Classical Dimer Research Articles

Exact solution to the quantum and classical dimer models on the spectre aperiodic monotiling

The decades-long search for a shape that tiles the plane only aperiodically under translations and rotations recently ended with the discovery of the “spectre” aperiodic monotile. In this setting we study the dimer model, in which dimers are placed along tile edges such that each vertex meets precisely one dimer. The complexity of the tiling combines with the dimer constraint to allow an exact solution to the model. The partition function is Z=2Nmystic+1 where Nmystic is the number of “mystic” tiles. We exactly solve the quantum dimer (Rokhsar-Kivelson) model in the same setting by identifying an eigenbasis at all interaction strengths V/t. We find that test monomers, once created, can be infinitely separated at zero energy cost for all V/t, constituting a deconfined phase in a (2+1)-dimensional bipartite quantum dimer model. Published by the American Physical Society 2024

Derivation of field theory for the classical dimer model using bosonization.

We derive a field theory for the two-dimensional classical dimer model by applying bosonization to Lieb's (fermionic) transfer-matrix solution. Our constructive approach gives results that are consistent with the well-known height theory, previously justified based on symmetry considerations, but also fixes coefficients appearing in the effective theory and the relationship between microscopic observables and operators in the field theory. In addition, we show how interactions can be included in the field theory perturbatively, treating the case of the double dimer model with interactions within and between the two replicas. Using a renormalization-group analysis, we determine the shape of the phase boundary near the noninteracting point, in agreement with results of Monte Carlo simulations.

ТЕОРЕТИЧЕСКОЕ ИССЛЕДОВАНИЕ АССОЦИАЦИИ АРИЛОВЫХ ПРОИЗВОДНЫХ МОЛОЧНОЙ КИСЛОТЫ

A theoretical study of the association of phenyl and ortho-substituted aryl derivatives of lactic acid was carried out. Two variants of hydrogen-bonded associates in the gas phase were calculated: non-classical, actually found in the crystals, and simulated classical dimers. The energy advantage of classical dimers and the non-equivalence of diastereotopic electron lone pairs at the carbonyl oxygen atom were shown.

Reentrance effect in the high-temperature critical phase of the quantum dimer model on the square lattice

We present a quantum Monte Carlo investigation of the finite-temperature phase diagram of the quantum dimer model on the square lattice. We use the sweeping cluster algorithm, which allows to implement exactly the dimer constraint, supplemented with a equal-time directed loop move that allows to sample winding sectors. We find a high-temperature critical phase with power-law correlations that extend down to the Rokshar-Kivelson point, in the vicinity of which a re-entrance effect in the lines of constant exponent is found. For small values of the kinetic energy strength, we find finite-temperature transitions to ordered states (columnar and staggered) which match those of interacting classical dimer models.

Statistical mechanics of dimers on quasiperiodic Ammann-Beenker tilings

We study classical dimers on two-dimensional quasiperiodic Ammann-Beenker (AB) tilings. Using the discrete scale-symmetry of quasiperiodic tilings, we prove that each infinite tiling admits ``perfect matchings'', where every vertex is touched by one dimer. We show the appearance of so-called monomer pseudomembranes. These are sets of edges, which collectively host exactly one dimer, which bound certain eightfold-symmetric regions of the tiling. Regions bounded by pseudomembranes are matched together in a way that resembles perfect matchings of the tiling itself. These structures emerge at all scales, suggesting the preservation of collective dimer fluctuations over long distances. We provide numerical evidence, via Monte Carlo simulations, of dimer correlations consistent with power laws over a hierarchy of different lengthscales. We also find evidence of rich monomer correlations, with monomers displaying a pattern of attraction and repulsion to different regions within pseudomembranes, along with signatures of deconfinement within certain annular regions of the tiling.

