Phase diagrams of SO($N$) Majorana-Hubbard models: Dimerization, internal symmetry breaking, and fluctuation-induced first-order transitions

We study the zero-temperature phase diagrams of Majorana-Hubbard models with $\mathrm{SO}(N$) symmetry on two-dimensional honeycomb and $\pi$-flux square lattices, using mean-field and renormalization group approaches. The models can be understood as real counterparts of the $\mathrm{SU}(N$) Hubbard-Heisenberg models and may be realized in Abrikosov vortex phases of topological superconductors, or in fractionalized phases of strongly frustrated spin-orbital magnets. In the weakly interacting limit, the models feature stable and fully symmetric Majorana semimetal phases. Increasing the interaction strength beyond a finite threshold for large $N$, we find a direct transition towards dimerized phases, which can be understood as staggered valence bond solid orders, in which part of the lattice symmetry is spontaneously broken and the Majorana fermions acquire a mass gap. For small to intermediate $N$, on the other hand, phases with spontaneously broken $\mathrm{SO}(N$) symmetry, which can be understood as generalized Néel antiferromagnets, may be stabilized. These antiferromagnetic phases feature fully gapped fermion spectra for even $N$, but gapless Majorana modes for odd $N$. While the transitions between Majorana semimetal and dimerized phases are strongly first order, the transitions between Majorana semimetal and antiferromagnetic phases are continuous for small $N\le 3$ and weakly first order for intermediate $N\ge 4$. The weakly first-order nature of the latter transitions arises from fixed-point annihilation in the corresponding effective field theory, which contains a real symmetric tensorial order parameter coupled to the gapless Majorana degrees of freedom, realizing interesting examples of fluctuation-induced first-order transitions.

Figure 1

(a) Honeycomb lattice with uniform hopping parameter $t_{ij}\equiv t$ and two sublattices $A$ and $B$, indicated by the black and white dots, respectively. (b) Same as (a), but for the $\pi$-flux square lattice with uniform (alternating) signs $t_{ij}\equiv t$ ($t_{ij}=\pm t$) along horizontal (vertical) bonds, as indicated by the solid (alternating solid and dashed) lines. (c) Majorana dispersion for $J=0$ in the honeycomb-lattice model with nodal points at $\vec{q}=\frac{2\pi }{3}(\pm \frac{1}{\sqrt{3}},1)$, using units in which the distance between nearest-neighbor sites is $a=1$. The shaded negative-energy band serves as a reminder for the fact that excitations in the “particle” and “hole” bands are not independent, as Majorana fermions constitute their own antiparticles. (d) Same as (c), but for the $\pi$-flux model with nodal points at $\vec{q}=(\pm \frac{\pi }{2},0)$.

Figure 2

Mean-field phase diagrams of $\mathrm{SO}\left(N\right)$ Majorana-Hubbard models as function of $J/t$ for different values of $N$ on (a) honeycomb and (b) $\pi$-flux square lattices. The red shading indicates the norm $\left|\rho \right|$ of the $\mathrm{SO}\left(N\right)$-breaking order parameter matrix $\rho ={\sum }_{a<b}{\varphi }^{ab}{L}^{ab}$, representing the generalized Néel antiferromagnetic order (AFM). This phase features a gapped (gapless) Majorana spectrum for even (odd) $N$, as illustrated in the insets. The blue shading indicates an anisotropy in the bond order parameters, $\mathrm{\Delta }\chi =max{\chi }_{ij}-min{\chi }_{ij}$, representing the gapped staggered dimerized phase (DIM). This phase can also be understood as a staggered valence bond solid, as illustrated in the inset. The white region corresponds to the symmetric gapless Majorana semimetal phase (MSM). The transitions involving the DIM phase are strongly first order, while the transition between MSM and AFM is continuous on the mean-field level. Renormalization group arguments presented in Sec. 4 indicate that the MSM-AFM transition remains continuous for $N\le 3$, but becomes weakly first order for $N\ge 4$ upon the inclusion of quantum fluctuations. For $N=3$ and large coupling $J/t\gtrsim 5.88$ in the honeycomb-lattice model, mean-field theory suggest a coexistence phase characterized by both antiferromagnetic and dimerized order [hatched region in (a)].

Figure 3

Evolution of fixed-point structure of Gross-Neveu-SO(4) field theory as function of dimension of Clifford algebra representation $d_{\gamma }$, analytically continued to noninteger values. Arrows denote flow towards infrared. (a) Bosonic case $d_{\gamma }=0$, featuring, besides the Gaussian fixed point G, three interacting fixed points; the O(4) vector fixed point at ${\lambda }_2=0$ and the fixed point B are bicritical, while the O(4) tensor fixed point at finite ${\lambda }_1>0$ and ${\lambda }_2<0$ is fully infrared stable, corresponding to a continuous phase transition. (b) Upon the successive inclusion of fermion fluctuations, the O(4) vector and O(4) tensor fixed points approach each other as function of increasing $0<d_{\gamma }<0.0164$. The counterpart of the Gaussian fixed point ${\mathrm{G}}^{\prime }$ has moved slightly towards finite interaction. Here, $d_{\gamma }=0.0150$, i.e., slightly below $d_{\gamma }^{\text{cr,1}}$. (c) For $0.0165<d_{\gamma }<1.4853$, the O(4) vector and O(4) tensor fixed points have annihilated, leaving behind a regime of slow flow in parameter space (gray shaded region). Here, $d_{\gamma }=1.450$, i.e., slightly below $d_{\gamma }^{\text{cr,2}}$. (d) For $d_{\gamma }>1.4854$, fixed points B and ${\mathrm{G}}^{\prime }$ have annihilated as well, leaving behind only the runaway flow towards ${\lambda }_1\to -\infty$ and ${\lambda }_2<\left|{\lambda }_1\right|$ (faint region marked as “unstable”), suggesting a weak first-order transition. Here, $d_{\gamma }=2$, corresponding to the physical situation in $d=2$ spatial dimensions.

Figure 4

Mean-field parameters of $\mathrm{SO}(N$) Majorana-Hubbard model as function of $J/t$ for different values of $N$ on (a,c) honeycomb and (b,d) $\pi$-flux square lattices. (a,b) Eigenvalues $\pm \tilde{\varphi }$ of antiferromagnetic order parameter matrix $\rho ={\sum }_{a<b}{\varphi }^{ab}{L}^{ab}$. (c,d) Dimer order parameters $\chi \left(\delta \right)\in \{\tilde{\chi }-\mathrm{\Delta }\chi ,\tilde{\chi }\}$ on different nearest-neighbor bonds $\delta$ in $\langle ij\rangle =\langle i,i+\delta \rangle$.