A Complex Fermionic Tensor Model in $d$ Dimensions

Abstract

In this note, we study a melonic tensor model in $d$ dimensions based on three-index Dirac fermions with a four-fermion interaction. Summing the melonic diagrams at strong coupling allows one to define a formal large-$N$ saddle point in arbitrary $d$ and calculate the spectrum of scalar bilinear singlet operators. For $d=2-\epsilon$ the theory is an infrared fixed point, which we find has a purely real spectrum that we determine numerically for arbitrary $d<2$, and analytically as a power series in $\epsilon$. The theory appears to be weakly interacting when $\epsilon$ is small, suggesting that fermionic tensor models in 1-dimension can be studied in an $\epsilon$ expansion. For $d>2$, the spectrum can still be calculated using the saddle point equations, which may define a formal large-$N$ ultraviolet fixed point analogous to the Gross-Neveu model in $d>2$. For $2<d<6$, we find that the spectrum contains at least one complex scalar eigenvalue (similar to the complex eigenvalue present in the bosonic tensor model recently studied by Giombi, Klebanov and Tarnopolsky) which indicates that the theory is unstable. We also find that the fixed point is weakly-interacting when $d=6$ (or more generally $d=4n+2$) and has a real spectrum for $6<d<6.14$ which we present as a power series in $\epsilon$ in $6+\epsilon$ dimensions.

28 pages, 11 figures; v2: added references

References in corpus (11)

• Uncolored Random Tensors, Melon Diagrams, and the SYK Models
• Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model
• Random tensor models in the large N limit: Uncoloring the colored tensor models
• More on Supersymmetric and 2d Analogs of the SYK Model
• Higher Spins in AdS and Twistorial Holography
• Three Dimensional View of the SYK/AdS Duality
• Bosonic Tensor Models at Large $N$ and Small $\epsilon$
• M-theory branes and their interactions
• Tensorial Gross-Neveu models
• Towards a Finite-$N$ Hologram
• Anomalous dimensions of leading twist conformal operators

Cited by in corpus (28)

• An introduction to the SYK model
• Space-Time in the SYK Model
• Prismatic Large $N$ Models for Bosonic Tensors
• 2PI effective action for the SYK model and tensor field theories
• Thouless time for mass-deformed SYK
• Symmetry Breaking in Coupled SYK or Tensor Models
• Phases Of Melonic Quantum Mechanics
• Supersymmetric Tensor Model at Large $N$ and Small $\epsilon$
• Line of fixed points in a bosonic tensor model
• Supersymmetric SYK Model with Global Symmetry
• On large $q$ expansion in the Sachdev-Ye-Kitaev model
• Large $N$ limit of irreducible tensor models: $O(N)$ rank-$3$ tensors with mixed permutation symmetry
• Phase diagram and fixed points of tensorial Gross-Neveu models in three dimensions
• On Melonic Supertensor Models
• Exact Solution of a Strongly Coupled Gauge Theory in 0+1 Dimensions
• Melonic Dominance in Subchromatic Sextic Tensor Models
• The tri-fundamental quartic model
• The F-theorem in the melonic limit
• Hints of unitarity at large $N$ in the $O(N{)}^3$ tensor field theory
• Conformal Symmetry and Composite Operators in the $O(N{)}^3$ Tensor Field Theory
• Remarks on a melonic field theory with cubic interaction
• $SO(3)$-invariant phase of the $O(N{)}^3$ tensor model
• Spectrum of a Gross-Neveu Yukawa model with flavor disorder in $d=3$
• Instability of complex CFTs with operators in the principal series
• Three-point functions of conserved currents in 3D CFT: general formalism for arbitrary spins
• A Renormalizable SYK-type Tensor Field Theory
• Sextic tensor model in rank $3$ at next-to-leading order
• Coupling renormalization flow in the strongly interacting regime of an asymptotically free quantum field theory in four dimensions