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Time Dependent Variational Principle for Tree Tensor Networks

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Pith's one-line read A single projector generalizes TDVP to every loop-free tensor network, not just matrix product states.

desk verdict A useful TDVP generalization for arbitrary loop-free tensor networks, with convincing numerics and a real but fixable index error in the vertical-subspace derivation that should be corrected before publication. read the letter →

arxiv 1908.03090 v3 pith:ZHROH27E submitted 2019-08-08 cond-mat.str-el

Daniel Bauernfeind , Markus Aichhorn This is my paper · ORCID
classification cond-mat.str-el
keywords time-dependentvariationalprincipletreetensornetworkstangentspaceprojectorForkProductStatesAndersonimpuritymodeloff-diagonalhybridizationreal-timeevolutionTrotterdecomposition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper extends the time-dependent variational principle (TDVP) from matrix product states to any finite, loop-free tensor network, which it calls a tree tensor network (TTN). Its central claim is that Eq. (15) gives the orthogonal projector onto the tangent space at any TTN state, so that time evolution stays on the tensor-network manifold as long as the Hamiltonian can be written in the same network structure. This matters because TDVP then offers a time-evolution method for Hamiltonians with long-range or off-diagonal couplings, where TEBD becomes difficult to implement. The authors demonstrate the method on Fork Tensor Product States for multi-orbital Anderson impurity models, showing that off-diagonal hybridizations relevant for spin-orbit coupling and lattice distortions can be treated.

What carries the argument

The central object is the tangent-space projector of Eq. (15), built from site tensors $T^{s_i}_{Q_i}$ orthogonalized toward each neighbor (written $T_N^{[q_k]}$) and from the mutually orthogonal link states $|q^{(i)}_k\rangle$ that each cut of the tree defines. This projector decomposes the global projected Schrödinger equation into one-site effective Hamiltonians $H_{(s_i Q_i),(s'_i Q'_i)}$ and link effective Hamiltonians $K_{(q^{(i)}_k q^{(j)}_k),(q^{(i)'}_k q^{(j)'}_k)}$; each local piece is integrated by exponentiating the effective Hamiltonian with Krylov methods. The sweeping order, from a chosen start leaf to a chosen end leaf, turns these local updates into a first-order TDVP step, and reversing the sweep with half time steps gives a second-order integrator. The two-site variant replaces the site and link updates by a two-site effective Hamiltonian followed by an SVD, allowing adaptive bond-dimension growth.

What would settle it

On a small non-binary tree (for example a star graph with three leaves), compute the dimension of the kernel of the linear map from tangent tensors $\{B^{s_i}_{Q_i}\}$ to the state $|\Theta[B]\rangle$. If this dimension is larger than the number of free parameters in the $X$-matrices allowed by Eq. (9) under Eq. (10) — or if a single gauge-equivalent-to-zero perturbation cannot be written in that form — then Eq. (15) is not the full tangent-space projector, and TDVP on such a tree would not be the exact manifold projection.

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Extended reading notes

Core claim

The paper establishes that for any finite loop-free tensor network, the tangent space at a state $|\psi[T]\rangle$ is spanned by local tensor variations, and the orthogonal projector onto it has the explicit form $P_{T|\psi[T]\rangle} = \sum_i \mathbb{1}_{s_i}\otimes\sum_{Q_i}|q^{(i)}_1\cdots q^{(i)}_{r_i}\rangle\langle q^{(i)}_1\cdots q^{(i)}_{r_i}| - \sum_{\langle i,j\rangle_{q_k}}\sum_{q_k q'_k} |q^{(j)'}_k\rangle\langle q^{(j)'}_k|\otimes |q^{(i)}_k\rangle\langle q^{(i)}_k|$ (Eq. 15). The derivation generalizes the MPS construction by parameterizing the vertical subspace with one matrix $X$ per link, fixing the gauge with $N-1$ constraints, and choosing any leaf as end point. Integrating the projected Schrödinger equation term by term with a Suzuki-Trotter breakup yields single-site and two-site TDVP sweeps; the paper verifies the scheme on impurity models, including a non-interacting case with off-diagonal hybridizations where it matches the exact Green's functions.

