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.

Understanding optical conductivity data in the optimally doped cuprates in the framework of quantum criticality requires a strongly-coupled quantum critical metal which violates hyperscaling. In the simplest scaling framework, hyperscaling violation can be characterized by a single non-zero exponent θ, so that in a spatially isotropic state in d spatial dimensions, the specific heat scales with temperature as T (d−θ)/z , and the optical conductivity scales with frequency as ω (d−θ−2)/z for ω T , where z is the dynamic critical exponent. We study the Ising-nematic critical point, using the controlled dimensional regularization method proposed by Dalidovich and Lee (Phys. Rev. B 88, 245106 (2013)). We find that hyperscaling is violated, with θ = 1 in d = 2. We expect that similar results apply to Fermi surfaces coupled to gauge fields in d = 2. 1 arXiv:1605.00657v1 [cond-mat.str-el] 2 May 2016 CONTENTS I. Introduction 3 II. Action and scaling analysis at tree level 5 III. Current-current correlation function and optical conductivity 7 A. Current-current correlation function at one-loop level 8 B. Two-loop self-energy correction to current-current correlation function 8 C. Two-loop vertex correction to current-current correlation function 9 D. Scaling behavior of optical conductivity and free energy 9 1. Scaling behavior of optical conductivity and free energy: General arguments 9 2. Scaling behavior of conductivity: Evaluation for fixed point theory 10 E. Pole contribution to conductivity 11 IV. Free energy at finite temperature 12 A. Contribution of free fermions 12 B. Contribution of free bosons 13 C. Interaction correction to the fermionic part of the free energy 13 D. Interaction correction to the bosonic part of the free energy 14 V. Conclusions 15 Acknowledgments 16 A. One-loop self-energies 16 B. Current vertex at one-loop level 18 C. Evaluation of current-current correlation function 20 1. Free fermion contribution 20 2. Self-energy correction 20 3. Vertex correction contribution 21 D. Particle current-momentum susceptibility 24 E. One-loop conductivity at finite temperature 25References 26where σ Q , α Q , κ Q are the frequency-dependent conductivities after the pole has been subtracted out. The residues of the pole are related exactly to static thermodynamic observables: these are the entropy density, S, the current-momentum correlator Q ≡ χ Jx,Px , and the momentum-momentum correlator M ≡ χ Px,Px . Combining Eqs. (1.2) and (1.3), we observe that the pole at ω = 0 does not appear in κ, and in the d.c. limit [20,27] In many cases, the σ Q , α Q , κ Q conductivities are not independent of each other, and obey identities connecting them at all frequencies [28]. This is the case in systems with Hamiltonians which are invariant under relativistic or Galilean transformations. However, our interest here is in systems which conserve P , but do not enjoy relativistic or Galilean invariance, and such systems have not been as extensively studied. In such situations, it appears that σ Q , α Q , and κ Q are independen...

We derive and evaluate one-loop functional flow equations for the effective interactions, selfenergy and gap function in spin-singlet superfluids. The flow is generated by a fermionic frequency cutoff, which is supplemented by an external pairing field to treat divergencies associated with the Goldstone boson. To parametrize the singular momentum and frequency dependences of the effective interactions, the Nambu interaction vertex is decomposed in charge, magnetic, and normal and anomalous pairing channels. The one-loop flow solves reduced (mean-field) models for superfluidity exactly, and captures also important fluctuation effects. The Ward identity from charge conservation is generally violated, but can be enforced by projecting the flow. Applying the general formalism to the two-dimensional attractive Hubbard model, we obtain detailed results on the momentum and frequency dependences of the effective interactions for weak and moderate bare interactions. The gap is reduced by fluctuations, with a stronger reduction at weaker interactions, as expected.

We derive an efficient and unbiased method for computing order parameters in correlated electron systems with competing instabilities. Charge, magnetic and pairing fluctuations above the energy scale of spontaneous symmetry breaking are taken into account by a functional renormalization group flow, while the formation of order below that scale is treated in mean-field theory. The method captures fluctuation driven instabilities such as d-wave superconductivity. As a first application we study the competition between antiferromagnetism and superconductivity in the ground state of the two-dimensional Hubbard model.Competing order is a ubiquitous phenomenon in twodimensional interacting electron systems. A most prominent example is the competition between antiferromagnetism and high temperature superconductivity in cuprate and iron pnictide compounds. Some of the ordering tendencies are fluctuation driven, and can therefore not be captured by mean-field (MF) theory. Numerical simulations of correlated electrons are still restricted to relatively small systems.For weak and moderate interaction strengths, the functional renormalization group (fRG) has been developed as an unbiased and sensitive tool to detect instabilities toward any kind of order in interacting electron models. 1 In that method, effective interactions, self-energies, and susceptibilities are computed from a differential flow equation, where the flow parameter Λ controls a scale-by-scale integration of fields in the underlying functional integral. Instabilities are signalled by divergences of effective interactions and susceptibilities at a critical energy scale Λ c . To complete the calculation and compute, for example, the size of the order parameters, one has to continue the flow below the scale Λ c , which requires the implementation of spontaneous symmetry breaking. This can be done either in a purely fermionic framework 2 or by introducing bosonic order parameter fields. 3 Both approaches have been applied already to interacting electron models, such as the two-dimensional Hubbard model with repulsive 3-5 and attractive 6-8 interactions.The flow in the symmetry-broken regime (Λ < Λ c ) is complicated considerably by the presence of anomalous interaction vertices. In complex problems, such as systems with several competing and possibly coexisting order parameters, or in multi-band systems, it can therefore be mandatory or at least desirable to simplify the integration of the scales below Λ c . A natural possibility is to treat the low-energy degrees of freedom (below Λ c ) in mean-field theory. The generation of instabilities and also the possible reduction of the critical scale by fluctuations is not affected by such a simplification. In the ground state, fluctuations below the critical scale are expected to influence the size of order parameters only mildly. This has been confirmed for the attractive and repulsive Hubbard model by several previous fRG studies. 3,5,6,8 A combination of an fRG flow for Λ > Λ c with a mean-field treatment of symme...

We derive an efficient method for treating renormalization contributions at two-loop level within the functional renormalization group in the one-particle irreducible formalism for fermions. It is based on a decomposition of the two-particle vertex in charge, magnetic and pairing channels. The method treats single-particle and collective excitations in all channels on equal footing, allows for the description of symmetry-breaking and captures collective mode fluctuation physics in the infrared. As a first application, we study the superfluid ground state of the two-dimensional attractive Hubbard model. We obtain superfluid gaps that are reduced by fluctuations in comparison to the one-loop approximation and demonstrate that the method captures the renormalization of the amplitude mode by long-range phase fluctuations.