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arxiv: 2605.10418 · v1 · submitted 2026-05-11 · ❄️ cond-mat.quant-gas · cond-mat.str-el

Recognition: 2 theorem links

· Lean Theorem

Bose-Fermi Mapping in Hubbard Models at Imaginary Chemical Potential and Phase-Induced Fermionization

Evangelos G. Filothodoros
Authors on Pith no claims yet

Pith reviewed 2026-05-12 05:19 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas cond-mat.str-el
keywords Fermi-Hubbard modelBose-Hubbard modelimaginary chemical potentialphase-induced fermionizationBCS-BEC crossoverlarge-N expansionthermal kernelBose-Fermi crossover
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The pith

A phase shift of π in imaginary chemical potential maps the attractive Fermi-Hubbard model onto the repulsive Bose-Hubbard model and converts the BCS-BEC crossover into a Bose-Fermi crossover.

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

The paper establishes that the partition functions of the two Hubbard models at finite temperature become related by the replacement θ → θ + π when the chemical potential is purely imaginary. This relation is obtained through a large-N expansion and rests on an analytic continuation of a thermal kernel that swaps bosonic and fermionic sectors. The mapping therefore converts the familiar BCS-BEC crossover of attractive fermions into a crossover in which repulsive bosons acquire fermion-like occupation numbers. A sympathetic reader would care because the construction shows that fermion-like behavior can appear at finite interaction strength purely through a thermodynamic phase twist, without requiring infinite repulsion or altered particle statistics.

Core claim

We find a mapping between the attractive Fermi-Hubbard model and the repulsive Bose-Hubbard model at finite temperature and at imaginary chemical potential μ = iθ. Using a large N-expansion we show that the partition functions are related by the simple shift θ → θ + π. This condition maps the BCS-BEC crossover of attractive fermions to a Bose-Fermi crossover of repulsive bosons. The thermal kernel g(βE, φ) whose analytic continuation satisfies g_B(βE, φ) = g_F(βE, φ + π) governs both sectors. The special angles φ = 2π/3, 4π/3 for fermions correspond to φ = π/3, 5π/3 for bosons and mark the boundaries of a universal thermal window. The phase φ acts as a statistical parameter so that fermion-l

What carries the argument

The thermal kernel g(βE, φ) together with its analytic continuation g_B(βE, φ) = g_F(βE, φ + π), which directly relates the bosonic and fermionic partition functions and induces the crossover mapping.

If this is right

  • The BCS-BEC crossover of attractive fermions is converted into a Bose-Fermi crossover in which repulsive bosons exhibit fermion-like occupation.
  • The special angles φ = 2π/3, 4π/3 (fermions) and φ = π/3, 5π/3 (bosons) bound a universal thermal window that applies to both models.
  • Fermionization arises at finite interaction strength through a thermodynamic effect generated by the imaginary chemical potential rather than by infinite repulsion.
  • The imaginary chemical potential functions as a statistical regulator that appears in the gap and number equations of the bosonic model.
  • The construction supplies a unified thermodynamic framework for crossovers in interacting lattice gases.

Where Pith is reading between the lines

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

  • Cold-atom experiments could test the mapping by preparing bosonic gases at imaginary chemical potentials and checking whether their occupation statistics reproduce the fermionic crossover.
  • The phase-twist mechanism may extend to other lattice models or to dimensions beyond the large-N limit, offering a tunable knob for statistical behavior without changing particle statistics.
  • Because the effect is thermodynamic rather than kinematic, it could appear in systems where direct hard-core constraints are difficult to impose.

Load-bearing premise

The analytic continuation of the thermal kernel that equates the bosonic kernel at phase φ to the fermionic kernel at phase φ + π remains valid, and the large-N expansion captures the exact relation between the partition functions without higher-order corrections that would alter the crossover.

What would settle it

Numerical or experimental measurement of the average occupation number or momentum distribution in a repulsive Bose-Hubbard system at imaginary chemical potential i(θ + π) that fails to match the corresponding quantities in the attractive Fermi-Hubbard system at iθ, especially near the crossover points identified by the special angles.

Figures

Figures reproduced from arXiv: 2605.10418 by Evangelos G. Filothodoros.

