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On the IR completion of geometries with hyperscaling violation

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Jyotirmoy Bhattacharya, Sera Cremonini, Annamaria Sinkovics

High Energy Physics - Theory · Submitted 8 Aug 2012 · 31 references

Pith's verdict: A clean, honest extension of Harrison-Kachru-Wang to nonzero hyperscaling violation; the analytic core is solid and the numerical window is fine-tuned but openly flagged as such.

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arXiv ID1208.1752 (version 3)
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CitationAnnamaria Sinkovics, Jyotirmoy Bhattacharya, Sera Cremonini. "On the IR completion of geometries with hyperscaling violation." arXiv:1208.1752 (2012). https://pith.science/paper/RXUKV5AY
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The paper

The paper claims

A failing holographic geometry is rescued by corrections that pin the dilaton at the horizon, leaving a stable two-dimensional anti-de Sitter throat in the deep infrared.

The reading

This paper asks what becomes of a class of holographic geometries that describe strongly interacting matter with two scaling exponents — a dynamical critical exponent $z$ and a hyperscaling violation exponent $θ$ — when the description is pushed into the deep infrared. In these solutions a scalar field, the dilaton, runs logarithmically and grows without bound toward the horizon, so the classical theory is being evaluated exactly where it can no longer be trusted; the authors call such geometries 'IR-incomplete.' They show that once the gauge kinetic function is allowed even a modest correction, the dilaton is pinned at a constant value near the horizon and the geometry settles into an exact $AdS_2×R^2$ throat. Numerically integrating the full equations, they obtain flows that begin at that throat, pass through a window in which both $z$ and $θ$ scaling hold, and end at $AdS_4$ in the ultraviolet. If this is right, the scaling geometry is not the final word about the ground state but a transient, and the true infrared is a two-dimensional anti-de Sitter factor whose entropy and stability properties are already much studied.

What carries the argument

The load-bearing device is the promotion of the gauge kinetic function to $f(φ)=e^{2αφ}+ξ_1+ξ_2e^{-2αφ}+…$, which acts as an effective potential that traps the dilaton at a constant $φ_H$ near the horizon. Two conditions, read off the Einstein and scalar equations for the $AdS_2×R^2$ ansatz, fix the value and slope of the correction at the horizon; because only $f$ and $∂_φf$ appear, the existence of the throat is a universal feature of a broad class of corrections rather than an artifact of the two-parameter expansion. The second piece of machinery is the classification of linear perturbations about the throat: only modes with $ν>0$ are irrelevant in the infrared, and the paper exhibits exactly two such modes, which is what makes a two-parameter shooting problem toward the ultraviolet possible. A third piece is the null-energy analysis in the appendix, which constrains the classical scaling exponents to $0≤β≤1$ and $γ≤1$ and is used to check that the chosen parameters correspond to sensible matter in the region where the corrections are negligible.

Extended reading notes

Core claim

The central claim is that the magnetically charged hyperscaling-violating solutions of Einstein-Maxwell-dilaton theory are completed in the infrared by an $AdS_2×R^2$ region, once the gauge kinetic function is deformed away from the pure exponential $f(φ)=e^{2αφ}$ to $f(φ)=e^{2αφ}+G(φ)$ with $G$ a generic correction. The authors first show that classically, with $G=0$, a constant-dilaton $AdS_2×R^2$ solution exists only in two special cases: $θ=2$ at finite $z$, which violates the entanglement-entropy area law $θ≤d-1$, or the $z→∞$ limit. Turning on the correction changes this: $AdS_2×R^2$ becomes an exact solution whenever the correction satisfies two algebraic conditions at the horizon, one fixing its value, $e^{2αφ_H}+G(φ_H)=2b_H^4L^2/Q_m^2$, and one fixing its slope, relating $2αe^{2αφ_H}+∂_φG(φ_H)$ to $-η/α$ times the same flux-to-geometry ratio. Because only $G$ and its first derivative enter, the mechanism is insensitive to the detailed shape of the correction. Linearizing about the throat, they identify two modes that are irrelevant in the infrared, with exponents $ν=1$ and a second $ν$ set by $α$, $η$ and the correction parameters. Integrating the full equations outward from the throat for a specific parameter set ($α=√3$, $ξ_1=0$, $ξ_2=1$, $V_0=3×10^4$, $c_1=10^{-4}$, $η=2√3$), they find a flow that leaves $AdS_2×R^2$, crosses an intermediate region with $θ=-2$ and $z=3/2$, and lands on $AdS_4$ with the dilaton settling at the minimum of the potential and falling off as $r^{-1}$.

