Ring frustration and factorizable correlation functions of critical spin rings

Tackling the critical transverse Ising ring with or without ring frustration, we establish the concept of nonlocality in a many-body system in the thermodynamic limit by calculating the nonlocal factors embedded in the factorizable correlation function. Through this exactly solvable prototype, we clearly show the intriguing difference between the periodic chains with odd and even numbers of lattice sites even in the thermodynamic limit. In the context of nonlocality, we also address the important application of finite-size scaling analysis by numerically working out the nonlocal factors of the isotropic $XY$ model and the spin-1/2 Heisenberg model.

Figure 1

Periodic spin chain. Ring frustration occurs if the number of spins is odd and the nearest-neighbor interactions are antiferromagnetic.

Figure 2

Nonlocal factors of the transverse Ising ring at its phase transition point.

Figure 3

$R_{r,N}$ with $N=101,1001$, and 10001 for the isotropic $XY$ model. The data collapse to the proposed scaling curve $B_2\left(\alpha \right)$ very accurately. The ratios, $\frac{R_{r,N}}{B_2\left(\alpha \right)}$, in the inset demonstrate how the data approach the curve $B_2\left(\alpha \right)$ with $N$ increasing.

Figure 4

$R_{r,N}$ with $N$ from 11 to 21 for the spin-1/2 Heisenberg model. The data collapse to the proposed scaling curve $B_2\left(\alpha \right)$ very well.