One-dimensional q-state Potts model with multi-site interactions

0.27 MB·

by  L. Turban

7 One-dimensional q-state Potts model with multi-site
1
0 interactions
2

n
a
Lo¨ıc Turban
J

1 Groupe de Physique Statistique, Institut Jean Lamour, Universit´e de Lorraine,
3
CNRS (UMR 7198), Vandœuvre l`es Nancy Cedex, F-54506,France

] E-mail: [email protected]
h
c
e
Abstract. A one-dimensional (1D) q-state Potts model with N sites, m-site
m
interaction K in a field H is studied for arbitrary values of m. Exact results for
-
t the partition function and the two-point correlation function are obtained at H = 0.
a
t The system in a field is shown to be self-dual. Using a change of Potts variables, it is
s
. mapped onto a standard 2D Potts model, with first-neighbour interactions K and H,
t
a onacylinderwithhelicalboundaryconditions(BC).The2DsystemhasalengthN/m
m
and a transverse size m. Thus the Potts chain with multi-site interactions is expected
- to develop a 2D critical singularity along the self-duality line, (eqK −1)(eqH −1)=q,
d
n when N/m→∞ and m→∞.
o
c
[

Keywords: Potts model, multi-site interactions, self-duality, helical boundary
1
conditions
v
8
5
0
9 Submitted to: J. Phys. A: Math. Theor.
0
.
1
1. Introduction
0
7
1
The standard Potts model is a lattice statistical model with pair interactions between
:
v
q-state variables attached to neighbouring sites [1,2]. Multi-site Potts models can be
i
X
constructed by extending to an arbitrary number of states existing multispin Ising
r
a models for which q = 2. In this way, a self-dual three-site Potts model on the triangular
lattice was introduced by Enting [3,4], which corresponds to the Baxter-Wu model [5,6]
when q = 2. Similarly, a 2D self-dual Potts model with m-site interactions in one
direction and n-site interactions in the other [7–9] follows from the Ising version with
n = 1 [10].
Multi-site interactions may be generated from two-site interactions in a position-
space renormalisation group transformation and thus have to be included in the initial
Hamiltonian. In this way Schick andGriffiths have introduced a three-state Potts model
on the triangular lattice with two- and three-site interactions [11]. For any value of q it
can been reformulated as a standard q-state Potts model with two-site interactions on a
Potts model with multi-site interactions 2
3-12 lattice [12]. When the three-site interactions are restricted to up-pointing triangles,
the model is self-dual [13,14] and related to a 20-vertex model [13,15].
Extending the results of Fortuin and Kasteleyn [16] for pair interactions, a
random-cluster representation for Potts models with multi-site interactions has been
introduced [17–19] and exploited in Monte Carlo simulations [20].
Multi-site interactions enter naturally when the site percolation process is
formulated as a Potts model in the limit q → 1 [21–25]. Various Potts multi-site
interactions have also been used to model conformational transitions in polypeptide
chains [26–29].
With s = 0,1,...,q − 1 denoting a q-state Potts variable attached to site j, a
j
multi-site interaction can take one of the following forms
m−1 m−1
(a) −K δ , (b) −Kδ s , (1.1)
sj,sj+1 q j+l
!
j=1 l=0
Y X
where δn,n′ is the standard Kronecker delta and δq(n) is a Kronecker delta modulo q.
When K > 0 the ground state is q-times degenerate in the first case (the standard one)
whereas the degeneracy depends on m and is given by qm−1 in the second case. As an
example, when q = m = 3 the degenerate ground states are the following ones:
000 000 012 210
(a) 111 , (b) 111 120 021 (1.2)
 
 222  222 201 102
In the present work we generalize for q-state Potts variables some results recently
 
obtained for the 1D Ising model with multispin interactions [30]. The Hamiltonian of
the q-state Potts chain takes the following form
m−1
−βH [{s}] = K qδ s −1 +H [qδ (s )−1] , β = (k T)−1. (1.3)
N q j+l q j B
" ! #
j l=0 j
X X X
where m > 1. We assume ferromagnetic interactions K ≥ 0 and H ≥ 0, too. The
Kronecker delta modulo q is given by:
q−1
1 2iπks 1 when s = 0 (mod q)
δ (s) = exp = . (1.4)
q
q q ( 0 otherwise
k=0 (cid:18) (cid:19)
X
Introducing the Potts spins [31,32]
2iπs
j
σ = exp , (1.5)
j
q
(cid:18) (cid:19)
the Hamiltonian in (1.3) can be rewritten as †:
q−1 m−1 q−1
−βH [{σ}] = K σk +H σk. (1.6)
N j+l j
j k=1 l=0 j k=1
XX Y XX
† One may also express the Potts interaction using clock angular variables (see appendix A).
