# Infinite-range correlations in 1D systems with continuous symmetry

Tobias Rindlisbacher<EMAIL_ADDRESS>University of Helsinki,
Department of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland

###### Abstract

$\operatorname{O}\mathopen{}\mathclose{{}\left(N}\right)$-symmetric lattice
scalar fields are considered, coupled to a chemical potential and source
terms. At the example of $N=2$, it is shown that such systems can even in
(0+1) dimensions produce infinite-range correlations and a non-zero vacuum
expectation value whenever the chemical potential assumes certain discrete
values. Different mechanisms for how the latter phenomena are produced are
discussed, depending on whether source terms are set to zero or non-zero
values. In the conclusion, the relation of these findings to the Mermin-Wagner
theorem is addressed.

$\operatorname{O}\mathopen{}\mathclose{{}\left(N}\right)$ scalar field;
chemical potential; finite density; long-range order; Mermin-Wagner theorem.

##  I Introduction

Mermin and Wagner showed in 1966 Mermin , using Bogoliubov’s inequality
Bogoliubov , that in one and two dimensions the ferromagnetic
(antiferromagnetic) isotropic Heisenberg model with finite-range interaction
cannot undergo spontaneous magnetization (sub-lattice magnetization) at any
non-zero temperature. Isotropic here refers to the internal space in which the
Heisenberg spins $\bar{s}_{x}=(s_{x}^{1},s_{x}^{2},s_{x}^{3})$ take value, and
means that if we write the interaction between two spins $\bar{s}_{x}$,
$\bar{s}_{y}$, located on sites $x$ and $y$, as
$\sum_{i=1}^{3}\,\alpha_{i}\mathchoice{\mathopen{}\mathclose{{}\left(x-y}\right)}{(x-y)}{(x-y)}{(x-y)}\,s_{x}^{i}\,s_{y}^{i}$,
then
$\alpha_{1}\mathchoice{\mathopen{}\mathclose{{}\left(r}\right)}{(r)}{(r)}{(r)}=\alpha_{2}\mathchoice{\mathopen{}\mathclose{{}\left(r}\right)}{(r)}{(r)}{(r)}=\alpha_{3}\mathchoice{\mathopen{}\mathclose{{}\left(r}\right)}{(r)}{(r)}{(r)}$
for all $r$. If, on the other hand, the couplings satisfy
$\alpha_{1}\mathchoice{\mathopen{}\mathclose{{}\left(r}\right)}{(r)}{(r)}{(r)}=\alpha_{2}\mathchoice{\mathopen{}\mathclose{{}\left(r}\right)}{(r)}{(r)}{(r)}\neq\alpha_{3}\mathchoice{\mathopen{}\mathclose{{}\left(r}\right)}{(r)}{(r)}{(r)}$,
so that the global
$\operatorname{O}\mathopen{}\mathclose{{}\left(3}\right)$-symmetry of the
isotropic case is reduced to
$\operatorname{O}\mathopen{}\mathclose{{}\left(2}\right)$, the authors mention
that the same line of reasoning would then only rule out spontaneous
magnetization (sub-lattice magnetization) in the $s^{1}$-$s^{2}$-plane (where
it would break the reduced
$\operatorname{O}\mathopen{}\mathclose{{}\left(2}\right)$-symmetry). The
authors also stressed, that the impossibility of spontaneous magnetisation
does not exclude the existence of other types of phase transitions in these
models. For the theorem to apply, it is important that long-range interactions
are sufficiently suppresssed Thouless ; Imry .
In 1967 Wegner Wegner  confirmed this for the ferromagnetic Heisenberg model
with reduced
$\operatorname{O}\mathopen{}\mathclose{{}\left(2}\right)$-symmetry ($\sim$XY-
model), by showing, based on a low-temperature expansion, that the spin-spin
correlation function in one and two dimensions always converges to zero at
sufficiently large distances. The Mermin-Wagner theorem has in the following
years been generalized to many other (non-relativistic) classical and quantum
systems, cf. Hohenberg ; Hamilton ; Moore ; Mermin2 ; Berezinsky1 ;
Berezinsky2 ; Kishore , to mention only a few. In 1973 Coleman Coleman  then
proved the analogue of the Mermin-Wager theorem Mermin  and the work of Wegner
Wegner  and Berezinsky Berezinsky1 ; Berezinsky2  for a relativistic scalar
field theory; showing that in two (i.e. $(1+1)$) dimensions, the vacuum
expectation value of a scalar field with a continuous global symmetry is
always zero, so that no Goldstone bosons can occur.

In this article, we consider non-linear
$\operatorname{O}\mathopen{}\mathclose{{}\left(N}\right)$ lattice models in
the presence of a chemical potential that couples to the conserved charge of a
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ sub-symmetry of
$\operatorname{O}\mathopen{}\mathclose{{}\left(N}\right)$. On an Euclidean
lattice, one can in these models for a discrete but infinite set of values of
the chemical potential even in the one-dimensional case (i.e. in (0+1)
dimensions) observe infinite range correlations and the formation of a non-
zero vacuum expectation value. From a field theoretic point of view, this
(0+1)-dimensional case is, of course, not particularly interesting, and it
does also not conflict with Coleman’s "no Goldstone bosons" theorem, as the
latter addresses only $(1+1)$ dimensional systems. However, the Euclidean
lattice formulations of our
$\operatorname{O}\mathopen{}\mathclose{{}\left(N}\right)$ spin models can also
be interpreted in a solid state physics context, with the the Euclidean action
$S$ playing the role of $\beta\,H$, i.e. of the product of the inverse
temperature $\beta=1/(k_{B}\,T)$ and the Hamiltonian $H$. From this point of
view, the formation of long-range order in a one-dimensional system seems to
conflict with the Mermin-Wagner theorem. As the one-dimensional lattice is in
this case spatial, the chemical potential parameter $\mu$ couples to a spatial
current and could be interpreted as a _negative resistance_.

The paper is organized as follows: in the following section, Sec. II, the
lattice model as well as its formulation in terms of dual flux-variables will
be introduced in detail. In Sec. III we then show analytically that on a one-
dimensional lattice, the model can for a discrete but infinite set of values
of the chemical potential develop infinite range correlations and a non-zero
vacuum expectation value. Sec. IV summarizes the findings and discusses their
relation to the Mermin-Wagner theorem Mermin .

##  II The model

The model of interest to us in the present work is the non-linear
$\operatorname{O}\mathopen{}\mathclose{{}\left(N}\right)$ spin model on a 1D
(one-dimensional) Euclidean lattice, coupled to a chemical potential and
source terms. In order to get an idea of how the lattice model is related to
the corresponding continuum theory, we will first write down the action in
$d=(d_{s}+1)$-dimensional continuous Minkowski space ($d_{s}$ being the number
of spatial dimensions). We then go through the steps of performing the Wick-
rotation to Euclidean time, and then putting the model on a
$d=(d_{s}+1)$-dimensional Euclidean lattice. In this way, we can keep track of
how the lattice parameters are related to their continuum counterparts and the
lattice spacing $a$.
So, in $(d_{s}+1)$-dimensional Minkowski space, the action for our non-linear
$\operatorname{O}\mathopen{}\mathclose{{}\left(N}\right)$-model reads:

$S_{M}[\phi]\,=\,\int\mathrm{d}^{d}x\,\Bigl{\\{}\Big{.}f_{\pi}^{2}\,\bigl{(}\big{.}\mathchoice{\mathopen{}\mathclose{{}\left(\partial_{\nu}+2\,\mathrm{i}\,\mu_{c}\,\tau_{12}\,\delta^{0}_{\nu}}\right)}{(\partial_{\nu}+2\,\mathrm{i}\,\mu_{c}\,\tau_{12}\,\delta^{0}_{\nu})}{(\partial_{\nu}+2\,\mathrm{i}\,\mu_{c}\,\tau_{12}\,\delta^{0}_{\nu})}{(\partial_{\nu}+2\,\mathrm{i}\,\mu_{c}\,\tau_{12}\,\delta^{0}_{\nu})}\phi_{c}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}\big{.}\bigr{)}^{\top}\\!g^{\nu\rho}\,\bigl{(}\big{.}\mathchoice{\mathopen{}\mathclose{{}\left(\partial_{\rho}+2\,\mathrm{i}\,\mu_{c}\,\tau_{12}\,\delta^{0}_{\rho}}\right)}{(\partial_{\rho}+2\,\mathrm{i}\,\mu_{c}\,\tau_{12}\,\delta^{0}_{\rho})}{(\partial_{\rho}+2\,\mathrm{i}\,\mu_{c}\,\tau_{12}\,\delta^{0}_{\rho})}{(\partial_{\rho}+2\,\mathrm{i}\,\mu_{c}\,\tau_{12}\,\delta^{0}_{\rho})}\phi_{c}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}\big{.}\bigr{)}\,-\,j\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}\phi_{c}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}\Big{.}\Bigr{\\}}\
,$ (II.1)

where $g^{\nu\rho}=\operatorname{diag}(1,-1,\ldots,-1)$ is the Minkowski space
metric, $\phi_{c}\in\operatorname{S}^{N-1}\subset\mathbb{R}^{N}$, and
$f_{\pi}$ is a (coupling) constant of mass dimension $(d-2)/2$, necessary to
render the action (in natural units) dimension less while
$\phi\in\operatorname{S}^{N-1}$ is itself dimensionless. The field
$j=(j^{1},\ldots,j^{N})\in\mathbb{R}^{N}$ is a $N$-vector of source terms. The
quantity $\tau_{12}$ that comes with the chemical potential $\mu_{c}$ is an
$\operatorname{O}\mathopen{}\mathclose{{}\left(N}\right)$-generator
$(\tau_{ij})\indices{{}^{a}_{b}}=-\mathrm{i}(\delta_{i}^{a}\delta_{j,b}-\delta_{j}^{a}\delta_{i,b})$,
so that $\mu_{c}$ couples to the conserved charge, corresponding to the
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$-symmetry of
rotations in the $\phi_{c}^{1}$-$\phi_{c}^{2}$-plane111The $2$ in front of the
chemical potential in (II.1) is convention.. The subscript c (for continuum)
in $\phi_{c}$ and $\mu_{c}$ is just there to avoid confusing these continuum
quantities with their lattice analogues that will be introduced shortly. If
the chemical potential is non-zero, the global
$\operatorname{O}\mathopen{}\mathclose{{}\left(N}\right)$-symmetry is
explicitly broken to
$\operatorname{O}\mathopen{}\mathclose{{}\left(N-2}\right)\\!\times\\!\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$.
By setting
$j\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}=J$, with
$J\in\mathbb{R}^{N}$ to a position-independent, non-zero value, the symmetry
can be reduced further. Setting the components $J^{1}$ or $J^{2}$ to non-zero
values can be problematic in combination with a non-zero chemical potential,
as these source components explicitly break the
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ symmetry in the
$\phi_{c}^{1}$-$\phi_{c}^{2}$-plane and thereby render the charge to which the
chemical potential couples non-conserved. However, as long as
$\tilde{J}=\sqrt{(J^{1})^{2}+(J^{2})^{2}}$ is small, the
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$-charge can still be
considered approximately conserved. Non-zero values for the components $J^{i}$
with $i\geq 3$ are on the other hand never a problem and will just break the
$\operatorname{O}\mathopen{}\mathclose{{}\left(N-2}\right)$ part of the global
symmetry further down to
$\operatorname{O}\mathopen{}\mathclose{{}\left(N-3}\right)$.
We now perform a Wick rotation, so that $\mathrm{i}\,S_{M}[\phi]\to-
S_{E}[\phi]$. With our sign convention for the Minkowski metric, this is
achieved by setting $x^{0}=-\mathrm{i}\,x^{d}$ (implying
$\mathrm{d}x^{0}=-\mathrm{i}\,\mathrm{d}x^{d}$ and
$\partial_{0}=\mathrm{i}\,\partial_{d}$), and the Euclidean action is then
obtained as:

$S_{E}[\phi]\,=\,\int\mathrm{d}^{d}x\,\Bigl{\\{}\Big{.}f_{\pi}^{2}\,\bigl{(}\big{.}\mathchoice{\mathopen{}\mathclose{{}\left(\partial_{\nu}+2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\right)}{(\partial_{\nu}+2\,\mu\,\tau_{12}\,\delta_{\nu,d})}{(\partial_{\nu}+2\,\mu\,\tau_{12}\,\delta_{\nu,d})}{(\partial_{\nu}+2\,\mu\,\tau_{12}\,\delta_{\nu,d})}\phi\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}\big{.}\bigr{)}^{\top}\\!g^{\nu\rho}\,\bigl{(}\big{.}\mathchoice{\mathopen{}\mathclose{{}\left(\partial_{\rho}+2\,\mu\,\tau_{12}\,\delta_{\rho,d}}\right)}{(\partial_{\rho}+2\,\mu\,\tau_{12}\,\delta_{\rho,d})}{(\partial_{\rho}+2\,\mu\,\tau_{12}\,\delta_{\rho,d})}{(\partial_{\rho}+2\,\mu\,\tau_{12}\,\delta_{\rho,d})}\phi\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}\big{.}\bigr{)}\,+\,j\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}\phi\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}\Big{.}\Bigr{\\}}\
,$ (II.2)

where $g^{\nu\rho}$ and
$\phi\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}$ are
now the Euclidean metric and field, respectively, with the latter being
obtained from the Minkowski space field by substituting
$\phi(-\mathrm{i}\,x^{d},x^{1},\ldots,x^{d-1})\to\phi(x^{1},\ldots,x^{d})$.
Finally, we put the theory on a lattice with finite lattice spacing $a$,
substituting $\int\mathrm{d}^{d}x\,\to\,a^{d}\,\sum\limits_{x}$ and

$\mathchoice{\mathopen{}\mathclose{{}\left(\partial_{\nu}+2\,\mu_{c}\,\tau_{12}\,\delta^{d}_{\nu}}\right)}{(\partial_{\nu}+2\,\mu_{c}\,\tau_{12}\,\delta^{d}_{\nu})}{(\partial_{\nu}+2\,\mu_{c}\,\tau_{12}\,\delta^{d}_{\nu})}{(\partial_{\nu}+2\,\mu_{c}\,\tau_{12}\,\delta^{d}_{\nu})}\phi_{c}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}\to\\\
\frac{1}{a}\mathchoice{\mathopen{}\mathclose{{}\left(\phi_{x+\smash{\widehat{\nu}}}-\operatorname{e}^{-2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\,\phi_{x}}\right)}{(\phi_{x+\smash{\widehat{\nu}}}-\operatorname{e}^{-2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\,\phi_{x})}{(\phi_{x+\smash{\widehat{\nu}}}-\operatorname{e}^{-2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\,\phi_{x})}{(\phi_{x+\smash{\widehat{\nu}}}-\operatorname{e}^{-2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\,\phi_{x})}\
,$ (II.3)

where on the right-hand side of (II.3), factors of $a$, resp. $1/a$ have been
absorbed into $\mu$ and $x$ by redefining $a\,\mu_{c}\to\mu$ and $x/a\to x$,
in order to render the quantities dimensionless. The coordinate $x$ on the
right-hand side of (II.3) takes then values in $\mathbb{Z}^{d}$, and the
relation between $\phi$ and $\phi_{c}$ is given by $\phi_{x}=\phi_{c}(a\,x)$
and $\phi_{x+\smash{\widehat{\nu}}}=\phi_{c}(a(x+e_{\nu}))$ with $e_{\nu}$
being the unit vector in $\nu$-direction. After setting also
$\beta=f_{\pi}^{2}\,a^{(d-2)}$ and $s=J\,a^{d}$, the Euclidean lattice action
then takes the standard form:

