Bosonization in $2+1$ dimensions via Chern-Simons bosonic particle-vortex duality

Dualities provide deep insight into physics by relating two seemingly distinct theories. Here we consider a duality between lattice fermions and bosons in ($2+1$) spacetime dimensions, relating free massive Dirac fermions to Abelian Chern-Simons Higgs (ACSH) bosons. To establish the duality, we represent the exact partition function of the lattice fermions in terms of the writhe of fermionic worldlines. On the bosonic side, the partition function is expressed in the writhe of the vortex loops of the particle-vortex dual of the ACSH Lagrangian. In the continuum and scaling limit, we show these to be identical. This result can be understood from the closed fermionic worldlines being direct mappings of the ACSH vortex loops, with the writhe keeping track of particle statistics.

Figure 1

(a) For this loop, the perimeter $L$ is equal to the number of lattice sites $2n=6$, the number of straight sections/sides is $k=4$, and the number of windings $I=1$. (b) Here, $2n=8$, $k=6$, and $I=1$ (the number of winding would not change with the orientation of the loop). (c) Here, $2n=16$, $I=2$, and $k=12$. (d) Nonallowed loop, since it consists of backtracking.

Figure 2

For a given $\tilde{C}$, (a) ${\mathrm{\Omega }}_i$ is the solid angle traced by ${\overrightarrow{e}}_i$ while traveling from $i$ to $i+1$. (b) ${\mathrm{\Omega }}^{\prime }$ is $4\pi -\mathrm{\Omega }$.

Figure 3

(a) $C$ is the worldline of a fermion and $\overrightarrow{e}$ is the unit tangent to $C$. (b) $\tilde{C}$ is the curve drawn by the tip of $\overrightarrow{e}$ while traveling on $C$.

Figure 4

(a) Here $\overrightarrow{e}$ is the tangent vector at point ${\overrightarrow{r}}_1$ on curve $C_1$, and $\epsilon \overrightarrow{a}$ is the distance between $C_2$ and $C_1$, where $\overrightarrow{a}·{\overrightarrow{r}}_1=0$. (b) If $C_2$ winds around $C_1$, $\overrightarrow{a}$ rotates around $C_1$. If we view $C_1$ and $C_2$ as edges of a ribbon, then $\overrightarrow{a}$ rotates as the ribbon twists. The expression $\frac{\overrightarrow{a}\times \overrightarrow{\dot{a}}{\epsilon }^2}{{\epsilon }^2}·\overrightarrow{e}$ can be regarded as the angular speed of the point $\epsilon \overrightarrow{a}$ around $\overrightarrow{e}$, and the expression for the twist as given in Eq. (35d) is the angular displacement of $\epsilon \overrightarrow{a}$ divided by $2\pi$.