# Intertwined Orders in Holography: Pair and Charge Density Waves

## TL;DR

This paper presents a holographic model where intertwined charge and pair density wave orders emerge from spontaneous symmetry breaking, with numerical solutions illustrating complex coexisting phases and effects of explicit translational symmetry breaking.

## Contribution

It introduces a novel holographic framework capturing intertwined orders, including pair and charge density waves, with numerical analysis of their coexistence and effects of explicit symmetry breaking.

## Key findings

- Realization of pair density wave features with scalar condensate oscillations
- Coexistence of superconducting and charge density wave orders
- Numerical backreacted geometries including axion effects

## Abstract

Building on [1], we examine a holographic model in which a U(1) symmetry and translational invariance are broken spontaneously at the same time. The symmetry breaking is realized through the St\"{u}ckelberg mechanism, and leads to a scalar condensate and a charge density which are spatially modulated and exhibit unidirectional stripe order. Depending on the choice of parameters, the oscillations of the scalar condensate can average out to zero, with a frequency which is half of that of the charge density. In this case the system realizes some of the key features of pair density wave order. The model also admits a phase with co-existing superconducting and charge density wave orders, in which the scalar condensate has a uniform component. In our construction the various orders are intertwined with each other and have a common origin. The fully backreacted geometry is computed numerically, including for the case in which the theory contains axions. The latter can be added to explicitly break translational symmetry and mimic lattice-type effects.