An efficient algorithm for translationally invariant finite-size periodic lattice systems in one spatial dimension

Self-organized Monte Carlo simulations of 2D Ising ferromagnet on the square lattice are performed. The essence of the suggested simulation method is an artificial dynamics consisting of the well-known single-spin-flip Metropolis algorithm supplemented by a random walk on the temperature axis. The walk is biased towards the critical region through a feedback based on instantaneous energy and magnetization cumulants, which are updated at every Monte Carlo step and filtered through a special recursion algorithm. The simulations revealed the invariance of the temperature probability distribution function, once some self-organized critical steady regime is reached, which is called here noncanonical equilibrium. The mean value of this distribution approximates the pseudocritical temperature of canonical equilibrium. In order to suppress finite-size effects, the self-organized approach is extended to multi-lattice systems, where the feedback basis on pairs of instantaneous estimates of the fourth-order magnetization cumulant on two systems of different size. These replica-based simulations resemble, in Monte Carlo lattice systems, some of the invariant statistical distributions of standard self-organized critical systems.

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Transfer Matrix Study of Finite-size Corrections in the 2D Ising Model

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2005-0003 ◽

2005 ◽

Vol 5 (1) ◽

pp. 72-85 ◽

Cited By ~ 1

Author(s):

J. Kaupužs

Keyword(s):

Correlation Function ◽

Ising Model ◽

Transfer Matrix ◽

Finite Size ◽

Lattice Systems ◽

Small Magnitude ◽

Amplitude Correction ◽

Matrix Algorithms ◽

Point Correlation ◽

Point Correlation Function

AbstractEffective exact transfer matrix algorithms have been developed to compute the two-point correlation function G(r) of the 2D Ising model on a square finite size lattice. Systems including up to 800 spins have been considered and corrections to the finite-size scaling at the critical point have been analysed.As a new result, we have found that the correlation function has a nontrivial amplitude correction of a very small magnitude.

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Distribution function of finite-size lattice systems per particle in the grand canonical ensemble

The Journal of Physical Chemistry ◽

10.1021/j100165a048 ◽

1991 ◽

Vol 95 (12) ◽

pp. 4853-4856

Author(s):

A. V. Vernov ◽

A. V. Andryushin ◽

V. I. Shimulis

Keyword(s):

Distribution Function ◽

Canonical Ensemble ◽

Finite Size ◽

Grand Canonical Ensemble ◽

Lattice Systems ◽

Grand Canonical

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NUMERICAL CALCULATION OF THE COMPLEX BERRY PHASE IN NON-HERMITIAN SYSTEMS

Acta Polytechnica ◽

10.14311/ap.2017.57.0470 ◽

2017 ◽

Vol 57 (6) ◽

pp. 470 ◽

Cited By ~ 13

Author(s):

Marcel Wagner ◽

Felix Dangel ◽

Holger Cartarius ◽

Jörg Main ◽

Günter Wunner

Keyword(s):

Berry Phase ◽

Tight Binding ◽

Analytical Calculation ◽

Periodic Lattice ◽

Topological Phases ◽

Lattice Systems ◽

One Dimensional ◽

Berry Phases ◽

Biorthogonal Basis ◽

Smoothing Procedure

We numerically investigate topological phases of periodic lattice systems in tight-binding description under the influence of dissipation. The effects of dissipation are effectively described by <em>PT</em>-symmetric potentials. In this framework we develop a general numerical gauge smoothing procedure to calculate complex Berry phases from the biorthogonal basis of the system's non-Hermitian Hamiltonian. Further, we apply this method to a one-dimensional <em>PT</em>-symmetric lattice system and verify our numerical results by an analytical calculation.

