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.
We study the Lindbladian dynamics of the Sachdev-Ye-Kitaev (SYK) model, where the SYK model is coupled to Markovian reservoirs with jump operators that are either linear or quadratic in the Majorana fermion operators. Here, the linear jump operators are nonrandom, while the quadratic jump operators are sampled from a Gaussian distribution. In the limit of large
N, where
N is the number of Majorana fermion operators, and also in the limit of large
N and
M, where
M is the number of jump operators, the SYK Lindbladians are analytically tractable, and we obtain their stationary Green's functions, from which we can read off the decay rate. For finite
N, we also study the distribution of the eigenvalues of the SYK Lindbladians.
We study the original Sachdev-Ye (SY) model in its Majorana fermion representation which can be called the two indices Sachdev-Ye-Kitaev (SYK) model. Its advantage over the original SY model in the $ SU(M) $ complex fermion representation is that it need no local constraints, so a $1/M $ expansion can be more easily performed. Its advantage over the 4 indices SYK model is that it has only two site indices $ J_{ij} $ instead of four indices $ J_{ijkl} $, so it may fit the bulk string theory better. By performing a $1/M $ expansion at $ N=\infty $, we show that a quantum spin liquid (QSL) state remains stable at a finite $ M $. The $ 1/M $ corrections are exactly marginal, so the system remains conformably invariant at any finite $ M $. The 4-point out of time correlation ( OTOC ) shows quantum chaos neither at $ N=\infty $ at any finite $ M $, nor at $ M=\infty $ at any finite $ N $. By looking at the replica off-diagonal channel, we find there is a quantum spin glass (QSG) instability at an exponentially suppressed temperature in $ M $. We work out a criterion for the two large numbers $ N $ and $ M $ to satisfy so that the QSG instability may be avoided. We speculate that at any finite $ N $, the quantum chaos appears at the order of $ 1/M^{0} $, which is the subleading order in the $ 1/M $ expansion. When the $ 1/N $ quantum fluctuations at any finite $ M $ are considered, from a general reparametrization symmetry breaking point of view, we argue that the eThis work may motivate future works to study the possible new gravity dual of the 2 indices SYK model.ffective action should still be described by the Schwarzian one, the OTOC shows maximal quantum chaos.
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev 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 non-perturbatively 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.
Through a mixture of analytic and numerical techniques, we explore the optimal approximation by a free Majorana state to individual disorder realizations of the Sachdev-Ye-Kitaev model, along with a generalization of it. We elucidate the properties of the known time-reversal symmetry breaking phase in the generalized model, finding strong evidence of "spin glass" order. For the Sachdev-Ye-Kitaev model itself, our results are inconclusive but suggest a similar order may be present at zero temperature.
The authors study in detail the quantum mechanical model of
N Majorana fermions with random interactions of a few fermions at a time (Sachdev-Ye-Kitaev model) in the large
N limit. At low energies, the system is strongly interacting and an emergent conformal symmetry develops. Performing technical calculations, the authors elucidate a number of properties of the model near the conformal point.