Phase transitions in three-dimensional Dirac semi-metals using Schwinger–Dyson equations

Abstract

We study the semi-metal/insulator quantum phase transition in three-dimensional Dirac semi-metals by solving a set of Schwinger–Dyson equations. We study the effect of an anisotropic fermion velocity on the critical coupling of the transition. We consider the influence of several different approximations that are commonly used in the literature and show that results for the critical coupling change considerably when some of these approximations are relaxed. Most importantly, the nature of the dependence of the critical coupling on the anisotropy depends strongly on the approximations that are used for the photon polarization tensor. On the one hand, this means that calculations that include full photon dynamics are necessary to answer even the basic question of whether the critical coupling increases or decreases with anisotropy. On the other hand, our results mean that it is possible that anisotropy could provide a mechanism to promote dynamical gap generation in realistic three-dimensional Dirac semi-metallic materials.

Graphical abstract

Appendices

Appendix A: One loop polarization tensor

The zero–zero component of the one loop polarization tensor is obtained from the integral

$$\begin{aligned}{} & {} \Pi _{00}(p_0,\vec p) = - N_\textrm{f} e^2 \int \frac{d^4k}{(2\pi )^4} \, \nonumber \\{} & {} \quad \textrm{Tr}\big (\gamma _0 S^{(0)}(k_0,\vec k) \gamma _0 S^{(0)}(q_0,\vec q)\big ) \, \end{aligned}$$

(A1)

where the bare fermion propagator is given in Eq. (4). After tracing over the Dirac indices, we obtain

$$\begin{aligned}{} & {} \Pi _{00}(p_0,\vec p) = - 4 N_\textrm{f} e^2 \int \frac{d^4k}{(2\pi )^4}\nonumber \\{} & {} \quad \frac{k_0 q_0 -v_1^2(k_1 q_1+k_2 q_2) - v_3^2 k_3q_3}{(k_0^2+v_1^2(k_1^2+k_2^2)+v_3^2 k_3^2)(q_0^2+v_1^2(q_1^2+q_2^2)+v_3^2 q_3^2)}. \nonumber \\ \end{aligned}$$

(A2)

We rewrite the denominator by introducing a Feynman parameter using

$$\begin{aligned} \frac{1}{d_k d_q} = \int _0^1 \textrm{d}x \frac{1}{(d_k(1-x)+d_q x)^2}, \end{aligned}$$

(A3)

where $d_k$ and $d_q$ indicate the two factors in the denominator of (A2). We then perform a shift of the integration variables using the definition $S_\mu = M_{\mu \nu }(K_\nu -x P_\nu )$.We define $M^2 = x(1-x) (p_0^2+v_1^2(p_1^2+p_2^2)+v_3^2 p_3^2)$ and write the result

$$\begin{aligned}{} & {} \Pi _{00}(p_0,\vec p) = \frac{ 4 N_f e^2 }{v_1^2 v_3} \int _0^1 dx\, \int \frac{d^4s}{(2\pi )^4}\nonumber \\{} & {} \quad \frac{[s_0^2-s_1^2-s_2^2-s_3^2]-x(1-x)[p_0^2-p_1^2 v_1^2-p_2^2v_2^2-p_3^2v_3^2] }{s_0^2+s_1^2+s_2^2+s_3^2 + M^2}.\nonumber \\ \end{aligned}$$

(A4)

We can now do the s integrals in spherical coordinates. Introducing a cutoff $\Lambda$ on the momentum $|\vec s|,$ we obtain the result in Eq. (14). We note that while a cutoff is needed to define the polarization tensor, it should not have a significant effect on the critical coupling. The point is that since dynamical gap generation is a low-energy phenomenon, the dominant contribution to the integral equations for the fermion dressing functions should come from the small momentum regime. We have verified numerically that this is true.

Appendix B: Numerical method

In this section, we give some details of our numerical method.

The integrals in Eqs. (16, 18, 20, 21), using either (24) or (25) for the polarization tensor, have an integrable singularity when $|\vec q| = |\vec k-\vec p| \rightarrow 0$. This means that the points in the domain of the integral where $\vec k \rightarrow \vec p$ must be treated carefully. We use Gauss–Legendre integration and divide the domain into regions so that all singular points are bracketed with equal sized intervals. This procedure allows for numerically accurate cancellations of large contributions on either side of the singularities. We also use a logarithmic scale for the momentum variables to increase sensititivity to the small momentum region where the dressing functions vary most strongly.

To calculate the critical coupling, we start from a set of data points that gives the condensate D(0) for different values of the coupling $\alpha$ (see Figs. 2, 3, 4 and 5). To obtain this data, we start with a large value of $\alpha$ for which the condensate is not zero, and reduce the coupling step by step until the condensate goes to zero (numerically $10^{-4}$). We invert the array of data points to obtain a numerical representation of $\alpha [D(0)]$, construct an interpolated function, and extrapolate to find the critical coupling $\alpha _c\equiv \alpha [0]$. To obtain an accurate result, it is clearly necessary to have a lot of data points for values of $\alpha$ that are close to the critical point, where the curve bends steeply downward. To do this in an efficient way, we dynamically adjust the step size as we approach the critical point. We start with the value $\Delta \alpha =1$ and, after the first two steps, the change in $\alpha$ is calculated from the results obtained from the previous two values as

$$\begin{aligned} \Delta \alpha = \left( \frac{1}{3}\frac{\alpha _{i-1}-\alpha _i}{\left( \ln [D(0)_{i-1}]-\ln [D(0)_i]\right) }\right) ^2. \end{aligned}$$

(B1)

This expression gives a large step size at large $\alpha$, when the value of the condensate changes slowly, and a smaller and smaller step size as the curve bends towards vertical. To prevent the step from becoming either too large or too small, we set a maximum step size of 1 and a minimum of 0.02.

In all cases, the data that we need to interpolate is very smooth, and different interpolation methods give results for the critical coupling that agree to very high precision. It is clear, however, that the accuracy of the result for the critical coupling does not depend on the accuracy of the interpolated function. The smallest $\alpha$ points on the curves in Figs. 2, 3, 4 and 5 take the longest to calculate, but if the last five or six points were missing, the extrapolated critical coupling would be much too small. One way to quantify the error in the extrapolated result for the critical coupling would be to remove the last calculated point and compare with the previous result. It the data stopped at a point where the curvature of the data was large, this would give a significantly different critical coupling. However, if the last few points in the data give a line that is fairly straight but not close to vertical, this method would indicate a small error even though the extrapolated critical alpha will not be very accurate. An alternative estimate is the absolute difference of the extrapolated result and the smallest value of $\alpha$ in the data set. The error calculated this way is related to the inverse slope of the data at small $\alpha$, because it will be small if the curve drops steeply to the horizontal axis, and large if the curve is fairly flat. In all cases, we have calculated the error both ways and taken the larger of the two results.

Fig. 8

The critical coupling versus the average number of internal grid points per dimension

The numerical solution of the SD equations involves four dimensional integrals and dressing functions that depend on four external variables. The total phase space therefore has a very large number of grid points, approximately $1.9 \times 10^9$. To verify that our integration and interpolation procedures are sufficiently accurate, we show in Fig. 8 a typical convergence plot. The graph shows the critical coupling, using the one loop polarization tensor for $\eta =0.4$ and $N_f=0.01$, as a function of the fourth root of the number of internal grid points (which gives the average value for one dimension). The graph indicates that good convergence is obtained for $\langle L \rangle > rsim 14$. In most of our calculations, we used about 14.8.