Quantum and Thermal Phase Transitions of the Triangular SU(3) Heisenberg Model under Magnetic Fields

We study the quantum and thermal phase transition phenomena of the SU(3) Heisenberg model on triangular lattice in the presence of magnetic fields. Performing a scaling analysis on large-size cluster mean-field calculations endowed with a density-matrix renormalization-group solver, we reveal the quantum phases selected by quantum fluctuations from the massively degenerate classical ground-state manifold. The magnetization process up to saturation reflects three different magnetic phases. The low- and high-field phases have strong nematic nature, and especially the latter is found only via a nontrivial reconstruction of symmetry generators from the standard spin and quadrupolar description. We also perform a semiclassical Monte Carlo simulation to show that thermal fluctuations prefer the same three phases as well. Moreover, we find that exotic topological phase transitions driven by the binding-unbinding of fractional (half-)vortices take place, due to the nematicity of the low- and high-field phases. Possible experimental realization with alkaline-earth-like cold atoms is also discussed.

Figure 1

Field dependences of the magnetization $M$ (blue circles) and the uniform scalar nematic order parameter $S_u$ (divided by $\sqrt{3}$, red triangles), obtained by the $\mathrm{CMF}+\mathrm{S}$ analysis. The classical (mean-field) value of $M$ is plotted together (dashed line). Left and right axes are shifted by $2/3$. The inset shows cluster-size scalings of the critical fields.

Figure 2

(a) Nonzero components of the spin and quadrupolar moments, obtained by the $\mathrm{CMF}+\mathrm{S}$ analysis in a fixed gauge with $⟨{\widehat{Q}}_A^{xy}⟩=⟨{\widehat{Q}}_A^{yz}⟩=0$. The inset shows the three-sublattice structure. (b) Spherical plots of $|⟨\mathit{S}|{\widehat{Q}}^{x^2-y^2}|\mathit{S}⟩|$ and its $\pi /2$ and $\pi$ rotations about $U(1{)}_{S^z}$ with $|\mathit{S}⟩$ being the spin coherent state pointing in the $\mathit{S}$ direction [23].

Figure 3

Thermal phase diagram obtained by the semiclassical Monte Carlo simulations. We also mark the critical fields $H_{c1}$ and $H_{c2}$ obtained by the $\mathrm{CMF}+\mathrm{S}$ method at the quantum ($T=0$) limit. The dashed lines are the sketches of the phase boundaries expected from the combination of the semiclassical Monte Carlo (valid at high temperatures) and $\mathrm{CMF}+\mathrm{S}$ (valid at $T=0$) results.

Figure 4

(a) Stiffness ${\rho }_{S^z}(T)$ along $H/J=2.0$, which shows a universal jump ${\rho }_{S^z}(T_c)=8k_B{T}_c/\pi$ at the LF-IF transition, except for a slight finite-size effect. (b) Vortex and antivortex with half-vorticity ${\rho }_v=\pm 1/2$ in the projected $(Q^{x^2-y^2},Q^{xy})$ plane. The right-hand panel is a schematic illustration of a topological half-vortex pair excitation on the background of a uniform quadrupolar order on, say, sublattice $A$. (c) Same as in (a) for ${\rho }_{P_{+}^z}(T)$ at the HF-paramagnetic transition along $H/J=3.5$.