$S\mathopen{}\mathclose{{}\left[\phi}\right]\,=\,-\sum\limits_{x}\biggl{\\{}\bigg{.}\frac{\beta}{2}\,\sum\limits_{\nu=1}^{d}\mathchoice{\mathopen{}\mathclose{{}\left(\phi_{x}\,\operatorname{e}^{2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\,\phi_{x+\smash{\widehat{\nu}}}+\phi_{x}\,\operatorname{e}^{-2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\,\phi_{x-\smash{\widehat{\nu}}}}\right)}{(\phi_{x}\,\operatorname{e}^{2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\,\phi_{x+\smash{\widehat{\nu}}}+\phi_{x}\,\operatorname{e}^{-2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\,\phi_{x-\smash{\widehat{\nu}}})}{(\phi_{x}\,\operatorname{e}^{2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\,\phi_{x+\smash{\widehat{\nu}}}+\phi_{x}\,\operatorname{e}^{-2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\,\phi_{x-\smash{\widehat{\nu}}})}{(\phi_{x}\,\operatorname{e}^{2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\,\phi_{x+\smash{\widehat{\nu}}}+\phi_{x}\,\operatorname{e}^{-2\,\mu\,\tau_{12}\,\delta_{\nu,d}}\,\phi_{x-\smash{\widehat{\nu}}})}+\mathchoice{\mathopen{}\mathclose{{}\left(s\cdot\phi_{x}}\right)}{(s\cdot\phi_{x})}{(s\cdot\phi_{x})}{(s\cdot\phi_{x})}\bigg{.}\biggr{\\}}\
.$ (II.4)

Note, that for $d=(d_{s}+1)=1$ (i.e. in the case of spatial dimensionality
$d_{s}=0$), the parameter $f_{\pi}$ has mass dimension -1/2 and its lattice
counterpart is given by $\beta=f_{\pi}^{2}/a$. If $f_{\pi}$ is non-zero, we
therefore have, that $(a\to 0)$ implies that $(\beta\to\infty)$, i.e. the
parameter $\beta$ has to diverge in the continuum limit.

###  II.1 Dual formulation

The action (II.4) is in general complex for non-zero values of the chemical
potential $\mu$, which gives rise to a so-called _sign-problem_ when trying to
evaluate the partition function,

$Z\,=\,\int\mathcal{D}\bigl{[}\phi\bigr{]}\,\operatorname{e}^{-S\mathopen{}\mathclose{{}\left[\phi}\right]}\
,$ (II.5)

numerically by means of Monte Carlo simulations: a complex action $S$ implies
that the Boltzmann weight $\operatorname{e}^{-S}$ is complex as well and
therefore lacks the probabilistic interpretation, required to do importance
sampling. This problem can be overcome by changing representation and
expressing (II.5) in terms of new, discrete variables. Such a change of
representation is carried out in detail in appendix 1, following the steps
described in Endres ; Bruckmann ; Rindlisbacher1 ; Rindlisbacher3 . After
dualization, the partition function (II.5) reads:

$Z\,=\,\sum\limits_{\mathopen{}\mathclose{{}\left\\{k,l,\chi,p,q,n}\right\\}}\prod\limits_{x}\biggl{\\{}\bigg{.}\biggl{(}\bigg{.}\prod\limits_{\nu=1}^{d}\frac{\beta^{\mathopen{}\mathclose{{}\left|k_{x,\nu}}\right|+2\,l_{x,\nu}+\sum_{i=3}^{N}\,\chi^{i}_{x,\nu}}}{\mathchoice{\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left|k_{x,\nu}}\right|+l_{x,\nu}}\right)}{(\mathopen{}\mathclose{{}\left|k_{x,\nu}}\right|+l_{x,\nu})}{(\mathopen{}\mathclose{{}\left|k_{x,\nu}}\right|+l_{x,\nu})}{(\mathopen{}\mathclose{{}\left|k_{x,\nu}}\right|+l_{x,\nu})}!\,l_{x,\nu}!\,\prod_{i=3}^{N}\chi^{i}_{x,\nu}!}\bigg{.}\biggr{)}\\\
\cdot\frac{(s^{+})^{\frac{1}{2}\mathchoice{\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left|p_{x}}\right|+p_{x}}\right)}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|+p_{x})}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|+p_{x})}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|+p_{x})}+q_{x}}(s^{-})^{\frac{1}{2}\mathchoice{\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left|p_{x}}\right|-p_{x}}\right)}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|-p_{x})}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|-p_{x})}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|-p_{x})}+q_{x}}\,\operatorname{e}^{2\,\mu\,k_{x,d}}}{\mathchoice{\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left|p_{x}}\right|+q_{x}}\right)}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|+q_{x})}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|+q_{x})}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|+q_{x})}!\,q_{x}!}\biggl{(}\bigg{.}\prod\limits_{i=3}^{N}\frac{(s^{i})^{n^{i}_{x}}}{n^{i}_{x}!}\bigg{.}\biggr{)}\,\delta\bigl{(}\big{.}p_{x}-\sum\limits_{\nu}\bigl{(}\big{.}k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu}\big{.}\bigr{)}\big{.}\bigr{)}\\\
\cdot
W\bigl{(}\big{.}A_{x}+\mathopen{}\mathclose{{}\left|p_{x}}\right|+2\,q_{x},\,B^{3}_{x}+n^{3}_{x},\,\ldots,\,B^{N}_{x}+n^{N}_{x}\big{.}\bigr{)}\bigg{.}\biggr{\\}}\
,$ (II.6)

with

$A_{x}=\sum\limits_{\nu}\mathchoice{\mathopen{}\mathclose{{}\left(\big{|}k_{x,\nu}\big{|}+\big{|}k_{x-\smash{\widehat{\nu}},\nu}\big{|}+2\bigl{(}\big{.}l_{x,\nu}+l_{x-\smash{\widehat{\nu}},\nu}\big{.}\bigr{)}}\right)}{(\big{|}k_{x,\nu}\big{|}+\big{|}k_{x-\smash{\widehat{\nu}},\nu}\big{|}+2\bigl{(}\big{.}l_{x,\nu}+l_{x-\smash{\widehat{\nu}},\nu}\big{.}\bigr{)})}{(\big{|}k_{x,\nu}\big{|}+\big{|}k_{x-\smash{\widehat{\nu}},\nu}\big{|}+2\bigl{(}\big{.}l_{x,\nu}+l_{x-\smash{\widehat{\nu}},\nu}\big{.}\bigr{)})}{(\big{|}k_{x,\nu}\big{|}+\big{|}k_{x-\smash{\widehat{\nu}},\nu}\big{|}+2\bigl{(}\big{.}l_{x,\nu}+l_{x-\smash{\widehat{\nu}},\nu}\big{.}\bigr{)})}\quad,\quad
B^{i}_{x}=\sum\limits_{\nu}\mathchoice{\mathopen{}\mathclose{{}\left(\chi^{i}_{x,\nu}+\chi^{i}_{x-\smash{\widehat{\nu}},\nu}}\right)}{(\chi^{i}_{x,\nu}+\chi^{i}_{x-\smash{\widehat{\nu}},\nu})}{(\chi^{i}_{x,\nu}+\chi^{i}_{x-\smash{\widehat{\nu}},\nu})}{(\chi^{i}_{x,\nu}+\chi^{i}_{x-\smash{\widehat{\nu}},\nu})}$
(II.7)

and

$W\bigl{(}\big{.}A,B^{3},\ldots,B^{N}\big{.}\bigr{)}=\frac{\Gamma\bigl{(}\big{.}\frac{2+A}{2}\big{.}\bigr{)}\,\prod\limits_{i=3}^{N}\frac{1+\mathchoice{\mathopen{}\mathclose{{}\left(-1}\right)}{(-1)}{(-1)}{(-1)}^{B^{i}}}{2}\Gamma\bigl{(}\big{.}\frac{1+\vphantom{B}\smash{B^{i}}}{2}\big{.}\bigr{)}}{2^{\nicefrac{{A}}{{2}}}\,\Gamma\bigl{(}\big{.}\frac{N+A+\sum_{i=3}^{N}B^{i}}{2}\big{.}\bigr{)}}\
,$ (II.8)

and where, for all sites $x$ and all links
$\mathchoice{\mathopen{}\mathclose{{}\left(x,\nu}\right)}{(x,\nu)}{(x,\nu)}{(x,\nu)}$:

$\displaystyle k_{x,\nu},p_{x}$ $\displaystyle\in\mathbb{Z}\ ,$ (II.9a)
$\displaystyle
l_{x,\nu},q_{x},\chi^{3}_{x,\nu},n^{3}_{x},\ldots,\chi^{N}_{x,\nu},n^{N}_{x}$
$\displaystyle\in\mathbb{N}_{0}\ .$ (II.9b)

The delta function at the end of the second line of (II.6) is just a Kronecker
delta,
$\delta\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}=\delta_{0,x}$,
and the new parameters $s^{\pm}=\frac{1}{\sqrt{2}}(s^{1}\mp\mathrm{i}\,s^{2})$
are the sources for the
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$-charged
$\phi^{\pm}=\frac{1}{\sqrt{2}}\mathchoice{\mathopen{}\mathclose{{}\left(\phi^{1}\pm\mathrm{i}\,\phi^{2}}\right)}{(\phi^{1}\pm\mathrm{i}\,\phi^{2})}{(\phi^{1}\pm\mathrm{i}\,\phi^{2})}{(\phi^{1}\pm\mathrm{i}\,\phi^{2})}$.
While in (II.5) the partition function was given in terms of an integral over
a continuum of configurations
$\mathopen{}\mathclose{{}\left\\{\phi_{x}}\right\\}_{x}$ with complex weights,
in the flux-representation (II.6) it is given in terms of an infinite sum over
configurations of discrete variables $\\{k,l,\chi^{i},p,q,n^{i}\\}$, which all
have real and non-negative weights222For a configuration to have a real and
non-negative weight, the weight factor
$W_{s}=\prod_{x}(s^{+})^{\frac{1}{2}\mathchoice{\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left|p_{x}}\right|+p_{x}}\right)}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|+p_{x})}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|+p_{x})}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|+p_{x})}+q_{x}}(s^{-})^{\frac{1}{2}\mathchoice{\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left|p_{x}}\right|-p_{x}}\right)}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|-p_{x})}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|-p_{x})}{(\mathopen{}\mathclose{{}\left|p_{x}}\right|-p_{x})}+q_{x}}$
has to be real and non-negative. That this is the case can be seen, by writing
$s^{\pm}=\frac{\tilde{s}\,\operatorname{e}^{\mp\mathrm{i}\phi_{s}}}{\sqrt{2}}$,
so that
$W_{s}=\tilde{s}^{\sum_{x}(\mathopen{}\mathclose{{}\left|p_{x}}\right|+2\,q_{x})}\,\operatorname{e}^{\mathrm{i}\phi_{s}\sum_{x}\,p_{x}}$,
and noting that due to the Kronecker delta on the second line of (II.6), we
have
$\sum_{x}\,p_{x}=\sum_{x,\nu}\mathchoice{\mathopen{}\mathclose{{}\left(k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu}}\right)}{(k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu})}{(k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu})}{(k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu})}=0$.
In the present work we will primarily be interested in the
$\mathchoice{\mathopen{}\mathclose{{}\left(0+1}\right)}{(0+1)}{(0+1)}{(0+1)}$-dimensional
case, which could also in the standard formulation be solved using transfer-
matrix methods, regardless of whether the action is real or complex. The
reformulation of the partition function in terms of flux-variables is
nevertheless useful as it simplifies the necessary computations considerably.
Even more important is, however, that as discussed in the next section, the
flux-variables have a direct physical meaning, which can be useful in
identifying processes underlying different thermodynamic properties.

###  II.2 Physical meaning of dual variables

The variables $k$, $l$ and $\chi^{i}$ live on the links of the lattice and are
called _flux-variables_. They can in general be interpreted as counting the
number of particle world lines passing along the links. So, for example
$\chi^{i}_{x,\nu}$, with
$i\in\mathopen{}\mathclose{{}\left\\{3,\ldots,N}\right\\}$, counts the number
of $\phi^{i}$-world lines traversing the link that connects the sites $x$ and
$x+\smash{\widehat{\nu}}$. Because the $\phi^{i}$-particles (with
$i\in\mathopen{}\mathclose{{}\left\\{3,\ldots,N}\right\\}$) are neutral, they
are their own anti-particles, so that their world lines don’t need an
orientation. For the $\phi^{\pm}$ particles, which carry a
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$-charge, the
situation is different: it matters whether a $\phi^{+}$ moves from $x$ to
$x+\smash{\widehat{\nu}}$ or from $x+\smash{\widehat{\nu}}$ to $x$, as the two
cases lead to different charge displacements. On the other hand, it doesn’t
make a difference whether a $\phi^{+}$ moves from $x$ to
$x+\smash{\widehat{\nu}}$, or a $\phi^{-}$ moves form
$x+\smash{\widehat{\nu}}$ to $x$, as the charge-displacement is in both cases
the same ($\mathcal{CT}$ symmetry is preserved). The $k$ and $l$ variables are
therefore picked in such a way, that $k_{x,\nu}$ counts the net-positive
charge that moves from site $x$ to site $x+\smash{\widehat{\nu}}$, whereas
$l_{x,\nu}$ counts the number of neutral pairs of $\phi^{\pm}$ world lines
that pass along the link between the two sites.
The remaining variables $p$, $q$ and $n^{i}$, with
$i\in\mathopen{}\mathclose{{}\left\\{3,\ldots,N}\right\\}$, live on the
lattice sites and are called _monomer numbers_. In analogy to the flux-
variables, the monomer numbers are organized so that $p_{x}$ counts the net-
positive charge on site $x$ and $q_{x}$ the number of neutral pairs of
$\phi^{\pm}$-monomers, while $n^{i}_{x}$ counts the number of
$\phi^{i}$-monomers on site $x$.
Due to the delta-function constraints in (II.6), the $k$\- and $p$-variables
are on each site $x$ subject to a conservation law:

