Quantum Field Theory in the Large N Limit: a review

Jean Zinn-Justin; Moshe Moshe

REVIEW

Quantum Field Theory in the Large N Limit: a review

Moshe Moshe, Jean Zinn-Justin

High Energy Physics - Theory · Submitted 13 Jun 2003 · 207 references

Pith's verdict: A 2003 review that still earns its citations: the model-equivalence derivations are on the page, and the authors are candid about where the 1/N method runs out of road.

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CitationJean Zinn-Justin, Moshe Moshe. "Quantum Field Theory in the Large N Limit: a review." arXiv:hep-th/0306133 (2003). https://pith.science/paper/OVKRWYGG

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Editor's note

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This is the Moshe–Zinn-Justin large N review, and it does what a review is supposed to do: it sets up the auxiliary-field formalism once (Eqs. 2.3–2.7), then applies the same machinery to scalars, the non-linear sigma model, CP(N-1), Gross–Neveu, the SUSY 3D model, and finite temperature without re-deriving the wheel each time. If you need one place to see why the (phi^2)^2 theory and the non-linear sigma model describe the same critical behaviour to all orders in 1/N, Section 3.1 is it: shift lambda, drop lambda^2 because [lambda]=2 makes it irrelevant below four dimensions, integrate, and Eq. (3.2) is the sigma model with a delta-function constraint. The CP(N-1)/abelian Higgs and Gross–Neveu/Gross–Neveu–Yukawa versions follow the same template, and each is derived rather than asserted. That is the real content, and it holds up on reading. The reader's strongest-claim pick is the right one, and the stress-test is right that the derivation does not depend on the sign of a(d). One correction to the stress-test's framing: the classifier reports 'no limitations found,' which is simply wrong. The authors flag the a(d)>0 problem themselves in Section 2.4 ('the sign is arbitrary' for simple lattice regularizations in d=3), the multicritical failure in Section 10, the Borel-summability question in Section 3.3, and the u phi^4 >> (m phi)^{4-d} condition in Section 3.1. That candour is a strength, not a soft spot. Genuine soft spots are small. Eq. (2.34) drops a factor of N: combining (2.21) and (2.33) gives sigma^2 = -(6N/u) tau, not -(6/u) tau. The exponent beta=1/2 is unaffected, so it is a normalization slip, but it will confuse a student working through the algebra. The higher-order exponents in Section 3.3 are quoted from refs [58–66] without derivation, which is normal for a review but means those numbers are not checkable here. And the obvious contextual caveat: this is a 2003 snapshot, so nothing on SYK, the modern bootstrap, or the GN conformal window. Also note the text I could read stops partway through Section 7, so Sections 8–10 and the appendices were not assessed. Who should read it: graduate students and anyone who needs the 1/N toolbox with the gap equations written out, not just cited. Worth a serious referee pass — the derivations are checkable and the pedagogy is the point.

Referee report

Significance

High. The review is comprehensive and pedagogical, and it derives its central claims explicitly: the saddle-point equations (2.10), the expansion of Ω_d(m) in (2.13), the exponent values ν = 1/(d-2) and η = 0, the λ propagator (2.55) and its p^ε behaviour (2.61), and the RG functions (2.70)-(2.72) all follow from the formalism set up in Section 2. The model-equivalence derivations in Sections 3.1, 3.5 and 4.6 are checkable and correctly state their conditions of validity. The review is also candid about the limitations of the large-N expansion: the scheme-dependence of a(d) in Section 2.4, the divergence of the 1/N series in Section 3.3, and the artifact of unbroken SUSY at leading order in Section 6.4 are all flagged. It is a useful reference for the field.

