Scaling in the Fan of an Unconventional Quantum Critical Point

We present results of extensive finite-temperature quantum Monte Carlo simulations on a SU(2) symmetric $S=1/2$ quantum antiferromagnet with four-spin interaction [A. W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007)]. Our simulations, which are free of the sign problem and carried out on lattices containing in excess of $1.6\times {10}^4$ spins, indicate that the four-spin interaction destroys the Néel order at an unconventional $z=1$ quantum critical point, producing a valence-bond solid paramagnet. Our results are consistent with the “deconfined quantum criticality” scenario.

Figure 1

Schematic of the proposed $T\mathrm{\text{−}}J/Q$ phase diagram of the JQ model. Large-scale finite-$T$ simulations presented here substantiate the following: (i) The $T=0$ Néel order present for $J/Q\gg 1$ is destroyed at a QCP ($J/Q\approx 0.038$); (ii) in the “quantum critical fan” there is scaling behavior characteristic of a $z=1$ QCP, (iii) an accurate estimate of the scaling dimension of the Néel field establishes that this transition is not in the O(3) universality class, and, (iv) the paramagnetic ground state for sufficiently small $J/Q$ is a VBS. In the QMC basis used here, the region with $Q<0$ is sign problematic. The inset shows how the $Q$ term is written in terms of bonds on a plaquette.

Figure 2

$T\to 0$ converged Néel (main) and VBS (inset) order parameters as a function of $1/L$. Dashed lines are quadratic fits that illustrate the finite condensate in the ordered phases. The solid (red) line is a fit to the form $y=c_1{x}^{c_2}$ (illustrated for $J/Q=0.040$), where $c_2=z+{\eta }_N$ is expected at the critical coupling. In fitting to the nine $L$ values for each $J/Q$, we find a minimum in the ${\chi }^2$ value (per degree of freedom) of 3.1 for $J/Q=0.040$, with $c_2\approx 1.35(1)$. For $J/Q=0.038$, the ${\chi }^2$ value is 3.9, with $c_2\approx 1.37(1)$. All other $J/Q$ produce much larger ${\chi }^2$ (greater than 10).

Figure 3

Criticality of the Néel field at $J=0.038$: collapse of the Néel structure factor ($S_N$) and susceptibility (${\chi }_N$) with $z=1$ and ${\eta }_N=0.35$, determining the universal functions ${\mathbb{X}}_S(x)$ and ${\mathbb{X}}_{\chi }(x)$ (up to nonuniversal scale factors on the $x$ and $y$ axes). The only fit parameter for both $S_N$ and ${\chi }_N$ is ${\eta }_N$, the anomalous dimension of the Néel field.

Figure 4

Finite-$T$ uniform susceptibility, for a $L=128$ system near $J_c$. Error bars are much smaller than the symbol size. For the region $0.08\le T\le 0.18$, the data is highly linear, and a straight-line fit for $J/Q=0.038$ (shown) intercepts the origin within error bars. Intercepts of straight-line fits for all data sets are in the inset. From the slope of the linear-$T$ behavior we obtain ${\mathcal{A}}_{\chi }/c^2=0.0412(2)$.

Figure 5

Zoom-in of the spin stiffness and susceptibility close to the expected critical point, taken for simulation cells of size $L=1/T$. Data in the inset is scaled to get the best collapse for the largest system sizes, which occurs for $J_c\sim 0.039(1)$ and $\nu \sim 0.68(4)$.

Figure 6

Scaling of ${\chi }_u$ and ${\rho }_s$ at $J=0.038\approx J_c$, with $z=1$ and $d=2$. These plots are the universal functions $\mathbb{Y}(x,0)$ and $\mathbb{Z}(x,0)$ up to the nonuniversal scale factor $c$ on the $x$ axis. The expected asymptotes (see text) are plotted as dashed lines $\mathbb{Y}(x\to 0,0)={\mathcal{A}}_{\rho }/x$ and $\mathbb{Z}(x\to \infty ,0)={\mathcal{A}}_{\chi }{x}^{d/z}$. From fits to the data, we find ${\mathcal{A}}_{\chi }/c^2=0.041(4)$ and ${\mathcal{A}}_{\rho }c=0.37(3)$, allowing us to estimate a universal model-independent number associated with the QCP, ${\mathcal{A}}_{\rho }\sqrt{{\mathcal{A}}_{\chi }}\approx 0.075(4)$.