Coupled dimer and bond-order-wave order in the quarter-filled one-dimensional Kondo lattice model

Motivated by the experiments on the organic compound $(Per)_{2}[Pt(mnt)_{2}]$, we study the ground state of the one-dimensional Kondo lattice model at quarter filling with the density matrix renormalization group method. We show a coupled dimer and bond-order-wave (BOW) state in the weak coupling regime for the localized spins and itinerant electrons, respectively. The quantum phase transitions for the dimer and the BOW orders occur at the same critical coupling parameter $J_{c}$, with the opening of a charge gap. The emergence of the combination of dimer and BOW order agrees with the experimental findings of the simultaneous Peierls and spin-Peierls transitions at low temperatures, which provides a theoretical understanding of such phase transition. We also show that the localized spins in this insulating state have quasi-long ranged spin correlations with collinear configurations, which resemble the classical dimer order in the absence of a magnetic order.

Interacting double dimer model on the square lattice

We study phases and transitions of the square-lattice double dimer model, consisting of two coupled replicas of the classical dimer model. As on the cubic lattice, we find a thermal phase transition from the Coulomb phase, a disordered but correlated dimer liquid, to a phase where fluctuations of the two replicas are closely synchronized with one another. Surprisingly, and in contrast to the cubic case, the phase boundary includes the noninteracting point, as we establish using a symmetry-based analysis of an effective height theory, indicating that infinitesimal coupling is sufficient to synchronize the double dimer model. In addition, we observe a novel antisynchronized phase when the coupling between replicas is repulsive, and use Monte Carlo simulations to establish the full phase diagram, including (anti)synchronized columnar and staggered phases, with interactions between parallel dimers in each replica.

A deep learning functional estimator of optimal dynamics for sampling large deviations

In stochastic systems, numerically sampling the relevant trajectories for the estimation of the large deviation statistics of time-extensive observables requires overcoming their exponential (in space and time) scarcity. The optimal way to access these rare events is by means of an auxiliary dynamics obtained from the original one through the so-called ‘generalised Doob transformation’. While this optimal dynamics is guaranteed to exist its use is often impractical, as to define it requires the often impossible task of diagonalising a (tilted) dynamical generator. While approximate schemes have been devised to overcome this issue they are difficult to automate as they tend to require knowledge of the systems under study. Here we address this problem from the perspective of deep learning. We devise an iterative semi-supervised learning scheme which converges to the optimal or Doob dynamics with the clear advantage of requiring no prior knowledge of the system. We test our method in a paradigmatic statistical mechanics model with non-trivial dynamical fluctuations, the fully packed classical dimer model on the square lattice, showing that it compares favourably with more traditional approaches. We discuss broader implications of our results for the study of rare dynamical trajectories.

Classical Dimers on Penrose Tilings

We study the classical dimer model on rhombic Penrose tilings, whose edges and vertices may be identified with those of a bipartite graph. We find that Penrose tilings do not admit perfect matchings (defect-free dimer coverings). Instead, their maximum matchings have a monomer density of $81-50\varphi\approx 0.098$ in the thermodynamic limit, with $\varphi=\left(1+\sqrt{5}\right)/2$ the golden ratio. Maximum matchings divide the tiling into a fractal of nested closed regions bounded by loops that cannot be crossed by monomers. These loops connect second-nearest neighbour even-valence vertices, each of which lies on such a loop. Assigning a charge to each monomer with a sign fixed by its bipartite sublattice, we find that each bounded region has an excess of one charge, and a corresponding set of monomers, with adjacent regions having opposite net charge. The infinite tiling is charge neutral. We devise a simple algorithm for generating maximum matchings, and demonstrate that maximum matchings form a connected manifold under local monomer-dimer rearrangements. We show that dart-kite Penrose tilings feature an imbalance of charge between bipartite sub-lattices, leading to a minimum monomer density of $\left(7-4\varphi\right)/5\approx 0.106$ all of one charge.

Assaying Homodimers of NF-κB in Live Single Cells.

