We introduce a family of tensor quantum-mechanical models based on irreducible rank-3 representations of Sp(N). In contrast to irreducible tensor models with 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 U(N)×O(N)×U(N) tensor quantum mechanics, the main difference being that the states are now singlets under Sp(N). We introduce character formulas that enumerate such singlets as a function of N, and compute their first values. We conclude with an explicit numerical diagonalization of the Hamiltonian in two simple examples: the symmetric model at N=1, and the antisymmetric traceless model at N=3.
We introduce a family of tensor quantum-mechanical models based on irreducible rank-3 representations of Sp(N). In contrast to irreducible tensor models with 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 U(N)×O(N)×U(N) tensor quantum mechanics, the main difference being that the states are now singlets under Sp(N). We introduce character formulas that enumerate such singlets as a function of N, and compute their first values. We conclude with an explicit numerical diagonalization of the Hamiltonian in two simple examples: the symmetric model at N=1, and the antisymmetric traceless model at N=3.
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 Sachdev-Ye-Kitaev (SYK) model with real fermions; it includes the h=2 zero mode. For the operators antisymmetric in the two indices, the scaling dimensions are the same as in the additional sector found in the complex tensor and SYK models; the lowest h=0 eigenvalue corresponds to the conserved O(N) charges. A class of singlet operators may be constructed from contracted combinations of m symmetric traceless or antisymmetric two-particle operators. Their two-point functions receive contributions from m melonic ladders. Such multiple ladders are a new phenomenon in the tensor model, which does not seem to be present in the SYK model. The more typical 2k-particle operators do not receive any ladder corrections and have quantized large N scaling dimensions k/2. We construct pictorial representations of various singlet operators with low k. For larger k, we use available techniques to count the operators and show that their number grows as 2kk!. As a consequence, the theory has a Hagedorn phase transition at the temperature which approaches zero in the large N limit. We also study the large N spectrum of low-lying operators in the Gurau-Witten model, which has O(N)6 symmetry. We argue that it corresponds to one of the generalized SYK models constructed by Gross and Rosenhaus. Our paper also includes studies of the invariants in large N tensor integrals with various symmetries.
We study quantum mechanical models in which the dynamical degrees of freedom are real fermionic tensors of rank 5 and higher. They are the nonrandom counterparts of the Sachdev-Ye-Kitaev (SYK) models where the Hamiltonian couples six or more fermions. For the tensors of rank 5, there is a unique O(N)5 symmetric sixth-order Hamiltonian leading to a solvable $\mathrm{large}\text{\ensuremath{-}}N$ 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 exhibits degeneracies related to the discrete symmetries of the gauged model. We also study quantum chaos properties of the tensor model and compare them with those of the q=6 SYK model. For q>6, there is a rapidly growing number of $O(N{)}^{q\ensuremath{-}1}$ invariant tensor interactions. We focus on those of them that are maximally single trace---their stranded diagrams stay connected when any set of $q\ensuremath{-}3$ colors is erased. We present a general discussion of why the tensor models with maximally single-trace interactions have $\mathrm{large}\text{\ensuremath{-}}N$ limits dominated by the melonic diagrams. We solve the $\mathrm{large}\text{\ensuremath{-}}N$ Schwinger-Dyson equations for the higher rank Majorana tensor models and show that they match those of the corresponding SYK models exactly. We also study other gauge invariant operators present in the tensor models.
We study the $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})\ifmmode\times\else\texttimes\fi{}O({N}_{3})$ symmetric quantum mechanics of 3-index Majorana fermions. When the ranks Ni are all equal, this model has a large N limit which is dominated by the melonic Feynman diagrams. We derive an integral formula which computes the number of group invariant states for any set of Ni. It is non-vanishing only when each Ni is even. For equal ranks the number of singlets exhibits rapid growth with N: it jumps from 36 in the O(4)3 model to 595 354 780 in the O(6)3 model. We derive bounds on the values of energy, which show that they scale at most as N3 in the large N limit, in agreement with expectations. We also show that the splitting between the lowest singlet and non-singlet states is of order 1/N. For N3=1 the tensor model reduces to $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})$ fermionic matrix quantum mechanics, and we find a simple expression for the Hamiltonian in terms of the quadratic Casimir operators of the symmetry group. A similar expression is derived for the complex matrix model with $SU({N}_{1})\ifmmode\times\else\texttimes\fi{}SU({N}_{2})\ifmmode\times\else\texttimes\fi{}U(1)$ symmetry. Finally, we study the N3=2 case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})\ifmmode\times\else\texttimes\fi{}U(1)$. All energies are again integers in appropriate units, and we derive a concise formula for the spectrum. The fermionic matrix models we studied possess standard 't Hooft large N limits where the ground state energies are of order N2, while the energy gaps are of order 1.
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 solving the vector, matrix, and tensor large N 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 N limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(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) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large N 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 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 solving the vector, matrix, and tensor large N 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 N limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(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) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large N tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.
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 of the paper, we contrast the random matrix aspects of some ungauged tensor models, the original SYK model, and a model due to Gross and Rosenhaus. The latter, even though disorder averaged, shows parallels with the Gurau-Witten model. In particular, the two models fall into identical Andreev ensembles as a function of N . In an appendix, we contrast the (expected) spectra of AdS2 quantum gravity, SYK and SYK-like tensor models, and the zeros of the Riemann Zeta function.