Nearest Particle Statistics and Renormalization

Enrico Arrigoni, Werner R. Hanke

We study the occurrence of SO(5) symmetry in the low-energy sector of two-chain Hubbard-like systems by analyzing the flow of the running couplings ( $g$-ology) under renormalization group in the weak-interaction limit. It is shown that SO(5) is asymptotically restored for low energies for rather general parameters of the bare Hamiltonian. This holds also with the inclusion of a next-nearest-neighbor hopping which explicitly breaks particle-hole symmetry if one accounts for a different single-particle weight for the quasiparticles of the two bands of the system. The physical significance of this renormalization SO(5) symmetry is discussed.

Uwe Claus Täuber

L Markhof, Björn Sbierski, V. Meden +1 more

The functional renormalization group (FRG) has been used widely to investigate phase diagrams, in particular the one of the two-dimensional Hubbard model. So far, the study of one-dimensional models has not attracted as much attention. We use the FRG to investigate the phases of a one-dimensional spinless tight-binding chain with nearest and next-nearest neighbor interactions at half filling. The phase diagram of this model has already been established with other methods, and phase transitions from a metallic phase to ordered phases take place at intermediate to strong interactions. The model is thus well suited to analyze the potential and the limitations of the FRG in this regime of interactions. We employ flow equations that are exact up to second order in the interaction, which implies that we take into account the frequency dependence of the two-particle vertex as well as the feedback of the dynamic self-energy. For intermediate nearest neighbor interactions, our scheme captures the phase transition from a metallic phase to a charge density wave with alternating occupation. The critical interaction, at which this transition occurs, is underestimated due to our approximations. Similarly, for intermediate next-nearest neighbor interactions, we observe a transition to a charge density wave with occupation pattern $...00110011...$. We show that taking into account a feedback of the two-particle vertex in the flow equation is essential for the detection of those phases.

Igor Viktorovich Tyutin

Heinz‐Otto Peitgen, Péter Richter

James H. Luscombe

The primary goal of modern theories of critical phenomena is the evaluation of critical exponents from first principles, implying the need to get beyond mean field theories which predict the same exponents independent of the nature of the system. This chapter focuses on a new approach emerged in the 1960s based on two ideas—scaling and renormalization. It presents the physical picture developed by L. P. Kadanoff that provides a conceptual basis for scaling and which has become part of the standard language of critical phenomena. Renormalization provides a way to calculate the free energy not involving a direct evaluation of the partition function—it's a new paradigm in statistical mechanics. The chapter analyzes a simplified, approximate set of recursion relations valid at high temperature for a model with nearest and next-nearest-neighbor interactions. In 1971 K. G. Wilson published two articles marking the introduction of the renormalization group to the study of critical phenomena.