On Archimedean lattices, the Ising model exhibits spontaneous ordering. Three examples of these lattices of the majority-vote model with noise are considered and studied through extensive Monte Carlo simulations. The order/disorder phase transition is observed in this system. The calculated values of the critical noise parameter are q_c=0.089(5), q_c=0.078(3), and q_c=0.114(2) for honeycomb, Kagome and triangular lattices, respectively. The critical exponents beta/nu, gamma/nu and 1/nu for this model are 0.15(5), 1.64(5), and 0.87(5); 0.14(3), 1.64(3), and 0.86(6); 0.12(4), 1.59(5), and 1.08(6) for honeycomb, Kagome and triangular lattices, respectively. These results differs from the usual Ising model results and the majority-vote model on so-far studied regular lattices or complex networks. The effective dimensionalities of the system D_{eff}= 1.96(5) (honeycomb), D_{eff} =1.92(4) (Kagome), and D_{eff}= 1.83(5) (triangular) for these networks are just compatible to the embedding dimension two.
An antiferromagnetic version of the well-known majority voter model on square and honeycomb lattices is proposed. Monte Carlo simulations give evidence for a continuous order-disorder phase transition in the stationary state in both cases. Precise estimates of the critical point are found from the combination of three cumulants, and our results are in good agreement with the reported values of the equivalent ferromagnetic systems. The critical exponents $1/ν$, $γ/ν$ and $β/ν$ were found. Their values indicate that the stationary state of the antiferromagnetic majority voter model belongs to th…
We calculate complex-temperature (CT) zeros of the partition function for the $q$-state Potts model on the honeycomb and kagomé lattices for several values of $q$. These give information on the CT phase diagrams. A comparison of results obtained for different boundary conditions and a discussion of some CT singularities are given. Among other results, our findings show that the Potts antiferromagnet with $q=4$ and $q=5$ on the kagomé lattice has no phase transition at either finite or zero temperature.
A critical dilute O($n$) model on the kagome lattice is investigated analytically and numerically. We employ a number of exact equivalences which, in a few steps, link the critical O($n$) spin model on the kagome lattice to the exactly solvable critical $q$-state Potts model on the honeycomb lattice with $q=(n+1)^2$. The intermediate steps involve the random-cluster model on the honeycomb lattice, and a fully packed loop model with loop weight $n'=\sqrt{q}$ and a dilute loop model with loop weight $n$, both on the kagome lattice. This mapping enables the determination of a branch of critical p…
We consider the $(2+1)$-d $SU(2)$ quantum link model on the honeycomb lattice and show that it is equivalent to a quantum dimer model on the Kagomé lattice. The model has crystalline confined phases with spontaneously broken translation invariance associated with pinwheel order, which is investigated with either a Metropolis or an efficient cluster algorithm. External half-integer non-Abelian charges (which transform non-trivially under the $\mathbb{Z}(2)$ center of the $SU(2)$ gauge group) are confined to each other by fractionalized strings with a delocalized $\mathbb{Z}(2)$ flux. The strand…
Phase transitions and crtical exponents in the six-vertex model on kagome lattices
College of Physics and Optoelectronics, Taiyuan University of Technology（太原理工大学物理与光电学院）；School of Science and Engineering of Mathematics and Physics, Anhui University of Technology（安徽工业大学数理科学与工程学院）；Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China（中国科学技术大学微尺度物质科学国家实验室和近代物理系）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）
Lattices that can be represented in a kagome-like form are shown to satisfy a universal percolation criticality condition, expressed as a relation between P_3, the probability that all three vertices in the triangle connect, and P_0, the probability that none connect. A linear approximation for P_3(P_0) is derived and appears to provide a rigorous upper bound for critical thresholds. A numerically determined relation for P_3(P_0) gives thresholds for the kagome, site-bond honeycomb, (3-12^2), and "stack-of-triangle" lattices that compare favorably with numerical results.
Here we consider the Ising-Heisenberg model in the expanded Kagomé lattice, also known as triangle-dodecagon (3-12) or star lattice. This model can still be understood as a decorated honeycomb lattice. Assuming that the Heisenberg spins are at the vertices of the triangle while other spins are of the Ising type. Thus, this model is equivalent to an effective Kagomé Ising lattice, through the decoration transformation technique. Thus this means that the model is exactly solvable so we can study the most relevant properties of this model. Like the phase diagram at zero temperature, exhibiting a …
We present and analyze low-temperature series and complex-temperature partition function zeros for the $q$-state Potts model with $q=4$ on the honeycomb lattice and $q=3,4$ on the triangular lattice. A discussion is given as to how the locations of the singularities obtained from the series analysis correlate with the complex-temperature phase boundary. Extending our earlier work, we include a similar discussion for the Potts model with $q=3$ on the honeycomb lattice and with $q=3,4$ on the kagomé lattice.
