On the possibility of Bose-Einstein condensation in lower dimensions in the thermodynamic limit

Shirish M Chitanvis

arxiv: 1907.06578 · v2 · pith:K36BENDYnew · submitted 2019-07-15 · ⚛️ physics.atom-ph · cond-mat.quant-gas

On the possibility of Bose-Einstein condensation in lower dimensions in the thermodynamic limit

Shirish M Chitanvis This is my paper

Pith reviewed 2026-05-24 20:57 UTC · model grok-4.3

classification ⚛️ physics.atom-ph cond-mat.quant-gas

keywords Bose-Einstein condensationlower dimensionsthermodynamic limitregularizationoptical trapsboson number densityphase transition

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The pith

Regularization keeps excited-state boson density finite in two and one dimensions, allowing Bose-Einstein condensation in the thermodynamic limit.

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

Standard arguments prohibit Bose-Einstein condensation in fewer than three dimensions because the excited-state population diverges in the thermodynamic limit, implying infinite number density. The paper applies regularization techniques to the relevant integrals and obtains a finite excited-state number density in two and one dimensions at low temperatures. A finite excited density removes the usual obstruction, so a macroscopic ground-state occupation becomes possible even as system size tends to infinity. The work proposes testing the idea by enlarging two-dimensional optical traps until the thermodynamic limit is approached.

Core claim

By applying regularization procedures to the expressions for the number of bosons in excited states, the number density remains finite in two and one dimensions at low temperatures. This finite density means that a macroscopic number of particles can occupy the ground state in the thermodynamic limit, allowing Bose-Einstein condensation to occur in lower dimensions.

What carries the argument

Regularization techniques applied to the momentum integrals that count excited-state bosons.

If this is right

Bose-Einstein condensation can occur in two-dimensional systems in the thermodynamic limit.
Bose-Einstein condensation can occur in one-dimensional systems in the thermodynamic limit.
Two-dimensional optical traps of increasing size can be used to approach the thermodynamic limit and observe the transition.
The standard no-BEC proofs in lower dimensions rely on an unregularized divergence that regularization removes.

Where Pith is reading between the lines

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

Similar regularization might change conclusions about other phase transitions that rely on divergent integrals in low dimensions.
Cold-atom experiments could search for condensation signatures in successively larger 2D traps to test the finite-density prediction.
The result suggests the d greater than 2 requirement for BEC may not be universal once divergences are handled by regularization.

Load-bearing premise

The chosen regularization procedure correctly removes the divergence without introducing unphysical artifacts or violating the thermodynamic-limit definition.

What would settle it

Measuring whether a condensate fraction appears in two-dimensional optical traps as trap size is increased toward the thermodynamic limit; failure to observe condensation in that limit would falsify the claim.

read the original abstract

Standard arguments state that Bose Einstein condensation (BEC) cannot occur in dimensions lower than three in the thermodynamic limit as the expressions for the number of bosons in the excited states are unbounded. These arguments imply that the number density is infinite, which is an extraordinary condition. As an alternative to this pedagogy, we explore the use of regularization techniques to show that the number density of bosons in the excited state is finite in two and one dimensions at low temperatures, making it possible to have a BEC transition in the thermodynamic limit. We suggest creating a two-dimensional optical traps of increasing sizes to test this hypothesis as the thermodynamic limit is approached.

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.

Desk Editor's Note private letter to a colleague

Regularization is invoked to claim finite excited-state density for BEC in d=1,2 but the derivation and thermodynamic-limit check are missing.

read the letter

The main takeaway is that the paper uses regularization to argue that the number of bosons in excited states can be finite in one and two dimensions at low temperatures, allowing for Bose-Einstein condensation in the thermodynamic limit. This goes against the standard result, but the supporting calculation is not provided in the abstract or framed as a detailed derivation here. The paper does well in noting that the conventional approach leads to infinite density, which is an extraordinary condition, and it offers regularization as an alternative pedagogy. It also makes a concrete suggestion for experimental verification using two-dimensional optical traps of increasing sizes to approach the thermodynamic limit. This shows some thought about how theory connects to experiment. However, the soft spots are significant. The central claim depends on a regularization technique that is not detailed, so it is impossible to check if it properly handles the infrared divergence without introducing volume-dependent cutoffs or other artifacts. The stress-test point about preserving the standard thermodynamic limit definition is key, and the paper would need to demonstrate that the regulator commutes with the V to infinity limit at fixed density. Without that, it does not overturn the established no-BEC theorems in low dimensions. There is also no comparison to how regularization is used in related contexts like the ideal Bose gas or critical phenomena. This paper might appeal to readers interested in pedagogical approaches to statistical mechanics or those working on ultracold atoms in low dimensions who are open to rethinking the dimensionality restriction. A reader looking for rigorous derivations or new predictions would not find much value here. I would not bring this to the next reading group, and I would not cite it in my own work. It does not seem to merit sending out for peer review, as the key step of the argument is not shown and the claim conflicts with well-established results without clear resolution.

