Competing states in the $S=1/2$ triangular-lattice $J_1$-$J_2$ Heisenberg model: a dynamical density-matrix renormalization group study

Abstract

Previous studies of the $S=1/2$ triangular-lattice $J_1$--$J_2$ Heisenberg antiferromagnet have inferred the existence of a non-magnetic ground-state phase for an intermediate range of $J_2$, but disagree concerning whether it is a gapped $\mathbb{Z}_2$ quantum spin liquid (QSL), a gapless (Dirac) QSL, or a weakly symmetry-broken phase. Using an improved dynamical density-matrix renormalization group method, we investigate the relevant intermediate $J_2$ regime for cylinders with circumferences from 6 to 9. Depending on the initial state and boundary conditions, we find two {\it distinct} variational states. The higher energy state is consistent with a Dirac QSL. In the lower-energy state, both the static and dynamical properties are qualitatively similar to the magnetically ordered state at $J_2=0$, suggestive of either a weakly magnetically ordered non-QSL or a gapped QSL proximate to a continuous transition to such an ordered state.

Figures (6)

• Figure 1: Top panel (a): Conjectured ground state phase diagram of the $S=1/2$$J_1$-$J_2$ Heisenberg model on the triangular lattice from previous studies j1j2-zhuj1j2-becca1j1j2-cenkej1j2-mccullochj1j2-donnaj1j2-campbellj1j2-donna2j1j2-tomj1j2-lauchlij1j2-imadaj1j2-hc, where an intermediate $0.07 \lesssim J_2\lesssim 0.16$ region was found with vanishing magnetic order, albeit with somewhat different phase boundaries and different putative QSLs. The middle point of this region $J_2=0.12$, and specifically what QSL if any occurs there, is the focus of this paper. Insets of (a) show the XC6 cylinder with $\vec{e}_1$ and $\vec{e}_2$ being the unit vectors along its length and circumference, respectively, and the first Brillouin Zone with the high symmetry points marked. The $Y$ point (midpoint between $\Gamma$ and $M$) is where the Dirac spinon node is located for a gapless Dirac QSL dirac-theorydirac-theory2. Middle panel: Strength of the spin-spin correlation $\langle \vec{S}_i\cdot \vec{S}_{i+\delta i}\rangle$ on three types of NN bonds (defined in (a)), averaged over the circumferential direction and plotted versus the length of the cylinder $l_x$, for four different states: (b) the 120$^\circ$ state at $J_2=0$, (c) the LE state at $J_2=0.12$, (d) the HE state at $J_2=0.12$ and (e) the striped state at $J_2=0.2$. The strengths of bonds 1 and 2 are identical except in (e). Bottom panel: The equal-time spin structural factor $S(\vec{q})$ for (f) the 120$^\circ$ state at $J_2=0$ that has dominant peaks at $K$, (g) the LE state at $J_2=0.12$ with weaker peaks at $K$, (h) the HE state at $J_2=0.12$ with still weaker peaks at both $K$ and $M$ (marked by red circles), and (i) the striped state at $J_2=0.2$ with peaks at $M$. The white dashed lines mark the boundaries of the first Brillouin zone.
• Figure 2: The low-energy DSSF $S(\vec{q},\omega)$ at a specific $\omega$ shown in the center of each figure, for the 120$^\circ$ state at $J_2=0$ (top row, (a)(b)(c)), the LE state at $J_2=0.12$ (middle row, (d)(e)(f)), and the HE state at $J_2=0.12$ (bottom row, (g)(h)(i)). Simulations were carried out on XC6-24 (left column, (a)(d)(g)), XC8-24 (middle column, (b)(e)(h)), and XC9-24 (right column, (c)(f)(i)), respectively. The color scale has an upper cutoff of 2.
• Figure 3: The DSSF $S(\vec{q},\omega)$ at or around high symmetry points $\vec{q}=K=(4\pi/3,0)$ and $\vec{q}=M=(\pi,\pi/\sqrt3)$ after symmetrization (see text), for the 120$^\circ$ state at $J_2=0$ (top row, (a)(b)(c)), the LE state at $J_2=0.12$ (middle row, (d)(e)(f)), and the HE state at $J_2=0.12$ (bottom row, (g)(h)(i)). Simulations were carried out on XC6-24 (left column, (a)(d)(g)), XC8-24 (middle column, (b)(e)(h)), and XC9-24 (right column, (c)(f)(i)), respectively.
• Figure 4: Correlation functions in the LE (top panels) and HE state (lower panels) in a XC9-24 cylinder, with $J_2=0.12$. Panels (a) and (d) plot the strength of the spin-spin correlation $\langle \vec{S}_i\cdot \vec{S}_{i+\delta i}\rangle$ on three types of NN bonds (defined in Fig. \ref{['fig:phd']}(a) in the main text), averaged over the circumferential direction and plotted versus the position along the cylinder $l_x$. Panels (b) and (e) show the equal-time spin structural factor $S(\vec{q})$. (c) and (f) show the the low-energy DSSF $S(\vec{q},\omega=0.2)$.
• Figure 5: Sign of the spin-spin correlation function $\langle \vec{S}_{\vec{r}_0} \cdot \vec{S}_{\vec{r}}\rangle/ |\langle \vec{S}_{\vec{r}_0} \cdot \vec{S}_{\vec{r}}\rangle|$ for (a) the LE state and (b) the HE state at $J_2=0.12$ on a XC6-24 cylinder at $m=2400$, with three columns subtracted on each edge. The reference site $\vec{r}_0$ is marked by the red circle. Green dots and yellow dots denote plus and minus signs at site $\vec{r}$, respectively. Those sites with a correlation amplitude smaller than $5\times 10^{-5}$ are excluded and marked by a cross, since their signs are susceptible to errors and can not be definitely determined. Outside the shaded vicinity of $\vec{r}_0$, the patterns of the correlation are different for the two states. The Fourier transform of the sign of the spin-spin correlation functions: $\tilde{S}(\vec{q})=\sum_{\vec{r}} e^{-i\vec{q} \cdot \vec{r}} \langle \vec{S}_{\vec{r}_0} \cdot \vec{S}_{\vec{r}}\rangle/ |\langle \vec{S}_{\vec{r}_0} \cdot \vec{S}_{\vec{r}}\rangle|$, for (c) the LE state and (d) the HE state. The color scale is renormalized and has a lower cutoff of 0.2 to filter out the fuzzy background due to contributions from short-ranged correlations.