A d-wave high temperature cuprate superconductor exhibits a nematic ordering transition at zero temperature. Near the quantum critical point, the coupling between gapless nodal quasiparticles and nematic order parameter fluctuation can result in unusual behaviors, such as extreme anisotropy of fermion velocities. We study the disorder effects on the nematic quantum critical behavior and especially on the flow of fermion velocities. The disorders that couple to nodal quasiparticles are divided into three types: random mass, random gauge field, and random chemical potential. A renormalization group analysis shows that random mass and random gauge field are both irrelevant and thus do not change the fixed point of extreme velocity anisotropy. However, the marginal interaction due to random chemical potential destroys this fixed point and makes the nematic phase transition unstable.

We analyze the effect of disorder on the weak-coupling instabilities of quadratic band crossing point (QBCP) in two-dimensional Fermi systems, which, in the clean limit, display interactiondriven topological insulating phases. In the framework of a renormalization group procedure, which treats fermionic interactions and disorder on the same footing, we test all possible instabilities and identify the corresponding ordered phases in the presence of disorder for both single-valley and two-valley QBCP systems. We find that disorder generally suppresses the critical temperature at which the interaction-driven topologically non-trivial order sets in. Strong disorder can also cause a topological phase transition into a topologically trivial insulating state.PACS numbers: 73.43. Nq, 71.55.Jv, 11.30.Qc The study of topological phases of matter is one of the most active research areas in contemporary condensed matter physics. The explanation of the quantum Hall effect in terms of the topological properties of the Landau levels [1,2] in the 1980's triggered an intense research effort in the theoretical prediction [3][4][5] and the experimental discovery [6,7] of a plethora of different topologically non-trivial quantum phases. In two-dimensional (2D) insulating systems only two distinct topological non-trivial phases can be realized according to the well-established classification of topological insulators and superconductors [8,9] : (i) the quantum anomalous Hall state (QAH) [3] with a time-reversal symmetry-broken ground state and topologically protected chiral edge states and (ii) the time-reversal invariant quantum spin Hall (QSH) state [4,5], which possesses helical edge states with counterpropagating electrons of opposite spins.In recent years, attention has gradually shifted from non-interacting topological states of matter towards interaction-driven topological phases:many-particle quantum ground-states in which chiral orbital currents or spin-orbit couplings are spontaneously generated by electronic correlations. These states of matter possess both conventional order, characterized by an order parameter and a broken symmetry, and protected edge states associated with a topological quantum number. Interactiondriven QAH and QSH phases were first conceived in the context of 2D honeycomb lattice Dirac fermions [10] assuming sufficiently strong electronic repulsions although more recent analytical and numerical works question the proposal for this particular model [11][12][13][14].On the contrary, it has been proposed that 2D systems with a quadratic band crossing point (QBCP) are unstable to electronic correlation because of the finite density of states at the Fermi level leading to the possibility of The relationship between fixed points QAH and QAH-II is provided in the inset figure of Fig. 2(a).weak-coupling interaction-driven topological insulating phases [15][16][17]. And, indeed, QAH and QSH phases generated by electronic repulsions occur both in the checkerboard lattice model [15,18], and in two-valley QB...

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...