The Peierls instability is the textbook mechanism for charge density wave formation. A one-dimensional metallic chain at half filling — one electron per two sites — spontaneously dimerizes: the atoms pair up, alternating their spacing, opening a gap at the Fermi level and lowering the total energy. The distortion doubles the unit cell and the metal becomes an insulator. The instability is universal for one-dimensional metals at half filling. The question is what happens away from half filling, when doping adds or removes electrons.
The standard expectation is that doping destroys the charge density wave. The Peierls gap opens at the Fermi level of the half-filled system. Doping moves the Fermi level away from the gap, reducing the energy gained from dimerization. At sufficient doping, the energy gain vanishes and the normal metallic state is restored. The CDW melts monotonically with doping.
Garcia-Ruiz, Hsu, Liu, and Mucha-Kruczynski (arXiv 2602.23208, February 2026) show that this monotonic picture fails when chains are coupled together. In finite-thickness arrays of identical chains — the minimal extension from one dimension toward two — the CDW response to doping depends on the stacking geometry and can be non-monotonic, bistable, or reentrant.
For parallel-coupled chains, doping creates a bistable regime where the normal metallic state and the dimerized CDW state coexist as local energy minima. The system can be trapped in either state depending on its history — the CDW is metastable even when the metal is the ground state, or vice versa. Hysteresis results: approaching the same doping level from different initial conditions leads to different phases.
For skew-coupled chains, doping produces reentrance. The CDW is stable at half filling, destroyed by initial doping as expected, but then reappears at higher doping levels. The order comes back. This is not the same CDW returning — the reentrant phase has the same symmetry (dimerization) but is stabilized by a different mechanism involving the interchain hybridization and the new Fermi surface geometry at higher doping. The phase diagram has an island of CDW order floating in a sea of normal metal, disconnected from the half-filling CDW by a metallic gap.
The physics is that interchain coupling creates multiple bands with different Fermi surface geometries at different fillings. The Peierls instability occurs when the Fermi surface has the right nesting properties — when states at the Fermi level are connected by the CDW wavevector. In a single chain, nesting is perfect at half filling and degrades monotonically with doping. In coupled chains, the multi-band structure creates new nesting conditions at fillings away from half, where a different band's Fermi surface happens to match the dimerization wavevector. The CDW reappears not because the original mechanism recovered but because a new mechanism activated.
Even the simplest models of coupled atomic chains — the absolute minimum extension beyond one dimension — produce phase diagrams controlled by the interplay of doping, lattice rigidity, and coupling geometry. The Peierls instability, taught as a clean one-parameter story (filling determines whether the CDW exists), is actually a multi-parameter landscape where order can appear, vanish, and return.