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Non-existence of Bose-Einstein condensation in Bose-Hubbard model in dimensions 1 and 2

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Pith's one-line read The paper proves that the one- and two-dimensional Bose–Hubbard model has no Bose–Einstein condensation at any positive temperature and any filling, provided particle jumps decay fast enough with distance.

desk verdict A rigorous, self-contained proof that the 1D/2D Bose-Hubbard model has no U(1) quasiaverage at positive temperature; the physics is expected, but the technical execution is solid and worth refereeing. read the letter →

arxiv 1908.09188 v2 pith:EALGCSSP submitted 2019-08-24 math-ph cond-mat.stat-mechmath.MP

Piotr Stachura , Wies{l}aw Pusz , Jacek Wojtkiewicz This is my paper · ORCID
classification math-phcond-mat.stat-mechmath.MP MSC 82B1082B2082B26
keywords Bose–HubbardmodelBose–EinsteincondensationBogoliubovinequalitylong-rangeorderlow-dimensionallatticesystemsthermodynamiclimitquantumstatisticalmechanicsno-ordertheorem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper turns a widely held piece of physics folklore into a theorem: in the Bose–Hubbard model on a one- or two-dimensional lattice, with translationally invariant hopping that falls off fast enough with distance and periodic boundary conditions, there is no Bose–Einstein condensation at any positive temperature or any filling. The proof uses a version of the Bogoliubov inequality adapted to bosonic operators, where the Hilbert space is infinite-dimensional even on a finite lattice, and passes carefully from finite-dimensional approximations to the thermodynamic limit. Along the way it proves that the particle density is bounded above and below uniformly in the lattice size, a fact the paper says it could not find proved elsewhere. If correct, the result justifies the common assumption that low-dimensional lattice bosons never condense and sharpens the question of what kind of order, if any, survives in two dimensions.

What carries the argument

The load-bearing object is an infinite-dimensional operator version of the Bogoliubov inequality, which bounds a thermal average of a commutator by a product of two other thermal averages. The paper applies it with $A=c^{\dagger}(k)$, a momentum creation operator, and $C=|\Lambda|^{-1/2}\sum_x e^{ik\cdot x}n_x$, a density-wave operator. The commutator $[C,A]=|\Lambda|^{-1/2}c^{\dagger}(0)$ makes the condensate order parameter appear, while the double commutator $[[C,H],C^{\dagger}]$ is controlled by the second-moment bound on hopping. After summing over momenta, the inequality reads $m_\Lambda$ times a momentum sum bounded by a constant depending on $\beta$ and the density; in $d=1,2$ that sum diverges as the symmetry-breaking field tends to zero, which squeezes $m_\Lambda$ to zero. A separate convexity argument applied to the logarithm of the partition function supplies the uniform density estimates needed to take that limit.

What would settle it

Exhibit a one- or two-dimensional lattice boson model with translationally invariant jump amplitudes violating the second-moment bound, for example $|t_z|\sim |z|^{-(d+1)}$, and show numerically or analytically that the condensate fraction stays nonzero as the symmetry-breaking field tends to zero and the lattice grows; that would show the fast-decay condition is not merely technical.

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Extended reading notes

Core claim

On the paper's own terms, the central result is that the order parameter $m_\Lambda(\lambda)=|\langle c^{\dagger}(0)\rangle|^2/|\Lambda|$ — the squared magnitude of the zero-momentum annihilation operator's thermal average per site, i.e. the condensate density — can be made arbitrarily small by taking the symmetry-breaking field $\lambda$ small and the lattice $\Lambda$ large, for dimensions $d=1$ and $d=2$. This holds for every filling and every positive temperature, under the assumptions that hopping is translationally invariant and satisfies the uniform second-moment bound $\sum_z |t_z||z|^2 \le M_2$. The proof obtains an operator Bogoliubov inequality, computes the relevant commutators with a density-wave operator and a momentum creation operator, sums the resulting bound over momenta, and shows that in one and two dimensions the momentum sum diverges as $\lambda\to0$, forcing the order parameter to zero.

