Lattice gas description of pyrochlore and checkerboard antiferromagnets in a strong magnetic field

Quantum Heisenberg antiferromagnets on pyrochlore and checkerboard lattices in a strong external magnetic field are mapped onto hard-core lattice gases with an extended exclusion region. The effective models are studied by the exchange Monte Carlo simulations and by the transfer-matrix method. The transition point and the critical exponents are obtained numerically for a square-lattice gas of particles with the second-neighbor exclusion, which describes a checkerboard antiferromagnet. The exact structure of the magnon crystal state is determined for a pyrochlore antiferromagnet.

Figure 1

Pyrochlore lattice of corner-sharing tetrahedra (a) and its two-dimensional analog, checkerboard lattice (b). Solid lines show the nearest-neighbor bonds of strength $J$. Dashed lines indicate possible further neighbor exchanges.

Figure 2

(Color online). Localized magnons on a checkerboard lattice. Thick lines indicate positions of one-magnon states with corresponding phases shown by + and −. The shaded area shows excluded volume around a central square-void state, which has to be respected for construction of multiparticle states. Four LMs in the upper-left part satisfy a close-packed condition (see the text).

Figure 3

(Color online). Examples of topologically different states of localized magnons: two-magnon subsector (a) and (b); three-magnon subsector (c)–(e); (f) shows an elementary defect state from the class (b).

Figure 4

(Color online). Effective lattice gas model for a checkerboard antiferromagnet. The dual square lattice is 45° rotated compared to the original checkerboard lattice in Fig. 2 and has twice fewer sites. Hard particles are shown by large light circles; small dark circles indicate the excluded sites around one particle. The presented close-packed structure with half filled rows randomly shifted in the horizontal direction has a nonzero Fourier harmonic at ${\mathbf{Q}}_1=(0,\pi )$.

Figure 5

(Color online). Dependence of the two order parameters Eq. (7) on logarithm of the fugacity $\ln z=\mu ∕T$ for three system sizes.

Figure 6

(Color online). Binder cumulant $U_4$ as a function of $\ln z=\mu ∕T$ for different cluster sizes $L$. The inset shows scaling of the derivative $d{U}_4∕d\left(\ln z\right)$ at the transition point $\ln z_c=4.56$ vs $L$.

Figure 7

(Color online). Cross section of a pyrochlore lattice perpendicular to the [111] axis. Tetrahedra from the same kagomé layer are drawn by solid lines. Thick line shows a localized magnon on one of the hexagons in the kagomé plane. Dashed lines indicate tetrahedra from the upper kagomé layer (shown only partially). Dots with numbers denote centers of hexagons in horizontal and tilted planes, which form a dual pyrochlore lattice.

Figure 8

(Color online). Close-packed structure of particles (large light circles) on a dual pyrochlore lattice. Thick (thin) lines represent tetrahedra from the lower (upper) kagomé layer. Small dark circles denote vertices of one truncated tetrahedron.

Figure 9

(Color online). Density of a lattice gas on a pyrochlore lattice as a function of $\ln z=\mu ∕T$. Arrow indicates the density of a close-packed structure $n_0=1∕12$. Inset: probability distribution for the particle density obtained for $N=1728$ cluster at $\ln z=5.2$.