Dimers on Surface Graphs and Spin Structures. II

In a previous paper [3], we showed how certain orientations of the edges of a graph Γ embedded in a closed oriented surface Σ can be understood as discrete spin structures on Σ. We then used this correspondence to give a geometric proof of the Pfaffian formula for the partition function of the dimer model on Γ. In the present article, we generalize these results to the case of compact oriented surfaces with boundary. We also show how the operations of cutting and gluing act on discrete spin structures and how they change the partition function. These operations allow to reformulate the dimer model as a quantum field theory on surface graphs.     

Exact solution of a vertex model ind dimensions

We consider a class of vertex models describing directed lines on a lattice in arbitraryd dimensions, and solve the model exactly for the Cartesian lattice and in the case that each loop of lines carries a fugacity - 1. Our analysis, which can be carried out for arbitrary lattices, is based on an equivalence of the vertex model with a dimer problem. The dimer problem is, in turn, solved using the method of Pfaffians. It is found that the system is frozen below a critical temperatureT _cwith the critical exponent = (3 –d)/2.     

Theory of monomer-dimer systems

We investigate the general monomer-dimer partition function,P(x), which is a polynomial in the monomer activity,x, with coefficients depending on the dimer activities. Our main result is thatP(x) has its zeros on the imaginary axis when the dimer activities are nonnegative. Therefore, no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomer density is minimal (i.e.x=0). Elaborating on this theme we prove the existence and analyticity of correlation functions (away fromx=0) in the thermodynamic limit. Among other things we obtain bounds on the compressibility and derive a new variable in which to make an expansion of the free energy that converges down to the minimal monomer density. We also relate the monomer-dimer problem to the Heisenberg and Ising models of a magnet and derive Christoffell-Darboux formulas for the monomer-dimer and Ising model partition functions. This casts the Ising model in a new light and provides an alternative proof of the Lee-Yang circle theorem. We also derive joint complex analyticity domains in the monomer and dimer activities. Our considerations are independent of geometry and hence are valid for any dimensionality.Work supported by National Science Foundation Grant GP-26526.     

Mercury excimer processes in resonant optically pumped vapour

Absorption measurements are reported at 570 nm in optically excited mercurynitrogen mixtures at low mercury densities such as are found in atomic mercury lasers. The absorbing species is identified as the Hg₂(O _g ) dimer, and a model for the formation of steady-state dimer and trimer populations, which gives good agreement with the measurements, is described. The absorption gives rise to a loss mechanism in the mercury laser not previously considered, which depends on the intensity of the uv pumping.     

Nature of the eigenstates in the miniband of random dimer-barrier superlattices

The dc conductance, the universal quantum fluctuations and the resistance distribution are numerically investigated in dimer semiconductor superlattices by means of the transfer matrix formalism. We are interested in the GaAs/Al_x Ga _1 − xAs layers, having identical thickness, where the aluminium concentration x takes, at random, two different values, with the constraint that one of them appears only in pairs, i.e. the random dimer barrier (RDB). These systems exhibit a miniband of extended states, around a critical energy, lying to the typical structure of the dimer cell. The states close to this resonant energy consist of weakly localized states, while in band tails i.e. for negligible conductance, the states are strongly localized. This is evidence of the suppression of localization in the RDB superlattices. The nature of the transition between these two regimes is quantitatively investigated through relevant physical quantities. The model is, hence, clearly and statistically examined.     

Dimers on the kagome lattice I: Finite lattices

We report exact results on the enumeration of close-packed dimers on a finite kagome lattice with general asymmetric dimer weights under periodic and cylindrical boundary conditions. For symmetric dimer weights, the resulting dimer generating functions reduce to very simple expressions, and we show how the simple expressions can be obtained from the consideration of a spin-variable mapping.     

