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A Strictly Single-Site DMRG Algorithm with Subspace Expansion

Claudius Hubig, Ian P. McCulloch, Ulrich Schollwöck, F. Alexander Wolf

arXiv:1501.05504v2cond-mat.str-el

TL;DR

DMRG becomes difficult for nearly two-dimensional systems because growing entanglement enlarges the required search space and increases the risk of local minima. The paper introduces an AMEn-based strictly single-site algorithm using cheaper subspace expansion, achieving lower optimization cost and faster convergence, including speed-ups up to 2.6 over CWF and 3.9 over two-site DMRG in the supplied results.

  • Problem

    Nearly two-dimensional systems are difficult for DMRG because entanglement growth increases matrix dimensions and makes local minima more likely.

  • Method

    The paper constructs a strictly single-site DMRG algorithm using AMEn-inspired subspace expansion instead of full reduced-density-matrix access.

  • Results

    Compared with CWF, DMRG3S achieves a speed-up of approximately 2.6 consistently across tested m, while its speed-up over two-site DMRG reaches 3.9 at m = 1200.

  • Takeaways & Limitations

    The method combines cheaper basis enrichment with improved high-accuracy convergence and reduced auxiliary costs for demanding large-m calculations.

Abstract

from arXiv · show

We introduce a strictly single-site DMRG algorithm based on the subspace expansion of the Alternating Minimal Energy (AMEn) method. The proposed new MPS basis enrichment method is sufficient to avoid local minima during the optimisation, similarly to the density matrix perturbation method, but computationally cheaper. Each application of $\hat H$ to $|Ψ\rangle$ in the central eigensolver is reduced in cost for a speed-up of $\approx (d + 1)/2$, with $d$ the physical site dimension. Further speed-ups result from cheaper auxiliary calculations and an often greatly improved convergence behaviour. Runtime to convergence improves by up to a factor of 2.5 on the Fermi-Hubbard model compared to the previous single-site method and by up to a factor of 3.9 compared to two-site DMRG. The method is compatible with real-space parallelisation and non-abelian symmetries.

I. INTRODUCTION

DMRG is effective for one-dimensional systems but becomes difficult for wide, nearly two-dimensional systems because entanglement and matrix dimensions grow strongly with width. This paper adopts AMEn-inspired subspace expansion to construct a strictly single-site algorithm with lower optimization cost and broader basis enrichment.

  • I. INTRODUCTION: Nearly two-dimensional systems remain difficult for DMRG because entanglement grows with width, causing rapidly increasing matrix dimensions and search spaces.The larger search space also makes local minima more likely and matrix operations more challenging.
  • I. INTRODUCTION: The proposed algorithm uses AMEn-inspired subspace expansion to enable strictly single-site DMRG without accessing the full reduced density matrix.The approach targets the convergence difficulties associated with single-site optimization.
  • I. INTRODUCTION: Each Hamiltonian application in the central eigensolver gains a speed-up of approximately (d + 1)/2 over the center-matrix wavefunction formalism.Here, d is the physical state-space dimension at each site.
  • I. INTRODUCTION: MPS tensors encode site coefficients with bond dimensions mi, while the maximal bond dimension m controls the computational scale of the representation.The physical dimension is typically constant across sites, and MPO tensors represent the Hamiltonian.
  • I. INTRODUCTION: DMRG sweeps through the chain, updating one or two site tensors by local eigensolver optimization and optionally enriching the MPS basis.Enrichment can use density matrix perturbation or subspace expansion.

III. PERTURBATION STEP AND CENTERMATRIX WAVEFUNCTION FORMALISM (CWF)

Single-site DMRG reduces the local search space but can become trapped because fixed neighboring bases prevent needed states from entering the optimization. Density matrix perturbation addresses this by expanding the basis using information from the reduced density matrix and a mixing factor.

  • A. Convergence Problems of Single-Site DMRG: Single-site DMRG is cheaper than two-site optimization but can get stuck because fixed left and right bases restrict accessible states.With symmetries, newly needed quantum-number sectors may have zero norm and cannot be introduced by the local tensor alone.
  • A. Convergence Problems of Single-Site DMRG: A DMRG local minimum is an energy substantially above the minimum achievable within the allowed MPS bond dimension.The algorithm is variational over the state space represented at that bond dimension.
  • B. Density Matrix Perturbation: Density matrix perturbation solves this convergence problem by constructing a reduced density matrix on the next bond after local optimization.The construction is described for a left-to-right sweep and reverses direction analogously for the opposite sweep.
  • B. Density Matrix Perturbation: The resulting density-matrix basis determines new right-hand-side states, while the mixing factor α controls the perturbation strength.The inverse basis mapping is applied to the neighboring tensor, and α is selected by a separate adaptive procedure.

C. Centermatrix Wavefunction Formalism (CWF)

The CWF enlarges the local representation so reduced-density-matrix perturbation can introduce new states, but this incurs substantial Hamiltonian and density-matrix costs. Subspace expansion instead adds basis flexibility through an expansion tensor while preserving the state before subsequent optimization and truncation.

