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REVIEW 2 major objections 5 minor 52 references

Large N vector models in the Hamiltonian framework

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The central claim is that the standard large-$N$ expansion of O($N$) vector models can be rederived in a Hamiltonian where the number of species fluctuates, making the large-$N$ ground state a Bose-Einstein condensate of $N$ extended…

desk verdict A careful Hamiltonian reformulation of known large N results with a fresh BEC picture; one concrete normalization typo in Eq. (3.1) should be fixed before publication. read the letter →

arxiv 2502.08031 v1 pith:P443BX46 submitted 2025-02-12 hep-th cond-mat.stat-mechquant-ph

Diego Barberena This is my paper · ORCID
classification hep-thcond-mat.stat-mechquant-ph
keywords largeNexpansionO(N)vectormodelfluctuatingformalismsecondquantizationBose-Einsteincondensationbilocalfieldsgapequationanomalousdimension
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 is trying to establish that the large-$N$ expansion of O($N$) vector models can be reproduced by a Hamiltonian formalism in which $N$ is not fixed but fluctuates. The method promotes the $N$ species to bosonic 'species' operators in a Fock space, so that a state with $N$ species is created by acting $N$ times on a vacuum; the large-$N$ saddle point then becomes a Bose-Einstein condensate of $N$ extended objects. Working on a lattice, the paper rederives the gap equation for the ground state, the negative-coupling bound-state masses in $d=2$ and $d=3$, and the leading anomalous dimension $\eta=8/(3\pi^2 N)$ in $d=2$. The payoff is a state-based, many-body picture of large-$N$ field theory in which O($N$)-invariant excitations organize naturally into bilocal fields, the language used in AdS$_4$/CFT$_3$ studies. A reader should care because this connects standard large-$N$ technology to condensed-matter intuition and to Hilbert-space methods such as tensor networks and quantum simulation.

What carries the argument

The central object is the fluctuating-$N$ (third-quantized) representation: a bosonic Fock space with species creation and annihilation operators $\hat\Psi_n^\dagger$, $\hat\Psi_n$ acting on $\bigoplus_N \mathrm{Sym}(\mathcal{H}^N)$, through which a single-species operator $O$ is replaced by $\sum_{nm}\langle m|O|n\rangle\hat\Psi_m^\dagger\hat\Psi_n$. The leading-order step is the condensate ansatz $|\hat\Psi\rangle\!\rangle\approx \hat F|F\rangle$, which makes the Hamiltonian $N$ times a c-number and produces the nonlinear gap equation for the condensate wavefunction $|F\rangle$. At next-to-leading order, orthogonal fluctuations $\delta\hat\Psi$ are kept; the NLO Hamiltonian is a set of coupled harmonic oscillators whose normal-mode frequencies give bound and scattering state masses. O($N$) invariance is imposed as an operator constraint that, to leading order in $1/N$, annihilates all modes except the two-particle (bilocal) modes $\hat\Psi_{kk'}$, and the correlators of these modes encode the anomalous dimension.

What would settle it

The claim would be settled by computing the order $N^{-2}$ correction to the anomalous dimension in $d=2$ using this Hamiltonian method and comparing the coefficient with the standard diagrammatic $1/N$ expansion; a mismatch would show that the two $1/N$ expansions are not equivalent.

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Extended reading notes

Core claim

On its own terms, the central claim is that the fluctuating-$N$ Hamiltonian, acting on the permutationally symmetric sector of the direct sum of $N$-species Hilbert spaces, is exactly equivalent to the fixed-$N$ Hamiltonian in that sector, and that its $1/N$ expansion reproduces the standard large-$N$ expansion of the path integral. The evidence is a sequence of recovered results: the self-consistent gap equation (3.14), bound states with $E^2=P^2+(1.92\,\tilde m)^2$ in $d=2$ and $E^2=P^2+(1.84\,\tilde m)^2$ in $d=3$ at negative coupling (3.45)--(3.46), and the anomalous dimension $\eta=8/(3\pi^2 N)$ (3.65). Within the formalism these results are read differently: the ground state is a condensate determined by a Gross-Pitaevskii-like nonlinear equation, and excitations about it are harmonic oscillators, with O($N$) invariance restricting the spectrum to bilocal modes.