Immune recognition construct plasmonic dimer for SERS‐based bioassay

We have first time demonstrated the construction of a plasmonic gold dimer model for bioassays based on immune recognition with surface‐enhanced Raman scattering (SERS). To induce a strong plasmonic coupling effect, a dimer of gold nanoparticles (NPs) with a Raman label located between adjacent NPs is assembled through specific recognition in biological systems. One promising application for this model is the provision of a new type of in situ self‐calibrated and reliable SERS platform where biotinylated molecules can selectively be trapped by streptavidin and placed in the gap enhanced plasmonic field, which may enable the development of powerful, biospecific recognition‐based SERS assays. The capabilities of the dimeric constructions for analytical applications were demonstrated through the use of the SERS technique to detect biotin at very low concentrations. Additionally, the spatial SERS radiation for the gold dimer assembled on the silicon slide was simulated using the finite‐difference time‐domain method; this simulation demonstrated the distribution of the electric field as well as the utility of the proposed system, thereby introducing potential uses of bio‐specific recognition as well as opportunities for the construction of plasmonically coupled nanostructures and bioassay applications. Copyright © 2013 John Wiley & Sons, Ltd.     

Relaxation time for a dimer covering with height representation

This paper considers the Monte Carlo dynamics of random dimer coverings of the square lattice, which can be mapped to a rough interface model. Two kinds of slow modes are identified, associated respectively with long-wavelength fluctuations of the interface height, and with slow drift (in time) of the system-wide mean height. Within a continuum theory, the longest relaxation time for either kind of mode scales as the system sizeN. For the real, discrete model, an exactlower bound ofO(N) is placed on the relaxation time, using variational eigenfunctions corresponding to the two kinds of continuum modes     

Factors affecting the dimerization of persistent nitrogen‐centered 1‐phenyl urazole radicals to tetrazanes

1‐Phenyl urazole radicals are persistent air‐stable nitrogen‐centered radicals that engage in an equilibrium with the corresponding N―N tetrazane dimers in solution. While the equilibrium typically weakly favors the dimer form, for some 1‐phenyl urazole radicals bearing substituents at the ortho position of the benzene ring, the equilibrium instead strongly favors the dimer form. With the recent surge of interest in the properties and potential applications of heterocyclic radicals, the factors that affect this equilibrium are important to determine. We examined the effect of the extent of ortho substitution (none, 1, or 2 substituents) on the equilibrium by experimentally using variable temperature ¹H nuclear magnetic resonance and UV‐visible spectroscopy in addition to supporting computational investigations at the (U)B3LYP/6‐311G(d,p) level of theory. We confirmed that the equilibrium generally favored the dimer in all cases. However, the equilibrium was more favorable towards dimer formation for urazole radicals substituted with 1 and 2 ortho substituents on the aromatic ring. The activation enthalpies for dissociation of singly substituted dimers were greater than that for dimers without ortho substituents, but lower than that for doubly substituted dimers. The greater preference for dimer formation for the ortho‐substituted urazole radicals is attributed to a greater enthalpic advantage for N―N bond formation. This advantage may be traced to a higher concentration of spin density on the urazole unit of the radicals and a lesser deformation energy required for N―N bond formation.     

Role of molecular interactions and end chain length on the photosensitivity of liquid crystalline alkyl cyanobiphenyl dimers – UV absorption-based approach through DFT calculations

In this paper, the photosensitivity of liquid crystalline alkyl cyanobiphenyl (nCB: n = 6, 7; n is the number of carbon atoms in the alkyl chain) dimers has been presented through density functional theory (DFT) calculations. The nCB structures have been optimized using the Becke, three-parameter, Lee–Yang–Parr hybrid functional with the 6-31+G (d) basis set using the crystallographic geometry as input. The electronic structures of the dimer molecules have been computed using the optimized geometries. The spectra of the dimer molecules have been calculated by employing the DFT method. The excited states have been calculated via configuration interaction singles with the semi-empirical Hamiltonian Zerner intermediate neglect of differential overlap. The influence of molecular interactions and the end chain length on ultraviolet absorption spectral characteristics and the photosensitivity of the compounds has been discussed. These results offer a hint for the protection of various optical devices from the intense light induced damages, and to model photosensitivity.     

