Axion-Induced Translation Symmetry Breaking

• Axion-induced translation symmetry breaking is defined by spatially linear axion fields that explicitly disrupt translation invariance, enabling momentum relaxation in holographic models.
• The mechanism mimics impurity or lattice effects, linking modified thermodynamic properties with observable changes in optical conductivity in s+p holographic superconductors.
• Increasing the k/T parameter suppresses p-wave order while favoring s-wave dominance, resulting in distinct phase boundaries and conductivity gap signatures.

Axion-induced translation symmetry breaking refers to the phenomenon where an axion field—typically one associated with a Peccei-Quinn (PQ) symmetry—explicitly and spontaneously breaks the translation (shift) invariance in space or field space, resulting in momentum relaxation and significant modifications to the dynamics of condensed matter systems. In the context of holographic models and gauge/gravity duality, this breaking is implemented via background axion fields with spatial dependence, which facilitates the construction of tractable models of strongly correlated phases with suppressed momentum conservation. The explicit paper of such a mechanism and its impact on phase competition and optical conductivity in s+p holographic superconductors is detailed in "Holographic s+p superconductors with axion induced translation symmetry breaking" (Chen et al., 2 Oct 2025).

1. Axion Fields and the Mechanism of Translation Symmetry Breaking

In holographic duality, translation symmetry can be broken by coupling the bulk gravitational theory to massless scalar (axion) fields φ^I with spatial profiles linear in the boundary directions: $\phi^I = k x^I \quad (I = x, y)$ Here, k is a parameter controlling the strength of momentum relaxation. The axion sector in the action,

$$S_G = \int d^4x\, \sqrt{-g}\left(\frac{R}{2} - \Lambda - \frac{X}{2}\right),\quad X = \frac{1}{2} g^{\mu\nu}\partial_\mu\phi^I \partial_\nu\phi^I,$$

becomes spatially inhomogeneous on the boundary, thus explicitly breaking continuous translational invariance. This mechanism models impurity or lattice-like effects in the dual condensed matter system, enabling finite DC conductivity and a more realistic description of strongly correlated phases.

In the black hole metric of AdS_4, the presence of axions modifies the background geometry: $$f(r) = \frac{r^2}{L^2}\left(1 - \frac{r_h^3}{r^3}\right) - \frac{k^2}{2L^2}\left(1 - \frac{r_h}{r}\right)$$ with Hawking temperature

$T = \frac{3 r_h - k^2/2r_h}{4\pi L^2}$

demonstrating the impact of k (and hence translation symmetry breaking) on thermodynamic quantities.

2. Holographic s+p Superconductor Model: Framework and Equations

The matter sector couples a complex scalar field Ψ (s-wave order) and a complex vector field ρ_μ (p-wave order) in the bulk: $$S_M = \int d^4x\, \sqrt{-g}\Big[ -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} - |D_\mu\Psi|^2 - m_s^2|\Psi|^2 - \frac{1}{2}\rho_{\mu\nu}^\dagger\rho^{\mu\nu} - m_p^2\rho_\mu^\dagger\rho^\mu \Big]$$ with covariant derivatives: $$D_\mu \Psi = \nabla_\mu \Psi - i q_s A_\mu \Psi,\qquad \rho_{\mu\nu} = \partial_\mu\rho_\nu - \partial_\nu\rho_\mu - i q_p (A_\mu\rho_\nu - A_\nu\rho_\mu)$$ The fields are taken in an ansatz appropriate for homogeneous, static condensates: $$A_t = \Phi(r),\qquad \Psi = \Psi_s(r),\qquad \rho_x = \Psi_p(r)$$ The system of coupled ODEs for condensates and potential reduces to: $$\begin{aligned} & \frac{2q_p^2 \Phi \Psi_p^2}{r^2 f} + \frac{2q_s^2 \Phi \Psi_s^2}{f} - \frac{2\Phi'}{r} - \Phi'' = 0 \ & \frac{m_s^2 \Psi_s}{f} - \frac{q_s^2 \Phi^2 \Psi_s}{f^2} - \frac{2\Psi_s'}{r} - \frac{f'\Psi_s'}{f} - \Psi_s'' = 0 \ & \frac{m_p^2 \Psi_p}{f} - \frac{q_p^2 \Phi^2 \Psi_p}{f^2} - \frac{f'\Psi_p'}{f} - \Psi_p'' = 0 \end{aligned}$$ Solution of these equations (numerically, in the probe limit) yields the phase structure and phase competition under variation of k/T and chemical potential μ.

