Thermodynamics of two-dimensional spin models with bimodal random-bond disorder

We use numerical linked cluster expansions to study the thermodynamic properties of the two-dimensional classical Ising, quantum $XY$, and quantum Heisenberg models with bimodal random-bond disorder on the square and honeycomb lattice. In all cases, the nearest-neighbor coupling between the spins takes values $\pm J$ with equal probability. We obtain the disorder-averaged (over all disorder configurations) energy, entropy, specific heat, and uniform magnetic susceptibility in each case. These results are compared with the corresponding ones in the clean models. Analytic expressions are obtained for low orders in the expansion of these thermodynamic quantities in inverse temperature.

Figure 1

Schematic of lattice models, square (left) and honeycomb (right), with bond disorder considered in this work. The red and blue bonds represent $J_{ij}=\pm J$. Black dots at vertices represent the spins. It is easy to see that the bond disorder causes frustration by trying to arrange the spins to minimize energy.

Figure 2

Clusters up to the fourth order in the site-based NLCE on the square lattice. The two three-site clusters have the same Hamiltonian. At fourth order, in addition to three clusters with identical Hamiltonian, two topologically new clusters appear—the closed loop and the “$\perp$.” Each topologically new cluster is diagonalized separately.

Figure 3

Spin-1/2 Ising model on the square lattice. Panels (a)–(d) show the energy, entropy, specific heat, and uniform susceptibility vs $T$, respectively. Solid lines depict disorder-averaged quantities, while dashed lines depict results for the clean system. Thin lines report bare results for the last two orders of the NLCE, while thick lines report the results of two resummation techniques. A thin continuous line following the results of the resummations reports results for a lower order of the same resummation technique, and it is used to gauge their stability. The dotted vertical line marks the position of the phase transition. The dashed-dotted line shows exact analytic results for the clean system in the thermodynamic limit.

Figure 4

Spin-1/2 Ising model on the honeycomb lattice. Panels (a)–(d) show the energy, entropy, specific heat, and uniform susceptibility vs $T$, respectively. Solid lines depict disorder-averaged quantities, while dashed lines depict results for the clean system. Thin lines report bare results for the last two orders of the NLCE, while thick lines report the results of two resummation techniques. A thin continuous line following the results of the resummations reports results for a lower order of the same resummation technique, and it is used to gauge their stability. Resummation results are not presented for the clean case as they do not extend the convergence to lower temperatures. The dotted vertical line marks the position of the phase transition.

Figure 5

Spin-1/2 $XY$ model on the square lattice. Panels (a)–(d) show the energy, entropy, specific heat, and uniform susceptibility vs $T$, respectively. Solid lines depict disorder-averaged quantities, while dashed lines depict results for the clean system. Thin lines report bare results for the last two orders of the NLCE, while thick lines report the results of two resummation techniques. A thin continuous line following the results of the resummations reports results for a lower order of the same resummation technique, and it is used to gauge their stability. The dotted vertical line marks the position of the phase transition.

Figure 6

Spin-1/2 $XY$ model on the honeycomb lattice. Panels (a)–(d) show the energy, entropy, specific heat, and uniform susceptibility vs $T$, respectively. Solid lines depict disorder-averaged quantities, while dashed lines depict results for the clean system. Thin lines report bare results for the last two orders of the NLCE, while thick lines report the results of two resummation techniques. A thin continuous line following the results of the resummations reports results for a lower order of the same resummation technique, and it is used to gauge their stability. Note that the results converge to temperatures similar to those in the square lattice.

Figure 7

Spin-1/2 Heisenberg model on the square lattice. Same as Fig. 5 but for the spin-1/2 Heisenberg model.

Figure 8

Spin-1/2 Heisenberg model on the honeycomb lattice. Same as Fig. 6 but for the spin-1/2 Heisenberg model.