Load-bearing premise

The load-bearing premise is that particle jumps between lattice sites become weak fast enough with distance that the total jump strength times squared distance stays bounded as the lattice grows; with longer-ranged jumps the argument no longer forces the condensate to zero.

Editorial extensions

If this is right

  • In one and two dimensions, the Bose–Hubbard model with finite-range or sufficiently fast-decaying hopping has zero condensate density at every positive temperature and every filling.
  • Models with a hard upper bound on site occupation inherit the no-condensation result, since they are a special case of the same estimates.
  • The grand-canonical particle density is bounded uniformly in the lattice size and in small symmetry-breaking fields, so the thermodynamic limit of these thermal averages is well behaved.
  • Any low-temperature ordering in two dimensions cannot be off-diagonal long-range order; if order exists it must be of a different kind, and the paper leaves the Kosterlitz–Thouless-type transition as an open problem.
  • The finite-dimensional approximation plus trace-class convergence strategy gives a template for rigorous no-order proofs in other lattice boson models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same commutator pair should work for any U(1)-symmetric lattice boson model whose hopping satisfies the same uniform second-moment bound, because only the commutator identities and that bound enter the proof.
  • The fast-decay condition is probably close to a threshold: a model with hopping decaying like $|z|^{-(d+2)}$ sits at the boundary of the second-moment condition, and numerics could test whether condensation reappears there.
  • The argument is inherently at positive temperature, so zero-temperature Bose–Einstein condensation in the two-dimensional Bose–Hubbard model is not excluded; the result and a possible $T=0$ superfluid can coexist.
  • Tracking the constants in the momentum sum would turn the order-parameter estimate into a quantitative rate as $\lambda\to0$, likely logarithmic in two dimensions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper presents a rigorous proof of the absence of Bose-Einstein condensation in the Bose-Hubbard model on d-dimensional cubic lattices for d=1 and 2, with periodic boundary conditions and translationally invariant hopping amplitudes satisfying a finite second-moment condition, at any positive temperature and for any filling. The proof combines a Bogoliubov inequality adapted to infinite-dimensional bosonic systems, finite-dimensional truncations with a convergence theorem for thermal averages, and a Mermin-Wagner-type estimate. The key result, Proposition 26, shows that the U(1) quasiaverage order parameter m_Λ(λ)=|⟨c†(0)⟩|²/|Λ| can be made arbitrarily small for large lattices and small symmetry-breaking fields. The paper also establishes uniform density bounds (Proposition 25) that are needed for the thermodynamic limit.

Significance. The result is a natural lattice analogue of the classical no-BEC theorem for continuum bosons in d≤2, and it fills a gap in the literature where the absence of BEC in the Bose-Hubbard model was widely believed but not rigorously established. The main technical contribution is the careful treatment of the infinite-dimensional bosonic Hilbert space: self-adjointness of the Hamiltonian via Kato-Rellich, trace-class Gibbs states, and convergence of finite-dimensional thermal averages to the true thermal averages. The proof is detailed and self-contained, with no fitted parameters or ad-hoc assumptions beyond the explicitly stated fast-decay condition on the hopping amplitudes. The uniform density bounds derived in Proposition 25 are also of independent interest.

minor comments (5)
  1. [Abstract and Section 6] The abstract and summary claim 'no Bose-Einstein condensation', but the proof controls the U(1) quasiaverage m_Λ(λ)=|⟨c†(0)⟩|²/|Λ|. The authors should add a brief discussion clarifying how this quasiaverage criterion relates to the standard Penrose-Onsager definition of condensation, since the logical connection is not immediate to all readers.
  2. [Section 5.1, Lemma 34] In Lemma 34 the small parameter is denoted λ, but in the application to Proposition 25 it corresponds to |λ|, the modulus of the symmetry-breaking field. This notational mismatch may confuse readers; consider renaming the parameter in Lemma 34 (e.g., to η) or adding an explicit remark that the lemma is applied with η=|λ|.
  3. [Throughout] There are frequent typos that should be corrected in a revision: 'anihilation' in Section 2, 'possess' in Section 2, 'Begolyubov' after Eq. (79), and 'Te get' in Section 5. A careful proofread is needed.
  4. [Theorem 3(ii), Eq. (18)] The lower bound on γ(u) in Eq. (18) is written in a convoluted and likely misprinted form, with a plus sign preceding a negative term. Consider simplifying or rederiving this expression for readability.
  5. [Section 5, Eq. (83)] The passage from the Riemann sum to the integral as N→∞ is asserted without proof. A one-sentence justification, noting that the integrand is continuous away from k=0 and that the relevant uniformity in λ follows from monotonicity in α, would improve the exposition.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the core derivation is self-contained; one non-load-bearing self-citation appears only as background.