Minor objections (7)

[Section 2.4, Eq. (2.45)] As printed, Eq. (2.45) defines u* = 6a(d)Λ^ε, but Eqs. (2.68), (2.73), (2.74) and (2.76) all require u* = 6Λ^ε/a(d). The rest of Section 2.4 is self-consistent with the latter; (2.45) should be corrected.
[Section 2.3, Eq. (2.34)] Eq. (2.34) writes σ² = -(6/u)τ, omitting the factor N that follows from combining (2.21) and (2.33); the correct relation is σ² = -N(6/u)τ.
[Section 3.2, Eq. (3.28)] The three expressions chained for X_1 in Eq. (3.28) do not appear to be equal as printed; at d=3 the first gives 1/(8π²), the second 2, and the third 8/π², and only the third reproduces the η₁ = 8/(3π²) quoted in Section 3.3 and X_1(3) = 8/π² quoted in Section 6.7. Since X_1 enters (3.32)-(3.36), (4.71)-(4.74) and (6.7), the chain should be checked.
[Section 2.4, the sign of a(d)] The authors acknowledge that a(d) is not guaranteed positive for d<4 and that for simple lattice regularizations in d=3 'the sign is arbitrary'. The review could state more explicitly which conclusions depend on a(d)>0 (the identification of the IR fixed point and the cancellation of corrections to scaling) and which do not (the leading exponents ν = 1/(d-2) and η = 0, which follow from the saddle-point structure). This would help the reader judge the reliability of the d=3 results.
[Section 3.3, 'The nature of the large N expansion'] The statement that the expansion is 'Borel summable for dimensions d<4' is a strong claim; a sentence on what is being summed (the 1/N series at fixed d, or the ε-expansion) and what the evidence is (ref. [68]) would help.
[Section 6.4, the 'infinite hysteresis' discussion] The claim that the system will not return from B to A when u is increased past u_c is a physical interpretation of the degenerate saddle points, not a derived result; the text hedges with 'most probably' in the following sentence, but the preceding sentence states it more strongly. A clearer marking of the conjecture would be appropriate.
[Sections 7-10] I could not assess Sections 7-10 in the version provided, since the text is truncated; the comments above are confined to Sections 1-6.

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The paper

The paper claims

A single bookkeeping trick — replacing $N$ coupled fields with one self-averaging composite field — solves $O(N)$ and $U(N)$ quantum field theories in the large-$N$ limit and reveals that theories renormalizable in different dimensions can,

The reading

This review sets out to show that a whole family of quantum field theories with $N$ fields in a vector representation becomes solvable when $N$ is taken large, and that the solution is not a toy: it reproduces critical exponents, phase diagrams, and bound-state spectra in every dimension between two and four, where ordinary perturbation theory has nothing to say. The organising idea is a probability-style argument: at large $N$ the invariant composite field $O(N)$ built out of the vector, $O(N)$-invariant $O(N)$, has small fluctuations, so it can be treated as the dynamical variable and the original fields integrated out. The authors' most consequential claim is that this procedure relates pairs of theories that look unrelated — the $(O(N))$ theory and the non-linear $O(N)$ model, the $O(N)$ model and the abelian Higgs model, the Gross–Neveu model and a Yukawa theory — showing they share the same long-distance physics despite being renormalizable in different dimensions. If the claim holds, dimensional crossovers (finite size, finite temperature, dimensional reduction) become calculable rather than merely conjectured, and the large-$N$ expansion becomes a check on the $4-d$ and $d-2$ expansions rather than a rival to them. The review is explicit that the method is a technique with its own limits, and it repeatedly insists that the $1/N$ expansion must be checked to be both infrared finite and renormalizable before its results can be trusted.

What carries the argument

The mechanism is the auxiliary-field (Hubbard–Stratonovich-type) identity $1 = N$ times the delta function inserted for the constraint $O(N)$, followed by a Gaussian integration over the vector field and a steepest-descent evaluation of the resulting functional integral. The Gaussian integration produces a trace-log term, $O(N)$ times the logarithm of the inverse propagator of the vector field in the presence of the multiplier; the multiplier's saddle-point value is the mass of the elementary quanta, and its propagator, computed from the bubble diagram, carries the anomalous dimension that makes the $1/N$ expansion renormalizable in every dimension. This is the object that does the work: it converts a strongly coupled vector theory into a weakly fluctuating scalar problem, and it is what allows the review to move between the linear $(O(N))$ theory, the non-linear $O(N)$ model, the $O(N)$ model, the Gross–Neveu model, and their supersymmetric versions without re-deriving anything from scratch. A parallel section shows that a Gaussian variational ansatz reproduces the same saddle-point equations in the large-$N$ limit, giving an independent handle on the same physics.