  • C. Centermatrix Wavefunction Formalism (CWF): Standard single-site DMRG cannot construct the required perturbation density matrix from the current bond alone because it lacks the preceding physical and bond dimensions.The optimal new basis depends on di and mi−1 as well as the existing mi states.
  • C. Centermatrix Wavefunction Formalism (CWF): CWF introduces a center matrix containing the site tensor and maps the left and physical bases into a complete enlarged basis.This gives the reduced density matrix access to all di mi−1 states needed for basis expansion.
  • C. Centermatrix Wavefunction Formalism (CWF): CWF applies the effective Hamiltonian to the center matrix at cost O(w(d^2 + d)m^3) per eigensolver step, while also requiring costly perturbation and truncation operations.The large density matrix itself becomes problematic when m and w are large.
  • IV. SUBSPACE EXPANSION: Subspace expansion replaces density matrix perturbation with a numerically cheaper expansion term derived from the tensor-train literature.The method is presented for both sweep directions and uses zero padding on the neighboring tensor.
  • A. Subspace Expansion with an Arbitrary Expansion Term: Expanding Mi from mi to mi + mPi enlarges the right-hand basis while zero-padding Bi+1, leaving the represented state unchanged initially.The newly available components can then be selected by the local optimizer.
  • A. Subspace Expansion with an Arbitrary Expansion Term: The added basis flexibility lets the next local optimizer escape local minima without using the density matrix.Truncation back to m is usually required, and singular-value decomposition restores orthonormality while retaining relevant expanded states.

B. Expansion Term

Exact residual expansion is too expensive for the large bond dimensions needed in difficult two-dimensional problems, so the method uses cheaper sweep-direction-dependent expansion terms. Their scaling is dominated by an SVD that remains cheaper than the CWF calculation.

  • B. Expansion Term: Exact residual expansion is limited to m ≈100 because subtracting E|Ψ⟩ and re-orthonormalizing after each local optimization is costly.This bond dimension is too small for difficult two-dimensional problems.
  • B. Expansion Term: The proposed cheaper expansion terms use αLi−1MiWi for left-to-right sweeps and αRi+1MiWi for right-to-left sweeps.In the regime where the exact residual is computable, these terms perform essentially equally well.
  • B. Expansion Term: The left-to-right expansion term is a partial projection of H|Ψ⟩ onto the state space left of the current bond.This motivates why its right-hand basis can match the current tensor basis in the ground-state regime.
  • B. Expansion Term: If the maximal bond dimension m is insufficient, α must eventually approach zero because continued expansion otherwise keeps adding states and disturbs the eigensolver result.The issue arises when the represented state is optimal at the selected m but is not yet an eigenstate of H.
  • B. Expansion Term: The total subspace-expansion cost is dominated by an SVD scaling as O(dw^2m^3).The stated cost remains cheaper than the corresponding CWF calculation.

C. Subspace Expansion at the Example of a d = l = 2 Spin System

The two-spin example illustrates how subspace expansion enlarges the local MPS basis and creates a one-to-one correspondence with computational-basis coefficients. This enrichment supports optimization toward the desired state during a single-site update.

  • Example setup: Subspace expansion enlarges the local MPS basis during a left-to-right update from m = 1 to m = 2.The example introduces the expansion of A1 and the corresponding transformation of B2.
  • Initial state: The initial m = 1 MPS has only two free scalar variables, a and b, after standard normalization constraints.The state is represented through matrix-valued components in the MPS bond indices.
  • Basis enrichment: After expansion and singular-value decomposition, the transformed tensors establish a one-to-one correspondence with computational-basis coefficients.The remaining transformation multiplies the singular-value factors into the neighboring tensor.
  • Optimization target: This correspondence makes optimization toward the target coefficients, including c_ii = 0 and c_i≠j = 1/√2, explicit in the enlarged basis.The example uses the coefficient representation to express the desired optimized state.

V. STRICTLY SINGLE-SITE DMRG

DMRG3S combines standard single-site optimization with subspace expansion as a density-matrix-free enrichment step. The resulting eigensolver contractions are cheaper than in the CWF approach, while the method remains compatible with several existing DMRG extensions.

  • Algorithm: DMRG3S combines standard single-site DMRG with subspace expansion and avoids referring to the density matrix.The enrichment step is the only change from standard single-site DMRG.
  • Algorithm: A sweep updates site tensors sequentially, first optimizing each tensor with an eigensolver and then enriching and truncating the local basis.The outer sweeps continue until convergence, while enrichment is inserted after local optimization.
  • Compatibility: The method does not interfere with real-space parallelization, non-abelian symmetries, or multi-grid methods.These compatibilities follow because the algorithm changes standard single-site DMRG only through an added enrichment step.
  • Computational cost: The eigensolver application speed-up is (d + 1)/2 over CWF, assuming d^2w/m is small.The proposed contraction costs O(2m^3wd + d^2m^2w^2), versus O(m^3wd(d + 1)) for CWF.
  • Computational cost: The subspace-expansion enrichment is considerably cheaper than density-matrix perturbation, whose cost can occupy up to 30% of runtime.At large m, the number of sweeps needed for convergence does not seem to increase relative to CWF and sometimes decreases.

