Classical Dimer Research Articles

First-Order versus Unconventional Phase Transitions in Three-Dimensional Dimer Models

We study the phase transition between the Coulomb liquid and the columnar crystal in the 3D classical dimer model, which was found to be continuous in the O(3) universality class. In addition to nearest-neighbor interactions which favor parallel dimers, further neighbor interactions are allowed in such a manner that the cubic symmetry of the original system remains intact. We show that the transition in the presence of weak additional, symmetry preserving interactions is first order. However, the universality class of the transition remains continuous when the additional interactions are weakly repulsive. In this way, we verify the existence of a multicritical point near the unperturbed transition, and we identify a critical line of unconventional transitions between the Coulomb liquid phase and the sixfold columnar phase.

Tubulin-Bound GTP Can not Pump Microtubule Coherence in Stable Microtubules: Towards a Revision of Microtubule Based Quantum Models of Mind

Neurons have a large fraction of stable brain microtubules forming the cytoskeleton of the cell, which provide a mechanical support for the extended dendrites and axonal arborizations and serve as railroads for molecular and vesicular transport. Except for the latter two functions it has been hypothesized that these stable microtubules might also act as quantum or classical computers, the function of which is based on electron hopping associated with kinking of the tubulin α/β-dimer. Hameroff, Tuszynski and others have supposed that the energy needed for such computation could be somehow delivered via cycles of tubulin bound GTP hydrolysis with subsequent GDP exchange for GTP. Here we review the microtubule biophysics and present structural data explaining why the proposed tubulin-bound GTP energized classical or quantum tubulin dimer computation is a fiction and cannot occur in stable microtubules. In addition, we point a flaw in Sataric-Tuszynski ferroelectric microtubule model and show a physical inconsistency in Hameroff-Penrose Orch OR based on the fact that the energy released from a single GTP molecule is 10 13 times greater compared to the gravitational energy needed to collapse the relevant number of tubulins for 25 ms.

Classical to quantum mapping for an unconventional phase transition in a three-dimensional classical dimer model

We study the transition between a Coulomb phase and a dimer crystal observed in numerical simulations of the three-dimensional classical dimer model, by mapping it to a quantum model of bosons in two dimensions. The quantum phase transition that results, from a superfluid to a Mott insulator at fractional filling, belongs to a class that cannot be described within the Landau-Ginzburg-Wilson paradigm. Using a second mapping, to a dual model of vortices, we show that the long-wavelength physics near the transition is described by a U(1) gauge theory with SU(2) matter fields.

Classical dimer model with anisotropic interactions on the square lattice

We discuss phase transitions and the phase diagram of a classical dimer model with anisotropic interactions defined on a square lattice. For the attractive region, the perturbation of the orientational order parameter introduced by the anisotropy causes the Berezinskii-Kosterlitz-Thouless transitions from a dimer-liquid to columnar phases. According to the discussion by Nomura and Okamoto for a quantum-spin chain system [J. Phys. A 27, 5773 (1994)], we proffer criteria to determine transition points and also universal level-splitting conditions. Subsequently, we perform numerical diagonalization calculations of the nonsymmetric real transfer matrices up to linear dimension specified by L=20 and determine the global phase diagram. For the repulsive region, we find the boundary between the dimer-liquid and the strong repulsion phases. Based on the dispersion relation of the one-string motion, which exhibits a twofold "zero-energy flat band" in the strong repulsion limit, we give an intuitive account for the property of the strong repulsion phase.

Coulomb gas transitions in three-dimensional classical dimer models

Close-packed, classical dimer models on three-dimensional, bipartite lattices harbor a Coulomb phase with power-law correlations at infinite temperature. Here, we discuss the nature of the thermal phase transition out of this Coulomb phase for a variety of dimer models which energetically favor crystalline dimer states with columnar ordering. For a family of these models, we find a direct thermal transition from the Coulomb phase to the dimer crystal. While some systems exhibit (strong) first-order transitions in correspondence with the Landau-Ginzburg-Wilson paradigm, we also find clear numerical evidence for continuous transitions. A second family of models undergoes two consecutive thermal transitions with an intermediate paramagnetic phase separating the Coulomb phase from the dimer crystal. We can describe all of these phase transitions in one unifying framework of candidate field theories with two complex Ginzburg-Landau fields coupled to a $U(1)$ gauge field. We derive the symmetry-mandated Ginzburg-Landau actions in these field variables for the various dimer models and discuss implications for their respective phase transitions.