Potts model with multi-site interactions 3
t t
j N
s j+3p=N
j t
N−2
t t
j N−1
s j+3p=N−1
j t
N
t t
j N−2
s j+3p=N−2
j t
N−1
Figure 1. t-variables entering into the expression (2.3) of s for m = 3 and different
j
valuesofthedistanceN−j fromtheendofthechain. Thet-variables,definedin(2.2),
are the sums of m s-variables (circles) with the convention s = 0 when i > N. The
i
second lines are subtracted from the first so that only s is remaining.
j
When q = 2, σ = ±1, k = 1 and the Ising multispin Hamiltonian studied in [30] is
j
recovered, which a posteriori justifies the choice of interaction (b) in (1.3).
The zero-field partition function of the Potts chain with m-site interaction K is
obtained for free BC in section 2 and for periodic BC in section 3. The periodic BC
result allowsadeterminationoftheeigenvalues ofTm whereTisthesite-to-sitetransfer-
matrix. The two-site correlation function is calculated in section 4. In section 5 the
system with periodic BC is shown to be self-dual when the external field H is turned on.
In section 6 the system with free BC is mapped onto a standard 2D Potts model with
first-neighbour interactions K and H, length N/m and transverse size m. The mapping
of 1D Potts models with m-site and n-site interactions is discussed in section 7. The
conclusion in section 8 is followed by 4 appendices.
2. Zero-field partition function for free BC
With free BC the zero-field Hamiltonian of a chain with N Potts spins, with m-site
interaction K, takes the following form
N−m+1 m−1
(f)
−βH [{s}] = K qδ s −1 (2.1)
N q j+l
" ! #
j=1 l=0
X X
when written interms of the Potts variables s . Let us introduce the new Potts variables
j
t = 0,...,q−1 defined as
j
m−1
t = s (mod q), j = 1,...,N , (2.2)
j j+l
l=0
X
Potts model with multi-site interactions 4
with the convention s = 0 when i > N in (2.2). Note that the relationship between old
i
and new variables is one-to-one with the inverse transformation given by (see figure 1):
p
s = (t −t ) (mod q), j +pm = N −l, l = 0,...,m−1. (2.3)
j rm+j rm+j+1
r=0
X
Using (2.2) in (2.1) one obtains a system of N −m+1 non-interacting Potts spins in a
field K with
N−m+1
(f)
−βH [{t}] = K [qδ (t )−1]. (2.4)
N q j
j=1
X
The canonical partition function is easily obtained and reads:
N−m+1 N
ZN(f) = Tr{t}e−βH(Nf)[{t}] = Trtj eK[qδq(tj)−1] Trtj 1
j=1 j=N−m+2
Y Y
= qm−1 e(q−1)K +(q −1)e−K N−m+1 . (2.5)
Note that although only N(cid:2) − m + 1 new vari(cid:3)ables enter into the expression of the
transformed Hamiltonian (2.4), one has to trace over the N Potts variables t in (2.5).
j
When q = 2 the Ising result (equation (2.6) in [30]) is recovered.
The free energy can be decomposed as follows
(f) (f)
F = −k T lnZ = Nf +F (m), (2.6)
N B N b s
where the bulk free energy per site
f = −k T ln e(q−1)K +(q −1)e−K , (2.7)
b B
does not depend on m wher(cid:2)eas the surface contri(cid:3)bution
exp[(q −1)K]+(q −1)exp(−K)
F (m) = (m−1)k T ln , (2.8)
s B
q
(cid:26) (cid:27)
is m-dependent.
