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![Kink of width λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} in a 1D chain that displaces the particles (black circles) from an equidistant grid (dotted circles). The long-range forces acting on a particular particle, e.g. in red, are computed (Color figure online)](publication/357147456/figure/fig5/AS:1102231172595721@1639803798797/Kink-of-width-ldocumentclass12ptminimal-usepackageamsmath-usepackagewasysym.png)
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We develop a new expansion for representing singular sums in terms of integrals and vice versa. This method provides a powerful tool for the efficient computation of large singular sums that appear in long-range interacting systems in condensed matter and quantum physics. It also offers a generalised trapezoidal rule for the precise computation of...
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A bstract
The late time behavior of OTOCs involving generic non-conserved local operators show exponential decay in chaotic many body systems. However, it has been recently observed that for certain holographic theories, the OTOC involving the U(1) conserved current for a gauge field instead varies diffusively at late times. The present work genera...
Citations
... It has further been applied to study long-range interacting hardcore bosons [22], and the melting of Devil's staircases in quantum magnets [21]. The Epstein zeta function forms the foundation for the Singular Euler-Maclaurin (SEM) expansion, a recent generalization of the Euler-Maclaurin summation formula to physically relevant long-range interactions on higher-dimensional lattices [11,10]. ...
Many-body interactions arise naturally in the perturbative treatment of classical and quantum many-body systems and play a crucial role in the description of condensed matter systems. In the case of three-body interactions, the Axilrod-Teller-Muto (ATM) potential is highly relevant for the quantitative prediction of material properties. The computation of the resulting energies in d-dimensional lattice systems is challenging, as a high-dimensional lattice sum needs to be evaluated to high precision. This work solves this long-standing issue. We present an efficiently computable representation of many-body lattice sums in terms of singular integrals over products of Epstein zeta functions. For three-body interactions in 3D, this approach reduces the runtime for computing the ATM lattice sum from weeks to minutes. Our approach further extends to a broad class of n-body lattice sums. We demonstrate that the computational cost of our method only increases linearly with n, evading the exponential increase in complexity of direct summation. The evaluation of 51-body interactions on a two-dimensional lattice, corresponding to a 100-dimensional sum, can be performed within seconds on a laptop. We discuss techniques for computing the arising singular integrals and compare the accuracy of our results against computable benchmarks, achieving full precision for exponents greater than the system dimension. Finally, we apply our method to study the stability of a three-dimensional lattice system with Lennard-Jones two-body interactions under the inclusion of an ATM three-body term at finite pressure, finding a transition from the face-centered-cubic to the body-centered-cubic lattice structure with increasing ATM coupling strength. This work establishes the mathematical foundation for an ongoing investigation into the influence of many-body interactions on the stability of matter.
... The Epstein zeta function has recently been used to derive rigorous error bounds in boundary integral equations [50]. It forms a key ingredient of the Singular Euler-Maclaurin expansion (SEM), a recent generalization of the Euler-Maclaurin summation formula to singular functions in multi-dimensional lattices [8,9]. The Epstein zeta function offers numerous applications in theoretical quantum physics and chemistry of technological relevance, especially in systems with longrange interactions. ...
The Epstein zeta function generalizes the Riemann zeta function to oscillatory lattice sums in higher dimensions. Beyond its numerous applications in pure mathematics, it has recently been identified as a key component in simulating exotic quantum materials. This work establishes the Epstein zeta function as a powerful tool in numerical analysis by rigorously investigating its analytical properties and enabling its efficient computation. Specifically, we derive a compact and computationally efficient representation of the Epstein zeta function and thoroughly examine its analytical properties across all arguments. Furthermore, we introduce a superexponentially convergent algorithm, complete with error bounds, for computing the Epstein zeta function in arbitrary dimensions. We also show that the Epstein zeta function can be decomposed into a power law singularity and an analytic function in the first Brillouin zone. This decomposition facilitates the rapid evaluation of integrals involving the Epstein zeta function and allows for efficient precomputations through interpolation techniques. We present the first high-performance implementation of the Epstein zeta function and its regularisation for arbitrary real arguments in EpsteinLib, a C library with Python and Mathematica bindings, and rigorously benchmark its precision and performance against known formulas, achieving full precision across the entire parameter range. Finally, we apply our library to the computation of quantum dispersion relations of three-dimensional spin materials with long-range interactions and Casimir energies in multidimensional geometries, uncovering higher-order corrections to known asymptotic formulas for the arising forces.