$\sum_{\nu}\mathchoice{\mathopen{}\mathclose{{}\left(k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu}}\right)}{(k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu})}{(k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu})}{(k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu})}\,\overset{!}{=}\,p_{x}\
,$ (II.10)

which is a lattice manifestation of Gauss’ law
$\partial_{\nu}j^{\nu}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}=\rho_{s}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}$,
with $k_{x,\nu}$ being related to the current density,
$j^{\nu}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}=\mathrm{i}\,\mathchoice{\mathopen{}\mathclose{{}\left(\phi^{+}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}(\partial^{\nu}\phi^{-}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)})-\phi^{-}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}(\partial^{\nu}\phi^{+}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)})}\right)}{(\phi^{+}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}(\partial^{\nu}\phi^{-}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)})-\phi^{-}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}(\partial^{\nu}\phi^{+}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}))}{(\phi^{+}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}(\partial^{\nu}\phi^{-}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)})-\phi^{-}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}(\partial^{\nu}\phi^{+}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}))}{(\phi^{+}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}(\partial^{\nu}\phi^{-}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)})-\phi^{-}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}(\partial^{\nu}\phi^{+}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}))}$,
and $p_{x}$ (which can be non-zero only if
$\mathopen{}\mathclose{{}\left|s^{\pm}}\right|>0$) being related to the
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$-breaking field
$\rho_{s}\mathchoice{\mathopen{}\mathclose{{}\left(x}\right)}{(x)}{(x)}{(x)}=\mathrm{i}\,(s^{+}\,\phi^{+}-s^{-}\,\phi^{-})$.
While showing, that for $|s^{\pm}|>0$, charge conservation is locally
violated, equation (II.10) also implies that for periodic lattices, we have:

$\sum_{x}\,p_{x}\,=\,\sum_{x,\nu}\mathchoice{\mathopen{}\mathclose{{}\left(k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu}}\right)}{(k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu})}{(k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu})}{(k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu})}\,=\,0\
,$ (II.11)

meaning that the overall charge carried by $\phi^{\pm}$-monomers must be zero
($\sim$ integral-form of Gauss’ law in a space without boundary). Note,
however, that in this quantum field theory context, the number of
$\phi^{\pm}$-monomers has nothing to do with the electric charge density in
the system! The latter, which in the continuum would be given by $j^{0}$ (or
$j^{d}$ in Euclidean space-time), is represented by the flux variables
$k_{x,d}$ that live on the time-like links.
The $l$\- and $q$-variables are completely unconstrained as they represent the
numbers of neutral pairs of $\phi^{\pm}$-world line segments or monomers,
which can at any point be created and annihilate. This is not the case for the
$\chi^{i}$\- and $n^{i}$-variables, which are on each site $x$ subject to the
_evenness-constraint_ ,

$\bigl{(}\big{.}n^{i}_{x}\,+\,\sum_{\nu}(\chi^{i}_{x,\nu}-\chi^{i}_{x-\smash{\widehat{\nu}},\nu})\big{.}\bigr{)}\bmod
2\,\overset{!}{=}\,0\ ,$ (II.12)

encoded in the site weight (II.8). This constraint reflects the fact that also
neutral particles can be created and annihilated only in pairs, so that a
world line, that enters a site, has either to continue and leave the site, or
to annihilate with a monomer.

##  III Results

For simplicity we discuss in the following the case of $N=2$, which
corresponds to an XY-model that is coupled to source terms and a chemical
potential for the $\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$
charge. The qualitative findings generalize to $N>2$, as well as to the case
of linear spin models with arbitrary on-site potentials.
With $N=2$, the site weight (II.8) simplifies to
$W\mathchoice{\mathopen{}\mathclose{{}\left(A}\right)}{(A)}{(A)}{(A)}=2^{-A/2}$.
The summations over the individual $l$\- and $q$-variables in the partition
function (II.6) then decouple, and the partition function simplifies to:

$Z\,=\,\sum\limits_{\mathopen{}\mathclose{{}\left\\{k,p}\right\\}}\prod\limits_{x}\biggl{\\{}\bigg{.}\biggl{(}\bigg{.}\prod\limits_{\nu=1}^{d}\,I_{k_{x,\nu}}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}\bigg{.}\biggr{)}\,I_{p_{x}}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}\,\operatorname{e}^{\mathrm{i}\,\phi_{s}\,p_{x}}\,\operatorname{e}^{2\,\mu\,k_{x,d}}\,\delta\bigl{(}\big{.}p_{x}-\sum\limits_{\nu}\bigl{(}\big{.}k_{x,\nu}-k_{x-\smash{\widehat{\nu}},\nu}\big{.}\bigr{)}\big{.}\bigr{)}\bigg{.}\biggr{\\}}\
,$ (III.1)

with
$I_{n}\mathchoice{\mathopen{}\mathclose{{}\left(\alpha}\right)}{(\alpha)}{(\alpha)}{(\alpha)}$
being the modified Bessel function of the first kind333The modified Bessel
function of the first kind can be written as
$I_{n}\mathchoice{\mathopen{}\mathclose{{}\left(\alpha}\right)}{(\alpha)}{(\alpha)}{(\alpha)}=\sum_{m=0}^{\infty}\frac{\mathchoice{\mathopen{}\mathclose{{}\left(\frac{\alpha}{2}}\right)}{(\frac{\alpha}{2})}{(\frac{\alpha}{2})}{(\frac{\alpha}{2})}^{n+2\,m}}{\mathchoice{\mathopen{}\mathclose{{}\left(n+m}\right)}{(n+m)}{(n+m)}{(n+m)}!\,m!}$,
which is precisely the form of the sums over $l$ and $q$ in (II.6) when $N=2$,
with the replacements ($m\to\mathopen{}\mathclose{{}\left|k}\right|$,
$\alpha\to\beta$) and ($m\to\mathopen{}\mathclose{{}\left|p}\right|$,
$\alpha\to\tilde{s}$). Note also that the factorials in the denominator of the
expression for the Bessel function are understood in terms of gamma-functions,
i.e.
$n!=\Gamma\mathchoice{\mathopen{}\mathclose{{}\left(1+n}\right)}{(1+n)}{(1+n)}{(1+n)}$,
so that
$I_{n}\mathchoice{\mathopen{}\mathclose{{}\left(\alpha}\right)}{(\alpha)}{(\alpha)}{(\alpha)}=I_{\mathopen{}\mathclose{{}\left|n}\right|}\mathchoice{\mathopen{}\mathclose{{}\left(\alpha}\right)}{(\alpha)}{(\alpha)}{(\alpha)}$
and we therefore don’t have to write the modulus in the subscript of
$I_{n}\mathchoice{\mathopen{}\mathclose{{}\left(\alpha}\right)}{(\alpha)}{(\alpha)}{(\alpha)}$.
and $\tilde{s}$ and $\phi_{s}$ are magnitude and phase of the sources, so that
$s^{\pm}=\frac{\tilde{s}\,\operatorname{e}^{\mp\,\mathrm{i}\,\phi_{s}}}{\sqrt{2}}$.

In (0+1) dimensions, i.e. when the lattice extends only in the Euclidean time
direction, the partition function (III.1) simplifies further. Setting for the
moment $\tilde{s}=0$ (which implies $p_{x}=0$), it reduces to a single sum:

$Z\,=\,\sum\limits_{k=-\infty}^{\infty}\mathchoice{\mathopen{}\mathclose{{}\left(I_{k}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}\,\operatorname{e}^{2\,\mu\,k}}\right)}{(I_{k}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}\,\operatorname{e}^{2\,\mu\,k})}{(I_{k}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}\,\operatorname{e}^{2\,\mu\,k})}{(I_{k}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}\,\operatorname{e}^{2\,\mu\,k})}^{N_{t}}\
,$ (III.2)

with $N_{t}$ being the temporal extent (number of sites) of the periodic
lattice. The summation variable $k$ represents the collective value of all the
$k_{x,d}$-variables in the system and each value of $k$ therefore represents a
distinct homogeneous (in terms of flux variables) configuration. As discussed
in Sec. II.2, the values of the $k_{x,d}$-variables can be identified with
quanta of the local charge density $n=j^{d}$ in a dual configuration and the
value of $k$ in (III.2) is therefore not just a label, but represents the
charge density carried by the corresponding homogeneous configuration. This
can be verified by taking the derivative of the logarithm of the partition
function with respect to $\mu$: in arbitrary dimensions, this yields the
charge density:

$\mathopen{}\mathclose{{}\left\langle
n}\right\rangle\,=\,\frac{1}{2\,V}\frac{\partial\log\mathchoice{\mathopen{}\mathclose{{}\left(Z}\right)}{(Z)}{(Z)}{(Z)}}{\partial\mu}\\\
=\,\frac{1}{V}\sum_{x}\mathopen{}\mathclose{{}\left\langle
k_{x,d}}\right\rangle_{dual}\,\overset{\mathchoice{\mathopen{}\mathclose{{}\left(d=(0+1)}\right)}{(d=(0+1))}{(d=(0+1))}{(d=(0+1))}}{=}\,\mathopen{}\mathclose{{}\left\langle
k}\right\rangle_{dual}\ ,$ (III.3)

and we see that in (0+1) dimensions, using the partition function (III.2), the
result is given by the expectation value of $k$.
The partition function (III.2) is therefore a sum over distinct homogeneous
(in terms of dual variables) configurations, which are labelled by the
integer-valued charge (density) $k$ that they carry.

###  III.1 Transitions between distinct ground states

Figure 1: The figure shows the charge density (top) and corresponding
susceptibility (bottom) as functions of the chemical potential $\mu$ for the
$\operatorname{O}\mathopen{}\mathclose{{}\left(2}\right)$-model in (0+1)
dimensions for $\tilde{s}=0$ (c.f. eq. (III.2)), $\beta=0.03$ and lattice
sizes $N_{t}=$16, 32, 64, 128, 1024. At subsequent critical values of
$\mu=\mu_{0,1}$, $\mu_{1,2}$, $\ldots$, the system undergoes transitions
between vacua carrying different integer-valued charge densities. The
transitions become apparently stronger with increasing system size, as one
would expect from (III.3).

For each fixed value of $\beta$, the modified Bessel function of the first
kind,
$I_{n}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}$,
decays super-exponentially as function of increasing
$\mathopen{}\mathclose{{}\left|n}\right|$ (otherwise, the sum in (III.2) would
diverge), and it is therefore evident, that for $\mu=0$, the term in (III.2)
with $k=0$ dominates if $N_{t}$ is sufficiently large. With increasing $\mu$,
however, the factor $\operatorname{e}^{2\,\mu\,k}$ will at some point be able
to compensate for the decrease in
$I_{k}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}$
when $k$ changes from zero to one, and if $\mu$ exceeds this point, the term
in (III.3) with $k=1$ will become the dominant one. As all the configurations
with different $k$ are homogeneous configurations (in terms of our flux
variables), each of them qualifies in the $(N_{t}\to\infty)$-limit as vacuum
state if its weight dominates the sum in (III.3). The transitions between
states with different index $k$ can therefore be understood as transitions
between different ground states which carry different integer-valued charge
density. These transitions occur whenever the value of $\mu$ is such that two
successive terms in (III.3) become degenerate, i.e. when we have for some
$k\in\mathbb{Z}$, that $\mu$ is such that:

$I_{k}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}\,\operatorname{e}^{2\,\mu\,k}\,=\,I_{k+1}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}\,\operatorname{e}^{2\,\mu\,\mathchoice{\mathopen{}\mathclose{{}\left(k+1}\right)}{(k+1)}{(k+1)}{(k+1)}}\
.$ (III.4)

Upon solving for $\mu$, this yields the critical value

$\mu_{k,k+1}\,=\,\frac{1}{2}\log\mathchoice{\mathopen{}\mathclose{{}\left(\frac{I_{k}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}}{I_{k+1}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}}}\right)}{(\frac{I_{k}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}}{I_{k+1}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}})}{(\frac{I_{k}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}}{I_{k+1}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}})}{(\frac{I_{k}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}}{I_{k+1}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}})}\
,$ (III.5)

for the transition between the states carrying charge densities $k$ and $k+1$,
respectively. That the system described by (III.2) indeed undergoes such
transitions between successive discrete states when the chemical potential is
increased, is illustrated in in the upper panel of Fig. 1 for systems of
different size $N_{t}$ and with $\beta=0.03$. The quantity in the lower panel
is the corresponding charge-susceptibility,

$\chi_{n}\,=\,V\,\bigl{(}\big{.}\langle\smash{n^{2}}\rangle-\langle\smash{n}\rangle^{2}\big{.}\bigr{)}\,=\,\frac{1}{4\,V}\,\frac{\partial^{2}\log\mathchoice{\mathopen{}\mathclose{{}\left(Z}\right)}{(Z)}{(Z)}{(Z)}}{\mathchoice{\mathopen{}\mathclose{{}\left(\partial\mu}\right)}{(\partial\mu)}{(\partial\mu)}{(\partial\mu)}^{2}}\\\
=\,V\,\bigl{(}\big{.}\langle\smash{k_{x,d}^{2}}\rangle_{dual}-\langle\smash{k_{x,d}}\rangle_{dual}^{2}\big{.}\bigr{)}\\\
\overset{\mathchoice{\mathopen{}\mathclose{{}\left(d=1}\right)}{(d=1)}{(d=1)}{(d=1)}}{=}\,N_{t}\bigl{(}\big{.}\langle\smash{k^{2}}\rangle_{dual}-\langle\smash{k}\rangle^{2}_{dual}\big{.}\bigr{)}\
,$ (III.6)

which, with increasing $N_{t}$, becomes more and more peaked at the transition
points $\mu_{k,k+1}$, $k\in\mathbb{Z}$.

Figure 2: The figure shows for the
$\operatorname{O}\mathopen{}\mathclose{{}\left(2}\right)$-model in (0+1)
dimensions with $\tilde{s}=0$ (c.f. eq. (III.2)), $\beta=0.03$ and lattice
sizes $N_{t}=16,32,64,128,1024$, scaling collapses for the charge density
(top) and the charge susceptibility (bottom) around the first critical point
$\mu=\mu_{0,1}$. The collapses are obtained according to (III.7) and (III.8).

Figure 3: Same as Fig. 2, but here the scaling collapses are performed at the
critical point $\mu=\mu_{2,3}$ instead of $\mu=\mu_{0,1}$.

In the neighbourhood of each critical point $\mu_{k,k+1}$, one can rescale
$\langle\smash{n}\rangle\mathchoice{\mathopen{}\mathclose{{}\left(\mu,N_{t}}\right)}{(\mu,N_{t})}{(\mu,N_{t})}{(\mu,N_{t})}$
and
$\chi_{n}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,N_{t}}\right)}{(\mu,N_{t})}{(\mu,N_{t})}{(\mu,N_{t})}$,
so that the curves for different $N_{t}$ coincide. This is achieved by setting
for each $k$:

$\mathopen{}\mathclose{{}\left\langle\tilde{n}}\right\rangle_{k}\mathchoice{\mathopen{}\mathclose{{}\left(\Delta\tilde{\mu}}\right)}{(\Delta\tilde{\mu})}{(\Delta\tilde{\mu})}{(\Delta\tilde{\mu})}=\mathopen{}\mathclose{{}\left\langle
n}\right\rangle\mathchoice{\mathopen{}\mathclose{{}\left(\mu_{k,k+1}+N_{t}^{-1}\Delta\tilde{\mu},N_{t}}\right)}{(\mu_{k,k+1}+N_{t}^{-1}\Delta\tilde{\mu},N_{t})}{(\mu_{k,k+1}+N_{t}^{-1}\Delta\tilde{\mu},N_{t})}{(\mu_{k,k+1}+N_{t}^{-1}\Delta\tilde{\mu},N_{t})}$
(III.7)

and

$\tilde{\chi}_{n,k}\mathchoice{\mathopen{}\mathclose{{}\left(\Delta\tilde{\mu}}\right)}{(\Delta\tilde{\mu})}{(\Delta\tilde{\mu})}{(\Delta\tilde{\mu})}=N_{t}^{-1}\chi_{n}\mathchoice{\mathopen{}\mathclose{{}\left(\mu_{k,k+1}+N_{t}^{-1}\Delta\tilde{\mu},N_{t}}\right)}{(\mu_{k,k+1}+N_{t}^{-1}\Delta\tilde{\mu},N_{t})}{(\mu_{k,k+1}+N_{t}^{-1}\Delta\tilde{\mu},N_{t})}{(\mu_{k,k+1}+N_{t}^{-1}\Delta\tilde{\mu},N_{t})}\
,$ (III.8)

where for each $k$, $\Delta\tilde{\mu}$ is, in terms of the original, unscaled
$\mu$, given by $\Delta\tilde{\mu}=N_{t}(\mu-\mu_{k,k+1})$. Examples for such
_scaling collapses_ are shown in Figs. 2-3 for $\beta=0.03$ and $\tilde{s}=0$,
i.e. for the same systems that were used in Fig. 1. As the rescaled charge-
susceptibility (III.8) is independent of $N_{t}$, the form of the rescaling
implies that the original $\chi_{n}$ converges towards a sum of Dirac delta-
functions when the thermodynamic limit
$\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}\to\infty}\right)}{(N_{t}\to\infty)}{(N_{t}\to\infty)}{(N_{t}\to\infty)}$
is taken at fixed $\beta$ (corresponding to taking the zero-temperature limit
at finite lattice spacing). The charge density
$\mathopen{}\mathclose{{}\left\langle n}\right\rangle$ becomes therefore
discontinuous in this limit and the transition is of first order.