Zero Temperature Phase Diagram of the Kitaev Model

We show that the zero-temperature phase diagram of the vortex free sector of the Kitaev model is in one-to-one correspondence with that of the classical dimer model on the same lattice. We find that the model generically has three distinct phases. On a honeycomb lattice with a 3 x 3 fundamental domain all three phases are accessible. As the couplings are varied, there are two distinct transitions. The new transition is one to a gapped phase that opens up in the interior of the B phase.

Gauge Theory Picture of an Ordering Transition in a Dimer Model

We study a phase transition in a 3D lattice gauge theory, a "coarse-grained" version of a classical dimer model. Duality arguments indicate that the dimer lattice theory should be dual to a XY model coupled to a gauge field with geometric frustration. The transition between a Coulomb phase with dipolar correlations and a long range ordered columnar phase is understood in terms of a Higgs mechanism. Monte Carlo simulations of the dual model indicate a continuous transition with exponents close but apparently different from those of the 3D XY model. The continuous nature of the transition is confirmed by a flowgram analysis.

SU(2)-Invariant Continuum Theory for an Unconventional Phase Transition in a Three-Dimensional Classical Dimer Model

We derive a continuum theory for the phase transition in a classical dimer model on the cubic lattice, observed in recent Monte Carlo simulations. Our derivation relies on the mapping from a three-dimensional classical problem to a two-dimensional quantum problem, by which the dimer model is related to a model of hard-core bosons on the kagome lattice. The dimer-ordering transition becomes a superfluid-Mott insulator quantum phase transition at fractional filling, described by an SU(2)-invariant continuum theory.

Correlations and order parameter at a Coulomb-crystal phase transition in a three-dimensional dimer model

The three-dimensional classical dimer model with interactions shows an unexpected continuous phase transition between an ordered dimer crystal and a Coulomb liquid. A detailed analysis of the critical dimer and monomer correlation functions points to a subtle interplay between the fluctuations of the crystal order parameter and the ``magnetic'' degrees of freedom present in the Coulomb phase. The distribution probability of the crystal order parameter suggests an emerging continuous $O(3)$ symmetry at the critical point.

Synthesis, Structural and Photophysical Evaluations of Urea Based Fluorescent PET Sensors for Anions

The design, synthesis and photophysical evaluation of four anthracene based photoinduced electron transfer (PET) sensors (3a–d) for anions is described. The [4π+4π] photodimerization product, 4, was also obtained from 3b, by slow evaporation from DMSO solution and its X-ray crystal structure determined. The structure of 4 showed the classical dimerization at the 9,10-positions of anthracene. Sensors 3a–d are all based on the use of charge neutral aryl urea receptors, where the recognition of anions such as acetate, phosphate and iodide, was due to the formation of strong hydrogen bonding interactions in DMSO. This anion recognition resulted in enhanced quenching of the anthracene excited state via electron transfer from the receptor; hence the emission was ‘switched on–off’. The sensing of fluoride was, however, found to be a two-step process, which involved initial hydrogen bonding interactions with the receptor, followed by deprotonation and the formation of bifluoride (). The changes in the emission spectra upon sensing of chloride and bromide were, however, minor. The photophysical properties of these sensors in the presence of various anions were further studied, including investigation of their excited state lifetimes and quantum yields, as well as detailed Stern–Volmer kinetic analysis with the aim of determining the dynamic and static quenching constants, k q and k s for the above anion recognition. These measurements indicated that while the anion dependant quenching was mostly by a dynamic process, some contribution from static quenching was also observed at lower anion concentrations.

Classical Dimers and Dimerized Superstructure in an Orbitally Degenerate Honeycomb Antiferromagnet

We discuss the ground state of the spin-orbital model for spin-one ions with partially filled t_{2g} levels on a honeycomb lattice. We find that the orbital degrees of freedom induce a spontaneous dimerization of spins and drive them into nonmagnetic manifold spanned by hard-core dimer (spin-singlet) coverings of the lattice. The cooperative "dimer Jahn-Teller" effect is introduced through a magnetoelastic coupling and is shown to lift the orientational degeneracy of dimers leading to a peculiar valence bond crystal pattern. The present theory provides a theoretical explanation of nonmagnetic dimerized superstructure experimentally seen in Li2RuO3 compound at low temperatures.