Load-bearing premise

The distinctive claim that an intermediate hyperscaling-violating window actually appears rests on tuning the two amplitudes of the infrared-irrelevant deformations onto a narrow locus, and the paper itself notes that the window grows only as one approaches a boundary of that two-dimensional parameter space beyond which solutions diverge in the ultraviolet.

Editorial extensions

If this is right

  • Hyperscaling-violating and Lifshitz-scaling geometries should be read as intermediate descriptions of a flow, not as candidate ground states: the deep infrared is instead an $AdS_2×R^2$ throat with a constant dilaton.
  • The existence of an $AdS_2$ factor in the infrared reduces to two algebraic constraints on the value and slope of any correction to the gauge kinetic function at the horizon, independently of whether that correction is classical or quantum in origin.
  • The same constraints can be written down for corrections to the scalar potential, so the mechanism extends beyond the specific deformation studied here.
  • Because the infrared endpoint carries an $AdS_2$ factor, the usual puzzle of extensive ground-state entropy for extremal charged black branes is inherited by this class of hyperscaling-violating solutions.
  • The construction gives an explicit template for embedding hyperscaling violation as an intermediate regime inside a supergravity or string theory flow that starts from $AdS_4$ and ends at $AdS_2×R^2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If only $f$ and its first derivative at the horizon matter, then the same infrared endpoint should be reachable from many ultraviolet completions, which suggests searching for it in known gauged supergravity truncations where the gauge kinetic function is fixed by supersymmetry rather than chosen by hand.
  • The fine tuning needed to expose the intermediate scaling window hints that hyperscaling violation may be rare in generic flows — visible only near the boundary of the space of infrared deformations — which would explain why such geometries have been hard to identify in explicit string compactifications.
  • Because the intermediate region is where the area law is violated for $θ>d-1$, a natural next check is whether the entanglement entropy computed along the full numerical flow shows a transition between area-law and violating behaviour at the radial scale where the scaling window opens and closes.
  • The appendix's null-energy bounds and the horizon conditions together give a compact checklist that could be run against any proposed holographic model of a compressible state with a hidden Fermi surface, without solving the full equations.
Original abstract

