We investigate a frustrated four-spin plaquette spin-boson model with competing nearest-neighbor and diagonal Ising couplings, where each spin is coupled to an independent bosonic bath. Combining a path-integral strong-coupling analysis with variational matrix product state simulations, we obtain the ground-state phase diagram. In the strong-dissipation limit we map the model onto a classical plaquette and derive analytic phase boundaries between ferromagnetic, Néel, and stripe phases. At intermediate dissipation we find a delocalized phase and two localized ordered phases with Néel and stripe character. We show that the localized phases are separated by a first-order line, while each is connected to the delocalized regime via a continuous (second-order) localization transition, and that these three boundaries meet at a quantum triple point. Analysis of spin correlations and reduced density matrices further reveals that entanglement concentrates on nearest-neighbor (diagonal) bonds in the Néel (stripe) phase, whereas in the delocalized regime intra-plaquette two-spin entanglement is strongly suppressed in favor of enhanced spin-bath correlations.
We investigate a frustrated four-spin plaquette spin-boson model with competing nearest-neighbor and diagonal Ising couplings, where each spin is coupled to an independent bosonic bath. Combining a path-integral strong-coupling analysis with variational matrix product state simulations, we obtain the ground-state phase diagram. In the strong-dissipation limit we map the model onto a classical plaquette and derive analytic phase boundaries between ferromagnetic, Néel, and stripe phases. At intermediate dissipation we find a delocalized phase and two localized ordered phases with Néel and stripe character. We show that the localized phases are separated by a first-order line, while each is connected to the delocalized regime via a continuous (second-order) localization transition, and that these three boundaries meet at a quantum triple point. Analysis of spin correlations and reduced density matrices further reveals that entanglement concentrates on nearest-neighbor (diagonal) bonds in the Néel (stripe) phase, whereas in the delocalized regime intra-plaquette two-spin entanglement is strongly suppressed in favor of enhanced spin-bath correlations.
Using the coupled-cluster method for high orders of approximation and Lanczos exact diagonalization we study the ground-state phase diagram of a quantum spin-21J1-J2 model on the square lattice with plaquette structure. We consider antiferromagnetic (J1>0) as well as ferromagnetic (J1<0) nearest-neighbor interactions together with frustrating antiferromagnetic next-nearest-neighbor interaction J2>0. The strength of interplaquette interaction $\ensuremath{\lambda}$ varies between $\ensuremath{\lambda}=1$ (which corresponds to the uniform J1-J2 model) and $\ensuremath{\lambda}=0$ (which corresponds to isolated frustrated 4-spin plaquettes). While on the classical level ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) both versions of models (i.e., with ferro- and antiferromagnetic J1) exhibit the same ground-state behavior, the ground-state phase diagram differs basically for the quantum case s=1/2. For the antiferromagnetic case (J1>0) N\'eel antiferromagnetic long-range order at small J2/J1 and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.47$ as well as collinear striped antiferromagnetic long-range order at large J2/J1 and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.30$ appear which correspond to their classical counterparts. Both semiclassical magnetic phases are separated by a nonmagnetic quantum paramagnetic phase. The parameter region, where this nonmagnetic phase exists, increases with decreasing $\ensuremath{\lambda}$. For the ferromagnetic case (J1<0) we have the trivial ferromagnetic ground state at small J2/∣J1∣. By increasing J2 this classical phase gives way for a semiclassical plaquette phase, where the plaquette block spins of length s=2 are antiferromagnetically long-range ordered. Further increasing of J2 then yields collinear striped antiferromagnetic long-range order for $\ensuremath{\lambda}\ensuremath{\gtrsim}0.38$, but a nonmagnetic quantum paramagnetic phase $\ensuremath{\lambda}\ensuremath{\lesssim}0.38$.
We study the half-filled single-band Hubbard model on the Shastry-Sutherland lattice with nearest-neighbor hopping t , incorporating (i) a frustrating diagonal hopping t ′ to probe the emergence of spin-liquid behavior, and (ii) an additional Heisenberg exchange J that explicitly stabilizes plaquette-singlet correlations. Using the variational cluster approximation (VCA) with exact diagonalization at zero temperature on an eight-site cluster, we map the phase diagrams in the ( U / t , t ′ / t ) and ( U / t , J / t ) parameter planes. In the ( U / t , t ′ / t ) plane, our analysis uncovers a competition between Néel antiferromagnetic (AFM) order with staggered magnetization and a nonmagnetic regime lacking conventional order, consistent with a nonmagnetic insulating phase that may be related to the spin-liquid-like regimes reported in previous studies. In contrast, the ( U / t , J / t ) phase diagram exhibits a competition between the Néel antiferromagnet and a solid valence-bonding valence bond plaquette (PS), where spins form localized singlet plaquettes. Most notably, we observe that the transition from the PS phase to the Néel AFM phase is a first-order phase transition, with a discontinuous jump of the order parameters at the phase boundary, in quantitative agreement with high-pressure heat-capacity measurements on SrCu 2 ( BO 3 ) 2 [J. Guo , ]. Thus, our VCA analysis underscores the subtle interplay of geometric frustration, quantum fluctuations, and dimensionality, and elucidates the crucial role of short-range correlations in stabilizing and competing ground states of frustrated quantum magnets.