Figure 1
Figure 1. Figure 1: Bose-Fermi crossover for F + CUV for the Bose Hubbard model at imaginary chemical potential and δu = 0. Lower T /t means entropy term −T S becomes less negative so U term dominates. Then the system spreads out to avoid repulsion and we have more fermion-like occupation. The blue region indicates enhanced fermionization (gB < 1), green indicates suppressed fermionization (gB > 1), and red dots at βθ = π/3, … view at source ↗
read the original abstract

We find a mapping between the attractive Fermi-Hubbard model and the repulsive Bose-Hubbard model at finite temperature and at imaginary chemical potential $\mu =i\theta$. We show, by using a large $N$-expansion, that the partition functions of the two models are related by a simple shift $\theta \to \theta + \pi$. This condition maps the BCS--BEC crossover of attractive fermions to a Bose--Fermi crossover (fermion-like occupation) of repulsive bosons. Central feature of this correspondence plays the thermal kernel $g(\beta E,\phi),$ whose analytic continuation $g_B(\beta E,\phi) = g_F(\beta E,\phi+\pi)$ governs the bosonic and fermionic sectors. Interestingly, we are able to find that the special angles $\phi = 2\pi/3,4\pi/3$ for fermions correspond to $\phi = \pi/3,5\pi/3$ for bosons, marking the boundaries of a universal thermal window. We further argue that the present mechanism shows that fermionization can occur at finite interaction strength through a thermodynamic effect induced by the imaginary chemical potential. This suggests that it is a new way of fermionization (not a change in statistics but a fermion-like behaviour) unlike the Tonks--Girardeau limit, where fermionization arises from an infinite repulsive interaction and anyonic or Floquet-engineered systems where transmutation emerges from modified statistics or dynamics. Essentially, the phase $\phi$ is a statistical parameter; by twisting the thermal phase, it generates fermion-like behaviour without hard-core constraints or infinite repulsion but only by using thermodynamics. We derive the gap equation and number equation for the bosonic model, highlighting the role of the imaginary chemical potential as a statistical regulator. Our results provide a unified framework for understanding crossovers in interacting lattice systems.

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

3 major / 2 minor

Summary. The manuscript proposes a mapping between the attractive Fermi-Hubbard model and the repulsive Bose-Hubbard model at finite temperature and imaginary chemical potential μ = iθ. Using a large-N expansion, the authors argue that the partition functions are related by the shift θ → θ + π because the thermal kernel satisfies g_B(βE, φ) = g_F(βE, φ + π). This relation is used to map the BCS-BEC crossover of fermions onto a Bose-Fermi crossover of bosons, with the special angles φ = 2π/3, 4π/3 (fermions) and φ = π/3, 5π/3 (bosons) marking the boundaries of a universal thermal window. The work further claims that this constitutes a new thermodynamic mechanism for phase-induced fermionization at finite interaction strength, distinct from the Tonks-Girardeau or anyonic cases, and derives the corresponding gap and number equations on the bosonic side.

Significance. If the central mapping and kernel continuation hold, the result would provide a unified thermodynamic framework for crossovers in lattice Hubbard systems and a novel route to fermion-like behavior induced purely by the imaginary-chemical-potential phase rather than by infinite repulsion or modified statistics. The explicit derivation of the bosonic gap and number equations is a concrete strength that could be checked numerically. However, the significance for physical N = 2 systems is currently limited by the leading-order large-N treatment and the absence of error estimates on the analytic continuation.

major comments (3)
  1. [Section on thermal kernel and large-N expansion (around the definition of g(βE, φ))] The central partition-function relation rests on the analytic continuation g_B(βE, φ) = g_F(βE, φ + π) of the thermal kernel. No explicit derivation from the Matsubara frequency sums or from the underlying path-integral representation is supplied, nor is the continuation verified against known limits (e.g., non-interacting case or high-temperature expansion). This assumption is load-bearing for the claimed θ → θ + π mapping.
  2. [Large-N expansion and partition-function relation] The large-N expansion is performed only to leading order. No bound or estimate is given for O(1/N) corrections that generically depend on both the phase θ and the interaction U; such terms would violate the exact shift relation for the physical spin-1/2 (N = 2) Hubbard models and could shift or destroy the claimed crossover boundaries at the special angles.
  3. [Derivation of gap and number equations; discussion of special angles] The identification of φ = 2π/3, 4π/3 (fermions) and φ = π/3, 5π/3 (bosons) as boundaries of the universal thermal window follows from the periodicity of the kernel but is not shown to be independent of the truncation order. The gap and number equations derived for the bosonic model should be re-examined at these angles to confirm that the crossover signatures survive beyond leading large-N.
minor comments (2)
  1. [Abstract and Introduction] The abstract and introduction could more clearly distinguish the proposed thermodynamic fermionization from existing mechanisms (Tonks-Girardeau, Floquet, anyonic) by emphasizing which observables are affected and which are not.
  2. [Throughout] Notation for the phase angles φ and the imaginary chemical potential θ is introduced without a compact summary table; a small table listing the special values for fermions and bosons would improve readability.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, providing the strongest honest defense of our results while indicating where the manuscript will be revised for clarity and rigor.

read point-by-point responses

  1. Referee: The central partition-function relation rests on the analytic continuation g_B(βE, φ) = g_F(βE, φ + π) of the thermal kernel. No explicit derivation from the Matsubara frequency sums or from the underlying path-integral representation is supplied, nor is the continuation verified against known limits (e.g., non-interacting case or high-temperature expansion). This assumption is load-bearing for the claimed θ → θ + π mapping.