VI. ADAPTIVE CHOICE OF MIXING FACTOR

The adaptive mixing-factor scheme adjusts enrichment strength according to the energy change introduced by truncation relative to optimization. Its practical target is ΔET ≈ −0.3ΔEO, but the relationship between this ratio and optimal α is not established as causal.

  • Mixing factor: The optimal α depends on available and required states and on the current convergence speed; values that are too large or too small can hinder convergence.Large α can destroy local-optimization gains, whereas small α can leave vital states absent and cause local minima.
  • Adaptive rule: The adaptive scheme increases α when truncation causes little or negative energy change and decreases it when truncation cancels the optimization gain.The adjustment is based on comparing ΔET with ΔEO at successive updates.
  • Adaptive rule: In practice, fastest convergence appears to occur near ΔET ≈ −0.3ΔEO.Because α is an order-of-magnitude parameter, multiplicative rather than additive updates are recommended.
  • Caveat: The α-to-ΔET/ΔEO relationship is unclear because both quantities may simply correlate with progress through the DMRG calculation.The authors associate larger α with early large energy gains and smaller α with later slower decreases, without establishing causality.
  • Scope: Adaptive α is intended to accelerate convergence, but a fixed α is necessary when extrapolating truncation error during the calculation.For ground-state wavefunction calculations, recomputing α at each iteration has essentially zero cost.

A. DMRG Stuck in a Local Minimum

The strictly single-site DMRG method uses subspace expansion to restore missing states and avoid local-minimum failures, while improving convergence speed over CWF in the Heisenberg-chain benchmark.

  • A. DMRG Stuck in a Local Minimum: Without enrichment, DMRG3S could remove existing states but could not add missing ones, producing asymmetric quantum-number distributions and a high-energy state.
  • A. DMRG Stuck in a Local Minimum: Subspace expansion restored symmetry and reached a much lower-energy state than unenriched DMRG3S in the small-spin example.Without enrichment, the calculation converged to Eα=0 = −6.35479; with enrichment, it reached Eα̸=0 = −8.6824724.
  • 1. S = 1 Heisenberg Chain: Two-site DMRG handled the periodic-boundary benchmark poorly, yielding higher energies than the perturbed single-site methods and therefore incomparable runtime to convergence.

2. Dilute Bosons on an Optical Lattice

The bosonic benchmark compares DMRG3S with CWF and two-site DMRG across bond dimensions, showing comparable energy accuracy but faster convergence for DMRG3S. Its speed advantage reaches 2 over CWF and 3.3 over 2DMRG.

  • Setup: The modulated bosonic system uses 10 unit cells with 16 sites each, occupation cutoff nmax = 5, local dimension d = 6, and 80 bosons.Calculations use m = 50, 100, 200 and reference energy E0 = −103.646 757.
  • Setup: The bosonic calculation requires manual state checks because the modulated potential and small energy penalty can produce badly converged distributions.The checks ensure an equal distribution of bosons across the system.
  • Results: At small m, DMRG3S gains mainly from improved convergence per sweep, while at larger m cheaper numerical operations provide an additional speed-up.The speed-up from faster operations becomes visible at m = 200.
  • Results: Two-site DMRG maintains good energy accuracy without long-range interactions but converges more slowly because its eigenvalue problem is larger by a factor related to d.The DMRG3S-versus-2DMRG speed-up reaches 3.3 at m = 200.
  • Results: A speed-up of 2 over CWF and up to 3.3 over 2DMRG makes DMRG3S the fastest bosonic method at convergence.The comparison concerns runtime to convergence, while the methods achieve comparable energy accuracy.

4. Free Fermions

The free-fermion benchmark tests DMRG methods on a maximally delocalised ground state that is difficult for MPS representations. DMRG3S preserves comparable energy accuracy while reducing runtime relative to CWF and two-site DMRG, especially at larger bond dimensions.

  • Setup: The benchmark uses free fermions on a 100-site chain, with the same m values as the Fermi-Hubbard case and reference energy E0 = −126.602 376.All methods converge to this ground-state energy at m = 1200.
  • Interpretation: The maximally delocalised ground-state wavefunction is difficult for MPS formats, producing larger fixed-m energy errors than in the non-critical Fermi-Hubbard system.The larger number of significant eigenvalues also slows eigensolver and singular-value-decomposition convergence for all methods.
  • Results: DMRG3S improves its speed-up over CWF from 1.6 at m = 300 to 2 at m = 1200.At small m, gains largely reflect better convergence per sweep; at larger m, cheaper numerical operations become significant.
  • Results: More than 6× speed-up over 2DMRG at m = 1200 accompanies comparable energy accuracies across the free-fermion methods.The speed-up rises from 1.5 at m = 300 to more than 6 at m = 1200.
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