We compute the thermodynamic properties of the Sachdev-Ye-Kitaev (SYK) models of fermions with a conserved fermion number, Q. We extend a previously proposed Schwarzian effective action to include a phase field, and this describes the low temperature energy and Q fluctuations. We obtain higherdimensional generalizations of the SYK models which display disordered metallic states without quasiparticle excitations, and we deduce their thermoelectric transport coefficients. We also examine the corresponding properties of Einstein-Maxwell-axion theories on black brane geometries which interpolate from either AdS 4 or AdS 5 to an AdS 2 × R 2 or AdS 2 × R 3 near-horizon geometry. These provide holographic descriptions of non-quasiparticle metallic states without momentum conservation. We find a precise match between low temperature transport and thermodynamics of the SYK and holographic models. In both models the Seebeck transport coefficient is exactly equal to the Q-derivative of the entropy. For the SYK models, quantum chaos, as characterized by the butterfly velocity and the Lyapunov rate, universally determines the thermal diffusivity, but not the charge diffusivity.

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 numerical studies of fermion and boson models with random all-to-all interactions (the SYK models). The high temperature expansion and exact diagonalization of the N -site fermion model are used to compute the entropy density: our results are consistent with the numerical solution of N = ∞ saddle point equations, and the presence of a non-zero entropy density in the limit of vanishing temperature.The exact diagonalization results for the fermion Green's function also appear to converge well to the N = ∞ solution. For the hard-core boson model, the exact diagonalization study indicates spin glass order. Some results on the entanglement entropy and the out-of-time-order correlators are also presented.

We study the nonequilibrium dynamics of an electronic model of competition between an unconventional charge density wave (a bond density wave) and $d$-wave superconductivity. In a time-dependent Hartree-Fock+BCS approximation, the dynamics reduces to the equations of motion of operators realizing the generators of SU(4) at each pair of momenta, ( $\boldsymbol{k}$, - $\boldsymbol{k}$ ), in the Brillouin zone. We also study the nonequilibrium dynamics of a quantum generalization of a O(6) nonlinear $\sigma$ model of competing orders in the underdoped cuprates [Hayward et al., Science $\boldsymbol{343}$, 1336 (2014)]. We obtain results, in the large $N$ limit of a O($N$) model, on the time dependence of correlation functions following a pulse disturbance. We compare our numerical studies with recent picosecond optical experiments. We find that, generically, the oscillatory responses in our models share various qualitative features with the experiments.Comment: 39 pages, 33 figures; v2: includes analysis previously submitted as arXiv:1401.7674. v3: published versio

No abstract