Load-bearing premise

The load-bearing premise is that the fluctuating-$N$ bosonic Fock space, together with its $1/N$ expansion, faithfully reproduces the permutationally symmetric sector of the fixed-$N$ O($N$) theory; this equivalence is adopted from earlier work and is not proven in the paper.

Editorial extensions

If this is right

  • The large-$N$ saddle point of the O($N$) model is identified with a Bose-Einstein condensate of $N$ extended objects, so the ground state and its excitations can be studied with the standard toolbox of BEC theory, including condensate depletion and Bogoliubov transformations.
  • O($N$)-invariant excitations at leading order in $1/N$ are bilocal modes $\hat\Psi_{kk'}$, giving a Hamiltonian-level derivation of why bilocal collective fields are the right variables for AdS$_4$/CFT$_3$ vector-model holography.
  • At negative coupling with dynamically generated mass $\tilde m$, there are bound states with $E^2=P^2+(1.92\,\tilde m)^2$ in $d=2$ and $E^2=P^2+(1.84\,\tilde m)^2$ in $d=3$, matching the known $1/N$ bound-state spectrum.
  • In $d=2$ the formalism reproduces the leading anomalous dimension $\eta=8/(3\pi^2 N)$ from an equal-time descendant correlator, and it also yields critical-exponent corrections such as $\alpha=1-32/(\pi^2 N)+O(N^{-2})$.
  • Finite-temperature analyses within the O($N$)-invariant sector show a crossover to non-condensation at temperatures of order $VT^2\sim N^{1/2}$, even though the free energy shows no corresponding feature.

Reading between the lines

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

  • The author does not pursue numerical checks, but the identification of the large-$N$ saddle with a condensate suggests a concrete test: truncate the fluctuating-$N$ Fock space to a modest $N$ and compare the low-lying spectrum with exact diagonalization of the fixed-$N$ O($N$) lattice model.
  • If the equivalence holds beyond leading order, the $1/N$ expansion of any O($N$)-invariant observable could be generated by Bogoliubov and perturbation theory on the condensate, potentially avoiding the need for diagrammatic resummations.
  • The finite-temperature crossover found in the O($N$)-invariant sector, although absent from the free energy, could be observable in the depletion of the condensate or in Renyi entropies of spatial subregions, since occupation of sp-states is a state-level quantity.
  • Because the species operators remain bosonic even when the underlying degrees of freedom are fermionic, the same framework should give a Hamiltonian derivation of Gross-Neveu dynamical mass generation and threshold bound states, an extension the author only sketches.
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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

2 major / 5 minor

Summary. The paper develops a 'fluctuating N' second-quantized Hamiltonian formalism for large N vector models, first in a 0+1d quantum-mechanical quartic oscillator and then for the lattice O(N) model in 2+1 and 3+1 dimensions. The leading-order saddle is interpreted as a Bose-Einstein condensate of N extended objects, and the next-to-leading-order Hamiltonian is a quadratic system whose normal modes are computed by resolvent/rank-one perturbation methods. The paper claims to reproduce the standard large N gap equation, the negative-coupling bound-state masses E^2 = P^2 + (1.92 m̃)^2 in d=2 and E^2 = P^2 + (1.84 m̃)^2 in d=3, the critical exponent η = 8/(3π^2 N) in d=2, and the finite-temperature free energy of the singlet sector, while emphasizing the bilocal nature of O(N)-invariant excitations.

