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The quantum dimer model is a low-energy effective model for many magnetic systems (materials) that are candidates for quantum spin liquids. It has strict local constraints which is described by gauge field theory. However, since constraints hinder the application of numerical algorithms, the phase diagrams of quantum dimer models are still controversial, even on a square lattice. The core controversy is whether the mixed state exists due to the restriction. In this paper, we give strong evidence to solve this dispute. With our sweeping cluster quantum Monte Carlo method, we studied the phase diagram of a large parameter region by introducing the definition of the pair correlation function and other supporting evidence to distinguish the mixed phase from the columnar phase with high precision. In particular, we find that the ground state belongs to the mixed phase for a vast parameter region.

We study two charge-ordered states in a $t\ensuremath{-}J\ensuremath{-}U$ model with the method of renormalized mean-field theory. Compared with the $t\ensuremath{-}J$ model, which excludes the existence of one site occupied by two electrons, finite Coulomb interaction $U$ will allow the appearance of doublons and produce more charge fluctuations. The charge density wave states accompanied by Cooper pair density modulations have similar features with the states found in the $t\ensuremath{-}J$ model. The difference is that the modulation of doublons exists, and it has the same wavelength and phase as holes. Charge-ordered states disappear at the $U/t\ensuremath{\le}9$ region when the system does not correspond to the strongly correlated resonating valence-bond state. This feature indicates that local Coulomb interactions are the origin of charge-ordered states observed in cuprates. In addition, we calculate the strength of superconducting coherence and the gap depth of the nodal pair-density-wave state based on the local density of states. These two quantities also vary with lattice sites periodically. The oscillations have same wave vector and phase as the modulation of holes. These results are consistent with recent experiments.

Quantum spin systems with strong geometric restrictions give rise to rich quantum phases such as valence bond solids and spin liquid states. However, the geometric restrictions often hamper the application of sophisticated numerical approaches. Based on the stochastic series expansion method, we develop an efficient and exact quantum Monte Carlo ``sweeping cluster'' algorithm which automatically satisfies the geometrical restrictions. Here we use the quantum dimer model as a benchmark to demonstrate the reliability and power of this algorithm. Comparing to existing numerical methods, we can obtain higher accuracy results for a wider parameter region and much more substantial system sizes.

To examine the validity of the scenario of the deconfined critical phenomena, we carry out a quantum Monte Carlo simulation for the SU($N$) generalization of the Heisenberg model with four-body and six-body interactions. The quantum phase transition between the SU($N$) N\'eel and valence-bond solid phases is characterized for $N=2,3,$ and $4$ on the square and honeycomb lattices. While finite-size scaling analysis works well up to the maximum lattice size ($L=256$) and indicates the continuous nature of the phase transition, a clear systematic change towards the first-order transition is observed in the estimates of the critical exponent $y \equiv 1/\nu$ as the system size increases. We also confirm the relevance of a squared valence-bond solid field $\Psi^2$ for the SU(3) model., Comment: 5 pages, 5 figures

We study generalizations of the singlet-sector amplitude-product (AP) states in the valence-bond basis of S=1/2 quantum spin systems. In the standard AP states, the weight of a tiling of the system into valence bonds (singlets of two spins) is a product of amplitudes depending on the length of the bonds. We here introduce correlated AP (CAP) states, in which the amplitude product is further multiplied by factors depending on two bonds connected to a pair of sites (here nearest neighbors). While the standard AP states can describe a phase transition between an antiferromagnetic (Neel) state and a valence-bond solid (VBS) in one dimension (which we also study here), in two dimensions it cannot describe VBS order. With the CAP states, Neel-VBS transitions are realized as a function of some parameter describing the bond correlations. We here study such phase transitions of CAP wave-functions on the square lattice. We find examples of direct first-order Neel-VBS transitions, as well as cases where there is an extended U(1) spin liquid phase intervening between the Neel and VBS states. In the latter case the transitions are continuous and we extract critical exponents and address the issue of a possible emergent U(1) symmetry in the near-critical VBS. We also consider variationally optimized CAP states for the standard Heisenberg model in one and two dimensions and the J-Q model in two dimensions, with the latter including four-spin interactions (Q) in addition to the Heisenberg exchange (J) and harboring VBS order for large Q/J. The optimized CAP states lead to significantly lower variational energies than the simple AP states for these models., 21 pages, 27 figures. v2: Expanded discussion of results

We investigate the entanglement properties of the valence-bond-solid (VBS) state defined on two-dimensional lattices, which is the exact ground state of the Affleck-Kennedy-Lieb-Tasaki model. It is shown that the entanglement entropy obeys an area law and the nonuniversal prefactor of the leading term is strictly less than $\mathrm{ln}2$. The analysis of entanglement spectra for various lattices reveals that the reduced density matrix associated with the VBS state is closely related to a thermal density matrix of a holographic spin chain, the spectrum of which is reminiscent of that of the spin-1/2 Heisenberg chain. This correspondence is further supported by comparing the entanglement entropy in the holographic spin chain with conformal field theory predictions.

We discuss ground-state projector simulations of a modified two-dimensional $S=1/2$ Heisenberg model in the valence bonds basis. Tuning matrix elements corresponding to the diagonal and off-diagonal terms in the quantum dimer model, we show that there is a quantum phase transition from the antiferromagnet into a columnar valence-bond solid (VBS). There are no signs of discontinuities, suggesting a continuous or very weak first-order transition. The ${\text{Z}}_{4}$-symmetric VBS order parameter exhibits an emergent U(1) symmetry as the phase transition is approached. We extract the associated length-scale governing the $\text{U}(1)\ensuremath{-}{\text{Z}}_{4}$ cross-over inside the VBS phase.

We study two-dimensional Heisenberg antiferromagnets with additional multispin interactions which can drive the system into a valence-bond-solid state. For standard SU(2) spins, we consider both four- and six-spin interactions. We find continuous quantum phase transitions with the same critical exponents. Extending the symmetry to $\text{SU}(N)$, we also find continuous transitions for $N=3$ and 4. In addition, we also study quantitatively the crossover of the order-parameter symmetry from ${Z}_{4}$ deep inside the valence-bond-solid phase to $U(1)$ as the phase transition is approached.

We study a variational wave function for the ground state of the two-dimensional S=1/2 Heisenberg antiferromagnet in the valence bond basis. The expansion coefficients are products of amplitudes h(x,y) for valence bonds connecting spins separated by (x,y) lattice spacings. In contrast to previous studies, in which a functional form for h(x,y) was assumed, we here optimize all the amplitudes for lattices with up to 32*32 spins. We use two different schemes for optimizing the amplitudes; a Newton/conjugate-gradient method and a stochastic method which requires only the signs of the first derivatives of the energy. The latter method performs significantly better. The energy for large systems deviates by only approx. 0.06% from its exact value (calculated using unbiased quantum Monte Carlo simulations). The spin correlations are also well reproduced, falling approx. 2% below the exact ones at long distances. The amplitudes h(r) for valence bonds of long length r decay as 1/r^3. We also discuss some results for small frustrated lattices.