Exact Boson-Fermion Duality on a 3D Euclidean Lattice

The idea of statistical transmutation plays a crucial role in descriptions of the fractional quantum Hall effect. However, a recently conjectured duality between a critical boson and a massless two-component Dirac fermion extends this notion to gapless systems. This duality sheds light on highly nontrivial problems such as the half-filled Landau level, the superconductor-insulator transition, and surface states of strongly coupled topological insulators. Although this boson-fermion duality has undergone many consistency checks, it has remained unproven. We describe the duality in a nonperturbative fashion using an exact UV mapping of partition functions on a 3D Euclidean lattice.

Figure 1

(a) The various terms that arise on a link $n\mu$ in the exact expansion of Grassmann fields in $Z_W$. From top to bottom, the contributions are hopping ${\overline{\chi }}_{n+\widehat{\mu }}(-{\sigma }^{\mu }-1){\chi }_n/2$, hopping ${\overline{\chi }}_n({\sigma }^{\mu }-1){\chi }_{n+\widehat{\mu }}/2$, double hopping plus interaction $(1+U)[{\overline{\chi }}_n({\sigma }^{\mu }-1){\chi }_{n+\widehat{\mu }}/2][{\overline{\chi }}_{n+\widehat{\mu }}(-{\sigma }^{\mu }-1){\chi }_n/2]$. (b) In a Grassmann integral, each fermion component must appear exactly once. Consider a conjugate pair of fermion components, say ${\chi }_{n\uparrow }$ and ${\overline{\chi }}_{n\uparrow }$. They either appear together in a mass term, or appear separately in two link terms. So the link terms always form closed loops. If this condition is not satisfied as in (c), the contribution vanishes by Grassmann algebra. Thus, all contributions to $Z_W$ manifestly satisfy Gauss’s law. (The lattice is 3D. We drew a 2D lattice for clarity.)

Figure 2

The gray dashed curve (section of parabola) contains the values of $M$, $U$ that satisfy Eq. (14) for some $T_c$. Suppose we have chosen some $M$, $U$ on the gray dashed curve, say the blue dot. Then as $T$ increases from 0 to $+\infty$, the corresponding $M^{\prime }$, $U^{\prime }$ trace up along the blue section of the parabola. The red line represents the phase boundary between the $C=1$ and $C=0$ phase regions in the vicinity of $M=3$, $U=0$. The slope is $\approx -8.8$ but we did not compute higher derivatives, so we do not know how the phase boundary looks further away from the point $M=3$, $U=0$. However, this is sufficient to show our choice of blue dot lies within the $C=1$ phase as desired.