Superuniversality of topological quantum phase transition and global phase diagram of dirty topological systems in three dimensions

The quantum phase transition between two clean, noninteracting topologically distinct gapped states in three dimensions is governed by a massless Dirac fermion fixed point, irrespective of the underlying symmetry class, and this constitutes a remarkably simple example of superuniversality. For a sufficiently weak disorder strength, we show that the massless Dirac fixed point is at the heart of the robustness of superuniversality. We establish this by considering both perturbative and nonperturbative effects of disorder. The superuniversality breaks down at a critical strength of disorder, beyond which the topologically distinct localized phases become separated by a delocalized diffusive phase. In the global phase diagram, the disorder controlled fixed point where superuniversality is lost, serves as a multicritical point, where the delocalized diffusive and two topologically distinct localized phases meet and the nature of the localization-delocalization transition depends on the underlying symmetry class. Based on these features, we construct the global phase diagrams of noninteracting, dirty topological systems in three dimensions. We also establish a similar structure of the phase diagram and the superuniversality for weak disorder in higher spatial dimensions. By noting that $1/r^2$ power-law correlated disorder acts as a marginal perturbation for massless Dirac fermions in any spatial dimension $d$, we have established a general renormalization group framework for addressing disorder driven critical phenomena for fixed spatial dimension $d>2$.

Figure 1

The phase diagrams of dirty superconductors in class DIII for three dimensions. The coupling constant $m$ is the disorder averaged Dirac mass and it corresponds to the chemical potential of the quasiparticles in normal state, and ${\mathrm{\Delta }}_{-}={\mathrm{\Delta }}_{45}-{\mathrm{\Delta }}_4$ is a combination of two underlying disorder couplings, ${\mathrm{\Delta }}_4$ and ${\mathrm{\Delta }}_{45}$. For the typical example of ${}^3\mathrm{He}$-B, ${\mathrm{\Delta }}_{45}$, and ${\mathrm{\Delta }}_4$ correspond to the strengths of random, real $s$-wave pairing and a random chemical potential for normal quasiparticles, respectively, while for a different physical system, they can represent realization of different random potentials. While the disorder couplings ${\mathrm{\Delta }}_{45}$ and ${\mathrm{\Delta }}_4$ are positive definite quantities, ${\mathrm{\Delta }}_{-}$ can acquire both positive and negative values. When ${\mathrm{\Delta }}_{-}$ is less than the critical strength ${\mathrm{\Delta }}_{-}^{*}$, the direct transition between two topologically distinct localized states [namely, the topological superconductor (TSC) and the normal superconductor (NSC)] is described by the massless, Dirac fermion fixed point (red dot) with a dynamic scaling exponent $z=1$ and a correlation/localization length exponent ${\nu }_M=1$ (scaling dimension of the Dirac mass $m$, describing a correlation/localization length ${\xi }_M\sim 1/\left|m\right|)$. The disorder controlled multicritical point is marked by the black dot, where the TSC, NSC and the diffusive phase DIII-DM meet. We note that the DIII-DM also describes a paired state, and not an ordinary diffusive metal or normal state. At the multicritical point, the correlation/localization length ${\xi }_M$ diverges with an exponent ${\nu }_M=2/3$ for a Gaussian white-noise disorder, while the correlation length exponent for disorder (the mean free path of the diffusive phase) ${\xi }_{\mathrm{\Delta }}$ diverges with an exponent ${\nu }_{\mathrm{\Delta }}$. For the Gaussian white-noise disorder and at one-loop level, we find ${\nu }_{\mathrm{\Delta }}=1$. The multicritical point is actually a projection of a line of fixed points on the $m-{\mathrm{\Delta }}_{-}$ plane, determined by ${\mathrm{\Delta }}_{-}={\mathrm{\Delta }}_{-}^{*}$ and the arbitrary value of ${\mathrm{\Delta }}_{+}={\mathrm{\Delta }}_{45}+{\mathrm{\Delta }}_4$. Along the line of fixed points, the correlation length exponents have universal values, while the dynamical exponent varies continuously depending on ${\mathrm{\Delta }}_{+}$ (or the ratio ${\mathrm{\Delta }}_{45}/{\mathrm{\Delta }}_4)$. The other four symmetry classes AIII, AII, CI, and CII also possess similar phase diagrams as a function of disorder strength and the Dirac mass. Dirty topological systems in higher spatial dimensions also exhibit similar structure of the phase diagram, irrespective of the underlying symmetry class.

