A new one-dimensional spin model with an exponential U(1) symmetry realizes a disorder-free quantum glass state, featuring a large number of ground states and finite entropy density.

Overall, the study defines a disorder-free quantum glass phase by exponential U(1) symmetry, extensive ground-state degeneracy, finite entropy density, nondecaying correlations, and distinctive boundary phenomena.

The Rokhsar-Kivelson line provides exact solvability and underpins the analytical framework, including a Holstein-Primakoff transformation to support numerical results.

Under open boundary conditions, an edge mode on the left edge shifts the energy spectrum and indicates boundary-sensitive behavior.

Despite randomness, the condensate lacks off-diagonal long-range order and maintains a bulk energy gap, defying Goldstone’s theorem expectations for broken continuous symmetries.

The condensate hosts about 2^L spontaneous symmetry-breaking ground states for a lattice of length L, driven by ~2^L exponential U(1) charge sectors, yielding an entropy density of ln 2.

The order parameter is the local in-plane spin whose orientation angles follow the chaotic Bernoulli map, making the order parameter effectively random and accompanied by nondecaying local autocorrelations.

This chaotic Bernoulli map, which generates random-in-plane spins, is central to the phase and signals an order-disorder interplay not captured by conventional glass or crystal classifications.

The exponential U(1) symmetry is defined by a charge Q = sum_j f_j n_j with site-dependent coefficients f_j; the charge decays as 2^(-j), and this symmetry underpins the phase and ground-state structure.

A first-order spontaneous symmetry-breaking transition into the condensate is supported by numerical analysis and the exactly solvable Rokhsar-Kivelson line, with the phase diagram mapped via exact diagonalization and DMRG.