#  Form Factors of the Tricritical Three-state Potts Model in its Scaling
Limit

Giuseppe<EMAIL_ADDRESS>SISSA and INFN, Sezione di Trieste, via
Bonomea 265, I-34136, Trieste, Italy  Marco<EMAIL_ADDRESS>Physics Department, University of Turin and INFN, Sezione di Torino, Via
Pietro Giuria 1, I-10125 Turin, Italy  Andrea<EMAIL_ADDRESS>corresponding author SISSA and INFN, Sezione di Trieste, via Bonomea 265,
I-34136, Trieste, Italy

###### Abstract

We compute the form factors of the order and disorder operators, together with
those of the stress-energy tensor, of the two-dimensional three-state Potts
model with vacancies along its thermal deformation of the critical point. At
criticality the model is described by the non-diagonal partition function of
the unitary minimal model $\mathcal{M}_{6,7}$ of conformal field theories and
is accompanied by an internal $S_{3}$ symmetry. Its off-critical thermal
deformation is an integrable massive theory which is still invariant under
$S_{3}$. The presence of infinitely many conserved quantities, whose spin
spectrum is related to the exceptional Lie algebra $E_{6}$, allows us to
determine the analytic $S$-matrix, the exact mass spectrum and the matrix
elements of local operators of this model in an exact non-perturbative way. We
use the spectral representation series of the correlators and the fast
convergence of these series to compute several universal ratios of the
Renormalization Group.

##  I Introduction

An interesting problem in theoretical physics consists of the study of two-
dimensional statistical models which exhibit a second order phase transition
for certain values of their coupling constants. As well known, the proper
language to frame the scaling limit of these microscopical critical models is
the one of Conformal Field Theories (CFT) Belavin et al. (1984) which, in two
dimensions, have the peculiar feature of the underlying infinite-dimensional
Virasoro algebra and the possibility to define consistent models employing
only a finite number of families of irreducible representations of this
algebra (the so-called degenerate fields). These ‘minimal models’ capture the
universal scaling behavior of the two-dimensional critical models and are
exactly solvable, in the sense that it is possible to obtain - at least in
principle - closed expressions for the multi-point correlation functions of
scaling operators, by exploiting properties of the Virasoro algebra and null-
vector conditions of the corresponding degenerate fieldsBelavin et al. (1984);
Friedan et al. (1984).

However, no general framework has yet been formulated for generic off-critical
models. Nonetheless powerful tools of analysis can be employed if the deformed
theory turns out to be integrable, i.e. if it has infinitely many commuting
conserved local quantities. When this happens, these conserved quantities
provide severe constraints on the structure of scattering amplitudes, which
are non zero only for the elastic processes and can be computed exactly Fateev
and Zamolodchikov (1990); Christe and Mussardo (1990a, b); Sotkov and Zhu
(1989); Zamolodchikov (1989); Mussardo (1992, 2020). In practice it suffices
to find all possible two-particle amplitudes, since all other scattering
amplitudes involving $n>2$ particles can be expressed in terms of their
${n(n-1)/2}$ two-particle scatterings.

Notable examples of these two-dimensional integrable field theories are the
ones described by the Ising model in a magnetic field (IMMF), the thermally
deformed tricritical Ising model and the thermally deformed three-state
tricritical Potts model (TPM). These models possess conserved charges whose
spins match the Coxeter exponents of the three exceptional algebras ($E_{8}$,
$E_{7}$ and $E_{6}$ respectively Zamolodchikov (1989); Christe and Mussardo
(1990a, b); Sotkov and Zhu (1989); Mussardo (1992); Acerbi et al. (1996);
Smirnov (1992); Mussardo (2020)).

Once the $S$-matrix of an integrable theory is known, it is possible to study
off-critical two-point correlation functions through the technique of form
factors (FFs) Karowski and Weisz (1978); Smirnov (1992); Mussardo (2020).
These quantities are defined as matrix elements of (semi)-local operators
between the vacuum of the theory and multi-particle states. The novel
contribution of this paper is the exact computation of FFs of the leading and
sub-leading order and disorder operators in the thermal deformation of the
TPM. We also compute some Universal Ratios of Renormalization Group (see
Fioravanti et al. (2000) and references therein), which can be useful for
numerical or experimental studies.

