Exotic Quantum Phase Transitions of $(2+1)d$ Dirac fermions

TL;DR

The paper addresses exotic quantum phase transitions in $16$ copies of $(2+1)d$ Majorana fermions on a bilayer honeycomb lattice, using determinant quantum Monte Carlo (d-QMC) to access nonperturbative effects. It uncovers two continuous quantum phase transitions: (i) a semimetal-to-symmetric trivial phase transition not captured by the Gross-Neveu model, related to the $\mathbb{Z}_{16}$ classification of the interacting $^3$He-B boundary; (ii) a QSH-to-trivial transition described by a bosonic $O(4)$ nonlinear sigma model with a $\Theta$-term, where the single-particle gap remains open while spin and charge gaps close. At the QSH-to-trivial critical point, the single-particle gap stays finite while spin and charge gaps close, signaling a bosonic critical theory. Analysis of the Green's function shows the topological invariant $C_s$ can change via zeros of $G$ in the gapped phase, linking the numerical results to a bosonic description.

Abstract

Using determinant quantum Monte Carlo (d-QMC) simulations, we demonstrate that an extended Hubbard model on a bilayer honeycomb lattice has two novel quantum phase transitions. The first is a quantum phase transition between the weakly interacting gapless Dirac fermion phase and a strongly interacting fully gapped and symmetric trivial phase, which cannot be described by the standard Gross-Neveu model. The second is a quantum critical point between a quantum spin Hall insulator with spin $S^z$ conservation and the previously mentioned strongly interacting fully gapped phase. At the latter quantum critical point the single particle excitations remain gapped, while spin and charge gap both close. We argue that the first quantum phase transition is related to the $\mathbb{Z}_{16}$ classification of the topological superconductor $^3\text{He-B}$ phase with interactions, while the second quantum phase transition is a topological phase transition described by a bosonic O(4) nonlinear sigma model field theory with a $Θ$-term.

Figures (9)

• Figure 1: The bilayer honeycomb lattice. In each layer, $t$ and $\lambda$ are the nearest- and next-nearest-neighbor hopping. The Hubbard interaction $U$ acts on each site, and the Heisenberg interaction $J$ acts across the layers.
• Figure 2: (Color online.) A schematic phase diagram of the bilayer honeycomb model. The red line is the phase boundary between the two QSH phases of opposite spin Hall conductivity, where both the single particle and the spin/charge gaps are closed. The blue line is the phase boundary between the QSH phase $\Theta=\pm2\pi$ and the trivial gapped phase $\Theta=0$, where the single particle gap remains open but the spin/charge gaps are closed. $U_c$ is the tricritical point, above which the topological number defined in Eq. \ref{['tknn']} changes inside the trivial phase (without gap closing) through the dashed line, also see Fig. \ref{['topo_number']}.
• Figure 3: The topological number defined in Eq. \ref{['tknn']} as a function of $\lambda$ for both models at $U = 2$. The topological number was calculated at the dots using DQMC data via the methods discussed in the Topological Number Calculation Methods appendix. This demonstrates that this topological number Eq. \ref{['tknn']} is nonzero even in the strongly interacting trivial phase.
• Figure 4: Single particle and spin gap for the 1d coupled chain model with $J/U = 2$. (a) When $\lambda = 0$, the system is gapped out immediately by an infinitesimal interaction with a gap of the form $e^{a - b/U}$ for small $U$ (dotted black line with $a = 2.60$ and $b = 2.65$). (b) When $\lambda = 0.25$, there are no phase transitions when $\lambda \neq 0$ and $U > 0$.
• Figure 5: Single particle gap, spin gap (gap for spin-1 excitation), and charge gap (gap for charge-2 excitation) on the bilayer honeycomb lattice with $J/U = 2$. (a) When $\lambda = 0$, there is a single continuous phase transition from a semimetal to a trivial insulator at $U_c \sim 1$, whose field theory also describes the phase transition of the boundary of 16 copies of the $^3\text{He-B}$ phase. (b) When $\lambda = 0.25$, only the spin and charge gap close at the continuous phase transition from an SPT to a trivial insulator (which is at $U_c \sim 1.5$ for $\lambda = 0.25$). We propose that this phase transition is described by a bosonic O(4) nonlinear sigma model field theory with a $\Theta$-term [Eq. \ref{['o4nlsm']}]. These gaps are calculated as explained in the Gap Calculation Methods appendix. This involves calculating gaps in finite systems of sizes up to 9x9 unit cells (with 4 sites each) and extrapolating to the infinite size limit. Error bars on all figures denote one standard deviation ($i.e.$$\approx 68 \%$ confidence).