Accurate Expansions of Internal Energy and Specific Heat of Critical Two-Dimensional Ising Model with Free Boundaries

X. Wu, R. Zheng, Nikolay Izmailian, W. Guo

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

The bond-propagation algorithm for the specific heat of the two dimensional Ising model is developed and that for the internal energy is completed. Using these algorithms, we study the critical internal energy and specific heat of the model on the square lattice and triangular lattice with free boundaries. Comparing with previous works (Phys Rev E 86:041149, 2012; Phys Rev E 87:022124, 2013), we reach much higher accuracy ( 10 −28 ) of the internal energy and specific heat, compared to the accuracy 10 −11 of the internal energy and 10 −9 of the specific heat reached in the previous works. This leads to much more accurate estimations of the edge and corner terms. The exact values of all edge and corner terms are therefore conjectured. The accurate forms of finite-size scaling for the internal energy and specific heat are determined for the rectangle-shaped square lattice with various aspect ratios and various shaped triangular lattice. For the rectangle-shaped square and triangular lattices and the triangle-shaped triangular lattice, there is no logarithmic correction terms of order higher than 1/S , with S the area of the system. For the triangular lattice in rhombus, trapezoid and hexagonal shapes, there exist logarithmic correction terms of order higher than 1/S for the internal energy, and logarithmic correction terms of all orders for the specific heat.
Original languageEnglish
Pages (from-to)106-150
JournalJournal of Statistical Physics
Volume155
Issue number1
DOIs
Publication statusPublished - 2014

Fingerprint

free boundaries
Specific Heat
internal energy
Free Boundary
Triangular Lattice
Ising model
Ising Model
specific heat
Internal
heat
expansion
Energy
Square Lattice
Term
Logarithmic
rectangles
Rectangle
Higher Order
Rhombus
Trapezium or trapezoid

Bibliographical note

The full text is available free from the link given. The published version is available from http://dx.doi.org/10.1007/s10955-014-0942-x .

Keywords

  • two dimensional Ising model
  • free boundary
  • bond propagation algorithm

Cite this

Accurate Expansions of Internal Energy and Specific Heat of Critical Two-Dimensional Ising Model with Free Boundaries. / Wu, X.; Zheng, R.; Izmailian, Nikolay; Guo, W.

In: Journal of Statistical Physics, Vol. 155, No. 1, 2014, p. 106-150.

Research output: Contribution to journalArticle

@article{3b417630e173498b90635674f4e26737,
title = "Accurate Expansions of Internal Energy and Specific Heat of Critical Two-Dimensional Ising Model with Free Boundaries",
abstract = "The bond-propagation algorithm for the specific heat of the two dimensional Ising model is developed and that for the internal energy is completed. Using these algorithms, we study the critical internal energy and specific heat of the model on the square lattice and triangular lattice with free boundaries. Comparing with previous works (Phys Rev E 86:041149, 2012; Phys Rev E 87:022124, 2013), we reach much higher accuracy ( 10 −28 ) of the internal energy and specific heat, compared to the accuracy 10 −11 of the internal energy and 10 −9 of the specific heat reached in the previous works. This leads to much more accurate estimations of the edge and corner terms. The exact values of all edge and corner terms are therefore conjectured. The accurate forms of finite-size scaling for the internal energy and specific heat are determined for the rectangle-shaped square lattice with various aspect ratios and various shaped triangular lattice. For the rectangle-shaped square and triangular lattices and the triangle-shaped triangular lattice, there is no logarithmic correction terms of order higher than 1/S , with S the area of the system. For the triangular lattice in rhombus, trapezoid and hexagonal shapes, there exist logarithmic correction terms of order higher than 1/S for the internal energy, and logarithmic correction terms of all orders for the specific heat.",
keywords = "two dimensional Ising model, free boundary, bond propagation algorithm",
author = "X. Wu and R. Zheng and Nikolay Izmailian and W. Guo",
note = "The full text is available free from the link given. The published version is available from http://dx.doi.org/10.1007/s10955-014-0942-x .",
year = "2014",
doi = "10.1007/s10955-014-0942-x",
language = "English",
volume = "155",
pages = "106--150",
journal = "Journal of Statistical Physics",
issn = "0022-4715",
publisher = "Springer Verlag",
number = "1",

}

TY - JOUR

T1 - Accurate Expansions of Internal Energy and Specific Heat of Critical Two-Dimensional Ising Model with Free Boundaries

AU - Wu, X.

AU - Zheng, R.

AU - Izmailian, Nikolay

AU - Guo, W.

N1 - The full text is available free from the link given. The published version is available from http://dx.doi.org/10.1007/s10955-014-0942-x .

PY - 2014

Y1 - 2014

N2 - The bond-propagation algorithm for the specific heat of the two dimensional Ising model is developed and that for the internal energy is completed. Using these algorithms, we study the critical internal energy and specific heat of the model on the square lattice and triangular lattice with free boundaries. Comparing with previous works (Phys Rev E 86:041149, 2012; Phys Rev E 87:022124, 2013), we reach much higher accuracy ( 10 −28 ) of the internal energy and specific heat, compared to the accuracy 10 −11 of the internal energy and 10 −9 of the specific heat reached in the previous works. This leads to much more accurate estimations of the edge and corner terms. The exact values of all edge and corner terms are therefore conjectured. The accurate forms of finite-size scaling for the internal energy and specific heat are determined for the rectangle-shaped square lattice with various aspect ratios and various shaped triangular lattice. For the rectangle-shaped square and triangular lattices and the triangle-shaped triangular lattice, there is no logarithmic correction terms of order higher than 1/S , with S the area of the system. For the triangular lattice in rhombus, trapezoid and hexagonal shapes, there exist logarithmic correction terms of order higher than 1/S for the internal energy, and logarithmic correction terms of all orders for the specific heat.

AB - The bond-propagation algorithm for the specific heat of the two dimensional Ising model is developed and that for the internal energy is completed. Using these algorithms, we study the critical internal energy and specific heat of the model on the square lattice and triangular lattice with free boundaries. Comparing with previous works (Phys Rev E 86:041149, 2012; Phys Rev E 87:022124, 2013), we reach much higher accuracy ( 10 −28 ) of the internal energy and specific heat, compared to the accuracy 10 −11 of the internal energy and 10 −9 of the specific heat reached in the previous works. This leads to much more accurate estimations of the edge and corner terms. The exact values of all edge and corner terms are therefore conjectured. The accurate forms of finite-size scaling for the internal energy and specific heat are determined for the rectangle-shaped square lattice with various aspect ratios and various shaped triangular lattice. For the rectangle-shaped square and triangular lattices and the triangle-shaped triangular lattice, there is no logarithmic correction terms of order higher than 1/S , with S the area of the system. For the triangular lattice in rhombus, trapezoid and hexagonal shapes, there exist logarithmic correction terms of order higher than 1/S for the internal energy, and logarithmic correction terms of all orders for the specific heat.

KW - two dimensional Ising model

KW - free boundary

KW - bond propagation algorithm

U2 - 10.1007/s10955-014-0942-x

DO - 10.1007/s10955-014-0942-x

M3 - Article

VL - 155

SP - 106

EP - 150

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -