Hyperscaling-Violating and Lifshitz Geometries

A hyperscaling-violating geometry converts two exponents, $z$ and $\theta$, into thermodynamic and correlation-function scaling. That economy is useful only after identifying the matter that supports the metric, checking the null-energy inequalities, and bounding the radial interval over which the solution is trustworthy. Acceptable exponents do not by themselves provide a nonsingular ultraviolet-complete spacetime.

Required background. Sparse Spectra, Large Gaps, and Semiclassical Bulk Criteria supplies the conditions under which a low-derivative bulk model is meaningful. Chemical Potential and Charged Black Branes supplies finite-density thermodynamics.

Helpful background. Critical Exponents, Scaling Relations, and Hyperscaling Caveats supplies the field-theory scaling language. Dangerously Irrelevant Couplings and Hyperscaling Violation explains how irrelevant couplings can set observable scales.

Scale-covariant geometry

With $d$ boundary spatial dimensions, use the radial convention in which the boundary lies toward $r\to0$:

$$ds^2=r^{2\theta/d}\left( \frac{dt^2}{r^{2z}}-\frac{L_r^2dr^2}{r^2}-\frac{d\mathbf x^2}{r^2} \right).$$

Under

$$r\to\lambda r, \qquad \mathbf x\to\lambda\mathbf x, \qquad t\to\lambda^z t,$$

the line element transforms as $ds\to\lambda^{\theta/d}ds$. Lifshitz scaling is $\theta=0$; relativistic AdS is $(z,\theta)=(1,0)$. The effective spatial dimension appearing in extensive thermodynamics is $d_{\mathrm{eff}}=d-\theta$.

These metrics commonly arise as scaling solutions of Einstein–Maxwell–dilaton actions,

$$S=\int d^{d+2}x\sqrt{\lvert g\rvert}\left[R+\frac12(\partial\phi)^2 -V_0e^{-\delta\phi}-\frac{Z_0}{4}e^{\gamma\phi}F^2\right],$$

where $(z,\theta)$ are functions of $(\gamma,\delta)$ and the electric or magnetic branch. A metric written without its supporting matter omits the equations that determine whether it is a solution.

Temperature and entropy scaling

Introduce a regular horizon through

$$ds^2=r^{2\theta/d}\left( \frac{f(r)dt^2}{r^{2z}}-\frac{L_r^2dr^2}{r^2f(r)}-\frac{d\mathbf x^2}{r^2} \right), \qquad f(r_h)=0 .$$

Scale covariance gives $T\propto r_h^{-z}$. The horizon area density gives

$$s=\frac{1}{4G_{d+2}}r_h^{\theta-d} \propto T^{(d-\theta)/z}.$$

This is the first application: a measured or computed entropy exponent fixes the ratio $(d-\theta)/z$, not $z$ and $\theta$ separately. A second independent observable is required to separate them. The same effective dimension controls the logarithmic entanglement-entropy violation at $\theta=d-1$ in appropriate geometries Dong et al. 2012.

Null-energy and completion tests

For the metric above and two-derivative matter satisfying the null-energy condition, independent null directions yield

$$(d-\theta)\bigl[d(z-1)-\theta\bigr]\ge0, \qquad (z-1)(d+z-\theta)\ge0 .$$

These are necessary, not sufficient. One must also check positive specific heat, reality and sign of the matter couplings, stability of fluctuations, and whether curvature and the effective string coupling remain controlled throughout the claimed interval.

An adversarial test follows the scaling solution toward its infrared endpoint. If a curvature invariant diverges, the dilaton runs to strong coupling, or tidal forces become large before the lowest temperature used in a fit, then the formal exponent cannot support that fit. A regular finite-temperature horizon may cloak a singular zero-temperature endpoint without resolving it.

In well-defined applications, the scaling metric is an intermediate region matched to a UV AdS solution and an IR completion. Goutéraux and Kiritsis classified such effective holographic theories and their acceptable extremal limits Goutéraux and Kiritsis 2011. The claim should name the matching scales: $\Lambda_{\mathrm{IR}}\ll T,\omega\ll\Lambda_{\mathrm{UV}}$. Outside that window, crossover data replace pure power laws.

Exercises

For $d=2$, $z=3/2$, and $\theta=1$, determine the entropy exponent and check the two null-energy inequalities.

Solution

$s\propto T^{(2-1)/(3/2)}=T^{2/3}$. The inequalities give $(1)[2(1/2)-1]=0$ and $(1/2)(2+3/2-1)=5/4>0$, so both are satisfied. Completion and stability checks are still required.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Dong, Xi, Sarah Harrison, Shamit Kachru, Gonzalo Torroba, and Huajia Wang. “Aspects of Holography for Theories with Hyperscaling Violation.” Journal of High Energy Physics 2012, 041 (2012). DOI.
• Goutéraux, Blaise, and Elias Kiritsis. “Generalized Holographic Quantum Criticality at Finite Density.” Journal of High Energy Physics 2011, 036 (2011). DOI.