The situation changes if we take the thermodynamic limit not by sending the
temperature to zero while keeping the lattice spacing fixed, but instead
sending the lattice spacing to zero while keeping the temperature fixed. As
$\beta$ is given by $\beta=f_{\pi}^{2}/a$, and $f_{\pi}$ has a physical
meaning (i.e. it should assume a constant value if one enters the scaling
window), we can do this by keeping $\kappa:=\beta/N_{t}=T\,f_{\pi}^{2}$
constant while sending $N_{t}$ to infinity. This is illustrated in Figs. 4-5,
which show for different values of the temperature (i.e. different values of
$\kappa=\beta/N_{t}$), how the charge density (III.3) and the corresponding
susceptibility (III.6) (multiplied by $\beta^{-1}$ to cancel the dependency on
the lattice spacing) as functions of $\beta\,\mu$ behave when $N_{t}$ is
changed: at low temperatures (cf. Fig. 4) one can again observe the formation
of plateaus at integer values of the charge density and for $(N_{t}\to\infty)$
the curves converge towards a fixed shape that represents the continuum limit.
In contrast to the situation we had in Fig. 1, the charge density as function
$\mu$ now remains smooth when $\kappa$ instead of $\beta$ is held fixed while
$N_{t}$ is sent to infinity.
Although decreasing the value of $\kappa$ (lowering the temperature) makes the
crossovers between the sectors of different integer-valued charge density
become more abrupt, the crossovers are turned into first order transitions
only if in the end also the zero-temperature limit is taken, i.e. when
$\kappa$ is sent to zero. For increasing $\kappa$, on the other hand (cf. Fig.
5), the step-like behavior of the charge density as function of $\mu$ slowly
disappears and turns into a linear behavior: $\langle\smash{n}\rangle\approx
2\,\mu\,\beta$.

Figure 4: The figure shows for the
$\operatorname{O}\mathopen{}\mathclose{{}\left(2}\right)$ model in (0+1)
dimensions the charge density (top) and charge susceptibility (bottom) as
function of $\beta\,\mu$ at fixed $T\,f_{\pi}^{2}=\beta/N_{t}=0.03$ for
various lattice sizes $N_{t}=16,64,256,1024,4096$. The continuum limit would
be obtained by sending $N_{t}\to\infty$.

Figure 5: Same as Fig. 4 but at higher temperatures
$T\,f_{\pi}^{2}=\beta/N_{t}=1.2$.

The $N_{t}$-dependency of charge density and charge susceptibility as function
of $\mu$ at fixed $\kappa=\beta/N_{t}=T\,f_{\pi}^{2}$ can be understood from
the behavior of $\beta\,\mu_{n,n+1}(\beta)$ as function of $\beta$. With the
expression for $\mu_{n,n+1}(\beta)$ from eq. (III.5), one finds the asymptotic
behavior of $\beta\,\mu_{n,n+1}(\beta)$ at small and large values of $\beta$
to be given by:

$\beta\,\mu_{n,n+1}(\beta)\,\approx\,\begin{cases}\frac{\beta}{2}\log\mathchoice{\mathopen{}\mathclose{{}\left(\frac{2\mathchoice{\mathopen{}\mathclose{{}\left(n+1}\right)}{(n+1)}{(n+1)}{(n+1)}}{\beta}}\right)}{(\frac{2\mathchoice{\mathopen{}\mathclose{{}\left(n+1}\right)}{(n+1)}{(n+1)}{(n+1)}}{\beta})}{(\frac{2\mathchoice{\mathopen{}\mathclose{{}\left(n+1}\right)}{(n+1)}{(n+1)}{(n+1)}}{\beta})}{(\frac{2\mathchoice{\mathopen{}\mathclose{{}\left(n+1}\right)}{(n+1)}{(n+1)}{(n+1)}}{\beta})}\
\text{if}\ \beta\ll 1\\\ \frac{2\,n+1}{4}\ \text{if}\ \beta\gg 1\end{cases}\
,$ (III.9)

with a maximum separating the two regions. Clearly, for the curves in Figs.
4-5 to converge, the $N_{t}$-values have to be sufficiently large, so that the
corresponding values of $\beta(N_{t})=\kappa\,N_{t}$ are all in the region
$\beta\gg 1$, where $\beta\,\mu_{n,n+1}(\beta)$ is essentially independent of
$\beta$. If the physical temperature is very low (i.e. if $\kappa$ is very
small) then $\beta(N_{t})$ can for small values of $N_{t}$ fall into the range
$\beta\ll 1$, where
$\beta\,\mu_{n,n+1}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}$
grows only logarithmically instead of linearly with $n$. As function of
$\beta\,\mu$, the successive transitions to higher and higher charge densities
then happen for these small systems at much lower values of $\beta\,\mu$ than
for the larger systems. This is the reason why in Fig. 4 the charge density
and susceptibility grow for $N_{t}=16$ much faster as function of $\beta\,\mu$
than for the larger values of $N_{t}$.

Finally, it can also be understood, why the step-like behavior of the charge
density disappears when $\kappa=\beta/N_{t}$ is sufficiently large. To this
end, we define the _weight ratios_ ,

$W_{n_{1},n_{2}}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta,N_{t}}\right)}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}=\biggl{(}\bigg{.}\frac{I_{n_{2}}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}\operatorname{e}^{2\,\mu\,n_{2}}}{I_{n_{1}}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}\operatorname{e}^{2\,\mu\,n_{1}}}\bigg{.}\biggr{)}^{N_{t}}\\\
=\operatorname{e}^{2\,N_{t}\,\operatorname{sgn}\mathchoice{\mathopen{}\mathclose{{}\left(n_{2}-n_{1}}\right)}{(n_{2}-n_{1})}{(n_{2}-n_{1})}{(n_{2}-n_{1})}\,\sum_{n^{\prime}=\min(n_{1},n_{2})}^{\max(n_{1},n_{2})-1}(\mu-\mu_{n^{\prime},n^{\prime}+1}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)})}\
,$ (III.10)

which tell us, as function of the parameters $\mu$, $\beta$ and $N_{t}$, how
strongly the term in the partition sum (III.2) with $k=n_{2}$ is suppressed,
compared to the term with $k=n_{1}$. Setting $n_{1}=0$ and $n_{2}=k$, the
partition function (III.2) can then be written as:

$Z=(I_{0}(\beta))^{N_{t}}\,\sum\limits_{k=-\infty}^{\infty}W_{0,k}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta,N_{t}}\right)}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}\
.$ (III.11)

If we now plug into expression (III.10) for the weight ratios, that according
to (III.9) we have $\mu_{n,n+1}(\beta)\approx(2\,n+1)/(4\,\beta)$ for
$\beta\gg 1$, expression (III.10) simplifies to:

$W_{0,k}\mathchoice{\mathopen{}\mathclose{{}\left(\beta\,\mu,\kappa}\right)}{(\beta\,\mu,\kappa)}{(\beta\,\mu,\kappa)}{(\beta\,\mu,\kappa)}\approx
c\mathchoice{\mathopen{}\mathclose{{}\left(\beta\,\mu,\kappa}\right)}{(\beta\,\mu,\kappa)}{(\beta\,\mu,\kappa)}{(\beta\,\mu,\kappa)}\operatorname{e}^{-\frac{\mathchoice{\mathopen{}\mathclose{{}\left(k-2\,\beta\,\mu}\right)}{(k-2\,\beta\,\mu)}{(k-2\,\beta\,\mu)}{(k-2\,\beta\,\mu)}^{2}}{2\,\kappa}}\
,$ (III.12)

with
$c(\beta\,\mu,\kappa)=\operatorname{e}^{\frac{\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu}\right)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}^{2}}{2\,\kappa}}$,
and the partition function (III.11) reduces to a sum over the values of a
shifted Gaussian with variance $\kappa=\beta/N_{t}$ at integer values. If
$\kappa$ is small, the Gaussian is strongly peaked around the $k$-value that
is closest to $(2\,\beta\,\mu)$, so that there is always only one $k$-value
that significantly contributes to the partition sum (unless $\beta\,\mu$ is
fine-tuned to be exactly in the middle between two successive $k$-values). If
$(2\,\beta\,\mu)$ changes from being closest to one $k$-value to being closest
to the next $k$-value, the peak moves very abruptly from the old to the new
dominant $k$, and the average $k$ changes in a step-like manner as function of
$\beta\,\mu$. This is not the case if $\kappa$ is sufficiently large, as then
the Gaussian is wide and also sub-dominant $k$-values contribute significantly
to the partition sum, which allows the average $k$ value to change much more
smoothly. The value of $\kappa$ at which the latter behavior can be expected
to set in, can be approximated by the value for which the width of the
$2\,\sigma$ interval of the Gaussian in (III.12) does no longer fit between
two successive $k$-values, i.e. if $2\times(2\,\sigma)\geq 1$, which is the
case, if $\kappa=\sigma^{2}\geq 1/8$.
This can be made more explicitly by noting that the sum in (III.11) reduces
for $\beta\gg 1$, using (III.12), to a _Jacobi theta function_ :

$\vartheta\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)}=\sum\limits_{k=-\infty}^{\infty}u^{2\,k}\,q^{k^{2}}\
,$ (III.13)

with
$q=\operatorname{e}^{-1/\mathchoice{\mathopen{}\mathclose{{}\left(2\,\kappa}\right)}{(2\,\kappa)}{(2\,\kappa)}{(2\,\kappa)}}$
and $u=\operatorname{e}^{\beta\,\mu/\kappa}$. We can then use one of the
_Jacobi identities_ for
$\vartheta\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)}$,
namely:

$\vartheta\mathchoice{\mathopen{}\mathclose{{}\left(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)}}\right)}{(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)})}{(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)})}{(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)})}=\alpha\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)}\,\vartheta\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)}\
,$ (III.14)

with

$q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)}=\operatorname{e}^{\pi^{2}/\log\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)}}=\operatorname{e}^{-2\,\pi^{2}\,\kappa}\
,$ (III.15a)
$u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)}=\operatorname{e}^{\mathrm{i}\,\pi\,\log\mathchoice{\mathopen{}\mathclose{{}\left(u}\right)}{(u)}{(u)}{(u)}/\log\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)}}=\operatorname{e}^{-\pi\,\mathrm{i}\,\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu}\right)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}}\
,$ (III.15b) and
$\alpha\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)}=\sqrt{-\tfrac{\log\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)}}{\pi}}\operatorname{e}^{\frac{\log^{2}\mathchoice{\mathopen{}\mathclose{{}\left(u}\right)}{(u)}{(u)}{(u)}}{\log\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)}}}=\frac{e^{-\frac{\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu}\right)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}^{2}}{2\,\kappa}}}{\sqrt{2\,\pi\,\kappa}}\
,$ (III.15c)

to write the partition function (III.11) as:

$Z\overset{\beta\gg
1}{=}\mathchoice{\mathopen{}\mathclose{{}\left(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})}}\right)}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})})}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})})}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})})}^{N_{t}}\vartheta\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)}\\\
=\mathchoice{\mathopen{}\mathclose{{}\left(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})}}\right)}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})})}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})})}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})})}^{N_{t}}\vartheta\mathchoice{\mathopen{}\mathclose{{}\left(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)}}\right)}{(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)})}{(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)})}{(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)})}/\alpha\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)}\\\
=\mathchoice{\mathopen{}\mathclose{{}\left(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})}}\right)}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})})}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})})}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})})}^{N_{t}}\,\sqrt{2\,\pi\,\kappa}\,e^{\frac{\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu}\right)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}^{2}}{2\,\kappa}}\\\
\cdot\vartheta\bigl{(}\big{.}\operatorname{e}^{-\pi\,\mathrm{i}\,\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu}\right)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}},\operatorname{e}^{-2\,\pi^{2}\,\kappa}\big{.}\bigr{)}\
.$ (III.16)

For sufficiently large $\kappa$, the theta function on the last line converges
towards unity as a small $q$ suppresses all terms with $k\neq 0$ in the
defining sum (III.13). The partition function then reduces to:

$Z\approx\mathchoice{\mathopen{}\mathclose{{}\left(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})}}\right)}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})})}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})})}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\kappa\,N_{t}}\right)}{(\kappa\,N_{t})}{(\kappa\,N_{t})}{(\kappa\,N_{t})})}^{N_{t}}\,\sqrt{2\,\pi\,\kappa}\,e^{\frac{\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu}\right)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}^{2}}{2\,\kappa}}\
,$ (III.17)

and the corresponding charge density becomes:

$\mathopen{}\mathclose{{}\left\langle
n}\right\rangle=\frac{1}{2\,N_{t}}\frac{\partial\log\mathchoice{\mathopen{}\mathclose{{}\left(Z}\right)}{(Z)}{(Z)}{(Z)}}{\partial\mu}=2\,\beta\,\mu\
,$ (III.18)

with the linear behavior we observed in Fig. 3.