Effects of “excessive” exciton interactions in polarized IR spectra of the hydrogen bond in 2-butynoic acid crystals: Proton transfer induced by dynamical co-operative interactions involving hydrogen bonds

In this article, we present the results of our study of polarized IR spectra of the hydrogen bond in crystals of 2-butynoic acid (CH 3C CCOOH) as well as in crystals of its deuterium derivative (CH 3C CCOOD). 2-Butynoic acid can exist in two polymorphous crystalline forms: in the “α” form, based on a classic dimer motif and in the “β” form, based on the catamer pattern of the hydrogen bond arrangement. By cooling the melted substance crystals of the “β” phase were obtained selectively. The polarized IR spectra of the hydrogen bond in the “β” form of 2-butynoic acid crystals were measured at room temperature and at the temperature of liquid nitrogen in the ν O–H and ν O–D band frequency ranges. In terms of the “strong-coupling” theory the fine structure patterns of the ν O–H and ν O–D polarized bands were quantitatively explained along with the dichroic and the H/D isotopic effects in the spectra. To interpret the main properties of the spectra the existence of a non-conventional effect concerning a self-stimulated concerted proton position rearrangements in the neighboring cells in the lattice had to be assumed. On the basis of the spectra of isotopically diluted crystalline samples of 2-butynoic acid it was suggested that a random distribution of protons and deuterons occurred in the open chains of the hydrogen bonded molecules. However, coordination in the mutual arrangement of protons and deuterons in the neighboring hydrogen bonds from the closely spaced molecular chains was found to be non-random. This fact was ascribed to dynamical co-operative interactions, most strongly involving hydrogen bonds from different chains in the modified lattice. These non-conventional interactions were responsible for appearance of the so-called H/D “self-organization” isotopic effects in the spectra.

Dynamical dimer correlations at bipartite and non-bipartite Rokhsar–Kivelsonpoints

We determine the dynamical dimer correlation functions of quantum dimer models at theRokhsar–Kivelson point on the bipartite square and cubic lattices and the non-bipartitetriangular lattice. On the basis of an algorithmic idea by Henley, we simulate astochastic process of classical dimer configurations in continuous time and perform astochastic analytical continuation to obtain the dynamical correlations in momentumspace and the frequency domain. This approach allows us to observe directlythe dispersion relations and the evolution of the spectral intensity within theBrillouin zone beyond the single-mode approximation. On the square lattice,we confirm analytical predictions related to soft modes close to the wavevectors(π,π) and (π,0) and further reveal the existence of shadow bands close to the wavevector(0,0). On the cubic lattice the spectrum is also gapless but here only a single soft mode at(π,π,π) is found, as predicted by the single-mode approximation. The soft mode has a quadraticdispersion at very long wavelength, but crosses over to a linear behavior very rapidly. Webelieve this to be the remnant of the linearly dispersing ‘photon’ of the Coulomb phase.Finally the triangular lattice is in a fully gapped liquid phase where the bottom of thedimer spectrum exhibits a rich structure. At the M point the gap is minimal and thespectral response is dominated by a sharp quasiparticle peak. On the other hand, at the Xpoint the spectral function is much broader. We sketch a possible explanation basedon the crossing of the coherent dimer excitations into the two-vison continuum.

Vacancy localization in the square dimer model

We study the classical dimer model on a square lattice with a single vacancy by developing a graph-theoretic classification of the set of all configurations which extends the spanning tree formulation of close-packed dimers. With this formalism, we can address the question of the possible motion of the vacancy induced by dimer slidings. We find a probability 57/4-10[sqrt2] for the vacancy to be strictly jammed in an infinite system. More generally, the size distribution of the domain accessible to the vacancy is characterized by a power law decay with exponent 9/8. On a finite system, the probability that a vacancy in the bulk can reach the boundary falls off as a power law of the system size with exponent 1/4. The resultant weak localization of vacancies still allows for unbounded diffusion, characterized by a diffusion exponent that we relate to that of diffusion on spanning trees. We also implement numerical simulations of the model with both free and periodic boundary conditions.

Criticality of a classical dimer model on the triangular lattice

We consider a classical interacting dimer model which interpolates between the square lattice case and the triangular lattice case by tuning a chemical potential in the diagonal bonds. The interaction energy simply corresponds to the number of plaquettes with parallel dimers. Using transfer matrix calculations, we find in the anisotropic triangular case a succession of different physical phases as the interaction strength is increased: a short-range disordered liquid dimer phase at low interactions, then a critical phase similar to the one found for the square lattice, and finally a transition to an ordered columnar phase for large interactions. Our results indicate that criticality and nonbipartiteness are compatible in a dimer model. For the isotropic triangular case, we have indications that the system undergoes a first-order phase transition to an ordered phase, without appearance of an intermediate critical phase.

