Lattice Green Function Research Articles

The Bałaban variational problem in the non-linear sigma model

The minimization of the action of a QFT with a constraint dictated by the block averaging procedure is an important part of Bałaban’s approach to renormalization. It is particularly interesting for QFTs with non-trivial target spaces, such as gauge theories or non-linear sigma models on a lattice. We analyze this step for the [Formula: see text] non-linear sigma model in two dimensions and demonstrate, in this case, how various ingredients of Bałaban’s approach play together. First, using variational calculus on Lie groups, the equation for the critical point is derived. Then, this non-linear equation is solved by the Banach contraction mapping theorem. This step requires detailed control of lattice Green functions and their integral kernels via random walk expansions.

Laplacian operator and its square lattice discretization: Green function vs. Lattice Green function for the flat 2-torus and other related 2D manifolds

Abstract The paper investigates the truncation error between the Green function and the lattice Green function (LGF) for the Laplacian operator defined on the 2-torus and its discretization on a regular square lattice. Extensions to the cylinder and the rectangular domain with free (or Neumann) boundary conditions are also proposed. In each of these instances, the Green function and its discrete analog are given in exact analytical closed-form allowing to infer accurate estimates as the lattice spacing tends to zero. As expected, it is shown that the continuum limit of the LGF coincides well with the Green function in every case. In particular, the issue of logarithmic singularity regularization of the Green function by the lattice discretization is addressed through two related application examples regarding the rectangular domain, and devoted to the computation of corner-to-corner resistance of an electrical conducting square and the mean first-passage time between the diagonally opposite vertices of a square for a standard Brownian motion, both derived considering the continuum limit.

An adaptive lattice Green's function method for external flows with two unbounded and one homogeneous directions

We solve the incompressible Navier-Stokes equations using a lattice Green's function (LGF) approach, including immersed boundaries (IB) and adaptive mesh refinement (AMR), for external flows with one homogeneous direction (e.g. infinite cylinders of arbitrary cross-section). We hybridize a Fourier collocation (pseudo-spectral) method for the homogeneous direction with a specially designed, staggered-grid finite-volume scheme on an AMR grid. The Fourier series is also truncated variably according to the refinement level in the other directions. We derive new algorithms to tabulate the LGF of the screened Poisson operator and viscous integrating factor. After adapting other algorithmic details from the fully inhomogeneous case [1], we validate and demonstrate the new method with transitional and turbulent flows over a circular cylinder at Re=300 and Re=12,000, respectively.

The starting vortices generated by bodies with sharp and straight edges in a viscous fluid

A two-dimensional body that moves suddenly in a viscous fluid can instantly generate vortices at its sharp edges. Recent work using inviscid flow theory, based on the Birkhoff–Rott equation and the Kutta condition, predicts that the ‘starting vortices’ generated by the sharp and straight edges of a body – i.e. the vortices formed immediately after motion begins – can be one of three distinct self-similar types. We explore the existence of these starting vortices for a flat plate and two symmetric Joukowski aerofoils immersed in a viscous fluid, using high-fidelity direct numerical simulations (DNS) of the Navier–Stokes equations. A lattice Green's function method is employed and simulations are performed for chord Reynolds numbers ranging from 5040 to 45 255. Vortices generated at the leading and trailing edges of the flat plate show agreement with the derived inviscid theory, for which a detailed assessment is reported. Agreement is also observed for the two symmetric Joukowski aerofoils, demonstrating the utility of the inviscid theory for arbitrary bodies. While this inviscid theory predicts an abrupt transition between the starting-vortex types, DNS shows a smooth transition. This behaviour occurs for all Reynolds numbers and is related to finite-time effects – there is a maximal time for which the (self-similar) starting vortices exist. We confirm the inviscid prediction that the leading-edge starting vortex of a flat plate can be suppressed dynamically. This has implications for the performance of low-speed aircraft such as model aeroplanes, micro air vehicles and unmanned air vehicles.

Electric-field-controlled electronic structures and quantum transport in monolayer InSe nanoribbons

Electronic structures and quantum transport properties of the monolayer InSe nanoribbons are studied by adopting the tight-binding model in combination with the lattice Green function method. Besides the normal bulk and edge electronic states, a unique electronic state dubbed as edge-surface is found in the InSe nanoribbon with zigzag edge type. In contrast to the zigzag InSe nanoribbon, a singular electronic state termed as bulk-surface is observed along with the normal bulk and edge electronic states in the armchair InSe nanoribbons. Moreover, the band gap, the transversal electron probability distributions in the two sublayers, and the electronic state of the topmost valence subband can be manipulated by adding a perpendicular electric field to the InSe nanoribbon. Further study shows that the charge conductance of the two-terminal monolayer InSe nanoribbons can be switched on or off by varying the electric field strength. In addition, the transport of the bulk electronic state is delicate to even a weak disorder strength, however, that of the edge and edge-surface electronic states shows a strong robustness against to the disorders. These findings may be helpful to understand the electronic characteristics of the InSe nanostructures and broaden their potential applications in two-dimensional nanoelectronic devices as well.