Extended reading notes

Core claim

The central claim is an equivalence, valid to all orders in the $1/N$ expansion, between the $O(N)$-symmetric $(O(N))$ field theory and the $O(N)$ non-linear $O(N)$ model. The derivation runs through the auxiliary-field representation: inserting $1 = N$ times a delta function for $O(N)$ and integrating over the original vector field turns the partition function into a functional integral over a Lagrange-multiplier field $O(N)$ and a composite field $O(N)$, with the vector field appearing only quadratically. In the critical domain, the $O(N)$ term in the effective action has dimension $4-d$ and is irrelevant for $d<4$, so it can be dropped; shifting $O(N)$ by its expectation value and integrating it back out returns the original vector field subject to a fixed-length constraint, which is exactly the non-linear $O(N)$ model. The same argument, applied to a gauged complex vector, relates the $O(N)$ model to the abelian Higgs model, and applied to fermions relates the Gross–Neveu four-fermion theory to the Gross–Neveu–Yukawa theory. Along the way the review solves the general $U(phi^2)$ scalar theory at leading order in $1/N$, obtaining critical exponents $beta = 1/2$, $nu = 1/(d-2)$, $eta = 0$, $delta = (d+2)/(d-2)$, the scaling equation of state, and the leading correction-to-scaling exponent $omega = 4-d$, all in arbitrary dimension $2<d<4$.

Load-bearing premise

The consistency of the large-$N$ results with ordinary renormalization-group behaviour at fixed dimension requires the regularization-dependent constant $a(d)$ to be positive, and the review itself states that for $d<4$ this positivity is not assured and that for simple lattice regularizations in $d=3$ the sign is arbitrary.

Editorial extensions

If this is right

Critical exponents and the scaling equation of state for $O(N)$-symmetric systems become calculable in every dimension $2<d<4$, not just near $d=4$ or near $d=2$, so the two epsilon expansions can be connected through a common answer.
The $(O(N))$ theory and the non-linear $O(N)$ model must give identical universal quantities; a scheme-dependence in the cutoff is the only place where their apparent difference in number of parameters can hide.
The Gross–Neveu and Gross–Neveu–Yukawa models must produce the same long-distance physics, including the mass ratio $m_sigma/m_psi = 2$ at leading order, which the review verifies independently in $d=2$ and $d=4$.
Chiral symmetry breaking in the Gross–Neveu model is always present in two dimensions and appears above a critical coupling $G_c = 1/Omega_d(0)$ in higher dimensions, with the mass vanishing as a power of $G - G_c$ for $d<4$.
Finite-size and finite-temperature crossovers between a $d$-dimensional theory and its $(d-1)$-dimensional reduction become tractable, since the same gap equation with quantized momenta or Matsubara frequencies replaces the infinite-volume one.

Reading between the lines

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

If the equivalence holds to all orders in $1/N$, then any lattice simulation that measures, say, the $O(N)$ model's correlation-length exponent at $N=10$ in $d=3$ is implicitly a test of the $(O(N))$ theory's exponent at the same $N$ and $d$ — a cross-check that does not require simulating both theories.
The review's insistence that the $1/N$ series is divergent but Borel summable for $d<4$ suggests that the practical value of the method is as an asymptotic expansion whose first few terms are reliable, not as a convergent scheme; a reader should expect the $1/N^2$ and $1/N^3$ coefficients quoted here to be useful only for large $N$.
The equivalence argument leans on dropping the $O(N)$ term as irrelevant for $d<4$; in exactly $d=4$ that term is marginal and only logarithmically suppressed, which is precisely where the review says the non-linear $O(N)$ model must be replaced by the $(O(N))$ theory to be understood perturbatively — a hint that the correspondence is cleanest strictly below four dimensions.
The variational derivation reproducing the same saddle-point equations suggests the large-$N$ results are the mean-field sector of a more general variational principle, so any improvement scheme (e.g. going beyond Gaussian trial states) would generate the $1/N$ corrections systematically rather than by diagrammatic brute force.

