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arxiv: 2503.15811 · v2 · submitted 2025-03-20 · ❄️ cond-mat.quant-gas · quant-ph

Reduced density matrix approach to one-dimensional ultracold bosonic systems

Mitchell J. Knight , Harry M. Quiney , Andy M. Martin This is my paper

Pith reviewed 2026-05-22 23:58 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas quant-ph
keywords reduced density matrixone-dimensional bosonsultracold atomsvariational methodground-state energyharmonic trapcontact interactioncrossover regime
0
0 comments X

The pith

The two-boson reduced density matrix can be variationally optimized to give accurate ground states for one-dimensional systems of up to ten thousand trapped bosons.

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

This paper develops a variational method that determines the two-boson reduced density matrix for one-dimensional harmonically trapped bosons with contact interactions. It computes ground-state energies, densities, and correlation functions for particle numbers from two to ten thousand. The results match exact solutions at small N and mean-field limits at large N while covering the intermediate crossover. A sympathetic reader would care because the approach avoids storing the full many-body wave function yet still produces structural observables across interaction strengths.

Core claim

The variational determination of the two-boson reduced density matrix for a one-dimensional system of N bosons in a harmonic trap with contact interaction yields ground-state energies and structural properties including the density and correlation functions. These quantities match the analytic case for N=2 and mean-field approaches for large N, collectively demonstrating the capacity of the method to accurately calculate ground-state properties for N from 2 to 10^4 across a large range of interaction strengths.

What carries the argument

The two-boson reduced density matrix optimized variationally subject to N-representability constraints, from which the energy and one-body observables are extracted.

If this is right

  • Ground-state energies agree with the exact analytic result for N=2 and with mean-field approaches for large N.
  • Density profiles and correlation functions, including the value when boson coordinates coincide, are accurately obtained.
  • The method covers the crossover regime between few and many bosons reliably.
  • It applies over a wide range of interaction strengths.

Where Pith is reading between the lines

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

  • The same matrix could be tested against exact solutions for the Lieb-Liniger gas to check consistency beyond the harmonic trap.
  • Extension to time-dependent or driven systems would require only minor changes to the constraint set if the static case holds.
  • The success at intermediate N suggests two-body correlations carry most of the energetic information even when particle number is neither small nor macroscopic.

Load-bearing premise

Optimizing only the two-boson reduced density matrix under N-representability constraints produces accurate results in the intermediate particle-number regime without significant errors from missing higher-order constraints.

What would settle it

Exact many-body calculations for N between 10 and 100 at moderate interaction strengths that deviate substantially from the reduced-density-matrix energies or densities would disprove the accuracy claim.

Figures

Figures reproduced from arXiv: 2503.15811 by Andy M. Martin, Harry M. Quiney, Mitchell J. Knight.

Figure 2
Figure 2. Figure 2: FIG. 2. The ground-state energy [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The ground-state density as a function of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The value of the correlation function [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

The variational determination of the two-boson reduced density matrix is described for a one-dimensional system of $N$ (where $N$ ranges from $2$ to $10^4$) harmonically trapped bosons interacting via contact interaction. The ground-state energies are calculated, and compared to existing methods in the field, including the analytic case (for $N=2)$ and mean-field approaches such as the one-dimensional Gross-Pitaevskii equation and its variations. Structural properties including the density and correlation functions are also derived, including the behaviour of the correlation function when boson coordinates coincide, collectively demonstrating the capacity of the reduced density matrix method to accurately calculate ground-state properties of bosonic systems comprising few to many bosons, including the cross-over region between these extremes, across a large range of interaction strengths.

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 presents a variational method based on optimization of the two-boson reduced density matrix (2-RDM) for N bosons (N=2 to 10^4) in a 1D harmonic trap with contact interactions. It reports ground-state energies compared to the exact N=2 analytic solution and 1D Gross-Pitaevskii variants, derives density profiles and correlation functions including g(0), and claims the approach accurately captures properties across few-body, many-body, and intermediate-N crossover regimes over a wide range of interaction strengths.

