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arxiv: 2605.11146 · v1 · submitted 2026-05-11 · ❄️ cond-mat.quant-gas · cond-mat.str-el· cond-mat.supr-con

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BCS-BEC crossover in trapped one-dimensional Fermi-Hubbard chains: entanglement and correlation signatures from DMRG and effective-pairing theory

G. Diniz , I. M. Carvalho , M. Sanino , F. Iemini , V. V. Fran\c{c}a
Authors on Pith no claims yet

Pith reviewed 2026-05-13 00:49 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas cond-mat.str-elcond-mat.supr-con
keywords BCS-BEC crossoverFermi-Hubbard modelone-dimensional chainsDMRGeffective-pairing theoryharmonic confinementsuperfluid correlationsentanglement
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The pith

Effective-pairing theory and DMRG simulations yield a unified picture of the BCS-BEC crossover in harmonically confined one-dimensional Fermi-Hubbard chains.

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

This paper aims to characterize the BCS-BEC crossover in one-dimensional Fermi-Hubbard chains under harmonic confinement. It combines numerical DMRG simulations with entanglement diagnostics and effective models for tightly bound pairs to show how confinement reshapes the crossover. The result is the emergence of insulating regions that coexist with superfluid correlations, unlike in homogeneous systems. Conditioned correlation functions are proposed to distinguish between BCS-like and BEC-like regimes based on their power-law decay. This provides a consistent framework that a reader interested in quantum simulation would value for understanding trapped strongly correlated fermions.

Core claim

The BCS-BEC crossover in harmonically confined one-dimensional Fermi-Hubbard chains is characterized by the formation of insulating regions coexisting with persistent superfluid correlations. Effective models describing the formation of tightly bound fermion pairs, together with DMRG and entanglement-based diagnostics, produce a consistent description. Within this framework, conditioned correlation functions exhibit power-law decays that distinguish BCS-like from BEC-like regimes.

What carries the argument

Conditioned correlation functions, whose power-law decay distinguishes the BCS-like and BEC-like regimes in the crossover, underpinned by effective-pairing theory for tightly bound pairs.

If this is right

  • Insulating regions coexist with superfluid correlations due to the harmonic confinement.
  • Conditioned correlation functions provide a diagnostic tool to identify BCS or BEC regimes.
  • The effective descriptions match DMRG results to give a unified view of the crossover.
  • Unconventional phases emerge that have no direct analog in homogeneous systems.

Where Pith is reading between the lines

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

  • This characterization could guide experimental probes of correlation functions in ultracold atom realizations of the model.
  • The approach might generalize to other forms of confinement or higher dimensions to reveal additional phase behaviors.
  • Persistent superfluid correlations in confined geometries suggest potential for studying transport in inhomogeneous quantum systems.

Load-bearing premise

That the effective-pairing theory remains accurate when applied to harmonically confined geometries without additional unstated corrections.

What would settle it

Numerical DMRG results that deviate from the power-law behaviors predicted by the conditioned correlation functions in the effective-pairing model for the trapped chains would falsify the distinction between regimes.

Figures

Figures reproduced from arXiv: 2605.11146 by F. Iemini, G. Diniz, I. M. Carvalho, M. Sanino, V. V. Fran\c{c}a.

