Deformations of Lifshitz holography

TL;DR

The paper analyzes Lifshitz holography with $z=2$ in Einstein–Proca gravity, focusing on a marginally relevant deformation that dynamically generates a scale $\Lambda$ and studying finite-temperature black holes in the regime $T \gg \Lambda^2$. It develops holographic renormalization for asymptotically Lifshitz geometries with logarithmic running, derives exact relations among thermodynamic quantities, and characterizes how $F$, $E$, and $\mathcal J^{\hat t}$ depend on $\log(\Lambda^2/T)$ via high-temperature expansions. A key result is the integrated first law $\mathcal F = \mathcal E - Ts$ together with an RG-invariant $K$ linking deformation data to boundary observables, illustrating crossover behavior from Lifshitz to IR physics. The work provides a framework for incorporating marginally relevant Lifshitz deformations in holographic models of quantum criticality with potential relevance to condensed matter systems.

Abstract

The simplest gravity duals for quantum critical theories with z=2 `Lifshitz' scale invariance admit a marginally relevant deformation. Generic black holes in the bulk describe the field theory with a dynamically generated momentum scale Lambda as well as finite temperature T. We describe the thermodynamics of these black holes in the quantum critical regime where T >> Lambda^2. The deformation changes the asymptotics of the spacetime mildly and leads to intricate UV sensitivities of the theory which we control perturbatively in Lambda^2/T.

Figures (7)

• Figure 1: A few scales of significance in our discussion and solutions. The first one is $\Lambda$, which is the scale dynamically generated by the marginally relevant mode. In particular $\Lambda=0$ when the marginally relevant mode is turned off. For the high temperature black hole solutions we are able to discuss most explicitly, the scale $\Lambda$ is behind the event horizon located at $r_+^{-1}$. To the right is the sliding UV cut-off beyond which the strict Lifshitz asymptopia is destroyed (for a given choice of $\mu$, which can be taken arbitrarily large).
• Figure 2: Left: $f r^4/f_0 r_+^4$ versus $\log r/r_+$. The dashed red line shows the function $f r_+^4/f_0 r^4$ as given in (\ref{['asymptotics']}), including higher order terms, for $f_0 r_+^4=27.72$. Right: $p r^2/p_0 r_+^2$ versus $\log r/r_+$, with the dotted line corresponding to $p_0 r_+^2=1.89$. The black line is the numerical curve and the black dots are the values used for fitting.
• Figure 3: Entropy density over $T$ versus $\log \Lambda^2/T$ for $h_0=0.9713$ to $h_0=0.9708$. Dots are data points, which are joined by straight lines.
• Figure 4: Plots of ${\cal F}/Ts$, ${\cal E}/Ts$ and ${\cal J}^{\hat{t}}/Ts$ as a function of $\log r/r_+$ for $h_0=.971$, corresponding to $\log (\Lambda^2/T) \approx -3210$. The near horizon region is dominated by IR effects while near the boundary divergences set in due to use of a truncated series of counterterms (\ref{['eq:coeffs']}). The quantities are well defined in the intermediate region, which can be made large by working with a sufficiently high order counterterm expansion. Black lines are numerical results while the red dashed line is a fit to the analytical renormalised expressions.
• Figure 5: Plots of ${\cal F}/Ts$ and ${\cal E}/Ts$ , and ${\cal F}/{\cal E}$, as functions of $\log (\Lambda/T^2)$. The range of $h_0$ is from $0.9712$ to $0.9698$, corresponding to $\log \Lambda^2/T$ from about $-12000$ to $-1150$. The dots are numerical results, while the lines are the fits in equations (\ref{['FoverSTfit']}), (\ref{['EoverSTfit']}), and (\ref{['FoverEfit']}).