Topological superconductivity of spin-$3/2$ carriers in a three-dimensional doped Luttinger semimetal

We investigate topological Cooper pairing, including gapless Weyl and fully gapped class DIII superconductivity, in a three-dimensional doped Luttinger semimetal. The latter describes effective spin-3/2 carriers near a quadratic band touching and captures the normal-state properties of the 227 pyrochlore iridates and half-Heusler alloys. Electron-electron interactions may favor non-$s$-wave pairing in such systems, including even-parity $d$-wave pairing. We argue that the lowest energy $d$-wave pairings are always of complex (e.g., $d+id$) type, with nodal Weyl quasiparticles. This implies $\varrho \left(E\right)\sim {\left|E\right|}^2$ scaling of the density of states (DoS) at low energies in the clean limit or $\varrho \left(E\right)\sim \left|E\right|$ over a wide critical region in the presence of disorder. The latter is consistent with the $T$ dependence of the penetration depth in the half-Heusler compound YPtBi. We enumerate routes for experimental verification, including specific heat, thermal conductivity, NMR relaxation time, and topological Fermi arcs. Nucleation of any $d$-wave pairing also causes a small lattice distortion and induces an $s$-wave component; this gives a route to strain-engineer exotic $s+d$ pairings. We also consider odd-parity, fully gapped $p$-wave superconductivity. For hole doping, a gapless Majorana fluid with cubic dispersion appears at the surface. We invent a generalized surface model with $\nu$-fold dispersion to simulate a bulk with winding number $\nu$. Using exact diagonalization, we show that disorder drives the surface into a critically delocalized phase, with universal DoS and multifractal scaling consistent with the conformal field theory (CFT) SO($n){}_{\nu }$, where $n\to 0$ counts replicas. This is contrary to the naive expectation of a surface thermal metal, and implies that the topology tunes the surface renormalization group to the CFT in the presence of disorder.

Figure 1

The temperature dependence of the superconducting gap for (a) $d_{x^2-y^2}$ and (b) $d_{3z^2-r^2}$ pairings, calculated with ${\lambda }_d=1$ and ${\omega }_D=0.02$. The two gaps have identical transition temperature ($T_c$) given by Eq. (2.24), but are generally different at any lower temperature $T<T_c$, with the zero-temperature values approximately given by Eq. (2.22).

Figure 2

Superconducting gap profiles for (a) $d_{x^2-y^2}$ and (b) $d_{3z^2-r^2}$ pairing on an isotropic Fermi sphere (realized for $m_1=m_2$), with the red curves representing the nodal lines of the gap (see Table 1). It is obvious that the two gap structures cannot be related by any SO(3) rotation. By contrast, any of the three $T_{2g}$ gaps can be obtained by an appropriate rotation of the $d_{x^2-y^2}$ gap, shown in (a).

Figure 3

Competition between $s+i{d}_{x^2-y^2}$ and $d_{x^2-y^2}+i{d}_{3z^2-r^2}$ pairings as a function of pairing strength ratio ${\lambda }_d/{\lambda }_s$, computed for a fixed ${\lambda }_s=0.4$. (a) Zero-temperature gap values in the $d+id$ channel (black squares), as well as the $s$ component (circles) and $d$ component (diamonds) of the $s+id$ solution. The shaded magenta region indicates the (unstable) $s+id$ phase with nonzero values of both $s$ and $d$ components. (b) The corresponding free energies in these two channels. The vertical dotted line indicates the position of a first-order phase transition from a pure $s$-wave pairing on the left, to the $(d+id)$ pairing on the right.

Figure 4

Distribution of the Abelian Berry curvature for the nodal Weyl superconductor arising due to $d_{x^2-y^2}+i{d}_{3z^2-r^2}$ ($E_g$) pairing. The source (with outward arrows) and sink (with inward arrows) are symmetrically placed about the four possible body-diagonal directions of the spherical Fermi surface (an octupolar arrangement). The net Berry curvature through any high-symmetry plane therefore vanishes and the paired state does not support any anomalous pseudospin or thermal Hall conductivity.

