Dangerously Irrelevant Couplings and Hyperscaling Violation

An RG-irrelevant coupling usually produces only vanishing corrections near a fixed point. It is dangerously irrelevant when an observable becomes singular if that coupling is set to zero. Such a variable can control the ordered amplitude or thermal transition line and invalidate naive one-scale hyperscaling. Hyperscaling violation from an extended set of gapless modes is a distinct phenomenon and should not be labeled dangerous irrelevance automatically.

Required background. Quantum-Critical Fans and Finite-Temperature Scaling supplies the one-scale free-energy hypothesis; Critical Surfaces, Crossover, and Corrections to Scaling supplies irrelevant eigenvalues and crossover fields. Helpful background. Hertz–Millis Theory and Landau Damping supplies the metallic quartic example.

A coupling that cannot be set to zero

Consider an LGW potential above its upper critical dimension,

$$V(\phi)=\frac r2\phi^2+\frac u{4!}\phi^4, \qquad u>0.$$

At the Gaussian fixed point $y_u=4-(d+z)<0$, so $u$ flows toward zero. Yet for $r<0$,

$$\phi_0^2=-\frac{6r}{u}.$$

The ordered amplitude is singular as $u\to0$. The coupling is irrelevant for critical two-point exponents but dangerous for the broken-symmetry state. A scaling form must retain it,

$$f_s(r,T,u)=b^{-(d+z)} \mathcal F(rb^{1/\nu},Tb^z,ub^{y_u}),$$

and the small final argument may enter $\mathcal F$ nonanalytically.

Thermal mass and the shift exponent

In a simple Hertz–Millis theory, thermal fluctuations generate

$$\delta r(T)\sim u\,T^{(d+z-2)/z},$$

up to marginal logarithms and model-dependent constants. On the ordered side the thermal transition satisfies $|r|\sim\delta r(T_c)$, hence

$$T_c\sim |r|^\psi, \qquad \psi=\frac{z}{d+z-2}.$$

The zero-temperature gap crossover instead has exponent $\nu z$; with Gaussian $\nu=1/2$, these generally differ above $d+z=4$. Millis derived this split for itinerant quantum critical points Millis 1993, §§ III–IV.

The formula is not universal outside its assumptions. The symmetry of the thermal transition, dimensional reduction, disorder, and nonanalytic fermion vertices can change it. A fitted phase-boundary exponent should therefore not be substituted directly for $\nu z$.

Hyperscaling and its violation exponent

Ordinary hyperscaling assumes one singular degree of freedom per correlation volume,

$$f_s\sim \xi^{-(d+z)} \quad\Longrightarrow\quad f_s(T,r=0)\sim T^{(d+z)/z}.$$

It is useful to parameterize a reduced effective spatial count by $\theta$,

$$f_s\sim \xi^{-(d+z-\theta)}, \qquad f_s(T)\sim T^{(d+z-\theta)/z}.$$

For a conventional Fermi surface, low-energy modes occupy a codimension-one manifold and thermodynamics often behaves as if $\theta=d-1$, yielding $C\sim T$ when $z=1$. In critical metals, patch anisotropy and interactions can modify the assignment; $\theta$ must be derived from scaling of the free energy, not inferred from the phrase “Fermi surface.”

A dangerous coupling and nonzero $\theta$ answer different questions. The former is an RG variable whose vanishing limit is singular; the latter counts how the singular free energy scales relative to volume. A theory may have either, both, or neither. Critical Fermi-surface scaling gives concrete examples of hyperscaling violation tied to an extended momentum manifold Senthil 2008, §§ II–IV.

Observable consequences and checks

Hyperscaling relations among $\alpha,\beta,\gamma,\nu,\eta$ can fail above the upper critical dimension because the order parameter depends on $u$. Finite-size scaling can also use a thermodynamic length different from $\xi$. In quantum-critical data, signatures include:

• a thermal line with $\psi\ne\nu z$;
• amplitudes singular in a nominally irrelevant coupling;
• entropy or specific heat inconsistent with $d+z$ correlated volumes;
• different scaling variables for static and dynamic observables.

None is decisive alone. Analytic background heat capacity can imitate a hyperscaling-violating power, and two nearby crossovers can imitate split exponents. A validity check fits a common RG structure to the phase boundary, zero-temperature gap, finite-size drift, and thermodynamics.

Exercises

1. For $d=3$, $z=2$, find $y_u$, $\nu z$, and the Hertz–Millis shift exponent.

Solution

$y_u=4-(d+z)=-1$. Gaussian $\nu=1/2$ gives $\nu z=1$. The shift exponent is $\psi=2/(3+2-2)=2/3$, demonstrating the split.

2. If $d=2$, $z=3$, and $\theta=1$, what is the critical specific-heat power?

Solution

$f_s\sim T^{(d+z-\theta)/z}=T^{4/3}$. Therefore $C_s\sim T^{(d-\theta)/z}=T^{1/3}$, assuming the scaling form and no larger regular term.

References

• Millis, A. J. “Effect of a Nonzero Temperature on Quantum Critical Points in Itinerant Fermion Systems.” Physical Review B 48 (1993): 7183–7196. DOI.
• Senthil, T. “Critical Fermi Surfaces and Non-Fermi Liquid Metals.” Physical Review B 78 (2008): 035103. DOI.