NF-κB is a family of heterodimers and homodimers which are generated from subunits encoded by five genes. The predominant classical dimer RelA:p50 is presumed to operate as “NF-κB” in many contexts. However, there are several other dimer species which exist and may even be more functionally relevant in specific cell types. Accurate characterization of stimulus-specific and tissue-specific dimer repertoires is fundamentally important for understanding the downstream gene regulation by NF-κB proteins. In vitro assays such as immunoprecipitation have been widely used to analyze subunit composition, but these methods do not provide information about dimerization status within the natural intracellular environment of intact live cells. Here we apply a live single cell microscopy technique termed Number and Brightness to examine dimers translocating to the nucleus in fibroblasts after pro-inflammatory stimulation. This quantitative assay suggests that RelA:RelA homodimers are more prevalent than might be expected. We also found that the relative proportion of RelA:RelA homodimers can be perturbed by small molecule inhibitors known to disrupt the NF-κB pathway. Our findings show that Number and Brightness is a useful method for investigating NF-κB dimer species in live cells. This approach may help identify the relevant targets in pathophysiological contexts where the dimer specificity of NF-κB intervention is desired.

Emergent Haldane phase in an alternating-bond Z3 parafermion chain

The Haldane phase represents one of the most important symmetry-protected states in modern physics. This state can be realized using spin-1 and $\mathrm{spin}\ensuremath{-}\frac{1}{2}$ Heisenberg models and bosonic particles. Here we explore the emergent Haldane phase in an alternating bond ${\mathbb{Z}}_{3}$ parafermion chain, which is different from the previous proposals from fundamental statistics and symmetries. We show that this emergent phase can also be characterized by a modified long-range string order, as well as fourfold degeneracy in the ground-state energies and entanglement spectra. This phase is protected by both the charge conjugate and parity symmetry, and the edge modes are shown to satisfy parafermionic statistics in which braiding of the two edge modes yields a $\frac{2\ensuremath{\pi}}{3}$ phase. This model also supports rich phases, including a topological ferromagnetic parafermion (FP) phase, trivial paramagnetic parafermion phase, classical dimer phase, and gapless phase. The boundaries of the FP phase are shown to be gapless and critical with central charge $c=4/5$. Even in the topological FP phase, it is also characterized by long-range string order; thus we observe a drop of string order across the phase boundary between the FP phase and the Haldane phase. This work opens a new way for finding of exotic topological phases with parafermions.

Crystal structure of CagV, the Helicobacter pylori homologue of the T4SS protein VirB8.

The VirB/D type IV secretion system (T4SS) plays an essential role in materials transport between host cells and pathogenic Helicobacter pylori and is considered the major pathogenic mediator of H.pylori-associated gastric disease. VirB8, an inner membrane protein that interacts with many other proteins, is a crucial component for secretory function. Here, we present a crystal structure of the periplasmic domain of CagV, the VirB8 counterpart in the H.pylori Cag-T4SS. The structure reveals a fold similar to that of other VirB8 members except for the absence of the α5 helix, a discontinuous β1 strand, a larger angle between the α2 and α3 helices, a more hydrophobic surface groove, but exhibits a different dimer interface. Whether the dimerization occurs in solution was proved by mutagenesis, size-exclusion chromatography and cross-linking assays. Unlike the classical dimerization mode, the interface of the CagV dimer is principally formed by several hydrogen bonds, which indicates instability of dimerization. The structure here demonstrates the difference in dimerization among VirB8 homologues and indicates the considerable compositional and functional diversity of them in T4SS. DATABASE: Coordinates and structure factors have been deposited in the Protein Data Bank under accession codes 6IQT.

Using the Predicted Structure of the Amot Coiled Coil Homology Domain to Understand Lipid Binding.

Angiomotins (Amots) are a family of adapter proteins that modulate cellular polarity, differentiation, proliferation, and migration. Amot family members have a characteristic lipid-binding domain, the coiled coil homology (ACCH) domain that selectively targets the protein to membranes, which has been directly linked to its regulatory role in the cell. Several spot blot assays were used to validate the regions of the domain that participate in its membrane association, deformation, and vesicle fusion activity, which indicated the need for a structure to define the mechanism. Therefore, we sought to understand the structure-function relationship of this domain in order to find ways to modulate these signaling pathways. After many failed attempts to crystallize the ACCH domain of each Amot family member for structural analysis, we decided to pursue homologous models that could be refined using small angle x-ray scattering data. Theoretical models were produced using the homology software SWISS-MODEL and threading software I-TASSER and LOMETS, followed by comparison to SAXS data for model selection and refinement. We present a theoretical model of the domain that is driven by alpha helices and short random coil regions. These alpha helical regions form a classic dimer interface followed by two wide spread legs that we predict to be the lipid binding interface.