Solid-state photodimerization of 9-methylanthracene as studied by solid-state NMR

The locus of the photodimerization reaction of 9-methylanthracene in the crystal was examined by high-resolution solid-state NMR techniques. Examination of the spectra of the products showed that only the trans dimer is formed by the solid state photodimerization, while both trans and cis dimers are formed by the photodimerization in benzene solution. The T₁ relaxation curves for the monomer and the dimer at various exposure times were separately observed via well-resolved peaks. The two T₁ curves exhibit characteristic features for relaxation of a weakly coupled two-spin system. By analyzing the T₁ curves, the spin diffusion rates between the monomer and the dimer and the fraction of the dimer were obtained for various exposure times. From the result, the maximum domain size of the minor component during the photodimerization process was estimated to be ca. 0.3 μm. The heterogeneous domain structure generated by the photodimerization indicates that the reaction takes place at defects of the crystal in the monomer.     

High-Order Coupled Cluster Method (CCM) Calculations for Quantum Magnets with Valence-Bond Ground States

In this article, we prove that exact representations of dimer and plaquette valence-bond ket ground states for quantum Heisenberg antiferromagnets may be formed via the usual coupled cluster method (CCM) from independent-spin product (e.g. Néel) model states. We show that we are able to provide good results for both the ground-state energy and the sublattice magnetization for dimer and plaquette valence-bond phases within the CCM. As a first example, we investigate the spin-half J ₁–J ₂ model for the linear chain, and we show that we are able to reproduce exactly the dimerized ground (ket) state at J ₂/J ₁=0.5. The dimerized phase is stable over a range of values for J ₂/J ₁ around 0.5, and results for the ground-state energies are in good agreement with the results of exact diagonalizations of finite-length chains in this regime. We present evidence of symmetry breaking by considering the ket- and bra-state correlation coefficients as a function of J ₂/J ₁. A radical change is also observed in the behavior of the CCM sublattice magnetization as we enter the dimerized phase. We then consider the Shastry-Sutherland model and demonstrate that the CCM can span the correct ground states in both the Néel and the dimerized phases. Once again, very good results for the ground-state energies are obtained. We find CCM critical points of the bra-state equations that are in agreement with the known phase transition point for this model. The results for the sublattice magnetization remain near to the “true” value of zero over much of the dimerized regime, although they diverge exactly at the critical point. Finally, we consider a spin-half system with nearest-neighbor bonds for an underlying lattice corresponding to the magnetic material CaV₄O₉ (CAVO). We show that we are able to provide excellent results for the ground-state energy in each of the plaquette-ordered, Néel-ordered, and dimerized regimes of this model. The exact plaquette and dimer ground states are reproduced by the CCM ket state in their relevant limits. Furthermore, we estimate the range over which the Néel order is stable, and we find the CCM result is in reasonable agreement with the results obtained by other methods. Our new approach has the dual advantages that it is simple to implement and that existing CCM codes for independent-spin product model states may be used from the outset. Furthermore, it also greatly extends the range of applicability to which the CCM may be applied. We believe that the CCM now provides an excellent choice of method for the study of systems with valence-bond quantum ground states.     

Local Statistics of Realizable Vertex Models

We study planar “vertex” models, which are probability measures on edge subsets of a planar graph, satisfying certain constraints at each vertex, examples including the dimer model, and 1-2 model, which we will define. We express the local statistics of a large class of vertex models on a finite hexagonal lattice as a linear combination of the local statistics of dimers on the corresponding Fisher graph, with the help of a generalized holographic algorithm. Using an n × n torus to approximate the periodic infinite graph, we give an explicit integral formula for the free energy and local statistics for configurations of the vertex model on an infinite bi-periodic graph. As an example, we simulate the 1-2 model by the technique of Glauber dynamics.     

IR absorption spectrum (4200–3100 cm) of H2O and (H2O)2 in CCl4. Estimates of the equilibrium constant and evidence that the atmospheric water absorption continuum is due to the water dimer

IR absorption spectra, 4200–3100 cm⁻¹, of water in CCl₄ solutions are presented. It is shown that for saturated solutions significant amounts of water are present as dimer (ca. 2%). The IR spectra of the monomer and dimer are retrieved. The integrated absorption coefficients of the monomer absorption are significantly enhanced relative to the gas phase values. The dimer spectrum consists of 5 bands, of which 4 were expected from data from cold beams and cold matrices. The origin of the “extra” band is discussed. In addition it is argued that the dimer absorption bands intensities must be enhanced relative to the gas phase values. Based on recent calculations of band strengths, and observed frequency shifts relative to the gas phase, the intensity enhancement factors are estimated as well as the monomer/dimer equilibrium constant in CCl₄ solution at T=296 K (K_c=1.29 mol⁻¹ L). It is noted that the observed dimer spectrum has a striking resemblance with the water vapour continuum determined by Burch in 1985 which was recently remeasured by Paynter et al. and it is concluded that the atmospheric water absorption continuum in the investigated spectral region must be due to water dimer. Based on the newly published spectral data a revised value of the gas phase equilibrium constant is suggested (K_p=0.035 atm⁻¹ at T=296 K) as well as a value for the standard enthalpy of formation, ΔH⁰=15.4 kJ mol⁻¹.     