Load-bearing premise

The argument depends on the assumption that every redundant direction of the tensor network is caught by exactly one matrix per link, with signs set by the chosen end point, and that the $N-1$ gauge conditions fix these matrices uniquely.

Editorial extensions

If this is right

  • Any Hamiltonian that admits a representation in the same TTN structure as the state can be time-evolved with TDVP, including long-range couplings that are hard for TEBD.
  • The single-site TDVP variant preserves exactly the conserved quantities of the Hamiltonian, because the evolution never leaves the tensor-network manifold.
  • The two-site variant provides a controlled truncation and can grow bond dimensions dynamically during the evolution.
  • For FTPS impurity solvers, off-diagonal hybridizations (for example from spin-orbit coupling or lattice distortions) can be included without reformulating TEBD.
  • Second-order convergence in the time step $\Delta t$ is observed, matching the numerical $\sim \Delta t^2$ scaling reported in the paper.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: a direct count of the kernel of the map $B \mapsto |\Theta[B]\rangle$ on a small non-binary tree would test whether Eq. (9) really captures all gauge directions; if it does not, the projector needs extra null vectors for high-degree nodes, though the FTPS demonstration would still stand.
  • Beyond the paper: because the projector is constructed from the current orthogonalized state, the same local update structure could be combined with a Hamiltonian-adapted sweeping order, possibly reducing the number of long-range terms each sweep must cover.
  • Beyond the paper: the non-interacting off-diagonal benchmark is the cleanest reported test; repeating it at finite interaction with a controlled impurity-impurity bond dimension would show whether the method's accuracy survives the interacting regime.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper generalizes the time-dependent variational principle (TDVP) to finite loop-free tensor networks (TTNs). The derivation follows the MPS route of Haegeman et al.: it defines a tangent-space representation, parameterizes the vertical subspace by one matrix per link (Eq. 9), imposes gauge-fixing constraints (Eq. 10), and solves a least-squares problem to obtain a tangent-space projector (Eq. 15). The projector is then integrated by sequential local site and link updates, yielding first- and second-order single-site and two-site sweeping algorithms. As an application, the authors implement TDVP for Fork Tensor Product States (FTPS) and use it to compute impurity Green's functions for multi-orbital Anderson impurity models with off-diagonal hybridizations, benchmarking against TEBD and exact diagonalization. The numerical results in Figs. 10–12 show good agreement and convergence.

Significance. If the central derivation is correct, the paper provides a genuinely useful framework: TDVP for arbitrary tree tensor networks, beyond the MPS and binary-tree cases, with the practical advantage that only a Hamiltonian representation in the same tensor-network structure is needed. The FTPS application targets a real problem in DMFT, namely off-diagonal hybridizations relevant for spin-orbit coupling and lattice distortions. The numerical tests are appropriate and include external benchmarks (TEBD, exact diagonalization) rather than self-referential checks, and the convergence study in Fig. 12 is a strength. The presentation is clear and the claimed generalizations are concrete. However, the derivation of the central projector contains a load-bearing error in the parameterization of the vertical subspace, which currently prevents the main theoretical claim from being accepted as stated. The algorithm may well be correct, but the paper's proof is not.

