We describe the non-equilibrium quench dynamics of the Sachdev-Ye-Kitaev models of fermions with random all-to-all interactions. These provide tractable models of the dynamics of quantum systems without quasiparticle excitations. The Kadanoff-Baym equations show that the final state is thermal, and their numerical analysis is consistent with a thermalization rate proportional to the absolute temperature of the final state. We also obtain an exact analytic solution of the quench dynamics in the large q limit of a model with q fermion interactions: in this limit, the thermalization of the fermion Green's function is instantaneous.

We show that a Fermi surface reconstruction due to spiral antiferromagnetic order may explain the rapid change in the Hall number as recently observed near optimal doping in cuprate superconductors [Badoux et al., Nature 531, 210 (2016)]. The single-particle spectral function in the spiral state exhibits hole pockets which look like Fermi arcs due to a strong momentum dependence of the spectral weight. Adding charge-density wave order further reduces the Fermi surface to a single electron pocket. We propose quantum oscillation measurements to distinguish between commensurate and spiral antiferromagnetic order. Similar results apply to certain metals in which topological order replaces antiferromagnetic order. Introduction.-Cuprate superconductors evolve from a Mott insulator to a correlated metal with increasing hole doping p. Long ago it was suggested that this evolution involves a quantum critical point (QCP) near optimal doping, and that the associated fluctuations are responsible for the high critical temperature for superconductivity [1][2][3]. The existence and nature of this QCP has not been clarified yet, because it is masked by superconductivity. Recently, the normal ground state became accessible by suppressing superconductivity with high magnetic fields. Near optimal doping in YBCO, Badoux et al. reported a rapid change of the Hall number n H = (R H e) −1 with doping [4]. A similar behavior consistent with a drastic drop of the charge carrier density upon lowering the doping was found shortly after in the Hall number and the resistivity of several cuprate materials [5,6]. These results suggest that a QCP at optimal doping is associated with the reconstruction of a large Fermi surface enclosing a volume corresponding to a density 1 + p of empty states (holes) at large doping, to small pockets with a volume corresponding to a hole-density p in the underdoped regime. Moreover, these experiments indicate that the QCP for the closing of the pseudogap [4,7] is distinct from that for the disappearance of charge order [8].

We analyze the competition of magnetism and superconductivity in the two-dimensional Hubbard model with a moderate interaction strength, including the possibility of incommensurate spiral magnetic order. Using an unbiased renormalization group approach, we compute magnetic and superconducting order parameters in the ground state. In addition to previously established regions of Néel order coexisting with d-wave superconductivity, the calculations reveal further coexistence regions where superconductivity is accompanied by incommensurate magnetic order.PACS numbers: 71.10. Fd, Introduction. The two-dimensional Hubbard model is a prototype system for competing order in layered transition metal oxide compounds. Shortly after the discovery of high temperature superconductivity in cuprates, it has been proposed as a key model describing the valence electrons in the copper-oxygen planes [1]. Indeed, the model exhibits the most prominent ordering phenomena observed in high-T c cuprates, namely antiferromagnetism and d-wave superconductivity [2].While the magnetic order is captured already by conventional mean-field theory [3], superconductivity is fluctuation-driven and hence more subtle. Nevertheless, the emergence of d-wave superconductivity in the 2D Hubbard model is nowadays well established [2]. In particular, unbiased evidence for superconductivity with sizable gaps at moderate interaction strengths has been obtained from functional renormalization group (fRG) calculations [4-6], and from embedded quantum cluster methods at intermediate and strong coupling [7][8][9][10][11][12][13].The magnetic order is not necessarily of commensurate Néel type, that is, with antiparallel spin orientation between adjacent sites. The possibility of magnetic order with generally incommensurate wave vectors away from (π, π) has been explored by several mean-field studies [14][15][16][17][18], and also by expansions in the limit of a small hole density, where fluctuation effects were taken into account [19][20][21]. Incommensurate magnetic order in the two-dimensional Hubbard model is also indicated by diverging interactions and susceptibilities at incommensurate momenta in fRG flows [4,22,23]. However, the competition and possible coexistence of incommensurate magnetism and superconductivity has not yet been analyzed [24]. To do this, one needs to access the ordered phase in a framework that captures the fluctuations which generate d-wave superconductivity, allowing at the same time for a high momentum space resolution to distinguish the incommensurate ordering wave vector from (π, π). The latter requirement is an obstacle for cluster methods, which have so far been restricted to commensurate

Using a recently developed renormalization group method for fermionic superfluids, we determine conditions for d-wave superconductivity in the ground state of the two-dimensional Hubbard model at moderate interaction strength, and we compute the pairing gap in the superconducting regime. A pairing instability signaled by a divergent flow in the Cooper channel leads to a superconducting state in all studied cases. The next-to-nearest neighbor hopping t plays a crucial role in the competition between antiferromagnetism and superconductivity. A sizable t is necessary to obtain a sizable pairing gap.