We study solutions to Einstein-Maxwell-dilaton gravity with a constant magnetic flux which describe, in the holographic AdS/CFT framework, field theories characterized by a dynamical critical exponent and a hyperscaling violation exponent. Such solutions are known to be IR-incomplete due to the presence of a running dilaton, which drives the theory towards strong coupling in the IR, where quantum corrections become important. After introducing generic corrections, in this note we examine the conditions for the emergence of an AdS_2 x R^2 region close to the horizon, which provides an IR-completion for the hyperscaling violating solutions. In the presence of these corrections, we construct explicit numerical solutions where the geometry flows from AdS_4 in the UV to AdS_2 x R^2 in the deep IR, with an intermediate region which exhbits both hyperscaling violation and Lifshitz scaling.We also provide constraints on the structure of Einstein-Maxwell-dilaton theories that admit such solutions, as well as an emergent AdS_2 x R^2 region in the infrared.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,862 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.31 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.33 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We study solutions to Einstein-Maxwell-dilaton gravity with a constant magnetic flux which describe, in the holographic AdS/CFT framework, field theories characterized by a dynamical critical exponent and a hyperscaling violation exponent. Such solutions are known to be IR-incomplete due to the presence of a running dilaton, which drives the theory towards strong coupling in the IR, where quantum corrections become important. After introducing generic corrections, in this note we examine the conditions for the emergence of an AdS_2 x R^2 region close to the horizon, which provides an IR-completion for the hyperscaling violating solutions. In the presence of these corrections, we construct explicit numerical solutions where the geometry flows from AdS_4 in the UV to AdS_2 x R^2 in the deep IR, with an intermediate region which exhbits both hyperscaling violation and Lifshitz scaling.We also provide constraints on the structure of Einstein-Maxwell-dilaton theories that admit such solutions, as well as an emergent AdS_2 x R^2 region in the infrared.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study solutions to Einstein-Maxwell-dilaton gravity with a constant magnetic flux which describe, in the holographic AdS/CFT framework, field theories characterized by a dynamical critical exponent and a hyperscaling violation exponent. contribution 0.66 0.71 0.04 asserted 0.99
s1 Such solutions are known to be IR-incomplete due to the presence of a running dilaton, which drives the theory towards strong coupling in the IR, where quantum corrections become important. background 0.94 0.06 0.04 asserted 1.00
s2 After introducing generic corrections, in this note we examine the conditions for the emergence of an AdS_2 x R^2 region close to the horizon, which provides an IR-completion for the hyperscaling violating solutions. contribution 0.99 0.83 0.34 asserted 0.54
s3 In the presence of these corrections, we construct explicit numerical solutions where the geometry flows from AdS_4 in the UV to AdS_2 x R^2 in the deep IR, with an intermediate region which exhbits both hyperscaling violation and Lifshitz scaling.We also provide constraints on the structure of Einstein-Maxwell-dilaton theories that admit such solutions, as well as an emergent AdS_2 x R^2 region in the infrared. result 0.51 0.95 0.79 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.84 · evidence from simulation or computation 0.95.

What was examined, and what was not

sources read
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not examined
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  • figures and tables
  • references and citing sentences
  • relations to other papers
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not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Plots of the metric functions and dilaton for the parameter set 1 (η = 0.1). Note that the red line represents a hyperscaling violating and Lifshitz-like scaling solution with {θ = −0.21, z = 4.9}, while the blue line represents our numerical solution. The fact that a hyperscaling violating regime emerges in the UV is clear from the matching of the two plots in that region. In all the three functions in
Figure 2. Plots of the metric functions and dilaton for the parameter set 2 (η = -0.1). As in the previous figure, the red line represents a hyperscaling violating and Lifshitz-like scaling solution with {θ = 0.19, z = 5.1}, while the blue line represents our numerical solution. The fact that the hyperscaling violating solution emerges in the UV is clear from the matching of the two plots in that region. log-log plots on the right hand side of Figures 1 and 2. Deep in the IR, we start with the AdS2 × R2 solution, for which a ′ (r) and b(r) are constants, as clearly visible in their respective
Figure 3. Plots of the metric functions and dilaton with the modified potential. Here the red lines represent a hyperscaling violating solution (with θ = −2 and z = 3/2) while the blue lines represents our numerical solution. The fact that the hyperscaling violating solution emerges in the intermediate region is clear from the matching of the two plots there. In the log-log plot for φ, the yellow line is the constant UV value φuv = 3.99. The numerical plot (blue line) approaches the yellow line like φuv − 2 × 105 r −1 , as expected. 5. Discussion Einstein-Maxwell-dilaton theories with simple
Figure 4. Plots of E(r) for two distinct values of ψ for the parameter set 1 (η = 0.1) and set 2 (η=-0.1). The fact that E(r) is positive for all values of r (for two distinct values of ψ) shows that the chosen set of parameters correspond to sensible matter. To test whether we have sensible matter in a theory with negative cosmological constant we generally impose the null energy condition E = nµnνT µν ≥ 0 , (A.1) – 24 –

Figures reproduced from arXiv:1208.1752 by the authors.

References
How the literature uses it

48 arXiv papers cite this in 69 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 41 sentences without a confident label

Compares with it 5 papers
  • Thus, while in both of these constructions the IR endpoint of the hyperscaling violating solutions is $AdS_2$, with the associated extensive ground state entropy, the origin of the latter is of a different nature – quantum mechanical in [ref] , and classical in the dyonic system studied in [ref] .].

