Non-Abelian Chern-Simons theory from a Hubbard-like model

Here, we provide a simple Hubbard-like model of spin-$1/2$ fermions that gives rise to the SU(2)-symmetric Thirring model that is equivalent, in the low-energy limit, to the Yang-Mills-Chern-Simons model. First, we identify the regime that simulates the SU(2) Yang-Mills theory. Then, we suitably extend this model so that it gives rise to the SU(2) Chern-Simons theory with level $k\ge 2$ that can support non-Abelian anyons. This is achieved by introducing multiple fermionic species and modifying the Thirring interactions, while preserving the SU(2) symmetry. Our proposal provides the means to theoretically and experimentally probe non-Abelian SU(2) level $k$ topological phases.

Figure 1

(Left) The honeycomb lattice with its unit cell consisting of two sites, $b$ and $w$. Fermions at a certain site tunnel to their neighboring and next-to-neighboring sites, with coupling $t$ and $t^{\prime }$, respectively. The phase factor ${\chi }_{\mathbf{i},\mathbf{j}}=\pm i$ in Hamiltonian (1) has $+$ sign when the link $⟨⟨i,j⟩⟩$ points along the directions of ${\mathbf{n}}_1$, ${\mathbf{n}}_2$ or ${\mathbf{n}}_1-{\mathbf{n}}_2$ and $-$ sign otherwise, where ${\mathbf{n}}_1=(3/2,\sqrt{3}/2)$ and ${\mathbf{n}}_2=(3/2,-\sqrt{3}/2)$. (Right) The energy dispersion relation along $p_y$ where both Fermi points, ${\mathbf{P}}_{\pm }$, reside for generic values of the couplings. The corresponding energy gaps, $\mathrm{\Delta }{E}_{\pm }$, can be independently tuned.

Figure 2

The fermionic interactions, given by grey lines, within a single unit cell that includes one black and one white site (see Fig. 1). (Left) The interactions for the single fermionic species model between populations $n_s^{\alpha }={\alpha }_s^{†}{\alpha }_s$ with $\alpha =b,w$, $s=\uparrow ,\downarrow$ and strength $\pm U$. (Right) The interactions for the two fermionic species model. We can consider this as a bilayered system with the interactions between the populations $n_{\beta s}^{\alpha }={\alpha }_{\beta s}^{†}{\alpha }_{\beta s}$ with $\alpha =b,w$, $s=\uparrow ,\downarrow$ and $\beta =1,2$ given explicitly by Hamiltonian (14).

Figure 3

Examples of a rectangular loops $K$ with area $A_K=L_1{L}_2$. (Left) When $L_1$ is small, of the order of the correlation length $\xi$, then the short-distance behavior of our model is given by (11). Then, confinement is manifested by the area law of the Wilson loop observable, $⟨W(K)⟩\approx e^{-\sigma A_K}$, with string tension $\sigma \sim g^{-4}$ [22]. (Right) When both $L_1$ and $L_2$ are large compared to the correlation length $\xi \sim g^2$ of the theory then the long distance behavior is given by (16). Then $⟨W(K)⟩=V_K(q)$, where $V_K(q)$ is the Jones polynomial of the loop $K$ with variable $q$. For the simple loop considered here it is $V_K(q)=1$. This is in stark contrast to the confining regime where $⟨W(K)⟩$ tends to zero as $A_K$ increases.