Theoretical constraints for the magnetic-dimer transition in two-dimensional spin models

From general arguments, that are valid for spin models with sufficiently short-range interactions, we derive strong constraints on the excitation spectrum across a continuous phase transition at zero temperature between a magnetic and a dimerized phase, that breaks the translational symmetry. From the different symmetries of the two phases, it is possible to predict, at the quantum critical point, a branch of gapless excitations, not described by standard semiclassical approaches. By using these arguments, supported by intensive numerical calculations, we obtain a rather convincing evidence in favor of a first-order transition from the ferromagnetic to the dimerized phase in the two-dimensional spin-half model with a four-spin ring-exchange interaction, recently introduced by A.W. Sandvik, S. Daul, R.R.P. Singh, and D.J. Scalapino [Phys. Rev. Lett. 89, 247201 (2002)].

Figure 1

(Color online) Comparison on a $6\times 6$ lattice between Lanczos and GFMC for the lowest energy gap. Upper panel: ${\Delta }^{+}(\pi ,0)$ (see text) [in units of $(K+J)$], as a function of $K∕J$. Lower panel: ${\Delta }_{SP}(\pi ,0)$ (see text) [in units of $(K+J)$], as a function of $K∕J$. Inset: direct comparison of the dynamical correlation function for the dimer operator calculated by the GFMC approach and by Lanczos.

Figure 2

(Color online) The dimer gap ${\Delta }_{SP}(\pi ,0)$ [in units of $(K+J)$] obtained by GFMC as a function of the size $N$ of the cluster for different values of $K∕J$. The lines are linear fits of the points. Inset: the extrapolated value in the thermodynamic limit as a function of $K∕J$.

Figure 3

(Color online) The spin-wave spectrum [in units of $(K+J)$] obtained by GFMC by using the operator $O_q=S_q^z$. The linear sizes of the cluster are $L=16$ and 18, and the lines are guides to the eye. The vector $q$ is taken along the $(1,0)$ direction. Upper inset: the behavior of the finite-size gap at the minimum momentum, i.e., $q_{\min }=(2\pi ∕L,0)$ as a function of $L$ for different values of $K∕J$. Lower inset: the thermodynamic limit of the gap at $q=(0,0)$ as a function of $K∕J$.

Figure 4

(Color online) The gap ${\Delta }^{+}(\pi ,0)$ [in units of $(K+J)$] calculated by GFMC as a function of the size $N$ of the cluster for different values of $K∕J$. The lines are linear fits of the points. Inset: the thermodynamic value of the gap as a function of $K∕J$.

Figure 5

(Color online) The gap ${\Delta }^z(\pi ,0)$ [in units of $(K+J)$] calculated by GFMC as a function of the size $N$ of the cluster for different values of $K∕J$. The lines are linear fits of the points. Inset: the thermodynamic value of the gap as a function of $K∕J$.