major comments (2)
  1. [Sec. 3.1, Eq. (9)] The proposed parameterization of the vertical subspace is incorrect, because it uses right multiplication by the same matrix X on both sides of each link. For a two-site TTN with endpoint N=2, tensors A_{s1,q} and C_{s2,q}, Eq. (9) gives B1 = A X and B2 = -C X. The zero-state condition following from Eq. (8) is B1 C^T + A B2^T = 0. Substituting yields A X C^T - A X^T C^T = A (X - X^T) C^T, which vanishes only for symmetric X. Taking A = C = I_2 and X with the single off-diagonal entry X_{12} = 1 gives |Theta[B]> = |1,2> - |2,1> != 0, so a non-symmetric X is mapped to a nonzero tangent vector. The correct gauge variation on the side of the link away from the endpoint requires the transpose: B2 = -C X^T (equivalently, left multiplication by X). Consequently, Eq. (9) does not reduce to the MPS kernel of Ref. [29], and the derivation of Eqs. (10)–(15) as written is unsupported. The kernel parameterization and the subsequent counting argument must be re-derived with the correct orientation of the X matrices on each side of every link; the final projector may be correct, but the proof needs to be fixed.
  2. [Sec. 4, Eqs. (16)–(22)] The Hamiltonian representation in the FTPS network is never constructed explicitly. The effective one-site and link Hamiltonians in Eqs. (16) and the two-site Hamiltonian in Eq. (18) presuppose a tensor-network (MPO-like) representation of H in the same FTPS geometry. For the off-diagonal hybridization Eq. (22), the paper only states that the matrix V can be chosen lower-triangular; it does not show how the terms c^†_{l0} c_{l'k} are encoded as operators on the TTN. Since the abstract and introduction emphasize that the method works whenever such a representation is available, and since the FTPS demonstration is a central part of the paper, the explicit construction (or a precise pointer to it) is needed for reproducibility and to substantiate the claim that the off-diagonal case is within the method's scope.
minor comments (4)
  1. [Sec. 3.1, Eq. (9) and Sec. 2.2] The notation T_N^{[q_l]} in Eq. (9) is confusing: the subscript N suggests the endpoint of the tree, but the text says this is the tensor on site i orthogonalized towards the neighbor along q_l. It should read T_i^{[q_l]} (or an equivalent site label). The same ambiguity appears in Sec. 2.2 when tensor 4 is denoted as T_N^{[q2]}.
  2. [Throughout] There are several typos and formatting issues: 'coefficientcs1···sN' in Sec. 2.1, 'Where∑<i,j>qk' after Eq. (15), and '10 −9' in the captions of Figs. 10 and 11 should be corrected to proper mathematical notation.
  3. [Sec. 4] In the algorithm description, 'for k = Nb : 1' should be written as a decreasing loop (e.g., 'k = N_b, N_b - 1, ..., 1'), and 'impurity tensors itself' should be 'impurity tensors themselves'.
  4. [Fig. 10 caption] The caption reports TDVP with dt = 0.1 and TEBD with dt = 0.01 and notes that TDVP generally allows larger steps. A brief explanation of why this is the case for the present model, beyond citing Ref. [50], would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the TTN tangent-space projector is derived from the Schrödinger equation and linear algebra, and the numerical tests are benchmarked against independent TEBD and exact-diagonalization results.

full rationale

The central derivation is self-contained. Equation (15) is obtained from the tangent-space parameterization in Eq. (7), the vertical-subspace characterization in Eq. (9), the gauge-fixing constraints in Eq. (10), and the minimization problem in Eqs. (12)-(14); no fitted quantity or external target is used to produce the projector. The single-site and two-site TDVP updates are then obtained by Trotterized integration of the projected Schrödinger equation, Eq. (6), which is again a direct construction rather than a restatement of an input. The numerical demonstrations compare TDVP against TEBD (Fig. 10) and against exact diagonalization in the non-interacting case (Fig. 11), with convergence checks in Fig. 12, so the reported agreement is verified against independent references rather than encoded in the derivation. Self-citations appear for the FTPS tensor-network construction (Refs. [18,19]) and for a prior performance comparison (Ref. [50]), but these are contextual or comparative and are not load-bearing for the claimed generalization of TDVP to arbitrary TTNs. A skeptical concern about whether Eq. (9) correctly parameterizes the vertical subspace for non-binary or high-degree trees is a mathematical-correctness issue, not a circularity issue: the equation is not derived from the final projector, and if it were false the derivation would be invalid rather than tautological. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard linear algebra (SVD, QR, Krylov exponentiation, Trotter-Suzuki breakup) and on the domain assumption that the Hamiltonian admits a tensor network representation with the same loop-free structure as the state. No free parameters are fitted to the numerical results, and no new physical entities are introduced.

