Multicritical Fermi Surface Topological Transitions

A wide variety of complex phases in quantum materials are driven by electron-electron interactions, which are enhanced through density of states peaks. A well-known example occurs at van Hove singularities where the Fermi surface undergoes a topological transition. Here we show that higher order singularities, where multiple disconnected leaves of Fermi surface touch all at once, naturally occur at points of high symmetry in the Brillouin zone. Such multicritical singularities can lead to stronger divergences in the density of states than canonical van Hove singularities, and critically boost the formation of complex quantum phases via interactions. As a concrete example of the power of these Fermi surface topological transitions, we demonstrate how they can be used in the analysis of experimental data on ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$. Understanding the related mechanisms opens up new avenues in material design of complex quantum phases.

Figure 1

(a) The $n=2$ 2D van Hove singularity in the form of a saddle point. The DOS $\nu$ diverges as $\ln \mu$. Red lines indicate FSs above and below the singularity. The critical FS is shown by the dotted black line. (b) This singularity occurs for example very close to the Fermi energy in the case of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ (green marker for the critical chemical potential) (c) The $n=3$ singularity that can occur at threefold symmetric points. (d) The band structure or FS of bilayer graphene that is very close to such a $n=3$ monkey saddle singularity. (e) The $n=4$ singularity at a fourfold symmetric point. (f) Schematic of the quasi-2D FS of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ in the $k_z=0$ plane at the Fermi energy (left side) and at ${\mu }_C$ (right side). The crucial bands that are close to the $n=4$ multicritical point are highlighted in red. The central pocket is a small perturbation (see main text). In order to emphasize the characteristic clover leaf FS we show an extended $k$-space picture beyond the BZ boundaries.

Figure 2

Schematic phase diagram of the relevant hole-like version of the effective dispersion relation (1). In ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ there is no control over the parameter $a>0$, so that only red, yellow, and green phases are accessible. There are two lines of LTs, a dashed-dotted line of a band edge type and a solid line corresponding to the transition of the neck-narrowing type. The $X_9$ singularity is located at the crossing of these two lines. The white line schematically shows the location of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ within this phase diagram with the diamond marking the location in zero field. (Right side) The three dimensional surfaces above are electron dispersions $ϵ=ϵ(k_x,k_y)$ in the vicinity of the singularity. The gray horizontal plane represents the critical energy of the singularity $ϵ=0$.

Figure 3

Schematic phase diagram of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ together with the crystal structure shown in the inset (see Ref. [6] for more details).

Figure 4

Result of the DFT calculation. (a) Here we show the schematic FS structure from Fig. 1 together with the $k_z$-projected DFT calculation to aid orientation. (b)–(f) Projected FSs for values of magnetization $\mu =0.0$, 0.2, 0.4, 0.5, 0.55 ${\mu }_B$ per unit cell as calculated by the DFT and centered at the $X$ point. The topological transitions are evident for the value of magnetization close to 0.5 and 0.55 ${\mu }_B$ per unit cell. The red arrow in (e), (f) connects the two nested parts of the characteristic clover leaf structure giving rise to the density wave in the $A$ phase.