Two-dimensional Lattice Model Research Articles

Excitonic Instability and Pseudogap Formation in Nodal Line Semimetal ZrSiS.

Electron correlation effects are studied in ZrSiS using a combination of first-principles and model approaches. We show that basic electronic properties of ZrSiS can be described within a two-dimensional lattice model of two nested square lattices. A high degree of electron-hole symmetry characteristic for ZrSiS is one of the key features of this model. Having determined model parameters from first-principles calculations, we then explicitly take electron-electron interactions into account and show that, at moderately low temperatures, ZrSiS exhibits excitonic instability, leading to the formation of a pseudogap in the electronic spectrum. The results can be understood in terms of Coulomb-interaction-assisted pairing of electrons and holes reminiscent of that of an excitonic insulator. Our finding allows us to provide a physical interpretation of the unusual mass enhancement of charge carriers in ZrSiS recently observed experimentally.

Noncommutative fiber products and lattice models

We establish a connection between the representation theory of certain noncommutative singular varieties and two-dimensional lattice models. Specifically, we consider noncommutative biparametric deformations of the fiber product of two Kleinian singularities of type A. Special examples are closely related to the Schrödinger algebra S(1), the affine Lie algebra A1(1), and a finite W-algebra associated to sl4.The algebras depend on two scalars and two polynomials that must satisfy the Mazorchuk–Turowska Equation (MTE), which we re-interpret as a quantization of the ice rule (local current conservation) in statistical mechanics. Solutions to the MTE, previously classified by the author and D. Rosso, can accordingly be expressed in terms of multisets of higher spin vertex configurations on a twisted cylinder.We first reduce the problem of describing the category of weight modules to the case of a single configuration L. Secondly, we classify all simple weight modules over the corresponding algebras A(L), in terms of the connected components of the cylinder minus L. Lastly, we prove that A(L) are crystalline graded rings (as defined by Nauwelaerts and Van Oystaeyen), and describe the center of A(L) explicitly in terms of L.Along the way we prove several new results about twisted generalized Weyl algebras and their simple weight modules.

Probing Conformal Invariant of Non-unitary Two-Dimensional Systems by Central Spin Decoherence

Universality classes of non-unitary critical theories in two-dimensions are characterized by the central charge. However, experimental determination of the central charge of a non-unitary critical theory has not been done before because of the intrinsic difficulty that complex parameters usually occur in non-unitary theory, which is not physical. Here we propose to extract the effective central charge of the non-unitary critical point of a two-dimensional lattice model from the quantum coherence measurement of a probe spin which is coupled to the lattice model. A recent discovery shows that quantum coherence of a probe spin which is coupled to a bath is proportional to the partition function of the bath with a complex parameter. Thus the effective central charge of a non-unitary conformal field theory may be extracted from quantum coherence measurement of a probe spin which is coupled to a bath. We have applied the method to the Yang-Lee edge singularity of the two-dimensional Ising model and extracted the effective central charge of the Yang-Lee edge singularity with good precision and tested other predictions of non-unitary conformal field theory. This work paves the way for the first experimental observation of the effective central charge of non-unitary conformal field theory.

Consistent second-order boundary implementations for convection-diffusion lattice Boltzmann method.

In this study, an alternative second-order boundary scheme is proposed under the framework of the convection-diffusion lattice Boltzmann (LB) method for both straight and curved geometries. With the proposed scheme, boundary implementations are developed for the Dirichlet, Neumann and linear Robin conditions in a consistent way. The Chapman-Enskog analysis and the Hermite polynomial expansion technique are first applied to derive the explicit expression for the general distribution function with second-order accuracy. Then, the macroscopic variables involved in the expression for the distribution function is determined by the prescribed macroscopic constraints and the known distribution functions after streaming [see the paragraph after Eq.(29) for the discussions of the "streaming step" in LB method]. After that, the unknown distribution functions are obtained from the derived macroscopic information at the boundary nodes. For straight boundaries, boundary nodes are directly placed at the physical boundary surface, and the present scheme is applied directly. When extending the present scheme to curved geometries, a local curvilinear coordinate system and first-order Taylor expansion are introduced to relate the macroscopic variables at the boundary nodes to the physical constraints at the curved boundary surface. In essence, the unknown distribution functions at the boundary node are derived from the known distribution functions at the same node in accordance with the macroscopic boundary conditions at the surface. Therefore, the advantages of the present boundary implementations are (i) the locality, i.e., no information from neighboring fluid nodes is required; (ii) the consistency, i.e., the physical boundary constraints are directly applied when determining the macroscopic variables at the boundary nodes, thus the three kinds of conditions are realized in a consistent way. It should be noted that the present focus is on two-dimensional cases, and theoretical derivations as well as the numerical validations are performed in the framework of the two-dimensional five-velocity lattice model.