Figure 1
Figure 1. Figure 1: Phase diagram predicted by the effective models [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic representation of the effective models derived from the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Spectrum ϵl of the confined tight-binding model with k = 0.002 and L = 100, where l indexes eigenstates by increasing energy. The blue dashed line shows a harmonic fit, while the black dashed lines mark the boundary between harmonic and localized states. Notice that the localized states are degenerate due to the system’s even symmetry. defines the positions j ∗ ± ≈ L 2 ± √ 1.3 k , at which the wavefunction… view at source ↗
Figure 4
Figure 4. Figure 4: shows the density profile along the chain obtained from the effective models and from DMRG in the BEC regime for two average densities. The results from Eq. (2) show a maximum percentage error of 5%, reflecting the approxima￾tion inherent to the mean-field approach; an exact solution would be expected to yield a smaller error. In comparison, the effective tight-binding model in Eq. (5) exhibits a maximum e… view at source ↗
Figure 6
Figure 6. Figure 6: Mean-field quasiparticle spectrum for U = −0.25t (black) and U = −2t (red), with fixed n = 1 and k = 0.002. The dashed lines delimitates the region where pairing is concentrated for U = −2t. The left side shows harmonic-like states, while the right contains localized states, as also seen in the confined tight￾binding chain in [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Critical densities nc for the superfluid–insulator transi￾tion obtained from DMRG calculations (circular dots) and from Eq. (6) (dashed line), for U = −10 and L = 200. C. Mean-field approach for the BCS-BEC crossover For given values of U, n, and k, the self-consistently diag￾onalization of the mean-field Hamiltonian (3) leads to HMF = X l h ξlγ † l,+γl,+ − ξlγ † l,−γl,− i , (7) where γ † l,± creates a Bog… view at source ↗
Figure 7
Figure 7. Figure 7: Bogoliubov coefficients along the chain for spin- [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Distribution p(r) = |PL/2(r)| P r′ |PL/2(r ′)| for r ≥ 0, obtained from Eq. (11) with fixed j = L/2 (center of the chain, minimizing boundary effects). Results are shown for k = 5×10−4 , n = 1, and L = 100. Several values of |U| are shown across the full crossover region to illustrate the behavior of the distribution beyond the lim￾iting regimes (colors indicated in the legend). The vertical dashed lines m… view at source ↗
Figure 9
Figure 9. Figure 9: Root-mean-square distance rrms of p(r) as a function of |U|/t on a log–log scale for several average densities n. Numerical results are shown as black circles, with densities indicated in each panel: n = 0.08 (a), n = 0.24 (b), n = 0.48 (c), n = 0.64 (d), n = 0.80 (e), and n = 1.00 (f). The green line shows a power-law fit rrms ∼ r1|U/t| −ν1 for weak interactions (|U|/t ≲ 1), characterized by a small expon… view at source ↗
Figure 10
Figure 10. Figure 10: Probability profiles (top panels) and entanglement [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: SL/2 as a function of n for several interaction strengths U (see color legend), with fixed L = 100 and k = 0.002. The dashed black line indicates the reference value SL/2 = 2 ln 2, cor￾responding to the noninteracting limit, which is used to distinguish between predominantly BEC-like and BCS-like regimes. For fixed n, the overall magnitude of SL/2 decreases as |U| increases, providing a more subtle signat… view at source ↗
Figure 12
Figure 12. Figure 12: Phase diagram as a function of the interaction strength [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
read the original abstract

Confined ultracold atoms in optical lattices provide a versatile platform for simulating lattice models of strongly correlated quantum systems, where pairing phenomena and superfluid phases can be explored under controlled conditions. While the crossover between the Bardeen-Cooper-Schrieffer (BCS) phase and the Bose-Einstein condensation (BEC) is well understood in homogeneous systems, spatial confinement breaks translational symmetry and reshapes correlation patterns, making the BCS-BEC identification in trapped geometries challenging and allowing unconventional phases to emerge with no direct analog in homogeneous systems. Here we present a characterization of the BCS-BEC crossover in harmonically confined one-dimensional Fermi-Hubbard chains. Our analysis combines Density Matrix Renormalization Group (DMRG) simulations and entanglement-based diagnostics with effective models describing the formation of tightly bound fermion pairs. This combined approach enables a detailed understanding of how the interplay between interactions and confinement reshapes the crossover, leading to insulating regions coexisting with persistent superfluid correlations. Within this framework, we further introduce conditioned correlation functions whose power-law decay allows a clear distinction between BCS-like and BEC-like regimes. The consistency between the effective descriptions and the numerical DMRG results yields a unified picture of the crossover in harmonically confined geometries.

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 characterizes the BCS-BEC crossover in harmonically confined one-dimensional Fermi-Hubbard chains. It combines DMRG simulations with entanglement diagnostics and effective models for tightly bound fermion pairs, introduces conditioned correlation functions to distinguish BCS-like from BEC-like regimes, and concludes that the agreement between numerics and effective descriptions produces a unified picture of the crossover, including coexistence of insulating regions and persistent superfluid correlations.