Original abstract

We review the solutions of O(N) and U(N) quantum field theories in the large $N$ limit and as 1/N expansions, in the case of vector representations. Since invariant composite fields have small fluctuations for large $N$, the method relies on constructing effective field theories for composite fields after integration over the original degrees of freedom. We first solve a general scalar $U(\phib^2)$ field theory for $N$ large and discuss various non-perturbative physical issues such as critical behaviour. We show how large $N$ results can also be obtained from variational calculations.We illustrate these ideas by showing that the large $N$ expansion allows to relate the $(\phib^2)^2$ theory and the non-linear $\sigma$-model, models which are renormalizable in different dimensions. Similarly, a relation between $CP(N-1)$ and abelian Higgs models is exhibited. Large $N$ techniques also allow solving self-interacting fermion models. A relation between the Gross--Neveu, a theory with a four-fermi self-interaction, and a Yukawa-type theory renormalizable in four dimensions then follows. We discuss dissipative dynamics, which is relevant to the approach to equilibrium, and which in some formulation exhibits quantum mechanics supersymmetry. This also serves as an introduction to the study of the 3D supersymmetric quantum field theory. Large $N$ methods are useful in problems that involve a crossover between different dimensions. We thus briefly discuss finite size effects, finite temperature scalar and supersymmetric field theories. We also use large $N$ methods to investigate the weakly interacting Bose gas. The solution of the general scalar $U(\phib^2)$ field theory is then applied to other issues like tricritical behaviour and double scaling limit.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 9,568 input tokens, $0.0004 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.31 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We review the solutions of O(N) and U(N) quantum field theories in the large $N$ limit and as 1/N expansions, in the case of vector representations. Since invariant composite fields have small fluctuations for large $N$, the method relies on constructing effective field theories for composite fields after integration over the original degrees of freedom. We first solve a general scalar $U(\phib^2)$ field theory for $N$ large and discuss various non-perturbative physical issues such as critical behaviour. We show how large $N$ results can also be obtained from variational calculations.We illustrate these ideas by showing that the large $N$ expansion allows to relate the $(\phib^2)^2$ theory and the non-linear $\sigma$-model, models which are renormalizable in different dimensions. Similarly, a relation between $CP(N-1)$ and abelian Higgs models is exhibited. Large $N$ techniques also allow solving self-interacting fermion models. A relation between the Gross--Neveu, a theory with a four-fermi self-interaction, and a Yukawa-type theory renormalizable in four dimensions then follows. We discuss dissipative dynamics, which is relevant to the approach to equilibrium, and which in some formulation exhibits quantum mechanics supersymmetry. This also serves as an introduction to the study of the 3D supersymmetric quantum field theory. Large $N$ methods are useful in problems that involve a crossover between different dimensions. We thus briefly discuss finite size effects, finite temperature scalar and supersymmetric field theories. We also use large $N$ methods to investigate the weakly interacting Bose gas. The solution of the general scalar $U(\phib^2)$ field theory is then applied to other issues like tricritical behaviour and double scaling limit.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We review the solutions of O(N) and U(N) quantum field theories in the large $N$ limit and as 1/N expansions, in the case of vector representations.	contribution 0.92	0.62	0.15	asserted 0.91
s1	Since invariant composite fields have small fluctuations for large $N$, the method relies on constructing effective field theories for composite fields after integration over the original degrees of freedom.	method 0.96	0.14	0.11	asserted 1.00
s2	We first solve a general scalar $U(\phib^2)$ field theory for $N$ large and discuss various non-perturbative physical issues such as critical behaviour.	contribution 0.88	0.85	0.07	asserted 0.99
s3	We show how large $N$ results can also be obtained from variational calculations.We illustrate these ideas by showing that the large $N$ expansion allows to relate the $(\phib^2)^2$ theory and the non-linear $\sigma$-model, models which are renormalizable in different dimensions.	contribution 0.40	0.89	0.12	asserted 1.00
s4	Similarly, a relation between $CP(N-1)$ and abelian Higgs models is exhibited.	result 0.81	0.81	0.04	asserted 1.00
s5	Large $N$ techniques also allow solving self-interacting fermion models.	contribution 0.59	0.42	0.05	asserted 0.81
s6	A relation between the Gross--Neveu, a theory with a four-fermi self-interaction, and a Yukawa-type theory renormalizable in four dimensions then follows.	result 0.88	0.76	0.06	asserted 0.99
s7	We discuss dissipative dynamics, which is relevant to the approach to equilibrium, and which in some formulation exhibits quantum mechanics supersymmetry.	contribution 0.88	0.42	0.04	asserted 0.72
s8	This also serves as an introduction to the study of the 3D supersymmetric quantum field theory.	contribution 0.39	0.36	0.03	asserted 0.94
s9	Large $N$ methods are useful in problems that involve a crossover between different dimensions.	contribution 0.32	0.26	0.06	asserted 0.87
s10	We thus briefly discuss finite size effects, finite temperature scalar and supersymmetric field theories.	contribution 0.98	0.69	0.04	asserted 0.93
s11	We also use large $N$ methods to investigate the weakly interacting Bose gas.	contribution 0.93	0.78	0.04	asserted 0.89
s12	The solution of the general scalar $U(\phib^2)$ field theory is then applied to other issues like tricritical behaviour and double scaling limit.	contribution 0.85	0.76	0.07	asserted 1.00

