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arxiv: 2501.02643 · v2 · submitted 2025-01-05 · ❄️ cond-mat.dis-nn

Random transverse and longitudinal field Ising chains

Tam\'as Pet\"o , Ferenc Igl\'oi , Istv\'an A. Kov\'acs This is my paper

Pith reviewed 2026-05-23 06:01 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn
keywords Ising chainrandom transverse fieldrandom longitudinal fieldstrong disorder renormalization groupinfinite disorder fixed pointseparatrixcorrelation length exponent
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0 comments X

The pith

In an Ising chain with both random transverse and longitudinal fields, renormalization flows reach disordered fixed points with correlation length diverging as exponent near 1 along the separatrix.

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

The paper examines the zero-temperature low-energy properties of an Ising chain with random bonds under simultaneous random transverse and longitudinal fields using the strong disorder renormalization group method. It establishes that the presence of both field types pulls all renormalization trajectories toward one of the two disordered fixed points. At the infinite disorder fixed point the relevant scaling direction lies along the separatrix, where the correlation length diverges with an exponent approximately equal to 1. This builds directly on earlier cases with only one field type present and shows how the combination of both fields changes the overall flow structure.

Core claim

In the simultaneous presence of both a longitudinal and transverse random field, the RG trajectories are attracted to one of the two disordered fixed-points and the relevant scaling direction at the infinite disorder fixed-point is along the separatrix, where the correlation-length is shown to diverge with an exponent ν_h ≈ 1.

What carries the argument

Strong disorder renormalization group flows in the plane of transverse and longitudinal random-field strengths, identifying two disordered fixed points and the separatrix that separates their basins of attraction.

If this is right

  • Without random longitudinal fields the model has trivial quantum-ordered and quantum-disordered fixed points plus a non-trivial infinite-disorder critical point.
  • Without random transverse fields the model follows the classical random-field Ising fixed point.
  • With both field types present every trajectory flows to a disordered fixed point rather than a critical point.
  • The correlation length diverges with exponent approximately 1 when the system is tuned exactly along the separatrix.

Where Pith is reading between the lines

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

  • Tuning the relative strength of the two random fields could move the system on or off the separatrix and thereby select which disordered phase is reached.
  • Measurements of correlation lengths in materials that realize both field components could provide a direct test of the predicted exponent value.
  • The result indicates that disorder in two orthogonal directions is sufficient to suppress any critical behavior that exists when only one direction is random.

Load-bearing premise

The strong disorder renormalization group procedure remains quantitatively accurate for the low-energy spectrum when both random field types are present simultaneously and does not require higher-order corrections.

What would settle it

Exact diagonalization or quantum Monte Carlo results on finite chains with both random field types that yield a correlation-length exponent clearly different from 1 would falsify the claim.

Figures

Figures reproduced from arXiv: 2501.02643 by Ferenc Igl\'oi, Istv\'an A. Kov\'acs, Tam\'as Pet\"o.

Figure 2
Figure 2. Figure 2: FIG. 2. Illustration of the SDRG decimation steps a) for [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Fraction of performed site (or field) decimations dur [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Upper panel: fraction of performed site (or field) dec [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The average of the absolute value of the log-excitation [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The middle of the cross-over region Γ [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Average magnetization moment (upper panel) and [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Average log-gap (main panel) and average mag [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Distribution of the log-gaps at ln [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Exponents along the special points, which define the [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Expected schematic RG phase-diagram for the [PITH_FULL_IMAGE:figures/full_fig_p011_14.png] view at source ↗
read the original abstract

Motivated by experimental results on compounds like ${\rm LiHo}_x{\rm Y}_{1-x}{\rm F}_4$, we consider an Ising chain with random bonds in the simultaneous presence of random transverse and longitudinal fields. We study the low-energy properties of the model at zero temperature by the strong disorder renormalization group (SDRG) method.In the absence of random longitudinal fields, the model showcases a trivial quantum-ordered and quantum-disordered fixed-point and a non-trivial infinite disorder critical point. In the absence of random transverse fields, the behavior is dictated by the classical random-field Ising fixed-point. In the simultaneous presence of both a longitudinal and transverse random field, the RG trajectories are attracted to one of the two disordered fixed-points and the relevant scaling direction at the infinite disorder fixed-point is along the separatrix, where the correlation-length is shown to diverge with an exponent $\nu_h \approx 1$.

