In this paper we introduce a simple field theoretic version of the Carrozza-Tanasa-Klebanov-Tarnopolsky (CTKT) "uncolored" holographic tensor model. It gives a more familiar interpretation to the previously abstract modes of the SYK or CTKT models in terms of momenta. We choose for the tensor propagator the usual Fermionic propagator of condensed matter, with a spherical Fermi surface, but keep the CTKT interactions. Hence our field theory can also be considered as an ordinary condensed matter model with a non-local and non-rotational invariant interaction. Using a multiscale analysis we prove that this field theory is just renormalizable to all orders of perturbation theory in the ultraviolet regime.
We introduce a family of tensor quantum-mechanical models based on irreducible rank-$3$ representations of $\mathrm{Sp}(N)$. In contrast to irreducible tensor models with $\mathrm{O}(N)$ symmetry, the fermionic tetrahedral interaction does not vanish and can therefore support a melonic large $N$ limit. The strongly-coupled regime has a very analogous structure as in the complex SYK model or in $\mathrm{U}(N)\times\mathrm{O}(N)\times\mathrm{U}(N)$ tensor quantum mechanics, the main difference being that the states are now singlets under $\mathrm{Sp}(N)$. We introduce character formulas that enu…
We discuss the two-particle irreducible (2PI) effective action for the SYK model and for tensor field theories. For the SYK model the 2PI effective action reproduces the bilocal reformulation of the model without using replicas. In general tensor field theories the 2PI formalism is the only way to obtain a bilocal reformulation of the theory, and as such is a precious instrument for the identification of soft modes and for possible holographic interpretations. We compute the 2PI action for several models, and push it up to fourth order in the $1/N$ expansion for the model proposed by Witten in…
This note fills a gap in the article with title above [1]. We provide the proof of Equation (82) of Lemma 5 in [1] and thereby complete its power counting analysis with a more precise next-to-leading-order estimate.
These notes are a short introduction to the Sachdev-Ye-Kitaev model. We discuss: SYK and tensor models as a new class of large N quantum field theories, the near-conformal invariance in the infrared, the computation of correlation functions, generalizations of SYK, and applications to AdS/CFT and strange metals.
We contrast some aspects of various SYK-like models with large-$N$ melonic behavior. First, we note that ungauged tensor models can exhibit symmetry breaking, even though these are 0+1 dimensional theories. Related to this, we show that when gauged, some of them admit no singlets, and are anomalous. The uncolored Majorana tensor model with even $N$ is a simple case where gauge singlets can exist in the spectrum. We outline a strategy for solving for the singlet spectrum, taking advantage of the results in arXiv:1706.05364, and reproduce the singlet states expected in $N=2$. In the second part …
The SYK model proposed by Sachdev, Ye, and Kitaev consists of Majorana fermions that interact randomly four at a time. The model develops a dense spectrum above the ground state, due to which the model becomes nearly conformal. This suggests that a holographic dual may exist, which makes the SYK model interesting in the study of quantum gravity. It has been found that the SYK model is similar to large $N$ tensor models: in both models, only the melonic diagrams survive in the large $N$ limit. In this thesis, we explore the large $N$ tensor model with $O(N)^3$ symmetry containing two flavors of…
We study a large $N$ tensor model with $O(N)^3$ symmetry containing two flavors of Majorana fermions, $ψ_1^{abc}$ and $ψ_2^{abc}$. We also study its random counterpart consisting of two coupled Sachdev-Ye-Kitaev models, each one containing $N_{\rm SYK}$ Majorana fermions. In these models we assume tetrahedral quartic Hamiltonians which depend on a real coupling parameter $α$. We find a duality relation between two Hamiltonians with different values of $α$, which allows us to restrict the model to the range of $-1\leq α\leq 1/3$. The scaling dimension of the fermion number operator $Q=iψ_1^{abc…
The Sachdev-Ye-Kitaev (SYK) model is a model of $q$ interacting fermions whose large N limit is dominated by melonic graphs. In this review we first present a diagrammatic proof of that result by direct, combinatorial analysis of its Feynman graphs. Gross and Rosenhaus have then proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, these flavors can be seen as reminiscent of the colors used in random tensor theory. Applying modern tools from random tensors to such a colored SYK model, all leading and next-to-leading orders diagra…
The Nambu-Goldstone (NG) bosons of the SYK model are described by a coset space Diff/$\mathbb{SL}(2,\mathbb{R})$, where Diff, or Virasoro group, is the group of diffeomorphisms of the time coordinate valued on the real line or a circle. It is known that the coadjoint orbit action of Diff naturally turns out to be the two-dimensional quantum gravity action of Polyakov without cosmological constant, in a certain gauge, in an asymptotically flat spacetime. Motivated by this observation, we explore Polyakov action with cosmological constant and boundary terms, and study the possibility of such a t…
In this paper, we derive the energy momentum tensor for the translation invariant noncommutative Tanasa scalar field model. The Wilson regularization procedure is used to improve this tensor and the local conservation property is recovered. The same question is addressed in the case where the Moyal star product is deformed including the tetrad fields. It provides with an extension of the recent work [J. Phys. A: Math. Theor. 43 (2010) 155202], regarding the computation and properties of the Noether currents to the renormalizable models.