Figure 5

Distribution of the Abelian Berry curvature for the nodal Weyl superconductor arising from a specific phase locking amongst $d_{xy},d_{xz}$, and $d_{yz}$ ($T_{2g}$) pairings, given by Eq. (3.10). Weyl node pairs are enumerated in Eq. (3.12). In the figure, the top and bottom pair of nodes represent set $\left(a\right)$, left and right pair set $\left(b\right)$, front and back pair set $\left(c\right)$, and the two diagonally opposite ones set $\left(d\right)$. The net Abelian Berry curvature of such a paired state does not vanish through any high-symmetry plane, and concomitantly the paired state exhibits nontrivial anomalous pseudospin and thermal Hall effects.

Figure 6

The distribution of the Abelian Berry curvature in the presence of $k_z(k_x+i{k}_y)$ pairing on the spherical Fermi surface. Notice that there is no flux line along the equator of the Fermi surface where the BdG quasiparticles support a nodal loop. The Berry flux through the $k_x-k_y$ plane is finite and consequently the paired state supports nonzero anomalous pseudospin and thermal Hall conductivities in the $xy$ plane, given by Eq. (3.27).

Figure 7

(a) The Feynman diagram contributing to the Landau free energy $f_{\mathrm{str}}\sim {\mathrm{\Phi }}_j{\mathrm{\Delta }}_j{\mathrm{\Delta }}_0$, capturing the nontrivial coupling amongst the $s$-wave and $d$-wave pairings and an external strain. Here solid lines represent fermions, wavy (dashed) lines $d$ ($s$)-wave pairing and the spiral line the strain field. (b) Scaling of the universal function $F\left({\omega }_D,t\right)$ on ${\omega }_D$ for various fixed values of $t$ (quoted in the figure), appearing in Eq. (4.6), where ${\omega }_D={\mathrm{\Omega }}_D/E_F,t=k_{B}T/E_F, {\mathrm{\Omega }}_D$ is the Debye frequency, $E_F$ is the Fermi energy, and $T$ is the temperature.

Figure 8

A qualitative phase diagram of a dirty thermal Weyl semimetal (ThWSM) at finite temperature ($T$). Here, $T_c$ is the superconducting transition temperature and $t_{*}(=T_{*}/T_C)$ is a crossover temperature above which the BdG-Weyl quasiparticles are not sharp. The red dot at $\delta =0$ represents the disorder-controlled ThWSM-thermal metal (ThMetal) quantum critical point, and the shaded region represents the associated quantum critical regime ($\delta$ is the reduced disorder strength, see text). The shape of the crossover boundaries at finite temperature are roughly determined by ${\left|\delta \right|}^{\nu z}\sim {\left|\delta \right|}^{1.5}$, leading to a wide quantum critical regime. For a more quantitative estimation of a similar phase diagram at finite temperature or energy, see Refs. [107, 108]. Scaling of the average density of states $\left[\varrho \right(E\left)\right]$ in various regimes of the phase diagram is displayed in the figure. Here the labeling of the temperature and disorder axes are qualitative.

Figure 9

A qualitative phase diagram of a dirty thermal double Weyl semimetal (WSM) or thermal line-node semimetal (LNSM) at finite temperature. Above a crossover temperature $t_{*}=T_{*}/T_C$, BdG quasiparticles are not sharp. The crossover boundary between the pseudoballistic semimetallic phase and the diffusive thermal metallic phase scales as $\sim \exp [-A/W]$, where $W$ denote the strength of disorder and $A(=0.35\text{here})$ is, however, a nonuniversal (material dependent) constant. Scaling of the density of states in various regimes of the phase diagram is quoted in the figure. For quantitative estimation of this phase diagram, see Ref. [108]. Here the labeling of the temperature and disorder axes are qualitative.

Figure 10

Various possible fits for the experimentally measured data of the penetration depth ($\mathrm{\Delta }\lambda$) in YPtBi as a function of the reduced temperature $t=T/T_c$ (replotted from Ref. [47] with permission). Here, $T_c=0.78$ K is the superconducting transition temperature in YPtBi. For details of these fittings and the corresponding physical picture, see Sec. 6.

