We study the quantum mechanics of three-index Majorana fermions ψ^{abc} governed by a quartic Hamiltonian with O(N)^{3} symmetry. Similarly to the Sachdev-Ye-Kitaev model, this tensor model has a solvable large-N limit dominated by the melonic diagrams. For N=4 the total number of states is 2^{32}, but they naturally break up into distinct sectors according to the charges under the U(1)×U(1) Cartan subgroup of one of the O(4) groups. The biggest sector has vanishing charges and contains over 165 million states. Using a Lanczos algorithm, we determine the spectrum of the low-lying states in this and other sectors. We find that the absolute ground state is nondegenerate. If the SO(4)^{3} symmetry is gauged, it is known from earlier work that the model has 36 states and a residual discrete symmetry. We study the discrete symmetry group in detail; it gives rise to degeneracies of some of the gauge singlet energies. We find all the gauge singlet energies numerically and use the results to propose exact analytic expressions for them.
We study the quantum mechanics of three-index Majorana fermions ψ^{abc} governed by a quartic Hamiltonian with O(N)^{3} symmetry. Similarly to the Sachdev-Ye-Kitaev model, this tensor model has a solvable large-N limit dominated by the melonic diagrams. For N=4 the total number of states is 2^{32}, but they naturally break up into distinct sectors according to the charges under the U(1)×U(1) Cartan subgroup of one of the O(4) groups. The biggest sector has vanishing charges and contains over 165 million states. Using a Lanczos algorithm, we determine the spectrum of the low-lying states in this and other sectors. We find that the absolute ground state is nondegenerate. If the SO(4)^{3} symmetry is gauged, it is known from earlier work that the model has 36 states and a residual discrete symmetry. We study the discrete symmetry group in detail; it gives rise to degeneracies of some of the gauge singlet energies. We find all the gauge singlet energies numerically and use the results to propose exact analytic expressions for them.
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.
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.
Certain models with rank-3 tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large N limit, where g2N3 is held fixed. In this limit the perturbative expansion in the quartic coupling constant, g, is dominated by a special class of ``melon'' diagrams. We study ``uncolored'' models of this type, which contain a single copy of real rank-3 tensor. Its three indices are distinguishable; therefore, the models possess O(N)3 symmetry with the tensor field transforming in the tri-fundamental representation. Such uncolored models also possess the large N limit dominated by the melon diagrams. The quantum mechanics of a real anticommuting tensor therefore has a similar large N limit to the model recently introduced by Witten as an implementation of the Sachdev-Ye-Kitaev (SYK) model which does not require disorder. Gauging the O(N)3 symmetry in our quantum mechanical model removes the nonsinglet states; therefore, one can search for its well-defined gravity dual. We point out, however, that the model possesses a vast number of gauge-invariant operators involving higher powers of the tensor field, suggesting that the complete gravity dual will be intricate. We also discuss the quantum mechanics of a complex 3-index anticommuting tensor, which has $U(N{)}^{2}\ifmmode\times\else\texttimes\fi{}O(N)$ symmetry and argue that it is equivalent in the large N limit to a version of SYK model with complex fermions. Finally, we discuss similar models of a commuting tensor in dimension d. While the quartic interaction is not positive definite, we construct the large N Schwinger-Dyson equation for the two-point function and show that its solution is consistent with conformal invariance. We carry out a perturbative check of this result using the $4\ensuremath{-}\ensuremath{\epsilon}$ expansion.
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.
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 Majorana fermions in the fundamental representation. Its quartic Hamiltonian depends on a real parameter β. We derive the kernels of the four point functions. With the spectra that we find from the kernels, we calculate the scaling dimensions of several types of conformal primaries. We also find a duality relation between two Hamiltonians of different values of β. This is not a perfect duality, because the normalization of energy scales with the transformation. Nevertheless, the ratios of the energies are the same, and the operator dimensions are preserved. In addition, we discover that for β>1 or β<0 the scaling dimensions of one of the conformal primaries become complex, rendering the model unstable.
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev (SYK) model. These are quantum mechanical models involving N Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The N=1 model with a single supercharge has unbroken supersymmetry at large N, but nonperturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large N equation, and also by performing numerical computations for small values of N. We also compute the large N spectrum of ``singlet'' operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an N=2 version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large N computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.