Time-evolving block decimation

For an open nearest-neighbor chain, group alternating bonds into even and odd Hamiltonians. Terms within one layer act on disjoint site pairs and commute, so that layer’s exponential factorizes into two-site gates. Even and odd layers generally do not commute with one another. A symmetric composition with a half step of one layer around a full step of the other gives a second-order approximation. Boundary terms must be assigned consistently; long-range couplings need a modified strategy rather than being silently omitted.

For sufficiently controlled local operators, the one-step operator error is cubic in the time step and the accumulated error at fixed final time is quadratic before truncation. The coefficient depends on commutators and energy scales. Halving the step should produce an approximately fourfold decrease of the leading error only after entering that asymptotic regime. This convergence statement does not include MPS compression, floating-point error, or a physical approximation already made when choosing the Hamiltonian.

Contract neighboring tensors with a gate, reshape the resulting center as a matrix across their bond, and perform a singular-value decomposition. Keeping every nonzero singular value preserves the gated state; retaining only a selected number introduces a Schmidt truncation. In a canonical environment, the omitted squared singular values have the one-cut meaning developed earlier. Renormalizing the MPS afterwards restores unit norm but cannot restore discarded correlations or undo a wrong gate sequence.

After a generic global quench, entanglement across cuts often grows with time, so maintaining a fixed truncation threshold can require rapidly increasing bond dimension. A converged short-time trace is therefore not evidence for arbitrary long-time accuracy. Monitor target observables under increasing dimension, and track norm, energy, conserved charges, and discarded weights as complementary diagnostics. A symmetry-preserving implementation can conserve a charge exactly while other correlations are already inaccurate, so a single conserved quantity is never a complete fidelity test.

Replacing the real-time phase exponential by $e^{-\delta\tau H}$ gives nonunitary imaginary-time evolution, where delta tau has inverse-energy units here. Higher-energy coefficients shrink relative to lower-energy ones. Repeated normalization prevents overall underflow and makes a state representation convenient, but the limiting state depends on nonzero overlap with the lowest accessible sector. Degenerate ground components are filtered together, while symmetries can prohibit reaching a lower state in another sector.

When studying real-time spectra, a finite observation window also limits frequency resolution. Windowing changes spectral features and must not be confused with an intrinsic decay rate. For either real or imaginary evolution, vary time step at fixed sufficiently large bond dimension, then vary dimension with a sufficiently small step, and finally examine physical size and final time. The pure two-state instrument precesses a phase coherently; it cannot apply two-site gates, create a tensor-chain light cone, or carry out normalized imaginary-time filtering.