# P9: Quasiparticle degradation in systems with quadratic band touchings

## Project Description

The stability of the semi-metallic state of fermions on the half-filled honeycomb lattice of graphene with respect to weak interactions is in accord with a vanishing density-of-states for the low-energy effective relativistic dispersion at the band-touching Dirac points [1]. This leads to a quantum critical point at a finite value of the interaction strength in, e.g., the half-filled Hubbard model on the honeycomb lattice [2,3]. This stability of the half-filled semi-metallic state towards a weak-coupling insulator is obtained already on the Hartree-Fock mean-field level [2]. In a Bernal-stacked honeycomb bilayer system the low-energy dispersion near the Dirac points exhibits, instead, quadratic band crossings [4,5]. It was generally believed that the resulting finite density-of-states would, in this case, lead to a Stoner-instability of the half-filled Hubbard model for arbitrarily weak finite interactions. This would again be captured, qualitatively, by Hartree-Fock theory [6]. However, a recent study, combining field-theoretical RG-arguments and quantum Monte Carlo simulations, suggests that this picture may not reflect the actual physics [7]. Namely, it has been argued that a low-energy effective linear dispersion may emerge from weak finite interactions, given that such a term is not symmetry-forbidden in the effective low-energy action. This dynamically generated linear dispersion will then hinder the immediate onset of the interaction-induced electronic instability, and, thus, result again in a finite-coupling critical point.

In this project, we will investigate this scenario within another model of interacting spinless fermions on the honeycomb lattice, which includes extended hopping terms to stabilize a quadratic-band touching point in the non-interacting limit at half-filling. In the presence of a nearest-neighbor interaction we expect a quantum phase transition to a gapped, charge-density-wave state in the strong coupling regime. Preliminary quantum Monte Carlo calculations show that this order does not persist down to the weakly-interacting region and this spinless fermion model, thus, exhibits another example of the scenario proposed in Ref. [7]. Due to the absence of a quantum Monte Carlo sign problem, we can efficiently study this model on relatively large system sizes. In particular, we plan to extend the investigations towards also exploring the quasiparticle properties within the semi-metallic regime and close to criticality, in terms of the electronic self-energy or the quasi-particle weight.

This numerical study will be complemented by an analytical treatment of this model based on higher-order perturbation theory and a RPA-analysis of the quasiparticle properties, similar to the recent study for the Hubbard model case reported in Ref. [8]. By such a complementary approach, we will be able to examine to what extent the degradation of quasi-particles, which was examined in Ref. [8], is relevant for the suppression of ordering tendencies and the emergence of such non-trivial quantum phase transitions in systems with quadratic band touchings.

[1] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. **81**, 109 (2009)

[2] S. Sorella and E. Tosatti, Europhys. Lett. **19**, 699 (1992)

[3] S. Sorella, Y. Otsuka, and S. Yunoki, Scientific Reports **2**, 992 (2012)

[4] K. Sun, H. Yao, E. Fradkin, and S. A. Kivelson, Phys. Rev. Lett. **103**, 046811 (2009)

[5] S. Uebelacker and C. Honerkamp, Phys. Rev. B **84**, 205122 (2011)

[6] T. C. Lang, Z.Y. Meng, M. M. Scherer, S. Uebelacker, F. F. Assaad, A. Muramatsu, C. Honerkamp, and S. Wessel, Phys. Rev. Lett. **109**, 126402 (2012)

[7] S. Pujari, T.C. Lang, G. Murthy, and R. K. Kaul, Phys. Rev. Lett. **117**, 086404 (2016)

[8] C. Honerkamp, Phys. Rev. B **96**, 245134 (2017)