Quantum-Critical Fans and Finite-Temperature Scaling

A quantum-critical fan is a finite-temperature crossover regime in which the correlation time is limited primarily by $1/T$ and observables follow scaling functions of the zero-temperature fixed point. Its boundaries are not phase boundaries in general. Demonstrating a fan requires consistent scaling across observables and systematic exclusion of ordinary incoherence, disorder, and hidden transition lines.

Required background. Quantum Phase Transitions and Competing Scales supplies $T^*\sim|r|^{\nu z}$ and finite-size variables. Helpful background. Model Selection and Parameter Inference supplies correlated-error fits and tests against alternative scaling forms.

Finite-temperature scaling

If hyperscaling holds and there is one relevant tuning field $r$, the singular free-energy density obeys

$$f_s(r,T,h)=T^{(d+z)/z} \Phi_f\!\left(\frac{r}{T^{1/(\nu z)}}, \frac{h}{T^{y_h/z}}\right).$$

Derivatives give thermodynamic responses. At $r=0$, for example, the singular specific heat scales as $C_s\sim T^{d/z}$, while an order-parameter susceptibility scales as

$$\chi(r,T)=T^{-(2-\eta)/z} \Phi_\chi\!\left(\frac{r}{T^{1/(\nu z)}}\right)$$

for the simplest order-parameter normalization. Conserved currents, Fermi surfaces, dangerous variables, and analytic backgrounds can change these powers or add dominant regular pieces.

Dynamic response takes the form

$$\chi''(\mathbf q,\omega,T) =T^{-(2-\eta)/z} \Phi''\!\left( \frac{\mathbf q-\mathbf Q}{T^{1/z}}, \frac{\omega}{T}, \frac{r}{T^{1/(\nu z)}}\right),$$

subject to metric factors and operator normalization. The appearance of $\omega/T$ is a consequence of single-scale dynamics, not a universal guarantee for every critical theory.

Locating the fan

The crossover condition is

$$T\gg T^*(r)\sim |r|^{\nu z},$$

while $T$ must remain below a microscopic upper cutoff and above any ordering, coherence, finite-size, disorder, or dangerously-irrelevant scale. Plotting straight lines by eye from a putative critical point is not a determination of these boundaries. One should extract a response-specific crossover criterion, propagate uncertainty in $g_c$, $\nu z$, and metric factors, and check whether several criteria converge toward the same zero-temperature point.

On an ordered side, a thermal transition may enter the fan. In a metal, superconductivity often hides the lowest-temperature normal state. A magnetic field used to suppress superconductivity introduces its own scaling field and orbital effects; the revealed regime is not automatically the zero-field critical metal.

Data collapse as a statistical test

For measured $O(r,T)$ with the convention $O\mapsto b^{-x_O}O$ used in this chapter, a collapse rewrites

$$T^{-x_O/z}O(r,T)=\Phi_O(r/T^{1/(\nu z)}).$$

The fit must include uncertainty in both axes, correlations among points, analytic backgrounds, and corrections such as

$$O=T^{x_O/z}\left[ \Phi_0(x)+uT^{|y_u|/z}\Phi_1(x)+\cdots\right]+O_{\mathrm{reg}}.$$

Flexible interpolation can make many exponents appear to collapse a small data set. Strong evidence uses held-out temperatures or tuning values, reports the range over which parameters are stable, and compares with alternative crossover models. Scaling analyses of the two-dimensional antiferromagnet illustrate how universal functions require both a continuum theory and calibrated amplitudes Chubukov, Sachdev, and Ye 1994, §§ III–VI.

What the fan does not identify

A fan identifies an organizing scale only to the extent that the scaling hypothesis is unique. It does not by itself identify order-parameter fields, prove quasiparticle destruction, or establish a transport bound. Linear resistivity, logarithmic heat capacity, and broad spectra can arise from distinct mechanisms. The most discriminating observables are those for which competing models predict different momentum structure, field dependence, amplitude ratios, or cross-correlations.

The scaling window can also be preasymptotic. Stable exponents across expanding ranges and consistency with zero-temperature finite-size or tuning behavior are stronger than a collapse at one decade.

Exercises

1. For $\nu z=1$, what combination of $r$ and $T$ should label fan contours?

Solution

The scaling variable is $x=r/T^{1/(\nu z)}=r/T$. Contours of fixed $x$ satisfy $T\propto|r|$ up to nonuniversal metric factors.

2. If $d=2$, $z=1$, and hyperscaling holds, what is the singular specific-heat power at criticality?

Solution

$f_s\sim T^{(d+z)/z}=T^3$. Two temperature derivatives give $C_s\sim T^{d/z}=T^2$. A larger analytic or noncritical contribution can mask this term.

References

• Chubukov, A. V., S. Sachdev, and J. Ye. “Theory of Two-Dimensional Quantum Heisenberg Antiferromagnets with a Nearly Critical Ground State.” Physical Review B 49 (1994): 11919–11961. DOI.