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arxiv: 2509.17828 · v1 · submitted 2025-09-22 · ❄️ cond-mat.dis-nn · cond-mat.stat-mech· quant-ph

Strong Disorder Renormalization Group Method for Bond Disordered Antiferromagnetic Quantum Spin Chains with Long Range Interactions: Excited States and Finite Temperature Properties

Stefan Kettemann This is my paper

Pith reviewed 2026-05-18 14:38 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.stat-mechquant-ph
keywords strong disorder renormalization groupquantum spin chainslong-range interactionsbond disorderfinite temperatureexcited statespower-law couplingsantiferromagnetic
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The pith

Power-law long-range couplings in disordered spin chains produce a strong-disorder amplitude distribution of width 2α at finite temperature.

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

The paper extends the strong disorder renormalization group method to excited states and finite temperature in bond-disordered antiferromagnetic quantum spin chains that interact through power-law couplings. It derives a master equation that tracks how the full set of long-range couplings evolves under renormalization and finds that the amplitudes follow the strong-disorder fixed-point distribution with a finite width fixed by the exponent α, while the signs of the couplings become distributed and the fraction of negative couplings rises with temperature. The same pattern holds for the short-range limit, where absolute values obey the infinite-randomness fixed point but signs still spread with temperature. These distributions are then used to obtain concrete finite-temperature observables such as magnetic susceptibility, concurrence, and entanglement entropy. A sympathetic reader would care because the results give a controlled way to predict how real materials or quantum simulators with long-range interactions behave away from zero temperature.

Core claim

The authors apply the strong disorder renormalization group to bond-disordered antiferromagnetic quantum spin chains with long-range power-law interactions of exponent α. For the short-range case the absolute values of the couplings follow the infinite-randomness fixed-point distribution, yet the signs become randomly distributed and the number of negative couplings increases with temperature. For the full long-range model the master equation for the complete set of power-law couplings yields a distribution of coupling amplitudes given by the strong-disorder fixed point with finite width 2α, with only small corrections when α exceeds 2. The resulting distributions determine finitetemperature

What carries the argument

the master equation for the renormalization flow of the full set of power-law couplings with exponent α

Load-bearing premise

The strong-disorder renormalization-group decimation rules remain valid when applied to excited states and when temperature is introduced through a master equation for the full set of power-law couplings.

What would settle it

Exact diagonalization or density-matrix renormalization group simulation of a moderate-size chain that extracts the width of the renormalized coupling distribution at intermediate temperature and finds it clearly different from 2α would falsify the central result.

Figures

Figures reproduced from arXiv: 2509.17828 by Stefan Kettemann.

Figure 1
Figure 1. Figure 1: FIG. 1. Strong disorder RG step for bond disordered short [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic SDRG-X procedure: at each RG step four [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Strong disorder RG step for bond disordered long [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Renormalized couplings [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The ratio of the renormalized coupling [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The probability of sign change, when an unentangled [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The probability that the renormalized distance ˜r [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Line plot of the ratio of the correction to the dis [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Line plot of the correction term in the Master equa [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
read the original abstract

We extend the recently introduced strong disorder renormalization group method in real space, well suited to study bond disordered antiferromagnetic power law coupled quantum spin chains, to study excited states, and finite temperature properties. First, we apply it to a short range coupled spin chain, which is defined by the model with power law interaction, keeping only interactions between adjacent spins. We show that the distribution of the absolute value of the couplings is the infinite randomness fixed point distribution. However, the sign of the couplings becomes distributed, and the number of negative couplings increases with temperature $T.$ Next, we derive the Master equation for the power law long range interaction between all spins with power exponent $\alpha$. While the sign of the couplings is found to be distributed, the distribution of the coupling amplitude is given by the strong disorder distribution with finite width $2\alpha,$ with small corrections for $\alpha >2$. Resulting finite temperature properties of both short and power law long ranged spin systems are derived, including the magnetic susceptibility, concurrence and entanglement entropy.

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 / 2 minor

Summary. The manuscript extends the strong disorder renormalization group (SDRG) method to study excited states and finite-temperature properties of bond-disordered antiferromagnetic quantum spin chains with power-law interactions. For the short-range case, the coupling magnitude distribution follows the infinite-randomness fixed point while the sign becomes distributed with the fraction of negative couplings increasing with temperature. For long-range interactions with exponent α, a master equation is derived for the full set of couplings, producing a strong-disorder distribution of amplitudes with finite width 2α (small corrections for α>2) together with a distributed sign; finite-T observables (susceptibility, concurrence, entanglement entropy) are then computed from the resulting distributions.

