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Register a new userQuantum discontinuity fixed point and renormalization group flow of the SYK model
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We determine the global renormalization group (RG) flow of the Sachdev-Ye-Kitaev (SYK) model. This flow allows for an understanding of the surprising role of critical slowing down at a quantum first-order transition in strongly-correlated electronic systems. From a simple truncation of the infinite hierarchy of the exact functional RG flow equations we identify several fixed points: Apart from a stable fixed point, associated with the celebrated non-Fermi liquid state of the model, we find another stable fixed point related to an integer-valence state. These stable fixed points are separated by a discontinuity fixed point with one relevant direction, describing a quantum first-order transition. Most notably, the fermionic spectrum continues to be quantum critical even at the discontinuity fixed point. This rules out a description of this quantum first-order transition in terms of a local effective Ising variable that is established for classical transitions. It reveals that quantum phase coexistence can be a genuine critical state of matter.
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We apply the finite-temperature renormalization-group (RG) to a model based on an effective action with a short-range repulsive interaction and a rotation invariant Fermi surface. The basic quantities of Fermi liquid theory, the Landau function and the scattering vertex, are calculated as fixed points of the RG flow in terms of the effective actions interaction function. The classic derivations of Fermi liquid theory, which apply the Bethe-Salpeter equation and amount to summing direct particle-hole ladder diagrams, neglect the zero-angle singularity in the exchange particle-hole loop. As a consequence, the antisymmetry of the forward scattering vertex is not guaranteed and the amplitude sum rule must be imposed by hand on the components of the Landau function. We show that the strong interference of the direct and exchange processes of particle-hole scattering near zero angle invalidates the ladder approximation in this region, resulting in temperature-dependent narrow-angle anomalies in the Landau function and scattering vertex. In this RG approach the Pauli principle is automatically satisfied. The consequences of the RG corrections on Fermi liquid theory are discussed. In particular, we show that the amplitude sum rule is not valid.
Quantum impurity models are the prototypical examples of quantum many-body dynamics which manifests in their spectral and transport properties. Single channel Anderson(and Kondo model) leads to the Fermi liquid ground state in the strong coupling regime which corresponds to a stable infrared fixed point at which the quantum impurity gets completely screened by the conduction electrons. Quantum impurity models with non-trivial density of states exhibit quantum phase transition and this quantum criticality lies in the universality class of local quantum critical systems. In this paper, we report first study of the flow equation renormalization of gapped Kondo model which has gapped density of states, the gap being at the Fermi level. Flow equation renormalization group method has proved to be one of the very robust renormalization methods to study Kondo physics both in equilibrium as well as in non-equilibrium. Here we have shown that this method can also be employed to study local quantum criticality. We have calculated the flow equations for the Kondo coupling and solved them for various values of the gap parameter and we find that there is suppression of Kondo divergence as gap is increased which signifies that as gap is increased, renormalization flow goes away from the strong coupling fixed point. We have also calculated the spin susceptibility and we find that as gap is increased, susceptibility goes over to the Curie behaviour and hence confirming the renormalization flow towards the local moment fixed point.
We present a comprehensive analysis of quantum fluctuation effects in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The superfluid order parameter, and fluctuations thereof, are implemented by a bosonic Hubbard-Stratonovich field, which splits into two components corresponding to longitudinal and transverse (Goldstone) fluctuations. Physical properties of the system are computed from a set of approximate flow equations obtained by truncating the exact functional renormalization group flow of the coupled fermion-boson action. The equations capture the influence of fluctuations on non-universal quantities such as the fermionic gap, as well as the universal infrared asymptotics present in every fermionic superfluid. We solve the flow equations numerically in two dimensions and compute the asymptotic behavior analytically in two and three dimensions. The fermionic gap Delta is reduced significantly compared to the mean-field gap, and the bosonic order parameter alpha, which is equivalent to Delta in mean-field theory, is suppressed to values below Delta by fluctuations. The fermion-boson vertex is only slightly renormalized. In the infrared regime, transverse order parameter fluctuations associated with the Goldstone mode lead to a strong renormalization of longitudinal fluctuations: the longitudinal mass and the bosonic self-interaction vanish linearly as a function of the scale in two dimensions, and logarithmically in three dimensions, in agreement with the exact behavior of an interacting Bose gas.
The phase diagram of the one-dimensional extended Hubbard model at half-filling is investigated by a weak coupling renormalization group method applicable beyond the usual continuum limit for the electron spectrum and coupling constants. We analyze the influence of irrelevant momentum dependent interactions on asymptotic properties of the correlation functions and the nature of dominant phases for the lattice model under study.
Using a leading algorithmic implementation of the functional renormalization group (fRG) for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model. In particular, we show that the recently introduced multiloop extension of the fRG flow equations for the self-energy and two-particle vertex allows for a precise match with the parquet approximation also for two-dimensional lattice problems. The refinement with respect to previous fRG-based computation schemes relies on an accurate treatment of the frequency and momentum dependences of the two-particle vertex, which combines a proper inclusion of the high-frequency asymptotics with the so-called truncated unity fRG for the momentum dependence. The adoption of the latter scheme requires, as an essential step, a consistent modification of the flow equation of the self-energy. We quantitatively compare our fRG results for the self-energy and momentum-dependent susceptibilities and the corresponding solution of the parquet approximation to determinant quantum Monte Carlo data, demonstrating that the fRG is remarkably accurate up to moderate interaction strengths. The presented methodological improvements illustrate how fRG flows can be brought to a quantitative level for two-dimensional problems, providing a solid basis for the application to more general systems.
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