Significance. If the formalism is valid, this is a valuable complementary derivation of large N vector-model results using Hamiltonian methods, with a concrete physical picture (Bose-Einstein condensation of extended objects) and a route to bilocal collective fields relevant to AdS_4/CFT_3. The paper is self-contained in its perturbative machinery: the second-quantization map (2.2)-(2.4) is exact on the permutationally symmetric subspace, the 1/N expansion is systematic, and the calculated quantities agree with known benchmarks (gap equation, bound-state masses, anomalous dimension, and free energy). The main strengths are the explicit derivations for the spectrum, the resolvent-based treatment of the ground-state energy, and the clean separation of the O(N) singlet sector. The manuscript also honestly states which extensions are left to future work.

major comments (2)
  1. [Section 3, Eq. (3.1) and Eq. (3.3)] The interaction term in Eq. (3.1) is written as (λ/N)∑_r [(1/N)∑_α (φ^α_r)^2]^2 = (λ/N^3)∑_r (∑_α (φ^α_r)^2)^2, which is subleading in 1/N and would make the large-N limit a free theory. The second-quantized Hamiltonian in Eq. (3.3), used for all subsequent calculations, contains λ/ˆN [⟨⟨Ψ|φ^2_r|Ψ⟩⟩]^2 = (λ/N)∑_r (∑_α (φ^α_r)^2)^2 on the N-species sector, which is the conventional normalization and is consistent with the quantum-mechanical model (2.1)-(2.3). As printed, Eq. (3.1) is internally inconsistent with the rest of the paper and prevents a reader from reproducing the gap equation (3.14), the bound-state masses (3.45)-(3.46), and the anomalous dimension (3.65). Please correct Eq. (3.1) by deleting the inner 1/N factor.
  2. [Section 3.2.2 and Appendix B.3] The claim that the O(N) invariance condition reduces the excitation spectrum to the bilocal modes Ψ_kk' (Eq. (3.35)) is stated without derivation. Appendix B.3 derives the constraint operator (B.21), but the projection onto sp-eigenstates that leads to Ψ_n|S⟩ = 0 for n ≠ kk' is not shown. Because the identification of the O(N) singlet sector with bilocal fields is a central claim of the paper (and underlies the use of the restricted Hamiltonian (3.36) for the bound-state calculation), this step needs to be presented explicitly or supported by a precise citation to Refs. [7,9] indicating where the projection is performed.
minor comments (5)
  1. [Section 3.1.1, Eq. (3.17)] The expression m′ = π/[λ(8λB2 + ~J)] should read m′ = π(8λB2 + ~J)/λ; the printed reciprocal is inconsistent with the small-m expansion of Eq. (3.16) and with the energy formula that follows in the same equation.
  2. [Section 4.1.1, after Eq. (4.16)] The text says 'When ζ ≪ 1 (low temperatures), the expression for ⟨n0⟩β tends to 1' but the correct regime for the ground-state occupation to approach 1 is ζ ≫ 1; the subsequent high-temperature expansion uses ζ ≪ 1.
  3. [Section 3.2.5, Eqs. (3.63)-(3.64)] The evaluation of the integral in Eq. (3.63) is summarized as 'after some algebra'; Appendix B.5 sets up the resolvent machinery but does not show the small-|k| asymptotic evaluation. Please provide a few intermediate steps so the logarithmic term in Eq. (3.64) can be checked.
  4. [Section 3.1, Eq. (3.8)] The notation δ_{k+k'} for the commutator is nonstandard; please write δ_{k+k',0} or equivalently δ_{k,-k'}.
  5. [Section 3.2.4, Eqs. (3.48)-(3.52)] The function B(z,P) is introduced for real l in Eq. (3.44) and then used as a complex function without comment; a brief note on the analytic continuation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No load-bearing circularity: the central results are derived from the fluctuating-N Hamiltonian and benchmarked externally, not fitted or assumed.

full rationale

Score 0. The derivation chain is self-contained given the fluctuating-N mapping adopted from Refs. [7,9], which is an external method citation with no author overlap and is not used to pin down numerical outputs. The O(N) results are obtained by solving the self-consistency/gap equation (3.14), diagonalizing the NLO oscillator Hamiltonian (3.36), and computing correlators; nothing is fitted to the target quantities. The dynamically generated mass m-tilde is fixed by the gap equation before the bound-state equation is solved, and the bound-state coefficients (1.92, 1.84) and anomalous dimension eta=8/(3 pi^2 N) emerge from the derived expressions (3.44)-(3.46) and (3.63)-(3.65). The paper compares, rather than identifies, its results with Refs. [37,38,43], and the benchmark values are external. No load-bearing step reduces by definition to its own inputs, and no self-citation chain is invoked to forbid alternatives. A display-level normalization mismatch between Eqs. (3.1) and (3.3) is noted in the skeptical review; that is an internal-consistency and correctness concern, not a circular reduction, and it does not affect this circularity score.

