Nonordinary edge criticality of two-dimensional quantum critical magnets

Based on large-scale quantum Monte Carlo simulations, we examine the correlations along the edges of two-dimensional semi-infinite quantum critical Heisenberg spin-$1/2$ systems. In particular, we consider coupled quantum spin-dimer systems at their bulk quantum critical points, including the columnar-dimer model and the plaquette-square lattice. The alignment of the edge spins strongly affects these correlations and the corresponding scaling exponents, with remarkably similar values obtained for various quantum spin-dimer systems. We furthermore observe subtle effects on the scaling behavior from perturbing the edge spins that exhibit the genuine quantum nature of these edge states. Our observations furthermore challenge recent attempts that relate the edge spin criticality to the presence of symmetry-protected topological phases in such quantum spin systems.

Figure 1

(a) CD lattice with an edge of dangling spins shown on the bottom edge (CD-D) and nondangling spins on the top edge (CD-N). (b) PS lattice with dangling (nondangling) edge spins on the bottom (top) edge, for the DAF-D and DAF-N cases, respectively, and with dangling (nondangling) edge spins on the top (bottom) edge, for the PAF-D and PAF-N cases, respectively. In both panels, the interdimer (intradimer) couplings $J\left(J_D\right)$ are indicated by black (bold red) lines, the edge (bulk) spins by open (full) symbols, and periodic boundary conditions by open lines. (c) Phase diagram of the PS lattice with the antiferromagnetic region (AF), the dimer phase (D) for $J<J_{DAF}\approx 0.6J_D$, and the plaquette phase (P) for $J>J_{PAF}\approx 1.1J_D$.

Figure 2

$C_{\parallel }\left(r\right)$ vs $\zeta \left(r\right)=sin(r\pi /L)L/\pi$ for the different edge spin configurations shown in Fig. 1.

Figure 3

$C_{\parallel }(L/2)$ (top panels), $C_{\perp }(L/2)$ (middle panels), and ${\chi }_s$ (bottom panels) as functions of $L$ for the different configurations of Fig. 1 along with finite-size fits (dashed lines).

Figure 4

Lateral correlations $C_{\parallel }\left(r\right)$ vs $\zeta \left(r\right)$ for the DAF-D configuration with different values of the edge coupling enhancement $\kappa$, as shown in the inset [left panel (a)], and for the DAF-D configuration with different values of the coupling $\kappa J_D$ to additional spins, shown in the inset by semifilled circles [right panel (b)]. In both panels the curves are labeled by the value of $\kappa$.