Holographic Translation Breaking: Explicit and Spontaneous

A spatially varying bulk field does not, by itself, diagnose spontaneous translation breaking. The distinction is made at the AdS boundary: an explicit lattice has a modulated nonnormalizable source, while a spontaneous density wave has a homogeneous source and a modulated normalizable response. That source–response assignment controls the Ward identity, phonon, pinning frequency, and dc transport.

Required background. Chemical Potential and Charged Black Branes supplies the finite-density state. Boundary Conditions, Alternate Quantization, and Deformations supplies the source–response distinction.

Helpful background. Symmetry Realization and Order Parameters supplies the order-parameter criterion. Explicit Breaking and Pseudo-Goldstone Modes supplies the pinned-mode language.

Boundary data and the Ward identity

For a scalar operator of dimension $\Delta$,

$$\Phi(z,x)=z^{d-\Delta}\lambda(x)+z^\Delta v(x)+\cdots, \qquad \langle\mathcal O(x)\rangle\propto v(x)$$

in standard quantization. A periodic $\lambda(x)$ explicitly breaks translations. A phase with constant $\lambda$ but periodic $v(x)$ breaks them spontaneously. The renormalized diffeomorphism Ward identity makes the distinction operational:

$$\nabla_\mu T^\mu{}_i =F_{i\mu}J^\mu+\langle\mathcal O\rangle\,\partial_i\lambda .$$

The last term is an external force only when the source varies. Normalizable bulk structure with $\partial_i\lambda=0$ can carry an order parameter without explicitly relaxing total momentum.

Explicit Q-lattice and source-free stripe

A Q-lattice uses a complex scalar

$$\Phi(r,x)=\varphi(r)e^{ikx}.$$

Its stress tensor is homogeneous, reducing the bulk equations to ordinary differential equations, but its leading boundary coefficient is proportional to $e^{ikx}$. It is therefore explicit translation breaking. Donos and Gauntlett used this construction to obtain metallic and insulating dc behavior from homogeneous black holes Donos and Gauntlett 2014.

For a spontaneous stripe, begin with homogeneous sources and find a finite-momentum normalizable zero mode at $T_c$. Continue it to a nonlinear modulated solution, verify that the modulated sources remain zero, and compare its free energy with the homogeneous saddle. Translating the stripe costs no free energy, so the fluctuation spectrum contains a gapless sliding mode when translations are exact.

These two constructions can have similar horizon profiles and similar resistivity curves. Their boundary data and low-frequency poles, not their visual bulk modulation, determine their symmetry class.

Weak pinning as the first application

Add a small commensurate source $\lambda_{\mathrm{pin}}(x)$ to the spontaneous phase. At long wavelengths the displacement field $u$ obeys, schematically,

$$\chi_{\pi\pi}\left(\partial_t^2+\Gamma\partial_t+\omega_0^2\right)u =\rho E+\cdots .$$

The optical response then contains poles satisfying

$$\omega^2+i\Gamma\omega-\omega_0^2=0 .$$

$\omega_0$ is the pinning frequency and vanishes with the explicit source; $\Gamma$ describes relaxation. In the weak-source regime, pseudo-Goldstone relations make $\omega_0^2$ linear in an explicit-breaking scale times an order-parameter scale. The proportionality is model- and normalization-dependent and should be tested rather than assumed. Holographic density-wave calculations exhibit a universal phase-relaxation structure in the weakly pinned regime Amoretti et al. 2019.

The calculation requires coupled metric, gauge, and scalar fluctuations with infalling horizon conditions. The source is held fixed while the retarded boundary matrix is constructed; allowing it to fluctuate changes the theory.

An ultraviolet adversary

Construct two backgrounds with nearly identical infrared geometry but different leading scalar data: one with periodic $\lambda(x)$, one with $\lambda$ constant and periodic $v(x)$. Similar entropy and dc conductivity do not make their symmetry realization equivalent. The explicit model has a force term in the Ward identity and no exact sliding Goldstone; the spontaneous model does.

As a second control, continuously turn off the pinning source. A genuine pseudo-Goldstone pole must approach the origin in the prescribed thermodynamic limit. If the gap remains finite, it comes from commensurability, finite volume, numerical boundary contamination, or another explicit deformation.

The result establishes explicit, spontaneous, or weakly pinned order in the specified holographic boundary theory. It does not identify a crystal, charge-density-wave material, or microscopic lattice mechanism.

Exercises

Why can the ansatz $\Phi=\varphi(r)e^{ikx}$ lead to homogeneous metric functions even though translations are explicitly broken?

Solution

The action has a global phase symmetry, and the scalar stress tensor depends on phase-invariant combinations such as $\lvert\Phi\rvert^2$ and $\lvert\partial_x\Phi\rvert^2=k^2\varphi^2$. These are independent of $x$, although the boundary source itself is periodic.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Amoretti, Andrea, Daniel Areán, Blaise Goutéraux, and Daniele Musso. “Universal Relaxation in a Holographic Metallic Density Wave Phase.” Physical Review Letters 123, 211602 (2019). DOI.
• Donos, Aristomenis, and Jerome P. Gauntlett. “Holographic Q-Lattices.” Journal of High Energy Physics 2014, 040 (2014). DOI.