A one-dimensional metallic chain at half-filling undergoes the Peierls instability: atoms pair up spontaneously, opening a gap at the Fermi level. The chain becomes an insulator. This is a textbook result — clean, one-dimensional, half-filled.
The question is what happens when you dope the chain (move away from half-filling) or couple it to neighboring chains (move toward three dimensions). Garcia-Ruiz, Hsu, Liu, and Mucha-Kruczynski (2026) show that the answer depends not on the coupling strength but on its geometry. Parallel-coupled chains exhibit bistability: a doping range where both the normal metallic state and the dimerized charge-density-wave state are energetically favorable simultaneously. The system can be in either phase depending on its history. Skew-coupled chains exhibit reentrance: the charge-density-wave order disappears with initial doping, then reappears at higher doping levels. The same perturbation — removing electrons — first destroys and then restores order, depending on how much you remove.
Both behaviors emerge from the same Hamiltonian with the same parameters. The only difference is the angle at which neighboring chains are stacked. Geometry selects the phase diagram. Not the interaction strength. Not the band structure. The spatial relationship between otherwise identical components determines whether the system remembers its ordered state (bistability) or rediscovers it after losing it (reentrance).
The general principle: when identical components are assembled into a structure, the geometry of assembly can matter more than the properties of the components. The same building blocks, connected differently, produce qualitatively different responses to the same perturbation. The answer to “what happens when you dope this material?” is not a property of the material. It is a property of how the material is stacked.