full rationale

The paper's derivation is not circular in any load-bearing sense. The order parameter m_Λ(λ) is defined independently as the normalized square of the thermal expectation of the zero-momentum annihilation operator, and it is not constructed from the conclusion it is meant to prove. The central tool, the Bogoliubov inequality, is first stated in finite-dimensional form (Eq. 38) as an external matrix inequality, and the paper then proves its own infinite-dimensional extension for the Bose-Hubbard Hamiltonian, including the needed finite-dimensional approximation and convergence results (Props. 15, 17, 21); it does not import these results from the authors' prior work. The double-commutator bound leading to Eq. 83 is obtained by direct commutator computations (Eqs. 69-71) together with the explicit translational-invariance and second-moment assumptions (76)-(77), which are stated as hypotheses rather than smuggled in as conclusions. The density bounds in Prop. 25 follow from convexity, trace inequalities, and explicit comparison Hamiltonians, with no fitted parameters. Proposition 26 follows from the divergence of the Riemann-sum/integral estimate (83) as λ→0, not from a normalization or definitional choice. The only self-citation is reference [22], mentioned in the introduction as background on operator versions of the Bogoliubov inequality in quantum rotor models; it is not used in the proof and is not load-bearing. Therefore no prediction reduces by construction to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof rests on standard functional analysis (Kato-Rellich theorem, trace-class perturbation results, Vitali-Porter convergence) and on the physical model assumptions: positive on-site interaction u>0, translationally invariant hopping with finite second moment, and periodic boundary conditions. No free parameters are fitted; all constants are derived. No new entities are introduced.

assumptions (5)
  • domain assumption The hopping matrix is translationally invariant on the torus: t_{x,x+z} = t_{y,y+z} = t_z for all x,y,z (Eq. 76).
    Used to rewrite the kinetic term in momentum space and to bound the double commutator by the density times the second moment of the hopping.
  • domain assumption The hopping has finite second moment: there exists M2 independent of the lattice size with ∑_z |t_z||z|² ≤ M2 (Eq. 77).
    Needed for the integral in Eq. (83) to diverge as λ→0 in d=1,2; without it the Mermin-Wagner mechanism can fail.
  • domain assumption The on-site interaction strength u is strictly positive (u>0) (Eq. 16).
    Ensures the unperturbed operator uN₂ has a trace-class Gibbs semigroup and makes the perturbation small relative to uN₂.
  • domain assumption The lattice is a finite cubic lattice with periodic boundary conditions, Λ = (Z_N)^d (Eqs. 58-60).
    The momentum-space representation and the Riemann-sum-to-integral limit rely on this geometry.
  • standard math Standard results from operator theory: Kato-Rellich theorem, trace-class perturbation theory, Ruskai's trace inequality, Vitali-Porter theorem (Props. 7, 9, 18, Lemma 19).
    These are unproved background results invoked in the proofs of self-adjointness and of convergence of thermal averages.

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Cite this review

Pith. "Pith review of Non-existence of Bose-Einstein condensation in Bose-Hubbard model in dimensions 1 and 2." pith.science (2026). https://pith.science/paper/EALGCSSP

@misc{pith2026190809188,
  author       = {Pith},
  title        = {Pith review of: Non-existence of Bose-Einstein condensation in Bose-Hubbard model in dimensions 1 and 2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EALGCSSP}},
  note         = {Machine review of arXiv:1908.09188}
}
read the original abstract

We apply the Bogoliubov inequality to the Bose-Hubbard model to rule out the possibility of Bose-Einstein condensation. The result holds in one and two dimensions, for any filling at any nonzero temperature. This result can be considered as complementary to analogous, classical result known for interacting bosons in continuum.

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