Significance. If the central claim holds with proper validation, the method would provide a scalable variational route to intermediate-N regimes where exact methods scale poorly and mean-field approximations lose accuracy; the explicit handling of N up to 10^4 and derivation of structural observables are potential strengths.

major comments (2)
  1. [Abstract] Abstract and methods description: no quantitative error metrics (e.g., relative energy deviations or overlap measures) are supplied for any N>2, and neither the explicit 2-RDM parameterization nor the enforced N-representability constraints (P, Q, G, or higher) are stated; this directly undermines assessment of the accuracy claim in the crossover regime.
  2. [Methods (variational 2-RDM section)] The variational procedure is described as direct minimization of an energy functional in the 2-RDM, yet without specification of which subset of N-representability conditions is imposed, it is impossible to determine whether the optimized 2-RDM remains consistent with the true ground state for 10 ≲ N ≲ 100 at intermediate coupling, as required by the central claim.
minor comments (2)
  1. Figures comparing energies or densities should include error bars or tabulated relative errors against benchmarks for multiple N and interaction strengths.
  2. Notation for the 2-RDM and the harmonic-oscillator units should be defined explicitly at first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We address each major comment point by point below and are prepared to make revisions to improve clarity and validation.

read point-by-point responses

  1. Referee: [Abstract] Abstract and methods description: no quantitative error metrics (e.g., relative energy deviations or overlap measures) are supplied for any N>2, and neither the explicit 2-RDM parameterization nor the enforced N-representability constraints (P, Q, G, or higher) are stated; this directly undermines assessment of the accuracy claim in the crossover regime.

    Authors: We agree that the manuscript would benefit from explicit quantitative error metrics for N>2 and from stating the 2-RDM parameterization and N-representability conditions. In the revised version we will add a dedicated subsection (or table) reporting relative energy deviations from available benchmarks (exact N=2 solution and, where feasible, other numerical references) across the N range, together with a clear statement of the 2-RDM parameterization and the specific N-representability conditions (P, Q, G) that are enforced during the variational optimization. revision: yes

  2. Referee: [Methods (variational 2-RDM section)] The variational procedure is described as direct minimization of an energy functional in the 2-RDM, yet without specification of which subset of N-representability conditions is imposed, it is impossible to determine whether the optimized 2-RDM remains consistent with the true ground state for 10 ≲ N ≲ 100 at intermediate coupling, as required by the central claim.

    Authors: We acknowledge the need for explicit specification. The revised manuscript will include a precise description of the 2-RDM parameterization employed and will state which N-representability conditions (P, Q, G) are imposed in the variational minimization. This addition will allow readers to assess consistency with the true ground state in the intermediate-N regime. revision: yes

Circularity Check

0 steps flagged

No circularity: direct variational minimization of 2-RDM energy functional

full rationale

The paper presents a variational optimization of the two-boson reduced density matrix to obtain ground-state energies and structural properties for trapped bosons. The energy is expressed directly as a functional of the 2-RDM, minimized subject to N-representability constraints, with results compared to independent benchmarks (exact N=2 analytic solution, Gross-Pitaevskii mean-field). No step reduces a reported quantity to a fitted parameter by construction, renames a known result, or relies on a load-bearing self-citation whose validity is internal to the present work. The central claim rests on the numerical performance of the constrained minimization against external references rather than any definitional equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit list of free parameters or new entities; the method rests on the domain assumption that 2-RDM variational optimization suffices for the N-body ground state.

axioms (1)
  • domain assumption The two-boson reduced density matrix can be variationally optimized under N-representability constraints to yield accurate ground-state observables for the full N-boson system.
    This is the central premise of the variational reduced-density-matrix approach described.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Representability for Quantum Theory beyond Particle-Number Conservation

    quant-ph 2026-04 unverdicted novelty 7.0

    A hierarchy of representability conditions for 2-RDMs in non-particle-number-conserving quantum systems is obtained from the polar cone of the p-positive cone, unified with conserving cases by adding particle-number variance.

Reference graph

Works this paper leans on

73 extracted references · 73 canonical work pages · cited by 1 Pith paper

  1. [1]

    ,xN)Ψ∗(x′ 1, x2,

    := N Z dx2dx3 · · ·dxN Ψ(x1, x2, . . . ,xN)Ψ∗(x′ 1, x2, . . . ,xN), (2) (2)D(x1, x2; x′ 1, x′

    work page

  2. [2]

    ,xN)Ψ∗(x′ 1, x′ 2, x3,

    := N(N − 1) Z dx3dx4 · · ·dxN Ψ(x1, x2, . . . ,xN)Ψ∗(x′ 1, x′ 2, x3, . . . ,xN). (3) The normalisations are by choice, with the unit nor- malisation also appearing prominently in the literature. Through the introduction of an orbital (i.e., single- particle) basis, {φj(xi)}, the 1- and 2-RDMs can be ex- pressed directly as the following spectral decomposi...