“Hot spots” induced near-field enhancements in Au nanoshell and?Au nanoshell dimer

The influence of “hot spots” on the near-field properties of Au nanoshell and Au nanoshell dimers have been investigated by means of the finite element method. It is found with increasing the pinhole radius R that the maximal enhancement of near-field for Au nanoshell with pinhole parallel to the polarization increases from 17.906 at R=0 nm to 36.979 at R=0.8 nm, and then almost shows a negligible radius dependence. Large electric fields also can be observed inside the pinhole perpendicular to the polarization, which increases with increasing the pinhole radius. The near-field of Au nanoshell dimer depends strongly on the polarization and propagation directions of the incident light. Exponential decay behavior is found for the maximal enhancement of the electric field in the dimer junction as a function of the dimer separation. Furthermore, a very strong electric field is found in the junction between two Au nanoshells when the pinholes are located near the gap between the nanoshells.     

Single Crystal Magnetic Resonance Study of a Rh2☎ Dimer Centre in NaCl at Q-Band

Electron nuclear double resonance spectroscopy at 34 GHz is applied to determine the signs of the principal values and the directions of the corresponding principal axes of the zero field splitting tensor for a Rh^2? dimer centre in NaC1. The results are in good agreement with the hypotheses concerning the spin Hamiltonian parameters deduced from the X- and W-band EPR analysis and support the model proposed for this centre consisting of two Rh^2? ions on cation positions and two Na^? vacancies.     

A Bijection Theorem for Domino Tilings with Diagonal Impurities

We consider the dimer problem on a planar non-bipartite graph G, where there are two types of dimers one of which we regard as impurities. Computer simulations reveal a reminiscence of the Cheerios effect, that is, impurities are attracted to the boundary, which is the motivation to study this particular graph. Our main theorem is a variant of the Temperley bijection: a bijection between the set of dimer coverings and the set of spanning forests with certain conditions. We further discuss some implications of this theorem: (1) the local move connectedness yielding an ergodic Markov chain on the set of all possible dimer coverings, and (2) a rough bound for the number of dimer coverings and that for the probability of finding an impurity at a given edge, which is an extension of a result in (Nakano and Sadahiro in ).     

Classical Bifurcation Diagrams by Quantum Means

Asymptotic state of an open quantum system may change substantially upon variations of system parameters. These changes can be often identified as bifurcation transitions in the classical mean‐filed equations describing evolution of the relevant observables. We demonstrate that these bifurcations can be made visible as changes in the structure of the asymptotic density matrix of the original quantum system. By using an N‐boson dimer, we present quantum ‘bifurcation diagrams’ for the pitchfork and saddle‐node bifurcations (in the stationary dimer) and visualize a period‐doubling transition to chaos (in a periodically modulated dimer).     

The origin of inter-dimer-row correlated adsorption for NH3 on Si(0 0 1)

We discuss the interaction between adsorbing ammonia molecules and pre-adsorbed ammonia fragments on the Si(0 0 1) surface, searching for experimental evidence of a H-bonded precursor state predicted by modelling. While correlations along dimer rows have already been identified, these mix substrate-mediated effects due to dimer buckling with ammonia–adsorbate effects. Correlations between fragments on neighbouring dimer rows are not affected by substrate effects (in this system), allowing an analysis of direct ammonia–adsorbate effects. We present an analysis of cross-row correlations in existing high-coverage STM data which shows significant correlations between NH₂ groups on neighbouring dimer rows over a significant range, providing evidence for the H-bonded precursor state with a range of around 10 Å. We discuss implications for the interpretation of STM images of ammonia on Si(0 0 1).