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M. Mueller et al. / Nuclear Physics B 914 (2017) 388–404 391
2.2. Periodic boundary conditions
When periodic boundary conditions are imposed, we map Lσ’s to Lτ’s without requiring
the condition τ
L
= σ
L
of the free boundary conditions. Since every configuration of τ’s can now
be made up from two configurations of σ ’s, this should be taken into account when relating
the partition functions expressed in terms of σ or τ . Explicitly, the transformations are now
given by τ
1
= σ
1
σ
2
, τ
2
= σ
2
σ
3
, ..., τ
L
= σ
L
σ
L+1
= σ
L
σ
1
, with an inverse relation of the form
σ
i
= σ
1
× τ
1
τ
2
τ
3
···τ
i−1
, and a direct consequence of the periodic boundary conditions is that
the constraint
L
i=1
τ
i
=
L
i=1
σ
2
i
= 1(7)
must be imposed on the τ -variables. This can be implemented in the partition function as
Z
1d, periodic
= 2
{τ }
exp
β
L
i=1
τ
i
δ
L
i=1
τ
i
, 1
, (8)
where the requisite factor of two takes account of the two-to-one σ -to-τ -mapping. Since
L
i=1
τ
i
=±1, it is possible to rewrite the Kronecker-δ function appearing in Eq. (8) as
Z
1d, periodic
=
{τ }
exp
β
L
i=1
τ
i
1 +
L
i=1
τ
i
(9)
which subsumes the factor of two. The partition function written in this form may now be
straightforwardly evaluated as the sum of two factorised terms,
Z
1d, periodic
=
⎡
⎣
L
i=1
τ
i
=±1
exp
(
βτ
i
)
+
L
i=1
τ
i
=±1
τ
i
exp
(
βτ
i
)
⎤
⎦
= 2
L
ch(β)
L
+ sh(β)
L
= 2
L
ch(β)
L
1 + th(β)
L
. (10)
The standard result for periodic boundary conditions, familiar from the transfer matrix calcu-
lation and numerous other approaches, is hence recovered. In the case of periodic boundary
conditions we can see that:
• The last spin, σ
L
, is included in the transformation,
• an additional f
actor of two appears in order to ensure the equivalence of the σ - and
τ -representations of the partition function,
• a constraint must be imposed on the product of all the τ -v
ariables resulting in two terms in
the partition function, corresponding to an additional correlation by comparison with free
boundary conditions.
The f
actor of two thus appears for different reasons in the τ -representation of the partition func-
tion in the free boundary case (summing over the last spin) and the periodic boundary case
(a two-to-one mapping between σ ’s and τ ’s).