Phases of quantum dimers from ensembles of classical stochastic trajectories

We study the connection between the phase behaviour of quantum dimers and the dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK) point a quantum dimer Hamiltonian is equivalent to the Markov generator of the dynamics of classical dimers. A less well understood fact is that away from the RK point the quantum-classical connection persists: in this case the Hamiltonian corresponds to a non-stochastic "tilted" operator that encodes the statistics of time-integrated observables of the classical stochastic problem. This implies a direct relation between the phase behaviour of quantum dimers and properties of ensembles of stochastic trajectories of classical dimers. We make these ideas concrete by studying fully packed dimers on the square lattice. Using transition path sampling - supplemented by trajectory umbrella sampling - we obtain the large deviation statistics of dynamical activity in the classical problem, and show the correspondence between the phase behaviour of the classical and quantum systems. The transition at the RK point between quantum phases of distinct order corresponds, in the classical case, to a trajectory phase transition between active and inactive dynamical phases. Furthermore, from the structure of stochastic trajectories in the active dynamical phase we infer that the ground state of quantum dimers has columnar order to one side of the RK point. We discuss how these results relate to those from quantum Monte Carlo, and how our approach may generalise to other problems.

Exact Solution of the Classical Dimer Model on a Triangular Lattice: Monomer–Monomer Correlations

We obtain an asymptotic formula, as $n\to\infty$, for the monomer-monomer correlation function $K_2(x,y)$ in the classical dimer model on a triangular lattice, with the horizontal and vertical weights $w_h=w_v=1$ and the diagonal weight $w_d=t>0$, where $x$ and $y$ are sites $n$ spaces apart in adjacent rows. We find that $t_c=\frac{1}{2}$ is a critical value of $t$. We prove that in the subcritical case, $0<t<\frac{1}{2}$, as $n\to\infty$, $K_2(x,y)=K_2(\infty)\left[1-\frac{e^{-n/\xi}}{n}\,\Big(C_1+C_2(-1)^n+\mathcal O(n^{-1})\Big)\right]$, with explicit formulae for $K_2(\infty)$, $\xi$, $C_1$, and $C_2$. In the supercritical case, $\frac{1}{2} < t < 1$, we prove that as $n\to\infty$, $K_2(x,y)=K_2(\infty)\Bigg[1- \frac{e^{-n/\xi}}{n}\, \Big(C_1\cos(\omega n+\varphi_1)+C_2(-1)^n\cos(\omega n+\varphi_2)+ C_3+C_4(-1)^n$ $+\mathcal O(n^{-1})\Big)\Bigg]$, with explicit formulae for $K_2(\infty)$, $\xi$, $\omega$, and $C_1$, $C_2$, $C_3$, $C_4$, $\varphi_1$, $\varphi_2$. The proof is based on an extension of the Borodin-Okounkov-Case-Geronimo formula to block Toeplitz determinants and on an asymptotic analysis of the Fredholm determinants in hand.

A Predicted Structure of the Angiomotin Lipid Binding Domain

Amots are a family of adapter proteins that modulate cellular polarity, differentiation, proliferation, and migration. Amot family members also have a characteristic lipid-binding domain, the coiled coil homology (ACCH) Domain that selectively targets the protein to membranes, which has been directly linked to its regulatory role in the cell. Therefore, we endeavored to understand the structure-function relationship of this domain with the desire to find ways to modulate these signaling pathways. After many failed attempts to crystallize the ACCH domain of each of the Amot family members for structural analysis, we decided to pursue homologous models that could be refined using small angle x-ray scattering data. Theoretical models were produced using the Zhang suite programs I-TASSER and LOMETS and then refined and analyzed using Coot and PyMol modeling software based on RMSD, C-score, TM-score, and template redundancy. Top models were then compared to SAXS data for further model selection and refinement. As a result, we present a theoretical model of the domain that is driven by alpha helices and short random coil regions. These alpha helical regions form a classic dimer interface followed by two wide spread legs that we predict to be the lipid binding interface. Finally, we validate the model presented with several lipid binding assays, which leads to a suggested mechanism that links ACCH lipid binding, membrane deformation, and vesicle fusion functions.

