In this paper, we consider the complete momentum-independent quartic order truncation for the effective average action of a real Abelian rank-3 tensorial group field theory. This complete truncation includes nonmelonic as well as double-trace interactions. In the usual functional renormalization group perspective, the inclusion of more operators that belong to the underlying theory space corresponds to an improvement of the truncation of the effective average action. We show that the inclusion of nonmelonic and double-trace operators in the truncation brings subtleties. In particular, we discuss the assignment of scaling dimensions to the nonmelonic sector and how the inclusion of double-trace operators considerably changes the results for critical exponents with respect to those obtained when they are not included. We argue that this is not a particular problem of the present model by comparing the results with a pure tensor model. We discuss how these issues should be investigated in future work.
In this paper, we consider the complete momentum-independent quartic order truncation for the effective average action of a real Abelian rank-3 tensorial group field theory. This complete truncation includes nonmelonic as well as double-trace interactions. In the usual functional renormalization group perspective, the inclusion of more operators that belong to the underlying theory space corresponds to an improvement of the truncation of the effective average action. We show that the inclusion of nonmelonic and double-trace operators in the truncation brings subtleties. In particular, we discuss the assignment of scaling dimensions to the nonmelonic sector and how the inclusion of double-trace operators considerably changes the results for critical exponents with respect to those obtained when they are not included. We argue that this is not a particular problem of the present model by comparing the results with a pure tensor model. We discuss how these issues should be investigated in future work.
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.
We present a truncation scheme of the effective-average-action approach of the nonperturbative renormalization group that allows for an accurate description of the critical regime as well as of correlation functions at finite momenta. The truncation is a natural modification of the standard derivative expansion that includes both all local correlations and two-point and four-point irreducible correlations to all orders in the derivatives. We discuss schemes for both the symmetric and the symmetry broken phase of the O(N) model and present results for D=3. All approximations are done directly in the effective average action rather than in the flow equations of irreducible vertices. The approach is numerically relatively easy to implement and yields good results for all N both for the critical exponents and for the momentum dependence of the two-point function.
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 consider the Wetterich exact renormalization group (RG) equation. Approximate closed equations are obtained from it, applying certain truncation schemes for the effective average action. These equations are solved either purely numerically or by certain extra truncations for the potential and related quantities, called the functional truncations. Traditionally, the functional truncations consist of truncated expansions in powers of ρ ˜ − ρ ˜ 0 , where ρ ˜ ∝ ϕ 2 , φ is the averaged order parameter, and ρ ˜ 0 corresponds to the minimum of the dimensionless potential u k ( ρ ˜ ) , depending on the infrared cut-off scale k . We propose a new approach of functional truncations, using the expansion u k ( ρ ˜ ) − u k ( 0 ) = ( 1 − s ) − μ u 1 , k s + u 2 , k s 2 + ⋯ , where s = ρ ˜ / ( ρ ˜ 0 + ρ ˜ ) , ρ ˜ 0 is an optimization parameter and µ is the exponent, describing the ρ ˜ → ∞ asymptotic. The newly developed method provides accurate estimates of the critical exponents η , ν and als
The standard nonperturbative approaches of renormalization group for tensor models are generally focused on a purely local potential approximation (i.e., involving only generalized traces and products of them) and are showed to strongly violate the modified Ward identities. This paper is a continuation of our recent contribution [Phys. Rev. D 101, 106015 (2020), intended to investigate the approximation schemes compatible with Ward identities and constraints between 2n-points observables in the large N limit. We consider separately two different approximations: In the first one, we try to construct a local potential approximation from a slight modification of the Litim regulator, so that it remains optimal in the usual sense, and preserves the boundary conditions in deep UV and deep IR limits. In the second one, we introduce derivative couplings in the truncations and show that the compatibility with Ward identities implies strong relations between $\ensuremath{\beta}$ functions, allowing one to close the infinite hierarchy of flow equations in the nonbranching sector, up to a given order in the derivative expansion. Finally, using an exact relation between correlations functions in large N limit, we show that strictly local truncations are insufficient to reach the exact value for the critical exponent, highlighting the role played by these strong relations between observables taking into account the behavior of the flow; and the role played by the multitrace operators, discussed in the two different approximation schemes. In both cases, we compare our conclusions to the results obtained in the literature and conclude that, at a given order, by taking into account the exact functional relations between observables like Ward identities in a systematic way, we can strongly improve the physical relevance of the approximation for an exact renormalization group equation.
Continuous phase transitions provide a natural setting for understanding universality. Near criticality, the correlation length diverges, microscopic details become increasingly irrelevant, and physical observables obey scaling laws characterized by universal critical exponents. Within the Functional Renormalization Group (FRG), this scale dependence is described through the Effective Average Action, which progressively incorporates fluctuations as the momentum scale is lowered. A first part of this thesis is developed within the Local Potential Approximation (LPA), where we study multicritical fixed points of scalar field theories. Their universal properties are compared with results obtained at higher orders in the derivative expansion. Particular attention is devoted to the relation between global fixed-point potentials and their local representations through finite polynomial truncations. We show that the convergence of such expansions is constrained by the analytic structure of the global solution, and that expanding around a suitable nonzero stationary point significantly improves the reconstruction of multicritical branches. Beyond the structure of individual fixed points, a central part of this thesis concerns RG irreversibility. In two dimensions, Zamolodchikov’s c-theorem describes the flow from the ultraviolet to the infrared through a monotonically decreasing c-function. Within the FRG framework, we develop a variational procedure to reconstruct RG trajectories between fixed points and integrate the corresponding c-function along them. The resulting quantity displays monotonic behaviour and becomes increasingly stable as the truncation order is raised, providing a manifestation of RG irreversibility within the approximation considered. Finally, close to the upper critical dimensions, the linearized FRG flow around the Gaussian fixed point is naturally organized in terms of Hermite polynomials. Using the epsilon-expansion, we derive a curvature-based candidate c-function and obtain an analytic explanation for the alternating curvature of successive multicritical fixed points and for their progressive accumulation around the Gaussian fixed point.