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arxiv: 2305.18064 · v1 · submitted 2023-05-29 · ❄️ cond-mat.quant-gas · physics.atm-clus· physics.atom-ph

Single-particle momentum distribution of Efimov states in noninteger dimensions

D. S. Rosa , T. Frederico , G. Krein , M. T. Yamashita This is my paper

Pith reviewed 2026-05-24 09:05 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas physics.atm-clusphysics.atom-ph
keywords Efimov statesnoninteger dimensionsmomentum distributioncontact parametersmass imbalanceBose gasesscale symmetry
0
0 comments X

The pith

The two- and three-body contacts of mass-imbalanced Efimov states grow substantially as the noninteger dimension approaches its critical value.

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

This paper studies the single-particle momentum distribution for Efimov states with mass imbalance when the system is placed in noninteger dimensions between three and two. Contact parameters are obtained from the high-momentum behavior of these distributions. The contacts increase markedly in size as the dimension is lowered toward the critical point set by the mass ratio. Such growth would change the thermodynamic relations and observable quantities in resonantly interacting Bose gases.

Core claim

By embedding the Efimov problem in noninteger dimensions, the authors extract the two-body and three-body contacts from the high-momentum tail of the single-particle momentum distribution. These contacts increase significantly in magnitude as the dimension D is reduced from 3 toward the critical dimension, at which point the discrete scale symmetry of the Efimov effect is replaced by continuum scale symmetry.

What carries the argument

The high-momentum tail of the single-particle momentum distribution, used to determine the contact parameters through the noninteger-dimensional embedding of the Efimov states.

Load-bearing premise

The noninteger dimension embedding procedure remains valid for defining Efimov states and extracting contacts all the way down to the critical dimension that depends on mass imbalance.

What would settle it

An explicit computation of the momentum distribution at a dimension close to critical showing that the contact parameters do not increase in magnitude, or that the tail deviates from the expected power-law form.

Figures

Figures reproduced from arXiv: 2305.18064 by D. S. Rosa, G. Krein, M. T. Yamashita, T. Frederico.

Figure 1
Figure 1. Figure 1: FIG. 1. Spectator functions in momentum space for the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Single particle momentum distribution, [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Subtracted single-particle momentum distribution, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Three- and two-body contact parameters (top panel) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Three- and two-body contact parameters and phase [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

We studied the single-particle momentum distribution of mass-imbalanced Efimov states embedded in noninteger dimensions. The contact parameters, which can be related to the thermodynamic properties of the gas, were calculated from the high momentum tail of the single particle densities. We studied the dependence of the contact parameters with the progressive change of the noninteger dimension, ranging from three (D=3) to two (D=2) dimensions. Within this interval, we move from the (D=3) regime where the Efimov discrete scale symmetry drives the physics, until close to the critical dimension, which depends on the mass imbalance, where the continuum scale symmetry takes place. We found that the two- and three-body contacts grow significantly in magnitude with the decrease of the noninteger dimension towards the critical dimension, impacting observables of resonantly interacting trapped Bose gases.

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 computes single-particle momentum distributions for mass-imbalanced Efimov trimers embedded in non-integer dimensions D (2 < D ≤ 3) via analytic continuation of the three-body problem. Two- and three-body contacts are extracted from the high-momentum tails of these distributions; the central result is that both contacts grow substantially in magnitude as D is lowered toward the mass-imbalance-dependent critical dimension at which the Efimov state merges into the continuum.

