Low-energy properties of two-dimensional quantum triangular antiferromagnets: Nonperturbative renormalization group approach

We explore low-temperature properties of quantum triangular Heisenberg antiferromagnets in two dimensions in the vicinity of the quantum phase transition at zero temperature. Using the effective field theory described by the $O\left(3\right)\times O\left(2\right)$ matrix Ginzburg-Landau-Wilson model and the nonperturbative renormalization group method, we clarify how quantum and thermal fluctuations affect long-wavelength behaviors in the parameter region where the systems exhibit a fluctuation-driven first order transition to a long-range ordered state. We show that at finite temperatures the crossover from a quantum ${\varphi }^6$ theory to a renormalized two-dimensional classical nonlinear sigma model region appears, and in this crossover region, massless fluctuation modes with linear dispersion a la spin waves govern low-energy physics. Our results are partly in good agreement with the recent experimental observations for the two-dimensional triangular Heisenberg spin system, $\mathrm{Ni}{\mathrm{Ga}}_2{\mathrm{S}}_4$.

Figure 1

Phase diagrams of quantum second order phase transition (top) and quantum first order transition (bottom). The vertical and horizontal axes are, respectively, temperature $T$ and a parameter $1∕\kappa$ which controls quantum fluctuations.

Figure 2

RG flows of dimensionless couplings. (a) Plots of $\kappa$, $\lambda \kappa$, and ${\lambda }_6{\kappa }^2$. (b) Plots of $\mu \kappa$, $\omega \kappa$, and ${\mu }_6{\kappa }^2$. Arrows indicate the directions of the RG flows. The crossover to the ${\varphi }^6$ theory appears in the intermediate regions.

Figure 3

Effective potentials $S_4+S_6$ as a function of $\sqrt{\widehat{\rho }}$ calculated for a particular set of scaling parameters $k$ under the condition $\widehat{\tau }=0$.

Figure 4

Plot of the critical scaling parameter $k_c$ versus temperature $T$.

Figure 5

A schematic phase diagram. The inserted plots are the renormalized potential versus the renormalized field $\sqrt{\widehat{\rho }}$.

Figure 6

RG flows of dimensionful couplings for some particular sets of the initial values of parameters. (a) Plots of $\tilde{\kappa }$, $\tilde{\lambda }$, and $\tilde{\mu }$. (b) Plots of $\tilde{\omega }$, ${\tilde{\lambda }}_6$, and ${\tilde{\mu }}_6$. The initial values of parameters are $\tilde{\kappa }(t=0)=0.3$, $\tilde{\lambda }\left(0\right)=0.5$, $\tilde{\mu }\left(0\right)=0.1$, $\tilde{\omega }\left(0\right)=-0.0001$, $\tilde{\omega }\left(0\right)∕c_3^2\left(0\right)=-0.5$, $c_1\left(0\right)=1.0$ (solid line); $\tilde{\kappa }\left(0\right)=0.3$, $\tilde{\lambda }\left(0\right)=0.3$, $\tilde{\mu }\left(0\right)=0.22$, $\tilde{\omega }\left(0\right)=-0.0001$, $\tilde{\omega }\left(0\right)∕c_3^2\left(0\right)=-0.5$, $c_1\left(0\right)=1.0$ (broken line); $\tilde{\kappa }\left(0\right)=0.3$, $\tilde{\lambda }\left(0\right)=0.3$, $\tilde{\mu }\left(0\right)=0.3$, $\tilde{\omega }\left(0\right)=-0.0001$, $\tilde{\omega }\left(0\right)∕c_3^2\left(0\right)=-0.5$, $c_1\left(0\right)=1.0$ (thin broken line); $\tilde{\kappa }\left(0\right)=0.3$, $\tilde{\lambda }\left(0\right)=0.3$, $\tilde{\mu }\left(0\right)=0.4$, $\tilde{\omega }\left(0\right)=-0.0001$, $\tilde{\omega }\left(0\right)∕c_3^2\left(0\right)=-0.5$, $c_1\left(0\right)=1.0$ (dotted line); $\tilde{\kappa }\left(0\right)=0.3$, $\tilde{\lambda }\left(0\right)=0.3$, $\tilde{\mu }\left(0\right)=0.5$, $\tilde{\omega }\left(0\right)=-0.0001$, $\tilde{\omega }\left(0\right)∕c_3^2\left(0\right)=-0.5$, $c_1\left(0\right)=1.0$ (dotted-and-broken line), and the initial values of both ${\lambda }_6$ and ${\mu }_6$ are equal to 0 in all calculations.

Figure 7

RG flows of dimensionful couplings. The initial values of parameters are $\tilde{\kappa }(t=0)=0.3$, $\tilde{\lambda }\left(0\right)=3.0$, $\tilde{\mu }\left(0\right)=3.0$, $\tilde{\omega }\left(0\right)=-0.0001$, $\tilde{\omega }\left(0\right)∕c_3^2\left(0\right)=-0.5$, $c_1\left(0\right)=1.0$, (solid line); $\tilde{\kappa }\left(0\right)=0.3$, $\tilde{\lambda }\left(0\right)=5.0$, $\tilde{\mu }\left(0\right)=5.0$, $\tilde{\omega }\left(0\right)=-0.0001$, $\tilde{\omega }\left(0\right)∕c_3^2\left(0\right)=-0.5$, $c_1\left(0\right)=1.0$ (broken line); $\tilde{\kappa }\left(0\right)=0.3$, $\tilde{\lambda }\left(0\right)=8.0$, $\tilde{\mu }\left(0\right)=8.0$, $\tilde{\omega }\left(0\right)=-0.0001$, $\tilde{\omega }\left(0\right)∕c_3^2\left(0\right)=-0.5$, $c_1\left(0\right)=1.0$ (thin broken line); $\tilde{\kappa }\left(0\right)=0.3$, $\tilde{\lambda }\left(0\right)=10.0$, $\tilde{\mu }\left(0\right)=10.0$, $\tilde{\omega }\left(0\right)=-0.0001$, $\tilde{\omega }\left(0\right)∕c_3^2\left(0\right)=-0.5$, $c_1\left(0\right)=1.0$ (dotted line); $\tilde{\kappa }\left(0\right)=0.3$, $\tilde{\lambda }\left(0\right)=12.0$, $\tilde{\mu }\left(0\right)=12.0$, $\tilde{\omega }\left(0\right)=-0.0001$, $\tilde{\omega }\left(0\right)∕c_3^2\left(0\right)=-0.5$, $c_1\left(0\right)=1.0$ (dotted-and-broken line), and the initial values of both ${\lambda }_6$ and ${\mu }_6$ are equal to 0 in all calculations.