Probing Quantum Frustrated Systems via Factorization of the Ground State

The existence of definite orders in frustrated quantum systems is related rigorously to the occurrence of fully factorized ground states below a threshold value of the frustration. Ground-state separability thus provides a natural measure of frustration: strongly frustrated systems are those that cannot accommodate for classical-like solutions. The exact form of the factorized ground states and the critical frustration are determined for various classes of nonexactly solvable spin models with different spatial ranges of the interactions. For weak frustration, the existence of disentangling transitions determines the range of applicability of mean-field descriptions in biological and physical problems such as stochastic gene expression and the stability of long-period modulated structures.

Figure 1

Analytical lower bound $f_c$ (solid blue line) and exact numerical value of the frustration compatibility threshold $f_t$ (dashed black line) as functions of the ratio $J_y/J_x$. GS factorization occurs if and only if $f<f_t$. For $f$ below the horizontal dotted line $f=\frac{1}{2}$ the magnetic order is single-step antiferromagnetic (SA), while for $f$ above it, it is dimerized antiferromagnetic (DA). Therefore, no factorized GS supports DA order, except at $J_y=0$.

Figure 2

Threshold value of the frustration, $f_t$, below which GS factorization occurs, for frustrated antiferromagnets with $r_{\max }=4$, as a function of $J_y/J_x$. Solid black line: $f_2=f$; $f_3=f^2$; $f_4=f^3$. Dashed red line: $f_2=f$; $f_3=f/2$; $f_4=f/3$. Dotted blue line: $f_2=f$; $f_3=f/2$; $f_4=f/4$.