assumptions (6)
  • standard math SVD and QR decompositions provide orthonormal bases for the orthogonalization and Schmidt truncation steps.
    Used in Sec. 2.2 and 2.3 to define orthogonality centers and truncations; this is standard linear algebra.
  • standard math The Trotter-Suzuki decomposition can split the sum of local terms in the tangent-space projector into a sequence of exponentials.
    Used in Sec. 3.2 and 3.3 to construct first- and second-order sweeps; standard and not specific to the paper.
  • standard math Krylov subspace methods approximate the action of the matrix exponential accurately enough for the effective Hamiltonians.
    Invoked in Sec. 3.2 after Eq. 17; the paper does not analyze the error introduced by Krylov exponentiation.
  • standard math For a finite tree with N sites and N-1 links, the local gauge constraints in Eq. 10 determine the L=N-1 link matrices uniquely.
    Counting argument in Sec. 3.1; it depends on the tree property that each non-root site has exactly one link toward the endpoint.
  • domain assumption The Hamiltonian admits a tensor network representation with the same loop-free structure as the state.
    Stated in the abstract and Sec. 4 as the main advantage of TDVP; the paper does not explicitly construct such a representation for the off-diagonal hybridization Hamiltonian.
  • domain assumption Every tensor in the TTN carries a physical index, or can be treated with a dummy single-state index.
    Invoked in Sec. 2.1 to simplify notation; harmless but an assumption about the network structure.

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Cite this review

Pith. "Pith review of Time Dependent Variational Principle for Tree Tensor Networks." pith.science (2026). https://pith.science/paper/ZHROH27E

@misc{pith2026190803090,
  author       = {Pith},
  title        = {Pith review of: Time Dependent Variational Principle for Tree Tensor Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZHROH27E}},
  note         = {Machine review of arXiv:1908.03090}
}
read the original abstract

We present a generalization of the Time Dependent Variational Principle (TDVP) to any finite sized loop-free tensor network. The major advantage of TDVP is that it can be employed as long as a representation of the Hamiltonian in the same tensor network structure that encodes the state is available. Often, such a representation can be found also for long-range terms in the Hamiltonian. As an application we use TDVP for the Fork Tensor Product States tensor network for multi-orbital Anderson impurity models. We demonstrate that TDVP allows to account for off-diagonal hybridizations in the bath which are relevant when spin-orbit coupling effects are important, or when distortions of the crystal lattice are present.

Figures

Figures reproduced from arXiv: 1908.03090 by the authors.

Figure 1
Figure 1. Example of a TTN with 7 tensors with different numbers of link-indices on each [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Gauge degree of freedom in tensor networks. At each link, one can insert an identity [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. With an SVD, we can orthogonalize a tensor towards one of its neighbors with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (10 more)
Figure 1
Figure 1. Figure 1: The property distinguishing a TTN from a general tensor network is that the graph [PITH_FULL_IMAGE:figures/full_fig_p004_1.png]
Figure 4
Figure 4. Figure 4: If the orthogonality center of the TTN depicted in Fig. 1 is placed on site 4, the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Definition of the vertical subspace for site 4 in Eq. 9 with end point site 7. The [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Formally, this result resembles the projection operator obtained for MPS [29]. The [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Update sequence to perform a first-order single-site TDVP time step from time [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Update sequence to perform a first-order two-site TDVP time step from time [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Graphical representation of a FTPS tensor network for a two orbital model. For [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: Comparison of the impurity greater Greens function [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Comparison of impurity greater Greens function [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: Error as a function of control parameters for the same AIM used in Fig. 11. We [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]

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