We review the fast developing subject of tensor models for the NAdS$_2$/NCFT$_1$ holographic correspondence. We include a brief review of the Sachdev-Ye-Kitaev (SYK) model and then focus on the associated quantum mechanical tensor models (GW and CTKT). We examine their main features and how they compare with SYK. To end, we discuss different extensions: the large $D$ limit of matrix-tensor models, the large $N$ expansion of symmetric/antisymmetric tensors, the use of probes, the construction of a bilocal action for tensors, some attempts to extend the above models to higher dimensions and a pr…
We study the effect of non-Gaussian average over the random couplings in a complex version of the celebrated Sachdev-Ye-Kitaev (SYK) model. Using a Polchinski-like equation and random tensor Gaussian universality, we show that the effect of this non-Gaussian averaging leads to a modification of the variance of the Gaussian distribution of couplings at leading order in N. We then derive the form of the effective action to all orders. An explicit computation of the modification of the variance in the case of a quartic perturbation is performed for both the complex SYK model mentioned above and t…
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 SY…
Various tensor models have been recently shown to have the same properties as the celebrated Sachdev-Ye-Kitaev (SYK) model. In this paper we study in detail the diagrammatics of two such SYK-like tensor models: the multi-orientable (MO) model which has an $U(N) \times O(N) \times U(N)$ symmetry and a quartic $O(N)^3$-invariant model whose interaction has the tetrahedral pattern. We show that the Feynman graphs of the MO model can be seen as the Feynman graphs of the $O(N)^3$-invariant model which have an orientable jacket. We then present a diagrammatic toolbox to analyze the $O(N)^3$-invarian…
Proposals are made to describe 1D, N = 4 supersymmetrical systems that extend SYK models by compactifying from 4D, N = 1 supersymmetric Lagrangians involving chiral, vector, and tensor supermultiplets. Quartic fermionic vertices are generated via integrals over the whole superspace, while 2(q-1)-point fermionic vertices are generated via superpotentials. The coupling constants in the superfield Lagrangians are arbitrary, and can be chosen to be Gaussian random. In that case, these 1D, N = 4 supersymmetric SYK models would exhibit Wishart-Laguerre randomness, which share the same feature among …
It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group $O(N)^{q-1}$ agrees with the large $N$ limit of the SYK model. In these notes we investigate aspects of the dynamics of the $O(N)^{q-1}$ theories that differ from their SYK counterparts. We argue that the spectrum of fluctuations about the finite temperature saddle point in these theories has $(q-1)\frac{N^2}{2}$ new light modes in addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories differ signif…
We compute the phase diagram of a $\text{U}(N)^{2}\times\text{O}(D)$ invariant fermionic planar matrix quantum mechanics [equivalently tensor or complex Sachdev-Ye-Kitaev (SYK) models] in the new large $D$ limit, dominated by melonic graphs. The Schwinger-Dyson equations can have two solutions describing either a high entropy, SYK black-hole-like phase, or a low entropy one with trivial IR behavior. In the strongly coupled region of the mass-temperature plane, there is a line of first order phase transitions between the high and low entropy phases. This line terminates at a new critical point …
We study quantum mechanical models in which the dynamical degrees of freedom are real fermionic tensors of rank five and higher. They are the non-random counterparts of the Sachdev-Ye-Kitaev (SYK) models where the Hamiltonian couples six or more fermions. For the tensors of rank five, there is a unique $O(N)^5$ symmetric sixth-order Hamiltonian leading to a solvable large $N$ limit dominated by the melonic diagrams. We solve for the complete energy spectrum of this model when $N=2$ and deduce exact expressions for all the eigenvalues. The subset of states which are gauge invariant exhibit dege…