Entanglement convertibility by sweeping through the quantum phases of the alternating bonds XXZ chain.

We study the entanglement structure and the topological edge states of the ground state of the spin-1/2 XXZ model with bond alternation. We employ parity-density matrix renormalization group with periodic boundary conditions. The finite-size scaling of Rényi entropies S2 and S∞ are used to construct the phase diagram of the system. The phase diagram displays three possible phases: Haldane type (an example of symmetry protected topological ordered phases), Classical Dimer and Néel phases, the latter bounded by two continuous quantum phase transitions. The entanglement and non-locality in the ground state are studied and quantified by the entanglement convertibility. We found that, at small spatial scales, the ground state is not convertible within the topological Haldane dimer phase. The phenomenology we observe can be described in terms of correlations between edge states. We found that the entanglement spectrum also exhibits a distinctive response in the topological phase: the effective rank of the reduced density matrix displays a specifically large “susceptibility” in the topological phase. These findings support the idea that although the topological order in the ground state cannot be detected by local inspection, the ground state response at local scale can tell the topological phases apart from the non-topological phases.

Emergence of cooperative dynamics in fully packed classical dimers.

The classical dimer model on the square lattice is a paradigmatic example of a system subject to strong local constraints. We study its behavior under local stochastic dynamics, by means of Monte Carlo simulations and theoretical arguments. We observe clear signatures of correlated dynamics in both global and local observables and over a broad range of time scales, indicating a breakdown of the simple continuum description that approximates well the statics. We show that this collective dynamics can be understood in terms of one-dimensional "strings" of high mobility, which govern both local and long-wavelength dynamical properties. We introduce a coarse-grained description of the strings, based on the Edwards-Wilkinson model, which leads to exact results in the limit of low string density and provides a detailed qualitative understanding of the dynamics in all flux sectors. We discuss the implications of our results for the dynamics of constrained systems more generally.

ChemInform Abstract: Atom‐Economical Dimerization Strategy by the Rhodium‐Catalyzed Addition of Carboxylic Acids to Allenes: Protecting‐Group‐Free Synthesis of Clavosolide A and Late‐Stage Modification.

Natural products of polyketide origin with a high level of symmetry, in particular C2-symmetric diolides as a special macrolactone-based product class, often possess a broad spectrum of biological activity. An efficient route to this important structural motif was developed as part of a concise and highly convergent synthesis of clavosolide A. This strategy features an atom-economic “head-to-tail” dimerization by the stereoselective rhodium-catalyzed addition of carboxylic acids to terminal allenes with the simultaneous construction of two new stereocenters. The excellent efficiency and selectivity with which the C2-symmetric core structures were obtained are remarkable considering the outcome under classical dimerization conditions. Furthermore, this approach facilitates late-stage modification and provides ready access to potential new lead structures.

Optimization-based Shrinking Dimer Method for Finding Transition States

We present a new and efficient optimization-based shrinking dimer (OSD) algorithm for finding transition states associated with a given energy landscape. The transition states are given by index-one saddle points of the energy surface. By constructing the rotation and translation steps in the classical dimer method under an optimization framework, we are able to take advantage of more powerful optimization methods to substantially speed up the computation of transition states. Specifically, the Barzilai--Borwein gradient method is proposed as an effective implementation of OSD. We show that the OSD method is the generalized formulation of the original shrinking dimer dynamics (SDD) and enjoys superlinear convergence. We test various numerical examples, including some standard lower dimensional test examples, a cluster of seven particles, seven-atom island on the (111) surface of an FCC crystal, and nucleation in phase transformations. The results demonstrate that the OSD method with the proposed Barzilai-...