Holographic Axion Models

• Holographic axion models are gravitational duals that couple shift-symmetric scalar fields with gauge sectors to mimic translation symmetry breaking and related emergent phenomena.
• They demonstrate key roles in driving metal–insulator transitions, unconventional conductivity, and anisotropic transport through controlled axion-gauge interactions.
• These models also extend to viscoelastic behavior, cosmological evolution, composite axion constructions, and quantum entanglement, highlighting their versatility.

Holographic axion models are a class of gravitational duals in anti-de Sitter (AdS) or related spacetimes, where one or more shift-symmetric scalar fields ("axions") are coupled to bulk gravity and, in many cases, to gauge sectors. These models, often constructed to mimic strong-coupling physics in condensed matter systems and the early universe, provide a controllable framework to paper emergent phenomena associated with the breaking of translation symmetry, viscoelasticity, metal-insulator transitions, nontrivial transport, and axion cosmology.

1. Fundamental Structure and Action

At their core, holographic axion models introduce one or more axion fields $\phi^I$ with a shift symmetry $\phi^I \rightarrow \phi^I + c^I$, breaking translations in the dual quantum field theory. The basic structure in $(d+1)$ bulk dimensions is: $$S = \int d^{d+1}x\,\sqrt{-g}\,\left[R - 2\Lambda - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} - V(X,Z) + \text{(axion-gauge couplings)}\right]$$ where

• $F_{\mu\nu}$ is the $U(1)$ field strength,
• $$X = \frac{1}{2}g^{\mu\nu}\partial_\mu\phi^I \partial_\nu\phi^I$$,
• $$Z = \det[g^{\mu\nu}\partial_\mu\phi^I\partial_\nu\phi^J]$$.

The breaking pattern (explicit vs spontaneous; isotropic vs anisotropic) is determined by the profile, potential $V(X,Z)$, and additional couplings such as those between axions and gauge fields.

Notable generalizations involve:

• Higher-derivative axion-gauge couplings, e.g., $-\frac{\mathcal{K}}{4} X F^2$ (Wang et al., 2021).
• Antisymmetric gauge–axion couplings, $-\frac{\mathcal{J}}{2}\mathrm{Tr}[X F]$ (Bai et al., 2023).
• Extension to $F(R)$ gravity coupled to axion matter for cosmology (Saharian et al., 2021, D'Onofrio, 2023).

2. Transport, Conductivity, and Metal–Insulator Transitions

DC Conductivity and Hall-like Response

With a standard linear axion model (e.g., $V(X) = X$), the DC conductivity is isotropic and independent of temperature after appropriate scaling: $\sigma_{\text{DC}} = 1 + \mu^2/\alpha^2$ in $d=3+1$ (Baggioli et al., 2021). More intricate behaviors arise for models with gauge–axion couplings. For example, including an antisymmetric term (Bai et al., 2023): $$S \supset -\frac{\mathcal{J}}{2}\,\mathrm{Tr}[X F],\qquad \mathrm{Tr}[X F] = \epsilon^{IJ} \partial_\mu\phi^I \partial_\nu\phi^J F^{\mu\nu}$$ produces nonzero, antisymmetric off-diagonal components of the conductivity tensor, even at zero magnetic field: $$\sigma_{xy} = -\sigma_{yx} \propto \mathcal{J}\mu u_h (\ldots)$$ This term acts as an "internal" Hall effect and induces time-reversal symmetry breaking, formally paralleling the effects of an external magnetic field, but now sourced by axion–gauge mixing.

Metal–Insulator Transitions (MIT)

Both in three and four bulk dimensions, gauge–axion couplings can induce a metal–insulator transition not accessible in conventional (linear-axion) models. In particular, for (Bai et al., 2023), the parameter space $(\mathcal{J},\alpha)$ contains phases where the zero-temperature conductivity vanishes, signaling an MIT:

• For $\mathcal{J}^2 < 1/6$, $\sigma_{xx}(0)>0$ (poor insulator).
• For $\mathcal{J}^2 \geq 1/6$, $\sigma_{xx}(0)\rightarrow 0$ at finite $\alpha$.

In $d=2+1$, even in the absence of gauge–axion coupling ($\mathcal{J}=0$), the DC conductivity is temperature-dependent, a property absent in higher dimensions. For nonzero $\mathcal{J}$, a disorder-driven MIT appears at $T=0$ (Li et al., 2022).

3. Viscoelasticity, Shear Modulus, and Anisotropy

Holographic axion models provide a realization of viscoelastic solids at strong coupling. When translation symmetry is broken spontaneously (polynomial axion potentials with $V(X)\sim X^n$, $n>5/2$ or with specific higher-derivative axion–gauge interactions as in (Wang et al., 2021)), the dual boundary theory develops a nonzero static shear modulus $G$ and propagating phonon modes: $$\omega(k) = \pm c_T k - i D_T k^2 + \ldots, \qquad c_T^2 = G/\chi_{PP}$$

In anisotropic models (Li et al., 14 Oct 2024), different choices for axion profiles $\phi^i = k_i x^i$ (with $k_x \neq k_y$) enable explicit control over both translational and rotational breaking. Key findings:

• Shear modulus $G$ is nonzero only if both directions break translations.
• $G$ is enhanced by anisotropy (larger $k_x$, $k_y$ disparity), while shear viscosity $\eta$ is doubly suppressed.
• All such models violate the KSS bound, $\eta/s<1/(4\pi)$, for any degree of anisotropy.