    work page

  3. [3]

    = X ij (1)Di j φi(x1) φ∗ j(x′ 1), (4) (2)D(x1, x2; x′ 1, x′

    work page

  4. [4]

    (5) 3 In Eqs

    = X ijkl (2)Dij klφi(x1)φj(x2) × φ∗ k(x′ 1)φ∗ l (x′ 2). (5) 3 In Eqs. (4) and (5) the tensors (1)Di j and (2)Dij kl can be expressed conveniently in second-quantised notation, providing the most common expressions for the 1- and 2-RDMs as found in the literature, (1)Di j = ⟨Ψ|ˆa† i ˆaj|Ψ⟩ , (6) (2)Dij kl = ⟨Ψ|ˆa† i ˆa† jˆalˆak|Ψ⟩ , (7) where ˆa† and ˆa ar...

    work page

  5. [5]

    3; it 7 0 1 2 3 40 0.1 0.2 0.3 0.4 a) β = 10 ρ(z) = (1)D(z;z) DG 1D GPE 1D NPSE 1 0 3 6 9 120 0.03 0.06 0.09 0.12 b) β = 10 3 z (az) ρ(z) = (1)D(z;z) DG 1D GPE 1D NPSE 1 FIG

    for the RDM methodology, very small ripples appear within the density, particular for panel b) in Fig. 3; it 7 0 1 2 3 40 0.1 0.2 0.3 0.4 a) β = 10 ρ(z) = (1)D(z;z) DG 1D GPE 1D NPSE 1 0 3 6 9 120 0.03 0.06 0.09 0.12 b) β = 10 3 z (az) ρ(z) = (1)D(z;z) DG 1D GPE 1D NPSE 1 FIG. 3. The ground-state density as a function ofz for two different values of β: β ...

    work page

  6. [6]

    The RDM methodology therefore duly cap- tures the ground-state energy across a range of extreme regimes, from small to largeN and β

    for large particle numbers and strong interaction strengths. The RDM methodology therefore duly cap- tures the ground-state energy across a range of extreme regimes, from small to largeN and β. Additionally, the RDM methodology accurately repro- duces the density as compared to the highly accurate mean-field descriptions forN = 103; this accuracy is re- t...

    work page

  7. [7]

    Feshbach, Ann

    H. Feshbach, Ann. Phys. (N. Y.)5, 357 (1958)

    work page 1958

  8. [8]

    M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science269, 198 (1995)

    work page 1995

  9. [9]

    Görlitz, J

    A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, Phys. Rev. Lett.87, 130402 (2001)

    work page 2001

  10. [10]

    Kinoshita, T

    T. Kinoshita, T. Wenger, and D. S. Weiss, Science305, 1125 (2004)

    work page 2004

  11. [11]

    Tonks, Phys

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

    work page 1936

  12. [12]

    Girardeau, Journal of Mathematical Physics1, 516 (1960)

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

    work page 1960

  13. [13]

    E. H. Lieb and W. Liniger, Phys. Rev.130, 1605 (1963)

    work page 1963

  14. [14]

    E. P. Gross, Il Nuovo Cimento20, 454 (1961)

    work page 1961

  15. [15]

    L. P. Pitaevskii, Sov. Phys. JETP13, 451 (1961). 9

    work page 1961

  16. [16]

    Salasnich, A

    L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A65, 043614 (2002)

    work page 2002

  17. [17]

    Bao and Y

    W. Bao and Y. Cai, arXiv preprint arXiv:1212.5341 (2012)

    work page Pith/arXiv arXiv 2012

  18. [18]

    Busch, B.-G

    T. Busch, B.-G. Englert, K. Rzażewski, and M. Wilkens, Found. Phys.28, 549 (1998)

    work page 1998

  19. [19]

    Krauth, Phys

    W. Krauth, Phys. Rev. Lett.77, 3695 (1996)

    work page 1996

  20. [20]

    Purwanto and S

    W. Purwanto and S. Zhang, Phys. Rev. E70, 056702 (2004)

    work page 2004

  21. [21]

    Vänskä, D

    T. Vänskä, D. Sundholm, and M. Lindberg, Phys. Rev. A 75, 023621 (2007)

    work page 2007

  22. [22]