Figure 11

Schematic phase diagram for the noninteracting 2D surface states of a class DIII bulk topological superconductor. The fixed point representing the clean surface band structure (red dot) is unstable in the presence of time-reversal preserving quenched disorder for any $\nu \ge 3$, where $\nu$ is the integer bulk winding number. The precise form of the clean limit depends on details. For a spin-1/2 bulk, one can have $\nu$ species of massless relativistic Majorana fermions, with disorder that enters as a non-Abelian gauge potential scattering between these [65]. For isotropic $p$-wave pairing in the LSM studied here with winding number $\nu =3$, the surface states in the hole-doped case consist of a single two-component surface fermion with cubic dispersion, see Fig. 12 and Eq. (7.9) [48], cf. Refs. [49, 52]. Our generalized surface theory in Eq. (7.23) has $\nu$-fold dispersion for the corresponding winding number. The disordered system should be described by a class DIII nonlinear sigma model with a Wess-Zumino-Novikov-Witten (WZNW) term. The WZNW term prevents Anderson localization [65, 127]. This theory has a stable thermal metal phase (green dot) and an unstable, critically delocalized fixed point. The latter (yellow dot) is governed by the SO($n){}_{\nu }$ CFT [65, 70]. Our numerical results are generally consistent with the SO($n){}_{\nu }$ theory, see Figs. 13–15, implying that the renormalization group trajectory away from the clean limit is fine-tuned by the topology to flow into the CFT (solid vertical flow), instead of flowing into the thermal metal (dashed flow). The same conclusion was reached for a model with $\nu =4$ in Ref. [52].

Figure 12

Surface Majorana fluid band structure for the Luttinger Hamiltonian with isotropic $p$-wave pairing. This is a class DIII, strong topological superconductor with winding number $\left|\nu \right|=3$ for pairing arising from either the conduction or valence bands. Results shown here are obtained from a lattice regularization and termination of Eq. (7.2); the momentum $k_x$ is measured in units of the lattice spacing $a$. The left panel (a) shows the cubic-dispersing two-dimensional surface states obtained for hole-doping, see Eq. (7.9). Only positive $k_x$ is depicted, since the results are symmetric under reflection about $k_x=0$. The BdG parameters are ${\mathrm{\Delta }}_p=1,{\lambda }_1=0.1,{\lambda }_2=0.5$, and $\mu =-1$. The right panel (b) shows a relativistic cone centered at $\mathrm{k}=0$ and a gapless ring in the electron-doped case. The parameters are the same as for (a), except that $\mu =+1$.

Figure 13

Numerical results for the surface states of the hole-doped Luttinger semimetal with isotropic $p$-wave pairing in the bulk and time-reversal symmetry preserving disorder on the surface. The winding number of the bulk is $\nu =3$. The left plot shows the multifractal spectrum [Eqs. (7.18)–(7.20)] for two typical lowest energy surface wave functions in fixed disorder realizations. The dotted red curves are the numerical results, while the solid blue curve is the analytical prediction from the SO($n){}_3$ CFT [Eqs. (7.20)–(7.22)]. The curves marked (i) and (ii) correspond to two different disorder strengths $\lambda$; the second one is shifted vertically for clarity. Since $\lambda$ has dimensions of 1/${\left(\text{length}\right)}^2$, it is measured in units of the squared momentum cutoff ${\mathrm{\Lambda }}^2$ (see text). The system size is a $109\times 109$ grid of momenta. Box sizes $b=1$ and 5 are used to extract $\tau \left(q\right)$ [see Eqs. (7.18) and (7.19)]. The right plot shows the integrated density of states $N\left(\varepsilon \right)$. In this case, both the clean limit and the SO($n){}_3$ theory predict $N\left(\varepsilon \right)\sim {\varepsilon }^{2/3}$ [Eq. (7.17)]. The full surface density of states is exhibited in the inset. For $\nu =3$, the effects of disorder are strong, as indicated by the analytical result for the universal multifractal spectrum (blue curves, left panel). It is almost “frozen” (a frozen state has $\tau \left(q\right)=0$ for $q>q_c$ [67, 130, 132, 133, 134]). This means that the typical wave function consists of a few rare peaks with arbitrarily large separation, see Fig. 1 in Ref. [67] for an example. We expect that finite size effects are quite severe in this case, responsible for the deviation between the analytical prediction and numerics. See Figs. 14 and 15 for higher $\nu$, which give much better agreement.