Unsupervised Discovery of Intermediate Phase Order in the Frustrated J1-J2 Heisenberg Model via Prometheus Framework
The spin-1/2J1-J2 Heisenberg model on the square lattice exhibits a debated intermediate phase between Néel antiferromagnetic and stripe ordered regimes, with competing theories proposing plaquette valence bond, nematic, and quantum spin liquid ground states. We apply the Prometheus variational autoencoder framework -- previously applied to classical (2D, 3D Ising) and quantum (disordered transverse field Ising) phase transitions -- to systematically explore the J1-J2 phase diagram using a multi-scale approach. For L=4, we employ exact diagonalization with full wavefunction analysis via quantum-aware VAE. For larger systems (L=6,8), we introduce a reduced density matrix (RDM) based methodology using DMRG ground states, enabling scaling beyond the exponential barrier of full Hilbert space representation. Through dense parameter scans of J2/J1∈[0,1] and comprehensive latent space analysis, we identify the structure factor S(π,π) and S(π,0) as the dominant order parameters discovered by the VAE, with correlations exceeding ∣r∣>0.97. The RDM-VAE approach successfully captures the Néel-to-stripe crossover near J2/J1≈0.5--0.6, demonstrating that local quantum correlations encoded in reduced density matrices contain sufficient information for unsupervised phase discovery. This work establishes a scalable pathway for applying machine learning to frustrated quantum systems where full wavefunction access is computationally prohibitive.
We examine the ground-state phase diagram and thermal phase transitions in a plaquettized fully frustrated bilayer spin-1/2 Heisenberg model. Based on a combined analysis from sign-problem free quantum Monte Carlo simulations, perturbation theory, and free-energy arguments, we identify a first-order quantum phase transition line that separates two competing quantum-disordered ground states with dominant singlet formations on interlayer dimers and plaquettes, respectively. At finite temperatures, this line extends to form a wall of first-order thermal transitions, which terminates in a line of thermal critical points. From a perturbative approach in terms of an effective Ising model description, we identify a quadratic suppression of the critical temperature scale in the strongly plaquettized region. Based on free-energy arguments we furthermore obtain the full phase boundary of the low-temperature dimer-singlet regime, which agrees well with the quantum Monte Carlo data.
We study the half-filled single-band Hubbard model on the Shastry–Sutherland lattice with nearest-neighbor hopping t, incorporating (i) a frustrating diagonal hopping t′ to probe the emergence of spin-liquid behavior, and (ii) an additional Heisenberg exchange J that explicitly stabilizes plaquette-singlet correlations. Using the variational cluster approximation (VCA) with exact diagonalization at zero temperature on an eight-site cluster, we map the phase diagrams in the (U/t,t′/t) and (U/t,J/t) parameter planes. In the (U/t,t′/t) plane, our analysis uncovers a competition between N\'eel antiferromagnetic (AFM) order with staggered magnetization and a nonmagnetic regime lacking conventional order, consistent with an algebraic spin-liquid (ASL) phase characterized by a gapped excitation spectrum and vanishing staggered magnetization. In contrast, the (U/t,J/t) phase diagram exhibits a competition between the N\'eel AFM and a solid valence-bonding valence bond plaquette (PS), where spins form localized singlet plaquettes. Most notably, we observe that the transition from the PS phase to the N\'eel AFM is a first-order phase transition, with a discontinuous jump of the order parameters at the phase boundary, in quantitative agreement with high-pressure heat-capacity measurements on SrCu2(BO3)2 [Guo \emph{et al.}, Commun. Phys. 8, 75 (2025)]. Thus, our VCA analysis underscores the subtle interplay of geometric frustration, quantum fluctuations, and dimensionality, and elucidates the crucial role of short-range correlations in stabilizing and competing ground states of frustrated quantum magnets.
We use a combination of analytical and numerical techniques to study the phase diagram of the frustrated Heisenberg model on the bilayer honeycomb lattice. Using the Schwinger-boson description of the spin operators followed by a mean-field decoupling, the magnetic phase diagram is studied as a function of the frustration coupling J2 and the interlayer coupling ${J}_{\ensuremath{\perp}}$. The presence of both magnetically ordered and disordered phases is investigated by means of the evaluation of ground-state energy, spin gap, local magnetization, and spin-spin correlations. We observe a phase with a spin gap and short-range N\'eel correlations that survives for nonzero next-nearest-neighbor interaction and interlayer coupling. Furthermore, we detect signatures of a reentrant behavior in the melting of the N\'eel phase and symmetry restoring when the system undergoes a transition from an on-layer nematic valence-bond crystal phase to an interlayer valence-bond crystal phase. We complement our work with exact diagonalization on small clusters and dimer-series expansion calculations, together with a linear spin-wave approach to study the phase diagram as a function of the spin S, the frustration, and the interlayer couplings.