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arXiv:1912.11104

Another proof of the $1/N$ expansion of the rank three tensor model with tetrahedral interaction

Valentin Bonzom

Published Dec 23, 2019• 2 citations
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Abstract

The rank three tensor model with tetrahedral interaction was shown by Carrozza and Tanasa to admit a $1/N$ expansion, dominated by melonic diagrams, and double tadpoles decorated with melons at next-to-leading order. This model has generated a renewed interest in tensor models because it has the same large $N$ limit as the SYK model. In contrast with matrix models, there is no method which would be able to prove the existence of $1/N$ expansions in arbitrary tensor models. The method used by Carrozza and Tanasa proves the existence of the $1/N$ expansion using two-dimensional topology, before identifying the leading order and next-to-leading graphs. However, another method was required for complex, rank three tensor models with planar interactions, which is based on flips. The latter are moves which cut two propagators of Feynman graphs and reglue them differently. They allow transforming graphs while tracking their orders in the $1/N$ expansion. Here we use this method to re-prove the results of Carrozza and Tanasa, thereby proving the existence of the $1/N$ expansion, the melonic dominance at leading order and the melon-decorated double tadpoles at next-to-leading order, all in one go.

DOI
10.48550/arxiv.1912.11104

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Another proof of the 1/N1/N1/N expansion of the rank three tensor model with tetrahedral interaction
The rank three tensor model with tetrahedral interaction was shown by Carrozza and Tanasa to admit a 1/N1/N1/N expansion, dominated by melonic diagrams, and double tadpoles decorated with melons at next-to-leading order. This model has generated a renewed interest in tensor models because it has the same large NNN limit as the SYK model. In contrast with matrix models, there is no method which would be able to prove the existence of 1/N1/N1/N expansions in arbitrary tensor models. The method used by Carrozza and Tanasa proves the existence of the 1/N1/N1/N expansion using two-dimensional topology, before identifying the leading order and next-to-leading graphs. However, another method was required for complex, rank three tensor models with planar interactions, which is based on flips. The latter are moves which cut two propagators of Feynman graphs and reglue them differently. They allow transforming graphs while tracking their orders in the 1/N1/N1/N expansion. Here we use this method to re-prove the results of Carrozza and Tanasa, thereby proving the existence of the 1/N1/N1/N expansion, the melonic dominance at leading order and the melon-decorated double tadpoles at next-to-leading order, all in one go.
Abstract We demonstrate that random tensors transforming under rank-5 irreducible representations of O(N)\mathrm {O}(N)O(N) O ( N ) can support melonic large N expansions. Our construction is based on models with sextic (5-simplex) interaction, which generalize previously studied rank-3 models with quartic (tetrahedral) interaction (Benedetti et al. in Commun Math Phys 371:55, 2019. arXiv:1712.00249 ; Carrozza in JHEP 06:039, 2018. arXiv:1803.02496 ). Beyond the irreducible character of the representations, our proof relies on recursive bounds derived from a detailed combinatorial analysis of the Feynman graphs. Our results provide further evidence that the melonic limit is a universal feature of irreducible tensor models in arbitrary rank.
This thesis focuses on renormalization of tensor field theories. Its first part considers a quartic tensor model with O(N)3O(N)^3O(N)3 symmetry and long-range propagator. The existence of a non-perturbative fixed point in any ddd at large NNN 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 NNN. We then compute 1/N1/N1/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 FFF-theorem. The next part of the thesis investigates sextic tensor field theories in rank 333 and 555. In rank 333, 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 555 is the Gaussian one. For the rank 333 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 NNN fixed point. The last part of the thesis investigates the class of model exhibiting a melonic large NNN limit. We prove that models with tensors in an irreducible representation of O(N)O(N)O(N) or Sp(N)Sp(N)Sp(N) in rank 555 indeed admit a large NNN limit. This generalization relies on recursive bounds derived from a detailed combinatorial analysis of Feynman graphs involved in the perturbative expansion of our model.
The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large NNN limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large NNN limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of O(N)O(N)O(N), O(N)2O(N)^2O(N)2, or O(N)3O(N)^3O(N)3 symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large NNN limits. In the latter example we review how the dominance of melonic diagrams follows when a special "tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large NNN limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(N)3O(N)^3O(N)3 symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with SU(N)2×O(N)×U(1)SU(N)^2\times O(N)\times U(1)SU(N)2×O(N)×U(1) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large NNN tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.
The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large NNN limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large NNN limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of O(N)O(N)O(N), O(N)2O(N)^2O(N)2, or O(N)3O(N)^3O(N)3 symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large NNN limits. In the latter example we review how the dominance of melonic diagrams follows when a special ``tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large NNN limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(N)3O(N)^3O(N)3 symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with SU(N)2×O(N)×U(1)SU(N)^2\times O(N)\times U(1)SU(N)2×O(N)×U(1) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large NNN tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.
Tensor models are natural generalizations of matrix models. The interactions and observables in the case of unitary invariant models are generalizations of matrix traces. Some notable interactions in the literature include the melonic ones, the tetrahedral one as well as the planar ones in rank three, or necklaces in even ranks. Here we introduce generalized melonic interactions which generalize the melonic and necklace interactions. We characterize them as tree-like gluings of quartic interactions. We also completely characterize the Feynman graphs which contribute to the large NNN limit. For a subclass of generalized melonic interactions called totally unbalanced interactions, we prove that the large NNN limit is Gaussian and therefore the Feynman graphs are in bijection with trees. This result further extends the class of tensor models which fall into the Gaussian universality class. Another key aspect of tensor models with generalized melonic interactions is that they can be written as matrix models without increasing the number of degrees of freedom of the original tensor models. In the case of totally unbalanced interactions, this new matrix model formulation in fact decreases the number of degrees of freedom, meaning that some of the original degrees of freedom are effectively integrated. We then show how the large NNN Gaussian behavior can be reproduced using a saddle point analysis on those matrix models.
A SYK–like model close to the colored tensor models has recently been proposed [1]. Building on results obtained in tensor models [2], we discuss the complete 1/N expansion of the model. We detail the two and four point functions at leading order. The leading order two point function is a sum over melonic graphs, and the leading order relevant four point functions are sums over dressed ladder diagrams. We then show that any order in the 1/N series of the two point function can be written solely in term of the leading order two and four point functions. The full 1/N expansion of arbitrary correlations can be obtained by similar methods.