    L. S. Cederbaum, O. E. Alon, and A. I. Streltsov, Phys. Rev. A 73, 043609 (2006)

    work page 2006

  23. [23]

    Bhowmik and O

    A. Bhowmik and O. E. Alon, J. Chem. Phys. 160, 044105 (2024), https://pubs.aip.org/aip/jcp/article- pdf/doi/10.1063/5.0176145/18702728/044105_1_5.0176145.pdf

    work page doi:10.1063/5.0176145/18702728/044105_1_5.0176145.pdf 2024

  24. [24]

    D. R. Harteee, Mathematical Proceedings of the Cam- bridge Philosophical Society24, 89 (1928)

    work page 1928

  25. [25]

    D. R. Harteee, Mathematical Proceedings of the Cam- bridge Philosophical Society24, 111 (1928)

    work page 1928

  26. [26]

    V. A. Fock, Zeitschrift für Physik61, 126 (1930)

    work page 1930

  27. [27]

    V. A. Fock, Zeitschrift für Physik62, 795 (1930)

    work page 1930

  28. [28]

    B. D. Esry, Phys. Rev. A55, 1147 (1997)

    work page 1997

  29. [29]

    Bergeman, Phys

    T. Bergeman, Phys. Rev. A55, 3658 (1997)

    work page 1997

  30. [30]

    Öhberg and S

    P. Öhberg and S. Stenholm, Phys. Rev. A 57, 1272 (1998)

    work page 1998

  31. [31]

    C. L. Benavides-Riveros, J. Wolff, M. A. L. Marques, and C. Schilling, Phys. Rev. Lett.124, 180603 (2020)

    work page 2020

  32. [32]

    D. A. Mazziotti, Phys. Rev. A65, 062511 (2002)

    work page 2002

  33. [34]

    Z. Zhao, B. J. Braams, M. Fukuda, M. L. Overton, and J. K. Percus, J. Chem. Phys.120, 2095 (2004)

    work page 2095

  34. [35]

    Nakata, H

    M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata, and K. Fujisawa, J. Chem. Phys.114, 8282 (2001)

    work page 2001

  35. [36]

    Fukuda, B

    M. Fukuda, B. J. Braams, M. Nakata, M. L. Overton, J. K. Percus, M. Yamashita, and Z. Zhao, Math. Pro- gram. 109, 553 (2007)

    work page 2007

  36. [37]

    Nakata, B

    M. Nakata, B. J. Braams, K. Fujisawa, M. Fukuda, J. K. Percus, M. Yamashita, and Z. Zhao, J. Chem. Phys.128, 164113 (2008)

    work page 2008

  37. [38]

    D. A. Mazziotti, Chem. Rev.112, 244 (2012)

    work page 2012

  38. [39]

    D. A. Mazziotti, Phys. Rev. Lett.93, 213001 (2004)

    work page 2004

  39. [40]

    D. A. Mazziotti, J. Chem. Phys.121, 10957 (2004)

    work page 2004

  40. [41]

    D. A. Mazziotti, Phys. Rev. Lett.106, 083001 (2011)

    work page 2011

  41. [42]

    Rosina and C

    M. Rosina and C. Garrod, J. Comput. Phys. 18, 300 (1975)

    work page 1975

  42. [43]

    DePrince, A

    I. DePrince, A. Eugene and D. A. Mazziotti, J. Chem. Phys. 132, 034110 (2010), https://pubs.aip.org/aip/jcp/article- pdf/doi/10.1063/1.3283052/15667691/034110_1_online.pdf

    work page doi:10.1063/1.3283052/15667691/034110_1_online.pdf 2010

  43. [44]

    Maradzike, M

    E. Maradzike, M. Hapka, K. Pernal, and A. E. De- Prince III, J. Chem. Theory Comput.16, 4351 (2020)

    work page 2020

  44. [45]

    A. E. DePrince III and D. A. Mazziotti, J. Phys. Chem. B 112, 16158 (2008)

    work page 2008

  45. [46]

    D. A. Mazziotti, Phys. Rev. A102, 052819 (2020)

    work page 2020

  46. [47]

    Head-Marsden and D

    K. Head-Marsden and D. A. Mazziotti, J. Phys. Chem. A 124, 4848 (2020)

    work page 2020

  47. [48]

    J. S. Anderson, M. Nakata, R. Igarashi, K. Fujisawa, and M. Yamashita, Computational and Theoretical Chem- istry 1003, 22 (2013)

    work page 2013

  48. [49]

