arXiv:cond-mat/9712253 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Large-q series expansion for the ground-state degeneracy of the q-state Potts antiferromagnet on the (3⋅12^{2}) lattice

Shan-Ho Tsai

Published Mar 1, 1998• 5 citations

Abstract

We calculate the large-$q$ series expansion for the ground-state degeneracy (= exponent of the ground-state entropy) per site of the $q$-state Potts antiferromagnet on the $(3\ensuremath{\cdot}{12}^{2})$ lattice, to order ${O(y}^{19})$, where $y=1/(q\ensuremath{-}1)$. We note a remarkable agreement, to ${O(y}^{18})$, between this series and a rigorous lower bound derived recently.

DOI: 10.1103/physreve.57.2686

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arXiv:cond-mat/9712253 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Large-q series expansion for the ground-state degeneracy of the q-state Potts antiferromagnet on the (3⋅12^{2}) lattice

Shan-Ho Tsai

Published Mar 1, 1998• 5 citations

arXiv HTML PDF

Abstract

We calculate the large-

q

series expansion for the ground-state degeneracy (= exponent of the ground-state entropy) per site of the

q

-state Potts antiferromagnet on the

(3\ensuremath{\cdot}{12}^{2})

lattice, to order

{O(y}^{19})

, where

y=1/(q\ensuremath{-}1)

. We note a remarkable agreement, to

{O(y}^{18})

, between this series and a rigorous lower bound derived recently.

DOI: 10.1103/physreve.57.2686

Related papers

Lower bounds on the ground-state entropy of the Potts antiferromagnet on slabs of the simple cubic lattice

Robert E. Shrock, Yan Xu

arXiv:1001.1958Physical Review EMar 31, 20102 citations

Ground state entropy of Potts antiferromagnets and the approach to the two-dimensional thermodynamic limit

Robert E. Shrock, Shan-Ho Tsai

We study the ground state degeneracy per site (exponent of the ground state entropy) $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ for the $q$ -state Potts antiferromagnet on infinitely long strips with width ${L}_{y}$ of 2D lattices \ensuremath{\Lambda} with free and periodic boundary conditions in the $y$ direction, denoted ${\mathrm{FBC}}_{y}$ and ${\mathrm{PBC}}_{y}.$ We show that the approach of $W$ to its 2D thermodynamic limit as ${L}_{y}$ increases is quite rapid; for moderate values of $q$ and ${L}_{y}\ensuremath{\simeq}4,$ $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ is within about 5% and ${O(10}^{\ensuremath{-}3})$ of the 2D value $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{(L}_{y}=\ensuremath{\infty}),q)$ for ${\mathrm{FBC}}_{y}$ and ${\mathrm{PBC}}_{y},$ respectively. The approach of $W$ to the 2D thermodynamic limit is proved to be monotonic (nonmonotonic) for ${\mathrm{FBC}}_{y}$ $({\mathrm{PBC}}_{y}).$ It is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states.We study the ground state degeneracy per site (exponent of the ground state entropy) $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ for the $q$-state Potts antiferromagnet on infinitely long strips with width ${L}_{y}$ of 2D lattices \ensuremath{\Lambda} with free and periodic boundary conditions in the $y$ direction, denoted ${\mathrm{FBC}}_{y}$ and ${\mathrm{PBC}}_{y}.$ We show that the approach of $W$ to its 2D thermodynamic limit as ${L}_{y}$ increases is quite rapid; for moderate values of $q$ and ${L}_{y}\ensuremath{\simeq}4,$ $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ is within about 5% and ${O(10}^{\ensuremath{-}3})$ of the 2D value $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{(L}_{y}=\ensuremath{\infty}),q)$ for ${\mathrm{FBC}}_{y}$ and ${\mathrm{PBC}}_{y},$ respectively. The approach of $W$ to the 2D thermodynamic limit is proved to be monotonic (nonmonotonic) for ${\mathrm{FBC}}_{y}$ $({\mathrm{PBC}}_{y}).$ It is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states.

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Shu-Chiuan Chang, Robert E. Shrock

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arXiv:cond-mat/0005236Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsOct 1, 200027 citations

We calculate the large-

q

series expansion for the ground-state degeneracy (= exponent of the ground-state entropy) per site of the

q

-state Potts antiferromagnet on the

(3\ensuremath{\cdot}{12}^{2})

lattice, to order

{O(y}^{19})

, where

y=1/(q\ensuremath{-}1)

. We note a remarkable agreement, to

{O(y}^{18})

, between this series and a rigorous lower bound derived recently.

We study the ground state degeneracy per site (exponent of the ground state entropy) $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ for the

q

-state Potts antiferromagnet on infinitely long strips with width

{L}_{y}

of 2D lattices \ensuremath{\Lambda} with free and periodic boundary conditions in the

y

direction, denoted

{\mathrm{FBC}}_{y}

and

{\mathrm{PBC}}_{y}.

We show that the approach of

W

to its 2D thermodynamic limit as

{L}_{y}

increases is quite rapid; for moderate values of

q

and ${L}_{y}\ensuremath{\simeq}4,$ $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ is within about 5% and ${O(10}^{\ensuremath{-}3})$ of the 2D value $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{(L}_{y}=\ensuremath{\infty}),q)$ for

{\mathrm{FBC}}_{y}

and

{\mathrm{PBC}}_{y},

respectively. The approach of

W

to the 2D thermodynamic limit is proved to be monotonic (nonmonotonic) for

{\mathrm{FBC}}_{y}

({\mathrm{PBC}}_{y}).

It is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states.