Unifying Interacting Nodal Semimetals: A New Route to Strong Coupling

We propose a general framework for constructing a large set of nodal-point semimetals by tuning the number of linearly ($d_L$) and (at most) quadratically ($d_Q$) dispersing directions. By virtue of such a unifying scheme, we identify a new perturbative route to access various strongly interacting non-Dirac semimetals with $d_Q>0$. As a demonstrative example, we relate a two-dimensional anisotropic semimetal with $d_L=d_Q=1$, describing the topological transition between a Dirac semimetal and a normal insulator, and its three-dimensional counterparts with $d_L=1$, $d_Q=2$. We address the quantum critical phenomena and emergence of non-Fermi liquid states with unusual dynamical structures within the framework of an $\epsilon$ expansion, where $\epsilon =2-d_Q$, when these systems reside at the brink of charge- or spin-density-wave orderings, or an $s$-wave pairing. Our results can be germane to two-dimensional uniaxially strained optical honeymcomb lattice, $\alpha \text{−}(\mathrm{BEDT}\text{−}\mathrm{TTF}{)}_2{\mathrm{I}}_3$.

Figure 1

Ten classes of nodal-point semimetals, distinguished by the number of linearly ($d_L$) and quadratically ($d_Q$) dispersing directions. DSM, ASM, and QBT denote Dirac, anisotropic, and quadratic band touching semimetals, respectively. The Yukawa coupling is marginal on the dashed line, separating the fixed points where it is relevant and irrelevant. Around this line (shaded region) the quantum fluctuations engendered by the Yukawa vertex are weak, and can be controlled by an $\epsilon$ expansion. The $d_Q=0$ line is special due to the presence of full spacetime Lorentz symmetry [18]. Lower inset: Material realizations of various semimetals.

Figure 2

A schematic finite temperature ($T$) phase diagram diagram of a two dimensional interacting ASM (red line), separating a DSM and a band insulator (BI). In a noninteracting system ($U=0$) the transition between these two phases takes place through a quantum critical point located at $\mathrm{\Delta }=0$ (gray circle). For sufficiently strong interaction ($U$) a broken symmetry phase (BSP) sets in through a multicritical point (red circle), where DSM, BI, ASM, and a BSP meet. Universality classes of various transitions across this fixed point are summarized in Table 1. The critical cone accommodates a quasiparticleless metallic state or NFL.

Figure 3

Schematic plot of the RG flow (red arrows) of $g(\ell )$ [see Eq. (7)]. (a) For $N_b=1$ (Ising) and 2 ($XY$) the noninteracting fixed (black asterisk) point is stable in $d=3$, while a stable quasiparticleless interacting fixed point (blue asterisk) is obtained only for $d<3$. (b) By contrast, for $N_b=3$ (Heisenberg) a stable interacting fixed point is already present in $d=3$, which shifts to a stronger coupling for $d<3$.