \otimes R_{j-1} \right)_{\bar{m}_{j},m_{j}}}_{\equiv \ket{d^{R}_{j}\otimes \bar{d}^{R,\phi}_{j}}} \notag \\ &= \frac{1}{2}\left( \left\lvert \ket{e^{L}_{j-1}\otimes e^{L,\phi}_{j-1}} + \ket{d^{R}_{j}\otimes \bar{d}^{R,\phi}_{j}} \right\rvert^{2} - \braket{e^{L}_{j-1}\otimes e^{L,\phi}_{j-1}|e^{L}_{j-1} \otimes e^{L,\phi}_{j-1}} - \braket{d^{R}_{j}\otimes \bar{d}^{R,\phi}_{j}|d^{R}_{j}\otimes \bar{d}^{R,\phi}_{j}} \right) \end{align}
and for the second summand
\begin{align} &\underbrace{\vphantom{\left(E_{j} \otimes E^{\phi}_{j} \right)_{\bar{m}_{j-1},m_{j-1}} \left(R_{j+1} \otimes R_{j+1} \right)}\left(L_{j-2} \otimes L_{j-2} \right)_{\bar{m}_{j-2},m_{j-2}} \left(D_{j-1} \otimes \bar{D}^{\phi}_{j-1} \right)}_{\equiv \bra{d^{L}_{j-1}\otimes \bar{d}^{L,\phi}_{j-1}}} \underbrace{\left(E_{j} \otimes E^{\phi}_{j} \right)_{\bar{m}_{j-1},m_{j-1}} \left(R_{j+1} \otimes R_{j+1} \right)}_{\equiv \ket{e^{R}_{j}\otimes e^{R,\phi}_{j}}} \notag \\ &= -\frac{1}{2}\left( \left\lvert \ket{d^{L}_{j-1}\otimes \bar{d}^{L,\phi}_{j-1}} - \ket{e^{R}_{j}\otimes e^{R,\phi}_{j}}\right\rvert^{2} - \braket{d^{L}_{j-1}\otimes \bar{d}^{L,\phi}_{j-1}|d^{L}_{j-1}\otimes \bar{d}^{L,\phi}_{j-1}} - \braket{e^{R}_{j}\otimes e^{R,\phi}_{j}|e^{R}_{j}\otimes e^{R,\phi}_{j}} \right) \;. \end{align}
We can bound the sums over the open bonds for the scalar products in terms of absolute values of expectation values of effective Hamiltonians $\hat{h}^{L/R,\mathrm{eff}}_{j}$ at site $j$
\begin{align} \sum_{\substack{ m_{j-1} \\ \bar{m}_{j-1}}}\braket{e^{L}_{j-1}\otimes e^{L,\phi}_{j-1}|e^{L}_{j-1} \otimes e^{L,\phi}_{j-1}} &\leq \sum_{\substack{ m_{j-1} \\ \bar{m}_{j-1}}} \left\lvert \ket{e^{L}_{j-1}\otimes e^{L,\phi}_{j-1}} \right\rvert^{2} = \left\lvert \bar{\psi}^{\mathrm{eff}}_{j-1}\hat{h}^{L,\mathrm{eff}}_{j-1}\psi^{\mathrm{eff}}_{j-1} \cdot \bar{\phi}^{\mathrm{eff}}_{j-1}\hat{h}^{L,\mathrm{eff}}_{j-1}\psi^{\mathrm{eff}}_{j-1} \right\rvert \\ \sum_{\substack{ m_{j} \\ \bar{m}_{j} }} \braket{d^{R}_{j}\otimes \bar{d}^{R,\phi}_{j}|d^{R}_{j}\otimes \bar{d}^{R,\phi}_{j}} &\leq \sum_{\substack{ m_{j} \\ \bar{m}_{j} }} \left\lvert \ket{d^{R}_{j}\otimes \bar{d}^{R,\phi}_{j}} \right\rvert^{2} = \left\lvert \bar{\phi}^{\mathrm{eff}}_{j}\left[\hat{h}^{R,\mathrm{eff}}_{j}\right]^2 \psi^{\mathrm{eff}}_{j} \right\rvert \\ \sum_{\substack{ m_{j-2} \\ \bar{m}_{j-2} }} \braket{d^{L}_{j-1}\otimes \bar{d}^{L,\phi}_{j-1}|d^{L}_{j-1}\otimes \bar{d}^{L,\phi}_{j-1}} &\leq \sum_{\substack{ m_{j-2} \\ \bar{m}_{j-2} }} \left\lvert \ket{d^{L}_{j-1}\otimes \bar{d}^{L,\phi}_{j-1}}\right\rvert^{2} = \left\lvert \bar{\phi}^{\mathrm{eff}}_{j-1} \left[\hat{h}^{L,\mathrm{eff}}_{j-1}\right]^{2}\psi^{\mathrm{eff}}_{j-1} \right\rvert \\ \sum_{\substack{ m_{j-1} \\ \bar{m}_{j-1} }} \braket{e^{R}_{j}\otimes e^{R,\phi}_{j}|e^{R}_{j}\otimes e^{R,\phi}_{j}} &\leq \sum_{\substack{ m_{j-1} \\ \bar{m}_{j-1} }} \left\lvert \ket{e^{R}_{j}\otimes e^{R,\phi}_{j}} \right\rvert^{2} = \left\lvert \bar{\psi}^{\mathrm{eff}}_{j}\hat{h}^{R,\mathrm{eff}}_{j}\psi^{\mathrm{eff}}_{j} \cdot \bar{\phi}^{\mathrm{eff}}_{j}\hat{h}^{R,\mathrm{eff}}_{j}\psi^{\mathrm{eff}}_{j} \right\rvert \end{align}