Significance. If the reported consistency is quantitatively robust, the work supplies useful diagnostics for pairing in inhomogeneous 1D systems and could inform ultracold-atom experiments. The combination of DMRG with an effective-pairing approach and the conditioned correlators is a constructive methodological contribution.

major comments (2)
  1. [Abstract and effective-model discussion] The central claim of a unified picture rests on consistency between DMRG and the effective-pairing theory in the trap. The effective model is described as treating formation of tightly bound pairs, yet the manuscript does not appear to incorporate or test position-dependent corrections arising from the spatially varying local chemical potential (e.g., position-dependent pair-binding energy or hopping). This assumption is load-bearing for the claim that the effective description remains accurate without additional trap-induced terms.
  2. [Abstract and results] The abstract and summary state consistency between DMRG and effective models but provide no quantitative measures of agreement, error bars on DMRG data, or finite-size scaling. Without these, it is difficult to assess whether the reported agreement confirms the effective theory or could be affected by uncontrolled approximations.
minor comments (2)
  1. [Abstract] The abstract would be strengthened by specifying the interaction and trap-strength regimes explored.
  2. [Figures] Figures presenting correlation functions or entanglement measures should include DMRG error estimates or convergence checks.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and positive assessment of the methodological contributions. We address the two major comments point by point below, indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract and effective-model discussion] The central claim of a unified picture rests on consistency between DMRG and the effective-pairing theory in the trap. The effective model is described as treating formation of tightly bound pairs, yet the manuscript does not appear to incorporate or test position-dependent corrections arising from the spatially varying local chemical potential (e.g., position-dependent pair-binding energy or hopping). This assumption is load-bearing for the claim that the effective description remains accurate without additional trap-induced terms.

    Authors: We appreciate the referee highlighting this point. The effective-pairing theory employed is derived under a local-density approximation in which pair-binding energies and hopping amplitudes are evaluated at the local chemical potential; trap-induced spatial variations are therefore incorporated at the mean-field level through the position-dependent density profile. In the parameter regime of tightly bound pairs (large attraction), the pair size is much smaller than the trap length scale, rendering higher-order gradient corrections subdominant. Nevertheless, we acknowledge that an explicit test of these corrections (e.g., via position-dependent renormalization of the effective model parameters) is not presented. In the revised manuscript we will add a dedicated paragraph in the effective-model section discussing the validity of the local approximation, including a brief estimate of the neglected terms and a comparison of results obtained with and without a simple position-dependent correction for a representative trap strength. revision: partial

  2. Referee: [Abstract and results] The abstract and summary state consistency between DMRG and effective models but provide no quantitative measures of agreement, error bars on DMRG data, or finite-size scaling. Without these, it is difficult to assess whether the reported agreement confirms the effective theory or could be affected by uncontrolled approximations.

    Authors: We agree that quantitative indicators of agreement would strengthen the presentation. DMRG calculations in one dimension are controlled by bond-dimension convergence; we will therefore include explicit error estimates (obtained from the difference between successive bond dimensions) on all plotted observables in the revised figures. In addition, we will add a supplementary figure or appendix panel showing finite-size scaling of the key correlation functions and entanglement measures for the largest system sizes accessible, demonstrating that the reported power-law exponents and crossover features remain stable. These additions will allow a more rigorous assessment of the consistency between DMRG and the effective theory. revision: yes

Circularity Check

0 steps flagged

No significant circularity; DMRG numerics provide independent benchmarks for effective-pairing theory

full rationale

The paper's central claim is the consistency between DMRG simulations (entanglement diagnostics, conditioned correlation functions) and effective-pairing models in harmonically confined 1D Fermi-Hubbard chains. DMRG supplies first-principles numerical data on the trapped system without reference to the effective theory, while the effective model is applied and compared rather than used to define or fit the target quantities. No quoted equations or sections reduce a prediction to a fitted input by construction, import uniqueness via self-citation chains, or smuggle ansatzes through prior work in a load-bearing manner. The derivation remains self-contained because the numerical results function as external validation of the model's applicability under confinement, with no definitional equivalence between inputs and outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that an effective-pairing model accurately captures pair formation under confinement; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Effective-pairing theory describes tightly bound fermion pairs in the presence of harmonic confinement.
    Invoked to enable comparison with DMRG results and to interpret the crossover.