Simulation of microstructure evolution coupled with fabrication parameters for two-phase ceramic tool materials

A computer simulation coupled with fabrication parameters for the sintering process of two-phase ceramic tool materials has been developed using a two-dimensional hexagon lattice model mapped from ...

Model of chiral spin liquids with Abelian and non-Abelian topological phases

We present a two-dimensional lattice model for quantum spin-1/2 for which the low-energy limit is governed by four flavors of strongly interacting Majorana fermions. We study this low-energy effective theory using two alternative approaches. The first consists of a mean-field approximation. The second consists of a Random Phase approximation (RPA) for the single-particle Green's functions of the Majorana fermions built from their exact forms in a certain one-dimensional limit. The resulting phase diagram consists of two competing chiral phases, one with Abelian and the other with non-Abelian topological order, separated by a continuous phase transition. Remarkably, the Majorana fermions propagate in the two-dimensional bulk, as in the Kitaev model for a spin liquid on the honeycomb lattice. We identify the vison fields, which are mobile (they are static in the Kitaev model) domain walls propagating along only one of the two space directions.

Two-dimensional waves in extended square lattice

We consider a two-dimensional square lattice model extended by additional not closed neighboring interactions. We assume the elastic forces between the masses in the lattice to be nonlinearly dependent on the spring elongations. First, we use an analysis of the linearized discrete equations to reveal the influence of additional interactions on the properties of the dispersion relation for longitudinal and shear plane waves. Then we develop an asymptotic procedure to obtain continuum two-dimensional non-linear equations to study the transverse instability of weakly non-linear localized plane longitudinal and shear waves. We find that the additional interactions used in the model may affect the sign of the amplitude of the plane strain waves (existence of compression (minus sign) or tensile (plus sign) plane waves) and their transverse stability.

On reductions of the discrete Kadomtsev–Petviashvili-type equations

The reduction by restricting the spectral parameters k and on a generic algebraic curve of degree is performed for the discrete AKP, BKP and CKP equations, respectively. A variety of two-dimensional discrete integrable systems possessing a more general solution structure arise from the reduction, and in each case a unified formula for the generic positive integer is given to express the corresponding reduced integrable lattice equations. The obtained extended two-dimensional lattice models give rise to many important integrable partial difference equations as special degenerations. Some new integrable lattice models such as the discrete Sawada–Kotera, Kaup–Kupershmidt and Hirota–Satsuma equations in extended form are given as examples within the framework.

Domino tilings of the expanded Aztec diamond

The expanded Aztec diamond is a generalized version of the Aztec diamond, with an arbitrary number of long columns and long rows in the middle. In this paper, we count the number of domino tilings of the expanded Aztec diamond. The exact number of domino tilings is given by recurrence relations of state matrices by virtue of the state matrix recursion algorithm, recently developed by the author to solve various two-dimensional regular lattice model enumeration problems.

Nature of continuous phase transitions in interacting topological insulators

We revisit the effects of the Hubbard repulsion on quantum spin Hall effects (QSHE) in two-dimensional quantum lattice models. We present both unbiased exact diagonalization and density-matrix renormalization group simulations with numerical evidences for a continuous quantum phase transition (CQPT) separating QSHE from the topologically trivial antiferromagnetic phase. Our numerical results suggest that, the nature of CQPT exhibits distinct finite-size scaling behaviors, which may be consistent with either Ising or XY universality classes for different time-reversal symmetric QSHE systems.

