Fermionic dualities with axial gauge fields

The dualities that map hard-to-solve, interacting theories to free, noninteracting ones often trigger a deeper understanding of the systems to which they apply. However, simplifying assumptions such as Lorentz invariance, low dimensionality, or the absence of axial gauge fields, limit their application to a broad class of systems, including topological semimetals. Here we derive several axial field theory dualities in $2+1$ and $3+1$ dimensions by developing an axial slave-rotor approach capable of accounting for the axial anomaly. Our $2+1$-dimensional duality suggests the existence of a dual, critical surface theory for strained three-dimensional nonsymmorphic topological insulators. Our $3+1$-dimensional duality maps free Dirac fermions to Dirac fermions coupled to emergent U(1) and Kalb-Ramond vector and axial gauge fields. Upon fixing an axial field configuration that breaks Lorentz invariance, this duality maps free to interacting Weyl semimetals, thereby suggesting that the quantization of the nonlinear circular photogalvanic effect can be robust to certain interactions. Our paper emphasizes how axial and Lorentz-breaking dualities improve our understanding of topological matter.

Figure 1

Schematic summary of the $2+1$-dimensional axial dualities discussed in this paper. (a) A theory of two helical massless fermions $\psi$ is coupled an external vector ($A_{\mu }$) and helical gauge fields ($A_{5,\mu }$). Physically, this can describe the double Dirac surface state of a strained nonsymmorphic topological insulator. Depending on the realization of time-reversal symmetry, this theory maps to two different extensions of ${\mathrm{QED}}_3$ of neutral fermions $f$, with mutual Chern-Simons coupling the two external fields with two emergent vector and helical fields, $a_{\mu }$ and $a_{5,\mu }$. (b) These dualities suggest the existence of dual surface theories for the double Dirac surface state of a strained nonsymmorphic topological insulator.

Figure 2

Schematic summary of the $3+1$-dimensional axial dualities discussed in this paper. (a) A massless Dirac fermion in $3+1$-dimensions $\psi$, coupled to an external vector ($A_{\mu }$) and chiral ($A_{5,\mu }$) gauge fields is dual to a neutral Dirac fermion $f$ coupled to vector and axial dynamical gauge fields ($a_{\mu }$ and $a_{5,\mu }$), and a vector and axial Kalb-Ramond fields ($B_{\rho \sigma }$ and $B_{5,\rho \sigma }$) through terms of the form ${\epsilon }^{\mu \nu \rho \sigma }{F}_{\mu \nu }{B}_{\rho \sigma }$, known as BF terms. The last term accounts for the chiral anomaly. (b) When $A_{5,\mu }$ is set to a constant ($-b_{\mu }$), Lorentz symmetry is broken and the noninteracting theory describes a Weyl semimetal with Weyl node separation set by $b_{\mu }$. Its dual is an interacting Weyl semimetal theory with a BF and mixed Carroll-Field-Jackiw terms. This duality suggests that an interacting Weyl theory can display a quantized photogalvanic effect.

Figure 3

Hall conductivity of a theory with two Weyl cones separated in the $k_z$ direction. For each $k_z$, the theory is equivalent to two $2+1$-dimensional gapped Dirac theories with $k_z$-dependent masses (dashed lines). The total 3D Hall conductivity is the integral of the solid curve and is proportional to the Weyl node separation, see Eq. (C3).