Single monkey-saddle singularity of a Fermi surface and its instabilities

Fermi surfaces can undergo sharp transitions under smooth changes of parameters. Such transitions can have a topological character, as is the case when a higher-order singularity, one that requires cubic or higher-order terms to describe the electronic dispersion near the singularity, develops at the transition. When time-reversal and inversion symmetries are present, odd singularities can only appear in pairs within the Brillouin zone. In this case, the combination of the enhanced density of states that accompanies these singularities and the nesting between the pairs of singularities leads to interaction-driven instabilities. We present examples of single $n=3$ (monkey-saddle) singularities when time-reversal and inversion symmetries are broken. We then turn to the question of what instabilities are possible when the singularities are isolated. For spinful electrons, we find that the inclusion of repulsive interactions destroys any isolated monkey-saddle singularity present in the noninteracting spectrum by developing Stoner or Lifshitz instabilities. In contrast, for spinless electrons and at the mean-field level, we show that an isolated monkey-saddle singularity can be stabilized in the presence of short-range repulsive interactions.

Figure 1

Under appropriate tuning of the staggered chemical potential $[M=\pm M_0^{}$ with $M_0^{}$ defined in Eq. (2.17b) with $t_1^{}=1$ and $t_2^{}=\frac{1}{4}]$, Hamiltonian (2.10) can be made to host a monkey-saddle singularity at either one of the ${\mathit{K}}_{\pm }^{}$ points. While the constant-energy contours of the monkey-saddle dispersion (2.1) are open, those in the Haldane model in (a) and (b) are closed due to the correction of order $\propto {\mathit{k}}^4$ to the monkey-saddle dispersion (2.1). The monkey saddle with its singular energy contour that is shaped like the boundary of a three-leaf clover (bold and black) is here realized at ${\mathit{K}}_{+}^{}$, while a simple maximum is realized at ${\mathit{K}}_{-}^{}$ higher up in energy.

Figure 2

Dispersions (2.11b) and density of states (DOS) $\nu \left(\varepsilon \right)=N^{-2}{\sum }_{\pm }{\sum }_{\mathit{k}}\delta (\varepsilon -{\varepsilon }_{\pm }^{}\left(\mathit{k}\right))$ for the single-particle tight-binding Hamiltonian (2.10). The choice $N=1001$ is made and each delta function entering the DOS is regularized by a normalized Gaussian of variance ${\sigma }_{\mathrm{Gaussian}}^2\sim N^{-2}$. Under tuning of parameters, where we set $t_1^{}=1$, Hamiltonian (2.10) displays changes in both the band geometry and the band topology. We seek to obtain a monkey saddle at ${\mathit{K}}_{+}^{}$ with an extremum at ${\mathit{K}}_{-}^{}$, located higher up in energy, in the upper band of Hamiltonian (2.10). Such a phase is automatically in the topological regime with nonvanishing Chern numbers for each filled band. This phase can be reached starting from the gapless, and time-reversal invariant Dirac semimetal in (a). As a staggered chemical potential ($M$) is turned on, a gap at both the ${\mathit{K}}_{\pm }^{}$ points appears as depicted in (b). The dispersions in the neighborhoods of ${\mathit{K}}_{+}^{}$ and ${\mathit{K}}_{-}^{}$ are symmetric because of time-reversal symmetry. In (c), a small time-reversal breaking next-nearest-neighbor hopping $t_2^{}$ that causes an asymmetry between the dispersion around ${\mathit{K}}_{+}^{}$ and that around ${\mathit{K}}_{-}^{}$ is turned on. As the strength of $t_2^{}$ is increased to $\left|M\right|/3\sqrt{3}$, the gap at ${\mathit{K}}_{+}^{}$ closes, as in (d). By increasing $t_2^{}$ further as in (e), the topological regime with nonvanishing Chern number is entered. Finally, increasing $t_2^{}$ to satisfy the condition $M=(t_1^2-18t_2^2)/\left(2\sqrt{3}{t}_2^{}\right)$, we obtain a monkey saddle at ${\mathit{K}}_{+}^{}$ and a simple maxima at ${\mathit{K}}_{-}^{}$, higher in energy. This is depicted in (f). The van Hove singularities in the DOS of (a)–(e) have become monkey-saddle singularities at ${\mathit{K}}_{+}^{}$ and van Hove singularities at ${\mathit{K}}_{-}^{}$ in (f).

