As shown in [1], two copies of the large N Majorana SYK model can produce spontaneous breaking of a Z2 symmetry when they are coupled by appropriate quartic terms. In this paper we similarly study two copies of the complex SYK model coupled by a quartic term preserving the U(1) x U(1) symmetry. We also present a tensor counterpart of this coupled model. When the coefficient α of the quartic term lies in a certain range, the coupled large N theory is nearly conformal. We calculate the scaling dimensions of fermion bilinear operators as functions of α. We show that the operator c$^†_{1i}$c2i, which is charged under the axial U(1) symmetry, acquires a complex dimension outside of the line of fixed points. We derive the large N Dyson-Schwinger equations and show that, outside the fixed line, this U(1) symmetry is spontaneously broken at low temperatures because this operator acquires an expectation value. We support these findings by exact diagonalizations extrapolated to large N.
As shown in [1], two copies of the large N Majorana SYK model can produce spontaneous breaking of a Z2 symmetry when they are coupled by appropriate quartic terms. In this paper we similarly study two copies of the complex SYK model coupled by a quartic term preserving the U(1) x U(1) symmetry. We also present a tensor counterpart of this coupled model. When the coefficient α of the quartic term lies in a certain range, the coupled large N theory is nearly conformal. We calculate the scaling dimensions of fermion bilinear operators as functions of α. We show that the operator c1i†c2i, which is charged under the axial U(1) symmetry, acquires a complex dimension outside of the line of fixed points. We derive the large N Dyson-Schwinger equations and show that, outside the fixed line, this U(1) symmetry is spontaneously broken at low temperatures because this operator acquires an expectation value. We support these findings by exact diagonalizations extrapolated to large N.
Abstract As shown in [1], two copies of the large N Majorana SYK model can produce spontaneous breaking of a Z2 symmetry when they are coupled by appropriate quartic terms. In this paper we similarly study two copies of the complex SYK model coupled by a quartic term preserving the U(1) × U(1) symmetry. We also present a tensor counterpart of this coupled model. When the coefficient α of the quartic term lies in a certain range, the coupled large N theory is nearly conformal. We calculate the scaling dimensions of fermion bilinear operators as functions of α. We show that the operator c1i†c2i c 1 i † c 2 i , which is charged under the axial U(1) symmetry, acquires a complex dimension outside of the line of fixed points. We derive the large N Dyson-Schwinger equations and show that, outside the fixed line, this U(1) symmetry is spontaneously broken at low temperatures because this operator acquires an expectation value. We support these findings by exact diagonalizations extrapolated to large N.
We study a 1+1 (space-time)-dimensional extension of the 0+1-dimensional Sachdev-Ye-Kitaev model for N Majorana fermions, with random all-to-all quartic interactions, averaged over disorder. At large interaction couplings and large N, the conformal symmetry of the effective action emerges, which is not broken spontaneously as in the original 0+1 SYK model. Two-point correlators are obtained from a coupling expansion of the Schwinger-Dyson equations. For N=4, the model can be mapped onto complex fermions and solved exactly via the bosonization technique, featuring two branches of excitations: a gapped ``pseudocharge'' mode and a gapless ``pseudospin'' mode. We give an approximate analytic form of the two-point correlators at large distances and zero temperature for N=4, which is adopted heuristically to evaluate an approximation to the large-N free energy in the zero temperature limit, numerically. The fact that this energy displays an absolute minimum at a finite value of the gap, though in a restricted range of parameters, suggests that a gapped branch of excitations is also present in our extension of the model at least in that range of parameters.
A bstract Motivated by SYK-like models describing near-BPS black holes in string/M-theory, we consider gauging the U(1) symmetry of the complex SYK model in the presence of a Wilson line with charge k . At a fixed background gauge field, solutions to the Schwinger-Dyson equations display vastly different properties from those at a fixed real chemical potential. In the partition function and the two-point function, the integral over the gauge field is performed either directly or via a large N saddle point approximation, and both results are consistent with exact diagonalization data. From the behaviour of the two-point function at large N , we deduce that the conformal symmetry at low energies is preserved at fixed κ = k / N = 0, but broken at κ ≠ 0. In addition, we find that there is maximal chaos for all k .
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.
The chiral SYK model [Lian The chiral SYK model, ] is a 1 + 1 -dimensional generalization of the Sachdev-Ye-Kitaev model with chiral Majorana fermions and homogeneous random interactions. In the large- N limit, the model admits an exact solution of the two-point function due to its scaling symmetry and exhibits a quantized thermal Hall conductance consistent with that of a 2 + 1 -dimensional gapped topological system. We study two chiral SYK systems coupled by a relevant quadratic interaction that explicitly breaks scaling and time-reversal symmetry. Working in the regime of weak intersystem coupling, we solve the Dyson-Schwinger equations perturbatively and obtain analytic expressions for two-point functions at finite temperature. Unlike the coupled SYK model in 0 + 1 dimensions, the 1 + 1 -dimensional chiral system does not develop a mass gap, and no thermal phase transition is observed. We show that the leading correction to the thermodynamic free energy is temperature independent, implying that the entropy density remains identical to that of two uncoupled chiral SYK systems. A real-time analysis of the retarded correlator reveals the emergence of massless collective bosonic modes propagating between the two subsystems at zero temperature, signaling quasiparticle tunneling without gap generation. Our results demonstrate a sharp qualitative distinction between relevant deformations of SYK models in zero and one spatial dimensions and highlight the robustness of gapless chiral edge dynamics against explicit scaling symmetry-breaking interactions.
In this thesis, we discuss the Sachdev-Ye-Kitaev (SYK) model and tensor models with similar properties. The SYK model is a quantum field theoretical model describing N interacting fermions, whose coupling constants are drawn from a Gaussian ensemble. Noteworthy properties of the SYK model include that it is analytically solvable in the large N limit, that it exhibits conformal symmetry at low energies and that it is maximally chaotic. These properties are remarkably similar to those of a 1 + 1 dimensional Schwarzschild black hole. It has been conjectured the SYK model is a holographic dual to the black hole. We introduce a set of Feynman rules for the SYK model. Using these rules, we show that in the large N limit the diagrams that contribute to the two-point function are all so-called iterated melonic diagrams. This allows us to derive a Schwinger-Dyson equation for the two-point function, which, in turn, can be solved exactly in the infrared limit. We also consider the four-point function. In the large N limit, the leading-order correction to the four-point function is given by so-called ladder diagrams. This allows us to derive an explicit expression for the four-point function. The SYK model can be generalized in a few different ways. In this thesis, we consider the generalization where the fermions act through q-fold interactions instead of quartic interactions present in the original SYK model. In particular, considerable simplifications can be achieved in the q → ∞ limit or q = 2 case, which we study. While the SYK model has many interesting properties, its random couplings limit its usability especially as a dual to a Schwarzschild black hole. We therefore also consider tensor models which do not have this drawback but manage to preserve the interesting properties of the SYK model. In the last chapter, we briefly inspect the chaotic behaviour of the SYK and tensor models and derive Lyapunov exponent for them. It can be shown that the expression saturates an upper bound for Lyapunov exponents of a large class of quantum systems, including large N systems.