Covering a graph or a lattice with non-overlapping dimers is a problem that has received considerable interest in areas such as discrete mathematics, statistical physics, chemistry and materials science. Yet, the problem of percolation on dimer-covered lattices has received little attention. In particular, percolation on lattices that are fully covered by non-overlapping dimers has not evidently been considered. Here, we propose a novel procedure for generating random dimer coverings of a given lattice. We then compute the bond percolation threshold on random and ordered coverings of the square and the triangular lattice, on the remaining bonds connecting the dimers. We obtain $p_c=0.367713(2)$ and $p_c=0.235340(1)$ for random coverings of the square and the triangular lattice, respectively. We observe that the percolation frustration induced as a result of dimer covering is larger in the low-coordination-number square lattice. There is also no relationship between the existence of long-range order in a covering of the square lattice, and its percolation threshold. In particular, an ordered covering of the square lattice, denoted by shifted covering in this work, has an unusually low percolation threshold, and is topologically identical to the triangular lattice. This is in contrast to the other ordered dimer coverings considered in this work, which have higher percolation thresholds than the random covering. In the case of the triangular lattice, the percolation thresholds of the ordered and random coverings are very close, suggesting the lack of sensitivity of the percolation threshold to microscopic details of the covering in highly-coordinated networks.
Restricted-valence random sequential adsorption~(RSA) is studied in its pure and disordered versions, on the square and triangular lattices. For the simplest case~(pure on the square lattice) we prove the absence of percolation for maximum valence $V_{\rm max}=2$. In other cases, Monte Carlo simulations are used to investigate the percolation threshold, universality class, and jamming limit. Our results reveal a continuous transition for the majority of the cases studied. The percolation threshold is computed through finite-size scaling analysis of seven properties; its value increases with th…
We study spin-1/2 Heisenberg antiferromagnets on a diamond-like-decorated square lattice. The diamond-like-decorated square lattice is a lattice in which the bonds in a square lattice are replaced with diamond units. The diamond unit has two types of antiferromagnetic exchange interactions, and the ratio $λ$ of the diagonal bond strength to that of the other four edges controls the frustration strength. For $0.974<λ<2$, the present system has a nontrivial macroscopic degeneracy, which is called the macroscopically degenerated tetramer-dimer (MDTD) states. The MDTD states are identical to the H…
Percolation and jamming phenomena are investigated for anisotropic sequential deposition of dimers (particles occupying two adjacent adsorption sites) on a square lattice. The influence of dimer alignment on the electrical conductivity was examined. The percolation threshold for deposition of dimers was lower than for deposition of monomers. Nevertheless, the problem belongs to the universality class of random percolation. The lowest percolation threshold (pc = 0.562) was observed for isotropic orientation of dimers. It was higher (pc = 0.586) in the case of dimers aligned strictly along one d…
The random sequential adsorption (RSA) model is a classical model in Statistical Physics for adsorption on two-dimensional surfaces. Objects are deposited sequentially at random and adsorb irreversibly on the landing site, provided that they do not overlap any previously adsorbed object. The kinetics of adsorption ceases when no more objects can be adsorbed (jamming state). Here, we investigate the role of post-relaxation on the jamming state and percolation properties of RSA of dimers on a two-dimensional lattice. We consider that, if the deposited dimer partially overlaps with a previously a…
Reduced dimensionality in layered quantum dimer magnets: Frustration vs. inhomogeneous condensates
受BaCuSi_2 O_6实验的启发，我们研究了具有层间阻挫的层状二聚体磁体的磁激发和量子相变。我们考虑了两种情况，（A）一个晶格与一个二聚体每个单元格和完美的层间挫折，和（B）一个扩大的单元格与不等层，有和没有完美的挫折。在所有情况下，在渐近低温下的临界行为是三维的，但相应的交叉尺度可能很小。情形（B）中的磁有序可以用两种凝聚体来讨论;值得注意的是，完全阻挫使得邻近效应无效。然后，有序转换将被一般地分裂，在可测量的属性中具有清晰的签名。利用广义键算符方法，计算了顺磁相和反铁磁相的低温磁性。基于现有的BaCuSi 2 O 6实验数据，我们提出在这种材料中实现具有不等效层和不完美挫折的情况（B），可能具有沿c轴的层间耦合的附加调制。
The dynamical percolation transition of two dimensional Axial Next Nearest Neighbour Ising (ANNNI) model to pulsed magnetic field has been studied by finite size scaling analysis (by Monte Carlo simulation) for various values of frustration parameters, pulse width and temperature (below the corresponding static transition temperature). It has been found that the size of the largest geometrical cluster shows a transition for a critical field amplitude. Although the transition points shift, the critical exponents remain invariant for a wide range of frustration parameters. It is also same as tha…
