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arXiv:2603.04481

Quantum field theories with many fields

Ludo Fraser-Taliente

Published Mar 4, 2026• 0 citations
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Abstract

The large-$N$ quantum field theories provide a window into the regime of strongly-coupled physics. Our principal object of study in this thesis is the large-$N$ family of melonic QFTs, which contain the Sachdev-Ye-Kitaev-like models, tensor models, and vector models. We begin with a review of this limit of a large number of degrees of freedom (large-$N$) as an approach to the solution of QFTs. Two toy models are used to clarify this approach: a zero-dimensional field theory and the flow of a generalized free field theory. Both models are solvable, and so we can explicitly demonstrate: using the former, the simplifications at large $N$; using the latter, the tools used to study scale-dependence of physics -- the renormalization group. We develop $\tilde{F}$-extremization, a simple method of solution for an arbitrary large-$N$ melonic QFT in its strongly-coupled limit. The infrared conformal field theories show remarkable simplicity, in that they are entirely solved by the requirement that they have as many degrees of freedom as possible, up to a simple constraint arising from the interaction between the fields. We measure the number of degrees of freedom of the conformal infrared theory via $\tilde{F}$, the universal part of the free energy. We then present the example of the quartic Yukawa model in continuous dimension. This model is considered as a tensor field theory, and solved for its conformal limit; we then illustrate its multiplicity of fixed points and their stability, as well as its operator spectrum, matching the data between the large-$N$ and dimensional expansions. These features reflect general characteristics of melonic conformal field theories: their existence, stability, and spectral characteristics. We conclude with future directions of exploration for the melonic theories.