... During the refereeing process of the manuscript, we became aware of the fact that the resummed couplings can be represented using the Epstein ζ-function [68,69]. Recently an efficient numerical implementation of the Epstein ζ-function was introduced [70][71][72][73]. This approach outperforms the brute force summation in speed and accuracy of the results and enables the evaluation of resummed couplings up to machine precision without any effort. ...
... With the recent development of efficient algorithms [70][71][72][73] for the calculation of the resummed couplings one shall be able to treat a larger number of unit-cells with larger sizes. Then the global optimizer on each unit cell will become the limiting factor of the calculations. ...
We analyse the ground-state quantum phase diagram of hardcore Bosons interacting with repulsive dipolar potentials. The bosons dynamics is described by the extended-Bose-Hubbard Hamiltonian on a two-dimensional lattice. The ground state results from the interplay between the lattice geometry and the long-range interactions, which we account for by means of a classical spin mean-field approach limited by the size of the considered unit cells. This extended classical spin mean-field theory accounts for the long-range density-density interaction without truncation. We consider three different lattice geometries: square, honeycomb, and triangular. In the limit of zero hopping the ground state is always a devil’s staircase of solid (gapped) phases. Such crystalline phases with broken translational symmetry are robust with respect to finite hopping amplitudes. At intermediate hopping amplitudes, these gapped phases melt, giving rise to various lattice supersolid phases, which can have exotic features with multiple sublattice densities. At sufficiently large hoppings the ground state is a superfluid. The stability of phases predicted by our approach is gauged by comparison to the known quantum phase diagrams of the Bose-Hubbard model with nearest-neighbour interactions as well as quantum Monte Carlo simulations for the dipolar case on the square and triangular lattice. Our results are of immediate relevance for experimental realisations of self-organised crystalline ordering patterns in analogue quantum simulators, e.g., with ultracold dipolar atoms in an optical lattice.
... Its applications span from the computation of electrostatic crystal potentials [18,19], over analytic number theory and statistical mechanics [42], to quantum field theory [17]. Two of the authors have recently developed the Singular Euler-Maclaurin expansion (SEM), a generalization of the 300-year-old Euler-Maclaurin summation formula to singular functions in higher dimensions [7,8] that uses the Epstein zeta function as a key element. This method has led to the prediction of two new phases in unconventional superconductors [10]. ...
This paper introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics, and have immediate applications in condensed matter physics and topological quantum physics. The challenge in their evaluation results from the combination of singular long-range interactions with the loss of translational invariance caused by the boundaries, rendering standard tools ineffective. Our work shows that these lattice sums can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums. We put forth a new representation of this zeta function together with a numerical algorithm that ensures exponential convergence across an extensive range of geometries. Notably, our method's runtime is influenced only by the complexity of the considered geometries and not by the number of particles, providing the foundation for efficient simulations of macroscopic condensed matter systems. We showcase the practical utility of our method by computing interaction energies in a three-dimensional crystal structure with 3×10^23 particles. Our method's accuracy is demonstrated through extensive numerical experiments. A reference implementation is provided online along with this article.
... We show that the discrete lattice problem can be separated into a term that describes its continuous analog, the continuum contribution, and a term that includes all information about the microstructure, the lattice contribution, hence demonstrating equivalence between lattice and continuum. To this end, we apply the recently developed singular Euler-Maclaurin (SEM) expansion [13][14][15], which generalizes the 300-year old Euler-Maclaurin summation formula, and extend it to nonlinear and multiatomic systems. The singular lattice sum is expressed in terms of an integral and a lattice contribution described by a differential operator, both of which are efficiently computable. ...
... The representation of the lattice contribution in Eq. (4) is called the SEM expansion, a full derivation of which is provided in Refs. [13][14][15]. For = R d the SEM operator D ( ) takes the particularly simple form ...
Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this paper, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analog, the integral contribution, and a term that fully resolves the microstructure, the lattice contribution. For any system dimension, for any lattice, for any power-law interaction, and for linear, nonlinear, and multiatomic lattices, we show that the lattice contribution can be described by a differential operator based on the multidimensional generalization of the Riemann zeta function, namely, the Epstein zeta function. We employ our representation in Fourier space to solve the important problem of long-range interacting unconventional superconductors. We derive a generalized Bardeen-Cooper-Schrieffer gap equation and find emerging exotic phases in two-dimensional superconductors with topological phase transitions. Finally, we utilize nonequilibrium Higgs spectroscopy to analyze the impact of long-range interactions on the collective excitations of the condensate. We show that the interactions can be used to fine tune the Higgs mode's stability, ranging from exponential decay of the oscillation amplitude up to complete stabilization. By providing a unifying framework for long-range interactions on a lattice, both classical and quantum, our research can guide the search for exotic phases of matter across different fields.
... We show that the discrete lattice problem can be separated into a term that describes its continuous analog, the continuum contribution, and a term that includes all information about the microstructure, the lattice contribution, hence demonstrating equivalence between lattice and continuum. To this end, we apply the recently developed Singular Euler-Maclaurin (SEM) expansion [13][14][15], which generalizes the 300-year old Euler-Maclaurin summation formula, and extend it to nonlinear and multiatomic systems. The singular lattice sum is expressed in terms of an integral and a lattice contribution described by a differential operator, both of which are efficiently computable. ...
... with D a differential operator of infinite order, D ( ) its truncation up to order 2 + 1, and ∆ the Laplacian. The representation of the lattice contribution in Eq. (4) is called the Singular Euler-Maclaurin (SEM) expansion, a full derivation of which is provided in [13][14][15]. For Ω = R d the SEM operator D ( ) takes the particularly simple form ...
Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this work, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analog, the integral contribution, and a term that fully resolves the microstructure, the lattice contribution. For any system dimension, any lattice, any power-law interaction, and for linear, nonlinear, and multi-atomic lattices, we show that the lattice contribution can be described by a differential operator based on the multidimensional generalization of the Riemann zeta function, namely the Epstein zeta function. We employ our representation in Fourier space to solve the important problem of long-range interacting unconventional superconductors. We derive a generalized Bardeen-Cooper-Schrieffer gap equation and find emerging exotic phases in two-dimensional superconductors with topological phase transitions. Finally, we utilize non-equilibrium Higgs spectroscopy to analyze the impact of long-range interactions on the collective excitations of the condensate. We show that the interactions can be used to fine-tune the Higgs mode's stability, ranging from exponential decay of the oscillation amplitude up to complete stabilization.
... The main tool for describing the difference between a sum and an integral in one dimension is the Euler-Maclaurin (EM) expansion (see [1] for a historic overview). For ∈ N 0 , a, b ∈ Z, δ 1 , δ 2 ∈ (0, 1] and f ∈ C +1 ([a + δ 1 , b + δ 2 ], C), we have [1,6,32] ...
... In our previous work [6], we have developed the singular Euler-Maclaurin (SEM) expansion that makes the EM expansion applicable to physically relevant summand functions in one-dimension, including functions that exhibit an algebraic singularity. In this paper, we extend our previous work and generalise the SEM expansion from one dimension to lattices in an arbitrary number of dimensions. ...
... The lack of convergence of the one-dimensional EM expansion for functions with singularities is a well known problem [2], which can be overcome by means of the 1D SEM expansion [6]. In the same way as the derivation of the 1D SEM expansion relies on the standard EM expansion in d = 1, we will make use of the EM expansion in higher dimensions in order to show the existence of mathematical objects, be it functions or distributions, that appear in the multidimensional SEM expansion. ...
We extend the classical Euler–Maclaurin (EM) expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for a precise and fast evaluation of singular sums that appear in multidimensional long-range interacting systems. We find that the approximation error decays exponentially with the expansion order for band-limited functions and that the runtime is independent of the number of particles. First, the EM summation formula is generalised to lattices in higher dimensions, assuming a sufficiently regular summand function. We then develop this new expansion further and construct the singular Euler–Maclaurin expansion in higher dimensions, an extension of our previous work in one dimension, which remains applicable and useful even if the summand function includes a singular function factor. We connect our method to analytical number theory and show that all operator coefficients can be efficiently computed from derivatives of the Epstein zeta function. Finally we demonstrate the numerical performance of the expansion and efficiently compute singular lattice sums in infinite two-dimensional lattices, which are of relevance in condensed matter, statistical, and quantum physics. An implementation in mathematica is provided online along with this article.
