Quantum-critical metal scaling

Let a dimensionless tuning distance measure how far the system lies from a proposed critical point in pressure, composition or field. A correlation length diverges with a critical exponent when a continuous transition is approached. A dynamical exponent relates that spatial length to a characteristic energy or inverse correlation time. Combining the two gives a crossover energy that tends to zero with a product of exponents. The underlying microscopic reference length and energy are essential for dimensional consistency.

Finite temperature introduces its own energy scale through Boltzmann constant. A quantum-critical regime is a range where thermal energy exceeds the tuning-induced crossover scale while remaining below microscopic cutoffs. This does not imply that no other scale exists: disorder, superconductivity, magnetic ordering and lattice effects can cut off the regime. First-order transitions, multiple nearby critical modes or inhomogeneous samples can also invalidate a simple one-parameter picture. Identifying the relevant tuning variable is part of the physical inference.

A proposed energy-over-temperature scaling form relates spectra measured at different temperatures through a common dimensionless argument and a temperature prefactor. The response amplitude needs a reference energy so its dimensions stay fixed when a noninteger power is used. To test collapse, compare matched values of energy divided by thermal energy and multiply the measured response by the compensating prefactor. Holding absolute energy fixed while temperature changes generally moves to a different point of the scaling function.

Neutron measurements on a specified heavy-fermion alloy near an antiferromagnetic quantum critical composition reported such magnetic spectral scaling over their measured range. That is evidence about that observable, wavevector region and sample, rather than a universal exponent for every metal. Background subtraction, resolution, uncertainty and the tested dynamic range affect the comparison. Competing scaling functions can look similar over a narrow interval, so a useful test includes deviations, parameter stability and independent tuning data instead of displaying only the best collapse.

For some critical theories obeying ordinary hyperscaling with one dynamical exponent, the singular free-energy density has a simple temperature power involving spatial dimension. Metals with an extended Fermi surface, dangerously irrelevant couplings or hyperscaling violation need not follow that naive power. It is therefore unsafe to infer a universal electronic specific-heat exponent merely from a bosonic scaling slogan. A concrete theory must state which modes contribute and which assumptions justify its thermodynamic scaling.

Linear resistivity is similarly not a direct measurement of a single-particle decay rate. A perfectly translation-invariant system can retain momentum and a dc transport singularity even when quasiparticles decay; momentum relaxation and current overlap must be considered. A phenomenological rate proportional to thermal energy is a useful scale comparison but not a proof of one mechanism. The quadratic-function lab supplies a Fermi-liquid reference power law only; it does not generate quantum-critical spectra, momentum relaxation or a metal's resistivity.