###  III.2 Correlation function and mass spectrum

Next, we would like to investigate how the two-point function of the
(0+1)-dimensional
$\operatorname{O}\mathopen{}\mathclose{{}\left(2}\right)$-model behaves as
function of $\mu$. The general form of one- and two-point functions in the
flux-variable formulation is derived in appendix 2. Here we will just need

$\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}\phi^{+}_{y}}\right\rangle\,=\,\frac{1}{Z}\frac{\partial^{2}Z}{\partial
s^{-}_{x}\partial
s^{+}_{y}}\,=\,\frac{Z_{2}^{-,+}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}{Z}\
,$ (III.19)

with $s^{\pm}_{x}$ being per-site sources444Note, that because of the
continuum field $\phi(x)$ being dimensionless (the dimensionality is absorbed
into $f_{\pi}$), the lattice fields $\phi^{\pm}_{x}$ do not explicitly scale
with the lattice spacing $a$. The local _per site sources_ , $s_{x}^{\pm}$, do
therefore also not explicitly scale with $a$; only the global $s^{\pm}$ do.
This can be seen by noting that
$\langle\smash{\phi^{\pm}}\rangle=\frac{1}{V}\frac{\partial Z}{\partial
s^{\pm}}$, while $\langle\smash{\phi_{x}^{\pm}}\rangle=\frac{\partial
Z}{\partial s^{\pm}_{x}}$, where $\langle\smash{\phi^{\pm}}\rangle$ and
$\langle\smash{\phi^{\pm}_{x}}\rangle$ are both dimensionless and in the case
of translation invariance, should even be equal,
$\langle\smash{\phi^{\pm}}\rangle=\langle\smash{\phi^{\pm}_{x}}\rangle$,
$\forall x$.. For the partition function (III.2) (or equivalently (III.11)),
this two-point function takes the simple form:

$\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}\phi^{+}_{y}}\right\rangle\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta,N_{t}}\right)}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}=\\\
\frac{\sum\limits_{k=-\infty}^{\infty}W_{0,k}^{1-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta,N_{t}}\right)}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}\,W_{0,k+1}^{s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta,N_{t}}\right)}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}}{2\,\sum\limits_{k=-\infty}^{\infty}W_{0,k}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta,N_{t}}\right)}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}}\
,$ (III.20)

with
$s(x,y)=\mathchoice{\mathopen{}\mathclose{{}\left(\mathchoice{\mathopen{}\mathclose{{}\left(y-x}\right)}{(y-x)}{(y-x)}{(y-x)}\bmod
N_{t}}\right)}{(\mathchoice{\mathopen{}\mathclose{{}\left(y-x}\right)}{(y-x)}{(y-x)}{(y-x)}\bmod
N_{t})}{(\mathchoice{\mathopen{}\mathclose{{}\left(y-x}\right)}{(y-x)}{(y-x)}{(y-x)}\bmod
N_{t})}{(\mathchoice{\mathopen{}\mathclose{{}\left(y-x}\right)}{(y-x)}{(y-x)}{(y-x)}\bmod
N_{t})}/N_{t}$, and where $\mathchoice{\mathopen{}\mathclose{{}\left(X\bmod
Y}\right)}{(X\bmod Y)}{(X\bmod Y)}{(X\bmod Y)}$ with $X\in\mathbb{Z}$ and
$Y\in\mathbb{N}$ is the modulo operation, that wraps $X$ around the interval
$\mathopen{}\mathclose{{}\left\\{0,\ldots,Y-1}\right\\}$. The
$W_{n_{1},n_{2}}(\mu,\beta,N_{t})$ are the weight ratios introduced in
(III.10). In order to avoid too lengthy expression we will often leave the
dependency of quantities on the parameters $\mu$, $\beta$ and $N_{t}$ implicit
in the following. Equation (III.20) simply states that if a $\phi^{-}$ is
inserted at site $x$ and a $\phi^{+}$ at site $y$, then, if the source
$\tilde{s}$ is set to zero, $\tilde{s}=0$, the two sites have to be connected
by a continuous charged flux-line.
For $s(x,y)<1/2$, it is useful to write (III.20) in the form:

$\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}\phi^{+}_{y}}\right\rangle=\frac{\sum\limits_{k=-\infty}^{\infty}W_{0,k}\,W_{k,k+1}^{s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}}{2\,\sum\limits_{k=-\infty}^{\infty}W_{0,k}}\\\
=\frac{\sum\limits_{k=-\infty}^{\infty}W_{0,k}\,\operatorname{e}^{2\,N_{t}\,\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{k,k+1}}\right)}{(\mu-\mu_{k,k+1})}{(\mu-\mu_{k,k+1})}{(\mu-\mu_{k,k+1})}\,s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}}{2\,\sum\limits_{k=-\infty}^{\infty}W_{0,k}}\
,$ (III.21a) where on the second line, we made use of the last line of
(III.10) to write the $W_{k,k+1}$ in terms of exponentials, from which one can
easily read off the masses of all the propagating modes. The restriction,
$s(x,y)<1/2$, ensures that the largest term in the sum in the numerator of
(III.21a) is the one with $k=n$ if
$\mu\in\mathopen{}\mathclose{{}\left(\mu_{n-1,n}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)},\mu_{n,n+1}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}}\right]$.
Similarly, one can for $s(x,y)>1/2$ write:
$\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}\phi^{+}_{y}}\right\rangle=\frac{\sum\limits_{k=-\infty}^{\infty}W_{0,k}\,W_{k,k-1}^{1-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}}{2\,\sum\limits_{k=-\infty}^{\infty}W_{0,k}}\\\
=\frac{\sum\limits_{k=-\infty}^{\infty}W_{0,k}\,\operatorname{e}^{-2\,N_{t}\,\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{k-1,k}}\right)}{(\mu-\mu_{k-1,k})}{(\mu-\mu_{k-1,k})}{(\mu-\mu_{k-1,k})}\,s\mathchoice{\mathopen{}\mathclose{{}\left(y,x}\right)}{(y,x)}{(y,x)}{(y,x)}}}{2\,\sum\limits_{k=-\infty}^{\infty}W_{0,k}}\
,$ (III.21b)

where the term with $k=n$ in the sum in the numerator of (III.21b) is this
time largest if
$\mu\in\mathopen{}\mathclose{{}\left[\mu_{n-1,n}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)},\mu_{n,n+1}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}}\right)$.
From eqns. (III.21a) and (III.21b) we can see that the spectrum contains a
massless mode whenever the chemical potential $\mu$ is set to one of the
critical values $\mu_{n,n+1}$, $n\in\mathbb{Z}$.

Figure 6: The figure shows for the one-dimensional
$\operatorname{O}\mathopen{}\mathclose{{}\left(2}\right)$-model with
$\tilde{s}=0$ and $\beta=0.03$ the zero-temperature $\phi^{\pm}$-masses from
eqn. (III.22a) and (III.22b) as functions of the chemical potential $\mu$.

At fixed $\beta$ and in the limit $(N_{t}\to\infty)$, the $\phi^{\pm}$-masses
can also be extracted in the standard way from the exponential decay of the
connected two-point function. To this end, we first note, that for
$\tilde{s}=0$ the
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$-current is conserved
and the $\phi^{\pm}$-worldlines, which carry a
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$-charge, therefore
need to be continuous. This means that
$\langle\smash{\phi_{x}^{-}\phi_{y}^{+}}\rangle$ is already a _connected two-
point function_ whose large distance behavior is dominated by the lightest
mode and not by any disconnected piece. For
$\mu\in\mathopen{}\mathclose{{}\left(\mu_{n-1,n}(\beta),\mu_{n,n+1}(\beta)}\right]$,
the $\phi^{+}$-mass is then obtained from (III.21a) with $1\ll t\ll t_{*}$ by
computing:

$m^{+}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta}\right)}{(\mu,\beta)}{(\mu,\beta)}{(\mu,\beta)}=\lim\limits_{\mathclap{N_{t}\to\infty}}\,\log\biggl{(}\bigg{.}\frac{\mathopen{}\mathclose{{}\left\langle\phi^{-}_{0}\phi^{+}_{t}}\right\rangle}{\mathopen{}\mathclose{{}\left\langle\phi^{-}_{0}\phi^{+}_{t+1}}\right\rangle}\bigg{.}\biggr{)}\\\
=-\lim\limits_{\mathclap{N_{t}\to\infty}}\,\log\mathchoice{\mathopen{}\mathclose{{}\left(W_{n,n+1}}\right)}{(W_{n,n+1})}{(W_{n,n+1})}{(W_{n,n+1})}/N_{t}\\\
=\,2\,\mathchoice{\mathopen{}\mathclose{{}\left(\mu_{n,n+1}-\mu}\right)}{(\mu_{n,n+1}-\mu)}{(\mu_{n,n+1}-\mu)}{(\mu_{n,n+1}-\mu)}\
,$ (III.22a) where $t_{*}$ is given by the value of $t$ for which the
amplitude of the lightest forward-propagating mode becomes smaller than the
amplitude of the lightest backward-propagating mode (if $\mu=0$, then
$t_{*}=N_{t}/2$). Similarly, using that
$\langle\smash{\phi^{+}_{0}\phi^{-}_{t^{\prime}}}\rangle=\langle\smash{\phi^{-}_{0}\phi^{+}_{N_{t}-t^{\prime}}}\rangle$,
the $\phi^{-}$-mass can for
$\mu\in\mathopen{}\mathclose{{}\left[\mu_{n-1,n}(\beta),\mu_{n,n+1}(\beta)}\right)$
be obtained from (III.21b) with $1\ll t^{\prime}\ll N_{t}-t_{*}$ as:
$m^{-}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta}\right)}{(\mu,\beta)}{(\mu,\beta)}{(\mu,\beta)}=\lim\limits_{\mathclap{N_{t}\to\infty}}\,\log\biggl{(}\bigg{.}\frac{\mathopen{}\mathclose{{}\left\langle\phi^{-}_{0}\phi^{+}_{N_{t}-t^{\prime}}}\right\rangle}{\mathopen{}\mathclose{{}\left\langle\phi^{-}_{0}\phi^{+}_{N_{t}-t^{\prime}-1}}\right\rangle}\bigg{.}\biggr{)}\\\
=-\lim\limits_{\mathclap{N_{t}\to\infty}}\,\log\mathchoice{\mathopen{}\mathclose{{}\left(W_{n-1,n}}\right)}{(W_{n-1,n})}{(W_{n-1,n})}{(W_{n-1,n})}/N_{t}\\\
=\,2\,\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n-1,n}}\right)}{(\mu-\mu_{n-1,n})}{(\mu-\mu_{n-1,n})}{(\mu-\mu_{n-1,n})}\
,$ (III.22b)

with $t^{\prime}$ kept fixed while $N_{t}$ is sent to infinity.
The masses (III.22a) and (III.22b) are shown in Fig. 6 for $\beta=0.03$.
With the expressions for the $\phi^{\pm}$-masses, we can now also write down
an expression for $t_{*}$: requiring that
$\operatorname{e}^{-m^{+}(\mu,\beta)\,t_{*}}=\operatorname{e}^{-m^{-}(\mu,\beta)\,(N_{t}-t_{*})}$
and solving this for $t_{*}$, we find:

$t_{*}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta,N_{t}}\right)}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}=\frac{m^{-}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta}\right)}{(\mu,\beta)}{(\mu,\beta)}{(\mu,\beta)}\,N_{t}}{m^{+}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta}\right)}{(\mu,\beta)}{(\mu,\beta)}{(\mu,\beta)}-m^{-}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta}\right)}{(\mu,\beta)}{(\mu,\beta)}{(\mu,\beta)}}\\\
=\frac{\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n-1,n}}\right)}{(\mu-\mu_{n-1,n})}{(\mu-\mu_{n-1,n})}{(\mu-\mu_{n-1,n})}\,N_{t}}{\mu_{n,n+1}-\mu_{n-1,n}}\
,$ (III.23)

where the $n$ is such that $\mu\in(\mu_{n-1,n},\mu_{n,n+1})$.

As the Euclidean two-point function (III.21) is a sum of pure exponentials
$\sim\operatorname{e}^{-m\,t}$, a zero mode implies the presence of a constant
piece in the two-point function. In the vicinity of a critical point
$\mu_{n,n+1}$ and for sufficiently large $N_{t}$, this constant piece can be
obtained from the value of the two-point function (III.20) at $t=t_{*}$ with
$t_{*}$ from (III.23). For $\mu$ slightly below $\mu_{n,n+1}$, one finds:

$\big{\langle}\phi^{-}_{0}\phi^{+}_{t_{*}}\big{\rangle}\approx\frac{W_{0,n}\,W_{n,n+1}^{s_{*}}+\ldots}{2\,\mathchoice{\mathopen{}\mathclose{{}\left(W_{0,n}+W_{0,n+1}+\ldots}\right)}{(W_{0,n}+W_{0,n+1}+\ldots)}{(W_{0,n}+W_{0,n+1}+\ldots)}{(W_{0,n}+W_{0,n+1}+\ldots)}}\\\
\approx\frac{\,\operatorname{e}^{2\,N_{t}\frac{\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n,n+1}}\right)}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n-1,n}}\right)}{(\mu-\mu_{n-1,n})}{(\mu-\mu_{n-1,n})}{(\mu-\mu_{n-1,n})}}{\mu_{n,n+1}-\mu_{n-1,n}}}}{2\,\mathchoice{\mathopen{}\mathclose{{}\left(1+\operatorname{e}^{2\,N_{t}\,\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n,n+1}}\right)}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}}}\right)}{(1+\operatorname{e}^{2\,N_{t}\,\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n,n+1}}\right)}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}})}{(1+\operatorname{e}^{2\,N_{t}\,\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n,n+1}}\right)}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}})}{(1+\operatorname{e}^{2\,N_{t}\,\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n,n+1}}\right)}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}})}}\
,$ (III.24a) and for $\mu$ slightly above $\mu_{n,n+1}$, correspondingly:
$\big{\langle}\phi^{-}_{0}\phi^{+}_{t_{*}}\big{\rangle}\approx\frac{W_{0,n+1}\,W_{n+1,n+2}^{s_{*}}+\ldots}{2\,\mathchoice{\mathopen{}\mathclose{{}\left(W_{0,n}+W_{0,n+1}+\ldots}\right)}{(W_{0,n}+W_{0,n+1}+\ldots)}{(W_{0,n}+W_{0,n+1}+\ldots)}{(W_{0,n}+W_{0,n+1}+\ldots)}}\\\
\approx\frac{\operatorname{e}^{2\,N_{t}\frac{\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n+1,n+2}}\right)}{(\mu-\mu_{n+1,n+2})}{(\mu-\mu_{n+1,n+2})}{(\mu-\mu_{n+1,n+2})}\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n,n+1}}\right)}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}}{\mu_{n+1,n+2}-\mu_{n,n+1}}}}{2\,\mathchoice{\mathopen{}\mathclose{{}\left(1+\operatorname{e}^{-2\,N_{t}\,\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n,n+1}}\right)}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}}}\right)}{(1+\operatorname{e}^{-2\,N_{t}\,\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n,n+1}}\right)}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}})}{(1+\operatorname{e}^{-2\,N_{t}\,\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n,n+1}}\right)}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}})}{(1+\operatorname{e}^{-2\,N_{t}\,\mathchoice{\mathopen{}\mathclose{{}\left(\mu-\mu_{n,n+1}}\right)}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}{(\mu-\mu_{n,n+1})}})}}\
,$ (III.24b)

where
$s_{*}=t_{*}\mathchoice{\mathopen{}\mathclose{{}\left(\mu,\beta,N_{t}}\right)}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}{(\mu,\beta,N_{t})}/N_{t}$
and we made again used of (III.10). For sufficiently large $N_{t}$, (III.24)
assumes at a critical point $\mu=\mu_{n,n+1}$, $n\in\mathbb{Z}$ the value
$1/4$ and decays rapidly as $\mu$ moves away from $\mu_{n,n+1}$.