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 paper applies the strong-disorder renormalization group (SDRG) to the one-dimensional random-bond Ising model in the simultaneous presence of random transverse and random longitudinal fields. It maps out the fixed-point structure in the limiting cases (pure transverse or pure longitudinal disorder) and claims that, when both disorder types are present, all RG trajectories flow to one of the two disordered fixed points; at the infinite-disorder fixed point the relevant scaling direction lies along the separatrix, producing a correlation-length exponent ν_h ≈ 1.

Significance. If the SDRG analysis is quantitatively reliable, the work supplies a concrete, falsifiable prediction (ν_h ≈ 1) for the scaling of the correlation length in a model directly relevant to LiHo_x Y_{1-x} F_4. The identification of an attractive separatrix between the two disordered fixed points is a non-trivial result that could guide future numerical or experimental tests.

major comments (2)
  1. [Methods / SDRG procedure] The central claim that the relevant direction at the infinite-disorder fixed point lies exactly along the separatrix (yielding ν_h ≈ 1) rests on the first-order SDRG decimation rules. No estimate or bound is given for the size of higher-order cluster corrections that appear when both transverse and longitudinal fields are retained after a decimation step; such corrections could shift the separatrix or renormalize ν_h by O(1).
  2. [Results / RG trajectories] The abstract and the description of the flow state that trajectories are attracted to one of the two disordered fixed points, yet no quantitative measure (e.g., distance to the fixed-point distributions or flow of the disorder strength) is supplied to demonstrate that the flow is not merely slow but asymptotically controlled by the claimed separatrix.
minor comments (1)
  1. [Abstract / Results] The value ν_h ≈ 1 is quoted without an accompanying error bar or statement of the numerical precision with which it was extracted from the SDRG iterations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our SDRG study of the random transverse and longitudinal field Ising chain. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the approximation and the flow analysis.

read point-by-point responses

  1. Referee: [Methods / SDRG procedure] The central claim that the relevant direction at the infinite-disorder fixed point lies exactly along the separatrix (yielding ν_h ≈ 1) rests on the first-order SDRG decimation rules. No estimate or bound is given for the size of higher-order cluster corrections that appear when both transverse and longitudinal fields are retained after a decimation step; such corrections could shift the separatrix or renormalize ν_h by O(1).

    Authors: We agree that the reported value ν_h ≈ 1 is obtained within the leading-order SDRG decimation rules and that no explicit bound on higher-order cluster corrections is provided. Such corrections are formally present when both field types survive a decimation step. In the strong-disorder limit the leading-order rules are expected to capture the dominant physics, consistent with prior SDRG applications, but the absence of a quantitative estimate is a genuine limitation of the current analysis. We will add a dedicated paragraph in the revised manuscript discussing the perturbative character of the SDRG and the expected irrelevance of higher-order terms. revision: partial

  2. Referee: [Results / RG trajectories] The abstract and the description of the flow state that trajectories are attracted to one of the two disordered fixed points, yet no quantitative measure (e.g., distance to the fixed-point distributions or flow of the disorder strength) is supplied to demonstrate that the flow is not merely slow but asymptotically controlled by the claimed separatrix.

    Authors: The manuscript presents the RG flow through explicit numerical iteration of the decimation rules and the resulting evolution of the bond and field distributions, which converge to the two disordered fixed points. While we do not report auxiliary metrics such as the L1 distance to the fixed-point distributions or the explicit decay of the disorder width, the observed convergence under repeated decimations already indicates asymptotic attraction to the separatrix. To make this more quantitative we will include, in the revised version, supplementary plots of the disorder strength versus RG scale for representative trajectories. revision: partial

Circularity Check

0 steps flagged

No circularity: SDRG flows and ν_h derived from iterative decimation on the Hamiltonian

full rationale

The paper applies the standard strong-disorder renormalization group procedure to the microscopic Ising Hamiltonian with random transverse and longitudinal fields. Fixed-point attraction and the separatrix scaling direction yielding ν_h ≈ 1 are obtained directly from the iterative decimation rules and numerical trajectory analysis; these quantities are not defined in terms of themselves, not obtained by fitting a parameter to a related observable, and not justified solely by self-citation. The derivation chain remains self-contained against the external benchmark of the SDRG algorithm itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the SDRG decimation procedure faithfully captures the infrared physics of the random-bond Ising chain with two independent random-field channels; no free parameters, invented entities, or additional axioms are stated in the abstract.

axioms (1)
  • domain assumption The strong disorder renormalization group method accurately describes the low-energy properties of the model.
    All conclusions are obtained exclusively via SDRG iterations.

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discussion (0)

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Reference graph

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