Gubser–Rocha Model Overview

• The Gubser–Rocha model is a charged Einstein–Maxwell–Dilaton holographic framework featuring a 4D asymptotically AdS black hole with nontrivial dilaton hair and linear temperature scaling of entropy.
• It enables analytic studies of finite-density systems and transport phenomena, notably yielding linear-in-T resistivity through momentum-relaxing extensions like linear axions.
• Flexible holographic interpretations via standard, alternate, and multitrace quantizations allow its application in exploring IR structures, entanglement measures, and bulk reconstructions.

The Gubser–Rocha model is a charged Einstein–Maxwell–Dilaton holographic model whose best-known realization is a four-dimensional asymptotically AdS black-hole background with nontrivial dilaton hair. It became a standard finite-density laboratory because it combines analytic control with a distinctive low-temperature sector: the extremal entropy density vanishes, the entropy is linear in temperature, and momentum-relaxing extensions can realize linear-in-$T$ resistivity. In later literature it has also been reinterpreted as a special point in a wider Einstein–Maxwell–Dilaton family, revisited through holographic renormalization and quantization of the scalar sector, generalized to dyonic and axio-dilatonic constructions, and used as a benchmark for entanglement-based bulk reconstruction and for analytic studies of black-hole interiors (Chagnet et al., 2022).

1. Definition and bulk construction

In one standard normalization, the model is a $4$-dimensional Einstein–Maxwell–Dilaton theory with bulk action

$$S_{\text{bulk}}=\frac{1}{2\kappa^2}\int d^4x \sqrt{-g}\, \left[ R - \frac{Z(\phi)}{4}F^2 - \frac12 (\partial\phi)^2 - V(\phi) \right],$$

with

$$Z(\phi)=e^{\phi/\sqrt{3}},\qquad V(\phi)=-6\cosh(\phi/\sqrt{3}).$$

The field content is the metric $g_{\mu\nu}$, a $U(1)$ gauge field $A_\mu$ with $F=dA$, and a dilaton $\phi$. Near $\phi=0$, the scalar mass is $4$0, which lies in the window allowing both standard and alternate quantization (Chagnet et al., 2022).

In a normalization often used in later condensed-matter applications, the same model is written as

$4$1

and the momentum-relaxing extension adds two linear axions,

$4$2

In that form the matter content consists of the metric, a Maxwell field $4$3, a dilaton $4$4, and two linear axions $4$5 that implement momentum relaxation (Ahn et al., 2024).

The model is also treated as a distinguished point in a broader $4$6-dimensional Einstein–Maxwell–Dilaton family. In that language the Gubser–Rocha point is the critical dilaton coupling

$4$7

This parametrization is used in higher-dimensional generalizations and in thermodynamic-topology analyses (Babaei-Aghbolagh et al., 20 Aug 2025).

Several works stress that the model is not merely bottom-up. It can be obtained top-down: for $4$8 from a consistent truncation of $4$9D supergravity on $$S_{\text{bulk}}=\frac{1}{2\kappa^2}\int d^4x \sqrt{-g}\, \left[ R - \frac{Z(\phi)}{4}F^2 - \frac12 (\partial\phi)^2 - V(\phi) \right],$$0, and for $$S_{\text{bulk}}=\frac{1}{2\kappa^2}\int d^4x \sqrt{-g}\, \left[ R - \frac{Z(\phi)}{4}F^2 - \frac12 (\partial\phi)^2 - V(\phi) \right],$$1 from type IIB string theory as a near-horizon D3-brane construction (Ahn et al., 2024).

2. Exact black-hole solution and thermodynamics

For the planar black hole, a standard ansatz is

$$S_{\text{bulk}}=\frac{1}{2\kappa^2}\int d^4x \sqrt{-g}\, \left[ R - \frac{Z(\phi)}{4}F^2 - \frac12 (\partial\phi)^2 - V(\phi) \right],$$2

with $$S_{\text{bulk}}=\frac{1}{2\kappa^2}\int d^4x \sqrt{-g}\, \left[ R - \frac{Z(\phi)}{4}F^2 - \frac12 (\partial\phi)^2 - V(\phi) \right],$$3 the AdS boundary and $$S_{\text{bulk}}=\frac{1}{2\kappa^2}\int d^4x \sqrt{-g}\, \left[ R - \frac{Z(\phi)}{4}F^2 - \frac12 (\partial\phi)^2 - V(\phi) \right],$$4 the horizon. The analytic Gubser–Rocha solution is