3. Zero-field partition function for periodic BC
Let us now evaluate the partition function for a periodic chain with N sites and m > 1.
To simplify the discussion we consider only the case where N is a multiple of m. Then
the Hamiltonian takes the following form
N=pm m−1
(p)
−βH [{s}] = K qδ s −1 , (3.1)
N=pm q j+l
" ! #
j=1 l=0
X X
with s = s . Making use of the change of variables (2.2), it can be rewritten as:
N+j j
N=pm
(p)
−βH [{t}] = K [qδ (t )−1]. (3.2)
N=pm q j
j=1
X
Potts model with multi-site interactions 5
With periodic BC the correspondence between {s} and {t} Potts configurations
is no longer one-to-one and the new variables have to satisfy a set of m − 1
constraints [30,33–35].
There are several {s} configurations leading to the same {t}. One of these
configurations, {s′}, is obtained by changing s into
j
s′ = s +∆ (mod q), j = 1,...,N . (3.3)
j j j
where the shifts ∆ = 0,...,q−1 have to satisfy some constraint. Let us first consider
j
t = m−1s (mod q). One can freely choose the first m−1 shifts (qm−1 choices)
1 l=0 l+1
and t keeps its value when ∆ is such that m−1∆ = 0 (mod q). Since t and t
1P m l=0 l+1 j j+1
have the shifts ∆ (l = 1,...,m−1 in common, the value of ∆ leaving t invariant is
j+l P j j
equal to the value of ∆ leaving t invariant. When N = pm a periodic repetition
j+m j+1
with period m of the first m shifts acting on {s} leaves {t} invariant. Thus there are
qm−1 Potts configurations {s′} leading to the same {t} ‡. When {s} is a ground-state
configuration, qm−1 gives the ground-state degeneracy.
In the following we shall make use of the Potts spin variables:
m−1
2iπt 2iπt
τ = exp j = σ , τ∗ = exp − j = τq−1, τ τ∗ = τq = 1. (3.4)
j q j+l j q j j j j
(cid:18) (cid:19) l=0 (cid:18) (cid:19)
Y
According to (1.4) one has:
q−1
2iπkt
Tr τk = exp j = qδ (k). (3.5)
τj j q q
tXj=0 (cid:18) (cid:19)
For later use, note that the Boltzmann factor
N=pm
e−βH(Np)[{t}] = e−K + e(q−1)K −e−K δq(tj) , (3.6)
j=1
Y (cid:2) (cid:0) (cid:1) (cid:3)
can be rewritten as
N=pm eqK −1 q−1
e−βH(Np)[{τ}] = e−NK 1+ τk
q j
" #
j=1 k=0
Y X
e−K N N=pm q−1
= eqK +q −1)+ eqK−1 τk , (3.7)
q j
" #
(cid:18) (cid:19) j=1 k=1
Y (cid:0) (cid:1)X
using (1.4) and (3.4),
Let us consider the product of Potts spins
p−1
P = τ τ∗ , i = 1,...,m−1. (3.8)
i rm+i rm+i+1
r=0
Y
Making use of
m−1
τ τ∗ = σ σ σ∗ σ∗ = σ σ∗ (3.9)
rm+i rm+i+1 rm+i rm+i+l rm+i+l (r+1)m+i rm+i (r+1)m+i
!
l=1
Y
‡ Note that the initial configuration, {s}, corresponding to ∆ =0 ∀j, is taken into account.
j
Potts model with multi-site interactions 6
and taking into account the periodic BC, one obtains the constraints
p−1
P = σ σ∗ = 1, i = 1,...,m−1, (3.10)
i rm+i (r+1)m+i
r=0
Y
to be satisfied by the τ-configurations in (3.8). When m > 2 other constraints can be
constructed, for instance from τ τ∗ , but these are automatically satisfied since
rm+i rm+i+2
they can be written as products of the fundamental ones: τ τ∗ τ τ∗ .
rm+i rm+i+1 rm+i+1 rm+i+2
1
Thus with the new Potts spin variables, taking the constraints into account, the
| {z }
partition function is given by:
m−1
ZN(p=)pm = qm−1Tr{τ}e−βH(Np)[{τ}] δPi,1. (3.11)
i=1
Y
To go further we need an explicit expression for the Kronecker delta, δ . Consider the
Pi,1
geometric series
q−1 1−Xq
f(X) = Xk = , (3.12)
1−X
k=0
X
it vanishes when X is a qth root of unity other than 1 and is equal to q when X = 1.
