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When $T\ll \Delta$, the argument of the inverse hyperbolic sine is large. Using $\mathrm{arsinh}z=\log (2z)+1/(4z^2)+O(z^{-4})$ givesThe mass remains the zero-temperature gap, with an exponentially small correction from thermally activated excitations.

When $T\gg \Delta$, expand around $z=1/2$. Ifis the golden ratio, then $\mathrm{arsinh}(1/2)=\log \phi$, andThe leading value $m/T=2\log \phi$ is universal. This is the quantum-critical regime: temperature is the only leading energy scale and the correlation length is of order $v/T$.

A quantum critical point is a continuous zero-temperature quantum phase transition reached by tuning a nonthermal parameter of a Hamiltonian, such as interaction strength or magnetic field. The ground state changes its long-distance structure, its correlation length diverges, and its characteristic excitation energy tends to zero in the thermodynamic limit.

For the Transverse-field Ising model, exchange favors aligned spins while the transverse field favors a competing orientation. At the critical ratio the gap closes; the dynamical critical exponent relates the diverging time and length scales. A finite sample rounds the singular behavior. Experiments at nonzero temperature can access a nearby quantum-critical regime, but a finite-temperature anomaly alone does not locate or prove a zero-temperature critical point.

The quantum-critical regime is the finite-temperature region in which temperature exceeds the energy scale generated by detuning from a quantum critical point. Temperature then controls leading masses, relaxation rates, and correlation lengths.

A quantum phase transition is a change of the ground state as a nonthermal control parameter is varied at zero temperature. Pressure, interaction strength or an applied magnetic field can tune the competing contributions to the Hamiltonian. The transition is driven by changes in the quantum ground state, whereas a thermal phase transition is driven by thermal occupation and entropy.

The Transverse-field Ising model illustrates the competition between aligned-spin order and a field favoring a different spin direction. A continuous transition has a quantum critical point, with a closing excitation gap and growing correlation scales; a quantum phase transition can also be first order. Experiments operate at finite temperature, where the nearby quantum-critical regime can reveal the influence of the zero-temperature transition without placing the sample exactly at zero.