    Spatially Modulated Instabilities of Geometries with Hyperscaling Violation · Introduction · 0.94

  • In fact the situation is similar to the relativistic case studied in [ref] where it was shown that by taking into account the quantum corrections the magnetically charged hyperscaling violating solution could be completed at IR by an emergent AdS$_2×R^d$ solution.

    Conformally Lifshitz solutions from Horava-Lifshitz Gravity · Hyperscaling Violating Metric · 0.83

  • Such flows can therefore be viewed as an interesting framework for resolving the singularity of the Lifshitz-like fixed point, differing from other singularity resolving flows [ref] .

    Boomerang RG flows in M-theory with intermediate scaling · Introduction · 0.90

Cites it as support 1 paper
Cites it as background 21 papers
  • The holographic system we use is an $AdS_2×R^2$ geometry (with a hard wall cutoff) supported by a magnetic field; this type of geometry is known to be the near-horizon region of extremal Reissner-Nordstrom black branes, as well as an endpoint of IR flows in more general Einstein-Maxwell-Dilaton (EMD) systems [ref] .

    Crystalline Scaling Geometries from Vortex Lattices · Introduction · 0.85

  • By now, there are several known ways to resolve this issue [ref] .

    What do non-relativistic CFTs tell us about Lifshitz spacetimes? · Lifshitz geodesics · 0.80

  • Recently, following earlier work on Lifshitz geometries [ref] , it has been suggested [ref] that hyperscaling IR geometries are destabilised due to the running dilaton.

    Bosonic Fractionalisation Transitions · Discussion · 0.89

What it cites · 25 sentences without a confident label

Builds on 1 paper
  • In this note, we would like to extend the analysis of [ref] to the more general class of geometries exhibiting both hyperscaling violation and Lifshitz-like scaling, which are generated by

    Resolving Lifshitz Horizons · Introduction · 1.00

Uses the method or tool of 1 paper
  • To this end, we will follow the strategy of [ref] and consider a toy model for the quantum-corrected version of the theory, by appropriately modifying the structure of the gauge kinetic function.

    Resolving Lifshitz Horizons · Introduction · 0.96

Compares with 5 papers
  • Thus, we should emphasize that our analysis in this note complements[The range for the lagrangian parameters $α,η$ in this note corresponds to Case II of [ref] .] that of [ref] , which stresses that turning on even a small amount of magnetic field can have a dramatic effect on the behavior of the system – and in particular on that of the entanglement entropy.

    Entangled Dilaton Dyons · Discussion · 0.87

  • Precisely the same type of flow was already seen in [ref] for the Lifshitz case.

    Resolving Lifshitz Horizons · Discussion · 0.88

  • As a simple check of our results, we note that the form of this solution is in agreement with the analogous one in [ref] .

    Holography of Dyonic Dilaton Black Branes · Lifshitz as a special case · 1.00

Cites as support 1 paper
  • Moreover, some of these hyperscaling violating solutions can be IR-completed already at the classical level, by turning on, in addition to an electric field, a small magnetic field [ref] .

    Entangled Dilaton Dyons · Discussion · 0.96

Cites as background 23 papers
  • While the fate of these geometries is equally interesting, in this note our focus will be on understanding the behavior of the strongly coupled, magnetic case (see [ref] and [ref] for related discussions in the context of Lifshitz systems without running couplings).

    The Benefits of Stress: Resolution of the Lifshitz Singularity · Introduction · 0.91

  • While the fate of these geometries is equally interesting, in this note our focus will be on understanding the behavior of the strongly coupled, magnetic case (see [ref] and [ref] for related discussions in the context of Lifshitz systems without running couplings).

    Lifshitz Singularities · Introduction · 0.91

  • Metrics of the form ([ref]) are exact solutions to gravitational theories coupled to a matter sector [ref] , with an abelian gauge field and a dilaton providing the simplest realization of the latter.

    Non-relativistic holography · Introduction · 0.93

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