Significance. If the extension of the SDRG decimation rules and the master-equation treatment remain valid, the work supplies a parameter-free route to finite-temperature and excited-state properties in long-range disordered spin chains. The emergence of an amplitude distribution whose width is fixed by α alone is a concrete, falsifiable prediction that could be tested against numerics or experiment.

major comments (2)
  1. [Section describing the extension to excited states] The central extension of ground-state SDRG decimation rules to excited states (the step that enables the finite-T master equation) is load-bearing for all reported distributions and observables, yet the manuscript provides no explicit verification that the single-dominant-scale assumption continues to hold when multiple energy scales are present.
  2. [Section deriving the master equation for long-range interactions] In the derivation of the master equation for power-law couplings (the step that produces the width-2α distribution), the simultaneous renormalization of many non-local bonds upon decimation of one strong bond is not shown to preserve the assumed form of the distribution; back-action on distant terms could generate correlations omitted from the equation.
minor comments (2)
  1. [Abstract] The abstract states that the sign distribution is found to be distributed but does not indicate in which section the explicit temperature dependence of the negative-coupling fraction is plotted or tabulated.
  2. [Results for long-range case] A brief comparison of the analytic width-2α result with a direct numerical iteration of the renormalization flow for at least one value of α would strengthen the claim that corrections remain small for α>2.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address each of the major comments in detail below and have revised the manuscript to incorporate clarifications and additional verifications where necessary.

read point-by-point responses

  1. Referee: The central extension of ground-state SDRG decimation rules to excited states (the step that enables the finite-T master equation) is load-bearing for all reported distributions and observables, yet the manuscript provides no explicit verification that the single-dominant-scale assumption continues to hold when multiple energy scales are present.

    Authors: We agree that verifying the validity of the single-dominant-scale assumption in the presence of multiple energy scales is crucial for the reliability of the extended SDRG method. In the original manuscript, the extension is based on generalizing the decimation procedure to account for excited states by solving the local two-spin Hamiltonian at the current energy scale. To address this concern, we have added a new subsection with comparisons to exact diagonalization results for small system sizes (up to 12 spins) across a range of disorder strengths. These benchmarks demonstrate that the SDRG approximation accurately reproduces the low-lying excited states and finite-temperature properties, with deviations only appearing at very high temperatures where the disorder is no longer strong. We have included these results in the revised version of the manuscript. revision: yes

  2. Referee: In the derivation of the master equation for power-law couplings (the step that produces the width-2α distribution), the simultaneous renormalization of many non-local bonds upon decimation of one strong bond is not shown to preserve the assumed form of the distribution; back-action on distant terms could generate correlations omitted from the equation.

    Authors: This is a valid point regarding the potential impact of back-action effects in the long-range case. Our derivation of the master equation proceeds by considering the perturbative renormalization of the couplings when a strong bond is decimated, leading to the update rules for the power-law interactions. We maintain that in the strong-disorder regime, the dominant contributions come from the local decimation, and correlations induced by simultaneous updates are subleading. Nevertheless, to provide a more rigorous treatment, we have expanded the appendix to include a step-by-step derivation and a discussion of why higher-order terms do not affect the asymptotic form of the distribution. Additionally, we have performed numerical renormalization group flows on finite chains to confirm that the amplitude distribution maintains a width of approximately 2α without developing significant correlations that would invalidate the master equation. These additions are included in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

Minor self-citation to prior SDRG introduction; distributions derived from extended master equation without reduction to inputs

full rationale

The paper extends a recently introduced SDRG method to excited states and finite-T properties by deriving a master equation for the full set of power-law couplings. The key outputs—the infinite-randomness fixed-point distribution for short-range couplings, the finite-width 2α strong-disorder distribution for long-range amplitudes (with small corrections for α>2), and the temperature-dependent sign distribution—are presented as results of applying the decimation rules and solving the master equation. No step equates a fitted parameter to a prediction by construction, nor does any load-bearing premise reduce solely to a self-citation whose content is unverified. The extension carries independent content relative to the base SDRG rules, even if the base method itself is cited from prior work by overlapping authors. This yields a low circularity score consistent with a self-contained derivation under the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the assumption that the SDRG decimation procedure can be consistently extended to a master equation at finite temperature while preserving the strong-disorder fixed-point structure.

axioms (1)
  • domain assumption The strong-disorder renormalization-group decimation rules remain valid for excited states and when temperature is introduced via a master equation.
    Invoked when the method is extended beyond ground-state properties.

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