pith-pipeline@v0.9.0 · 5557 in / 1141 out tokens · 39581 ms · 2026-05-13T00:49:53.428099+00:00 · methodology

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Works this paper leans on

84 extracted references · 84 canonical work pages · 1 internal anchor

  1. [1]

    Micnas, J

    R. Micnas, J. Ranninger, and S. Robaszkiewicz, Rev. Mod. Phys.62, 113 (1990)

    work page 1990

  2. [2]

    Paiva, R

    T. Paiva, R. Scalettar, M. Randeria, and N. Trivedi, Phys. Rev. Lett.104, 066406 (2010)

    work page 2010

  3. [3]

    Paiva, R

    T. Paiva, R. R. dos Santos, R. T. Scalettar, and P. J. H. Denteneer, Phys. Rev. B69, 184501 (2004)

    work page 2004

  4. [4]

    Xu, and W

    J.K.Chin, D.E.Miller, Y.Liu, C.Stan, W.Setiawan, C.San- ner, K. Xu, and W. Ketterle, Nature443, 961 (2006)

    work page 2006

  5. [5]

    C. F. Chan, M. Gall, N. Wurz, and M. Köhl, Phys. Rev. Res. 2, 023210 (2020)

    work page 2020

  6. [6]

    R. A. Fontenele, N. C. Costa, R. R. dos Santos, and T. Paiva, Phys. Rev. B105, 184502 (2022)

    work page 2022

  7. [7]

    Mizukami, M

    Y. Mizukami, M. Haze, O. Tanaka, K. Matsuura, D. Sano, J. Böker, I. Eremin, S. Kasahara, Y. Matsuda, and T. Shibauchi, Communications Physics6, 10.1038/s42005- 023-01289-8 (2023)

    work page doi:10.1038/s42005- 2023

  8. [8]

    Palestini and G

    F. Palestini and G. C. Strinati, Phys. Rev. B89, 224508 (2014)

    work page 2014

  9. [9]

    Stöferle, H

    T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, Phys. Rev. Lett.92, 130403 (2004)

    work page 2004

  10. [10]

    Keller, W

    M. Keller, W. Metzner, and U. Schollwöck, Phys. Rev. B60, 3499 (1999)

    work page 1999

  11. [11]

    Nozières and S

    P. Nozières and S. Schmitt-Rink, Journal of Low Temperature Physics59, 195 (1985)

    work page 1985

  12. [12]

    A. N. Kocharian, C. Yang, and Y. L. Chiang, Phys. Rev. B 59, 7458 (1999)

    work page 1999

  13. [13]

    Tonks, Phys

    L. Tonks, Phys. Rev.50, 955 (1936)

    work page 1936

  14. [14]

    Vignolo and A

    P. Vignolo and A. Minguzzi, Journal of Physics B: Atomic, Molecular and Optical Physics34, 4653 (2001)

    work page 2001

  15. [15]

    A. J. Leggett, inModern Trends in the Theory of Cond Mat- ter, Lecture Notes in Physics, Vol. 115, edited by A. Pękalski and J. A. Przystawa (Springer Berlin Heidelberg, 1980) pp. 13–27

    work page 1980

  16. [16]

    Shiba, Progress of Theoretical Physics48, 2171 (1972)

    H. Shiba, Progress of Theoretical Physics48, 2171 (1972)

    work page 1972

  17. [17]

    Q. Chen, J. Stajic, S. Tan, and K. Levin, Physics Reports 412, 1 (2005)

    work page 2005

  18. [18]

    Randeria and E

    M. Randeria and E. Taylor, Annual Review of Condensed Matter Physics5, 209 (2014)

    work page 2014

  19. [19]

    Roati, C

    G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Za- ccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008)

    work page 2008

  20. [20]

    W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner, Nature462, 74 (2009)

    work page 2009

  21. [21]

    M. Gall, C. F. Chan, N. Wurz, and M. Köhl, Phys. Rev. Lett. 124, 010403 (2020)

    work page 2020

  22. [22]

    Hartke, B

    T. Hartke, B. Oreg, C. Turnbaugh, N. Jia, and M. W. Zwier- lein, Science381, 82 (2023)

    work page 2023

  23. [23]

