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 the Journées de physique mathématique Lyon 2019: Random tensors and SYK models.
The four dimensional spherical model is treated as a perturbation of the three dimensional model. An exact resummation of the expansion is possible and the perturbed critical properties are compared with the predictions of the scaling theoretical approach to the problem. Both methods are found to yield the correct dependence on the strength of the perturbation, which can also be found from the exact solution of the model. A more general method of resummation is discussed and applied to the expansion of both the four and five dimensional models. The effects of perturbations which do not alter the critical exponents are also investigated.
We study the spectrum of the large
N quantum field theory of bosonic rank-3 tensors, the quartic interactions of which are such that the perturbative expansion is dominated by the melonic diagrams. We use the Schwinger-Dyson equations to determine the scaling dimensions of the bilinear operators of arbitrary spin. Using the fact that the theory is renormalizable in
d=4, we compare some of these results with the $4\ensuremath{-}\ensuremath{\epsilon}$ expansion, finding perfect agreement. This helps elucidate why the dimension of operator ${\ensuremath{\phi}}^{abc}{\ensuremath{\phi}}^{abc}$ is complex for
d<4: the large
N fixed point in $d=4\ensuremath{-}\ensuremath{\epsilon}$ has complex values of the couplings for some of the
O(N)3 invariant operators. We show that a similar phenomenon holds in the
O(N)2 symmetric theory of a matrix field ${\ensuremath{\phi}}^{ab}$, where the double-trace operator has a complex coupling in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. We also study the spectra of bosonic theories of rank-$q\ensuremath{-}1$ tensors with ${\ensuremath{\phi}}^{q}$ interactions. In dimensions
d>1.93, there is a critical value of
q, above which we have not found any complex scaling dimensions. The critical value is a decreasing function of
d, and it becomes 6 in $d\ensuremath{\approx}2.97$. This raises a possibility that the large
N theory of rank-5 tensors with sextic potential has an IR fixed point which is free of perturbative instabilities for
2.97<d<3. This theory may be studied using renormalized perturbation theory in $d=3\ensuremath{-}\ensuremath{\epsilon}$.
We construct large-
n (spherical) limits for a series of interesting quantum phase transitions in disordered systems, including quantum ferromagnets and spin glasses, superconducting thin films with and without an external magnetic field, the dirty boson problem, the fractional quantum Hall effect, and Nelson's model of flux lines in high-temperature superconductors. The spherical limit always produces a random matrix inversion problem with self-consistency conditions, which then must be solved numerically. We present preliminary results for the dirty boson problem in two spatial dimensions.
The spherical limit of strongly commensurate dirty bosons is studied perturbatively at weak disorder and numerically at strong disorder in two dimensions (2D). We argue that disorder is not perfectly screened by interactions and consequently that the ground state in the effective Anderson localization problem always remains localized. As a result there is only a gapped Mott insulator phase in the theory. Comparisons with other studies and the parallel with disordered fermions in 2D are discussed. We conjecture that while for the physical cases N = 2 (XY) and N = 1 (Ising) the theory should have the ordered phase, it may not for N = 3 (Heisenberg).