Since P in (3.8) is a qth root of unity, the constraint can be written as (cf. (1.4))
i
q−1 q−1 p−1
1 1
δ = Pk = τk τq−k , (3.13)
Pi,1 q i q rm+i rm+i+1
k=0 k=0 r=0
X XY
where (3.4) has been used. The partition function in (3.11) now takes the following
form:
m−1 q−1 p−1
ZN(p=)pm = Tr{τ}e−βH(Np)[{τ}] 1+ Pik , Pik = τrkm+iτrqm−+ki+1. (3.14)
!
i=1 k=1 r=0
Y X Y
The first product has the following expansion:
m−1 q−1
1+ Pk = 1+ Pk + PkPk′ + PkPk′Pk′′ +···+ Pq−1.(3.15)
i i i i′ i i′ i′′ i
i=1 k=1 ! i,k i<i′,k,k′ i<i′<i′′,k,k′,k′′ i
Y X X X X Y
The expression of Pk in (3.14) is periodic with period m. There are two consecutive
i
Potts spins contributing to the product for each period and the sum of their exponents
vanishes modulo q. Besides 1 the expansion (3.15) generates terms containing from
l = 2 to m spins for each period with m possible spatial configurations {α }. These
l l
spatial configurations are labelled by the l spin exponents, each varying from 1 to q−1
(cid:0) (cid:1)
with a sum which remains vanishing modulo q in the products, due to the Potts spins
properties (3.4). As shown in appendix B, for l spins the number ν of allowed exponent
l
distributions is given by:
1
ν = (q −1)l +(−1)l(q−1) . (3.16)
l
q
(cid:2) (cid:3)
Potts model with multi-site interactions 7
Combining these results, the expansion can be written as
(m)
m−1 q−1 m l νl
1+ Pk = 1+ Ξβl , (3.17)
i αl
!
Yi=1 Xk=1 Xl=2 αXl=1βXl=1
where Ξβl is a product for each period of l Potts spins in configuration α with an
αl l
exponent distribution β .
l
The partition function in (3.14) splits in two parts:
(m)
m l νl
ZN(p=)pm = Tr{τ}e−βH(Np)[{τ}]+ Tr{τ}e−βH(Np)[{τ}]Ξβαll . (3.18)
A Xl=2 αXl=1βXl=1 B
In A, according to (3.5), each of the pm factors in (3.7) contributes to the trace by:
| {z } | {z }
q−1
e−K
Tr eqK +q −1+ eqK −1 τk = e−K eqK +q −1 . (3.19)
q τj j
" #
k=1
(cid:0) (cid:1)X (cid:0) (cid:1)
In B, for each period, Ξβl contains l supplementary spin terms of the form τk′ with
αl j
k′ = 1,2,...,q − 1. Thus the trace involves p(m − l) factors given by (3.19) and pl
factors of the form
q−1
e−K
Tr eqK+q −1 τk′+ eqK−1 τk+k′ = e−K eqK−1 , (3.20)
q τj j j
" #
k=1
(cid:0) (cid:1) (cid:0) (cid:1)X (cid:0) (cid:1)
where the non-vanishing contribution comes from the term q − k′ in the sum over k
according to (3.5). Collecting the different contributions to the partition function, we
finally obtain
m
m
Z(p) = ν e(q−1)K+(q−1)e−K p(m−l) e(q−1)K−e−K pl
N=pm l l
l=0 (cid:18) (cid:19)
X (cid:2) (cid:3) (cid:2) (cid:3)
m m eqK −1 pl
= e(q−1)K+(q−1)e−K N 1+ ν , (3.21)
l
l eqK+q−1
" #
l=2 (cid:18) (cid:19) (cid:18) (cid:19)
(cid:2) (cid:3) X
where ν , given by (3.16), is such that ν = 1 and ν = 0. For q = 2
l 0 1
1 when l is even
ν = (3.22)
l
( 0 when l is odd
and the Ising result, equation (3.13) in [30], is recovered.