where we defined
\begin{equation} \hat{h}^{L,\mathrm{eff}}_{j} = L_{j-1} W _{j}\quad \text{and} \quad \left[\hat{h}^{L,\mathrm{eff}}_{j}\right]^{2} = \left(L_{j-1}\otimes L_{j-1}\right) \left(W_{j} \otimes W_{j}\right) \end{equation}
and in a similiar way $\hat{h}^{R,\mathrm{eff}}_{j}$ replacing $L_{j-1}\rightarrow R_{j+1}$. The formal absolute values $\lvert \cdot \rvert$ on the right side can be estimated by replacing the $L_{j}/R_{j}$ tensors with fractions of the overall energy expectation value. This is a valid approximation as long as there are no interactions connecting the left/right contracted MPS-MPO-MPS tensor networks with sites to the right/left of them which was exactly the condition for non-vanishing contributions to the commutator. We hence compare only squares of single-site expectation values with either one or two MPO site tensors sandwiched between the effective site tensors at sites $j-1,j$. For a large system with smoothly varying site tensors the differences at neighboring sites are negligible so that the commutator can be estimated to
\begin{align} \braket{\phi|\left[\hat{H}^{\hat \Pi^{\ket{\psi}}_{j-1}}, \hat{H}^{\hat \Pi^{\ket{\psi}}_{j}} \right]|\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)} &\leq \frac{1}{2}\left( \left\lvert \ket{e^{L}_{j-1}\otimes e^{L,\phi}_{j-1}} + \ket{d^{R}_{j}\otimes \bar{d}^{R,\phi}_{j}} \right\rvert^{2} + \left\lvert \ket{d^{L}_{j-1}\otimes \bar{d}^{L,\phi}_{j-1}} - \ket{e^{R}_{j}\otimes e^{R,\phi}_{j}}\right\rvert^{2} \right) \notag \\ &\leq \frac{1}{2}\left(\left\lvert \ket{e^{L}_{j-1}\otimes e^{L,\phi}_{j-1}}\right\rvert^{2} + \left\lvert \ket{d^{R}_{j}\otimes \bar{d}^{R,\phi}_{j}} \right\rvert^{2} \right) \;. \end{align}
The last expression can be estimated easily by the coupling strength of the nearest-neighbor interaction term which we denote by $\Gamma$. The contributions from the boundary tensors $L_{j}/R_{j}$ are bounded by the square of the system size and cancel with the prefactors $\frac{1}{L}$ of each projected Hamiltonian. We hence conclude this analysis by the somewhat straightforward statement that in general the error at second order in $\delta$ scales with $L\cdot \Gamma^2$. However, there is one special situation which simplifies the preceeding arguments drastically namely if we set $\ket{\phi}=\ket{\psi^{\hat \Pi^{\ket{\psi}}_{j}}(t)}$. Then the commutator compares only local overlaps between site tensors sandwiched between either one or two MPO tensors on neighboring sites which scales as $\frac{1}{L}$. Therefore equal-time observables are evolved with very high precision $\sim \frac{1}{L^2}$ and the contribution due to nearest-neighbor interactions is strongly suppressed.
