TEBD

Time-evolving block decimation (TEBD) approximates a local propagator by small gates and applies each gate to an MPS, compressing afterward. Its two defining errors are independent: splitting a noncommuting Hamiltonian and discarding Schmidt components. Reducing the time step removes the former but can increase the number of truncations.

Required background. MPS compression gives the local update, and split-operator propagation gives the second-order product formula.

Even and odd bond evolution

For an open nearest-neighbor chain, let $H=H_{\rm e}+H_{\rm o}$, where each group contains mutually disjoint bond terms. Include on-site terms exactly once, distributing them among adjacent bonds or treating them in a separate splitting. For time-independent $H$,

$$U_2(\Delta t)= e^{-iH_{\rm e}\Delta t/(2\hbar)} e^{-iH_{\rm o}\Delta t/\hbar} e^{-iH_{\rm e}\Delta t/(2\hbar)}.$$

Each layer is a product of two-site gates. The local operator error is $O(\Delta t^3)$ and, at fixed final time under suitable boundedness assumptions, the global splitting error is $O(\Delta t^2)$. The relevant constants involve commutators, so a small dimensionless $\Delta t\|H\|/\hbar$ alone is a conservative and often uninformative criterion for a long chain.

For time-dependent terms, midpoint evaluation can maintain second-order accuracy when combined with the symmetric sequence, but discontinuous pulses must be split at their discontinuities.

Applying a gate in canonical gauge

Move the orthogonality center to the active bond and form $\Theta_{\alpha st\beta}$. Apply

$$\Theta'_{\alpha uv\beta} =\sum_{st}G_{uv,st}\Theta_{\alpha st\beta}.$$

Reshape with row $(\alpha,u)$ and column $(v,\beta)$, SVD, truncate, and restore the canonical arrangement. Local orthonormality is essential: only then does the SVD tail measure state-norm loss. Avoid dividing by tiny old Schmidt coefficients in implementations that can instead move centers by QR or SVD.

In real time, each exact gate is unitary. Renormalizing after truncation keeps the state norm at one but hides discarded probability, so record the pre-renormalization loss separately.

An entangling two-site test

Take $H=JX_1X_2$, $|\psi(0)\rangle=|00\rangle$, and $\theta=Jt/\hbar$. Exact evolution gives

$$|\psi(t)\rangle=\cos\theta\,|00\rangle-i\sin\theta\,|11\rangle.$$

There is only one bond and hence no splitting error. With $D=2$, the algorithm should reproduce this state to gate-exponential and roundoff accuracy. Applying the entire gate for $\theta=\pi/4$ once and then compressing to $D=1$ discards half the Schmidt weight. Repeated smaller gates with rank-one compression after each gate follow a different approximate path. This test isolates compression from Trotter error and checks the complex phase of the off-diagonal amplitude.

A separate three-site test with noncommuting adjacent bonds is needed to test the expected second-order slope.

Convergence at a fixed final time

First increase $D$ until the observable is insensitive to compression at the finest time steps of interest. Compare $\Delta t$, $\Delta t/2$, and $\Delta t/4$; the ratio of successive observable differences should approach four when a nonzero second-order leading error dominates. Repeat with larger $D$ to ensure the slope is not a cancellation of two errors.

For time-independent $H$, monitor energy drift, but do not expect exact energy conservation from split gates. Compare local observables, connected correlations, and entanglement growth. The usable simulation time ends when those quantities lose convergence even if the norm remains exactly one.

Imaginary-time TEBD replaces $-i\Delta t/\hbar$ by a negative real inverse-energy step. Gates are then nonunitary, normalization is intentional, and the ground-state projection needs its own convergence in projection time and step size.

Exercises

Why does a smaller time step not guarantee a better fixed-$D$ TEBD trajectory?

Solution

It decreases splitting error but increases the number of gate applications and SVD projections before the same final time. The sequence of projections changes the trajectory and can accumulate compression error. Convergence must therefore vary both $D$ and $\Delta t$, rather than interpreting the time-step limit of a restricted manifold as the exact evolution.

References

• Schollwöck, Ulrich, “The density-matrix renormalization group in the age of matrix product states,” Annals of Physics 326, 96–192 (2011), arXiv:1008.3477 — canonical gauges, contractions, and finite-chain algorithms.
• Vidal, Guifré, “Efficient simulation of one-dimensional quantum many-body systems,” Physical Review Letters 93, 040502 (2004), doi:10.1103/PhysRevLett.93.040502 — adaptive gate evolution of compressed states.