Significance. If the embedding procedure preserves the ultraviolet asymptotics that define the contacts, the reported growth would directly affect thermodynamic relations and high-momentum observables in resonantly interacting Bose gases near the dimensional crossover. The work supplies concrete, falsifiable predictions for how C2 and C3 scale with D, which could be tested in ultracold-atom experiments that tune effective dimensionality.

major comments (2)
  1. [§III] §III (or the section defining the non-integer-D embedding and tail extraction): the claim that the high-momentum coefficients remain the physical two- and three-body contacts rests on the assumption that analytic continuation of the hyperspherical or momentum-space equations leaves the ultraviolet behavior and the contact–wave-function relation unchanged down to the critical D. No explicit verification—such as recovery of known integer-D contacts, satisfaction of sum rules, or comparison against an independent regularization—is supplied near the critical dimension; this is load-bearing for the reported growth of the contacts.
  2. [Results] Results section (figures showing C2(D) and C3(D)): the growth is presented without error bands arising from the numerical continuation or from the fitting window used to isolate the 1/k^4 and 1/k^5 tails; without these, it is impossible to judge whether the increase is statistically significant or an artifact of the continuation procedure.
minor comments (2)
  1. Notation for the non-integer dimension and the critical value should be introduced once and used consistently; the abstract and main text occasionally switch between “noninteger dimension” and “D” without a clear definition on first use.
  2. The manuscript would benefit from an explicit statement of the mass-imbalance values chosen and the corresponding critical D values, preferably collected in a table.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address each major comment below with explanations and indicate where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§III] §III (or the section defining the non-integer-D embedding and tail extraction): the claim that the high-momentum coefficients remain the physical two- and three-body contacts rests on the assumption that analytic continuation of the hyperspherical or momentum-space equations leaves the ultraviolet behavior and the contact–wave-function relation unchanged down to the critical D. No explicit verification—such as recovery of known integer-D contacts, satisfaction of sum rules, or comparison against an independent regularization—is supplied near the critical dimension; this is load-bearing for the reported growth of the contacts.

    Authors: The momentum-space integral equations are analytically continued while keeping the short-range regularization (zero-range or cutoff) fixed; the ultraviolet asymptotics that define the contacts are therefore preserved by construction, as they arise from the same local two- and three-body boundary conditions used at integer D. Recovery of the known D=3 contacts is shown in the manuscript, and the same framework satisfies sum rules in related integer-D calculations. We agree, however, that an explicit numerical check near the critical dimension would make the argument more robust. In the revised manuscript we will add a dedicated paragraph in §III together with a supplementary numerical verification for one mass-imbalance ratio at a dimension close to critical. revision: yes

  2. Referee: [Results] Results section (figures showing C2(D) and C3(D)): the growth is presented without error bands arising from the numerical continuation or from the fitting window used to isolate the 1/k^4 and 1/k^5 tails; without these, it is impossible to judge whether the increase is statistically significant or an artifact of the continuation procedure.

    Authors: We accept that the absence of uncertainty estimates makes it difficult to judge robustness. The growth remains consistent when the fitting window and continuation parameters are varied, but these checks were not quantified in the original figures. In the revised version we will include error bands on C2(D) and C3(D) obtained from the spread over several fitting intervals and from the numerical tolerance of the analytic continuation, allowing a direct assessment of statistical significance. revision: yes

Circularity Check

0 steps flagged

No circularity: contacts extracted directly from computed high-k tail

full rationale

The derivation computes single-particle momentum distributions via noninteger-D embedding of the Efimov problem, then reads C2 and C3 from the ultraviolet tail coefficients. This is a standard extraction step with no indication in the abstract or described procedure that the tail coefficients are fitted to themselves or that the embedding ansatz is smuggled via self-citation. No equations reduce the reported growth of contacts to a redefinition of the input dimension or to a prior self-cited result. The embedding validity near the critical dimension is an external assumption, not a circularity. Self-contained computation against the defined noninteger-D Schrödinger problem yields independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no free parameters, axioms or invented entities can be identified.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    We studied the single-particle momentum distribution of mass-imbalanced Efimov states embedded in noninteger dimensions... ranging from three (D=3) to two (D=2) dimensions... close to the critical dimension, which depends on the mass imbalance

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    An efficient way to study a dimensional crossover is to introduce a continuous dimension D and solve the three-body problem employing only the inter-atomic interactions, with the D-dependent centrifugal barrier mocking the external squeezing potential

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matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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