$$S_{\text{bulk}}=\frac{1}{2\kappa^2}\int d^4x \sqrt{-g}\, \left[ R - \frac{Z(\phi)}{4}F^2 - \frac12 (\partial\phi)^2 - V(\phi) \right],$$5

$$S_{\text{bulk}}=\frac{1}{2\kappa^2}\int d^4x \sqrt{-g}\, \left[ R - \frac{Z(\phi)}{4}F^2 - \frac12 (\partial\phi)^2 - V(\phi) \right],$$6

$$S_{\text{bulk}}=\frac{1}{2\kappa^2}\int d^4x \sqrt{-g}\, \left[ R - \frac{Z(\phi)}{4}F^2 - \frac12 (\partial\phi)^2 - V(\phi) \right],$$7

The continuous parameters in this analytic family are $$S_{\text{bulk}}=\frac{1}{2\kappa^2}\int d^4x \sqrt{-g}\, \left[ R - \frac{Z(\phi)}{4}F^2 - \frac12 (\partial\phi)^2 - V(\phi) \right],$$8 and $$S_{\text{bulk}}=\frac{1}{2\kappa^2}\int d^4x \sqrt{-g}\, \left[ R - \frac{Z(\phi)}{4}F^2 - \frac12 (\partial\phi)^2 - V(\phi) \right],$$9 (Chagnet et al., 2022).

The thermodynamic variables are

$$Z(\phi)=e^{\phi/\sqrt{3}},\qquad V(\phi)=-6\cosh(\phi/\sqrt{3}).$$0

At low temperature the entropy vanishes linearly,

$$Z(\phi)=e^{\phi/\sqrt{3}},\qquad V(\phi)=-6\cosh(\phi/\sqrt{3}).$$1

so the model has vanishing zero-temperature entropy density, unlike extremal AdS–Reissner–Nordström (Chagnet et al., 2022).

The linear-axion extension preserves analytic control at $$Z(\phi)=e^{\phi/\sqrt{3}},\qquad V(\phi)=-6\cosh(\phi/\sqrt{3}).$$2. With

$$Z(\phi)=e^{\phi/\sqrt{3}},\qquad V(\phi)=-6\cosh(\phi/\sqrt{3}).$$3

the metric and matter fields can be written as

$$Z(\phi)=e^{\phi/\sqrt{3}},\qquad V(\phi)=-6\cosh(\phi/\sqrt{3}).$$4

with exact $$Z(\phi)=e^{\phi/\sqrt{3}},\qquad V(\phi)=-6\cosh(\phi/\sqrt{3}).$$5 solution

$$Z(\phi)=e^{\phi/\sqrt{3}},\qquad V(\phi)=-6\cosh(\phi/\sqrt{3}).$$6

The horizon is at $$Z(\phi)=e^{\phi/\sqrt{3}},\qquad V(\phi)=-6\cosh(\phi/\sqrt{3}).$$7, and

$$Z(\phi)=e^{\phi/\sqrt{3}},\qquad V(\phi)=-6\cosh(\phi/\sqrt{3}).$$8

This extension is the version most often used in transport studies with finite DC resistivity (Ahn et al., 2023).

3. Quantization, IR structure, and holographic interpretation

A central later development is the claim that the usual analytic black hole does not exhaust the model’s solution space or its holographic interpretations. Because $$Z(\phi)=e^{\phi/\sqrt{3}},\qquad V(\phi)=-6\cosh(\phi/\sqrt{3}).$$9, the near-boundary scalar expansion is

$g_{\mu\nu}$0

and both standard and alternate quantization are allowed. In standard quantization one has

$g_{\mu\nu}$1

with trace Ward identity

$g_{\mu\nu}$2

In alternate quantization with multitrace deformation

$g_{\mu\nu}$3

the source is

$g_{\mu\nu}$4

For the analytic Gubser–Rocha family,

$g_{\mu\nu}$5

so $g_{\mu\nu}$6 and $g_{\mu\nu}$7 are not independent (Chagnet et al., 2022).

This leads to two related conclusions. First, the analytic Gubser–Rocha black hole is only a $g_{\mu\nu}$8-parameter subspace inside a larger $g_{\mu\nu}$9-parameter family of black holes, the extra parameter being an independent scalar boundary datum. Second, the same bulk solution can have multiple holographic interpretations depending on the choice of quantization and boundary terms. Only in the special multitrace quantization

$U(1)$0

does the analytic family define a sourceless conformal state; in generic standard or alternate quantization it is instead an explicitly sourced, non-conformal deformation. Correspondingly, the proper first law is

$U(1)$1

not merely a two-variable relation in $U(1)$2 (Chagnet et al., 2022).

On the IR side, the model is often described as a holographic realization of a “semi-local quantum liquid” type geometry, effectively related to an AdS$U(1)$3-like sector. In the language of hyperscaling-violating/Lifshitz scaling, the relevant limit is

$U(1)$4

This structure underlies the characteristic low-temperature transport scalings that made the model attractive for strange-metal applications (Ahn et al., 2023).

A further refinement is that the IR should not be viewed as a single nontrivial AdS$U(1)$5 fixed point. Because of the admissible scalar quantizations and the marginal multitrace deformation, the model instead realizes a one-parameter family of locally critical IR theories; this motivates the characterization of the dual as a local quantum critical phase rather than an isolated quantum critical point (Chagnet et al., 2022).

4. Transport phenomenology and the strange-metal debate

The model’s prominence in condensed-matter holography comes largely from transport. In its original translationally invariant form, the charged finite-density state has infinite DC conductivity because momentum is conserved. The linear-axion extension breaks translations homogeneously and isotropically through

$U(1)$6

or, in an equivalent notation,

$U(1)$7

and thereby produces finite resistivity (Ahn et al., 2023).

The attraction of the linear-axion Gubser–Rocha model is that it naturally yields

$U(1)$8

over a broad low-temperature regime, while the entropy implies linear specific heat. In holographic terms, this scaling is tied to the semi-local critical IR geometry rather than to quasiparticle transport (Ahn et al., 2023).

At finite magnetic field, the DC conductivities are horizon quantities. In the linear-axion model they can be written as

$U(1)$9

These formulas made the model a standard platform for magnetotransport tests (Ahn et al., 2023).

The main controversy is that linear-in-$A_\mu$0 resistivity does not by itself reproduce the broader strange-metal phenomenology. Direct numerical computation, general EMD scaling analysis, and a hydrodynamic argument all show that the simple linear-axion model fails to realize the experimentally characteristic separation

$A_\mu$1

Instead, at low temperature it gives

$A_\mu$2

The obstruction is structural: the model has a single momentum-relaxation timescale and, at low $A_\mu$3, an approximately temperature-independent charge density, so Hall and longitudinal transport remain too tightly linked (Ahn et al., 2023).

Dyonic and $A_\mu$4-dual extensions preserve part of the original transport appeal while changing the magnetic response. In one analytically solvable dyonic extension with momentum relaxation, the resistivity remains linear in $A_\mu$5 at low temperature, but strong momentum relaxation and strong magnetic field produce explicit deviations from linearity. In the same model, the Hall angle is linear in $A_\mu$6 in both the low- and high-temperature regimes for fixed momentum dissipation strength, and the Nernst signal is bell-shaped as a function of magnetic field even when momentum relaxation is strong (Ge et al., 2023).

The Gubser–Rocha background has also been used as the normal state for doped holographic superconductors. In that setting the normal phase again exhibits resistivity proportional to temperature under momentum dissipation, while the superconducting phase diagram develops a dome in the temperature–doping plane; the paper emphasizes that the coupling between the two gauge fields is crucial for dome formation (Zhao et al., 2023).