Lattice Green functions for pedestrians: Exponential decay

The exponential decay of lattice Green functions is one of the main technical ingredients of the Bałaban’s approach to renormalization. We give here a self-contained proof, whose various ingredients were scattered in the literature. The main sources of exponential decay are the Combes–Thomas method and the analyticity of the Fourier transforms. They are combined using a renormalization group equation and the method of images.

Velocity gradient analysis of a head-on vortex ring collision

We simulate the head-on collision between vortex rings with circulation Reynolds numbers of 4000 using an adaptive, multiresolution solver based on the lattice Green's function. The simulation fidelity is established with integral metrics representing symmetries and discretization errors. Using the velocity gradient tensor and structural features of local streamlines, we characterize the evolution of the flow with a particular focus on its transition and turbulent decay. Transition is excited by the development of the elliptic instability, which grows during the mutual interaction of the rings as they expand radially at the collision plane. The development of antiparallel secondary vortex filaments along the circumference mediates the proliferation of small-scale turbulence. During turbulent decay, the partitioning of the velocity gradients approaches an equilibrium that is dominated by shearing and agrees well with previous results for forced isotropic turbulence. We also introduce new phase spaces for the velocity gradients that reflect the interplay between shearing and rigid rotation and highlight geometric features of local streamlines. In conjunction with our other analyses, these phase spaces suggest that, while the elliptic instability is the predominant mechanism driving the initial transition, its interplay with other mechanisms, e.g. the Crow instability, becomes more important during turbulent decay. Our analysis also suggests that the geometry-based phase space may be promising for identifying the effects of the elliptic instability and other mechanisms using the structure of local streamlines. Moving forward, characterizing the organization of these mechanisms within vortices and universal features of velocity gradients may aid in modelling turbulent flows.

Lattice Green/Neumann function for the 2D Laplacian operator defined on square lattice on cylinders, tori and other geometries with some applications

The lattice Green functions for the discrete planar Laplacian defined on regular square lattice wrapped around cylinders and tori are rigorously defined and obtained in an exact analytic form. The method of images well-known in potential theory is implemented to derive for many other geometries with free boundaries (semi-infinite or finite cylinders and strips, rectangle) the related exact lattice Green-Neumann functions needed to readily solve discrete Neumann problems or, via a Neumann-to-Dirichlet mapping, discrete Dirichlet problems for these flat square lattices. Some applications are thus proposed as explicit expressions of two-point resistances for related resistor networks, and some probability-based characteristics regarding the associated Pòlya’s random walks.

Watermelons on the half-plane

We study the watermelon probabilities in the uniform spanning forests on the two-dimensional semi-infinite square lattice near either the open or closed boundary to which the forests can or cannot be rooted, respectively. We derive universal power laws describing the asymptotic decay of these probabilities with the distance between the reference points growing to infinity, as well as their non-universal constant prefactors. The obtained exponents match with the previous predictions made for the related dense polymer models using the Coulomb gas technique and conformal field theory, as well as with the lattice calculations made by other authors in different settings. We also discuss the logarithmic corrections some authors argued to appear in the watermelon correlation functions on the infinite lattice. We show that the full account for diverging terms of the lattice Green function, which ensures the correct probability normalization, provides the pure power law decay in the case of semi-infinite lattice with the closed boundary studied here, as well as in the case of infinite lattice discussed elsewhere. The solution is based on the all-minors generalization of the Kirchhoff matrix tree theorem, the image method and the developed asymptotic expansion of the Kirchhoff determinants.

Fractional saturable impurity

We examine analytically and numerically the effect of fractionality on a saturable bulk and surface impurity embedded in a one-dimensional lattice. We use a fractional Laplacian introduced previously by us, and by the use of lattice Green functions we are able to obtain the bound state energies and amplitude profiles, as a function of the fractional exponent $s$ and saturable impurity strength $\ensuremath{\chi}$ for both surface and bulk impurity. The transmission is obtained in closed form as a function of $s$ and $\ensuremath{\chi}$, showing strong deviations from the standard case, at small fractional exponent values. The self-trapping of an initially localized excitation is qualitatively similar for the bulk and surface mode, but in all cases complete confinement is obtained at $s\ensuremath{\rightarrow}0$, as shown theoretically and observed numerically.

Efficient lattice Green's function method for bounded domain problems

AbstractThe lattice Green's function method (LGFM) is the discrete counterpart of the continuum boundary element method and is a natural approach for solving intrinsically discrete solid mechanics problems that arise in atomistic‐continuum coupling methods. Here, the LGFM is extended to problems in a bounded domain with applied boundary conditions. An efficient controlled coarse‐graining method is introduced to significantly reduce the number of atomistic degrees of freedom on the outer boundary, and thus the size of the dense Green's function matrices involved while preserving the high accuracy of the solution. The method is demonstrated on various example problems to show reductions in computational costs and errors versus reference solutions.

Planar potential flow on Cartesian grids

Potential flow has many applications, including the modelling of unsteady flows in aerodynamics. For these models to work efficiently, it is best to avoid Biot–Savart interactions. This work presents a grid-based treatment of potential flows in two dimensions and its use in a vortex model for simulating unsteady aerodynamic flows. For flows consisting of vortex elements, the treatment follows the vortex-in-cell approach and solves the streamfunction–vorticity Poisson equation on a Cartesian grid after transferring the circulation from the vortices onto the grid. For sources and sinks, an analogous approach can be followed using the scalar potential. The combined velocity field due to vortices, sinks and sources can then be obtained using the Helmholtz decomposition. In this work, we use several key tools that ensure the approach works on arbitrary geometries, with and without sharp edges. First, the immersed boundary projection method is used to account for bodies in the flow and the resulting body-forcing Lagrange multiplier is identified as the bound vortex sheet strength. Second, sharp edges are treated by decomposing the vortex sheet strength into a singular and non-singular part. To enforce the Kutta condition, the non-singular part can then be constrained to remove the singularity introduced by the sharp edge. These constraints and the Poisson equation are formulated as a saddle-point system and solved using the Schur complement method. The lattice Green's function is used to efficiently solve the discrete Poisson equation with unbounded boundary conditions. The method and its accuracy are demonstrated for several problems.

Multi-resolution lattice Green's function method for incompressible flows

We propose a multi-resolution strategy that is compatible with the lattice Green's function (LGF) technique for solving viscous, incompressible flows on unbounded domains. The LGF method exploits the regularity of a finite-volume scheme on a formally unbounded Cartesian mesh to yield robust and computationally efficient solutions. The original method is spatially adaptive, but challenging to integrate with embedded mesh refinement as the underlying LGF is only defined for a fixed resolution. We present an ansatz for adaptive mesh refinement, where the solutions to the pressure Poisson equation are approximated using the LGF technique on a composite mesh constructed from a series of infinite lattices of differing resolution. To solve the incompressible Navier-Stokes equations, this is further combined with an integrating factor for the viscous terms and an appropriate Runge-Kutta scheme for the resulting differential-algebraic equations. The parallelized algorithm is verified through with numerical simulations of vortex rings, and the collision of vortex rings at high Reynolds number is simulated to demonstrate the reduction in computational cells achievable with both spatial and refinement adaptivity.

Causal versus local GW+EDMFT scheme and application to the triangular-lattice extended Hubbard model

Using the triangular-lattice extended Hubbard model as a test system, we compare $GW$+EDMFT results for the recently proposed self-consistency scheme with causal auxiliary fields to those obtained from the standard implementation which identifies the impurity Green's functions with the corresponding local lattice Green's functions. Both for short-ranged and long-ranged interactions we find similar results, but the causal scheme yields slightly stronger correlation effects at half-filling. We use the two implementations of $GW$+EDMFT to compute spectral functions and dynamically screened interactions in the parameter regime relevant for 1$T$-TaS$_2$. We address the question whether or not the sample used in a recent photoemission study [Phys. Rev. Lett. 120, 166401 (2018)] was half-filled or hole-doped.

Asymptotic behaviour of the lattice Green function

The lattice Green function, i.e., the resolvent of the discrete Laplace operator, is fundamental in probability theory and mathematical physics. We derive its long distance behaviour using the Laplace method applied to an integral representation involving modified Bessel functions. Our emphasis is on the decay of the massive lattice Green function in the vicinity of the massless (critical) case, and the recovery of Euclidean isotropy in the massless limit.

