Fig. 4. Magnetic dynamics.

Inverse bosonic dynamic susceptibility χ versus ω_n in three different regions, at U = 3, 6, 8, corresponding a in the FM phase, b in PG and SC phases, and c disordered phase. a log-log plot for various system size L = 6, 8, 10, 12, 14, each of which includes various β = 12, 16, 20, 24. Red line is a quadratic line of χ⁻¹ ~ ω² for low frequency part ω_n < 1. b log-log plot for various β = 10, 16, 20, 24, 30 with L = 12. At temperature T = 0.1 (β = 10), the fermions are in the quantum critical regime, the bosonic susceptibility is the linear function of ω, as indicated by the orange line. When temperature gets lower, the fermion goes into PG phase, prompting the bosonic scaling behavior to deriviate from linear function. And upon entering the SC phase, the χ⁻¹ ~ ω² (the blue line, a guide to the eye) as in the FM phase. c Bosonic susceptibility in the disordered phase at U = 8 plotted for system size L = 6, 8, 10, 12 with various β = 12, 16, 20, 24. At high frequency all data points successfully merge together, as indicated by the red line quadratic in ω for ω > 1. At low frequency, the χ⁻¹ ~ ω as indicated by the blue line, which is a guide to the eye, due to the non-conserved rotor order parameter.