    J. R. Hammond and D. A. Mazziotti, Phys. Rev. A73, 062505 (2006)

    work page 2006

  49. [50]

    Verstichel, H

    B. Verstichel, H. van Aggelen, W. Poelmans, and D. Van Neck, Phys. Rev. Lett.108, 213001 (2012)

    work page 2012

  50. [51]

    M. J. Knight, H. M. Quiney, and A. M. Martin, New J. Phys. 24, 053004 (2022)

    work page 2022

  51. [52]

    Gidofalvi and D

    G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 69, 042511 (2004)

    work page 2004

  52. [53]

    A. J. Coleman, Rev. Mod. Phys.35, 668 (1963)

    work page 1963

  53. [54]

    5,659(1961)

    C.GarrodandJ.K.Percus,J.Math.Phys. 5,659(1961)

    work page 1961

  54. [55]

    D. A. Mazziotti, Phys. Rev. A85, 062507 (2012)

    work page 2012

  55. [56]

    D. A. Mazziotti, Phys. Rev. Lett.108, 263002 (2012)

    work page 2012

  56. [57]

    D. A. Mazziotti, Phys. Rev. A74, 032501 (2006)

    work page 2006

  57. [58]

    Husimi, Proc

    K. Husimi, Proc. Phys.-Math. Soc. Jap. Ser. 3.22, 264 (1940)

    work page 1940

  58. [59]

    Löwdin, Phys

    P.-O. Löwdin, Phys. Rev.97, 1474 (1955)

    work page 1955

  59. [60]

    ter Haar, Rep

    D. ter Haar, Rep. Prog. Phys.24, 304 (1961)

    work page 1961

  60. [61]

    McWeeny, Rev

    R. McWeeny, Rev. Mod. Phys.32, 335 (1960)

    work page 1960

  61. [62]

    J. E. Mayer, Phys. Rev.100, 1579 (1955)

    work page 1955

  62. [63]

    R. H. Tredgold, Phys. Rev.105, 1421 (1957)

    work page 1957

  63. [64]

    Vandenberghe and S

    L. Vandenberghe and S. Boyd, SIAM Rev.38, 49 (1996)

    work page 1996

  64. [65]

    Gärtner and J

    B. Gärtner and J. Matousek,Approximation Algorithms and Semidefinite Programming(Springer-Verlag, 2012)

    work page 2012

  65. [66]

    Yamashita, K

    M. Yamashita, K. Fujisawa, K. Nakata, M. Nakata, M. Fukuda, K. Kobayashi, and K. Goto, A high- performance software package for semidefinite programs: SDPA 7, Tech. Rep. (Research Report B-460, Depart- ment of Mathematical and Computing Science, Tokyo Institute of Technology, Tokyo, 2010)

    work page 2010

  66. [67]

    Yamashita, K

    M. Yamashita, K. Fujisawa, M. Fukuda, K. Kobayashi, K. Nakata, and M. Nakata, Latest developments in the SDPA family for solving large-scale SDPs, inHandbook on Semidefinite, Conic and Polynomial Optimization , edited by M. F. Anjos and J. B. Lasserre (Springer US, Boston, MA, 2012) pp. 687–713

    work page 2012

  67. [68]

    X.-Y. Zhao, D. Sun, and K.-C. Toh, SIAM Journal on Optim. 20, 1737 (2010)

    work page 2010

  68. [69]

    L. Yang, D. Sun, and K.-C. Toh, Math. Program. Optim. 7, 331 (2015)

    work page 2015

  69. [70]

    Sun, K.-C

    D. Sun, K.-C. Toh, Y. Yuan, and X.-Y. Zhao, Optim. Methods Softw. 35, 87 (2020)

    work page 2020

  70. [71]

    D. Sun, J. Sun, and L. Zhang, Math. Program.114, 349 (2008)

    work page 2008

  71. [72]

    Meade, L

    B. Meade, L. Lafayette, G. Sauter, and D. Tosello, Spar- tan hpc-cloud hybrid: Delivering performance and flexi- bility (2017)

    work page 2017

  72. [73]

    V. I. Yukalov and M. D. Girardeau, Laser Physics Letters 2, 375 (2005)

    work page 2005

  73. [74]

    Ferrier-Barbut, H

    I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Phys. Rev. Lett.116, 215301 (2016)

    work page 2016