Topological superconductivity in EuS/Au/superconductor heterostructures

In a recent work [S. Manna, P. Wei, Y. Xie, K. T. Law, P. A. Lee, and J. S. Moodera, Proc. Natl. Acad. Sci. 117, 8775 (2020)], signatures of a pair of Majorana bound states (MBSs) were found in an experimental platform formed by EuS islands deposited on top of a gold surface which was made superconducting through proximity coupling to a superconductor. In this paper, we provide a theoretical understanding for how MBSs can be formed in EuS/Au/superconductor heterostructures. We focus on the strip geometry where a narrow ferromagnetic strip is deposited on a planar structure. We first explicitly map out the topological phase diagram of the EuS/Au/superconductor heterostructure using the lattice Green's function method. Importantly, we find that the chemical potential step between the region with and without EuS covering is a crucial ingredient for the creation of MBS of this setup. Next, we focus on the Bogoliugov quasiparticles that are bound to the region under the EuS by Andreev reflections from the surrounding superconductors. Moreover, we obtain the topological regimes analytically using the scattering matrix method. Notably, we confirm that the normal reflections induced by the chemical potential step are essential for creating finite topological regimes. Furthermore, the area of the topological regimes shows periodic oscillation as a function of chemical potential as well as the sample width. We conclude by showing that the feromagnetic strip geometry holds a number of advantages over other quasi-one-dimensional schemes that have been proposed.

Scattering of two particles in a one-dimensional lattice

This study concerns the two-body scattering of particles in a one-dimensional periodic potential. A convenient ansatz allows for the separation of center-of-mass and relative motion, leading to a discrete Schr\"odinger equation in the relative motion that resembles a tight-binding model. A lattice Green's function is used to develop the Lippmann-Schwinger equation, and ultimately derive a multiband scattering $K$ matrix which is described in detail in the two-band approximation. Two distinct scattering lengths are defined according to the limits of zero relative quasimomentum at the top and bottom edges of the two-body collision band. Scattering resonances occur in the collision band when the energy is coincident with a bound state attached to another higher or lower band. Notably, repulsive on-site interactions in an energetically closed lower band lead to collision resonances in an excited band.

An Accurate Integral Equation Formulation to Scattering of Periodic Grating Structures

Broadband wave transmission grating structures have potentials in many engineering applications. Theoretical predictions of the transmission spectra and near field characteristics of such structures are crucial for the understanding and design of wave functional gratings. However, accurate theoretical modeling of the wave interactions with such periodic grating structures turns out to be difficult, especially at near grazing incidence angles. In this paper, an efficient and accurate method to solve the scattering of periodic grating structures is proposed based upon integral equation formulations utilizing periodic lattice Green's functions. The method of moments is used to discretize the integral equation with the pulse basis function and the point matching scheme. The imaginary wavenumber extraction technique and the integral transformation approach are combined to efficiently and accurately evaluate the period Green's function. Meanwhile, an over-determined testing scheme on the integral equation is devised to overcome the intrinsic internal resonance of surface integral equations. Such strategy has significantly improved the numerical accuracy of the proposed approach. Calculation results are compared with Comsol simulations for various scenarios, and the proposed approach is found to be predicting physically sound results when Comsol fails at certain frequencies and incident angles. The proposed method is applicable to grating periodic structures of various shapes and it provides a useful reference and tool for researchers and designers.

Spanning tree generating functions for infinite periodic graphs L and connections with simple closed random walks on L

A spanning tree generating function T(z) for infinite periodic vertex-transitive (vt) lattices L vt has been proposed by Guttmann and Rogers (2012 J. Phys. A: Math. Theor. 45 494001). Their spanning tree constants are then given by T(z = 1). Here an extended T e (z) is constructed, relaxing the previous vt condition to q-regular lattices L and likewise leading to z L = T e (1). Further, in the vt case the method to derive T e yields a new integral formula for the lattice Green function. As examples, spanning tree generating functions for all the eleven vt Archimedean (surprisingly, easier to obtain from T e than from T) and two relevant non-vt, martini and the (4, 82) covering/medial, lattices are derived. The importance of T e (and T)—beyond just being a tool to calculate z L —is illustrated with the proof that the free energy of the random walk loop soup model (defined from closed random walks, the loops, over L) can be written in terms of T e (z). From such result, it is shown that the system critical point—existing only for d = 1 and d = 2—is directly related to z L .

Revisiting the discrete planar Laplacian: exact results for the lattice Green function and continuum limit

The paper deals with the discrete Laplacian on a uniform infinite square lattice. The definition of its fundamental solution or lattice Green function (LGF) is clarified as the Fourier coefficients of a certain generalized periodic function g. Such a functional must be regularized and gives the LGF up to a constant equal to \({<}g{>}\), the mean value of g. For \({<}g{>}=0\), the LGF may be expressed in an exact analytic form in terms of hypergeometric and gamma functions. The continuum limit of the LGF is finally studied requiring an appropriate renormalization of \({<}g{>}\) in order to obtain the logarithmic Coulomb potential.