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Register a new userLarge $N_f$ quantum field theory
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J.A. Gracey
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We review the development of the large $N$ method, where $N$ indicates the number of flavours, used to study perturbative and nonperturbative properties of quantum field theories. The relevant historical background is summarized as a prelude to the introduction of the large $N$ critical point formalism. This is used to compute large $N$ corrections to $d$-dimensional critical exponents of the universal quantum field theory present at the Wilson-Fisher fixed point. While pedagogical in part the application to gauge theories is also covered and the use of the large $N$ method to complement explicit high order perturbative computations in gauge theories is also highlighted. The usefulness of the technique in relation to other methods currently used to study quantum field theories in $d$-dimensions is also summarized.
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We consider Quantum Electrodynamics in $2{+}1$ dimensions with $N_f$ fermionic or bosonic flavors, allowing for interactions that respect the global symmetry $U(N_f/2)^2$. There are four bosonic and four fermionic fixed points, which we analyze using the large $N_f$ expansion. We systematically compute, at order $O(1/N_f)$, the scaling dimensions of quadratic and quartic mesonic operators. We also consider Quantum Electrodynamics with minimal supersymmetry. In this case the large $N_f$ scaling dimensions, extrapolated at $N_f{=}2$, agree quite well with the scaling dimensions of a dual supersymmetric Gross-Neveu-Yukawa model. This provides a quantitative check of the conjectured duality.
We study $mathcal{N}=1$ supersymmetric three-dimensional Quantum Electrodynamics with $N_f$ two-component fermions. Due to the infra-red (IR) softening of the photon, $ep$-scalar and photino propagators, the theory flows to an interacting fixed point deep in the IR, $p_E ll e^2 N_f/8$, where $p_E$ is the euclidean momentum and $e$ the electric charge. At next-to-leading order in the $1/N_f$-expansion, we find that the flow of the dimensionless effective coupling constant $overline{al}$ is such that: $overline{al} ra 8/big(N_f ,(1+C/N_f)big) approx (8/N_f)(1-0.4317/N_f)$ where $C= 2,(12-pi^2)/pi^2$. Hence, the non-trivial IR fixed point is stable with respect to quantum corrections. Various properties of the theory are explored and related via a mapping to the ones of a $mathcal{N}=1$ model of super-graphene. In particular, we derive the interaction correction coefficient to the optical conductivity of super-graphene, $C_{rm sg} = (12-pi^2)/(2pi) = 0.3391$, which is six times larger than in the non-supersymmetric case, $C_{rm g} = (92-9pi^2)/(18pi) = 0.0561$.
We study the emergence of Nambu-Goldstone modes due to broken translation symmetry in field theory. Purely spontaneous breaking yields a massless phonon which develops a mass upon introducing a perturbative explicit breaking. The pseudo-phonon mass agrees with Gell Mann-Oakes-Renner relations. We analyze the simplest possible theories featuring gradient Mexican hats and describing space-dependent order parameters. We comment on homogeneous translation breaking and the connections with holographic Q-lattices.
I derive an exact integral expression for the ratio of shear viscosity over entropy density $frac{eta}{s}$ for the massless (critical) O(N) model at large N with quartic interactions. The calculation is set up and performed entirely from the field theory side using a non-perturbative resummation scheme that captures all contributions to leading order in large N. In 2+1d, $frac{eta}{s}$ is evaluated numerically at all values of the coupling. For infinite coupling, I find $frac{eta}{s}simeq 0.42(1)times N$. I show that this strong coupling result for the viscosity is universal for a large class of interacting bosonic O(N) models.
Applying recursive renormalization group transformations to a scalar field theory, we obtain an effective quantum gravity theory with an emergent extra dimension, described by a dual holographic Einstein-Klein-Gordon type action. Here, the dynamics of both the dual order-parameter field and the metric tensor field originate from density-density and energy-momentum tensor-tensor effective interactions, respectively, in the recursive renormalization group transformation, performed approximately in the Gaussian level. This linear approximation in the recursive renormalization group transformation for the gravity sector gives rise to a linearized quantum Einstein-scalar theory along the $z-$directional emergent space. In the large $N$ limit, where $N$ is the flavor number of the original scalar fields, quantum fluctuations of both dynamical metric and dual scalar fields are suppressed, leading to a classical field theory of the Einstein-scalar type in $(D+1)$-spacetime dimensions. We show that this emergent background gravity describes the renormalization group flows of coupling functions in the UV quantum field theory through the extra dimension. More precisely, the IR boundary conditions of the gravity equations correspond to the renormalization group $beta$-functions of the quantum field theory, where the infinitesimal distance in the extra-dimensional space is identified with an energy scale for the renormalization group transformation. Finally, we also show that this dual holographic formulation describes quantum entanglement in a geometrical way, encoding the transfer of quantum entanglement from quantum matter to classical gravity in the large $N$ limit. We claim that this entanglement transfer serves as a microscopic foundation for the emergent holographic duality description.
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