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

No free parameters are fitted to data. The axioms are standard or explicitly adopted from prior work. No new physical entities are postulated.

assumptions (5)
  • domain assumption The permutationally symmetric sector of the N-species Hilbert space is faithfully represented by the bosonic field operators Ψ̂, and the 1/N expansion about the condensate reproduces the standard large N expansion.
    Adopted from Refs. [7,9] and used throughout Sections 2 and 3. If this equivalence fails, the whole formalism collapses.
  • domain assumption At leading order in 1/N, only one sp-state |F⟩ is macroscopically occupied (single condensate), with no fragmentation.
    Used to derive the LO Hamiltonian in Eq. (2.8) and the O(N) model LO in Section 3.1. Fragmented condensates are mentioned but not analyzed.
  • domain assumption The lattice Hamiltonian (3.1) is a valid discretization of the O(N) model, and the continuum limit is obtained when the mass parameter m is much smaller than the lattice scale.
    The entire field theory analysis is performed on a lattice; the paper assumes this faithfully represents the continuum theory.
  • domain assumption For λ<0, the theory is studied in the metastable vacuum defined by the non-perturbative gap solution m̃, following the PT-symmetric framework of Refs. [23-25].
    Used to compute bound states and ground state energy at negative coupling; the author notes metastability in Appendix B.2.
  • standard math Standard canonical commutation relations and second-quantization identities (e.g., Eq. (2.4)) are used without proof.
    These are standard results from many-body quantum theory (Ref. [8]).

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Pith. "Pith review of Large N vector models in the Hamiltonian framework." pith.science (2026). https://pith.science/paper/P443BX46

@misc{pith2026250208031,
  author       = {Pith},
  title        = {Pith review of: Large N vector models in the Hamiltonian framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P443BX46}},
  note         = {Machine review of arXiv:2502.08031}
}
abstract

We present a fluctuating $N$ formalism, based on second-quantization, to describe large $N$ vector models from field theory using Hamiltonian methods. We first present the method in the simpler setting of a quantum mechanical system with quartic interactions, and then apply these techniques to the $O(N)$ model in $2+1$ and $3+1$ dimensions. We recover various known results, such as the gap equation determining the ground state of the system, the presence of bound states at negative coupling and the leading order contribution to critical exponents, and provide an interpretation of the large $N$ path integral saddle point as a Bose-Einstein condensate of extended objects in the presence of a non-local interaction. In the large $N$ limit, this formalism leads naturally to a description of elementary $O(N)$ symmetric excitations in terms of bilocal fields, which are at the basis of $\text{AdS}_4/\text{CFT}_3$ studies of the $O(N)$ model and Vasiliev gravity.

Figures

Figures reproduced from arXiv: 2502.08031 by the authors.

Figure 2.1
Figure 2.1. The operators Ψˆ x and Ψˆ † x map a state with N species to a state with N − 1 and N + 1 species, respectively. Operators that are symmetric under permutations and that do not change the number of species, e.g. P i pˆ 2 i , can be represented in terms of Ψˆ x, Ψˆ † x when acting on permutationally symmetric states. components as a “species”. The operators ˆxα and ˆpβ form canonical pairs, i.e. [ˆxα, pˆβ] = iδαβ, and… view at source ↗
Figure 3.1
Figure 3.1. Terms in the gap equation Eq. (3.14) for d = 2 (a) and d = 3 (b). Right hand side (RHS) of Eq. (3.14) is shown in purple and has universal non analyticities as m → 0 (small m approximation is shown in dashed black). Left hand side (LHS) is shown in green. The height of the parabolic profile is controlled by −J˜ and its curvature is controlled by λ (curvature increases as λ → 0). Solutions relevant to field theory ar… view at source ↗

