Quasiuniversality from all-in–all-out Weyl quantum criticality in pyrochlore iridates

We identify an exotic quasiuniversal behavior near the all-in–all-out Weyl quantum critical point in three-dimensional Luttinger semimetals, such as the pyrochlore iridates $R_2{\mathrm{Ir}}_2{\mathrm{O}}_7$, with $R$ a rare-earth element. The quasiuniversal behavior is characterized by power laws with exponents that vary slowly over several orders of magnitude in energy or length. However, in contrast to the quasiuniversality discussed in the context of deconfined criticality, the present case is characterized by a genuinely universal ultra-low-temperature behavior. In this limit, the pertinent critical exponents can be computed exactly within a renormalization group analysis. Experimental implications for the pyrochlore iridates are outlined.

Figure 1

Schematic finite-temperature phase diagram of pyrochlore iridates $R_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ near the quantum critical point (QCP) between the Weyl semimetal with all-in–all-out (AIAO) order, sketched in the inset, and the symmetric Luttinger semimetal. $r$ indicates a nonthermal tuning parameter, such as chemical doping or hydrostatic pressure. Between the nonuniversal high-temperature regime and the genuinely universal ultra-low-temperature regime, there is a large quasiuniversal regime, characterized by power laws with slowly varying exponents.

Figure 2

Zero-temperature phase diagram of effective model as a function of cubic anisotropy $\delta$ and interaction strength $g^2/r$, from mean-field theory. Coloring indicates the magnitude of the AIAO order parameter $\langle \varphi \rangle$. With the onset of order, the quadratic band touching point of the Luttinger semimetal splits into four pairs of Weyl points in the AIAO Weyl semimetal (see insets). The transition is continuous (discontinuous) for $\delta \le {\delta }_0$ ($\delta >{\delta }_0$), with ${\delta }_0\approx 0.0624$. Dashed lines delimit the region in which metastable states exist.

Figure 3

Fixed-point structure for $N=1$ on the critical hypersurface $r=0$ in (a) the parameter space spanned by $\delta$, $g^2$, and $e^2$, (b) the $g^2\text{−}{e}^2$ plane for fixed $\delta =-1$, (c) the $\delta \text{−}{g}^2$ plane for fixed $e^2=9\epsilon /16$, and (d) the $\delta \text{−}{e}^2$ plane for fixed $g^2=0$. Black (gray) dots and lines indicate fixed points and separatrices, respectively, located within (projected onto) the respective planes. Arrows denote flow towards the infrared, with their coloring indicating the flow velocity. There is a unique quantum critical fixed point at finite coupling, labeled as QCP. It corresponds to the continuous transition between the symmetric Luttinger semimetal and the AIAO-ordered Weyl semimetal. The symmetric state is described by the Luttinger-Abrikosov-Beneslavskii fixed point at $g^2=0$, labeled as LAB. The dots labeled as A, ${\mathrm{A}}^{\prime }$, ${\mathrm{QCP}}_0$, and G correspond to unstable interacting and Gaussian, respectively, fixed points, further discussed in the SM.

Figure 4

Renormalization group flow on the critical hypersurface $r=0$ from the ultraviolet scale, corresponding to $b^{-\epsilon }=1$, to a deep infrared scale, corresponding to $b^{-\epsilon }={10}^{-6}$, starting from different microscopic parameters. Each curve corresponds to a numerical integration of the flow equations using the initial conditions $(\delta ,g^2,e^2)=(-0.5,0.7391,0.9098+\mathrm{\Delta })$ for $b^{-\epsilon }=1$, with $\mathrm{\Delta }=-0.7,-0.6,-0.4,0,0.8,2.4,5.6$ from purple to red. The starting values of $(g^2,e^2)$ have been chosen to satisfy the pseudo-fixed-point conditions for $\delta =-0.5$ when $\mathrm{\Delta }=0$. (a) Flow trajectories in the parameter space spanned by $\delta$, $g^2$, and $e^2$, illustrating the crossover from the nonuniversal regime, characterized by independent trajectories, to the quasiuniversal regime, in which these collapse onto a single curve. Deviations of (b) anisotropy parameter $\delta$ and effective exponents (c) ${\eta }_{\psi }=2-z$ and (d) ${\eta }_{\varphi }=2-1/\nu$ from their respective critical values as a function of renormalization group scale $b^{-\epsilon }$. The quasiuniversal regime emerges at $b^{-\epsilon }\lesssim {10}^{-3}$, as indicated by the dashed line, and manifests itself in $(g^2,e^2)$-independent, but anisotropy-dependent, drifting exponents. Note that the flow is still significantly away from its ultra-low-energy limit even at $b^{-\epsilon }={10}^{-6}$, as evidenced by the finite deviation $\delta -{\delta }_{★}\gtrsim 0.1$ in (b).