We consider the problem of counting the number of 3-colourings of the edges (bonds) of the 4-8 lattice and the 3-12 lattice. These lattices are Archimedean with coordination number 3, and can be regarded as decorated versions of the square and honeycomb lattice, respectively. We solve these edge-colouring problems in the infinite-lattice limit by mapping them to other models whose solution is known. The colouring problem on the 4-8 lattice is mapped to a completely packed loop model with loop fugacity n=3 on the square lattice, which in turn can be mapped to a six-vertex model. The colouring p…
A novel model, devised to describe spontaneous chirality synchronization in complex liquids and liquid crystals, is proposed and studied. Segments of ribbon-like molecular columns with left- or right-handed 180degree twist lie on the bonds of a honeycomb lattice so that three ribbons meet in a vertex of the hexagonal honeycomb. The energy of each vertex is a minimum if the three ribbons have the same chirality, -E, and is +E otherwise, and the ground state is homochiral, i.e. all ribbons have the same hand. The energy levels for two vertices linked by a single ribbon are either -2E, 0 and +2 E…
This article investigates phonons and elastic response in randomly diluted lattices constructed by combining (via the addition of next-nearest bonds) a twisted kagome lattice, with bulk modulus $B=0$ and shear modulus $G>0$, with either a generalized untwisted kagome lattice with $B>0$ and $G>0$ or with a honeycomb lattice with $B>0$ and $G=0$. These lattices exhibit jamming-like critical end-points at which $B$, $G$, or both $B$ and $G$ jump discontinuously from zero while the remaining moduli (if any) begin to grow continuously from zero. Pairs of these jamming points are joined by lines of …
An alternative definition for the order-parameter is proposed, for a family of non-equilibrium spin models with up-down symmetry on honeycomb lattices, and which depends on two parameters. In contrast to the usual definition, our proposal takes into account that each site of the lattice can be associated with a local temperature which depends on the local environment of each site. Using the generalised voter motel as a test case, we analyse the phase diagram and the critical exponents in the stationary state and compare the results of the standard order-parameter with the ones following from o…
We study the thermodynamics of Ising spins on the triangular kagome lattice (TKL) using exact analytic methods as well as Monte Carlo simulations. We present the free energy, internal energy, specific heat, entropy, sublattice magnetizations, and susceptibility. We describe the rich phase diagram of the model as a function of coupling constants, temperature, and applied magnetic field. For frustrated interactions in the absence of applied field, the ground state is a spin liquid phase with integer residual entropy per spin $s_0/k_B={1/9} \ln 72\approx 0.4752...$. In weak applied field, the sys…
We present a Markov-chain Monte Carlo algorithm of "worm"type that correctly simulates the O(n) loop model on any (finite and connected) bipartite cubic graph, for any real n>0, and any edge weight, including the fully-packed limit of infinite edge weight. Furthermore, we prove rigorously that the algorithm is ergodic and has the correct stationary distribution. We emphasize that by using known exact mappings when n=2, this algorithm can be used to simulate a number of zero-temperature Potts antiferromagnets for which the Wang-Swendsen-Kotecky cluster algorithm is non-ergodic, including the 3-…
We study the percolation critical surface of the kagome lattice in which each triangle is allowed an arbitrary connectivity. Using the method of critical polynomials, we find points along this critical surface to high precision. This kagome hypergraph contains many unsolved problems as special cases, including bond percolation on the kagome and $(3,12^2)$ lattices, and site percolation on the hexagonal, or honeycomb, lattice, as well as a single point for which there is an exact solution. We are able to compute enough points along the critical surface to find a very accurate fit, essentially a…
The recently fabricated two-dimensional magnetic materials Cu9X2(cpa)6.xH2O (cpa=2-carboxypentonic acid; X=F,Cl,Br) have copper sites which form a triangular kagome lattice (TKL), formed by introducing small triangles (``a-trimers'') inside of each kagome triangle (``b-trimer''). We show that in the limit where spins residing on b-trimers have Ising character, quantum fluctuations of XXZ spins residing on the a-trimers can be exactly accounted for in the absence of applied field. This is accomplished through a mapping to the kagome Ising model, for which exact analytic solutions exist. We deri…
Tensor network investigation of the monomer-dimer model on the honeycomb lattice
Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area（粤港澳大湾区量子科学中心）；Department of Physics, City University of Hong Kong（香港城市大学物理系）；City University of Hong Kong Shenzhen Research Institute（香港城市大学深圳研究院）
We consider dimers on the star lattice (aka the 3-12, Fisher, expanded kagome or triangle-honeycomb lattice). We show that dimer coverings on this lattice have Z_2 arrow and pseudo-spin representations analogous to those for the kagome lattice, and use these to construct an exactly solvable quantum dimer model (QDM) with a Rokhsar-Kivelson (RK) ground state. This QDM, first discussed by Moessner and Sondhi from a different point of view, is the star-lattice analogue of a kagome-lattice QDM analyzed by Misguich et al. We give a detailed analysis of various properties of the classical equal-weig…
We report on a systematic study of two dimensional, periodic, frustrated Ising models with a quantum dynamics introduced via a transverse magnetic field. The systems studied are the triangular and kagome lattice antiferromagnets, fully frustrated models on the square and hexagonal (honeycomb) lattices, a planar analog of the pyrochlore antiferromagnet, a pentagonal lattice antiferromagnet as well as a two quasi one-dimensional lattices that have considerable pedagogical value. All of these exhibit a macroscopic degeneracy at T=0 in the absence of the transverse field, which enters as a singula…