    Guarrera, N

    V. Guarrera, N. Fabbri, L. Fallani, C. Fort, K. M. R. van der Stam, and M. Inguscio, Phys. Rev. Lett.100, 250403 (2008)

    work page 2008

  24. [24]

    C. T. Schmiegelow, H. Kaufmann, T. Ruster, J. Schulz, V. Kaushal, M. Hettrich, F. Schmidt-Kaler, and U. G. Poschinger, Phys. Rev. Lett.116, 033002 (2016)

    work page 2016

  25. [25]

    Bissbort, D

    U. Bissbort, D. Cocks, A. Negretti, Z. Idziaszek, T. Calarco, F. Schmidt-Kaler, W. Hofstetter, and R. Gerritsma, Phys. Rev. Lett.111, 080501 (2013)

    work page 2013

  26. [26]

    Zwerger, Journal of Optics B: Quantum and Semiclassical Optics5, S9 (2003)

    W. Zwerger, Journal of Optics B: Quantum and Semiclassical Optics5, S9 (2003)

    work page 2003

  27. [27]

    P.T.Brown, E.Guardado-Sanchez, B.M.Spar, E.W.Huang, 11 T. P. Devereaux, and W. S. Bakr, Nature Physics16, 26 (2020)

    work page 2020

  28. [28]

    Hartke, B

    T. Hartke, B. Oreg, N. Jia, and M. Zwierlein, Phys. Rev. Lett. 125, 113601 (2020)

    work page 2020

  29. [29]

    Greiner, O

    M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature415, 39 (2002)

    work page 2002

  30. [30]

    Gemelke, X

    N. Gemelke, X. Zhang, C.-L. Hung, and C. Chin, Nature460, 995 (2009)

    work page 2009

  31. [31]

    N.A.Boidi, K.Hallberg, A.Aharony,andO.Entin-Wohlman, Phys. Rev. B109, L041404 (2024)

    work page 2024

  32. [32]

    Aucar Boidi, A

    N. Aucar Boidi, A. Aharony, O. Entin-Wohlman, K. Hallberg, and C. R. Proetto, Journal of Physics: Condensed Matter37, 385602 (2025)

    work page 2025

  33. [33]

    Hu, J.-J

    J.-H. Hu, J.-J. Wang, G. Xianlong, M. Okumura, R. Igarashi, S. Yamada, and M. Machida, Phys. Rev. B82, 014202 (2010)

    work page 2010

  34. [34]

    Guidini, L

    A. Guidini, L. Flammia, M. V. Milošević, and A. Perali, Journal of Superconductivity and Novel Magnetism29, 711 (2016)

    work page 2016

  35. [35]

    Kudla, D

    S. Kudla, D. M. Gautreau, and D. E. Sheehy, Phys. Rev. A 91, 043612 (2015)

    work page 2015

  36. [36]

    Orso, Phys

    G. Orso, Phys. Rev. Lett.98, 070402 (2007)

    work page 2007

  37. [37]

    Juillet, F

    O. Juillet, F. Gulminelli, and P. Chomaz, Phys. Rev. Lett. 92, 160401 (2004)

    work page 2004

  38. [38]

    Blume, Reports on Progress in Physics75, 046401 (2012)

    D. Blume, Reports on Progress in Physics75, 046401 (2012)

    work page 2012

  39. [39]

    von Stecher, C

    J. von Stecher, C. H. Greene, and D. Blume, Phys. Rev. A 76, 053613 (2007)

    work page 2007

  40. [40]

    Jiménez-García, R

    K. Jiménez-García, R. L. Compton, Y.-J. Lin, W. D. Phillips, J. V. Porto, and I. B. Spielman, Phys. Rev. Lett.105, 110401 (2010)

    work page 2010

  41. [41]

    G. A. Canella and V. V. França, Physica A536, 122646 (2019)

    work page 2019

  42. [42]

    Pauletti, M

    T. Pauletti, M. A. G. Silva, G. A. Canella, and V. V. França, Physica D: Nonlinear Phenomena642, 133009 (2024)

    work page 2024

  43. [43]

    V. L. Campo, Phys. Rev. A92, 013614 (2015)

    work page 2015

  44. [44]

    Capelle and V

    K. Capelle and V. L. C. Jr., Physics Reports528, 91 (2013)

    work page 2013

  45. [45]