Algorithm
Common Helper Functions
First, let us define some common helper functions. These are identical to those used in standard
DMRG ground-state search algorithms. In addition, we need a local exponential Lanczos solver. For a
more detailed description of solver used to evaluate the local exponentials, see
the page on the global Krylov method.
[NB: The CONTRACT-LEFT and CONTRACT-RIGHT functions defined here are identical to those on the
TDVP page. Once page includes are available, merge them]
\begin{align} & \texttt{CONTRACT-LEFT}(L_{j-1}, W_j, A_j) \; \{ \\ & \quad L_{j; m_j}^{\bar{m}_j, w_j} = \sum_{ \sigma_{j}, \sigma^\prime_{j}, w_{j-1}, m_{j-1}, \bar{m}_{j-1} } L_{j-1; m_{j-1}}^{\bar{m}_{j-1}, w_{j-1}} \bar{A}^{\sigma_{j}}_{j;\bar{m}_{j-1}, \bar{m}_{j}} W_{j; w_{j-1}, w_{j}}^{\sigma_{j}, \sigma^\prime_{j}} A^{\sigma^\prime_{j}}_{j; m_{j-1}, m_{j}} \\ & \} \\ & \nonumber \\ & \texttt{CONTRACT-RIGHT}(R_{j+1}, W_j, B_j) \; \{ \\ & \quad R_{j; m_{j-1}}^{\bar{m}_{j-1}, w_{j-1}} = \sum_{ \sigma_{j}, \sigma^\prime_{j}, w_j, m_j, \bar{m}_j } \bar{B}^{\sigma_{j}}_{\bar{m}_{j-1}, \bar{m}_{j}} W_{j; w_{j-1}, w_{j}}^{\sigma_{j}, \sigma^\prime_{j}} B^{\sigma^\prime_{j}}_{j; m_{j-1}, m_{j}} R_{j+1; m_j}^{\bar{m}_j, w_j} \\ & \} \\ & \nonumber \\ & \texttt{INITIALISE}(\textrm{MPO } \{ W_j \}_{j=1}^L, \textrm{MPS } \{ M_j \}_{j=1}^L) \; \{ \\ & \quad L_{0;m_0}^{\bar{m}_0, w_0} \gets 1 \textrm{ and } R_{L+1;m_L}^{\bar{m}_L,w_L} \gets 1 \\ & \quad \textrm{Right-normalize } \{ M_j \}_{j=1}^L \to \{ B_j \}_{j=1}^L \textrm{ from right to left } \\ & \quad \textbf{for } j \in [L, 2] \{ \\ & \quad \quad R_j \gets \texttt{CONTRACT-RIGHT}(R_{j+1}, W_j, B_j) \\ & \quad \} \\ & \quad \textbf{return } L_0, \; \{ R_j \}_{j=2}^{L+1}, \; \{ B_j \}_{j=1}^L \\ & \} \\ & \nonumber \\ & \texttt{TIMESTEP}(\delta, \; L_0, \; \{ R_j \}_{j=2}^{L+1}, \; \{ W_j \}, \; \{ M_1, \; B_j \}_{j=2}^L) \; \{ \\ & \quad \{ L_j \}_{j=0}^{L-1}, \; \{ A_j, M_L \}_{j=1}^{L-1} \gets \texttt{SWEEP-RIGHT}(\frac{\delta}{2}, \; L_0, \; \{ R_j \}_{j=2}^L, \; \{ W_j \}_{j=1}^L, \; \{ M_1, B_j \}_{j=2}^L) \\ & \quad \{ R_j \}_{j=2}^{L+1}, \; \{ M_1, B_j \}_{j=2}^L \gets \texttt{SWEEP-LEFT}(\frac{\delta}{2}, \; R_{L+1}, \; \{ L_j \}_{j=0}^{L-1}, \; \{ W_j \}_{j=1}^L, \; \{ A_j, M_L \}_{j=1}^{L-1}) \\ & \} \\ \end{align}
Local 2-site Krylov method/Time-Step Targeting Method
