Interacting topological insulator and emergent grand unified theory

Motivated by the Pati-Salam grand unified theory [J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974)], we study $(4+1)d$ topological insulators with $\mathrm{SU}\left(4\right)\times \mathrm{SU}{\left(2\right)}_1\times \mathrm{SU}{\left(2\right)}_2$ symmetry, whose $(3+1)d$ boundary has 16 flavors of left-chiral fermions, which form representations $(\mathbf{4},\mathbf{2},\mathbf{1})$ and $(\overline{\mathbf{4}},\mathbf{1},\mathbf{2})$. The key result we obtain is that, without any interaction, this topological insulator has a $\mathbb{Z}$ classification, namely, any quadratic fermion mass operator at the $(3+1)d$ boundary is prohibited by the symmetries listed above; while under interaction, this system becomes trivial, namely, its $(3+1)d$ boundary can be gapped out by a properly designed short-range interaction without generating nonzero vacuum expectation value of any fermion bilinear mass, or in other words, its $(3+1)d$ boundary can be driven into a “strongly-coupled symmetric gapped (SCSG) phase.” Based on this observation, we propose that after coupling the system to a dynamical $\mathrm{SU}\left(4\right)\times \mathrm{SU}{\left(2\right)}_1\times \mathrm{SU}{\left(2\right)}_2$ lattice gauge field, the Pati-Salam GUT can be fully regularized as the boundary states of a $(4+1)d$ topological insulator with a thin fourth spatial dimension, the thin fourth dimension makes the entire system generically a $(3+1)d$ system. The mirror sector on the opposite boundary will not interfere with the desired GUT, because the mirror sector is driven to the SCSG phase by a carefully designed interaction and is hence decoupled from the GUT.

Figure 1

Regularizing the GUT on a lattice with three extended dimensions $x_{1,2,3}$ and a compactified dimension $x_4$. The light sector (GUT) and the mirror sector are separated in the $x_4$ dimension, as two $3d$ boundaries of a $4d$ TI. The mirror sector is decoupled from GUT due to interaction, whose strength varies with $x_4$.

Figure 2

(a) Schematic phase diagram of the $4d$ TI with $\mathrm{SU}\left(4\right)\times \mathrm{SU}{\left(2\right)}_1\times \mathrm{SU}{\left(2\right)}_2$ symmetry under interaction. There exist a critical interaction strength ${\Delta }_c$, above which the topological-to-trivial transition can be circumvented. (b) and (c) The O(4) monopole core levels along a path connecting the $4d$ TI to the trivial insulator, parameterized by the reduced mass $m^{\prime }$. The effective Hamiltonian in the monopole core reads $H=H_{\text{free}}+H_{\text{int}}$, where $H_{\text{int}}$ is taken from (b) Eq. (18) or (c) Eq. (19). The 16-dimensional Hilbert space split according to $\mathrm{SU}\left(4\right)$ representations as $\mathbf{16}=\mathbf{1}+{\mathbf{1}}^{\prime }+\mathbf{4}+\overline{\mathbf{4}}+\mathbf{6}$ with the unique ground state $|\mathbf{1}\rangle +|{\mathbf{1}}^{\prime }\rangle$ (marked out in red). The dashed line marks out the $m^{\prime }=0$ critical point, where degeneracy is avoided by interaction.

Figure 3

Energy spectrum of $\mathrm{SO}\left(n\right)$ invariant Yukawa interaction ($n=1$ case labeled by ${\mathbb{Z}}_2$), which is constructed by taking the last $n$ components of ${\Phi }^a$ and coupling them to a $n$-component real boson field.