    A. E. Feiguin and F. Heidrich-Meisner, Phys. Rev. B76, 220508 (2007)

    work page 2007

  46. [46]

    Toschi, M

    A. Toschi, M. Capone, and C. Castellani, Phys. Rev. B72, 235118 (2005)

    work page 2005

  47. [47]

    Amaricci, A

    A. Amaricci, A. Privitera, and M. Capone, Phys. Rev. A89, 053604 (2014)

    work page 2014

  48. [48]

    Bauer, A

    J. Bauer, A. C. Hewson, and N. Dupuis, Phys. Rev. B79, 214518 (2009)

    work page 2009

  49. [49]

    Sanino, I

    M. Sanino, I. M. Carvalho, and V. V. França, APL Quantum 22, 026129 (2025)

    work page 2025

  50. [50]

    S. R. White, Phys. Rev. Lett.69, 2863 (1992)

    work page 1992

  51. [51]

    U.Schollwöck,AnnalsofPhysics326,96(2011),january2011 Special Issue

    work page 2011

  52. [52]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, SciPost Phys. Codebases , 4 (2022)

    work page 2022

  53. [53]

    M. D. Girardeau, Journal of Mathematical Physics1, 516 (1960)

    work page 1960

  54. [54]

    Thisquasiparticlebehavesasaspin-zerohard-coreboson.Op- erators on different sites commute, while double occupancy of the same site is forbidden due to the underlying fermionic structure of the pair

    work page

  55. [55]

    Bardeen, L

    J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957)

    work page 1957

  56. [56]

    Salasnich and F

    L. Salasnich and F. Toigo, Phys. Rev. A86, 023619 (2012)

    work page 2012

  57. [57]

    Fanto, Y

    P. Fanto, Y. Alhassid, and G. F. Bertsch, Phys. Rev. C96, 014305 (2017)

    work page 2017

  58. [58]

    J. A. Sheikh, J. Dobaczewski, P. Ring, L. M. Robledo, and C. Yannouleas, Journal of Physics G: Nuclear and Particle Physics48, 123001 (2021)

    work page 2021

  59. [59]

    T. Ying, R. T. Scalettar, and R. Mondaini, Phys. Rev. B105, 115116 (2022)

    work page 2022

  60. [60]

    R. T. Scalettar, E. Y. Loh, J. E. Gubernatis, A. Moreo, S. R. White, D. J. Scalapino, R. L. Sugar, and E. Dagotto, Phys. Rev. Lett.62, 1407 (1989)

    work page 1989

  61. [61]

    Since a hole with momentum⃗ qis equivalent to a particle with momentum−⃗ q, the Bogoliubov quasiparticle acting on the effective vacuum encodes a BCS- like pair structure

    Note thatαandβshow similar oscillations, leading to related momentum distributions. Since a hole with momentum⃗ qis equivalent to a particle with momentum−⃗ q, the Bogoliubov quasiparticle acting on the effective vacuum encodes a BCS- like pair structure

    work page

  62. [62]

    G. C. Strinati, P. Pieri, G. Röpke, P. Schuck, and M. Urban, Physics Reports738, 1 (2018)

    work page 2018

  63. [63]

    Ghosal, M

    A. Ghosal, M. Randeria, and N. Trivedi, Phys. Rev. B65, 014501 (2001)

    work page 2001

  64. [64]

    The resulting state can therefore be interpreted as a superpo- sition of different Cooper-pair configurations

    work page

  65. [65]

    Zawadzki, G

    K. Zawadzki, G. A. Canella, V. V. França, and I. D’Amico, Advanced Quantum Technologies7, 2300237 (2024)

    work page 2024

  66. [66]

    M. F. V. Oliveira, F. A. B. F. de Moura, M. L. Lyra, and G. M. A. Almeida, Entanglement dynamics of delocalized in- teracting particles (2026), arXiv:2604.18960 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  67. [67]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys.82, 277 (2010)

    work page 2010

  68. [68]

    Amico, R

    L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys.80, 517 (2008)

    work page 2008

  69. [69]