Numerical calculation on a two-step subdiffusion behavior of lateral protein movement in plasma membranes.

A two-step subdiffusion behavior of lateral movement of transmembrane proteins in plasma membranes has been observed by using single-molecule experiments. A nested double-compartment model where large compartments are divided into several smaller ones has been proposed in order to explain this observation. These compartments are considered to be delimited by membrane-skeleton "fences" and membrane-protein "pickets" bound to the fences. We perform numerical simulations of a master equation using a simple two-dimensional lattice model to investigate the heterogeneous diffusion dynamics behavior of transmembrane proteins within plasma membranes. We show that the experimentally observed two-step subdiffusion process can be described using fence and picket models combined with decreased local diffusivity of transmembrane proteins in the vicinity of the pickets. This allows us to explain the two-step subdiffusion behavior without explicitly introducing nested double compartments.

Variational cluster approach to superconductivity and magnetism in the Kondo lattice model

We investigate in detail antiferromagnetic (AF) and superconducting (SC) phases as well as their coexistence in the two-dimensional Kondo lattice model on a square lattice, which is a paradigmatic model for heavy fermion materials. The results presented are mainly obtained using the variational cluster approximation (VCA) and are complemented by analytical findings for the equations of motion of pairing susceptibilities. A particularly interesting aspect is the possibility to have s-wave SC near half filling as reported by Bodensiek \textit{et al.} [Phys. Rev. Lett. \textbf{110}, 146406 (2013)]. When doping the system, we identify three regions which correspond to an AF metallic phase with small Fermi surface at weak coupling, an AF metal with a different Fermi surface topology at intermediate coupling, and a paramagnetic metal with a large Fermi surface at strong coupling. The transition between these two AF phases is found to be discontinuous at lower fillings, but turns to a continuous one when approaching half-filling. In the quest for s-wave superconductivity, only solutions are found which possess mean-field character. No true superconducting solutions caused by correlation effects are found in the s-wave channel. In contrast, we clearly identify robust d-wave pairing away from half-filling. However, we show that only by treating antiferromagnetism and superconductivity on equal footing artificial superconducting solutions at half-filling can be avoided. Our VCA findings support scenarios previously identified by variational Monte Carlo approaches and are a starting point for future investigations with VCA and further approaches such as cluster-embedding methods.

Maximal independent sets on a grid graph

An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number of maximal independent sets of vertices on a complete rectangular grid graph. More precisely, we provide a recursive matrix-relation producing the partition function with respect to the number of vertices. The asymptotic behavior of the maximal hard square entropy constant is also provided. We adapt the state matrix recursion algorithm, recently invented by the author to answer various two-dimensional regular lattice model problems in enumerative combinatorics and statistical mechanics.

Two-dimensional lattice model for the surface states of topological insulators

The surface states in three-dimensional (3D) topological insulators (TIs) can be described by a two-dimensional (2D) continuous Dirac Hamiltonian. However, there exists the Fermion doubling problem when putting the continuous 2D Dirac equation into a lattice model. In this letter, we introduce a Wilson term with a zero bare mass into the 2D lattice model to overcome the difficulty. By comparing with a 3D Hamiltonian, we show that the modified 2D lattice model can faithfully describe the low-energy electrical and transport properties of surface states of 3D TIs. So this 2D lattice model provides a simple and cheap way to numerically simulate the surface states of 3D TI nanostructures. Based on the 2D lattice model, we also establish the wormhole effect in a TI nanowire by a magnetic field along the wire and show the surface states being robust against disorder. The proposed 2D lattice model can be extensively applied to study the various properties and effects, such as the transport properties, Hall effect, universal conductance fluctuations, localization effect, etc.. So it paves a new way to study the surface states of the 3D topological insulators.

Determination of the conformal-field-theory central charge by the Wang-Landau algorithm.

