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-
N melonic expansion of tensor models. For the first model, invariant under
U(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)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). A matrix model invariant under
U(N)×U(N2), with close properties, is also studied. For the other models, with symmetry groups
U(N)3 and
O(N)5, a non-melonic coupling (the "wheel") with an optimal scaling in
N drives us to a generalized melonic expansion. The kinetic terms are taken of short- and long-range, and we analyze perturbatively, at large-
N, 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
4/3 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.
Abstract We revisit the Amit-Roginsky (AR) model in the light of recent studies on Sachdev-Ye-Kitaev (SYK) and tensor models, with which it shares some important features. It is a model of N scalar fields transforming in an N-dimensional irreducible representation of SO(3). The most relevant (in renormalization group sense) invariant interaction is cubic in the fields and mediated by a Wigner 3jm symbol. The latter can be viewed as a particular rank-3 tensor coupling, thus highlighting the similarity to the SYK model, in which the tensor coupling is however random and of even rank. As in the SYK and tensor models, in the large-N limit the perturbative expansion is dominated by melonic diagrams. The lack of randomness, and the rapidly growing number of invariants that can be built with n fields, makes the AR model somewhat closer to tensor models. We review the results from the old work of Amit and Roginsky with the hindsight of recent developments, correcting and completing some of their statements, in particular concerning the spectrum of the operator product expansion of two fundamental fields. For 5.74 < d < 6 the fixed-point theory defines a real CFT, while for smaller d complex dimensions appear, after a merging of the lowest dimension with its shadow. We also introduce and study a long-range version of the model, for which the cubic interaction is exactly marginal at large N , and we find a real and unitary CFT for any d < 6, both for real and imaginary coupling constant, up to some critical coupling.
This thesis focuses on renormalization of tensor field theories. Its first part considers a quartic tensor model with
O(N)3 symmetry and long-range propagator. The existence of a non-perturbative fixed point in any
d at large
N 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
N. We then compute
1/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
F-theorem. The next part of the thesis investigates sextic tensor field theories in rank
3 and
5. In rank
3, 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
5 is the Gaussian one. For the rank
3 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
N fixed point. The last part of the thesis investigates the class of model exhibiting a melonic large
N limit. We prove that models with tensors in an irreducible representation of
O(N) or
Sp(N) in rank
5 indeed admit a large
N limit. This generalization relies on recursive bounds derived from a detailed combinatorial analysis of Feynman graphs involved in the perturbative expansion of our model.
We study the spectrum of the large
N quantum field theory of bosonic rank-3 tensors, the quartic interactions of which are such that the perturbative expansion is dominated by the melonic diagrams. We use the Schwinger-Dyson equations to determine the scaling dimensions of the bilinear operators of arbitrary spin. Using the fact that the theory is renormalizable in
d=4, we compare some of these results with the $4\ensuremath{-}\ensuremath{\epsilon}$ expansion, finding perfect agreement. This helps elucidate why the dimension of operator ${\ensuremath{\phi}}^{abc}{\ensuremath{\phi}}^{abc}$ is complex for
d<4: the large
N fixed point in $d=4\ensuremath{-}\ensuremath{\epsilon}$ has complex values of the couplings for some of the
O(N)3 invariant operators. We show that a similar phenomenon holds in the
O(N)2 symmetric theory of a matrix field ${\ensuremath{\phi}}^{ab}$, where the double-trace operator has a complex coupling in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. We also study the spectra of bosonic theories of rank-$q\ensuremath{-}1$ tensors with ${\ensuremath{\phi}}^{q}$ interactions. In dimensions
d>1.93, there is a critical value of
q, above which we have not found any complex scaling dimensions. The critical value is a decreasing function of
d, and it becomes 6 in $d\ensuremath{\approx}2.97$. This raises a possibility that the large
N theory of rank-5 tensors with sextic potential has an IR fixed point which is free of perturbative instabilities for
2.97<d<3. This theory may be studied using renormalized perturbation theory in $d=3\ensuremath{-}\ensuremath{\epsilon}$.
Abstract We propose a method to represent the path integral over gauge fields as a tensor network. We introduce a trial action with variational parameters and generate gauge field configurations with the weight defined by the trial action. We construct initial tensors with indices labelling these gauge field configurations. We perform the tensor renormalization group (TRG) with the initial tensors and optimize the variational parameters. As a first step to the TRG study of non-Abelian gauge theory in more than two dimensions, we apply this method to three-dimensional pure SU(2) gauge theory. Our result for the free energy agrees with the analytical results in the weak and strong coupling regimes.