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From $G_2$ to $SO(8)$: Emergence and reminiscence of supersymmetry and triality

Zhi-Qiang Gao, Congjun Wu

TL;DR

The paper develops a (1+1)D continuum model with $G_2$ symmetry for four-component fermions and analyzes it with one-loop RG to uncover a rich phase diagram comprising four gapped and five gapless phases. The gapped sectors flow to four stable $SO(8)$ Gross-Neveu fixed points organized by emergent triality, while the gapless sector includes three $SO(7)$ critical phases, one $G_2$ critical phase described by a product of Ising and tricritical Ising CFTs, and a Luttinger liquid; the $G_2$ critical phase exhibits an emergent spacetime supersymmetry with total central charge $c= rac{6}{5}$. The lattice realization with Hubbard-type interactions breaks the triality to a duality between two $SO(7)$ sectors, and the supersymmetric $G_2$ critical phase leaves behind a boson-fermion degeneracy reminiscence rather than a full quantum-mechanical SUSY. Embedding between continuum and lattice descriptions clarifies which continuum fixed points appear along lattice axes, showing that positive $u=v$ corresponds to the $G_2$-TCI critical point and negative $u=v$ to the $SO(7)_M$ GN critical point, with Ising-type transitions occurring along lines like $u=-7v$; the work reveals how hidden $SO(8)$ triality and $G_2$-related criticality can emerge from a $G_2$-invariant lattice model, offering insights into exotic multicritical phenomena in strongly correlated systems.

Abstract

We construct a (1+1)-dimension continuum model of 4-component fermions incorporating the exceptional Lie group symmetry $G_2$. Four gapped and five gapless phases are identified via the one-loop renormalization group analysis. The gapped phases are controlled by four different stable $SO(8)$ Gross-Neveu fixed points, among which three exhibit an emergent triality, while the rest one possesses the self-triality, i.e., invariant under the triality mapping. The gapless phases include three $SO(7)$ critical ones, a $G_2$ critical one, and a Luttinger liquid. Three $SO(7)$ critical phases correspond to different $SO(7)$ Gross-Neveu fixed points connected by the triality relation similar to the gapped SO(8) case. The $G_2$ critical phase is controlled by an unstable fixed point described by a direct product of the Ising and tricritical Ising conformal field theories with the central charges $c=\frac{1}{2}$ and $c=\frac{7}{10}$, respectively, while the latter one is known to possess spacetime supersymmetry. In the lattice realization with a Hubbard-type interaction, the triality is broken into the duality between two $SO(7)$ symmetries and the supersymmetric $G_2$ critical phase exhibits the degeneracy between bosonic and fermionic states, which are reminiscences of the continuum model.

From $G_2$ to $SO(8)$: Emergence and reminiscence of supersymmetry and triality

TL;DR

The paper develops a (1+1)D continuum model with symmetry for four-component fermions and analyzes it with one-loop RG to uncover a rich phase diagram comprising four gapped and five gapless phases. The gapped sectors flow to four stable Gross-Neveu fixed points organized by emergent triality, while the gapless sector includes three critical phases, one critical phase described by a product of Ising and tricritical Ising CFTs, and a Luttinger liquid; the critical phase exhibits an emergent spacetime supersymmetry with total central charge . The lattice realization with Hubbard-type interactions breaks the triality to a duality between two sectors, and the supersymmetric critical phase leaves behind a boson-fermion degeneracy reminiscence rather than a full quantum-mechanical SUSY. Embedding between continuum and lattice descriptions clarifies which continuum fixed points appear along lattice axes, showing that positive corresponds to the -TCI critical point and negative to the GN critical point, with Ising-type transitions occurring along lines like ; the work reveals how hidden triality and -related criticality can emerge from a -invariant lattice model, offering insights into exotic multicritical phenomena in strongly correlated systems.

Abstract

We construct a (1+1)-dimension continuum model of 4-component fermions incorporating the exceptional Lie group symmetry . Four gapped and five gapless phases are identified via the one-loop renormalization group analysis. The gapped phases are controlled by four different stable Gross-Neveu fixed points, among which three exhibit an emergent triality, while the rest one possesses the self-triality, i.e., invariant under the triality mapping. The gapless phases include three critical ones, a critical one, and a Luttinger liquid. Three critical phases correspond to different Gross-Neveu fixed points connected by the triality relation similar to the gapped SO(8) case. The critical phase is controlled by an unstable fixed point described by a direct product of the Ising and tricritical Ising conformal field theories with the central charges and , respectively, while the latter one is known to possess spacetime supersymmetry. In the lattice realization with a Hubbard-type interaction, the triality is broken into the duality between two symmetries and the supersymmetric critical phase exhibits the degeneracy between bosonic and fermionic states, which are reminiscences of the continuum model.