    In 1D, BEC particles behave as spinless fermions, as shown in Ref. [53]. Therefore, in the BEC regime the number of degrees of freedom is approximately half that of theU= 0case at the same density

    work page

  70. [70]

    K. G. Wilson, Phys. Rev. B4, 3174 (1971)

    work page 1971

  71. [71]

    K. G. Wilson and J. Kogut, Physics Reports12, 75 (1974)

    work page 1974

  72. [72]

    W. C. de Freitas da Silva, A. R. Medeiros-Silva, R. Mondaini, V. V. França, and T. Paiva, Phys. Rev. A112, 062436 (2025)

    work page 2025

  73. [73]

    G. Zürn, A. N. Wenz, S. Murmann, A. Bergschneider, T. Lompe, and S. Jochim, Phys. Rev. Lett.111, 175302 (2013)

    work page 2013

  74. [74]

    Koepsell, S

    J. Koepsell, S. Hirthe, D. Bourgund, P. Sompet, J. Vijayan, G. Salomon, C. Gross, and I. Bloch, Phys. Rev. Lett.125, 010403 (2020)

    work page 2020

  75. [75]

    Holten, L

    M. Holten, L. Bayha, K. Subramanian, S. Brandstetter, C. Heintze, P. Lunt, P. M. Preiss, and S. Jochim, Nature 606, 287 (2022). 12 SUPPLEMENTARY MATERIAL

    work page 2022

  76. [76]

    ATOMIC LIMIT SOLUTION Let us here discuss the case whent= 0, known as atomic limit. In this limit, the total Hamiltonian can be written as: H0 = X j t.kj2 (nj,↑ +n j,↓)− |U|n j,↑nj,↓ .(S1) This Hamiltonian is diagonal in the occupation basis and the energy of each site depends only on the number of electronsnj in each site, which can be written as ϵj(nj =...

    work page

  77. [77]

    EFFECTIVE MODEL IN STRONGLY COUPLED LIMIT VIA LÖWDIN PERTURBATION THEORY For simplicity, let us consider the initial Hamiltonian (1), but after an translationj0 = 0in the|U| ≫tregime. Once that is required a high energy to break one electronic pair in this system, we can also extract the effective model via well known Löwdin Perturbation Theory method, th...

    work page

  78. [78]

    −|U| X l nl,↑nl,↓ # ϕ† q |0⟩ ≈ −|U|(1−2δ 2),(S19) 14 and ⟨0|ϕ q

    EFFECTIVE TIGHT-BINDING MODEL FOR BEC QUASIPARTICLES An interesting case with an easily obtainable analytical solution is the two-site (dimmer) negative-UFermi-Hubbard model, whose Hamiltonian can be written as: HD =t.kj 2(n1 +n 2) + 2Vjn2 − |U|[n 1↑n1↓ +n 2↑n2↓]−t h c† 1,↑c2,↑ +c † 1,↓c2,↓ + h.c. i .(S9) Here,V j =t.k j+ 1 2 . Focusing on the two particl...

    work page

  79. [79]

    MEAN-FIELD APPROACH Using Wick’s theorem on the interaction term of the Hubbard model, one can express the quartic operator as: ni↑ni↓ ≈ ⟨n i↑⟩n i↓ +n i↑ ⟨ni↓⟩ − ⟨c† i↑ci↓⟩c † i↓ci↑ − ⟨c† i↓ci↑⟩c † i↑ci↓ +⟨c i↓ci↑⟩c † i↑c† i↓ +⟨c † i↑c† i↓⟩c i↓ci↑ − ⟨ni↑⟩⟨ni↓⟩+⟨c † i↑ci↓⟩⟨c† i↓ci↑⟩ − ⟨ci↓ci↑⟩⟨c† i↑c† i↓⟩.(S27) Once we consider nonmagnetic solutions,⟨c† i↑...

    work page

  80. [80]

    S1 (LEFT), as|U|increases the particle density along the chain becomes increasingly concentrated near the center, eventually forming a smaller, effectively fully filled region

    Influence of the interaction strength and Confinement effect As shown in Fig. S1 (LEFT), as|U|increases the particle density along the chain becomes increasingly concentrated near the center, eventually forming a smaller, effectively fully filled region. This behavior is consistent with the enhancement of the effective confinement strength for|U|>0. To un...

    work page

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