Flat band and Planckian metal

TL;DR

This paper explores an extended SYK model that demonstrates flat bands and their role in the linear resistivity observed in strange metals, linking microscopic theory with phenomenological ideas.

Contribution

It provides a microscopic derivation of flat bands in an interacting model, supporting the idea that flat bands cause linear resistivity in strange metals.

Findings

Identification of flat bands in an extended SYK model.

Support for flat bands causing linear temperature dependence of resistivity.

Connection between microscopic flat band formation and phenomenological models.

Abstract

We discuss the recent extension of the Sachdev-Ye-Kitaev (SYK) microscopic model by Patel and Sachdev in arXiv:1906.03265, which demonstrates the characteristic features of the Khodel-Shaginyan fermion condensate -- the existence of the finite region of momenta, where the energy of electrons is exactly zero (the flat band). The microscopic derivation of the flat band in this interacting model supports the original idea of Khodel and Shaginyan based on the phenomenological approach. It also suggests that it is the flat band, which is responsible for the linear dependence of resistivity on temperature in "strange metals".

Full text

There are different potential sources of the formation of the electronic flat band with zero energy, see e.g. Ref. Volovik2018 . In particular it can be formed due to electron-electron interaction. The flat band formed by interaction has been first discussed by Khodel and Shaginyan (KS) in 1990 KhodelShaginyan1990 , who used the phenomenological Landau theory of Fermi liquid, see also Volovik1991 ; Nozieres1992 ; Volovik1994 ; HeikkilaVolovik2016 and Fig. 1. This dispersionless energy spectrum has a singular density of states. As a result the superconducting gap and transition temperature are proportional to the coupling constant instead of the exponential suppression in conventional metals with Fermi surfaces. For nuclear systems the linear dependence of the superconducting gap on the coupling constant has been found by Belyaev Belyaev1961 . In a more rigourous manner the flat band induced by interaction has been obtained in Refs.Yudin2014 ; Lee2009 . Experimentally the merging of levels at the Fermi surface due to interaction has been reported in Refs. Dolgopolov2014 ; Dolgopolov2017

In twisted bilayer graphene there is indication that interaction leads to the further flattening of the spectrum Marchenko2018 ; Carr2018 in addition to the geometrical/topological flattening caused by the magic angle twist Cao2018a ; Cao2018b .

In recent paper by Patel and SachdevPatelSachdev2019 the lattice extension of the Sachdev-Ye-Kitaev (SYK) model has been used to study the problem of the ”bad metal” with the universal linear dependence of resistivity on temperature Legros2018 ; Nakajima2019 ; Cao2019 ; Brown2019 . However, it appears that signatures of the KS flat band in Figure 1 (right panel) are very similar to those in Figs. 2a and 3a from Patel and Sachdev (PS)PatelSachdev2019 . Indeed, Fig. 2a from the PS paper shows the occupancy $n({\bf p})$, which exhibits the same behavior as $n({\bf p})$ in Fig. 1 (right panel), with the finite region where $0<n({\bf p})<1$. According to Khodel-Shaginyan, in this region the quasiparticle energy should be zero. And this is clearly seen from the electron spectral density shown in Fig. 3a of the PS paper. So one may conclude that the extended SYK model provides another possible realization of the KS flat band.

That is why the extended SYK model can be used for studying different properties of the materials which experience formation of the KS flat band, including possibly the ”bad metal” behavior. In this model, the universal linear dependence of resistivity on temperature has been obtained PatelSachdev2019 in the regime, where the signatures of the flat band are transparent. From that one may conclude that the phenomenon of Planck metal or bad metal is the consequence of the Khodel-Shaginyan flat band emerging in this model. The idea that the flat band may serve as the origin of the ”strange metal” behavior has been suggested earlier, see e.g. Ref. Shaginyan2013 , and recent papers Khodel2018 ; Shaginyan2019 .

This work has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694248).