    Authors: We agree that an explicit derivation of the kernel relation will improve the manuscript. In the revised version we will derive g(βE, φ) directly from the Matsubara frequency sums for both the bosonic and fermionic cases, showing that the shift g_B(βE, φ) = g_F(βE, φ + π) follows from the structure of the sums under analytic continuation in the phase. We will also verify the relation explicitly in the non-interacting limit (U = 0), where the partition functions reduce to known closed forms, and in the high-temperature expansion by expanding the kernel in powers of β. revision: yes

  2. Referee: The large-N expansion is performed only to leading order. No bound or estimate is given for O(1/N) corrections that generically depend on both the phase θ and the interaction U; such terms would violate the exact shift relation for the physical spin-1/2 (N = 2) Hubbard models and could shift or destroy the claimed crossover boundaries at the special angles.

    Authors: The analysis is performed at leading order in the large-N expansion, where the mapping and the resulting Bose-Fermi crossover are exact. For finite N the O(1/N) terms may introduce corrections that depend on θ and U, so the shift relation is not exact for N = 2. The leading-order result nevertheless captures the dominant thermodynamic mechanism of phase-induced fermionization. In the revision we will add an explicit discussion of this limitation, clarifying that the results are exact only in the large-N limit and should be regarded as a qualitative guide for physical N = 2 systems. revision: partial

  3. Referee: The identification of φ = 2π/3, 4π/3 (fermions) and φ = π/3, 5π/3 (bosons) as boundaries of the universal thermal window follows from the periodicity of the kernel but is not shown to be independent of the truncation order. The gap and number equations derived for the bosonic model should be re-examined at these angles to confirm that the crossover signatures survive beyond leading large-N.

    Authors: The special angles are fixed by the periodicity of the thermal kernel g(βE, φ), which originates from the Matsubara sum and is independent of both the interaction U and the order of the large-N truncation. In the revised manuscript we will re-evaluate the bosonic gap and number equations explicitly at φ = π/3 and 5π/3, confirming that the fermion-like occupation and crossover signatures remain intact within the leading-order framework. We will note that quantitative details may receive corrections at higher orders, but the boundaries themselves are protected by the kernel periodicity. revision: partial

standing simulated objections not resolved
  • Explicit bounds or estimates for the magnitude of O(1/N) corrections as functions of θ and U cannot be provided without performing a next-to-leading-order calculation, which lies outside the scope of the present work.

Circularity Check

0 steps flagged

No significant circularity; mapping derived via large-N expansion

full rationale

The central claim is obtained by applying a large-N expansion to the partition functions of the attractive Fermi-Hubbard and repulsive Bose-Hubbard models, yielding the relation Z_F(θ) ~ Z_B(θ + π) as a derived result rather than an input. The thermal kernel property g_B(βE, φ) = g_F(βE, φ + π) is stated as following from analytic continuation within that expansion, not as a self-definition or fitted ansatz. No self-citations, uniqueness theorems from prior author work, or renaming of known results appear in the derivation chain. The gap and number equations are derived for the bosonic side after the mapping is established. The paper is self-contained as a theoretical expansion result with no load-bearing reduction to its own outputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the large-N expansion for relating the two models and on the analytic continuation of the thermal kernel; these are not independently derived or benchmarked in the provided abstract.

free parameters (2)
  • θ (imaginary chemical potential phase)
    Parameter introduced to enable the mapping; the central relation is a shift by π whose value is chosen to connect the models.
  • φ (phase angles 2π/3, 4π/3 for fermions and π/3, 5π/3 for bosons)
    Special values identified as boundaries of the universal thermal window; appear selected to mark the crossover regimes.
axioms (2)
  • domain assumption Large-N expansion accurately relates the partition functions of the Fermi and Bose Hubbard models
    Invoked to obtain the simple shift relation between the models.
  • ad hoc to paper The thermal kernel admits the analytic continuation g_B(βE, φ) = g_F(βE, φ + π)
    Central step that directly produces the mapping and the fermion-like occupation for bosons.

pith-pipeline@v0.9.0 · 5644 in / 1679 out tokens · 49605 ms · 2026-05-12T05:19:26.239364+00:00 · methodology

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Reference graph

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