We present a simple method to estimate the central charge of the conformal field theory corresponding to a critical point of a two-dimensional lattice model from Monte Carlo simulations. The main idea is to use the Wang-Landau flat-histogram algorithm, which allows us to obtain the free energy of a lattice model on a torus as a function of torus radii. The central charge is calculated with good precision from a free-energy scaling at the critical point. We apply the method to the Ising, tricritical Ising (Blume-Capel), Potts, and site-diluted Ising models, and we also discuss an estimation of the conformal weights.

Quantum critical properties of a metallic spin-density-wave transition

We report on numerically exact determinantal quantum Monte Carlo simulations of the onset of spin-density wave (SDW) order in itinerant electron systems captured by a sign-problem-free two-dimensional lattice model. Extensive measurements of the SDW correlations in the vicinity of the phase transition reveal that the critical dynamics of the bosonic order parameter are well described by a dynamical critical exponent z = 2, consistent with Hertz-Millis theory, but are found to follow a finite-temperature dependence that does not fit the predicted behavior of the same theory. The presence of critical SDW fluctuations is found to have a strong impact on the fermionic quasiparticles, giving rise to a dome-shaped superconducting phase near the quantum critical point. In the superconducting state we find a gap function that has an opposite sign between the two bands of the model and is nearly constant along the Fermi surface of each band. Above the superconducting $T_c$ our numerical simulations reveal a nearly temperature and frequency independent self energy causing a strong suppression of the low-energy quasiparticle spectral weight in the vicinity of the hot spots on the Fermi surface. This indicates a clear breakdown of Fermi liquid theory around these points.

Branes and integrable lattice models

This is a brief review of my work on the correspondence between four-dimensional [Formula: see text] supersymmetric field theories realized by brane tilings and two-dimensional integrable lattice models. I explain how to construct integrable lattice models from extended operators in partially topological quantum field theories, and elucidate the correspondence as an application of this construction.

Higher-Order Airy Scaling in Deformed Dyck Paths

We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, ‘jumps’ orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with respect to their half-length, area and number of jumps. This represents the first example of an exactly solvable two-dimensional lattice vesicle model showing a higher-order multicritical point. Applying the generalized method of steepest descents, we see that the associated two-variable scaling function is given by the logarithmic derivative of a generalized (higher-order) Airy integral.

Competing Orders in a Nearly Antiferromagnetic Metal.

We study the onset of spin-density wave order in itinerant electron systems via a two-dimensional lattice model amenable to numerically exact, sign-problem-free determinantal quantum MonteCarlo simulations. The finite-temperature phase diagram of the model reveals a dome-shaped d-wave superconducting phase near the magnetic quantum phase transition. Above the critical superconducting temperature, an extended fluctuation regime manifests itself in the opening of a gap in the electronic density of states and an enhanced diamagnetic response. While charge density wave fluctuations are moderately enhanced in the proximity of the magnetic quantum phase transition, they remain short ranged. The striking similarity of our results to the phenomenology of many unconventional superconductors points a way to a microscopic understanding of such strongly coupled systems in a controlled manner.

Critical Casimir interactions and colloidal self-assembly in near-critical solvents.

A binary solvent mixture close to critical demixing experiences fluctuations whose correlation length, ξ, diverges as the critical point is approached. The solvent-mediated (SM) interaction that arises between a pair of colloids immersed in such a near-critical solvent can be long-ranged and this so-called critical Casimir interaction is well-studied. How a (dense) suspension of colloids will self-assemble under these conditions is poorly understood. Using a two-dimensional lattice model for the solvent and hard disks to represent the colloids, we perform extensive Monte Carlo simulations to investigate the phase behaviour of this model colloidal suspension as a function of colloid size and wettability under conditions where the solvent reservoir is supercritical. Unlike most other approaches, where the solvent is modelled as an implicit background, our model employs an explicit solvent and treats the suspension as a ternary mixture. This enables us to capture important features, including the pronounced fractionation of the solvent in the coexisting colloidal phases, of this complex system. We also present results for the partial structure factors; these shed light on the critical behaviour in the ternary mixture. The degree to which an effective two-body pair potential description can describe the phase behaviour and structure of the colloidal suspension is discussed briefly.