Figure 2

Quantum critical fan for the direct quantum phase transition between topologically distinct localized states. In the absence of disorder, the topologically distinct insulating states possess sharp spectral gaps, described by the Dirac mass. In the presence of disorder, there is no sharp spectral gap, and the inverse of the disorder averaged Dirac mass describes the localization length $\left({\xi }_M\sim 1/\left|m\right|\right)$ for the subgap states. The tuning parameter $x$ is a combination of disorder couplings and at $x=x_c$ the disorder averaged Dirac mass vanishes, which determines the phase boundary between two localized states. In the weak-disorder regime for all five Altland-Zirnbauer symmetry classes, the average Dirac mass behaves as $\overline{m}=m\left(0\right)[1-x/x_c]$, where $m\left(0\right)$ is the Dirac mass in the absence of disorder. For class DIII, $x={\mathrm{\Delta }}_{-}={\mathrm{\Delta }}_{45}-{\mathrm{\Delta }}_4$ and $x_c\approx 6m\left(0\right)m^{*}/\left({\hslash }^2{\mathrm{\Lambda }}^2\right)$, where $\mathrm{\Lambda }$ is the momentum cutoff for the Dirac theory. The quantum critical behavior of massless Dirac fermion is observed for $k_{B}T\gg {|x-x_c|}^{z{\nu }_M}$, with dynamic scaling exponent $z=1$, and the localization length exponent ${\nu }_M=1$. This picture is only applicable for sufficiently weak disorder. For strong enough disorder, the massless Dirac fermion can undergo a semimetal to metal transition, and there will be no direct transition between two topologically distinct localized states. Here, $T_0=\hslash v\mathrm{\Lambda }/k_B$ corresponds to a microscopic energy scale, where $v$ is the velocity of the massless Dirac fermions and $k_B$ is the Boltzmann constant.

Figure 3

Feynmann diagrams arising from disorder averaged quartic interactions. (a) Disorder induced fermion self-energy, (b) disorder vertex correction, (c) ladder and (d) crossing type diagrams. At a one-loop level, the diagrams of class (a) and (b) are UV divergent for any $\alpha >0$ and contribute to the beta function. By contrast, the diagrams of class (c) and (d) at the one-loop level are UV convergent for all $\alpha >0$ and do not contribute to the RG flow equations of the disorder couplings. Therefore the one-loop beta function of a disorder vertex for any $\alpha >0$ is determined by the anomalous scaling dimension of an appropriate fermion bilinear. For example, if we consider a disorder $V_j\left(\mathbf{x}\right){\mathrm{\Psi }}^{†}{\widehat{M}}_j\mathrm{\Psi }$, where ${\widehat{M}}_j$ is a $4\times 4$ matrix, the one-loop beta function of the disorder coupling will be determined by the anomalous dimension ${\eta }_j$. For $\alpha =0$, both classes of diagrams are UV divergent and can contribute to the beta function as found within $d=2+\epsilon$ expansion scheme.

Figure 4

RG flows for class DIII (a) in the plane ${\mathrm{\Delta }}_{-}-{\mathrm{\Delta }}_{+}$ and (b) in the plane $m-{\mathrm{\Delta }}_{-}$. We have chosen the Gaussian white-noise distributions for the mass disorder and axial disorder couplings given by ${\mathrm{\Delta }}_4$ and ${\mathrm{\Delta }}_{45}$, respectively, and we also note that ${\mathrm{\Delta }}_{\pm }={\mathrm{\Delta }}_{45}\pm {\mathrm{\Delta }}_4$. (a) The solid vertical line at ${\mathrm{\Delta }}_{-}^{*}=3/8$ describes the line of fixed points controlling the semimetal-metal and metal-insulator transitions. (b) The black dot is a projection of the line of fixed points. The dot always describes the clean, massless Dirac fixed point. Our calculations suggest similar flow diagrams for class AIII.

Figure 5

Quantum critical fan for the semimetal-metal transition controlled by the multicritical point, in the presence of only random axial chemical potential. The depicted quantum critical fan follows the analytical formula in Eq. (82) obtained for the Gaussian white-noise distribution. For $\delta <0$, it describes the crossover between the non-Fermi liquid quantum critical region (QC/NFL) and the massless Dirac quantum critical region (QC/DSM). By contrast, for $\delta >0$, we have crossovers between the QC/NFL region and the diffusive metal (DM). If we consider axial chemical potential following the Gaussian white-noise distribution, the thermal conductivity in QC/NFL, QC/DSM, and DM regions follow the power laws $\kappa \sim T^{5/3},\kappa \sim T^2$, and $\kappa \sim T$, respectively. A similar fan can be drawn for the direct transition between two localized states occurring at this fixed point.

Figure 6

Feynmann diagrams needed for calculating anomalous scaling dimension of a fermion bilinear. The vertex diagrams can be obtained by differentiating the self-energy with respect to the appropriate source term as a consequence of the Ward identity.