3. Thermodynamic Potential and Phase Competition

The grand potential (thermodynamic potential) is calculated, in the probe limit, from the regularized matter action as: $$\Omega_m = \frac{V_2}{T}\left[ -\frac{\mu\rho}{2} - \int_{r_h}^\infty \left(\frac{q_p^2\Phi^2\Psi_p^2}{f} + \frac{q_s^2 r^2\Phi^2\Psi_s^2}{f} \right) dr \right]$$ Phase competition is understood via comparison of $\Omega_m$ for (i) pure s-wave condensate, (ii) pure p-wave condensate, and (iii) s+p coexistence. Increasing k/T is found to raise μ_c/T for both s-wave and p-wave transitions, but more rapidly so for the p-wave. As a consequence, the p-wave phase region shrinks with increasing k/T, and in the large k/T regime only the s-wave solution remains thermodynamically favorable.

4. Phase Diagram and Role of Charge Ratio

The k–μ phase diagram, as determined by the solutions to the equations above, shows a rich structure including:

• Normal phase (no superconductivity)
• Pure s-wave condensate phase
• Pure p-wave condensate phase
• s+p coexistence phase

Second-order phase boundaries are identified numerically. The p-wave phase window narrows and disappears as k/T increases, while above a critical k/T, only s-wave and coexisting s+p (with dominant s-wave) phases are thermodynamically viable. Additionally, a larger minimum ratio of the charges q_p/q_s is needed to maintain the s+p phase as k/T grows, with the precise functional dependence determined numerically.

Phase	k/T (low)	k/T (high)	Condition
Pure s-wave	✓	✓	always present
Pure p-wave	✓	✗	shrinks/disappears
s+p coexistence	narrow	very narrow	larger q_p/q_s required
Normal	boundary	boundary	at low μ

5. Optical Conductivity and Experimental Signatures

The AC response is studied by solving linearized equations for a perturbation δA_y = A_y(r) e^{–iωt}: $$A_y'' + \frac{f'}{f} A_y' + \left[\frac{\omega^2}{f^2} - \frac{2q_p^2\Psi_p^2}{r^2 f} - \frac{2q_s^2\Psi_s^2}{f}\right]A_y = 0$$ Conductivity is extracted from boundary expansions: $$\sigma_{yy}(\omega) = -i \frac{A^{(1)}}{\omega A^{(0)}}$$ Gap frequency (ω_g) in the real part of the conductivity increases monotonically with k/T. A pronounced kink in ω_g as a function of k/T is identified in correspondence with the s+p coexistence region. This feature may serve as a distinctive experimental indicator of multi-condensate superconductivity in systems where translation symmetry breaking is induced via axion-like fields.

6. Physical Implications and Universality

Axion-induced translation symmetry breaking provides a robust and controllable mechanism for modeling finite conductivity in holographic superconductors, enabling a unified description of both momentum relaxation and phase competition between distinct superconducting orders. The suppression of p-wave order relative to s-wave order with increasing k/T is a universal feature in this framework, reflecting the sensitivity of vector order to momentum relaxation. The existence and precise boundaries of the coexistent s+p phase are controlled by both k/T and the charge ratio q_p/q_s; their interplay dictates the experimental observability of s+p superconductivity.

The optical conductivity signature—particularly the gap frequency kink and its k/T dependence—offers a concrete methodology for empirical detection and discrimination of coexisting superfluid orders in strongly correlated electron systems.

7. Summary Table

Effect of Parameter	Increasing k/T	Increasing q_p/q_s
μ_c/T (transition)	Increases (all phases)	Lowers p-wave threshold
s-wave stability	Less suppressed	Less sensitive
p-wave stability	More suppressed	Stabilized if q_p/q_s large
s+p coexistence phase	Shrinks, higher q_p/q_s	Easier to realize for larger q_p/q_s
Optical gap frequency	Increases, kink at s+p boundary	-

The use of axions to induce translation symmetry breaking provides essential control over the phase structure and transport properties of holographic superconductors, with implications for modeling and detecting complex order in strongly correlated systems (Chen et al., 2 Oct 2025).