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.49.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 6.** Here µ − µc = 1 (sets the mass scale) and u/uc= varies between 1.6 and 1.2. There are two degenerate O(N) symmetric SUSY vacua with ϕ = 0 at mψ = mϕ = M+ = (µ − µc)/(u/uc + 1) and at mψ = mϕ = −M− = (µ − µc)/(u/uc − 1). 0 < u < uc and µ > µc, the system has a non-degenerate O(N) symmetric ground state with bosons and fermions of mass M = M+. As µ − µc changes sign (0 < u < uc fixed), two degenerate ground states appear. Either M = 0 and ϕ 2 = −N(µ−µc)/u or the system stays in an O(N) symmetric ground state

**Figure 9.** Region IV : The energy density W(m, ϕ) = 1 N E(m, ϕ) as given in Eq. (6.49) as a function of the boson mass (m) and A, where A 2 = ϕ 2 /uc. Here µ−µc = −1 , u/uc = 0.2. As seen here there are two distinct degenerate phases. One is an ordered phase (ϕ 6= 0) with a massless boson and fermion, the other is a symmetric phase (ϕ = 0) with a massive (m = |M−|) boson and fermion. with a mass |M−| for the bosons and fermions. Similarly, when one goes from µ < µc to µ > µc at u > uc = 4π the O(N) symmetry is restored but there are two degenerated ground states to choose from M = M±. In

Showing 3 of the paper's 9 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-th/0306133 by the authors.

References

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from the “improved” energy momentum tensor ˜Tµν. The divergences of these currents are proportional to the trace of the energy mo mentum tensor ˜Tµµ . They are: Sµ = xλ ˜Tµλ and Kµν = x2 ˜Tµν − 2xµxλ ˜Tνλ. (A1.1) There are two reasonably well understood mechanisms for bre aking these sym- metries in a theory which is scale and conformal invariant at tree ...

View on Pith

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Pith read 14 of the 312 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool.

Uses its method or tool 2 of 21 papers with this label

We will now calculate the partition function and hence free energy of this interacting Bosonic system in thermal equilibrium, using a large $N$ approximation [ref] .

Renyi Entropy of Interacting Thermal Bosons in Large $N$ Approximation (2020) · $𝒰(N)$ model and large-N field theory · 0.96
Following this paper's analysis for the Gross-Neveu-Yukawa model, integrated out only (N-1) fermions to obtain the fermionic contribution to the action.

Transition between algebraic and $\mathbb{Z}_2$ quantum spin liquids at large $N$ (2018) · 0.94 · paraphrased: the licence does not allow quoting

How the literature uses it

138 arXiv papers cite this in 217 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 105 sentences without a confident label

Uses its method or tool 11 papers

Renormalization of the saddle point condition then can be achieved by the non-perturbative renormalization conditions [ref]

What if $\phi^4$ theory in 4 dimensions is non-trivial in the continuum? · Calculation · 0.90
[ref] We first introduce an auxiliary field $α(x)$ to rewrite the action with source as [equation] In the large $N$ limit, let $Δm_0^2$ be the saddle point value of $i α$, which is determined by [equation] $ϕ^N = √(N) v$ gives [equation] Eqs. ([ref]) and ([ref]) are Eqs. (2.7) and (2.8) of Ref. [ref] .