Orphan nuclear receptor NGFI‐B forms dimers with nonclassical interface

The orphan receptor nerve growth factor-induced B (NGFI-B) is a member of the nuclear receptor's subfamily 4A (Nr4a). NGFI-B was shown to be capable of binding both as a monomer to an extended half-site containing a single AAAGGTCA motif and also as a homodimer to a widely separated everted repeat, as opposed to a large number of nuclear receptors that recognize and bind specific DNA sequences predominantly as homo- and/or heterodimers. To unveil the structural organization of NGFI-B in solution, we determined the quaternary structure of the NGFI-B LBD by a combination of ab initio procedures from small-angle X-ray scattering (SAXS) data and hydrogen-deuterium exchange followed by mass spectrometry. Here we report that the protein forms dimers in solution with a radius of gyration of 2.9 nm and maximum dimension of 9.0 nm. We also show that the NGFI-B LBD dimer is V-shaped, with the opening angle significantly larger than that of classical dimer's exemplified by estrogen receptor (ER) or retinoid X receptor (RXR). Surprisingly, NGFI-B dimers formation does not occur via the classical nuclear receptor dimerization interface exemplified by ER and RXR, but instead, involves an extended surface area composed of the loop between helices 3 and 4 and C-terminal fraction of the helix 3. Remarkably, the NGFI-B dimer interface is similar to the dimerization interface earlier revealed for glucocorticoid nuclear receptor (GR), which might be relevant to the recognition of cognate DNA response elements by NGFI-B and to antagonism of NGFI-B-dependent transcription exercised by GR in cells.

Asymptotics of Block Toeplitz Determinants and the Classical Dimer Model

We compute the asymptotics of a block Toeplitz determinant which arises in the classical dimer model for the triangular lattice when considering the monomer-monomer correlation function. The model depends on a parameter interpolating between the square lattice (t = 0) and the triangular lattice (t = 1), and we obtain the asymptotics for 0 < t ≤ 1. For 0 < t < 1 we apply the Szegö Limit Theorem for block Toeplitz determinants. The main difficulty is to evaluate the constant term in the asymptotics, which is generally given only in a rather abstract form.

Condensates of strongly interacting atoms and dynamically generated dimers

In a system of atoms with large positive scattering length, weakly-bound diatomic molecules (dimers) are generated dynamically by the strong interactions between the atoms. If the atoms are modeled by a quantum field theory with an atom field only, condensates of dimers cannot be described by the mean-field approximation because there is no field associated with the dimers. We develop a method for describing dimer condensates in such a model based on the one-particle-irreducible (1PI) effective action. We construct an equivalent 1PI effective action that depends not only on the classical atom field but also on a classical dimer field. The method is illustrated by applying it to the many-body behavior of bosonic atoms with large scattering length at zero temperature using an approximation in which the 2-atom amplitude is treated exactly but irreducible $N$-atom amplitudes for $N \ge 3$ are neglected. The two 1PI effective actions give identical results for the atom superfluid phase, but the one with a classical dimer field is much more convenient for describing the dimer superfluid phase. The results are also compared with previous work on the Bose gas near a Feshbach resonance.

Quantum surface diffusion of vibrationally excited molecular dimers

We consider the thermally activated quantum diffusion of a molecular dimer in a periodic surface potential by means of a time-dependent wave packet method. We show that the potential energy surface resulting from the interplay of intradimer and dimer-surface interactions can lead to resonant states and predict high tunneling probabilities at specific, below barrier, energies that depend also on the initial vibrational state of the dimer. For soft molecular bonds, we show that the chaotic dynamical regime of classical dimers is mirrored, in the quantum case, by the tunneling induced mixing of vibrational states. The knowledge of the transmission coefficient is used to formulate an approximate description of quantum thermal diffusion by defining an effective temperature-dependent activation energy that can be compared to the classical case.

Classical dimers with aligning interactions on the square lattice

We present a detailed study of a model of close-packed dimers on the square lattice with an interaction between nearest-neighbor dimers. The interaction favors parallel alignment of dimers, resulting in a low-temperature crystalline phase. With large-scale Monte Carlo and transfer matrix calculations, we show that the crystal melts through a Kosterlitz-Thouless phase transition to give rise to a high-temperature critical phase, with algebraic decays of correlations functions with exponents that vary continuously with the temperature. We give a theoretical interpretation of these results by mapping the model to a Coulomb gas, whose coupling constant and associated exponents are calculated numerically with high precision. Introducing monomers is a marginal perturbation at the Kosterlitz-Thouless transition and gives rise to another critical line. We study this line numerically, showing that it is in the Ashkin-Teller universality class, and terminates in a tricritical point at finite temperature and monomer fugacity. In the course of this work, we also derive analytic results relevant to the noninteracting case of dimer coverings, including a Bethe ansatz (at the free fermion point) analysis, a detailed discussion of the effective height model, and a free field analysis of height fluctuations.