Paper Structure

This paper contains 14 sections, 62 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The model Hamiltonians
  3. The minimal $G_2$ symmetric lattice model
  4. The coarse-grained continuum model in (1+1)D
  5. Renormalization group flows and fixed planes
  6. The $SO(7)$ symmetric fixed planes
  7. The $G_2$ critical fixed planes
  8. Phase diagrams in fixed bodies
  9. Embedding of the lattice model
  10. Conclusions
  11. Fermion bilinears and majorana operators in the lattice model
  12. RG equations
  13. Bosonization of the interaction on $SO(7)$ symmetric fixed planes
  14. RG for the lattice model

Figures (5)

  • Figure 1: The RG flows and phase diagram on the $SO(7)_A$ symmetric fixed plane. The horizontal and vertical axes are labeled by $g$ and $y$, respectively. Phase boundaries are marked by thick lines. The $SO(8)$ GN, $SO(8)_A$ GN, and $SO(7)_A$ GN fixed points (defined as $g\rightarrow +\infty$ points of the fixed flows) are marked by brown, blue, and crimson dots, respectively. The yellow dot stands for the free Majorana fixed point. "LL" denotes the Luttinger liquid phase. The transition between the $SO(8)$ GN phase and the $SO(8)_A$ GN phase is described by the critical $SO(7)_A$ GN phase in the Ising universality class. Phase structures in the $SO(7)_{B/M}$ fixed planes can be obtained via the triality mapping. The embedding of the $u$-axis in the lattice model phase diagram is colored in red.
  • Figure 2: The RG flows and phase diagram on the $G_2$ critical fixed planes spanned by the $G_2$ TCI fixed flow and the $SO(7)_M$ critical GN fixed flow. The horizontal and vertical axes are labeled by $g$ and $t_M$, respectively. The $SO(7)_M$, $G_2$ TCI, and free Majorana fixed points are denoted by the indigo, green, and yellow dots, respectively. Phase structures in other $G_2$ critical fixed planes are similar. The embedding of the $u=v$ line in the lattice model is colored in red.
  • Figure 3: The phase diagrams in the fixed bodies (a) $\tilde{t}=t_A+t_B+t_M=0$ and (b) $t_A=t_B$. The fixed flows are marked as arrows, and the critical surfaces are colored. In (a), $SO(8)_A$ GN, $SO(8)_B$ GN, $SO(8)_M$ GN, $G_2$ TCI, and free Majorana fixed points are denoted by crimson, orange, purple, green, and yellow dots, respectively. The transition between each two of the three gapped $SO(8)_i$ GN phases is through either the $G_2$ TCI phase or the Luttinger liquid (LL) critical phase. The triality relation is manifest in the $D_3$ symmetry of the phase diagram. In (b), the $SO(8)$ GN, $SO(7)_M$ GN, $SO(8)_M$ GN, $G_2$ TCI, and free Majorana fixed points are denoted by brown, indigo, purple, green, and yellow dots, respectively. The transition between the $SO(8)$ GN and the $SO(8)_M$ GN phases is through the critical $SO(7)_M$ GN phase in the Ising universality class. The embedding of the $u=v$ axis in the lattice model phase diagram is shown in red, whose positive and negative semi-axes are embedded in the $G_2$ TCI and the critical $SO(7)_M$ GN phases, respectively.
  • Figure 4: (a) The lattice model phase diagram. There are totally three gapped phases, belonging to $SO(8)_A$ GN, $SO(8)_B$ GN, and $SO(8)$ GN phases in the continuum model. The phase transition between $SO(8)_A$ GN and $SO(8)_B$ GN phases, marked as the green segment, is located at $u=v>0$, and described by $G_2$ TCI phase in the continuum model. The phase transition between $SO(8)_A$ ($SO(8)_B$) GN and $SO(8)$ GN phases, represented by the blue (turquoise) segment, is located at $u=-7v$ ($v=-7u$) and belongs to the critical $SO(7)_A$ ($SO(7)_B$) GN phase in the continuum model. Inside the $SO(8)$ GN phase there is a branch cut colored in indigo, on which the system becomes gapless. It is located at $u=v<0$ and described by the critical $SO(7)_M$ GN phase in the continuum model. (b) Embedding of the lattice model phase diagram into the continuum model phase diagram. RG flows of the continuum model in body $4g-2t_A-2t_B+t_M=0$ (not a fixed body) spanned by $SO(7)_{A}$ and $SO(7)_B$ symmetric fixed planes ($P_{A}$ and $P_B$) are shown in the right. On each $SO(7)$ symmetric fixed plane the RG flows are the same as figure \ref{['fig:plane']} (a) ensured by the triality. The phase boundaries are marked as thick segments. $P_A$ and $P_B$, as well as RG flows on them, are colored in crimson and orange, respectively. The lattice model phase diagram is shown in the left part. The $u/v$-axis is embedded in $P_{A/B}$ as a straight line.
  • Figure 5: RG flow on $SO(7)$ symmetric fixed planes.