Analytic form of the effective potential in the large N limit of a real scalar theory in four dimensions · The large $N$ limit of the $O(N)$ linear sigma model · 0.90
More specifically, we study quenches of scalar fields on a de Sitter spacetime using the non-perturbative large-$N$ approximation [ref] .

Non-linear effects in early Universe cosmology · Section · 1.00

Uses its data or values 3 papers

In $d=2+$, the Gross-Neveu model has a UV fixed point [ref] ; g^* =2πN , note that here $N=2 Ñ$, where the extra $2$ accounts for the dimension of spinors.

On Instantons in Gross-Neveu and Gross-Neveu-Yukawa models · Instanton solutions · 0.97
We also list the best available BMW results [ref] and some results obtained by other methods [ref] that can be considered the `world's best' values, as well as experimental values for the Ising model [ref] .

Effect of the quartic gradient terms on the critical exponents of the Wilson-Fisher fixed point in $O(N)$ models · Short overview on the present status of the determination of the critical exponents of the WF FP in the functional RG fr · 0.90
While they are available as functions of $d$ [ref] , we will just state them for the physically interesting case $d=3$: Δ_ϕ_i=1/2+ 4/3 π^2 N- 256/27 π^4 N^2+ … , Δ_ϕ^2 = 2 - 32 /3 π^2 N + 32 (16- 27 π^2)/27 π^4 N^2+ … .

TASI Lectures on Large $N$ Tensor Models · Some Applications of the Vector and Matrix Large $N$ Limits · 0.90

Compares with it 4 papers

These results are in agreement with results in [ref] .

$\epsilon$-Expansion in the Gross-Neveu Model from Conformal Field Theory · Anomalous dimension of $ψ$, $ψ̅$ and $ψ̅ψ$ · 0.99
[equation] which again agrees with the general analysis presented in <ref>, upto some normalization and contact terms which can be attributed different regulation used in [ref] .

Revisiting AdS/CFT at a finite radial cut-off · Large $N$ limit of $O(N)$ Wilson-Fisher model · 0.99
Nonetheless, the theory exhibits a remarkable similarity with the well-known $O(N)$ vector model [ref] that we elaborate on in more detail below, and offers a new perspective on the nascent NN-QFT correspondence that may be fruitfully developed further.

The edge of chaos: quantum field theory and deep neural networks · Introduction · 0.89

Cites it as support 2 papers

The explicit solution of this model in the large $N$ limit [ref] or via integrability [ref] confirms that the theory is gapped and no SSB occurs.

Spontaneous symmetry breaking on surface defects · Examples · 0.99
The existence of a critical transition for finite values of $ε$, and in particular in three dimensions, and for sufficiently large values of $N$ is confirmed by nonperturbative (with respect to the couplings) computations in the large-$N$ limit [ref] .

Three-dimensional Abelian and non-Abelian gauge Higgs theories · Three-dimensional Abelian Higgs field theory in the large-$N$ limit · 0.97

Cites it as background 75 papers

We consider the supersymmetric action [ref] for the Wess-Zumino model in $(2+1)d$ in the fundamental of $O(2N)$, with $2N$ real scalars and $2N$ real fermions.

Large $N$ Wess-Zumino model at finite temperature and large chemical potential in $3d$ · Wess-Zumino model with chemical potential($μ$) and large $μ$ · 0.81
In this section we review the relevant pieces in the large $N$ analysis of the $O(N)$ model as described in [ref] (see also the review [ref] and more recently [ref] ), which will also serve to establish some notation.

Notes on Resonances and Unitarity from Celestial Amplitudes · Effective Description of the O(N) Model · 0.98
The Lagrangian is [ref] [equation] where $ℛ$ is the Ricci scalar.