So far we have only looked at the behavior of the two-point function at fixed
$\beta$, in which case, as we have seen in the previous section, the limit
$(N_{t}\to\infty)$ gives rise to the appearance of first-order transitions at
the very same values of the chemical potential, for which the zero modes and
the constant piece appear in the two-point function (c.f. eq. (III.24)). Do
these zero modes and constant pieces survive when the limit $(N_{t}\to\infty)$
is taken at fixed $\kappa=\beta/N_{t}=T\,f_{\pi}^{2}$ instead of fixed
$\beta$, in which case there are no first order transitions?

As we have seen already in the previous section, the behavior of the system as
function of the chemical potential changes when we go from low to sufficiently
high temperatures. This should also be reflected in a change of behavior in
the two-point function. To analyse the latter, we use again expression
(III.12) for the weight ratios
$W_{n_{1},n_{2}}\mathchoice{\mathopen{}\mathclose{{}\left(\beta\,\mu,\kappa}\right)}{(\beta\,\mu,\kappa)}{(\beta\,\mu,\kappa)}{(\beta\,\mu,\kappa)}$
at large $\beta$ and write the two-point function as:

$\mathopen{}\mathclose{{}\left\langle\phi_{x}^{-}\phi_{y}^{+}}\right\rangle\mathchoice{\mathopen{}\mathclose{{}\left(\beta\,\mu,\kappa}\right)}{(\beta\,\mu,\kappa)}{(\beta\,\mu,\kappa)}{(\beta\,\mu,\kappa)}=\frac{\sum\limits_{k=-\infty}^{\infty}W_{0,k}\,W_{k,k+1}^{s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}}{2\,\sum\limits_{k=-\infty}^{\infty}W_{0,k}}\\\
\overset{\beta\gg
1}{=}\frac{\sum\limits_{k=-\infty}^{\infty}\operatorname{e}^{-\frac{\mathchoice{\mathopen{}\mathclose{{}\left(k-2\,\beta\,\mu}\right)}{(k-2\,\beta\,\mu)}{(k-2\,\beta\,\mu)}{(k-2\,\beta\,\mu)}^{2}}{2\,\kappa}}\,\operatorname{e}^{-\frac{\mathchoice{\mathopen{}\mathclose{{}\left(2\,k+1-4\,\beta\,\mu}\right)}{(2\,k+1-4\,\beta\,\mu)}{(2\,k+1-4\,\beta\,\mu)}{(2\,k+1-4\,\beta\,\mu)}\,s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}{2\,\kappa}}}{2\,\sum\limits_{k=-\infty}^{\infty}\operatorname{e}^{-\frac{\mathchoice{\mathopen{}\mathclose{{}\left(k-2\,\beta\,\mu}\right)}{(k-2\,\beta\,\mu)}{(k-2\,\beta\,\mu)}{(k-2\,\beta\,\mu)}^{2}}{2\,\kappa}}}\\\
=\frac{\sum\limits_{k=-\infty}^{\infty}\operatorname{e}^{-\frac{\mathchoice{\mathopen{}\mathclose{{}\left(k-2\,\beta\,\mu}\right)}{(k-2\,\beta\,\mu)}{(k-2\,\beta\,\mu)}{(k-2\,\beta\,\mu)}^{2}}{2\,\kappa}}\,\operatorname{e}^{-\frac{\beta\,m_{k}\,s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}{\kappa}}}{2\,\sum\limits_{k=-\infty}^{\infty}\operatorname{e}^{-\frac{\mathchoice{\mathopen{}\mathclose{{}\left(k-2\,\beta\,\mu}\right)}{(k-2\,\beta\,\mu)}{(k-2\,\beta\,\mu)}{(k-2\,\beta\,\mu)}^{2}}{2\,\kappa}}}\
.$ (III.25)

with mass terms

$\beta\,m_{k}\mathchoice{\mathopen{}\mathclose{{}\left(\beta\,\mu,\kappa}\right)}{(\beta\,\mu,\kappa)}{(\beta\,\mu,\kappa)}{(\beta\,\mu,\kappa)}=\frac{2\,k+1-4\,\beta\,\mu}{2}\
\forall\,k\in\mathbb{Z}\ .$ (III.26)

As long as the temperature is sufficiently small, i.e. for
$\kappa=\beta/N_{t}=f_{\pi}\,T<1/8$, the Gaussian weights in numerator and
denominator of (III.25) are sufficiently narrow to give dominant weight to the
terms for which $k$ is closest to the value $2\,\beta\,\mu$. At a critical
point

$\beta\,\mu=\beta\,\mu_{n,n+1}\overset{\beta\gg 1}{=}(2\,n+1)/4\ ,\
n\in\mathbb{Z}\ ,$ (III.27)

the value of $2\,\beta\,\mu$ is located precisely in the middle between two
successive $k$-values. The Gaussian weights for the terms with $k=n$ and
$k=n+1$ are then equal, and the masses corresponding to these two dominant
modes assume the values $\beta\,m_{n}=0$ and $\beta,m_{n+1}=1$, meaning that
the long-distance behavior of the system is dominated by a zero mode.

In Fig. 7 the $N_{t}$-dependency of the rescaled masses $\beta\,m^{\pm}$ as
functions of $\beta\,\mu$ is illustrated for $\kappa=0.00003$, corresponding
to a low but non-zero temperature. The spectrum in the uppermost panel with
$N_{t}=1024$ then corresponds to a system at $\beta=\kappa\,N_{t}\approx 0.03$
and looks therefore (up to the rescaling of the axes by a factor of $\beta$)
identical to the spectrum shown in Fig. 6. However, if now $N_{t}$ is
increased while $\kappa$ is kept fixed, $\beta$ has to increases too and the
critical values of $\beta\,\mu$ change according to (III.9): as long as
$\beta=\kappa\,N_{t}\ll 1$, the critical values of $\beta\,\mu$ grow like
$\beta\,\mu_{n,n+1}(\beta)\approx\frac{\beta}{2}\log\bigl{(}\big{.}\frac{2(n+1)}{\beta}\big{.}\bigr{)}$
as function of increasing $\beta=\kappa\,N_{t}$, but as soon as $N_{t}$ is
sufficiently large, so that $\beta=\kappa\,N_{t}\gg 1$, the critical values of
$\beta\,\mu$ converge towards their $\beta$-independent asymptotic values
$\beta\,\mu_{n,n+1}(\beta)\approx\frac{2\,n+1}{4}$, $n\in\mathbb{Z}$. The
critical values then become uniformly spaced an the mass-spectrum, which
depends on the difference between successive critical values, becomes periodic
in $\beta\,\mu$, as illustrated in the lower-most panel of Fig. 7. At this
point, a further increase of $N_{t}$ leaves the spectrum unchanged.

For $\kappa$ bigger than $1/8$, the Gaussian weights in (III.25) can no longer
sufficiently project on the terms for which $k$ is closest to $2\,\beta\,\mu$
and the long-distance behavior of the system at a critical value of $\beta\mu$
will not necessarily be dominated by a zero mode any more. By expressing the
two-point function in terms of theta functions, we can write:

$\mathopen{}\mathclose{{}\left\langle\phi_{x}^{-}\phi_{y}^{+}}\right\rangle\mathchoice{\mathopen{}\mathclose{{}\left(\beta\,\mu,\kappa}\right)}{(\beta\,\mu,\kappa)}{(\beta\,\mu,\kappa)}{(\beta\,\mu,\kappa)}\overset{\beta\gg
1}{=}\frac{\mathchoice{\mathopen{}\mathclose{{}\left(u^{2}\,q}\right)}{(u^{2}\,q)}{(u^{2}\,q)}{(u^{2}\,q)}^{s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}\vartheta\mathchoice{\mathopen{}\mathclose{{}\left(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q}\right)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}}{2\,\vartheta\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)}}\\\
=\frac{\mathchoice{\mathopen{}\mathclose{{}\left(u^{2}\,q}\right)}{(u^{2}\,q)}{(u^{2}\,q)}{(u^{2}\,q)}^{s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}\alpha\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)}\,\vartheta\mathchoice{\mathopen{}\mathclose{{}\left(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q}\right)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)}}\right)}{(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q}\right)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)})}{(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q}\right)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)})}{(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q}\right)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)})}}{2\,\alpha\mathchoice{\mathopen{}\mathclose{{}\left(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q}\right)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}{(u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)},q)}\,\vartheta\mathchoice{\mathopen{}\mathclose{{}\left(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)}}\right)}{(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)})}{(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)})}{(u^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(u,q}\right)}{(u,q)}{(u,q)}{(u,q)},q^{\prime}\mathchoice{\mathopen{}\mathclose{{}\left(q}\right)}{(q)}{(q)}{(q)})}}\\\
=\operatorname{e}^{-\frac{s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}\,\mathchoice{\mathopen{}\mathclose{{}\left(1-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}\right)}{(1-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}{(1-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}{(1-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}}{2\,\kappa}}\\\
\cdot\frac{\vartheta\mathchoice{\mathopen{}\mathclose{{}\left(\operatorname{e}^{-\pi\,\mathrm{i}\,\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}\right)}{(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}{(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}{(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}},\operatorname{e}^{-2\,\pi^{2}\,\kappa}}\right)}{(\operatorname{e}^{-\pi\,\mathrm{i}\,\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}\right)}{(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}{(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}{(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}},\operatorname{e}^{-2\,\pi^{2}\,\kappa})}{(\operatorname{e}^{-\pi\,\mathrm{i}\,\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}\right)}{(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}{(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}{(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}},\operatorname{e}^{-2\,\pi^{2}\,\kappa})}{(\operatorname{e}^{-\pi\,\mathrm{i}\,\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}\right)}{(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}{(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}{(2\,\beta\,\mu-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}},\operatorname{e}^{-2\,\pi^{2}\,\kappa})}}{2\,\vartheta\mathchoice{\mathopen{}\mathclose{{}\left(\operatorname{e}^{-\pi\,\mathrm{i}\,\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu}\right)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}},\operatorname{e}^{-2\,\pi^{2}\,\kappa}}\right)}{(\operatorname{e}^{-\pi\,\mathrm{i}\,\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu}\right)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}},\operatorname{e}^{-2\,\pi^{2}\,\kappa})}{(\operatorname{e}^{-\pi\,\mathrm{i}\,\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu}\right)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}},\operatorname{e}^{-2\,\pi^{2}\,\kappa})}{(\operatorname{e}^{-\pi\,\mathrm{i}\,\mathchoice{\mathopen{}\mathclose{{}\left(2\,\beta\,\mu}\right)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}{(2\,\beta\,\mu)}},\operatorname{e}^{-2\,\pi^{2}\,\kappa})}}\
,$ (III.28)

where
$q=\operatorname{e}^{-1/\mathchoice{\mathopen{}\mathclose{{}\left(2\,\kappa}\right)}{(2\,\kappa)}{(2\,\kappa)}{(2\,\kappa)}}$,
$u=\operatorname{e}^{\beta\,\mu/\kappa}$,
$u_{2}\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}=u\,q^{s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}$
and
$s(x,y)=\mathchoice{\mathopen{}\mathclose{{}\left(\mathchoice{\mathopen{}\mathclose{{}\left(y-x}\right)}{(y-x)}{(y-x)}{(y-x)}\bmod
N_{t}}\right)}{(\mathchoice{\mathopen{}\mathclose{{}\left(y-x}\right)}{(y-x)}{(y-x)}{(y-x)}\bmod
N_{t})}{(\mathchoice{\mathopen{}\mathclose{{}\left(y-x}\right)}{(y-x)}{(y-x)}{(y-x)}\bmod
N_{t})}{(\mathchoice{\mathopen{}\mathclose{{}\left(y-x}\right)}{(y-x)}{(y-x)}{(y-x)}\bmod
N_{t})}/N_{t}$, and on the second line, the transformations (III.15) were
used. For sufficiently large $\kappa$, the theta functions on the last line of
(III.28) converge towards unity and the two-point function reduces therefore
to:

$\mathopen{}\mathclose{{}\left\langle\phi_{x}^{-}\phi_{y}^{+}}\right\rangle\mathchoice{\mathopen{}\mathclose{{}\left(\kappa}\right)}{(\kappa)}{(\kappa)}{(\kappa)}\approx\frac{1}{2}\,\operatorname{e}^{-\frac{s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}\,\mathchoice{\mathopen{}\mathclose{{}\left(1-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)}}\right)}{(1-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}{(1-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}{(1-s\mathchoice{\mathopen{}\mathclose{{}\left(x,y}\right)}{(x,y)}{(x,y)}{(x,y)})}}{2\,\kappa}}\
,$ (III.29)

which is independent of $\beta\,\mu$ but depends only on the temperature
$\sim\kappa$.
To summarize: the expression (III.25) for the two-point function illustrates
that zero modes will always be present in the large-$N_{t}$ limit when $\mu$
is set to one of the critical points $\mu_{n,n+1}$, $n\in\mathbb{Z}$,
regardless of whether $\kappa$ is small or large. However, if $\kappa$, which
sets the temperature, is sufficiently large, the system will be completely
dominated by thermal modes and the zero-modes will no-longer give relevant
contributions to expectation values at the critical points.

Figure 7: The figures show for the one-dimensional
$\operatorname{O}\mathopen{}\mathclose{{}\left(2}\right)$-model with
$\tilde{s}=0$ and $\kappa=\beta/N_{t}=0.00003$ the rescaled
$\phi^{\pm}$-masses, $\beta\,m^{\pm}$, as functions of $\beta\,\mu$ for
different values of $N_{t}$.

###  III.3 Explicit symmetry breaking

We have seen in the previous section that the
$\operatorname{O}\mathopen{}\mathclose{{}\left(2}\right)$ model, coupled to a
chemical potential $\mu$, can in (0+1) dimensions develop some sort of long-
range order when $\mu$ approaches certain critical values $\mu_{n,n+1}$,
$n\in\mathbb{Z}$. In this section we would now like to check whether this long
range order also gives rise to the formation of a non-zero condensate
$\langle\smash{\phi^{\pm}}\rangle$.

To measure a condensate directly, one needs to break the global
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ symmetry explicitly
by setting $\tilde{s}$ in (III.1) to a non-zero value. This is necessary
because of the fact, that the partition function (III.1), although
reformulated in terms of flux-variables, still represents a path-integral that
sums over all possible configurations of the lattice field $\phi^{\pm}_{x}$,
which means, that if the global symmetry is spontaneously broken, the path-
integral still sums over all possible ways in which this spontaneous breaking
can happen. In expectation values of observables like
$\phi^{\pm}=\frac{1}{\sqrt{2}}\operatorname{e}^{\pm\,\mathrm{i}\,\theta}$,
which depend on the direction in which the symmetry gets broken, the
contributions from configurations that differ only by this direction would
therefore exactly cancel each other. To avoid such cancellations, one has to
single out a particular direction in which the symmetry should break, which is
most easily done by coupling $\phi^{\pm}_{x}$ to a non-zero source
$s^{\pm}=\frac{\tilde{s}\,\operatorname{e}^{\mp\,\mathrm{i}\,\phi_{s}}}{\sqrt{2}}$.
The value of the partition function does not depend on the choice of the
direction that is singled out and the choice of the phase $\phi_{s}$ of the
source is therefore irrelevant; only the magnitude $\tilde{s}$ of the source
matters. The value of $\phi_{s}$ will only enter in expectation values of
observables that are not invariant under the global
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ symmetry.

