Tsallis scaling and the long-range Ising chain: A transfer matrix approach
R. F. S. Andrade and S. T. R. Pinho
Instituto de Física, Universidade Federal da Bahia, 40210-340 Salvador, Brazil
Received 8 June 2004; revised manuscript received 3 December 2004; published 25 February 2005
A numerically efficient transfer matrix TM approach is introduced to investigate the long-range Ising spin
chain. Results obtained within this procedure are primarily used to verify the Tsallis scaling hypothesis for
long-range systems with an power-law decay of the interaction constants, both in the extensive 1 and
nonextensive 1 regimes. Results for finite-size systems, taking into account all interactions between spins
up to 24 sites apart, show that the conjecture is satisfied with a very good precision less than 0.004% for all
temperature intervals. This TM procedure is further used to investigate several other thermodynamic and
critical properties of this system, and it may also be extended to similar one-dimensional long-range systems.
DOI: 10.1103/PhysRevE.71.026126 PACS numbers: 05.50.+q, 05.70.Fh, 75.10.Pq
I. INTRODUCTION
The long-range Ising chain constitutes a classical chal-
lenge that has attracted the attention of physicists for many
decades. In its more common version, each spin
i
interacts
with all other spins on the chain mediated by coupling con-
stants J
r
= J / r
, where r is the distance between the interact-
ing spins measured in integer number of lattice spacings.
Despite the absence of a closed solution, much is known
about this system. Existence theorems for phase transitions
have been obtained by a large number of authors, e.g., 1–4.
On the other hand, sophisticated numerical schemes have led
to very precise estimates for values of the critical tempera-
ture, T
c
, and critical exponents, as reported, e.g., in 5–9.
This large number of contributions have indicated that, for
2, the system shows only a disordered phase, 2200T 4.
Phase transition at finite temperature is found for 1 2
1,2, with the presence of an ordered phase when T T
c
. For
0 1, the system is nonextensive 10 and it has a single
ordered phase 2200T. Classical mean-field exponents are found
when 1 1.5 4, while for = 2 a discontinuous magne-
tization at T
c
is observed 2.
The purpose of this work is to investigate the validity of a
conjecture raised by Tsallis 11 to the Ising long-range
chain, using a first-principles solution that comes as close as
possible to the exact one. Our solution is provided by a trans-
fer matrix TM approach that leads to numerical results tak-
ing into account the long-range interaction between spins up
to a certain distance g apart. It has been optimized regarding
both the required space to store the energy values for all
distinct spin configurations and the successive increase in the
value of g as well as avoiding the numerical evaluation of the
TM largest eigenvalue.
In 1995, Tsallis conjectured a universal scaling scheme
for thermodynamic functions that should be valid for a large
class of both extensive and nonextensive long-range models.
For the later models, the energy per degree of freedom di-
verges, so that the usual intensive energies are devoid of
significance. The Tsallis scaling conjecture TS10,11
states that any intensive energylike thermodynamic property,
e.g., in the case of magnetic systems, the free Gibbs energy,
f T , H , N = FT , H , N / N, of a finite system of N constitu-
ents, in a d-dimensional space, with long-range interaction
decaying with the distance r between particles as 1 / r
, can
be properly described by an N-independent function
f
˜
T
˜
= T/N
˜
, H
˜
= H/N
˜
= f T, H, N/N
˜
1
with the help of a scaling variable N
˜
, defined as
N
˜
=
N
1-/d
- /d
1- /d
. 2
When N → , we obtain limit values for N
˜
as a function of d
and ,
N
˜
=
/d
/d -1
if /d 1
ln N if /d =1
1
1- /d
N
1-/d
if 0 /d 1.
3
According to the same conjecture, other intensive thermody-
namic functions depending on the temperature T, the mag-
netic field H, and also on N, like the entropy s, specific heat
c, and magnetization m, admit N-independent related func-
tions defined as
s ˜T
˜
, H
˜
= sT, H, N ,
c ˜ T
˜
, H
˜
= cT, H, N ,
m ˜ T
˜
, H
˜
= mT, H, N . 4
TS takes into account the divergence in the definition of
usual intensive quantities, as the free energy f , for nonexten-
sive models / d 1, and generalizes the ad hoc normaliza-
tion procedure to treat mean-field models, which simply con-
sists of replacing the single-coupling constant for all pairs of
particles J by J / N. This particular value is recovered in defi-
nition 2 when → 0. For extensive long-range models
/ d 1, the usual intensive energies as f will reach an
N-independent value when N → , but the scaling procedure
shows a much faster convergence to f
˜
, valid for finite-size
systems.
PHYSICAL REVIEW E 71, 026126 2005
1539-3755/2005/712/0261268/$23.00 ©2005 The American Physical Society 026126-1