The large charge expansion at large N · $$ model · 0.93

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Some sections are directly inspired from J. Zinn-Justin , lectures given at 11th Taiwan Spring School , Taipei 1997, hep-th/9810198

View on Pith Pith review arXiv 1997

[2] [2]

For a general background with analogous notation see J. Z inn-Justin, Quan- tum Field Theory and Critical Phenomena , Clarendon Press (Oxford 1989, fourth ed. 2002)

View on Pith 1989

[3] [3]

As shown by H.E. Stanley, Phys. Rev. 176 (1968) 718, the large N -limit of the classical N -vector model coincides with the spherical model solved in ref. [4]

View on Pith 1968

[4] [4]

T.H. Berlin and M. Kac, Phys. Rev. 86 (1952) 821

View on Pith 1952

[5] [5]

The modern formulation of the RG ideas is due to: K.G. Wilson, Phys. Rev. B4 (1971) 3174, ibidem 3184; and presented in an expanded form in [6]

View on Pith 1971

[6] [6]

K.G. Wilson and J. Kogut, Phys. Rep. 12C (1974) 75

View on Pith 1974

[7] [7]

The idea of the ε-expansion is due to K.G. Wilson and M.E. Fisher, Phys. Rev. Lett. 28 (1972) 240

View on Pith 1972

[8] [8]

Early work on calculating critical properties for large N includes R. Abe, Prog. Theor. Phys. 48 (1972) 1414; 49 (1973) 113, 1074, 1877 and refs. [9–12]

View on Pith 1972

[9] [9]

S.K. Ma, Phys. Rev. Lett. 29 (1972) 1311 ; Phys. Rev. A7 (1973) 2172

View on Pith 1972

[10] [10]

M. Suzuki, Phys. Lett. 42A (1972) 5; Prog. Theor. Phys. 49 (1973) 424, 1106, 1440

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[11] [11]

R.A. Ferrel and D.J. Scalapino, Phys. Rev. Lett. 29 (1972) 413

View on Pith 1972

[12] [12]

K.G. Wilson, Phys. Rev. D7 (1973) 2911

View on Pith 1973

[13] [13]

The spin–spin correlation in zero ﬁeld is obtained in M.E. Fisher and A. Aharony, Phys. Rev. Lett. 31 (1973) 1238 and refs. [14,15]

View on Pith 1973

[14] [14]

A. Aharony, Phys. Rev. B10 (1974) 2834

View on Pith 1974

[15] [15]

R. Abe and S. Hikami, Prog. Theor. Phys. 51 (1974) 1041

View on Pith 1974

[16] [16]

The contribution of order 1 /N to the equation of state is given in E. Br´ ezin and D.J. Wallace,Phys. Rev. B7 (1973) 1967

View on Pith 1973

[17] [17]

The exponent ω has been calculated to order 1 /N by S.K. Ma, Phys. Rev. A10 (1974) 1818

View on Pith 1974

[18] [18]

See also the review of S.K. Ma in Phase Transitions and Critical Phenomena vol. 6, C. Domb and M.S. Green eds. (Academic Press, London 19 76)

View on Pith

[19] [19]

The study of the large N limit by the steepest descent method is explained in 177 E. Br´ ezin, J.C. Le Guillou and J. Zinn-Justin, contributio n to Phase Tran- sitions and Critical Phenomena vol. 6, C. Domb and M.S. Green eds. (Aca- demic Press, London 1976). It is applied to more general scal ar ﬁeld theories in [20]

View on Pith 1976

[20] [20]

M.B. Halpern, Nucl. Phys. B173 (1980) 504

View on Pith 1980

[21] [21]

For a Hartree–Fock variational approach to large N theories and large N QFT at ﬁnite temperature see W.A. Bardeen and M. Moshe, Phys. Rev. D28 (1983) 1372 and Phys. Rev. D34 (1986) 1229

View on Pith 1983

[22] [22]

The non-linear σ-model is discussed in the spirit of this review in E. Br´ ezin and J. Zinn-Justin, Phys. Rev. Lett. 36 (1976) 691; Phys. Rev. B14 (1976) 3110

View on Pith 1976

[23] [23]

Lattice calculations of the non-linear σ model with the large N expansion are reported in M. Campostrini, P. Rossi, Phys. Lett. B242 (1990) 81 and in [24]

View on Pith 1990

[24] [24]

P. Biscari, M. Campostrini, P. Rossi, Phys. Lett. B242 (1990) 225

View on Pith 1990

[25] [25]

The CP (N − 1) model is discussed in two dimensions with the large N expansion in M. L¨ uscher,Phys. Lett. 78B (1978) 465 and in refs. [26–29]

View on Pith 1978

[26] [26]

A. D’Adda, P. Di Vecchia and M. L¨ uscher, Nucl. Phys. B146 (1978) 63; B152 (1979) 125

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