The $q$-state Potts model has been studied in great detail in the literature -
see e.g. Wu (1982) for a review. A generalization of this model, called
‘dilute’ due to the presence of vacancies, has been studied in Nienhuis et al.
(1979). Its lattice realization has an Hamiltonian, invariant under the
permutation group $S_{3}$, of the form

$\displaystyle\mathcal{H}=-\sum_{\langle
i,j\rangle}t_{i}t_{j}\left(\mathcal{K}+\mathcal{F}\delta_{s_{i},s_{j}}\right)+\mathcal{D}\sum_{i}t_{i}\,\,\,,$
(1)

where at each site $i$ is associated a Potts ‘spin’ ${s_{i}=\\{0,1,2\\}}$ and
a vacancy variable ${t_{i}=\\{0,1\\}}$. In the latter equation $\mathcal{K}$
and $\mathcal{F}$ are interaction parameters associated to vacancies and Potts
spins respectively, while $\mathcal{D}$ plays the role of chemical potential
associated to vacancies. The usual Potts model is retrieved when the chemical
potential $\mathcal{D}$ is negative and large in absolute value. Since the TPM
possesses only two relevant $S_{3}$ invariant operators Friedan et al. (1984),
the energy $\epsilon$ and the vacancy $t$, it is retrieved from Eq. (1) in the
limit of vanishing $\mathcal{K}$. The TPM is also exactly solvable in a
honeycomb lattice and proven to be self-dual, i.e. there is a mapping between
its disordered (high-temperature) and ordered (low-temperature) phases
Nienhuis (1991).

The content of this paper is organised as follows. In Section II we recall few
basic properties of minimal models of CFT, useful to set mostly the notation,
followed by a thorough discussion of the TPM, in particular the operator
content and the fusion rules of this model. Section III is dedicated to the
thermal deformation of the TPM, regarded as an integrable field theory; the
scattering theory and the mass spectrum are discussed there. The FFs are
introduced in Section IV (their analytic properties are gathered in Appendix
A). In Section V we review the FFs of the stress-energy tensor, originally
obtained in Acerbi et al. (1996). The FFs of the order and disorder operators
of the thermal deformation of the TPM are presented in Section VI. The two-
point correlation functions and the universal ratios of the renormalization
group are discussed in Section VII. Some preliminary results coming from the
Monte Carlo simulations of the model are presented in Section VIII. Our
conclusions are finally summarised in Section IX.

##  II Critical TPM: the CFT Approach

As well known, minimal models $\mathcal{M}_{p,q}$ of CFT are characterized by
the following properties Belavin et al. (1984); Friedan et al. (1984):

  1. i)

their central charge $c$ is always less than one and can be parameterized by
two coprime integers $p,q$ (with $q>p$)

$c(p,q)=1-6\frac{(p-q)^{2}}{pq}.$ (2)

  2. ii)

there is a finite number of families of degenerate fields, both in the
analytic and anti-analytic sector of the CFT. In the analytic sector, a
degenerate primary field $\phi_{r,s}(z)$ has conformal dimension

$\Delta_{r,s}=\frac{(pr-qs)^{2}-(q-p)^{2}}{4pq},$ (3)

with $1\leq r\leq q-1$ and $1\leq s\leq p-1$, coincident to that of the anti-
analytic counterpart $\overline{\phi}_{r,s}(\overline{{z}})$:
$\overline{\Delta}_{r,s}=\Delta_{r,s}$. Unitary minimal models are retrieved
by setting in the previous formulas $m=q=p+1$ with $m\geq 3$.

  3. iii)

correlation functions of all conformal fields satisfy a set of linear
differential equations that can be exactly solved, yielding in turn the
structure constants of the operator algebra.

  4. iv)

the modular-invariant partition function can be determined exactly Cappelli et
al. (1987); Cardy (1986a, b).

Another important feature of minimal models is that the operator product
expansion (OPE) algebra of degenerate fields, in the analytic and anti-
analytic sector separately, is ‘closed’: the OPE of two degenerate fields can
be written in terms of degenerate fields and their descendants. In particular,
the operators appearing in the OPE are obtained by iterative multiplication of
the fields $\phi_{2,1}$ and $\phi_{1,2}$. The general formula, which does not
give information about the structure constants, is

$\phi_{r_{1},s_{1}}\times\phi_{r_{2},s_{2}}=\sum_{r_{3}=\\{\absolutevalue{r_{2}-r_{1}}+1,\absolutevalue{r_{2}-r_{1}}+3,\dots\\}}^{\min(r_{1}+r_{2}-1,2q-1-r_{1}-r_{2})}\sum_{s_{3}=\\{\absolutevalue{s_{2}-s_{1}}+1,\absolutevalue{s_{2}-s_{1}}+3,\dots\\}}^{\min(s_{1}+s_{2}-1,2p-1-s_{1}-s_{2})}\phi_{r_{3},s_{3}}.$
(4)

Finding the modular invariant partition functions of a given universality
class associated to a minimal model is equivalent to determining the operator
content of the model. The problem of the modular-invariant partition functions
for unitary theories was first addressed by Cardy Cardy (1986a, b), and the
solution for minimal models with periodic boundary conditions was found soon
after by Cappelli, Itzykson and Zuber Cappelli et al. (1987).

The conformal dimensions of the fields in $\mathcal{M}_{6,7}$ are summarized
in Table 1. The universality class $\mathcal{M}_{6,7}$ contains two algebras
of fields which close under the OPE: the pentacritical $\varphi^{10}$ Landau-
Ginzburg field theory and the TPM.

