Generic “unnecessary” quantum critical points with minimal degrees of freedom

We explore generic “unnecessary” quantum critical points with minimal degrees of freedom. These quantum critical points can be avoided with strong enough symmetry-allowed deformations of the Hamiltonian, but these deformations are irrelevant perturbations below certain threshold at the quantum critical point. These quantum critical points are hence unnecessary, but also unfine-tuned (generic). The previously known examples of such generic unnecessary quantum critical points involve at least eight Dirac fermions in both two and three spatial dimensions. In this work we seek for examples of generic unnecessary quantum critical points with minimal degrees of freedom. In particular, in three-dimensional space, we identify two examples of such generic unnecessary quantum critical points. The first example occurs in a $3d$ interacting topological insulator, and it is described by two $(3+1)d$ massless Dirac fermions in the infrared limit; the second example occurs in a $3d$ topological superconductor, and it is formally described by only one $(3+1)d$ massless Dirac fermion.

Figure 1

The schematic phase diagram and renormalization group flow of the generic unnecessary QCP. The horizontal axis is the tuning parameter across the QCP, in the examples discussed in this work it corresponds to the mass of either two or one $(3+1)d$ Dirac fermions; the vertical axis is the interaction, which is perturbatively irrelevant at the QCP in the noninteracting limit. There exists a adiabatic curve (dashed curve) in the entire phase diagram that connects the two sides of the QCP.

Figure 2

The coupled wire construction of the $2d$ edge state Hamiltonian Eq. (11) of the noninteracting $3d$ TI with $\mathrm{U}\left(1\right)\times Z_n\times \mathcal{P}$ symmetry. Each wire represents the $1d$ edge state of a $2d$ TI with $\mathrm{U}\left(1\right)\times Z_n$ symmetry, which is described by Eq. (1). Each wire is composed of a pair of counter-propagating $1d$ fermion modes that carry opposite $Z_n$ charges. Symmetry allowed tunnellings between the wires will drive the system into the $2d$ edge state Hamiltonian Eq. (11).

Figure 3

The coupled wire construction for the symmetric gapped edge state with $n=3$. Each wire is the boundary state of a $2d$ layer, and the wire with index $(-1)$ can be adiabatically deformed into a wire with index $(+2)$ under interaction without any transition in the $2d$ layer, due to the ${\mathbb{Z}}_3$ classification of the layer. Hence, the wires can be regrouped and gapped out by interactions that preserves all the symmetries.

Figure 4

Illustration of the symmetric gapped edge state with $n=5$. Each wire with index $(-1)$ can be deformed into a wire with index $(+4)$ under interaction, and again the wires can be regrouped and gapped out by interactions that preserve all the symmetries.