We study the dynamical properties of the fully frustrated Ising model. Due to the absence of disorder the model, contrary to spin glass, does not exhibit any Griffiths phase, which has been associated to non-exponential relaxation dynamics. Nevertheless we find numerically that the model exhibits a stretched exponential behavior below a temperature T_p corresponding to the percolation transition of the Kasteleyn-Fortuin clusters. We have also found that the critical behavior of this clusters for a fully frustrated q-state spin model at the percolation threshold is strongly affected by frustrat…
The ground-state phases of alternating-bond spin-1 diamond chains are investigated. Each ground state consists of an array of spin clusters separated by singlet dimers owing to an infinite number of local conservation laws. If no singlet dimers are present, the ground state is equivalent to that of a spin chain with infinite length.For strong frustration, we find a series of quantum phase transitions as in the case of alternating-bond mixed diamond chains with spins 1 and 1/2. For intermediate frustration, we find the nonmagnetic Haldane or dimer phases according to whether the bond alternatio…
Jamming and percolation transitions in the standard random sequential adsorption of particles on regular lattices are characterized by a universal set of critical exponents. The universality class is preserved even in the presence of randomly distributed defective sites that are forbidden for particle deposition. However, using large-scale Monte Carlo simulations by depositing dimers on the square lattice and employing finite-size scaling, we provide evidence that the system does not exhibit such well-known universal features when the defects have spatial long-range (power-law) correlations. T…
We present a numerical study on an interacting monomer-dimer model with nearest neighbor repulsion on a square lattice, which possesses two symmetric absorbing states. The model is observed to exhibit two nearby continuous transitions: the $Z_2$ symmetry-breaking order-disorder transition and the absorbing transition with directed percolation criticality. We find that the symmetry-breaking transition shows a non-Ising critical behavior, and that the absorbing phase becomes critical, in the sense that the critical decay of the dimer density observed at the absorbing transition persists even wit…
We study the ground-state phase diagram of a Heisenberg model with spin $S=\frac{1}{2}$ on a diamond-like decorated square lattice. A diamond unit has two types of antiferromagnetic exchange interactions, and the ratio $λ$ between the length of the diagonal bond and that of the other four edges determines the strength of frustration. It has been pointed out [J. Phys. Soc. Jpn {\bf85}, 033705 (2016)] that the so-called tetramer-dimer states, which are expected to be stabilized in an intermediate region of $λ_{\rm c}<λ<2$, are identical to the square-lattice dimer covering states, which ignited …
Percolation and criticality of systems with competing interactions on Bethe lattices: limitations and potential strengths of cluster schemes
Sorbonne Université（索邦大学）；Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Duke University（杜克大学）；Università degli Studi di Napoli “Federico II”（那不勒斯费德里科二世大学）
We study a general class of easy-axis spin models on a lattice of corner sharing even-sided polygons with all-to-all interactions within a plaquette. The low energy description corresponds to a quantum dimer model on a dual lattice of even coordination number with a multi dimer constraint. At an appropriately constructed frustration-free Rokhsar-Kivelson (RK) point, the ground state wavefunction can be exactly mapped onto a classical vertex model on the dual lattice. When the dual lattice is bipartite, the vertex models are bonded and are self dual under Wegner's duality, with the self dual po…
The nature of the randomness-induced quantum spin liquid state, the random-singlet state, is investigated in two dimensions (2D) by means of the exact-diagonalization and the Hams-de Raedt methods for several frustrated lattices, e.g., the triangular, the kagome and the J_1-J_2 square lattices. Properties of the ground state, the low-energy excitations and the finite-temperature thermodynamic quantities are investigated. The ground state and the low-lying excited states consist of nearly isolated singlet-dimers, clusters of resonating singlet-dimers, and orphan spins. Low-energy excitations ar…
Magnetization plateaus, spin-canted orders and field-induced transitions in a spin-1/2 Heisenberg antiferromagnet on a distorted diamond-decorated honeycomb lattice
Institute of Physics, Faculty of Science, P. J. Šafárik University（帕日·沙法里克大学理学院物理研究所）
Long-range frustration in minimal vertex cover problem on random graphs
College of Science, University of Shanghai for Science and Technology（上海理工大学理学院）；School of Data Science and Engineering, East China Normal University（华东师范大学数据科学与工程学院）；CoCoLab, School of Data Science and Engineering, South China Normal University（华南师范大学数据科学与工程学院）
Spin-1/2 Heisenberg diamond-like decorated honeycomb lattice in a magnetic field from the perspective of localized magnons and exact diagonalization
Faculty of Science, P. J. Šafárik University（帕·约·沙法里克大学理学院）；Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy Paris Université（CY塞尔吉巴黎大学，CNRS UMR 8089理论物理与建模实验室）
