Dynamical signatures of symmetry-broken and liquid phases in an $S$ = $\frac{1}{2}$ Heisenberg antiferromagnet on the triangular lattice

We present the dynamical spin structure factor of the antiferromagnetic spin-$\frac{1}{2} J_1\text{−}{J}_2$ Heisenberg model on a triangular lattice obtained from large-scale matrix-product state simulations. The high frustration due to the combination of antiferromagnetic nearest- and next-nearest-neighbor interactions yields a rich phase diagram. We resolve the low-energy excitations both in the ${120}^{\circ }$ ordered phase and in the putative spin-liquid phase at $J_2/J_1=0.125$. In the ordered phase, we observe an avoided decay of the lowest magnon branch, demonstrating the robustness of this phenomenon in the presence of gapless excitations. Our findings in the spin-liquid phase chime with the field-theoretical predictions for a gapless Dirac spin liquid, in particular the picture of low-lying monopole excitations at the corners of the Brillouin zone. We comment on possible practical difficulties of distinguishing proximate liquid and solid phases based on the dynamical structure factor.

Figure 1

Phase diagram of the $J_1\text{−}{J}_2$ antiferromagnetic Heisenberg model on a triangular lattice. At the Heisenberg point $J_2=0$, the ground state exhibits a ${120}^{\circ }$ order (a). Around $\frac{J_2}{J_1}\approx 0.07$, there is a transition into a candidate quantum spin-liquid state (b). In the illustration, we show the dispersion of the spinons at half-filling in the mean-field solution of the Dirac spin liquid discussed in Sec. F in the Supplemental Material [38]. For larger next-nearest-neighbor couplings, a stripe-ordered phase emerges (c). The blue arrows at $J_2=0$ and $\frac{J_2}{J_1}=0.125$ denote the points in the different regimes where we compute the spectral function.

Figure 2

Comparison of the static spin structure factor $\chi \left(\mathbf{k}\right)$ on a cylinder with $L_y=6$ from numerical DMRG results (top row) and analytical calculations (bottom row). (a) DMRG result for the ${120}^{\circ }$ ordered phase at $J_2=0$ and (b) the corresponding linear spin-wave theory. A cubic interpolation scheme along the circumference has been used for plotting. The right column shows the static structure factor in the QSL phase at $\frac{J_2}{J_1}=0.125$ for (c) DMRG and (d) the analytic result from the Dirac spin-liquid parton theory. The cylinder geometry used is $6\times 51$ for panels (a)–(c) and $6\times 50$ for (d).

Figure 3

Upper row: The spectral function in the ordered phase with only nearest-neighbor coupling ($J_2=0$) along different paths in the Brillouin zone as indicated in the inset in (d). The golden dot-dashed line shows the result for the lowest-excitation energy obtained from the quasiparticle ansatz. Lower row: Results from linear spin-wave theory (LSWT) to be compared to the MPS data in the panels above. The dashed bright line denotes the single-magnon branch. The red dotted line shows the minimum of the two-magnon continuum at $\mathbf{q}=\mathbf{K}$, whereas the red dot-dashed line corresponds to the local minimum for $\mathbf{q}={\mathbf{K}}^{\prime }$. The resulting two-magnon continuum is given by the hatched area. We indicate again the lowest-energy mode obtained from the quasiparticle ansatz and plot it additionally with a Gaussian broadening of $\sigma =0.1$ and weighted with the corresponding spectral weights (cf. Sec. B in the Supplemental Material [38]). Note that the finite-size gap determined at $K$ from the quasiparticle ansatz has been taken as an energy offset for the linear spin-wave results. We used system sizes $6\times 126$ and bond dimensions up to $\chi =1500$.

Figure 4

Spectral function in the candidate QSL phase at $\frac{J_2}{J_1}=0.125$ along the same momentum cuts as in Fig. 3 obtained on a cylinder geometry with $L_y=6$ and $L_x=51$. The dot-dashed line denotes the corresponding lowest-energy mode from the quasiparticle ansatz. We used bond dimension $\chi =2000$ for these simulations.