  In this paper, we analyze the quantum criticality of the Rabi-Stark model at
finite ratios of the qubit and cavity frequencies in terms of the energy gap,
the order parameter, as well as the fidelity, if the Stark coupling strength is
the same as the cavity frequency. The critical exponents are derived
analytically. The energy gap and the length critical exponents are different
from those in the quantum Rabi model and the Dicke model. The finite size
scaling analysis for the order parameter and the fidelity susceptibility is
also performed. The universal scaling behaviors are demonstrated and several
finite size exponents can be then extracted. Furthermore, universal critical
behavior can be also established in terms of the bosonic Hilbert space
truncation number, and the corresponding critical scaling exponents are found.
Interestingly, the critical correlation length exponents in terms of the
photonic truncation number as well as the equivalently effective length scales
are different in the Rabi-Stark model and the quantum Rabi model, suggesting
they belong to different universality classes. The second-order quantum phase
transition is convincingly corroborated in the Rabi-Stark model at finite
frequency ratios, by contrast, it only emerges at the infinite frequency ratio
in the original quantum Rabi model without the Stark coupling.
