Abstract
The Lee-Yang circle theorem revolutionized our understanding of phase
transitions in ferromagnetic systems by showing that the complex zeros of
partition functions lie on the unit circle, with criticality arising as these
zeros approach the real axis in the thermodynamic limit. However, in frustrated
systems such as antiferromagnets and spin glasses, the zeros deviate from this
structure, making it challenging to extend the Lee-Yang theory to disordered
systems. In this work, we establish a new circle theorem for two-dimensional
Ising spin glasses, proving that the square of the partition function exhibits
zeros densely packed along the unit circle. Numerical simulations on the square
lattice confirm our theoretical predictions, demonstrating the validity of the
circle law for quenched disorder. Furthermore, our results uncover a
finite-temperature crossover in $\pm J$ spin glasses, characterized by the
emergence of a spectral gap in the angular distribution of zeros. This result
extends the Lee-Yang framework to disordered systems, offering new insights
into spin-glass criticality.