... We show that the discrete lattice problem can be separated into a term that describes its continuous analogue, the continuum contribution, and a term that includes all information about the microstructure, the lattice contribution, hence demonstrating an equivalence between lattice and continuum. To this end, we apply the recently developed Singular Euler-Maclaurin (SEM) expansion [13][14][15], which generalizes the 300-year old Euler-Maclaurin summation formula, and extend it to nonlinear and multiatomic systems. The singular lattice sum is expressed in terms of an integral and a lattice contribution described by a differential operator, both of which are efficiently computable. ...
... with D a differential operator of infinite order, D ( ) its truncation up to order 2 + 1, and ∆ the Laplacian. The representation of the lattice contribution in Eq. (4) is called the Singular Euler-Maclaurin (SEM) expansion, a full derivation of which is provided in [13][14][15]. For Ω = R d the SEM operator D ( ) takes the particularly simple form ...
Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this work, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analogue, the integral contribution, and a term that fully resolves the microstructure, the lattice contribution. For any system dimension, any lattice, any power-law interaction and for linear, nonlinear, and multi-atomic lattices, we show that the lattice contribution can be described by a differential operator based on the multidimensional generalization of the Riemann zeta function, namely the Epstein zeta function. We determine the conditions under which this contribution becomes particularly relevant, demonstrating the existence of quasi scale-invariant lattice contributions in a wide range of fundamental physical phenomena. Our representation provides a broad set of tools for studying the analytical properties of the system and it yields an efficient numerical method for the evaluation of the arising lattice sums. We benchmark its performance by computing classical forces and energies in three important physical examples, in which the standard continuum approximation fails: Skyrmions in a two-dimensional long-range interacting spin lattice, defects in ion chains, and spin waves in a three-dimensional pyrochlore lattice with dipolar interactions. We demonstrate that our method exhibits the accuracy of exact summation at the numerical cost of an integral approximation, allowing for precise simulations of long-range interacting systems even at macroscopic scales. Finally, we apply our analytical tool set to the study of quantum spin lattices and derive anomalous quantum spin wave dispersion relations due to long-range interactions in arbitrary dimensions.
This work discusses two periods of transformation of the Proto-Kartvelian population: the first (L–XXV centuries BC) when the entire population spoke one Proto-Kartvelian language and lived in a relatively large area; the second period – (XXV–X centuries BC), when the population divided into three parts: Proto-Svan; speaking the Colchian-Georgian language and the third part was
scattered on the European continent.
The second period is described by two different mathematical models: a part of the Proto-Kartvelian speaking population went to Europe and slowly began the process of their assimilation on the European continent. The unknown function that determines the number of Proto-Kartvelian speaking people in Europe at the time is described by a Pearl - Verhulst-type mathematical model with variable coefficients that also take the assimilation process into account. The analytical
solution of the Cauchy problem is found in quadratures.
The population that remained primarily in former Asia and the Caucasus region was gradually divided into two groups: those who spoke the Proto-Svan and those who spoke Colchian-Georgian languages. To describe their interference and development, a mathematical model is used, which is described by a nonlinear dynamic system with nonlinear terms of self-limitation and takes into
account the unnatural reduction of the Colchian-Georgian population as a result of hostilities with neighboring peoples. For a dynamic system without nonlinear terms of self-constraint, in the case of certain relationships between variable coefficients, the first integral was found, by means of which the Bernoulli equation with variable coefficients was obtained for one of the unknown functions. In the case of constant coefficients of the dynamic system, for certain dependences
between the coefficients, the dynamic system follows the system of Lotka–Volterra equations, with corresponding periodic solutions.
For the general mathematical model (nonlinear terms of self-limitation and unnatural reduction of the Colchian-Georgian population due to hostilities with neighboring peoples) in two cases of certain interdependencies between constant coefficients, it is shown that the divergence of an unknown vector-function in the physically meaningful first quarter of the phase plane changes the sign when passing through some half-direct one. Taking into account the principle (theorem) of Bendixson, theorems have been proved, on the variability of the divergence of the vector field and the existence of closed trajectories in some singly connected domain of the point located on this half-direct (starting point of the trajectory).
Thus, for the general dynamic system, with some dependencies between constant coefficients, it is shown that there is no assimilation of the Proto-Svan population by the Colchian-Georgian population and these two populations (Colchian-Georgian and Proto-Svan) coexist peacefully in virtually the same region due to the transformation of the Proto-Kartvelian population.