5. Entanglement, reconstruction, and interior geometry

The model has become a useful testing ground for holographic information measures. In one study of Einstein–Maxwell–Dilaton gravity, the Gubser–Rocha background was compared systematically with RN–AdS for strip entangling regions. Despite the singular nature of the $A_\mu$7 limit, the holographic entanglement entropy, mutual information, and entanglement of purification remain nonsingular. The most unusual result is that in a low-temperature regime the holographic entanglement entropy decreases as temperature increases, opposite to the behavior found in most holographic models. The same work found

$A_\mu$8

suggesting that mutual information and entanglement of purification capture different aspects of mixed-state entanglement (Fu et al., 2020).

The model has also served as a benchmark for machine-learning reconstruction of bulk geometry from boundary entanglement entropy. In that context it is important because the spatial warp factor is nontrivial,

$A_\mu$9

so one must reconstruct both $F=dA$0 and $F=dA$1, not just a single blackening function. The small-strip expansion contains a finite constant term

$F=dA$2

and for the Gubser–Rocha solution this gives

$F=dA$3

which is a direct diagnostic of $F=dA$4. The reconstruction study reported that the full continuous metric functions $F=dA$5 and $F=dA$6 can be recovered from strip entanglement data plus thermal entropy, and it further observed that the geometry extracted from a half-filled tight-binding chain resembles the Gubser–Rocha metric, speculating that the similarity is due to the metallic property of both systems (Ahn et al., 2024).

The model is now also central to analytic studies of black-hole interiors. Within a wider Einstein–Maxwell–Dilaton family, the Gubser–Rocha point is the critical value $F=dA$7, and its interior is Kasner rather than timelike. For the $F=dA$8 Gubser–Rocha model one has

$F=dA$9

and near the singularity the metric is characterized by Kasner exponents

$\phi$0

satisfying

$\phi$1

In this class there is no inner horizon. The same analysis proposes a new CMC-based complexity functional whose late-time growth rate depends explicitly on the Kasner exponents, and shows that the thermal $\phi$2-function has a power-law decay near the singularity fixed by the same data (Areán et al., 2024).

In momentum-relaxing Einstein–Maxwell–Axion–Dilaton theories based on the model, the complexity-equals-action calculation yields another information-theoretic diagnostic: the growth rate violates Lloyd’s bound at finite times, but at late times saturates a bound that depends on the strength of momentum relaxation (Babaei-Aghbolagh et al., 2021).

6. Extensions, variants, and nomenclature

A major recent branch of the literature studies an $\phi$3-type Gubser–Rocha model, in which the dilaton is promoted together with an axion into an axio-dilaton

$\phi$4

and the bulk theory is organized to be invariant under an $\phi$5-duality transformation. This model admits an analytic dyonic black-brane solution with momentum relaxation and a grand potential

$\phi$6

From it one obtains

$\phi$7

so the system is diamagnetic. In a specific neutral limit, the same model exhibits spontaneous breaking of a global $\phi$8 symmetry with a complex order parameter dual to the axio-dilaton, and it retains finite Hall conductivity even at zero external magnetic field (Ishigaki et al., 2024).

The neutral $\phi$9-type model with momentum dissipation has then been analyzed as a genuine critical system. With the appropriate boundary action it undergoes a continuous phase transition at

$\phi=0$0

and the static critical exponents were reported as

$\phi=0$1

These match mean-field percolation rather than ordinary Landau mean-field theory. Quasinormal-mode analysis further shows that dynamical stability agrees with thermodynamic stability and that a dissipative Nambu–Goldstone mode emerges in the broken phase (Ishigaki, 16 May 2025).

In a different recent reinterpretation, thermodynamic topology classifies the Gubser–Rocha point $\phi=0$2 inside a broad Einstein–Maxwell–Dilaton family as belonging to the new class

$\phi=0$3

with total winding number

$\phi=0$4

This is presented as a robust branch-structure property of the model and its higher-dimensional analogues (Babaei-Aghbolagh et al., 20 Aug 2025).

A recurring source of confusion is nomenclature. The Gubser–Rocha model is a holographic Einstein–Maxwell–Dilaton black-brane construction for strongly coupled finite-density matter. It is not the same object as Gubser flow in heavy-ion hydrodynamics, which is a conformal, boost-invariant, radially expanding solution of relativistic hydrodynamics with $\phi=0$5 symmetry. The conceptual overlap is limited to the broader Gubser literature and the use of conformal structure; technically they are distinct frameworks (Naik et al., 2 Jun 2025).