Quantum Criticality of Two-Dimensional Quantum Magnets with Long-Range Interactions

We study the critical breakdown of two-dimensional quantum magnets in the presence of algebraically decaying long-range interactions by investigating the transverse-field Ising model on the square and triangular lattice. This is achieved technically by combining perturbative continuous unitary transformations with classical Monte Carlo simulations to extract high-order series for the one-particle excitations in the high-field quantum paramagnet. We find that the unfrustrated systems change from mean-field to nearest-neighbor universality with continuously varying critical exponents. In the frustrated case on the square lattice the system remains in the universality class of the nearest-neighbor model independent of the long-range nature of the interaction, while we argue that the quantum criticality for the triangular lattice is terminated by a first-order phase transition line.

Figure 1

Illustration of the square (left) and triangular (right) lattice with basis vectors ${\mathbf{e}}_{\mathbf{1}}$ and ${\mathbf{e}}_{\mathbf{2}}$.

Figure 2

Critical point ${\lambda }_{\mathrm{c}}$ (upper panel) and exponent $z\nu$ (lower panel) are shown as squares (triangles) for the ferromagnetic LRTFIM on the square (triangular) lattice. Error bars represent the standard deviation of nondefective dlogPadé extrapolants. Shaded areas correspond to MF (left) and nearest-neighbor (NN) (right) universality, which are known exactly in these $\alpha$ ranges [48]. Upper panel: Thick lines indicate the quantum-critical points ${\lambda }_c=0.16421$ [49] (square lattice) and ${\lambda }_c=0.105$ [50] (triangular lattice) for the nearest-neighbor TFIM. MF results are given as dot-dashed lines and the quantum Monte Carlo data as red triangles (see Ref. [34]). Lower panel: The upper (lower) dashed (dot-dashed) line refers to $z\nu \approx 0.63$ [47] ($z\nu =0.5$) of the nearest-neighbor TFIM (in MF).

Figure 3

Critical point ${\lambda }_c$ (upper panel) and exponent $z\nu$ (lower panel) are shown as squares (triangles) for the antiferromagnetic LRTFIM on the square (triangular) lattice. Error bars indicate the standard deviation of nondefective dlogPadé extrapolants. Upper panel: Thick lines correspond to ${\lambda }_c=-0.16421(1)$ [49] (${\lambda }_c=-0.305$ [9, 10]) for the TFIM on the square (triangular) lattice. Dot-dashed lines show MF results for the transition between the polarized and Néel (left) and $\sqrt{3}\times \sqrt{3}$ (right) order. Lower panel: Thick lines refer to the 3D-Ising exponent $z\nu \approx 0.63$ [47] (left) and 3D-$XY$ exponent $z\nu \approx 0.67$ [57] (right). The dot-dashed line refers to the MF exponent $z\nu =0.5$.