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arxiv: 1906.09077 · v1 · pith:BVYZ7ARQnew · submitted 2019-06-21 · ❄️ cond-mat.str-el

Comparison of MPS based real time evolution algorithms for Anderson Impurity Models

Daniel Bauernfeind , Markus Aichhorn , Hans Gerd Evertz This is my paper

Pith reviewed 2026-05-25 18:48 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Anderson impurity modelmatrix product statestime evolutionTDVPTEBDstar geometryWilson chain
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The pith

For Anderson Impurity Models the adapted TEBD algorithm in star geometry is faster than TDVP in chain geometry at equal accuracy.

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

The paper compares matrix product state methods for real-time evolution of Anderson Impurity Models, testing TDVP and TEBD on both star and Wilson chain bath geometries. It shows that TDVP on chains gives high precision while the adapted TEBD on stars runs substantially faster, so that for any target accuracy the star version wins on speed. Errors remain independent of system size in all cases except TEBD on chains, contrary to the naive expectation that all methods would scale linearly with size. This comparison matters because impurity models require repeated time evolutions and any reduction in cost per run directly expands the reachable parameter space. The authors conclude that the adapted star-geometry TEBD is therefore the most efficient overall choice.

Core claim

The authors demonstrate through detailed comparisons that while TDVP in the Wilson chain geometry is highly precise, the specially adapted TEBD in star geometry is significantly faster, leading to superior performance for a given accuracy level, and thus the most efficient approach for solving impurity problems.

What carries the argument

The specially adapted TEBD algorithm that handles long-range coupling terms in the star-geometry bath representation for MPS time evolution.

If this is right

  • TDVP in chain geometry yields high precision at higher computational cost.
  • TEBD in star geometry provides a better speed-accuracy tradeoff for impurity problems.
  • Errors stay independent of system size for all algorithm-geometry pairs except TEBD on chains.
  • The combination of bath representation and evolution algorithm determines overall performance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The speed gain could allow longer real-time simulations or larger bath sizes in the same computational budget.
  • Similar long-range adaptations might improve efficiency for other models that feature nonlocal couplings.
  • Verification on wider ranges of interaction strengths and temperatures would test whether the ranking persists.

Load-bearing premise

The adaptation of TEBD to star geometry correctly handles long-range couplings without introducing unaccounted errors, and the reported error scalings hold beyond the tested system sizes and parameters.

What would settle it

A direct run on a larger bath showing that TEBD-star error grows with system size or that its wall-clock time advantage vanishes relative to TDVP-chain would falsify the efficiency claim.

Figures

Figures reproduced from arXiv: 1906.09077 by Daniel Bauernfeind, Hans Gerd Evertz, Markus Aichhorn.

Figure 1
Figure 1. Figure 1: FIG. 1. Graphical representation of an MPS used for an AIM [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Application of the time evolution operator of Eq. (12) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Maximum error for times 20 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: (N=29) beginning at times t > 60 is due to such a reflection. Third, as a function of bath size, only the error of TEBD￾C shows the expected linear scaling with N as also shown in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Matrix elements [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fig.3. The bond dimensions of the MPS were not restricted [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

We perform a detailed comparison of two Matrix Product States (MPS) based time evolution algorithms for Anderson Impurity Models. To describe the bath, we use both the star-geometry as well as the commonly employed Wilson chain geometry. For each bath geometry, we use either the Time Dependent Variational Principle (TDVP) or the Time Evolving Block Decimation (TEBD) to perform the time evolution. To apply TEBD for the star-geometry, we use a specially adapted algorithm that can deal with the long-range coupling terms. Analyzing the major sources of errors, one expects them to be proportional to the system size for all algorithms. Surprisingly, we find errors independent of system size except for TEBD in chain geometry. Additionally, we show that the right combination of bath representation and time evolution algorithm is important. While TDVP in chain geometry is a very precise approach, TEBD in star geometry is much faster, such that for a given accuracy it is superior to TDVP in chain geometry. This makes the adapted version of TEBD in star geometry the most efficient method to solve impurity problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript compares TDVP and TEBD time-evolution algorithms within the MPS framework for Anderson impurity models, employing both star and Wilson-chain bath geometries. It reports that truncation and other errors are independent of system size except for TEBD on the chain, and concludes that the specially adapted TEBD algorithm on the star geometry is the most efficient choice for a target accuracy because it is substantially faster than TDVP on the chain while remaining comparably accurate.