Figure 3

The mean-field phase diagram at zero temperature in the coupling space spanned by the filling fraction $n_e^{}$, the onsite repulsive interaction $U$, and the nearest-neighbor repulsive interaction $V$ is obtained from solving numerically the mean-field equations (4.6) (all energies are measured in units of $t_1^{}$). Dashed red lines show the approximate phase boundaries in (a), (b), and (c). The yellow solid lines shows the approximate phase boundaries in the thermodynamic limit in (a) and (c). (a) Two-dimensional cut for the values taken by $\overline{m}$ when $V=0$ in units of $t_1^{}$. A Stoner instability towards itinerant ferromagnetism takes place for any nonvanishing Hubbard interaction $U>U_{\mathrm{c}}^{}$ for given $n_e^{}$. The minimum value $U_{\mathrm{c},\min }^{}$ taken by $U_{\mathrm{c}}^{}$ occurs for $n_e^{}=n_{e,\mathrm{ms}}^{}$. The nonvanishing value of $U_{\mathrm{c},\min }^{}$ above which ferromagnetism is established when the filling fraction is fine tuned to the monkey saddle, i.e., $n_e^{}=n_{e,\mathrm{ms}}^{}$, is due to a finite-size effect that cuts off the diverging DOS. In the thermodynamic limit, $U_{\mathrm{c},\min }^{}\to 0$ at $n_e^{}=n_{e,\mathrm{ms}}^{}$. (b) Two-dimensional cut for the values taken by $\overline{m}$ when $V=0.1$ in units of $t_1^{}$. A nonvanishing $V>0$ has two effects. It increases the minimum value of $U_{\mathrm{c}}^{}$ from (a) to a value that remains nonvanishing in the thermodynamic limit. The position of the minimum value of $U_{\mathrm{c}}^{}$ from (a) is shifted along the $n_e^{}$ axis. (c) Two-dimensional cut for the values taken by $\overline{m}$ when $n_e^{}=n_{e,\mathrm{ms}}^{}$. The critical value $U_{\mathrm{c}}^{}$ above which ferromagnetism sets in is an increasing function of $V$. (d) Two-dimensional cut for the values taken by $\overline{\chi }$ when $n_e^{}=n_{e,\mathrm{ms}}^{}$. The value of $\overline{\chi }$ is nonvanishing everywhere.

Figure 4

The regularized mean-field DOS as a function of the deviation $n_e^{}-n_{e,\mathrm{ms}}^{}$ of the filling fraction $n_e^{}$ from the monkey-saddle filling fraction $n_{\mathrm{e},\mathrm{ms}}^{}$ is plotted for different values of the interaction strengths $U\ge 0$ and $V\ge 0$. The $\delta$ functions in the mean-field DOS are regularized by normalized Gaussians of variance ${\sigma }_{\mathrm{Gaussian}}^2\sim 10/N^2$ with $N=501$. In (a), $U$ is increased holding $V=0$. The single regularized peak in the noninteracting DOS is split into two peaks by a nonvanishing $U$. Contrary to the height of the monkey-saddle peak of the regularized noninteracting DOS, the height of these secondary peaks remains finite in the limit ${\sigma }_{\mathrm{Gaussian}}^{}\to 0$. In (b), $V$ is increased holding $U=0$. The single peak in the regularized noninteracting DOS is translated to the left by a nonvanishing $V$. This observation can be explained by $V>0$ inducing quadratic perturbations to the monkey-saddle dispersion (2.1) at the mean-field level. These quadratric perturbations turn the monkey-saddle singularity into a central local extremum surrounded by three van Hove saddle singularities.

Figure 5

Dependencies of the uniform magnetization $\overline{m}$, uniform bond density $\overline{\chi }$, and chemical potential $\overline{\mu }$ along one-dimensional cuts in coupling space at zero temperature as is explained in the text.

Figure 6

Dependency of the uniform magnetization $\overline{m}$ along one-dimensional cuts in coupling space at zero temperature as is explained in the text.

Figure 7

Renormalized hopping amplitude ${\overline{t}}_1^{}$ that is obtained by solving the self-consistent mean-field equations (4.9) at zero temperature. The red dashed line shows the points in the parameter space for which the condition (4.9c) is met. For any interaction strength $V$, there is a fine-tuned value $\overline{\delta }$ of $\delta$ for which an interacting monkey-saddle singularity appears in the mean-field spectrum at the filling $n_e^{}=n_{e,\mathrm{ms}}^{}/2$.