The complex Sachdev-Ye-Kitaev (cSYK) model is a charge-conserving model of randomly interacting fermions. The interaction term can be chosen such that the model exhibits chiral symmetry. Then, depending on the charge sector and the number of interacting fermions, level spacing statistics suggests a fourfold categorization of the model into the three Wigner-Dyson symmetry classes. In this work, inspired by previous findings for the Majorana Sachdev-Ye-Kitaev model, we embed the symmetry classes of the cSYK model in the Altland-Zirnbauer framework and identify consequences of chiral symmetry originating from correlations across different charge sectors. In particular, we show that for an odd number of fermions, the model hosts exact many-body zero modes that can be combined into a generalized fermion that does not affect the system's energy. This fermion directly leads to quantum-mechanical supersymmetry that, unlike explicitly supersymmetric cSYK constructions, does not require fine-tuned couplings, but only chiral symmetry. Signatures of the generalized fermion, and thus supersymmetry, include the long-time plateau in time-dependent correlation functions of fermion-parity-odd observables: The plateau may take nonzero value only for certain combinations of the fermion structure of the observable and the system's symmetry class. We illustrate our findings through exact diagonalization simulations of the system's dynamics.
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.
The Sachdev-Ye-Kitaev (SYK) model, in its simplest form, describes
k Majorana fermions with random all-to-all four-body interactions. We consider the SYK model in the framework of a many-body Altland-Zirnbauer classification that sees the system as belonging to one of eight (real) symmetry classes depending on the value of
kmod8. We show that, depending on the symmetry class, the system may support exact many-body zero modes with the symmetries also dictating whether these may have a nonzero contribution to Majorana fermions, i.e., single-particle weight. These zero modes appear in all but two of the symmetry classes. When present, they leave clear signatures in physical observables that go beyond the threefold (Wigner-Dyson) possibilities for level spacing statistics studied earlier. Signatures we discover include a zero-energy peak or hole in the single-particle spectral function, depending on whether symmetries allow or forbid zero modes to have single-particle weight. The zero modes are also shown to influence the many-body dynamics, where signatures include a nonzero long-time limit for the out-of-time-order correlation function. Furthermore, we show that the extension of the four-body SYK model by quadratic terms can be interpreted as realizing the remaining two complex symmetry classes; we thus demonstrate how the entire tenfold Altland-Zirnbauer classification may emerge in the SYK model.
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.
We present a complete symmetry classification of the Sachdev-Ye-Kitaev (SYK) model with
N = 0, 1 and 2 supersymmetry (SUSY) on the basis of the Altland-Zirnbauer scheme in random matrix theory (RMT). For
N = 0 and 1 we consider generic q-body interactions in the Hamiltonian and find RMT classes that were not present in earlier classifications of the same model with q = 4. We numerically establish quantitative agreement between the distributions of the smallest energy levels in the
N = 1 SYK model and RMT. Furthermore, we delineate the distinctive structure of the
N = 2 SYK model and provide its complete symmetry classification based on RMT for all eigenspaces of the fermion number operator. We corroborate our classification by detailed numerical comparisons with RMT and thus establish the presence of quantum chaotic dynamics in the
N =2 SYK model. We also introduce a new SYK-like model without SUSY that exhibits hybrid properties of the
N = 1 and
N = 2 SYK models and uncover its rich structure both analytically and numerically.
We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model,
N Majorana fermions with infinite range interactions in
0+1 dimensions. We have found that, close to the ground state $E\ensuremath{\approx}0$, discrete symmetries alter qualitatively the spectral properties with respect to the non-supersymmetric SYK model. The average spectral density at finite
N, which we compute analytically and numerically, grows exponentially with
N for $E\ensuremath{\approx}0$. However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above $E\ensuremath{\approx}0$, the spectral density grows exponentially with the energy. Deep in the quantum regime, corresponding to the first
O(N) eigenvalues, the average spectral density is universal and well described by random matrix ensembles with chiral and superconducting discrete symmetries. The dynamics for $E\ensuremath{\approx}0$ is investigated by level fluctuations. Also in this case we find excellent agreement with the prediction of chiral and superconducting random matrix ensembles for eigenvalue separations smaller than the Thouless energy, which seems to scale linearly with
N. Deviations beyond the Thouless energy, which describes how ergodicity is approached, are universally characterized by a quadratic growth of the number variance. In the time domain, we have found analytically that the spectral form factor
g(t), obtained from the connected two-level correlation function of the unfolded spectrum, decays as
1/t2 for times shorter but comparable to the Thouless time with
g(0) related to the coefficient of the quadratic growth of the number variance. Our results provide further support that quantum black holes are ergodic and therefore can be classified by random matrix theory.