Chiral magnetic conductivity in an interacting lattice model of parity-breaking Weyl semimetal

We report on the mean-field study of the chiral magnetic effect (CME) in static magnetic fields within a simple model of parity-breaking Weyl semimetal given by the lattice Wilson-Dirac Hamiltonian with constant chiral chemical potential. We consider both the mean-field renormalization of the model parameters and nontrivial corrections to the CME originating from resummed ladder diagrams with arbitrary number of loops. We find that onsite repulsive interactions affect the chiral magnetic conductivity almost exclusively through the enhancement of the renormalized chiral chemical potential. Our results suggest that nontrivial corrections to the chiral magnetic conductivity due to interfermion interactions are not relevant in practice since they only become important when the CME response is strongly suppressed by the large gap in the energy spectrum.

Figure 1

The $CP$-breaking mass term $m_i$ in (23) as a function of interaction potential $U$ for different values of the bare chiral chemical potential ${\mu }_A^{\left(0\right)}$.

Figure 2

Mean-field phase diagram of our model in the $m^{\left(0\right)}-U$-plane for different values of the bare chiral chemical potential ${\mu }_A$. The points mark the boundaries of the Aoki phase.

Figure 3

Volume dependence of the mean-field phase diagram. The Aoki fingers are volume dependent and seem to vanish as $L_s\to \infty$. This is only relevant if ${\mu }_A^{\left(0\right)}=0$. At finite ${\mu }_A^{\left(0\right)}$, the fingers are not present and phase diagrams for different lattice volumes lie on top of each other. In the figure, the phase boundaries for ${\mu }_A^{\left(0\right)}=0.10$ were shifted by $-0.8$ in $U$ for better visibility.

Figure 4

Renormalized chiral chemical potential ${\mu }_A$ as a function of the interaction potential $U$ for different values of the bare chiral chemical potential ${\mu }_A^{\left(0\right)}$.

Figure 5

Phase diagram in the parameter space of bare mass $m^{\left(0\right)}$ and the interaction potential $U$ at different temperatures. The chiral chemical potential is fixed at ${\mu }_A^{\left(0\right)}=0.1$.

Figure 6

Numerical results for the chiral magnetic conductivity ${\sigma }_{\text{CME}}\left(k\right)$ at characteristic values of $m_r$ for different values of the interaction potential $U$ (lines with points). Clockwise from top left: $m_r=0.00,-2.00,-4.00$, and $-2.20$. In all cases ${\mu }_A^{\left(0\right)}=0.05$ and $L_s=50$. For comparison, we also plot the result (28) for free continuum Dirac fermions with Pauli-Villars regularization and with the renormalized chiral chemical potential ${\mu }_A$ (lines of the same style as for numerical data, but with no points on them).

Figure 7

Numerical estimate ${\sigma }_{\max }$ of the asymptotic value of ${\sigma }_{\mathrm{CME}}\left(k\right)$ at ${\mu }_A\ll k\ll 1$ as a function of bare mass $m^{\left(0\right)}$ and the interaction potential $U$ at fixed bare chiral chemical potential ${\mu }_A^{\left(0\right)}=0.05$. The red crosses mark the border of the Aoki phase and the black points mark the lines of constant $m_r=0,-2,-4,-6$.

Figure 8

Numerical estimate ${\sigma }_{\max }$ of the asymptotic values of ${\sigma }_{\mathrm{CME}}\left(k\right)$ at ${\mu }_A\ll k\ll 1$ as a function of the renormalized chiral chemical potential ${\mu }_A$ for different values of the bare mass and the interaction potential. The points are connected with straight lines to guide the eye. For small ${\mu }_A, {\sigma }_{\max }$ is a linear function of the renormalized chiral chemical potential.

Figure 9

The ratio of the estimated asymptotic value ${\sigma }_{\max }$ of ${\sigma }_{\mathrm{CME}}\left(k\right)$ at ${\mu }_A\ll k\ll 1$ to the renormalized chiral chemical potential ${\mu }_A$. The black dots mark the lines of constant renormalized mass $m_r=-6,-4,-2$, and 0, respectively, and the red crosses indicate the border of the Aoki phase for ${\mu }_A^{\left(0\right)}=0.0$.

Figure 10

Relative importance of nontrivial corrections to ${\sigma }_{\mathrm{CME}}\left(k\right)$ due to interfermion interactions. Solid lines with symbols correspond to the contribution of tree diagram to ${\sigma }_{\mathrm{CME}}\left(k\right)$ [first summand in (19)] and dashed lines with the same symbols to the contribution of loop diagrams [second summand in (19)]. Since the contribution of loop diagrams turns out to be numerically very small, we multiply it by a factor ${10}^2$ ($m_r=-2.0$) or ${10}^4$ ($m_r=0.0$).

Figure 11

The ratio of the contribution of tree-level diagram [first term in (19)] to the contribution of the loop diagrams [second term in (19)] to the chiral magnetic conductivity ${\sigma }_{\mathrm{CME}}\left(k\right)$ at $k=6$ and ${\mu }_A^{\left(0\right)}=0.05$ as a function of the bare mass $m^{\left(0\right)}$ and interaction potential $U$.

Figure 12

Numerical results for ${\sigma }_{\mathrm{CME}}\left(k\right)$ at finite temperatures. Point styles denote different sets of model parameters ($m_r$ and $U$) and line styles different values of temperature. The bare chiral chemical potential is ${\mu }_A^{\left(0\right)}=0.15$ everywhere.

Figure 13

Electric current density $j_3\left(\rho \right)$ in the direction of the magnetic field as a function of the radial coordinate $\rho$ for different values of the dimensionless quantity ${\mu }_{A}R$, where $R$ is the radius of the region with nonzero magnetic field. For convenience, we show the ratio of $j_3\left(\rho \right)$ to the conventional value $j_3=\frac{{\mu }_{A}B}{2{\pi }^2}$ which follows from (1) and (2).