A priori prediction of the shared bond dimension of bundled matrix product states for many low-lying excitations

Multi-target DMRG with block Lanczos stores the $g$ lowest-lying eigenstates of a one-dimensional local Hamiltonian as a single bundled matrix product state carrying an excitation index. Empirically the low-lying excitations share bond-basis states so heavily that a bond dimension far below the worst case $m \propto g$ — for example a bundle of one hundred excitations converging at bond dimensions of order a few hundred — suffices, yet no way exists to predict when this compactification succeeds for a given Hamiltonian. Establish an a priori criterion, computable from intrinsic properties of the Hamiltonian (spectral gap structure, low-energy density of states, symmetry-sector content), that predicts the bond dimension $m(g)$ required to represent $g$ excitations simultaneously to fixed accuracy. The direct follow-up by Baker and Seif proved qualitative theorems (large energy differences force dissimilar density matrices; similar same-sector states share the bond basis) only in an ultra-local limit and left the quantitative predictor open; subsequent multitarget-MPS work remains algorithmic and empirical. The core remains open.

Background

A matrix product state (MPS) compresses a one-dimensional quantum many-body wavefunction into a product of site-local tensors whose auxiliary bond indices are contracted; the maximum dimension $m$ of those indices, the bond dimension, controls both accuracy and cost (the number of parameters grows as $O(N d m^2)$ for $N$ sites of local dimension $d$). The density matrix renormalization group (DMRG) is the standard variational optimizer of such networks, and it excels for ground states because they obey an entanglement area law: across any cut, the Schmidt spectrum — the eigenvalues of the reduced density matrix obtained by tracing out one part of the system — decays quickly, so a small $m$ captures the state to high accuracy (see Schollwöck's review for the framework). Excited states are harder to reach: the usual penalty method adds a term that raises the energy of already-computed lower states, which degrades conditioning and mishandles degeneracies, while single-excitation ansätze describe one quasiparticle at a time in the thermodynamic limit. Baker, Foley and Sénéchal therefore introduced a multi-targeted DMRG: all $g$ targeted eigenstates live in one bundled MPS carrying an extra index of range $g$ cataloging the excitations, and the local update is a block Lanczos step — a block version of the Lanczos Krylov-subspace iteration that refines a whole set of vectors against the Hamiltonian at once — followed by a gauge decomposition (SVD, QR, or a polar form) that re-orthonormalizes the block. A related block-Krylov eigensolver in tensor-train format — the mathematics name for the matrix product ansatz — had been given earlier by Dolgov et al. using the locally optimal block preconditioned conjugate gradient method (LOBPCG) of Knyazev; the multi-target method is simpler, resolves degeneracies without enlarging the matrix product operator, and treats $g \geq 100$ excitations at once. The striking empirical fact is compactness of the shared bond basis: the low-lying excitations of the models studied share so many Schmidt states across every bond that a bond dimension far smaller than the worst case $m \propto g$ — one independent bond basis per state — suffices; on a 10-site spin-1/2 Heisenberg chain a bundle of $g = 100$ excitations — roughly one tenth of the model's total Hilbert space — converges at bond dimensions of order a few hundred. An exact-diagonalization study of the 30-site spin-1/2 Heisenberg model shows the mechanism and its limits: the exact bundle of the lowest states across the 16 distinct quantum-number (total $S^z$) sectors requires bond dimension about 700, yet truncating to $m = 200$ still resolves 8 excitations — and non-monotonically, $m = 1000$ resolves only 7, because singular values belonging to differently weighted symmetry sectors truncate unevenly. The authors state the natural question — how the required bond dimension scales with the number of bundled excitations — observe that the linear law is the logical worst case while in practice the bond dimension is often far lower, leave open an a priori criterion predicting which regime a given Hamiltonian falls into, and note that near-degenerate singular values and uneven cross-sector weighting complicate any truncation rule for the bundled density matrix. This is not merely a practicality: DMRG cost grows polynomially in $m$, so choosing $m$ too small silently loses excitations while choosing it too large wastes the entire computation budget, and the same compactness question governs which symmetry sectors can be bundled at all.

Problem Statement

For a local Hamiltonian $H$ on a one-dimensional lattice and a target accuracy $\varepsilon$, determine — and predict a priori — how the bond dimension $m(g)$ required for a bundled matrix product state to represent the $g$ lowest-lying eigenstates of $H$ simultaneously scales with $g$. The criterion must be computable before solving the multi-target eigenproblem, from intrinsic properties of $H$ such as its spectral gap structure and low-energy density of states and its symmetry-sector decomposition, and it must demarcate the compact regime in which the excitations share bond-basis states so that $m(g)$ stays far below the worst case, from the worst-case regime $m \propto g$ in which each added excitation essentially requires its own independent bond basis. Admissible forms of answer include quantitative bounds on $m(g)$ proved under explicitly stated assumptions on $H$, or a validated predictive criterion demonstrated across model classes with different behavior (such as gapped and critical, fermionic and bosonic, with varied symmetry content), or a rigorous demonstration that no criterion computable a priori from intrinsic properties of $H$ can exist.

References

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