We study the phase diagram of a one-dimensional spin quantum breakdown model, which has an exponential U(1) symmetry with charge unit decaying as 2^{-j} with site position j. By exact diagonalization and density matrix renormalization group, we show that the model with spin S≥2 exhibits an exponential U(1) spontaneous symmetry-breaking (SSB) phase dubbed a quantum breakdown condensate. It exhibits a bulk gap violating the Goldstone theorem and an edge mode only on the left edge if in open boundary condition. In a length L lattice, the condensate has O(2^{L}) number of SSB ground states originating from the O(2^{L}) number of exponential U(1) charge sectors, leading to a finite entropy density ln2. This enforces a first-order SSB phase transition into this phase, as observed numerically and verified in the large S limit on an exactly solvable Rokhsar-Kivelson line. The condensate has an SSB order parameter being the local in-plane spin, which points in angles related by the chaotic Bernoulli (dyadic) map and thus is effectively random. Moreover, we show the condensate exhibits nondecaying local autocorrelations and does not have an off-diagonal long-range order. The quantum breakdown condensate thus behaves as a disorder-free quantum glass and is beyond the existing classifications of phases of matter.
In the present work, a fermionic infinite-range Ising spin glass (SG) model with a transverse field Γ is studied. The model is analyzed in a one-lattice structure (quantum SG) and in a two-sublattice structure (SG/antiferromagnetism (AF) competition). In these two cases, the exchange interactions are random variables that follow Gaussian probability distributions (SK interactions). For the two-sublattice model, spins of distinct sublattices can couple with an SK-type interaction with an AF average. For a general case of this model, the disordered interactions can also occur with spins of same sublattices. The problem is formulated in a Grassmann path integral formalism within the static ansatz. The replica method is applied to treat the disorder, in which the replica symmetry (RS) and one-step replica symmetry breaking (1P-RSB) approximations are used. The one-lattice model results provide evidences that the 1S-RSB can be used to improve the physical description of the ordered quantum SG phase treated by the replica method. In the two-sublattice model, a parallel magnetic field H breaks the symmetry of the sublattices. It destroys the antiferromagnetic (AF) order, but it can favor the nonergodic mixed phase (SG+AF) characterizing an asymmetric RSB region. In this region, intra-sublattice disordered interactions V increase the difference between the RSB solutions of each sublattice. The freezing temperature (Tf ) shows a higher increase with H when V enhances. A discontinuous phase transition from the replica symmetry (RS) solution to the RSB solution can appear with the presence of an intra-sublattice ferromagnetic average coupling. The Γ field introduces a quantum spin flip mechanism that suppresses the magnetic orders leading them to quantum critical points. Results suggest that the quantum effects are not able to restore the RS solution. Therefore, the presence of both fields, Γ and H, produces effects that are in competition on Tf for instance. However, in the asymmetric RSB region, Γ can produce a stable RS solution at any finite temperature for a particular sublattice while the other sublattice still presents RSB solution for the special case in which only the intra-sublattice spins couple with disordered interactions.
Quantum impurity models provide a paradigmatic setting for studying Kondo screening, boundary criticality, and impurity entropy. While these phenomena are well understood in unitary systems, their fate in non-Hermitian many-body settings remains largely unexplored. We study a
PT-symmetric quantum impurity model consisting of a unitary SU(2)
1 Wess--Zumino--Witten bulk attached to two impurity spins through complex-conjugate Kondo couplings. Using an integrable lattice realization solved by the Bethe Ansatz and benchmarked against finite-temperature matrix-product-state calculations, we determine the impurity free energy, entropy, and the corresponding
g-function. We identify three distinct impurity phases. In the Kondo-screened phase, the spectrum remains real and the impurity entropy decreases monotonically from
ln4 in the ultraviolet to
0. Remarkably, this monotonic flow persists despite the nonunitary boundary interaction, beyond the standard domain of the
g-theorem. The Yu--Shiba--Rusinov phase instead exhibits spontaneous
PT-symmetry breaking and loss of boundary conformal invariance, while the local-moment phase displays cyclic renormalization-group flow, with identical impurity entropy and
g-value in the ultraviolet and infrared limits.