4. Hydrodynamic Modes and Diffusive Channels

The spectrum of collective excitations is intimately tied to the number and type of axion sectors:

• A single set of axions yields a standard viscoelastic spectrum: two phonon modes (one longitudinal, one transverse) and a single diffusion mode.
• In the presence of multiple axion species (Xia et al., 27 May 2024), e.g.,

$$\mathcal{W}(X_1, ..., X_\mathcal{N}) = \sum_{a=1}^{\mathcal{N}} m_a^2 (X_a)^{M_a},$$

the count of diffusion modes increases: there is always exactly two phonons (from broken translations), but each extra axion species supplies an additional diffusive pole to the hydrodynamic spectrum.

High-temperature scaling of $G$ is additive in the number of axion species ($G\sim \sum m_a^2 T^{-(2M_a-3)/(2M_a-1)}$), while at low temperature the sector with the largest $M_a$ dominates. The viscosity decreases monotonically with increasing axion number.

5. Holographic Axion Models in Cosmology and Modified Gravity

Axion–holographic constructions have been extended to $F(R)$ gravity and viscous cosmologies (Saharian et al., 2021, D'Onofrio, 2023, Brevik et al., 2023). These models provide:

• Autonomous dynamical system analyses for axion–holographic dark energy cosmologies, revealing fixed points associated with matter-dominated, dark-energy–dominated, and saddle behaviors.
• Analytic relations for holographic IR cutoffs in terms of particle or future event horizons, determined self-consistently from cosmic evolution.
• Unified frameworks in which early-time dustlike axion matter evolution, late-time viscous/holographic acceleration, and smooth cosmological bounce solutions are all encoded via the cutoff function $L_{IR}(t)$ and the EoS for bulk viscous fluids.
• Instabilities associated with de Sitter attractors in $F(R)$+axion, interpreted as necessary for successful exit from inflation or transition to dark energy domination.

A distinguishing feature in these cosmologies is the mapping of dark matter, dark energy, and viscosity into a generalized, cutoff-dependent holographic density, allowing wide flexibility in reproducing observed cosmic histories.

6. Composite and High-Quality Axion Realizations via Holography

Holographic methods provide a geometric solution to the axion quality problem (Cox et al., 2019, Cox et al., 2021, Lee et al., 2021). Here, PQ symmetry breaking occurs on an IR brane, while explicit breaking is confined to the UV. The bulk axion zero-mode profile is localized near the IR, exponentially suppressing dangerous UV-induced mass terms for large scaling dimensions $\Delta$: $$m_a^{\text{(UV)}} \sim e^{-(\Delta-4)kL} \cdot z_{IR}^{-1}$$ This sequestering mechanism ensures that gravitational violations of PQ symmetry are suppressed, making the axion an excellent candidate for the solution to the strong CP problem.

Multi-brane and doubly composite constructions can decouple the axion decay constant from the electroweak scale. The spectrum generically includes both a light axion and heavier Kaluza-Klein excitations, with the decay constants set by the geometric separation of branes in the bulk. Such setups naturally accommodate light sterile neutrinos as partially composite fermions, with careful control over neutrino mass hierarchies and axion–neutrino couplings (Cox et al., 2021).

The "holographic QCD axion" (Bigazzi et al., 2019) extends this by embedding the axion into the Witten–Sakai–Sugimoto background, realizing a composite KSVZ axion with all relevant couplings and masses computable in the dual gravity description. Notable is the "rising" axion mass at high temperature due to D0-instanton effects, contrasting with the dilute instanton prediction for QCD.

7. Quantum Information, Entanglement, and Holographic Axions

Entanglement measures in holographic axion backgrounds reveal rich structure, sensitive to the specifics of translation breaking and the nature of charge/axion–gauge couplings (Huang et al., 2019, Cheng et al., 2021):

• Holographic entanglement entropy (HEE) is dominated by thermal contributions in large subsystems and increases monotonically with the strength of symmetry breaking.
• Mutual information (MI) and entanglement of purification (EoP) can act as finer probes for quantum correlations, with EoP in particular showing a strictly monotonic response to axion–gauge couplings and disentangling transitions as system parameters are varied.
• The entanglement wedge cross-section (EWCS) increases monotonically with axion–gauge coupling strength and is comparatively robust against contamination by thermal entropy, supporting its interpretation as a refined quantum entanglement measure.

These findings confirm that the axion sector imparts nontrivial quantum information structure to the dual theory, with direct implications for quantum phase transitions and the organization of correlations in strange metals and quantum critical states.

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Holographic axion models, through their flexible structure and systematic realizations of translation and (an)isotropic symmetry breaking, have become a central tool in the paper of quantum transport, strong-coupling viscoelasticity, cosmology, and quantum information in dual strongly coupled quantum systems. Across diverse realizations—from black brane transport to cosmological evolution and composite axion phenomenology—these models have elucidated mechanisms of emergent collective dynamics, encoded bounds on transport and hydrodynamic coefficients, and enabled geometric solutions to outstanding problems in axion physics and cosmology.