We study the $t\text{\ensuremath{-}}J$ model with four holes on a 32-site square lattice using exact diagonalization. This system corresponds to doping level
x=81. At the ``realistic'' parameter
J∕t=0.3, holes in the ground state of this system are unbound. They have short range repulsion due to the lowering of the kinetic energy. There is no antiferromagnetic spin order and the electron momentum distribution function resembles hole pockets. Furthermore, we show evidence that in case antiferromagnetic order exists, holes form
d-wave bound pairs and there is mutual repulsion among hole pairs. This presumably will occur at low doping levels. This scenario is compatible with a checkerboard-type charge density state proposed to explain the ``
81 anomaly'' in the lanthanum cuprate family, except that it is the ground state only when the system possesses strong antiferromagnetic order.
We determine the phase diagram of a generalized Shastry-Sutherland model, using a combination of dimer- and quadrumer-boson methods and numerical exact diagonalization techniques. Along special lines in the parameter space the model reduces to the standard Shastry-Sutherland model, the 1/5th depleted square lattice and the two-dimensional plaquette square lattice model. We study the evolution of the ordered phases found in the latter two unfrustrated models under the effect of frustration. Furthermore we present exact diagonalization results for the Shastry-Sutherland model on clusters with up to 32 sites, supporting the existence of an intermediate gapped valence bond crystal phase with plaquette long-range order.
Using numerical results from density matrix renormalization group (DMRG) calculations for the t-J model, on systems as large as 10\ifmmode\times\else\texttimes\fi{}7, we examine the structure of the one and two hole ground states in ladder systems and in two dimensional clusters. A simple theoretical framework is used to explain why holes bind in pairs in two-dimensional antiferromagnets. For the case J/t=0.5, which we have studied, the hole pairs reside predominantly on a 2\ifmmode\times\else\texttimes\fi{}2 core plaquette with the probability that the holes are on diagonal sites greater than nearest-neighbor sites. There is a strong singlet bond connecting the spins on the two remaining sites of the plaquette. We find that a general characteristic of dynamic holes in an antiferromagnet is the presence of frustrating antiferromagnetic bonds connecting next-nearest-neighbor sites across the holes. Pairs of holes bind in order to share the frustrating bonds. At low doping, in addition to hole pairs, there are two additional low-energy structures which spontaneously form on certain finite systems. The first is an undoped L\ifmmode\times\else\texttimes\fi{}2 spin-liquid region, or ladder. The second is a hole moving along a one dimensional chain of sites. At higher doping we expect that hole pairing is always favored.
Exact results in frustrated quantum many-body systems are rare, especially in dimensions higher than one. The Shastry-Sutherland (SS) model stands out as a rare example of a two-dimensional spin system with an exactly solvable dimer singlet ground state. In this work, we introduce a three-dimensional analogue of the SS lattice, constructed by deforming the pyrochlore lattice to preserve the local SS geometry. Despite the dimensional increase and altered topology, the ground-state phase diagrams of classical Ising and Heisenberg spins, remain analytically tractable and closely follow their 2D counterparts, including the existence of a 1/3 magnetization plateau and umbrella states. Most notably, for quantum spins S = 1/2, the dimer singlet state survives as an exact ground state over a finite region of the phase diagram. We argue, using exact diagonalization, that the singlet phase is stabilized beyond its 2D counterpart, suggesting enhanced robustness in three dimensions. These results offer a rare, controlled platform to explore the impact of dimensionality on quantum frustration, exact solvability, and potential spin liquid behavior in 3D, with relevance to emergent topological and magnetic phases.
We investigate the one-dimensional t - J - J ′ model by the numerical diagonalization where a few holes are doped in the half-filled band. By introducing a finite-size scaling, we estimate the spin gap at J ′ / J =0.5 and J / t =2. When one hole is doped, the spin gap decreases to 0.190 J in comparison with 0.236 J which is the spin gap of the half-filled band. The spin gap further reduces to 0.141 J for two-hole doping and to about 0.05 J for three-hole doping. This result shows that the value of the spin gap as a function of the electron density is discontinuous at half-filling.
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 SrCu
2(BO
3)
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.