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The droplet formation-dissolution transition in different ensembles: Finite-size scaling from two perspectives

10.21468/scipostphys.5.6.062 ◽

2018 ◽

Vol 5 (6) ◽

Author(s):

Franz Paul Spitzner ◽

Johannes Zierenberg ◽

Wolfhard Janke

Keyword(s):

Canonical Ensemble ◽

Methodological Approach ◽

Cluster Formation ◽

Finite Size ◽

Droplet Formation ◽

Experimental Investigations ◽

Finite Size Scaling ◽

Lattice Systems ◽

Fixed Temperature ◽

Size Scaling

The formation and dissolution of a droplet is an important mechanism related to various nucleation phenomena. Here, we address the droplet formation-dissolution transition in a two-dimensional Lennard-Jones gas to demonstrate a consistent finite-size scaling approach from two perspectives using orthogonal control parameters. For the canonical ensemble, this means that we fix the temperature while varying the density and vice versa. Using specialised parallel multicanonical methods for both cases, we confirm analytical predictions at fixed temperature (rigorously only proven for lattice systems) and corresponding scaling predictions from expansions at fixed density. Importantly, our methodological approach provides us with reference quantities from the grand canonical ensemble that enter the analytical predictions. Our orthogonal finite-size scaling setup can be exploited for theoretical and experimental investigations of general nucleation phenomena – if one identifies the corresponding reference ensemble and adapts the theory accordingly. In this case, our numerical approach can be readily translated to the corresponding ensembles and thereby proves very useful for numerical studies of equilibrium cluster formation, in general.

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Wave polarization and dynamic degeneracy in a chiral elastic lattice

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2019.0313 ◽

2019 ◽

Vol 475 (2232) ◽

pp. 20190313 ◽

Cited By ~ 1

Author(s):

G. Carta ◽

I. S. Jones ◽

N. V. Movchan ◽

A. B. Movchan

Keyword(s):

Dynamic Response ◽

Shear Waves ◽

Low Frequency ◽

Elementary Cell ◽

Periodic Lattice ◽

Wave Polarization ◽

Lattice Systems ◽

Systematic Analysis ◽

Practical Applications ◽

Elastic Lattice

This paper addresses fundamental questions arising in the theory of Bloch–Floquet waves in chiral elastic lattice systems. This area has received a significant attention in the context of ‘topologically protected’ waveforms. Although practical applications of chiral elastic lattices are widely appreciated, especially in problems of controlling low-frequency vibrations, wave polarization and filtering, the fundamental questions of the relationship of these lattices to classical waveforms associated with longitudinal and shear waves retain a substantial scope for further development. The notion of chirality is introduced into the systematic analysis of dispersive elastic waves in a doubly-periodic lattice. Important quantitative characteristics of the dynamic response of the lattice, such as lattice flux and lattice circulation, are used in the analysis along with the novel concept of ‘vortex waveforms’ that characterize the dynamic response of the chiral system. We note that the continuum concepts of pressure and shear waves do not apply for waves in a lattice, especially in the case when the wavelength is comparable with the size of the elementary cell of the periodic structure. Special critical regimes are highlighted when vortex waveforms become dominant. Analytical findings are accompanied by illustrative numerical simulations.

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Static Kinks in Chains of Interacting Atoms

Condensed Matter ◽

10.3390/condmat5020035 ◽

2020 ◽

Vol 5 (2) ◽

pp. 35

Author(s):

Haggai Landa ◽

Cecilia Cormick ◽

Giovanna Morigi

Keyword(s):

Power Law ◽

Gordon Equation ◽

Ion Trap ◽

Finite Size ◽

Coulomb Repulsion ◽

Periodic Lattice ◽

Sine Gordon Equation ◽

Integral Term ◽

Power Law Exponent ◽

A Chain

We theoretically analyse the equation of topological solitons in a chain of particles interacting via a repulsive power-law potential and confined by a periodic lattice. Starting from the discrete model, we perform a gradient expansion and obtain the kink equation in the continuum limit for a power-law exponent n ≥ 1 . The power-law interaction modifies the sine-Gordon equation, giving rise to a rescaling of the coefficient multiplying the second derivative (the kink width) and to an additional integral term. We argue that the integral term does not affect the local properties of the kink, but it governs the behaviour at the asymptotics. The kink behaviour at the center is dominated by a sine-Gordon equation and its width tends to increase with the power law exponent. When the interaction is the Coulomb repulsion, in particular, the kink width depends logarithmically on the chain size. We define an appropriate thermodynamic limit and compare our results with existing studies performed for infinite chains. Our formalism allows one to systematically take into account the finite-size effects and also slowly varying external potentials, such as for instance the curvature in an ion trap.

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