The essential technique of TEBD is to write the time evolution operator $U(\delta t) = e^{-iH\delta t}$ (where $\hbar$ is considered to be $1$), for a small time $\delta t$ using Suzuki-Trotter (ST) decomposition. If the Hamiltonian of interest contains only the nearest-neighbor interactions, then the Hamiltonian can be re-written in terms of local two site terms as Here all $H_{odd}$ or $H_{even}$ terms commute among themselves but, $h_{(2i-1), 2i}$ and $ h_{2i, (2i+1)}$ do not commute in general since they share a common site $2i$. This non-commutative nature gives rise to an error in the ST decomposed time evolution operator. The first order Suzuki-Trotter (ST(1)) decomposed $U(\delta t)$ can be written as So, the time evolved wavefunction becomes In the MPS description of the wavefunction, the two site operators in $H_{odd}$ or $H_{even}$ can be operated independently since they commute with each other. After operating a two-site operator, the local dimension increases by $d^2$ times and the MPS description vanishes on those two sites. This tells us that at every time step, the dimension increases exponentially. However, applying the SVD technique, one can reduce the dimension by truncating the singular value spectrum as before and also restore the MPS description after each time step. Now, discretizing the total time of evolution $t = N\delta t$, the repeated operation of the Eq. \ref{eq:tebd} on the initial state can evolve the state to time $t$. An example of this method of time evolution is depicted in figure for $N=2$. The ST decomposition described above works if the Hamiltonian only has the nearest-neighbor terms. In general, if the Hamiltonian has the next-nearest-neighbor terms, the ST decomposition of the evolution operator is done with three-site local operators instead of two.

Note that during every step of time evolution, entanglement builds up in the system. In the process of truncation in SVD, the error also builds up in the wave-function. Apart from this, errors also arise from the ST decomposition, which is of the order of $\delta t^2$. This error can be reduced by considering a very short time step $\delta t$. The error in ST decomposition can be further reduced by considering higher-order ST decomposition since in ST($n$), the error comes of the order of $\delta t^{n+1}$. In the following, an ST($2$) decomposition is shown. Using small time steps and higher-order ST decomposition, this error can also be significantly minimized. Nevertheless, considering all these sources of errors, the accessible time scale allows us to study the system of interest efficiently. There is also a TEBD algorithm for infinite systems (iTEBD).

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