Quantum field theories with many fields

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.

Figures (15)

• Figure 1.1: Flow in the infinite-dimensional theory space. A QFT has some UV definition (which may not be a CFT: perhaps a lattice or stringy UV completion); then as we make measurements at lower and lower energy scales $E$ the best description of reality changes, so we have a QFT($E$); the number of effective degrees of freedom of that QFT, encoded by $C$, should decrease. At asymptotically low scales (in the IR) we find a fixed point of the flow, which may be empty -- or it may be a conformal field theory, which has a number of degrees of freedom measured by $\tilde{F}$. In some scenarios, we also find a CFT as we approach the UV.
• Figure 2.1: The propagator \ref{['eq:0dPropUnreno']}, connected four-point \ref{['eq:0dPhi4cUnreno']} and six-point \ref{['eq:phii6c']} functions, and the free energy $F=-\log \ref{['eq:0dphi4PFevaluated']}$ for the unrenormalized$N=1$$\phi^4$ model ($Z_\phi =1$ fixed). All correlators vanish as $\lambda_b\to\infty$, and the free energy diverges (i.e. $Z\to 0$).
• Figure 2.2: The renormalized four-point coupling against the bare coupling $\lambda_b$. That is, here we set $Z_\phi(\lambda)$ and $\lambda_b(\lambda)$ such that $\langle\phi\phi\rangle =1$ and $\langle\phi^4\rangle_c =- \frac{3}{N} \lambda \delta_{(ij}\delta_{kl)}$. We note that for increasing $N$, the four-point function converges very well to the $N=\infty$ result \ref{['eq:lamvsLam0nInfty']}.
• Figure 2.3: Observables of the renormalized $0$d $\phi^4$ model are fixed after we have set $Z_\phi(\lambda)$ and $\lambda_b(\lambda)$ such that $G^{(2)}=1$ and $G^{(4)}_c=-\frac{3\lambda}{N}$. In each case, we show the $N\to\infty$ limits (zero in the six-point function case), as well as the strong coupling limit at $\lambda=\lambda_{\text{max}}(N)$ (the latter makes it clear why each line suddenly ends).
• Figure 2.4: Schematic $4-\epsilon$ expansion (with $N=1$) and large-$N$ expansion for the critical $\mathrm{O}(N)$$\phi^4$ CFT. The vertical axis represents the renormalized coupling Antunes:2022vtbAntunes:2024mfb. The top line is the UV free field theory; the diagonal line is the IR Wilson-Fisher CFT -- RG flow proceeds downwards. The IR coupling strength becomes weak as we take $d \to 4$ or $N\to\infty$: we can expand about each to find perturbative solutions. We find unitary CFTs for integer $(d,N)$, indicated with black dots; the black lines are unitary QFTs. Non-unitary QFT/CFTs exist in any $d,N$. In $4$d the IR and UV CFTs are the same, and for $N=1$ in $2$d (the blue dot) the CFT is the minimal model $\mathcal{M}_{3,4}$.