Figure 14

Same as Fig. 13, but for the generalized surface model in Eq. (7.23) with $\nu =5$. Numerical results are shown as red dotted curves, while analytical predictions (blue solid curves) for $\tau \left(q\right)$ and $N\left(\varepsilon \right)$ obtain from the SO($n){}_5$ CFT. Box sizes $b=3$ and 6 are used to extract $\tau \left(q\right)$. The disorder strength $\lambda$ formally has units of $1/{\left(\text{length}\right)}^6$, hence proportional to the sixth power of the momentum cutoff $\mathrm{\Lambda }$. The absolute disorder strength is of the same order as in Fig. 13, with the same system size. The termination threshold $q_c=\sqrt{6}\simeq 2.45$ [see Eq. (7.21)].

Figure 15

Same as Fig. 13, but for the generalized surface model in Eq. (7.23) with $\nu =7$. Numerical results are shown as red dotted curves, while analytical predictions (blue solid curves) for $\tau \left(q\right)$ and $N\left(\varepsilon \right)$ obtain from the SO($n){}_7$ CFT. Box sizes $b=3$ and 6 are used to extract $\tau \left(q\right)$. The disorder strength $\lambda$ formally has units of $1/{\left(\text{length}\right)}^{10}$, hence proportional to the tenth power of the momentum cutoff $\mathrm{\Lambda }$. The absolute disorder strength is of the same order as in Fig. 13, with the same system size. The termination threshold $q_c=\sqrt{10}\simeq 3.16$ [see Eq. (7.21)].

Figure 16

Weyl superconductors that obtain via $d+id$ combinations of $T_{2g}$ and $E_g$ local pairings. Each figure shows the double- or single-Weyl nodes (block dots) that arise from the nodal-loop intersections. The pairings are (a) $d_{xy}+i{d}_{x^2-y^2}$, (b) $d_{xy}+i{d}_{3z^2-r^2}$, (c) $d_{xz}+i{d}_{x^2-y^2}$, (d) $d_{yz}+i{d}_{x^2-y^2}$, (e) $d_{xz}+i{d}_{3z^2-r^2}$, and (f) $d_{yz}+i{d}_{3z^2-r^2}$. Properties of these states are enumerated in Table 2. Nodal loops shown in green, red, blue, brown, and purple, respectively, corresponds to the ones associated with the $d_{xy},d_{xz},d_{yz},d_{x^2-y^2}$, and $d_{3z^2-r^2}$ pairings, respectively. Equations for these loops appear in the rightmost column of Table 1. Notice that $d_{xz/yz}+i{d}_{x^2-y^2}$ also supports a pair of gapless points at the north and south poles of the Fermi surface. These nodes do not possess any topological invariant and thus their existence is purely accidental. Here all momentum axes are measured in units of the Fermi momenta $k_F$.

Figure 17

Scaling of the function $F(x,y)$, defined in Eq. (H7), with $x$ for various choices of $y$. Here, $x=\mu /E_{\mathrm{\Lambda }}$ and $y=T/E_{\mathrm{\Lambda }}$ are, respectively, dimensionless chemical potential and temperature, with $E_{\mathrm{\Lambda }}={\mathrm{\Lambda }}^2/\left(2m\right)$, and $t\equiv y$. The function $F$ determines the strength of the coupling between the $s$-wave and $d$-wave pairing with lattice distortion or electronic nematicity, determined through $f_{\mathrm{str}}^{\text{Lutt}}$ shown in Eq. (H6).