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Works this paper leans on

52 extracted references · 45 canonical work pages

  1. [1]

    Klebanov, F

    I.R. Klebanov, F. Popov and G. Tarnopolsky, Tasi lectures on large n tensor models, 1808.09434

    arXiv

  2. [2]

    Hooft, A planar diagram theory for strong interactions, Nuclear Physics B 72 (1974) 461

    G. Hooft, A planar diagram theory for strong interactions, Nuclear Physics B 72 (1974) 461

    work page 1974

  3. [3]

    Maldacena, The large-n limit of superconformal field theories and supergravity, International Journal of Theoretical Physics 38 (1999) 1113

    J. Maldacena, The large-n limit of superconformal field theories and supergravity, International Journal of Theoretical Physics 38 (1999) 1113

    work page 1999

  4. [4]

    Witten, Anti de sitter space and holography, Adv

    E. Witten, Anti de sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253

    work page 1998

  5. [5]

    Klebanov and A

    I. Klebanov and A. Polyakov, Ads dual of the critical o(n) vector model, Physics Letters B 550 (2002) 213

    work page 2002

  6. [6]

    Moshe and J

    M. Moshe and J. Zinn-Justin, Quantum field theory in the large n limit: a review, Physics Reports 385 (2003) 69

    work page 2003

  7. [7]

    Maslov and O.Y

    V.P. Maslov and O.Y. Shvedov, Large-n expansion as a semiclassical approximation to the third-quantized theory, Phys. Rev. D 60 (1999) 105012

    work page 1999

  8. [8]

    Altland and B.D

    A. Altland and B.D. Simons, Condensed Matter Field Theory, Cambridge University Press, 2 ed. (2010)

    work page 2010

Show all 52 references
  1. [9]

    Shvedov, Large-N theory from the axiomatic point of view, Journal of Mathematical Physics 42 (2001) 4197

    O.Y. Shvedov, Large-N theory from the axiomatic point of view, Journal of Mathematical Physics 42 (2001) 4197

    2001

  2. [10]

    Das and A

    S.R. Das and A. Jevicki, Large-n collective fields and holography, Phys. Rev. D 68 (2003) 044011

    2003

  3. [11]

    Aharony, S.M

    O. Aharony, S.M. Chester and E.Y. Urbach, A derivation of ads/cft for vector models, Journal of High Energy Physics 2021 (2021) 208

    2021

  4. [12]

    Aharony, S.M

    O. Aharony, S.M. Chester, T. Sheaffer and E.Y. Urbach, Explicit holography for vector models at finite n, volume and temperature, Journal of High Energy Physics 2023 (2023) 16

    2023

  5. [13]

    de Mello Koch, Gravitational dynamics from collective field theory, Journal of High Energy Physics 2023 (2023) 151

    R. de Mello Koch, Gravitational dynamics from collective field theory, Journal of High Energy Physics 2023 (2023) 151. 30

    2023

  6. [14]

    Dalmonte and S

    M. Dalmonte and S. Montangero, Lattice gauge theory simulations in the quantum information era, Contemporary Physics 57 (2016) 388

    2016

  7. [15]

    Berenstein and G

    D. Berenstein and G. Hulsey, Bootstrapping simple qm systems, 2108.08757

    arXiv

  8. [16]

    Lawrence, Semidefinite programs at finite fermion density, Phys

    S. Lawrence, Semidefinite programs at finite fermion density, Phys. Rev. D 107 (2023) 094511

    2023

  9. [17]

    Halimeh, M

    J.C. Halimeh, M. Aidelsburger, F. Grusdt, P. Hauke and B. Yang, Cold-atom quantum simulators of gauge theories, 2310.12201

    arXiv

  10. [18]

