Twisted Bilayer graphene (TBG) is known to feature isolated and relatively flat bands near charge neutrality, when tuned to special magic angles. However, different criteria for the magic angle such as the vanishing of Dirac speed, minimal bandwidth or maximal band gap to higher bands typically give different results. Here we study a modified continuum model for TBG which has an infinite sequence of magic angles θ at which, we simultaneously find that (i) the Dirac speed vanishes (ii) absolutely flat bands appear at neutrality and (iii) bandgaps to the excited bands are maximized. When parameterized in terms of α ∼ 1/θ, they recur with the simple periodicity of ∆α ≃ 3/2, which, beyond the first magic angle, differs from earlier calculations. Further, in this model we prove that the vanishing of the Dirac velocity ensures the exact flatness of the band and show that the flat band wave functions are related to doubly-periodic functions composed of ratios of theta functions. Also, using perturbation theory up to α 8 we capture important features of the first magic angle θ ≈ 1.09 • (α ≈ 0.586), which precisely explains the numerical results. Finally, based on our model we discuss the prospects for observing the second magic angle in TBG.

Certain models with rank-3 tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large N limit, where g 2 N 3 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 anti-commuting tensor therefore has a similar large N limit to the model recently introduced by Witten as an implementation of the Sachdev-YeKitaev (SYK) model which does not require disorder. Gauging the O(N ) 3 symmetry in our quantum mechanical model removes the non-singlet 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 anti-commuting tensor, which has U (N ) 2 × 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 − expansion.

We study deformations of three-dimensional large N CFTs by double-trace operators constructed from spin s single-trace operators of dimension ∆. These theories possess UV fixed points, and we calculate the change of the 3-sphere free energy δF = F UV − F IR . To describe the UV fixed point using the dual AdS 4 space we modify the boundary conditions on the spin s field in the bulk; this approach produces δF in agreement with the field theory calculations. If the spin s operator is a conserved current, then the fixed point is described by an induced parity invariant conformal spin s gauge theory. The low spin examples are QED 3 (s = 1) and the 3-d induced conformal gravity (s = 2). When the original CFT is that of N conformal complex scalar or fermion fields, the U (N ) singlet sector of the induced 3-d gauge theory is dual to Vasiliev's theory in AdS 4 with alternate boundary conditions on the spin s massless gauge field. We test this correspondence by calculating the leading term in δF for large N . We show that the coefficient of log N in δF is equal to the number of spin s − 1 gauge parameters that act trivially on the spin s gauge field. We discuss generalizations of these results to 3-d gauge theories including Chern-Simons terms and to theories where s is half-integer. We also argue that the Weyl anomaly a-coefficients of conformal spin s theories in even dimensions d, such as that of the Weyl-squared gravity in d = 4, can be efficiently calculated using massless spin s fields in AdS d+1 with alternate boundary conditions. Using this method we derive a simple formula for the Weyl anomaly a-coefficients of the d = 4 Fradkin-Tseytlin conformal higher-spin gauge fields. Similarly, using alternate boundary conditions in AdS 3 we reproduce the well-known central charge c = −26 of the bc ghosts in 2-d gravity, as well as its higher-spin generalizations.arXiv:1306.5242v3 [hep-th]

We continue the study, initiated in arXiv:1404.1094, of the O(N ) symmetric theory of N + 1 massless scalar fields in 6 − dimensions. This theory has cubic interaction terms. We calculate the 3-loop beta functions for the two couplings and use them to determine certain operator scaling dimensions at the IR stable fixed point up to order 3 . We also use the beta functions to determine the corrections to the critical value of N below which there is no fixed point at real couplings. The result suggests a very significant reduction in the critical value as the dimension is decreased to 5. We also study the theory with N = 1, which has a Z 2 symmetry under φ → −φ. We show that it possesses an IR stable fixed point at imaginary couplings which can be reached by flow from a nearby fixed point describing a pair of N = 0 theories. We calculate certain operator scaling dimensions at the IR fixed point of the N = 1 theory and suggest that, upon continuation to two dimensions, it describes a non-unitary conformal minimal model.

We describe numerous properties of the Sachdev-Ye-Kitaev model for complex fermions with N 1 flavors and a global U(1) charge. We provide a general definition of the charge in the (G, Σ) formalism, and compute its universal relation to the infrared asymmetry of the Green function. The same relation is obtained by a renormalization theory. The conserved charge contributes a compact scalar field to the effective action, from which we derive the many-body density of states and extract the charge compressibility. We compute the latter via three distinct numerical methods and obtain consistent results. Finally, we present a two dimensional bulk picture with free Dirac fermions for the zero temperature entropy.In the infrared (IR), the spectral asymmetry is characterized by the long-time behavior of the Green function G β=∞ (±τ ) ∼ ∓e ±πE τ −2∆ for and τ J −1 , (1.7)or equivalently the small frequency behavior G β=∞ (±iω) ∼ ∓ie ∓iθ ω 2∆−1 for and 0 < ω J , (1.8)