Let T be the transfer matrix from |σ σ ...σ i to |σ σ ...σ i. As
j j+1 j+m−2 j+1 j+2 j+m−1
discussed in appendix C, its mth power is real and symmetric. The real eigenvalues
of Tm, ω , and their degeneracy, g , can be deduced from the expression (3.21) of the
l l
partition function (see (C.6)).
Potts model with multi-site interactions 8
4. Zero-field correlation function
In this section the zero-field correlation function is obtained for free BC and m > 1.
The correlations between the Potts variables at sites i and i′ are evaluated by taking
the thermal average of the following expression:
qδq(si −si′)−1
. (4.1)
(q−1)
It is equal to one when the two sites are in the same state and has a vanishing average
in a fully disordered system. Making use of (1.4) and (1.5) the numerator in (4.1) can
be expressed in terms of Potts spins as:
q−1 q−1
qδq(si −si′)−1 = exp 2iπk(sqi −si′) = σikσi∗′k. (4.2)
k=1 (cid:20) (cid:21) k=1
X X
Let us first suppose that i′ = i+rm. Taking into account (3.9) one may write
r−1
σiσi∗+rm = τr′m+iτr∗′m+i+1, (4.3)
r′=0
Y
and the correlation function takes the following form:
G(f)(i,i+rm) = qδq(si −si+rm)−1 = Tr e−βH(Nf)[{τ}] q−1 r−1 τk τ∗k . (4.4)
N (q−1) {τ} (q−1)Z(f) r′m+i r′m+i+1
(cid:28) (cid:29) N k=1r′=0
X Y
Following the same steps that led to (3.7), the Boltzmann factor for free BC can be
written as
e(q−1)K +(q −1)e−K N−m+1 N−m+1 eqK −1 q−1
e−βH(Nf)[{τ}] = 1+ τk′
q eqK +q −1 j
(cid:20) (cid:21) j=1 " k′=1 #
Y X
Z(f) N−m+1 eqK −1 q−1
= N 1+ τk′ (4.5)
qN eqK +q −1 j
j=1 " k′=1 #
Y X
(f)
where the expression of Z in (2.5) has been used §. Inserting this expression in (4.4),
N
one obtains:
q−1 r−1
1
G(f)(i,i+rm) = Tr τk τ∗k ×
N qN(q −1) {τ} r′m+i r′m+i+1
k=1 r′=0
X Y
N−m+1 eqK −1 q−1
× 1+ τk′ , (4.6)
eqK +q −1 j
" #
j=1 k′=1
Y X
The trace over {τ} contains r factors with j = r′m+i of the form
q−1
eqK −1 eqK −1
Tr τk + τk+k′ = q , (4.7)
τr′m+i r′m+i eqK +q −1 r′m+i eqK +q −1
" k′=1 #
X
§ Takingthe traceoverthe N Pottsspinsin(4.5),allthe termsintheproductinvolvingτ vanishand
j
the trace over 1 gives qN.
Potts model with multi-site interactions 9
theonlynon-vanishingcontributioncomingfromthesecondtermfork′ = q−k according
to (3.5). The same result is obtained for the r factors with j = r′m+i+1 and k′ = k.
The trace over the remaining N − 2r Potts spins contributes a factor qN−2r, the sum
over k gives q −1, so that, finally:
eqK −1 2r rm
(f)
G (i,i+rm) = = exp − . (4.8)
N eqK +q −1 ξ
(cid:20) (cid:21) (cid:18) (cid:19)
As expected, this expression can be rewritten in terms of transfer matrix
eigenvalues (C.6) as (ω /ω )r. The correlation length, given by
2 0
m eqK +q −1 −1
ξ = ln , (4.9)
2 eqK −1
(cid:20) (cid:18) (cid:19)(cid:21)
diverges at the zero-temperature critical point when K → ∞.
