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For the convention used in Ising-chain field derivative and transverse polarization, the polarization per site at the critical field $g=1$ isIndeed, $(g-\cos k)/(2\sqrt{1+g^2-2g\cos k})$ becomes $|\sin (k/2)|/2$, whose Brillouin-zone average is $1/\pi$. The full-polarization value in this spin normalization is $1/2$, so criticality does not mean complete alignment with the transverse field.

This expectation value is the response along the imposed field obtained by differentiating the ground-state energy. It is distinct from the spontaneous order along the exchange axis. Confusing those two directions would give the wrong interpretation of a finite transverse polarization at the quantum critical point.

The analytic continuation from bosonic imaginary frequency to a retarded frequency is $i{\omega }_n\mapsto \omega +i{0}^{+}$. ThusWriting $E_k=\sqrt{v^2{k}^2+m(T{)}^2}$ and using the Sokhotski–Plemelj formula gives, in this sign convention,The opposite overall convention for the retarded Green function reverses this sign; the corresponding spectral function is conventionally chosen positive at positive frequency.

Near the quantum critical point, $m/T$ is a function only of $\Delta /T$. The Green function has the scaling formwithThis is quantum critical scaling with dynamical critical exponent $z=1$ and leading large-$N$ anomalous dimension $\eta =0$.

Heavy fermion materials are correlated metals whose low-energy quasiparticles have exceptionally large effective masses. Partly localized $f$-electron moments hybridize with broader conduction states. The Kondo effect and coherence below the Kondo-lattice coherence temperature produce a narrow low-energy band, so a small energy change corresponds to a large momentum-space state density.

For the electronic heat capacity of a heavy Fermi liquid, the resulting density of states gives a large low-temperature electronic specific-heat coefficient,and often an enhanced magnetic susceptibility. Here $g$ includes both spins and must be expressed per sample, mole or volume consistently with $\gamma$. A coherent Fermi-liquid regime often has $\rho ={\rho }_0+A{T}^2$ with an enhanced coefficient $A$. Above coherence, scattering by local moments can instead give a Kondo-like resistivity upturn.

The competition between moment screening and intersite magnetic interactions can yield a heavy paramagnetic metal, magnetic order, quantum-critical behaviour or unconventional superconductivity. Ce- and U-based compounds supply familiar examples. “Heavy” describes the emergent electronic quasiparticles, not the ionic mass; the simple heavy-band picture need not survive at a quantum critical point.^[ref]

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.