With $\tilde{s}>0$, the 1D partition function for the
$\operatorname{O}\mathopen{}\mathclose{{}\left(2}\right)$-model does not
reduce to a single sum as in equation (III.2); the non-zero value of
$\tilde{s}$ breaks charge conservation, so that $k$-variables on different
links can now take on different values, at the cost of producing monomers and
thereby changing the site weight $I_{p_{x}}(\tilde{s})$. One then arrives at
the following expression:

$Z=\sum\limits_{\mathopen{}\mathclose{{}\left\\{k}\right\\}}\,\prod\limits_{x}\biggl{\\{}\bigg{.}I_{k_{x}}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}\,e^{2\,\mu\,k_{x}}\\\
\cdot
I_{\mathchoice{\mathopen{}\mathclose{{}\left(k_{x}-k_{x-1}}\right)}{(k_{x}-k_{x-1})}{(k_{x}-k_{x-1})}{(k_{x}-k_{x-1})}}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}\,\operatorname{e}^{\mathrm{i}\,\phi_{s}\,\mathchoice{\mathopen{}\mathclose{{}\left(k_{x}-k_{x-1}}\right)}{(k_{x}-k_{x-1})}{(k_{x}-k_{x-1})}{(k_{x}-k_{x-1})}}\bigg{.}\biggr{\\}}\\\
=\,\operatorname{tr}\bigl{(}\big{.}T_{0}^{N_{t}}\big{.}\bigr{)}\ ,$ (III.30)

where we have summed over the $p$-variables in order to get rid of the delta-
function constraints in (III.1), and after the second equality sign, $N_{t}$
is (as usual) the temporal system size. The _transfer matrix_ , $T_{\delta}$,
can for example be chosen as:

$\mathchoice{\mathopen{}\mathclose{{}\left(T_{\delta}}\right)}{(T_{\delta})}{(T_{\delta})}{(T_{\delta})}_{ab}=\\\
I_{a}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}\,\operatorname{e}^{2\,\mu\,a}\,\frac{I_{\mathchoice{\mathopen{}\mathclose{{}\left(a-b-\delta}\right)}{(a-b-\delta)}{(a-b-\delta)}{(a-b-\delta)}}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}\,\operatorname{e}^{\mathrm{i}\,\phi_{s}\,\mathchoice{\mathopen{}\mathclose{{}\left(a-b-\delta}\right)}{(a-b-\delta)}{(a-b-\delta)}{(a-b-\delta)}}}{2^{\mathopen{}\mathclose{{}\left|\delta}\right|/2}}\
,$ (III.31)

with $\delta\in\mathopen{}\mathclose{{}\left\\{-1,1,0}\right\\}$, depending on
whether the site to which the transfer matrix is associated contains an
external $\phi^{-}$, $\phi^{+}$ or no external field (cf. the discussion in
appendix 2 and the appearance of Kronecker-deltas in the delta-function
constraints for the one- and two-point partition functions).
The one- and two-point functions can be expressed in terms of the transfer
matrix as:

$\mathopen{}\mathclose{{}\left\langle\phi^{\pm}}\right\rangle\,=\,\frac{\operatorname{tr}\bigl{(}\big{.}T_{\pm
1}\,T_{0}^{N_{t}-1}\big{.}\bigr{)}}{\operatorname{tr}\bigl{(}\big{.}T_{0}^{N_{t}}\big{.}\bigr{)}}\
,$ (III.32)

and

$\mathopen{}\mathclose{{}\left\langle\phi^{-}_{0}\phi^{+}_{t}}\right\rangle=\\\
\begin{cases}\frac{1}{2}&\text{if}\quad t=0\\\
\frac{\operatorname{tr}\mathchoice{\mathopen{}\mathclose{{}\left(T_{-1}T_{0}^{t-1}T_{1}^{\vphantom{N_{t}}}T_{0}^{N_{t}-t-1}}\right)}{(T_{-1}T_{0}^{t-1}T_{1}^{\vphantom{N_{t}}}T_{0}^{N_{t}-t-1})}{(T_{-1}T_{0}^{t-1}T_{1}^{\vphantom{N_{t}}}T_{0}^{N_{t}-t-1})}{(T_{-1}T_{0}^{t-1}T_{1}^{\vphantom{N_{t}}}T_{0}^{N_{t}-t-1})}}{\operatorname{tr}\mathchoice{\mathopen{}\mathclose{{}\left(T_{0}^{N_{t}}}\right)}{(T_{0}^{N_{t}})}{(T_{0}^{N_{t}})}{(T_{0}^{N_{t}})}}&\text{if}\quad
t>0\end{cases}\ .$ (III.33)

Figure 8: The figure shows the $\phi^{\pm}$-condensate as function of the
chemical potential $\mu$ for the (0+1)-dimensional
$\operatorname{O}\mathopen{}\mathclose{{}\left(2}\right)$-model with
$\tilde{s}=0.01$ and $\beta=0.03$, obtained from eq. (III.32) for various
values of $N_{t}$. The curves for $N_{t}=1024$ and $N_{t}=4096$ are
essentially on top of each other.

In Fig. 8, we show (III.32) as function of $\mu$ when evaluated numerically
with $\tilde{s}=0.01$ and $\beta=0.03$ for various values of $N_{t}$. As can
be seen: at the critical points, $\mu=\mu_{n,n+1}$, $n\in\mathbb{Z}$, the
condensate, $\langle\smash{\phi^{\pm}}\rangle$, peaks and approaches for
sufficiently large $N_{t}$ a finite value. A comparison of the value of the
condensate and the magnitude of the constant piece of the two-point function
(III.33) at these critical points further shows, that:

$\lim\limits_{t\to\infty}\mathopen{}\mathclose{{}\left\langle\phi^{-}_{0}\phi^{+}_{t}}\right\rangle\,=\,\mathopen{}\mathclose{{}\left\langle\phi^{-}}\right\rangle\mathopen{}\mathclose{{}\left\langle\phi^{+}}\right\rangle\
.$ (III.34)

However, comparing the numerically obtained result for this constant piece in
the two point function for $\tilde{s}>0$ with the value we found above in
(III.24) for $\tilde{s}=0$, reveals, that the value for $\tilde{s}>0$ is only
about half the value obtained with $\tilde{s}=0$.

In order to better understand this behavior, we determine analytically the
leading contributions to the one- and two-point functions at the critical
points $\mu_{n,n+1}$, $n\in\mathbb{Z}$.
For $\mu=\mu_{n,n+1}$, the factors
$I_{k_{x}}\mathchoice{\mathopen{}\mathclose{{}\left(\beta}\right)}{(\beta)}{(\beta)}{(\beta)}\,\operatorname{e}^{2\,\mu\,k_{x}}$
in (III.30) are also for $\tilde{s}>0$ maximized if
$k_{x}\in\mathopen{}\mathclose{{}\left\\{n,n+1}\right\\}$. If $\tilde{s}$ is
not too large, we can then assume that the one- and two-point functions,
(III.32) and (III.33), are well approximated by taking into account only the 2
by 2 blocks of the transfer-matrix $(T_{\delta})_{ab}$ from (III.31) for which
$a,b\in\mathopen{}\mathclose{{}\left\\{n,n+1}\right\\}$.
For the two-point function at time separation $t$ one then finds:

$\displaystyle\big{\langle}\phi^{-}_{0}\phi^{+}_{t}\big{\rangle}$
$\displaystyle\mathchoice{\mathopen{}\mathclose{{}\left(\mu=\mu_{n,n+1},N_{t}}\right)}{(\mu=\mu_{n,n+1},N_{t})}{(\mu=\mu_{n,n+1},N_{t})}{(\mu=\mu_{n,n+1},N_{t})}$
$\displaystyle\overset{\phantom{\text{\eqref{eq:binarymodelpartf}}}}{\approx}\,\frac{\sum\limits_{\mathopen{}\mathclose{{}\left\\{k_{i}\in\mathopen{}\mathclose{{}\left\\{n,n+1}\right\\}}\right\\}_{\forall
i}}\,\bigl{(}\big{.}T_{-1}\big{.}\bigr{)}_{k_{N_{t}-1},k_{0}}\bigl{(}\big{.}T_{0}\big{.}\bigr{)}_{k_{0},k_{1}}\ldots\bigl{(}\big{.}T_{1}\big{.}\bigr{)}_{k_{t-1},k_{t}}\bigl{(}\big{.}T_{0}\big{.}\bigr{)}_{k_{t},k_{t+1}}\ldots}{\sum\limits_{\mathopen{}\mathclose{{}\left\\{k_{i}\in\mathopen{}\mathclose{{}\left\\{n,n+1}\right\\}}\right\\}_{\forall
i}}\,\bigl{(}\big{.}T_{0}\big{.}\bigr{)}_{k_{N_{t}-1},k_{0}}\ldots\bigl{(}\big{.}T_{0}\big{.}\bigr{)}_{k_{N_{t}-2},k_{N_{t}-1}}}$
$\displaystyle\overset{\begin{subarray}{1}\text{\eqref{eq:o2fluxrep1dtransfermat}}\\\
\text{\eqref{eq:o2critmu1d}}\end{subarray}}{\approx}\,\frac{c\mathchoice{\mathopen{}\mathclose{{}\left(t,I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}\right)}{(t,I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}{(t,I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}{(t,I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}\,c\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}-t,I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}\right)}{(N_{t}-t,I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}{(N_{t}-t,I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}{(N_{t}-t,I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}}{4\,c\mathchoice{\mathopen{}\mathclose{{}\left(N_{t},I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}\right)}{(N_{t},I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}{(N_{t},I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}{(N_{t},I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}}$
$\displaystyle\overset{\text{\eqref{eq:binarymodelpartf}}}{=}\,\frac{1}{8}\biggl{(}\bigg{.}1+\frac{\cosh\bigl{(}\big{.}{\mathchoice{\mathopen{}\mathclose{{}\left(\frac{N_{t}}{2}-t}\right)}{(\frac{N_{t}}{2}-t)}{(\frac{N_{t}}{2}-t)}{(\frac{N_{t}}{2}-t)}\log\mathchoice{\mathopen{}\mathclose{{}\left(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}}\right)}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}}\big{.}\bigr{)}}{\cosh\bigl{(}\big{.}{\frac{N_{t}}{2}\underbrace{{{\log\mathchoice{\mathopen{}\mathclose{{}\left(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}}\right)}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}}}}_{\mathclap{\begin{subarray}{c}-\mathchoice{\mathopen{}\mathclose{{}\left(\log(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+\frac{\mathrm{d}I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{\mathrm{d}\tilde{s}})-\log(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-\frac{\mathrm{d}I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{\mathrm{d}\tilde{s}})}\right)}{(\log(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+\frac{\mathrm{d}I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{\mathrm{d}\tilde{s}})-\log(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-\frac{\mathrm{d}I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{\mathrm{d}\tilde{s}}))}{(\log(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+\frac{\mathrm{d}I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{\mathrm{d}\tilde{s}})-\log(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-\frac{\mathrm{d}I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{\mathrm{d}\tilde{s}}))}{(\log(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+\frac{\mathrm{d}I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{\mathrm{d}\tilde{s}})-\log(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-\frac{\mathrm{d}I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{\mathrm{d}\tilde{s}}))}\\\
\approx-2\frac{\mathrm{d}\log\mathchoice{\mathopen{}\mathclose{{}\left(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}\right)}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}{(I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}}{\mathrm{d}\tilde{s}}=-\tilde{s}+\mathcal{O}(\tilde{s}^{3})\end{subarray}}}}\big{.}\bigr{)}}\bigg{.}\biggr{)}\,\approx\,\frac{1}{8}\biggl{(}\bigg{.}1+\frac{\cosh\bigl{(}\big{.}\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}/2-t}\right)}{(N_{t}/2-t)}{(N_{t}/2-t)}{(N_{t}/2-t)}\,\tilde{s}\big{.}\bigr{)}}{\cosh\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}\,\tilde{s}/2}\right)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}}\bigg{.}\biggr{)}\
,$ (III.35)

where the function
$c\mathchoice{\mathopen{}\mathclose{{}\left(L,A,B}\right)}{(L,A,B)}{(L,A,B)}{(L,A,B)}$
is given by

$c\mathchoice{\mathopen{}\mathclose{{}\left(L,A,B}\right)}{(L,A,B)}{(L,A,B)}{(L,A,B)}\,=\,\sum\limits_{m=0}^{\lfloor
L/2\rfloor}\,\binom{L}{2\,m}\,A^{L-2\,m}\,B^{2\,m}\\\
=\,\frac{1}{2}\mathchoice{\mathopen{}\mathclose{{}\left(\mathchoice{\mathopen{}\mathclose{{}\left(A+B}\right)}{(A+B)}{(A+B)}{(A+B)}^{L}\,+\,\mathchoice{\mathopen{}\mathclose{{}\left(A-B}\right)}{(A-B)}{(A-B)}{(A-B)}^{L}}\right)}{(\mathchoice{\mathopen{}\mathclose{{}\left(A+B}\right)}{(A+B)}{(A+B)}{(A+B)}^{L}\,+\,\mathchoice{\mathopen{}\mathclose{{}\left(A-B}\right)}{(A-B)}{(A-B)}{(A-B)}^{L})}{(\mathchoice{\mathopen{}\mathclose{{}\left(A+B}\right)}{(A+B)}{(A+B)}{(A+B)}^{L}\,+\,\mathchoice{\mathopen{}\mathclose{{}\left(A-B}\right)}{(A-B)}{(A-B)}{(A-B)}^{L})}{(\mathchoice{\mathopen{}\mathclose{{}\left(A+B}\right)}{(A+B)}{(A+B)}{(A+B)}^{L}\,+\,\mathchoice{\mathopen{}\mathclose{{}\left(A-B}\right)}{(A-B)}{(A-B)}{(A-B)}^{L})}\
,$ (III.36)

and represents the partition function for a system of $L\in\mathbb{N}$ binary
degrees of freedom, in which the two distinct states have weights $A$ and $B$,
respectively, and where the states with weight $B$ must always be occupied by
an even number of degrees of freedom. The reason for the appearance of the
function
$c\mathchoice{\mathopen{}\mathclose{{}\left(L,A,B}\right)}{(L,A,B)}{(L,A,B)}{(L,A,B)}$
in (III.35) is explained in Figs. 9-10; in short: to each site of our
(0+1)-dimensional lattice corresponds one binary d.o.f. and the two possible
states of the d.o.f. correspond to whether the two $k$-variables that touch at
the site do ($\sim$state with weight $A$) or do not ($\sim$state with weight
$B$) have the same value.