The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large $N$ 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 $N$ limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of $O(N)$, $O(N)^2$, or $O(N)^3$ symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for so…
Tensor models and tensor field theories admit a $1/N$ expansion and a melonic large $N$ limit which is simpler than the planar limit of random matrices and richer than the large $N$ limit of vector models. They provide examples of analytically tractable but non trivial strongly coupled quantum field theories and lead to a new class of conformal field theories. We present a compact introduction to the topic, covering both some of the classical results in the field, like the details of the $1/N$ expansion, as well as recent developments. These notes are loosely bases on four lectures given at th…
We study the operators in the large $N$ tensor models, focusing mostly on the fermionic quantum mechanics with $O(N)^3$ symmetry which may be either global or gauged. In the model with global symmetry we study the spectra of bilinear operators, which are in either the symmetric traceless or the antisymmetric representation of one of the $O(N)$ groups. In the symmetric traceless case, the spectrum of scaling dimensions is the same as in the SYK model with real fermions; it includes the $h=2$ zero-mode. For the operators anti-symmetric in the two indices, the scaling dimensions are the same as i…
This paper aims at giving a novel approach to investigate the behavior of the renormalization group flow for tensorial group field theories to all orders of the perturbation theory. From an appropriate choice of the kinetic kernel, we build an infinite family of just-renormalizable models, for tensor fields with arbitrary rank $d$. Investigating the large $ d$ limit, we show that the self-energy melonic amplitude is decomposed as a product of loop-vertex functions, depending only on dimensionless mass. The corresponding melonic amplitudes may be mapped as trees in the so-called Hubbard-Straton…
The SYK model has a wormhole-like solution after averaging over the fermionic couplings in the nearly $AdS_2$ space. Even when the couplings are fixed the contribution of these wormholes continues to exist and new saddle points appear which are interpreted as "half-wormholes". In this paper, we will study the fate of these wormholes in a model without quenched disorder namely a tensor model with $O(N)^{q-1}$ gauge symmetry whose correlation function and thermodynamics in the large $N$ limit are the same as that of the SYK model. We will restate the factorization problem linked with the wormhol…
Various combinatorially non-local field theories are known to be renormalizable. Still, explicit calculations of amplitudes are very rare and restricted to matrix field theory. In this contribution I want to demonstrate how the BPHZ momentum scheme in terms of the Connes-Kreimer Hopf algebra applies to any combinatorially non-local field theory which is renormalizable. This algebraic method improves the understanding of known results in noncommutative field theory in its matrix formulation. Furthermore, I use it to provide new explicit perturbative calculations of amplitudes in tensorial field…
Nucleation of Sachdev-Ye-Kitaev Clusters in One Spatial Dimension
A. Alikhanyan National Science Laboratory（阿·阿里哈尼扬国家科学实验室）；Max-Planck-Institut für Physik komplexer Systeme（马克斯·普朗克复杂系统物理研究所）；Department of Physics, University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校物理系）
We continue the constructive program about tensor field theory through the next natural model, namely the rank five tensor theory with quartic melonic interactions and propagator inverse of the Laplacian on $U(1)^5$. We make a first step towards its construction by establishing its power counting, identifiying the divergent graphs and performing a careful study of (a slight modification of) its RG flow. Thus we give strong evidence that this just renormalizable tensor field theory is non perturbatively asymptotically free.
