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Interacting Topological Insulator and Emergent Grand Unified Theory

Yi-Zhuang You, Cenke Xu

TL;DR

The paper develops a framework to regularize the Pati-Salam GUT on a lattice by embedding it on the 3d boundary of a 4d interacting topological insulator with $SU(4)\times SU(2)_1\times SU(2)_2$ symmetry. It shows that the noninteracting boundary has a $\mathbb{Z}$ classification, but appropriately designed interactions drive the bulk TI to a trivial phase, enabling the boundary to enter a strongly coupled symmetric gapped (SCSG) phase and decouple the mirror sector. Through defect-proliferation and monopole/vortex-core gapping (via $SO(6)$/$SO(7)$-invariant four-fermion interactions and FK-type constructions), the authors construct explicit interacting lattice Hamiltonians that realize the Pati-Salam gauge structure on the boundary when coupled to a dynamical lattice gauge field, using a thin fourth dimension to keep the IR physics effectively 3d. This cross-pertilization between interacting TI theory and high-energy lattice regularization provides a route to circumvent fermion-doubling while preserving chiral gauge couplings, with potential implications for QFT regularization and beyond.

Abstract

Motivated by the Pati-Salam Grand Unified Theory, we study $(4+1)d$ topological insulators with $SU(4) \times SU(2)_1 \times SU(2)_2$ symmetry, whose $(3+1)d$ boundary has 16 flavors of left-chiral fermions, which form representations $(\mathbf{4}, \mathbf{2}, \mathbf{1})$ and $(\bar{\mathbf{4}}, \mathbf{1}, \mathbf{2})$. 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 $SU(4) \times SU(2)_1 \times SU(2)_2$ lattice gauge field, the Pati-Salam GUT can be fully regularized as the boundary states of a $(4+1)d$ topological insulator with a {\it 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 {\it 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.

Interacting Topological Insulator and Emergent Grand Unified Theory

TL;DR

The paper develops a framework to regularize the Pati-Salam GUT on a lattice by embedding it on the 3d boundary of a 4d interacting topological insulator with symmetry. It shows that the noninteracting boundary has a classification, but appropriately designed interactions drive the bulk TI to a trivial phase, enabling the boundary to enter a strongly coupled symmetric gapped (SCSG) phase and decouple the mirror sector. Through defect-proliferation and monopole/vortex-core gapping (via /-invariant four-fermion interactions and FK-type constructions), the authors construct explicit interacting lattice Hamiltonians that realize the Pati-Salam gauge structure on the boundary when coupled to a dynamical lattice gauge field, using a thin fourth dimension to keep the IR physics effectively 3d. This cross-pertilization between interacting TI theory and high-energy lattice regularization provides a route to circumvent fermion-doubling while preserving chiral gauge couplings, with potential implications for QFT regularization and beyond.

Abstract

Motivated by the Pati-Salam Grand Unified Theory, we study topological insulators with symmetry, whose boundary has 16 flavors of left-chiral fermions, which form representations and . The key result we obtain is that, without any interaction, this topological insulator has a classification, namely any quadratic fermion mass operator at the boundary is prohibited by the symmetries listed above; while under interaction this system becomes trivial, namely its 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 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 lattice gauge field, the Pati-Salam GUT can be fully regularized as the boundary states of a topological insulator with a {\it thin} fourth spatial dimension, the thin fourth dimension makes the entire system generically a system. The mirror sector on the opposite boundary will {\it 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.

Paper Structure

This paper contains 14 sections, 44 equations, 3 figures, 1 table.

Table of Contents

  1. 1. Introduction
  2. 2. A toy model
  3. 3. Pati-Salam GUT
  4. 4. Summary
  5. Supplemental Material
  6. A. Review of SCSG Phase and Interacting TI in Lower Dimensions
  7. A1. SCSG Phase in $0d$: Boundary of $1d$ Systems
  8. A2. SCSG phase in $1d$: boundary of $2d$ systems
  9. A3. SCSG phase in $2d$: boundary of $3d$ systems
  10. B. Decomposition and Reconstruction of the Interaction
  11. B1. The $0d$ Case: Fidkowski-Kitaev Interaction and its Variants
  12. B2. Higher Dimensions: Generic Yukawa Interaction
  13. C. Pati-Salam Model in Majorana Fermion Basis
  14. D. Classification of $4d$ Free Fermion Topological Insulators

Figures (3)

  • Figure 1: (Color online.) 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 $\mathrm{SU}(4)\times\mathrm{SU}(2)_1\times\mathrm{SU}(2)_2$ symmetry under interaction. There exist a critical interaction strength $\Delta_c$, above which the topological-to-trivial transition can be circumvented. (b,c) The O(4) monopole core levels along a path connecting the $4d$ TI to the trivial insulator, parameterized by the reduced mass $m'$. The effective Hamiltonian in the monopole core reads $H=H_\text{free}+H_\text{int}$, where $H_\text{int}$ is taken from (b) Eq. \ref{['eq: int core']} or (c) Eq. \ref{['eq: int fact']}. The 16-dimensional Hilbert space split according to $\mathrm{SU}(4)$ representations as $\mathbf{16}=\mathbf{1}+\mathbf{1}'+\mathbf{4}+\bar{\mathbf{4}}+\mathbf{6}$ with the unique ground state $|\mathbf{1} \rangle+|\mathbf{1}' \rangle$ (marked out in red). The dashed line marks out the $m'=0$ critical point, where degeneracy is avoided by interaction.
  • Figure 3: Energy spectrum of $\mathrm{SO}(n)$ 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.