DOI
10.48550/arxiv.2603.04481

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The large-NNN quantum field theories provide a window into the regime of strongly-coupled physics. Our principal object of study in this thesis is the large-NNN family of melonic QFTs, which contain the Sachdev-Ye-Kitaev-like models, tensor models, and vector models. We begin with a review of this limit of a large number of degrees of freedom (large-NNN) as an approach to the solution of QFTs. Two toy models are used to clarify this approach: a zero-dimensional field theory and the flow of a generalized free field theory. Both models are solvable, and so we can explicitly demonstrate: using the former, the simplifications at large NNN; using the latter, the tools used to study scale-dependence of physics -- the renormalization group. We develop F~\tilde{F}F~-extremization, a simple method of solution for an arbitrary large-NNN melonic QFT in its strongly-coupled limit. The infrared conformal field theories show remarkable simplicity, in that they are entirely solved by the requirement that they have as many degrees of freedom as possible, up to a simple constraint arising from the interaction between the fields. We measure the number of degrees of freedom of the conformal infrared theory via F~\tilde{F}F~, the universal part of the free energy. We then present the example of the quartic Yukawa model in continuous dimension. This model is considered as a tensor field theory, and solved for its conformal limit; we then illustrate its multiplicity of fixed points and their stability, as well as its operator spectrum, matching the data between the large-NNN and dimensional expansions. These features reflect general characteristics of melonic conformal field theories: their existence, stability, and spectral characteristics. We conclude with future directions of exploration for the melonic theories.
This thesis focuses on renormalization of quantum field theories. Its first part considers three tensor models in three dimensions, a Fermionic quartic with tensors of rank-3 and two Bosonic sextic, of ranks 3 and 5. We rely upon the large-NNN melonic expansion of tensor models. For the first model, invariant under U(N)3U(N)^3U(N)3, we obtain the RG flow of the two melonic couplings and the vacuum phase diagram, from a reformulation with a diagonalizable matrix intermediate field. The discrete chiral symmetry breaks spontaneously and we compare with the three-dimensional Gross-Neveu model. Beyond the massless U(N)3U(N)^3U(N)3 symmetric phase, we also observe a massive phase of same symmetry and another where the symmetry breaks into U(N2)×U(N/2)×U(N/2)U(N^2)\times U(N/2)\times U(N/2)U(N2)×U(N/2)×U(N/2). A matrix model invariant under U(N)×U(N2)U(N)\times U(N^2)U(N)×U(N2), with close properties, is also studied. For the other models, with symmetry groups U(N)3U(N)^3U(N)3 and O(N)5O(N)^5O(N)5, a non-melonic coupling (the "wheel") with an optimal scaling in NNN drives us to a generalized melonic expansion. The kinetic terms are taken of short- and long-range, and we analyze perturbatively, at large-NNN, the RG flows of the sextic couplings up to four loops. Only the rank-3 model displays non-trivial fixed points (two real Wilson-Fisher-like in the short-range case and a line of fixed points in the other). We finally obtain the real conformal dimensions of the primary bilinear operators. In the second part, we establish the first results of perturbative multi-scale renormalization for a quartic scalar field on critical Galton-Watson trees, with a long-range kinetic term. At criticality, an emergent infinite spine provides a space of effective dimension 4/34/34/3 on which to compute averaged correlation fonctions. This approach formalizes the notion of a QFT on a random geometry. We use known probabilistic bounds on the heat-kernel on a random graph reviewed in detail.
This thesis focuses on renormalization of tensor field theories. Its first part considers a quartic tensor model with O(N)3O(N)^3O(N)3 symmetry and long-range propagator. The existence of a non-perturbative fixed point in any ddd at large NNN is established. We found four lines of fixed points parametrized by the so-called tetrahedral coupling. One of them is infrared attractive, strongly interacting and gives rise to a new kind of CFT, called melonic CFTs which are then studied in more details. We first compute dimensions of bilinears and OPE coefficients at the fixed point which are consistent with a unitary CFT at large NNN. We then compute 1/N1/N1/N corrections. At next-to-leading order, the line of fixed points collapses to one fixed point. However, the corrections are complex and unitarity is broken at NLO. Finally, we show that this model respects the FFF-theorem. The next part of the thesis investigates sextic tensor field theories in rank 333 and 555. In rank 333, we found two IR stable real fixed points in short range and a line of IR stable real fixed points in long range. Surprisingly, the only fixed point in rank 555 is the Gaussian one. For the rank 333 model, in the short-range case, we still find two IR stable fixed points at NLO. However, in the long-range case, the corrections to the fixed points are non-perturbative and hence unreliable: we found no precursor of the large NNN fixed point. The last part of the thesis investigates the class of model exhibiting a melonic large NNN limit. We prove that models with tensors in an irreducible representation of O(N)O(N)O(N) or Sp(N)Sp(N)Sp(N) in rank 555 indeed admit a large NNN limit. This generalization relies on recursive bounds derived from a detailed combinatorial analysis of Feynman graphs involved in the perturbative expansion of our model.
In this thesis, we discuss the Sachdev-Ye-Kitaev (SYK) model and tensor models with similar properties. The SYK model is a quantum field theoretical model describing N interacting fermions, whose coupling constants are drawn from a Gaussian ensemble. Noteworthy properties of the SYK model include that it is analytically solvable in the large N limit, that it exhibits conformal symmetry at low energies and that it is maximally chaotic. These properties are remarkably similar to those of a 1 + 1 dimensional Schwarzschild black hole. It has been conjectured the SYK model is a holographic dual to the black hole. We introduce a set of Feynman rules for the SYK model. Using these rules, we show that in the large N limit the diagrams that contribute to the two-point function are all so-called iterated melonic diagrams. This allows us to derive a Schwinger-Dyson equation for the two-point function, which, in turn, can be solved exactly in the infrared limit. We also consider the four-point function. In the large N limit, the leading-order correction to the four-point function is given by so-called ladder diagrams. This allows us to derive an explicit expression for the four-point function. The SYK model can be generalized in a few different ways. In this thesis, we consider the generalization where the fermions act through q-fold interactions instead of quartic interactions present in the original SYK model. In particular, considerable simplifications can be achieved in the q → ∞ limit or q = 2 case, which we study. While the SYK model has many interesting properties, its random couplings limit its usability especially as a dual to a Schwarzschild black hole. We therefore also consider tensor models which do not have this drawback but manage to preserve the interesting properties of the SYK model. In the last chapter, we briefly inspect the chaotic behaviour of the SYK and tensor models and derive Lyapunov exponent for them. It can be shown that the expression saturates an upper bound for Lyapunov exponents of a large class of quantum systems, including large N systems.
Quantum Field Theory (QFT) is the language that describes a wide spectrum of physics. However, it is notoriously hard at strong coupling regime. We approach this problem in an old Quantum Mechanical method - keep a finite number of states and diagonalize the Hamiltonian as a finite-size matrix. To study a QFT, we take the Hamiltonian to be the Conformal Field Theory as the Ultraviolet fixed point of the theory's Renormalization Group Flow, deformed by a relevant operator. We use a recent framework known as the Lightcone Conformal Truncation (LCT), where we use conformal basis and lightcone quantization. As an application of the method, we study the two dimensional Supersymmetric (SUSY) Gross-Neveu-Yukawa Model. The model is expected to have a critical point in the universality class of tri-critical Ising model, a massive phase and a massless SUSY-breaking phase. We use the LCT to compute the spectrum and the spectral density of the theory at all couplings and map the entire phase diagram.
The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large NNN limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large NNN limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of O(N)O(N)O(N), O(N)2O(N)^2O(N)2, or O(N)3O(N)^3O(N)3 symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large NNN limits. In the latter example we review how the dominance of melonic diagrams follows when a special ``tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large NNN limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(N)3O(N)^3O(N)3 symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with SU(N)2×O(N)×U(1)SU(N)^2\times O(N)\times U(1)SU(N)2×O(N)×U(1) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large NNN tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.
The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large NNN limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large NNN limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of O(N)O(N)O(N), O(N)2O(N)^2O(N)2, or O(N)3O(N)^3O(N)3 symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large NNN limits. In the latter example we review how the dominance of melonic diagrams follows when a special "tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large NNN limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(N)3O(N)^3O(N)3 symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with SU(N)2×O(N)×U(1)SU(N)^2\times O(N)\times U(1)SU(N)2×O(N)×U(1) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large NNN tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.