\begin{align} & \texttt{SWEEP-RIGHT}(\delta, \; L_0, \; \{ R_j \}_{j=3}^{L+1}, \; \{ W_j \}_{j=1}^L, \; \{M_1, B_j \}_{j=2}^L) \; \{ \\ & \quad A_1, C_{\underline{1}} \gets M_1 \textrm{ via QR} \\ & \quad M_2 \gets C_{\underline{1}} \cdot B_2 \\ & \quad P^{\sigma_1}_{\bar{m}_{0}, m_1} \gets A^{\sigma_1}_{1; m_0, m_1} \quad \quad \textrm{// entry-for-entry copy, only label of $m_0$ adapted} \\ & \quad \textbf{for } j \in [1, L-1] \; \{ \\ & \quad \quad T_{j,j+1} \gets \sum_{m_j} A_{j;m_{j-1}, m_j}^{\sigma_j} M_{j+1;m_j, m_{j+1}}^{\sigma_{j+1}} \\ & \quad \quad T_{j,j+1} \gets \mathrm{exp}(-\mathrm{i}\delta \hat H^{\mathrm{eff}}_{(j,j+1)}) T_{j,j+1} \textrm{ using } \hat H^{\mathrm{eff}}_{(j,j+1)} \equiv L_{j-1} \cdot W_j \cdot W_{j+1} \cdot R_{j+2} \\ & \quad \quad \textbf{if } j \neq L-1 \; \{ \\ & \quad \quad \quad A^\prime_j \gets T_{j,j+1} \textrm{ via SVD and truncation, $S$ and $V$ discarded } \\ & \quad \quad \quad L_j \gets \texttt{CONTRACT-LEFT}(L_{j-1}, W_j, A^\prime_j) \\ & \quad \quad \quad \textrm{Delete } R_{j+2} \\ & \quad \quad \quad P^{\prime \bar{m}_j}_{m_j} \gets \sum_{\sigma_j \bar{m}_{j-1}} P^{\sigma_j}_{\bar{m}_{j-1}, m_j} \cdot \bar{A}^{\prime \sigma_j}_{j; \bar{m}_{j-1}, \bar{m}_j} \quad \textrm{// complete projector} \\ & \quad \quad \quad M^{\sigma_{j+1}}_{j+1; \bar{m}_j, m_{j+1}} \gets \sum_{m_j} P^{\prime \bar{m}_j}_{m_j} \cdot M^{\sigma_{j+1}}_{j+1; m_j, m_{j+1}} \quad \quad \textrm{// project $M_{j+1}$ into new left basis} \\ & \quad \quad \quad A_{j+1}, C_{\underline{j+1}} \gets M_{j+1} \textrm{ via QR } \\ & \quad \quad \quad M_{j+2} \gets C_{\underline{j+1}} \cdot B_{j+2} \\ & \quad \quad \quad P^{\sigma_{j+1}}_{\bar{m}_{j}, m_{j+1}} \gets \sum_{m_j} P^{\prime \bar{m}_{j}}_{m_j} \cdot A^{\sigma_j}_{j+1; m_j,m_{j+1}} \\ & \quad \quad \} \textbf{ else } \{ \\ & \quad \quad \quad A^\prime_j, C_{\underline{j}}, B_{j+1} \gets T_{j,j+1} \textrm{ via SVD and truncation}\\ & \quad \quad \quad M_{j+1} \gets C_{\underline{j}} \cdot B_{j+1} \\ & \quad \quad \} \\ & \quad \quad A_j \gets A^\prime_j \\ & \quad \} \\ & \quad \textbf{return } \{ L_j \}_{j=0}^{L-2}, \{ A_j, M_L \}_{j=1}^{L-1} \\ & \} \\ & \nonumber \\ & \texttt{SWEEP-LEFT}(\delta, \; R_{L+1}, \; \{ L_j \}_{j=0}^{L-2}, \; \{ W_j \}_{j=1}^L, \; \{ A_j, M_L \}_{j=1}^{L-1}) \; \{ \\ & \quad B_L, C_{\underline{L-1}} \gets M_L \textrm{ via QR} \\ & \quad M_{L-1} \gets A_{L-1} \cdot C_{\underline{L-1}} \\ & \quad P^{\sigma_L}_{m_{L-1}, \bar{m}_L} \gets B^{\sigma_L}_{L; m_{L-1}, m_L} \quad \quad \textrm{// entry-for-entry copy, only label of m_L adapted} \\ & \quad \textbf{for }j \in [L, 2] \; \{ \\ & \quad \quad T_{j-1,j} \gets \sum_{m_{j-1}} A_{j-1;m_{j-2}, m_{j-1}}^{\sigma_{j-1}} M_{j;m_{j-1}, m_{j}}^{\sigma_{j}} \\ & \quad \quad T_{j-1,j} \gets \mathrm{exp}(-\mathrm{i} \delta \hat H^{\mathrm{eff}}_{(j-1,j)}) T_{j-1,j} \textrm{ using } \hat H^{\mathrm{eff}}_{(j-1,j)} \equiv L_{j-2} \cdot W_{j-1} \cdot W_{j} \cdot R_{j+1} \\ & \quad \quad \textbf{if } j \neq 2 \; \{ \\ & \quad \quad \quad B^\prime_{j} \gets T_{j-1,j} \textrm{ via SVD and truncation, $U$ and $S$ discarded} \\ & \quad \quad \quad R_j \gets \texttt{CONTRACT-RIGHT}(R_{j+1}, W_j, B_j) \\ & \quad \quad \quad \textrm{Delete } L_{j-2} \\ & \quad \quad \quad P^{\prime \bar{m}_{j-1}}_{m_{j-1}} \gets \sum_{\sigma_j \bar{m_j}} P^{\sigma_j}_{m_{j-1}, \bar{m}_j} \bar{B}^{\prime \sigma_j}_{j; \bar{m}_{j-1}, \bar{m}_j} \quad \quad \textrm{// complete projector} \\ & \quad \quad \quad M^{\sigma_{j-1}}_{j-1; m_{j-2}, \bar{m}_{j-1}} \gets \sum_{m_{j-1}} P^{\prime \bar{m}_{j-1}}_{m_{j-1}} M^{\sigma_{j-1}}_{j-1; m_{j-2}, m_{j-1}} \\ & \quad \quad \quad B_{j-1}, C_{\underline{j-2}} \gets M_{j-1} \textrm{ via QR} \\ & \quad \quad \quad M_{j-2} \gets M_{j-2} \cdot C_{\underline{j-2}} \\ & \quad \quad \quad P^{\sigma_{j-1}}_{m_{j-2}, \bar{m}_{j-1}} \gets \sum_{m_{j-1}} P^{\prime \bar{m}_{j-1}}_{m_{j-1}} B^{\sigma_{j-1}}_{j-1; m_{j-2}, m_{j-1}} \\ & \quad \quad \} \textbf{ else } \{ \\ & \quad \quad \quad A_{j-1}, C_{\underline{j-1}}, B_j \gets T_{j-1,j} \textrm{ via SVD and truncation} \\ & \quad \quad \quad M_{j-1} \gets A_{j-1} \cdot C_{\underline{j-1}} \\ & \quad \quad \} \\ & \quad \quad B_j \gets B^\prime_j \\ & \quad \} \\ & \quad \textbf{return } \{ R_j \}_{j=2}^{L+1}, \{ M_1, B_j \}_{j=2}^{L} \\ & \} \end{align}
The content of this page is based on Time-evolution methods for matrix-product states by S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck and C. Hubig and is licensed under the CC-BY 4.0 license.
References
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- Time-Step Targeting Time-Dependent and Dynamical Density Matrix Renormalization Group Algorithms with ab Initio Hamiltonians, Enrico Ronca, Zhendong Li, Carlos A. Hoyos, Garnet Kin-Lic Chan, J. Chem. Theory Comput. 13, 5560 - 5571 (2017)
- Time evolution of one-dimensional Quantum Many Body Systems, Salvatore R. Manmana, Muramatsu,Alejandro, Reinhard M. Noack, AIP Conference Proceedings 789, 269-278 (2005)
- Coherent matter waves emerging from Mott-insulators, K Rodriguez, S R Manmana, M Rigol, R M Noack, A Muramatsu, New Journal of Physics 8, 169 (2006)
- Time evolution of Matrix Product States, J. J. Ripoll, New J. Phys. 8, 305 (2006)
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