Along the same line, one finds the leading contributions to the one-point
function at a critical point $\mu_{n,n+1}$ to be:

$\big{\langle}\phi^{\pm}\big{\rangle}\mathchoice{\mathopen{}\mathclose{{}\left(\mu=\mu_{n,n+1},N_{t}}\right)}{(\mu=\mu_{n,n+1},N_{t})}{(\mu=\mu_{n,n+1},N_{t})}{(\mu=\mu_{n,n+1},N_{t})}\,\approx\,\frac{\sum\limits_{\mathopen{}\mathclose{{}\left\\{k_{i}\in\mathopen{}\mathclose{{}\left\\{n,n+1}\right\\}}\right\\}_{\forall
i}}\,\bigl{(}\big{.}T_{\pm
1}\big{.}\bigr{)}_{k_{N_{t}-1},k_{0}}\bigl{(}\big{.}T_{0}\big{.}\bigr{)}_{k_{0},k_{1}}\ldots\bigl{(}\big{.}T_{0}\big{.}\bigr{)}_{k_{N_{t}-2},k_{N_{t}-1}}}{\sum\limits_{\mathopen{}\mathclose{{}\left\\{k_{i}\in\mathopen{}\mathclose{{}\left\\{n,n+1}\right\\}}\right\\}_{\forall
i}}\,\bigl{(}\big{.}T_{0}\big{.}\bigr{)}_{k_{N_{t}-1},k_{0}}\ldots\bigl{(}\big{.}T_{0}\big{.}\bigr{)}_{k_{N_{t}-2},k_{N_{t}-1}}}\\\
=\,\frac{c_{odd}\mathchoice{\mathopen{}\mathclose{{}\left(N_{t},I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}\right)}{(N_{t},I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}{(N_{t},I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}{(N_{t},I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}}{2^{3/2}\,c\mathchoice{\mathopen{}\mathclose{{}\left(N_{t},I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}\right)}{(N_{t},I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}{(N_{t},I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}{(N_{t},I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})},I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})})}}\,=\,-\frac{1}{2^{3/2}}\tanh\mathchoice{\mathopen{}\mathclose{{}\left({\frac{N_{t}}{2}\log\mathchoice{\mathopen{}\mathclose{{}\left(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}}\right)}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}}}\right)}{({\frac{N_{t}}{2}\log\mathchoice{\mathopen{}\mathclose{{}\left(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}}\right)}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}})}{({\frac{N_{t}}{2}\log\mathchoice{\mathopen{}\mathclose{{}\left(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}}\right)}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}})}{({\frac{N_{t}}{2}\log\mathchoice{\mathopen{}\mathclose{{}\left(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}}\right)}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}{(\frac{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}-I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}}{I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}+I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}})}})}\\\
\approx\,2^{-3/2}\tanh\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}\,\tilde{s}/2}\right)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}\
,$ (III.37)

where
$c\mathchoice{\mathopen{}\mathclose{{}\left(L,A,B}\right)}{(L,A,B)}{(L,A,B)}{(L,A,B)}$
is again given by (III.36), and the new function,

$c_{odd}\mathchoice{\mathopen{}\mathclose{{}\left(L,A,B}\right)}{(L,A,B)}{(L,A,B)}{(L,A,B)}\,=\\\
\sum\limits_{m=0}^{\lfloor\mathchoice{\mathopen{}\mathclose{{}\left(L-1}\right)}{(L-1)}{(L-1)}{(L-1)}/2\rfloor}\,\binom{L}{2\,m+1}\,A^{L-\mathchoice{\mathopen{}\mathclose{{}\left(2\,m+1}\right)}{(2\,m+1)}{(2\,m+1)}{(2\,m+1)}}\,B^{2\,m+1}\\\
=\,\frac{1}{2}\mathchoice{\mathopen{}\mathclose{{}\left(\mathchoice{\mathopen{}\mathclose{{}\left(A+B}\right)}{(A+B)}{(A+B)}{(A+B)}^{L}\,-\,\mathchoice{\mathopen{}\mathclose{{}\left(A-B}\right)}{(A-B)}{(A-B)}{(A-B)}^{L}}\right)}{(\mathchoice{\mathopen{}\mathclose{{}\left(A+B}\right)}{(A+B)}{(A+B)}{(A+B)}^{L}\,-\,\mathchoice{\mathopen{}\mathclose{{}\left(A-B}\right)}{(A-B)}{(A-B)}{(A-B)}^{L})}{(\mathchoice{\mathopen{}\mathclose{{}\left(A+B}\right)}{(A+B)}{(A+B)}{(A+B)}^{L}\,-\,\mathchoice{\mathopen{}\mathclose{{}\left(A-B}\right)}{(A-B)}{(A-B)}{(A-B)}^{L})}{(\mathchoice{\mathopen{}\mathclose{{}\left(A+B}\right)}{(A+B)}{(A+B)}{(A+B)}^{L}\,-\,\mathchoice{\mathopen{}\mathclose{{}\left(A-B}\right)}{(A-B)}{(A-B)}{(A-B)}^{L})}\
,$ (III.38)

has a similar meaning as
$c\mathchoice{\mathopen{}\mathclose{{}\left(L,A,B}\right)}{(L,A,B)}{(L,A,B)}{(L,A,B)}$,
but the state with weight $B$ has now to be occupied always by an odd instead
of even number of degrees of freedom.

Figure 9: The figure shows how the function (III.36) enters the denominator of
(III.35). The $x$-axis represents the different sites of the periodic
(0+1)-dimensional lattice on which (III.30) is defined, and the $y$-axis
display the values of the $k$-variables between the sites. The chemical
potential is set to the critical value $\mu=\mu_{n,n+1}$, so that $k_{x}=n$
$\forall x$ and $k_{x}=n+1$ $\forall x$ are equally likely and have maximum
weight. However, as $\tilde{s}>0$, the $k$-variables don’t have to be all the
same, but can vary between the two values $n$ and $n+1$, while other values
are highly suppressed. Now, assume that we are initially in the configuration
where $k_{x}=n$ $\forall x$ (left-hand panel). As all $k$-variables are equal,
all site-weights in (III.30) assume the value
$I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}$.
Next, we modify this configuration by picking some _shift-points_ , e.g.
$\mathopen{}\mathclose{{}\left\\{x_{1},\ldots,x_{4}}\right\\}$, at which the
values of incoming and outgoing $k$-variable changes form $n$ to $n+1$ (upward
shift-point) or from $n+1$ back to $n$ (downward shift-point). As the lattice
is periodic, these shift-points have to appear in pairs. The modified
configuration has lower weight than the original one, as the site-weights for
the shift-points
$\mathopen{}\mathclose{{}\left\\{x_{1},\ldots,x_{4}}\right\\}$ in (III.30) are
reduced from
$I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}$
to
$I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}$;
but, there are $\binom{N_{t}}{4}$ ways of picking the four shift-points,
$\mathopen{}\mathclose{{}\left\\{x_{1},\ldots,x_{4}}\right\\}$, and therefore
$\binom{N_{t}}{4}$ such modified configurations which are distinct but have
the same weight. Similarly, if the number of shit-points is not four, but more
generally $2\,m$, with $m\in\mathopen{}\mathclose{{}\left\\{0,1,\ldots,\lfloor
N_{t}/2\rfloor}\right\\}$, there will be $\binom{N_{t}}{2\,m}$ configurations
with a weight in which $2\,m$ site-weights have value
$I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}$
instead of
$I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}$.
The total weight of all the modified configurations that can be obtained in
this way from the config. with $k_{x}=n$ $\forall x$ (including the config.
with $k_{x}=n$ $\forall x$ itself) is therefore: $\sum_{m=0}^{\lfloor
N_{t}/2\rfloor}\,\binom{N_{t}}{2\,m}\,I_{0}^{N_{t}-2\,m}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}\,I_{1}^{2\,m}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}$.
In the same way, one can modify the configuration with $k_{x}=n+1$ $\forall x$
(right-hand panel). One then gets the set of configurations which are the
mirror-images of the ones obtained from the config. with $k_{x}=n$ $\forall
x$, (i.e. the values $n$ and $n+1$ of the $k$-variables are interchanged), but
as they are equally many mirror-images, the total weight is again given by:
$\sum_{m=0}^{\lfloor
N_{t}/2\rfloor}\,\binom{N_{t}}{2\,m}\,I_{0}^{N_{t}-2\,m}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}\,I_{1}^{2\,m}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}$.

Figure 10: The figure illustrates how the function (III.36) enters the
enumerator of (III.35) and we proceed in a similar way as in Fig. 9. The
chemical potential is set to the critical value $\mu=\mu_{n,n+1}$, so that all
$k$-variables have value $n$ or $n+1$, and we start with the maximum-weight
configuration, displayed in left-hand panel, in which the external $\phi^{-}$
at site $z_{1}$ requires the incoming and outgoing $k$-variables to have
values $n$ and $n+1$, respectively, while the $\phi^{+}$ at site $z_{2}$
forces incoming and outgoing $k$-variables to have values $n+1$ and $n$. As in
Fig. 9, we can now pick an even number of lattice sites which become _shift-
points_ at which the values of incoming and outgoing $k$-variables have to be
$n$ and $n+1$, respectively (upward shift-point), or $n+1$ and $n$ (downward
shift-point), unless the shift-point coincides with $z_{1}$ or $z_{2}$, in
which case incoming and outgoing $k$-variable have to have the the same value
instead of distinct ones. In the right-hand panel, we have again choose four
such shift-points,
$\mathopen{}\mathclose{{}\left\\{x_{1},\ldots,x_{4}}\right\\}$, and obtain a
modified configuration for which four site-weights are changed form
$I_{0}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}$
to
$I_{1}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}$.
But, in contrast to the situation in Fig. 9, the presence of the external
fields on sites $z_{1}$ and $z_{2}$, limits the number of ways in which one
can pick the four shift-points,
$\mathopen{}\mathclose{{}\left\\{x_{1},\ldots,x_{4}}\right\\}$: $z_{1}$ and
$z_{2}$ split the system into two parts and in order to have
$k_{x}\in\mathopen{}\mathclose{{}\left\\{n,n+1}\right\\}$ $\forall x$, the
number of shift-points must be even in both of these parts separately.
Therefore, if we neglect the complications that arise at the boundaries
between the two parts, the total weight of all configurations that can be
obtained by adding shift-points to the maximum weight configuration in the
left-hand panel, is approximately given by:
$\bigl{(}\big{.}\sum_{m=0}^{\lfloor\Delta z/2\rfloor}\,\binom{\Delta
z/2}{2\,m}\,I_{0}^{\Delta
z/2-2\,m}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}\,I_{1}^{2\,m}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}\big{.}\bigr{)}\bigl{(}\big{.}\sum_{m=0}^{\lfloor\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}-\Delta
z}\right)}{(N_{t}-\Delta z)}{(N_{t}-\Delta z)}{(N_{t}-\Delta
z)}/2\rfloor}\,\binom{\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}-\Delta
z}\right)}{(N_{t}-\Delta z)}{(N_{t}-\Delta z)}{(N_{t}-\Delta
z)}/2}{2\,m}\,I_{0}^{\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}-\Delta
z}\right)}{(N_{t}-\Delta z)}{(N_{t}-\Delta z)}{(N_{t}-\Delta
z)}/2-2\,m}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}\,I_{1}^{2\,m}\mathchoice{\mathopen{}\mathclose{{}\left(\tilde{s}}\right)}{(\tilde{s})}{(\tilde{s})}{(\tilde{s})}\big{.}\bigr{)}$,
where $\Delta
z=\mathchoice{\mathopen{}\mathclose{{}\left(z_{2}-z_{1}}\right)}{(z_{2}-z_{1})}{(z_{2}-z_{1})}{(z_{2}-z_{1})}$.

From (III.35) and (III.37) we can finally obtain an expression for the
_connected_ two-point function at a critical point $\mu_{n,n+1}$:

$\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}\phi^{+}_{y}}\right\rangle_{conn.}\mathchoice{\mathopen{}\mathclose{{}\left(\mu=\mu_{n,n+1},N_{t}}\right)}{(\mu=\mu_{n,n+1},N_{t})}{(\mu=\mu_{n,n+1},N_{t})}{(\mu=\mu_{n,n+1},N_{t})}\\\
=\mathchoice{\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}\phi^{+}_{y}}\right\rangle-\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}}\right\rangle\mathopen{}\mathclose{{}\left\langle\phi^{+}_{y}}\right\rangle}\right)}{(\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}\phi^{+}_{y}}\right\rangle-\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}}\right\rangle\mathopen{}\mathclose{{}\left\langle\phi^{+}_{y}}\right\rangle)}{(\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}\phi^{+}_{y}}\right\rangle-\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}}\right\rangle\mathopen{}\mathclose{{}\left\langle\phi^{+}_{y}}\right\rangle)}{(\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}\phi^{+}_{y}}\right\rangle-\mathopen{}\mathclose{{}\left\langle\phi^{-}_{x}}\right\rangle\mathopen{}\mathclose{{}\left\langle\phi^{+}_{y}}\right\rangle)}\mathchoice{\mathopen{}\mathclose{{}\left(\mu=\mu_{n,n+1},N_{t}}\right)}{(\mu=\mu_{n,n+1},N_{t})}{(\mu=\mu_{n,n+1},N_{t})}{(\mu=\mu_{n,n+1},N_{t})}\\\
\approx\frac{1}{8}\mathchoice{\mathopen{}\mathclose{{}\left(\frac{\cosh\bigl{(}\big{.}\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}/2-t}\right)}{(N_{t}/2-t)}{(N_{t}/2-t)}{(N_{t}/2-t)}\,\tilde{s}\big{.}\bigr{)}}{\cosh\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}\,\tilde{s}/2}\right)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}}+1-\tanh^{2}(N_{t}\,\tilde{s}/2)}\right)}{(\frac{\cosh\bigl{(}\big{.}\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}/2-t}\right)}{(N_{t}/2-t)}{(N_{t}/2-t)}{(N_{t}/2-t)}\,\tilde{s}\big{.}\bigr{)}}{\cosh\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}\,\tilde{s}/2}\right)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}}+1-\tanh^{2}(N_{t}\,\tilde{s}/2))}{(\frac{\cosh\bigl{(}\big{.}\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}/2-t}\right)}{(N_{t}/2-t)}{(N_{t}/2-t)}{(N_{t}/2-t)}\,\tilde{s}\big{.}\bigr{)}}{\cosh\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}\,\tilde{s}/2}\right)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}}+1-\tanh^{2}(N_{t}\,\tilde{s}/2))}{(\frac{\cosh\bigl{(}\big{.}\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}/2-t}\right)}{(N_{t}/2-t)}{(N_{t}/2-t)}{(N_{t}/2-t)}\,\tilde{s}\big{.}\bigr{)}}{\cosh\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}\,\tilde{s}/2}\right)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}}+1-\tanh^{2}(N_{t}\,\tilde{s}/2))}\\\
\overset{N_{t}\tilde{s}\gg
1}{\approx}\frac{\cosh\bigl{(}\big{.}\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}/2-t}\right)}{(N_{t}/2-t)}{(N_{t}/2-t)}{(N_{t}/2-t)}\,\tilde{s}\big{.}\bigr{)}}{8\,\cosh\mathchoice{\mathopen{}\mathclose{{}\left(N_{t}\,\tilde{s}/2}\right)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}{(N_{t}\,\tilde{s}/2)}}\,\
,$ (III.39)

where on the last line we used that $\tanh^{2}(x)$ quickly approaches unity
for $x>1$, so that the constant piece in
$\langle\smash{\phi^{-}_{x}\phi^{+}_{y}}\rangle$ gets cancelled. The reason
why this cancellation happens only for sufficiently large $N_{t}\tilde{s}$ is,
that the fields $\phi^{\pm}$ are, as mentioned earlier, not invariant under
global