Referee Report

2 major / 0 minor

Summary. The manuscript argues that standard proofs forbidding Bose-Einstein condensation (BEC) in d ≤ 2 in the thermodynamic limit rely on divergent expressions for the excited-state number density; it proposes that regularization techniques can render this density finite at low temperatures, thereby permitting a BEC transition. An experimental test using two-dimensional optical traps of increasing size is suggested.

Significance. If a regularization procedure can be shown to produce a finite excited-state density while strictly preserving the thermodynamic limit (V → ∞ at fixed n with μ → 0^−) without introducing volume-dependent artifacts, the result would challenge a foundational result in quantum statistics. The manuscript supplies no such explicit derivation, cutoff scheme, or comparison to the standard integral, so the significance cannot be assessed from the given text.

major comments (2)

Abstract: the assertion that 'regularization techniques' yield finite excited-state density supplies neither the explicit regulator, the cutoff procedure, nor a direct comparison against the standard unbounded integral ∫ d^d k / (e^{β(ε_k−μ)}−1) as μ→0^−. The central claim therefore rests on an unshown calculation.
Abstract (and implied main text): no demonstration is given that the chosen regularization commutes with the V→∞ limit at fixed density and μ→0^−. Without this, the procedure may introduce an implicit infrared scale tied to system size, violating the thermodynamic-limit definition used in the standard no-BEC proofs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for greater explicitness in our presentation. We agree that the manuscript as submitted is concise and lacks the detailed derivations requested; we will revise it to supply these while preserving the original conceptual focus.

read point-by-point responses

Referee: Abstract: the assertion that 'regularization techniques' yield finite excited-state density supplies neither the explicit regulator, the cutoff procedure, nor a direct comparison against the standard unbounded integral ∫ d^d k / (e^{β(ε_k−μ)}−1) as μ→0^−. The central claim therefore rests on an unshown calculation.

Authors: We acknowledge that the abstract and the brief manuscript do not contain the explicit regulator, cutoff scheme, or side-by-side comparison with the standard integral. In the revised manuscript we will add a new section that specifies the regularization (e.g., a momentum-space cutoff or dimensional regularization), derives the resulting finite excited-state density for d = 2 and d = 1 at low T, and directly contrasts it with the divergent ∫ d^d k / (e^{β(ε_k−μ)}−1) expression as μ → 0^−, thereby making the central claim fully explicit. revision: yes
Referee: Abstract (and implied main text): no demonstration is given that the chosen regularization commutes with the V→∞ limit at fixed density and μ→0^−. Without this, the procedure may introduce an implicit infrared scale tied to system size, violating the thermodynamic-limit definition used in the standard no-BEC proofs.

Authors: We agree that it is necessary to verify that the regularization commutes with the thermodynamic limit. In the revision we will include an explicit argument showing that the regularized density remains finite, volume-independent, and free of implicit infrared cutoffs once V → ∞ is taken at fixed n with μ → 0^−; this will be contrasted with the standard proofs to confirm that the thermodynamic-limit definition is preserved while the excited-state density stays finite. revision: yes

Circularity Check

0 steps flagged

No circularity: regularization applied as external technique without self-referential reduction

full rationale

The paper's central move is to apply regularization techniques to the standard Bose integral for excited-state occupation, yielding a claimed finite density in d=1,2. The abstract presents this as an alternative to the usual divergence argument without quoting or exhibiting any equation in which the regulator is defined in terms of the target finite n_ex, any fitted parameter renamed as a prediction, or any load-bearing self-citation. Because the provided text contains no explicit derivation chain that reduces the result to its own inputs by construction, and the regularization is invoked as a standard external procedure rather than an ansatz smuggled from prior author work, the claim remains independent of the patterns that would trigger a positive circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5627 in / 1003 out tokens · 20656 ms · 2026-05-24T20:57:41.589799+00:00 · methodology

discussion (0)

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