Dirac Spin Liquid on the Spin-$1/2$ Triangular Heisenberg Antiferromagnet

We study the spin liquid candidate of the spin-$1/2$ $J_1\text{−}{J}_2$ Heisenberg antiferromagnet on the triangular lattice by means of density matrix renormalization group (DMRG) simulations. By applying an external Aharonov-Bohm flux insertion in an infinitely long cylinder, we find unambiguous evidence for gapless $U(1)$ Dirac spin liquid behavior. The flux insertion overcomes the finite size restriction for energy gaps and clearly shows gapless behavior at the expected wave vectors. Using the DMRG transfer matrix, the low-lying excitation spectrum can be extracted, which shows characteristic Dirac cone structures of both spinon-bilinear and monopole excitations. Finally, we confirm that the entanglement entropy follows the predicted universal response under the flux insertion.

Figure 1

Geometry of cylinders in real space and momentum points in the first Brillouin zone. (a) Three different cylindrical geometries, YC4-0, YC4-1, or YC4-2 [39], which correspond to identifying site $x$ with site $a$, $b$, or $c$, respectively. We also insert an Aharonov-Bohm flux in the hole of the cylinder, which modifies the spin exchange terms in Hamiltonian Eq. (1) across the boundary (labeled by dashed lines). (b) The black solid line shows the first Brillouin zone of the triangular lattice while the brown rectangle is the magnetic Brillouin zone due to the $\pi$ flux in each unit cell seen by spinons. All characteristic points are labeled by $(k_1,k_2)$ modulo $2\pi$. Two Dirac points (black dots) of spinons are located at $\mathbf{Q}=(\pi /2,\pi /2)$ and $-\mathbf{Q}$ (for details see [39]). The gauge invariant excitations, e.g. fermion bilinears and monopoles, are locating at high symmetric points, including $\mathrm{\Gamma }=(0,0)$; $M_1=(\pm \pi ,0)$ (violet hexagon), $M_2=(0,\pm \pi )$ (blue hexagon), $M_3=\pm (\pi ,\pi )$ (dark cyan hexagon), $K_{+}=(-2\pi /3,2\pi /3)$ (red left triangle), $K_{-}=(2\pi /3,-2\pi /3)$ (magenta right triangle); $X_{+}=(-\pi /3,\pi /3)$ (orange left triangle), $X_{-}=(\pi /3,-\pi /3)$ (gold right triangle).

Figure 2

Spin excitation gap. (a) Energy gap ${\mathrm{\Delta }}_{S^z=0}$ and (b) ${\mathrm{\Delta }}_{S^z=1}$ as a function of the inserted flux $\theta$ at $J_2/J_1=0.12$ for YC8-0 (blue circles), YC10-0 (blue squares), YC8-1 (black diamonds), and YC10-1 (black pentagon) cylinder geometries. The data are collected using DMRG bond dimension $m=4096$ for YC8/10-0 and 6144 for YC8/10-1. (For details, please see the Supplemental Material [39]).

Figure 3

Correlation length spectrum. Inverse correlation length $1/{\xi }_{S^z=1}$ as a function of the flux $\theta$ (left column), momentum $k_1$ (middle column), and momentum $k_2$ (right column) for the cylinder (a) YC8-0, (b) YC10-0, with $m=6144$ and (c) YC8-1, (d) YC10-1 with $m=12288$. The lowest-lying excitations contain a spinon pair at $M$ points and monopole excitations at $K_{\pm }$ points, the former is denoted by the blue hexagon while the latter is denoted by the red left triangle and magenta right triangle. It is an artifact of finite bond dimension that the correlation length is not diverging at the Dirac point, and it becomes more severe for the larger system sizes (see Supplemental Material [39] for more discussion).

Figure 4

Scaling behavior of the entanglement entropy. Entanglement entropy $\mathcal{S}$ (blue circles) as a function of flux $\theta$ for (a) YC8-0 ($m=12288$) and (b) YC10-0 ($m=8192$) cylinder geometries. Inset: Fit around minima to Eq. (2), where $f(\theta )={\sum }_{n=1}^{N_f}\ln |2\sin [s(\theta -{\theta }_n^c)/2]|$. We fit the formula with data labeled by red circles.