Significance. If the central efficiency claim is substantiated, the work supplies a concrete, practical recommendation for method selection in real-time impurity simulations. The direct numerical head-to-head comparisons constitute a strength; they allow quantitative statements about speed versus accuracy that are more useful than isolated benchmarks.

major comments (2)
  1. [Abstract] Abstract: the headline claim that 'the adapted version of TEBD in star geometry [is] the most efficient method' is load-bearing and rests on the assumption that the adaptation for all-to-one long-range couplings introduces no additional truncation or approximation error beyond standard MPS bounds. No explicit error bound, decomposition into local gates, or cross-check against an exact solver for the identical Hamiltonian is referenced.
  2. [Abstract] Abstract (error-scaling paragraph): the statement that 'one expects [errors] to be proportional to the system size for all algorithms' yet 'find[s] errors independent of system size except for TEBD in chain geometry' is surprising and central to the efficiency conclusion. Without a description of the precise error metric, the time-step protocol, or the bond-dimension convergence tests used to establish size independence, it is impossible to judge whether the reported scaling is physical or an artifact of the adaptation.
minor comments (1)
  1. [Abstract] The abstract would be clearer if it stated the range of system sizes, interaction strengths, and time intervals over which the size-independent error result was observed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline claim that 'the adapted version of TEBD in star geometry [is] the most efficient method' is load-bearing and rests on the assumption that the adaptation for all-to-one long-range couplings introduces no additional truncation or approximation error beyond standard MPS bounds. No explicit error bound, decomposition into local gates, or cross-check against an exact solver for the identical Hamiltonian is referenced.

    Authors: The adaptation of TEBD to the star geometry decomposes each long-range coupling into a finite sequence of two-site gates that can be applied exactly within the MPS framework; this decomposition is described in the methods section and does not introduce truncation beyond the standard bond-dimension cutoff. All efficiency comparisons are performed on identical Hamiltonians, with accuracy assessed via the same observables. We agree that the abstract would be clearer with an explicit reference to this decomposition and will revise it to point to the relevant section. revision: yes

  2. Referee: [Abstract] Abstract (error-scaling paragraph): the statement that 'one expects [errors] to be proportional to the system size for all algorithms' yet 'find[s] errors independent of system size except for TEBD in chain geometry' is surprising and central to the efficiency conclusion. Without a description of the precise error metric, the time-step protocol, or the bond-dimension convergence tests used to establish size independence, it is impossible to judge whether the reported scaling is physical or an artifact of the adaptation.

    Authors: The error metric is the maximum absolute deviation of the impurity Green's function (or occupation) from a reference run at high bond dimension. A fixed time step is used throughout, chosen small enough that further reduction does not change results within the target tolerance; for each system size the bond dimension is increased until the observable converges to within a preset threshold. These protocols and the resulting size-independent errors (except for TEBD on the chain) are shown in the results section and supplementary figures. We will add a concise statement of the error metric and convergence procedure to the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity: results from direct numerical benchmarks on MPS time evolution

full rationale

The paper performs a numerical comparison of TDVP and TEBD algorithms applied to Anderson impurity models in star and chain geometries. All claims about relative efficiency, error scaling, and superiority of the adapted TEBD-star method rest on explicit runtime and error measurements for specific system sizes and parameters, not on any derivation that reduces to fitted inputs, self-citations, or ansatzes. No load-bearing step invokes a uniqueness theorem, renames a known result, or defines a quantity in terms of its own output. The adaptation for long-range terms is described as a practical implementation detail whose fidelity is assessed by the same benchmark comparisons; no equation or result is shown to be equivalent to its inputs by construction. This is a standard self-contained empirical study.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard assumptions about the applicability of MPS to these models and the correctness of the adapted TEBD algorithm for long-range terms.

axioms (1)
  • domain assumption The Anderson impurity model can be accurately represented using matrix product states for the tested geometries.
    This is a standard assumption in the field of tensor network simulations for quantum many-body systems.

pith-pipeline@v0.9.0 · 5732 in / 1161 out tokens · 30444 ms · 2026-05-25T18:48:32.149855+00:00 · methodology

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Reference graph

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