Let us now consider the case where i′ −i is not a multiple of m. Using the Potts
spin variables (1.5) and (3.4) the inverse transformation in (2.3) translates into:
p
σi = τr′m+iτr∗′m+i+1, i+l = N −pm, l = 0,...,m−1.(4.10)
r′=0
Y
In the same way let
p−r
σi∗′ = τr∗′m+i′τr′m+i′+1, i′+l′ = N−pm+rm, l′ = 0,...,m−1,(4.11)
r′=0
Y
with, in both cases, τ = 1 when j > N. Since i′−i = rm+l−l′, we need l 6= l′. In the
j
product σ σ∗, the last factor contributed by σ is either τ∗ or τ when l = 0 whereas
i i′ i N−l+1 N
for σj∗′ it is either τN−l′+1 or τN∗ when l′ = 0. Thus these factors cannot all disappear
in the product when l 6= l′. At least one of them leads to a vanishing trace over {τ} in
the correlation function since the product over j in the Boltzmann factor (4.5) ends at
N −m+1. It follows that:
G(f)(i,i′) = 0, i′ −i 6= rm. (4.12)
N
When m = 2 and q = 2 this argument no longer applies. With m = 2 the τ and the τ∗
j j
always appear twice inthe product σ σ∗ for values ofj ≥ i′. Accordingly, the correlation
i i′
function does not vanish since τ2 = τ∗2 = 1 when q = 2. The difference between q = 2
j j
andq > 2 when m = 2 canbe understoodby looking atthe behaviour ofthecorrelations
in the ground state. For q = 2 there are 2 degenerate ground states which, using Potts
variables, are given by 00000... and 11111... so that h2δq(si − si′) − 1i = 1. When
q = 3, for example, there are 3 degenerate ground states, 00000..., 12121... and
21212..., leading to h3δq(si −si′)−1i = 0 when i′ −i is odd.
5. Self-duality under external field
In this section standard methods [4,7,36] are used to show that the Potts chain with
multi-site interactions and periodic BC is self-dual under external field.
Potts model with multi-site interactions 10
s~ a s~
i+1 i+3/2
s s s s s s s
i i+1 i+2 i i+1 i+2 i+3
s~ s~ s~ s~ s~ s~ s~
i−1 i i+1 i−3/2 i−1/2 i+1/2 i+3/2
s s
i i
m=3 m=4
Figure2. PositionofthedualPottsvariables(squares)enteringinthedefinitions(5.6)
of u and v relative to the original ones (circles) for odd and even values of m.
i i
According to (1.3), the partition function is given by:
N m−1
Z(p)(K,H) = e−N(K+H)Tr exp qKδ s exp[qHδ (s )] . (5.1)
N {s} q j+l q j
" !#
j=1 l=0
Y X
Introducing the auxiliary function
C(X,x) = eqX −1+qδ (x), (5.2)
q
one obtains the identity:
q−1
1 2iπxy
eqXδq(y) = 1+ eqX −1 δ (y) = C(X,x)exp . (5.3)
q
q q
x=0 (cid:18) (cid:19)
(cid:0) (cid:1) X
Thus the partition function can be rewritten as:
e−N(K+H) N q−1 q−1
(p)
Z (K,H) = Tr C(K,u )C(H,v )
N qN {s} j j
Yj=1uXj=0vXj=0
m−1
1 2iπ
× exp v s +u s . (5.4)
j j j j+l
q q
" !#
l=0
X
Regrouping the factors containing s in the last exponential and reordering the sums,
i
one obtains
e−N(K+H) N N 1 q−1 2iπs w
(p) i i
Z (K,H) = Tr C(K,u )C(H,v ) exp
N qN {u,v} j j q q
Yj=1 Yi=1 sXi=0 (cid:18) (cid:19)
e−N(K+H) N N
= Tr C(K,u )C(H,v ) δ (w ), (5.5)
qN {u,v} j j q i
j=1 i=1
Y Y
where w stands for v + m−1u .
i i l=0 i−l
P