    Bauer, Z

    C.W. Bauer, Z. Davoudi, A.B. Balantekin, T. Bhattacharya, M. Carena, W.A. de Jong et al., Quantum simulation for high-energy physics, PRX Quantum 4 (2023) 027001

    2023

  11. [19]

    Leggett, Quantum Liquids: Bose condensation and Cooper pairing in condensed-matter systems, Oxford University Press (09, 2006), 10.1093/acprof:oso/9780198526438.001.0001

    A.J. Leggett, Quantum Liquids: Bose condensation and Cooper pairing in condensed-matter systems, Oxford University Press (09, 2006), 10.1093/acprof:oso/9780198526438.001.0001

    2006

  12. [20]

    Rubakov, On third quantization and the cosmological constant, Physics Letters B 214 (1988) 503

    V. Rubakov, On third quantization and the cosmological constant, Physics Letters B 214 (1988) 503

    1988

  13. [21]

    Robles-P´ erez and P.F

    S. Robles-P´ erez and P.F. Gonz´ alez-D´ ıaz, Quantum state of the multiverse, Phys. Rev. D 81 (2010) 083529

    2010

  14. [22]

    Prosen, Third quantization: a general method to solve master equations for quadratic open fermi systems, New Journal of Physics 10 (2008) 043026

    T. Prosen, Third quantization: a general method to solve master equations for quadratic open fermi systems, New Journal of Physics 10 (2008) 043026

    2008

  15. [23]

    W.-Y. Ai, C.M. Bender and S. Sarkar, PT -symmetric−gφ4 theory, Phys. Rev. D 106 (2022) 125016

    2022

  16. [24]

    P. Romatschke, A solvable quantum field theory with asymptotic freedom in (3+1) dimensions, International Journal of Modern Physics A 38 (2023) 2350157 [https://doi.org/10.1142/S0217751X23501579]

    2023 doi

  17. [25]

    Romatschke, Quantum field theory in large n wonderland: Three lectures, 2310.00048

    P. Romatschke, Quantum field theory in large n wonderland: Three lectures, 2310.00048

    arXiv

  18. [26]

    Gross and A

    D.J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10 (1974) 3235

    1974

  19. [27]

    Ferrell and D.J

    R.A. Ferrell and D.J. Scalapino, Statistical mechanics of one-dimensional ginzburg-landau fields. ii. a test of the screening approximation n−1 expansion), Phys. Rev. A 9 (1974) 846

    1974

  20. [28]

    A.J. Bray, Statistical mechanics of one-dimensional ginzburg-landau fields: Feynman graph evaluation of the screening approximation (n-1 expansion), Journal of Physics A: Mathematical, Nuclear and General 7 (1974) 2144. 31

    1974

  21. [29]

    V.S.P. A . D. Dolgov, V. L. Eletskii, New approach to perturbation theory for a discrete spectrum (anharmonic oscillator), Journal of Experimental and Theoretical Physics 52 (1980) 861

    1980

  22. [30]

    Koudinov and M.A

    A.V. Koudinov and M.A. Smondyrev, 1/n-expansion for the anharmonic oscillator, Czechoslovak Journal of Physics B 32 (1982) 556

    1982

  23. [31]

    F. Hioe, D. Macmillen and E. Montroll, Quantum theory of anharmonic oscillators: Energy levels of a single and a pair of coupled oscillators with quartic coupling, Physics Reports 43 (1978) 305

    1978

  24. [32]

    Susskind and J

    L. Susskind and J. Glogower, Quantum mechanical phase and time operator, Physics Physique Fizika 1 (1964) 49

    1964

  25. [33]

    Kromminga and M

    A.J. Kromminga and M. Bolsterli, Perturbation theory of many-boson systems, Phys. Rev. 128 (1962) 2887

    1962

  26. [34]

    Bardeen and M

    W.A. Bardeen and M. Moshe, Phase structure of the O( n) vector model, Phys. Rev. D 28 (1983) 1372

    1983

  27. [35]

    Giombi, R

    S. Giombi, R. Huang, I.R. Klebanov, S.S. Pufu and G. Tarnopolsky, o(n) model in 4<d< 6: Instantons and complex cfts, Phys. Rev. D 101 (2020) 045013

    2020

  28. [36]

    Romatschke, C.-W

    P. Romatschke, C.-W. Su and R. Weller, Mass from nothing, 2405.00088

    arXiv

  29. [37]

    Abbott, J.S

    L.F. Abbott, J.S. Kang and H.J. Schnitzer, Bound states, tachyons, and restoration of symmetry in the 1 N expansion, Phys. Rev. D 13 (1976) 2212

    1976

  30. [38]

    Romatschke, What if ϕ4 theory in 4 dimensions is non-trivial in the continuum?, Physics Letters B 847 (2023) 138270

    P. Romatschke, What if ϕ4 theory in 4 dimensions is non-trivial in the continuum?, Physics Letters B 847 (2023) 138270

    2023

  31. [39]

    Abe and S

    R. Abe and S. Hikami, Breakdown of some scaling law relations in 1n expansion, Physics Letters A 42 (1973) 419

    1973

  32. [40]

    Abe and S

    R. Abe and S. Hikami, Discontinuities of critical amplitude for specific heat, Physics Letters A 45 (1973) 11

    1973

  33. [41]

    Aharony, Scaling function for two-point correlations

    A. Aharony, Scaling function for two-point correlations. ii. expansion to order 1 n, Phys. Rev. B 10 (1974) 2834

    1974

  34. [42]

    Wegner, Corrections to scaling laws, Phys

    F.J. Wegner, Corrections to scaling laws, Phys. Rev. B 5 (1972) 4529

    1972

  35. [43]

    Kleinert and V

    H. Kleinert and V. Schulte-Frohlinde, Critical Properties of ϕ4-Theories, WORLD SCIENTIFIC (2001), 10.1142/4733

    2001 doi

  36. [44]

    Grossmann and M

    S. Grossmann and M. Holthaus, Microcanonical fluctuations of a bose system’s ground state occupation number, Phys. Rev. E 54 (1996) 3495. 32

    1996

  37. [45]

    Holthaus, E

    M. Holthaus, E. Kalinowski and K. Kirsten, Condensate fluctuations in trapped bose gases: Canonical vs. microcanonical ensemble, Annals of Physics 270 (1998) 198

    1998

  38. [46]

    Kocharovsky, V.V

    V.V. Kocharovsky, V.V. Kocharovsky, M. Holthaus, C. Raymond Ooi, A. Svidzinsky, W. Ketterle et al., Fluctuations in ideal and interacting bose–einstein condensates: From the laser phase transition analogy to squeezed states and bogoliubov quasiparticles, vol. 53 of Advances In...

    2006

  39. [47]

    Shenker and X

    S.H. Shenker and X. Yin, Vector models in the singlet sector at finite temperature, 1109.3519

    arXiv

  40. [48]

    Yoon, Finite temperature holography in higher spin theory/vector model, Ph.D

    J. Yoon, Finite temperature holography in higher spin theory/vector model, Ph.D. thesis, Brown University, 2016

    2016

  41. [49]

    Grable and M

    S. Grable and M. Weiner, A fully solvable model of fermionic interaction in 3 + 1d, Journal of High Energy Physics 2023 (2023) 17

    2023

  42. [50]

    Jevicki, X

    A. Jevicki, X. Liu and J. Zheng, Symmetries and the hilbert space of large n extended states, Universe 10 (2024)

    2024

  43. [51]

    Coleman, More about the massive schwinger model, Annals of Physics 101 (1976) 239

    S. Coleman, More about the massive schwinger model, Annals of Physics 101 (1976) 239

    1976

  44. [52]

    Dempsey, I.R

    R. Dempsey, I.R. Klebanov, S.S. Pufu, B.T. Søgaard and B. Zan, Phase diagram of the two-flavor schwinger model at zero temperature, Phys. Rev. Lett. 132 (2024) 031603. A Quantum Mechanical Model A.1 Stationary points of the multispecies QM Hamiltonian In this appendix we look ...

    2024

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