Dimensional crossover of effective orbital dynamics in polar distorted ${}^3\mathrm{He}\text{−}\mathrm{A}$: Transitions to antispacetime

Topologically protected superfluid phases of ${}^3\mathrm{He}$ allow one to simulate many important aspects of relativistic quantum field theories and quantum gravity in condensed matter. Here we discuss a topological Lifshitz transition of the effective quantum vacuum in which the determinant of the tetrad field changes sign through a crossing to a vacuum state with a degenerate fermionic metric. Such a transition is realized in polar distorted superfluid ${}^3\mathrm{He}\text{−}\mathrm{A}$ in terms of the effective tetrad fields emerging in the vicinity of the superfluid gap nodes: the tetrads of the Weyl points in the chiral A-phase of ${}^3\mathrm{He}$ and the degenerate tetrad in the vicinity of a Dirac nodal line in the polar phase of ${}^3\mathrm{He}$. The continuous phase transition from the $A$-phase to the polar phase, i.e., the transition from the Weyl nodes to the Dirac nodal line and back, allows one to follow the behavior of the fermionic and bosonic effective actions when the sign of the tetrad determinant changes, and the effective chiral spacetime transforms to antichiral “anti-spacetime.” This condensed matter realization demonstrates that while the original fermionic action is analytic across the transition, the effective action for the orbital degrees of freedom (pseudo-EM) fields and gravity have nonanalytic behavior. In particular, the action for the pseudo-EM field in the vacuum with Weyl fermions (A-phase) contains the modulus of the tetrad determinant. In the vacuum with the degenerate metric (polar phase) the nodal line is effectively a family of $2+1\mathrm{d}$ Dirac fermion patches, which leads to a non-analytic $(B^2-E^2{)}^{3/4}$ QED action in the vicinity of the Dirac line.

Figure 1

Schematic illustration of the Lifshitz transition through the polar phase in polar distorted ${}^3\mathrm{He}\text{−}\mathrm{A}$, where right-handed fermions transform to left-handed and vice versa. (a) ${}^3\mathrm{He}\text{−}\mathrm{A}$ has two (spin-degenerate) Weyl points of opposite chirality (red stars) at ${\mathbf{W}}_{\pm }=\pm p_F\mathbf{l}$ with triads ${\mathbf{e}}_a^{\pm }$. (b) As the triad ${\mathbf{e}}_2$ is varied adiabatically through zero, the polar phase with a nodal line (red) emerges from the poles when ${\mathbf{e}}_2=0$. (c) For the adiabatic evolution ${\widehat{\mathbf{e}}}_2\to -{\widehat{\mathbf{e}}}_2$ or a $SO(3)$-rotation of the whole orbital part $\widehat{\mathbf{l}}\to -\widehat{\mathbf{l}}$ with respect to the laboratory frame, the chirality of the Weyl points interchanges as well.

Figure 2

Dependence of the effective vector potential of $2+1\mathrm{d}$ Dirac fermions on their position on the nodal line in Eq. (20). This field is caused by the texture of the polar phase order parameter $\widehat{\mathbf{m}}(\mathbf{r})={\widehat{\mathbf{m}}}_0+\delta \widehat{\mathbf{m}}(\mathbf{r})$. This vector is perpendicular to the nodal line, and thus the deformation $\widehat{\mathbf{m}}(\mathbf{r})$ tilts the plane of the nodal line. Such deformation corresponds to the Goldstone mode propagating in the polar phase. In terms of the effective $2+1\mathrm{d}$ QED, this is the photon mode.