TDVP

The time-dependent variational principle (TDVP) evolves an MPS by projecting the Schrödinger vector field onto the directions available within its variational manifold. This makes Hamiltonians represented as general MPOs accessible without splitting them into disjoint gates. It also introduces a representation error that can survive perfect norm and energy conservation.

Required background. MPS canonical gauges and MPO environments supply the local coordinates. TEBD provides a useful comparison.

Projection of the Schrödinger vector field

Take a normalized MPS $|\psi\rangle$ as a point on the smooth manifold $\mathcal M$ of unnormalized MPS with fixed bond ranks. Let $P_T(\psi)$ be the orthogonal projector onto that complex tangent space, including the direction of $\psi$ itself. TDVP solves

$$i\hbar\frac{d|\psi\rangle}{dt} =P_T(\psi)H|\psi\rangle.$$

It chooses the available tangent vector closest in Hilbert norm to $H|\psi\rangle$. The omitted component

$$r_{\rm proj}=(I-P_T)H|\psi\rangle$$

is a local-in-time measure of inaccessible dynamics. Integrating the projected equation accurately does not make this residual vanish.

For the full unnormalized fixed-rank MPS tangent space, $\psi$ is tangent, so $P_T\psi=\psi$. With time-independent Hermitian $H$, exact projected evolution conserves norm and energy: $P_T=P_T^\dagger$ makes $\langle H\psi|P_T|H\psi\rangle$ real, hence $d\langle H\rangle/dt=0$. Finite integrators and inexact local solves add their own errors.

Projector splitting as a sweep

Canonical gauges decompose the tangent projector into local site projectors minus overlapping bond projectors. A projector-splitting integrator evolves a center tensor with its effective Hamiltonian, shifts the center, and evolves the intervening bond tensor with the compensating sign. The backward bond substep corrects double counting of tangent directions; it does not represent physical backward time for the whole state.

A symmetric forward/backward sweep gives a second-order integration scheme when local exponentials are solved accurately. Use Krylov exponential actions rather than dense effective matrices. Tighten local exponential tolerances independently from the global step.

One-site TDVP fixes virtual dimensions and symmetry sectors. Two-site updates temporarily enlarge the local variational space and can grow bonds, then compress by SVD. That compression generally spoils exact energy conservation. Near rank-deficient states, unused directions require care; avoid unstable formulas involving inverses of tiny Schmidt values.

A conservation test that still gives wrong dynamics

Let $H=JX_1X_2$ and start from $|00\rangle$. On the rank-one product-state manifold, tangent vectors change one spin at a time. The exact direction $H|00\rangle=J|11\rangle$ is orthogonal to that tangent space. Thus one-site fixed-rank TDVP remains at $|00\rangle$.

Its norm is one and its energy is zero for all time. Nevertheless the exact solution is $\cos(Jt/\hbar)|00\rangle-i\sin(Jt/\hbar)|11\rangle$, which becomes entangled. Here $\|r_{\rm proj}\|=|J|$ detects the error immediately. A two-site enlargement to rank two recovers the missing direction.

Converging an observable

Vary the integration step at fixed sufficiently large rank, then increase rank at the accepted step. Compare one-site and two-site trajectories where both are feasible. Measure local observables, correlations, projection diagnostics where available, and Schmidt tails. A stable conserved energy is a useful implementation check but an incomplete accuracy test.

Long-range terms are naturally handled through the MPO, but their operator bond dimension and environment contraction cost still matter. TDVP does not make every long-range Hamiltonian inexpensive.

Exercises

Show that the exact energy derivative vanishes for projected evolution when $H$ is time independent.

Solution

Using $\dot\psi=-iP_TH\psi/\hbar$ gives $d\langle H\rangle/dt=2\operatorname{Re}\langle H\psi|\dot\psi\rangle$. This is $2\operatorname{Re}[-i\langle H\psi|P_T|H\psi\rangle/\hbar]=0$ because the expectation of an orthogonal projector is real. The argument concerns the projected differential equation, not an arbitrary discretization or subsequent truncation.

References

• Haegeman, Jutho, Christian Lubich, Ivan Oseledets, Bart Vandereycken, and Frank Verstraete, “Unifying time evolution and optimization with matrix product states,” Physical Review B 94, 165116 (2016), arXiv:1408.5056 — tangent-space projector splitting.
• 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.