5 | $5$ | $\frac{22}{7}$ | $\frac{12}{7}$ | $\frac{5}{7}$ | $\frac{1}{7}$ | $0$
---|---|---|---|---|---|---
4 | $\frac{23}{8}$ | $\frac{85}{56}$ | $\frac{33}{56}$ | $\frac{5}{56}$ | $\frac{1}{56}$ | $\frac{3}{8}$
3 | $\frac{4}{3}$ | $\frac{10}{21}$ | $\frac{1}{21}$ | $\frac{1}{21}$ | $\frac{10}{21}$ | $\frac{4}{3}$
2 | $\frac{3}{8}$ | $\frac{1}{56}$ | $\frac{5}{56}$ | $\frac{33}{56}$ | $\frac{85}{56}$ | $\frac{23}{8}$
1 | $0$ | $\frac{1}{7}$ | $\frac{5}{7}$ | $\frac{12}{7}$ | $\frac{22}{7}$ | $5$
s $\uparrow$, r $\rightarrow$ | 1 | 2 | 3 | 4 | 5 | 6
Table 1: Kač table of the universality class $\mathcal{M}_{6,7}$.

In correspondence to the minimal model $\mathcal{M}_{6,7}$ there are then two
modular-invariant partition functions: a diagonal one, which corresponds to
the $\varphi^{10}$ Landau-Ginzburg theory,

$\displaystyle Z^{(1)}$
$\displaystyle=\absolutevalue{\chi_{0}}^{2}+\absolutevalue{\chi_{\frac{1}{7}}}^{2}+\absolutevalue{\chi_{\frac{5}{7}}}^{2}+\absolutevalue{\chi_{\frac{12}{7}}}^{2}+\absolutevalue{\chi_{\frac{22}{7}}}^{2}+\absolutevalue{\chi_{5}}^{2}+\absolutevalue{\chi_{\frac{3}{8}}}^{2}+\absolutevalue{\chi_{\frac{1}{56}}}^{2}+\absolutevalue{\chi_{\frac{5}{56}}}^{2}+$
(5)
$\displaystyle\quad+\absolutevalue{\chi_{\frac{33}{56}}}^{2}+\absolutevalue{\chi_{\frac{85}{56}}}^{2}+\absolutevalue{\chi_{\frac{23}{8}}}^{2}+\absolutevalue{\chi_{\frac{4}{3}}}^{2}+\absolutevalue{\chi_{\frac{10}{21}}}^{2}+\absolutevalue{\chi_{\frac{1}{21}}}^{2}$

and an off-diagonal one relative to the TPM

$Z^{(2)}=\absolutevalue{\chi_{0}+\chi_{5}}^{2}+\absolutevalue{\chi_{\frac{1}{7}}+\chi_{\frac{22}{7}}}^{2}+\absolutevalue{\chi_{\frac{5}{7}}+\chi_{\frac{12}{7}}}^{2}+2\absolutevalue{\chi_{\frac{4}{3}}}^{2}+2\absolutevalue{\chi_{\frac{10}{21}}}^{2}+2\absolutevalue{\chi_{\frac{1}{21}}}^{2}.$
(6)

Here $\chi_{\Delta}$ is the character of the irreducible representation of the
Virasoro algebra of weight $\Delta$, whose explicit formulas can be found in
Rocha-Caridi and Wallach (1984).

Using the notation of Table 2 for the various fields, it is possible to show
that the fusion rules of Table 3 are compatible with a $S_{3}$ internal
symmetry, so that operators can be organized according to irreducible
representations of this discrete group, i.e. either in doublets ($\mathbf{s}$,
$\mathbf{Z}$ and $\mathbf{A}$) or in the one-dimensional neutral
representation ($\mathbbm{1}$, $\epsilon$, $t$, $\mathbf{X}$, $\mathbf{Y}$ and
$\mathbf{B}$). In the model there are no operators which transform according
to the alternating representation.

$\phi_{r,s}$ $(\Delta,\bar{\Delta})$ Operator Interpretation $S_{3}$
representation $\phi_{1,1}\times\bar{\phi}_{1,1}$ $\left(0,0\right)$
$\mathbbm{1}$ Identity $1$ $\phi_{3,3}\times\bar{\phi}_{3,3}$
$\left(\frac{1}{21},\frac{1}{21}\right)$ $\mathbf{s}$ Leading Order $2$
$\phi_{2,1}\times\bar{\phi}_{2,1}$ $\left(\frac{1}{7},\frac{1}{7}\right)$
$\epsilon$ Energy $1$ $\phi_{2,3}\times\bar{\phi}_{2,3}$
$\left(\frac{10}{21},\frac{10}{21}\right)$ $\mathbf{Z}$ Subleading Order $2$
$\phi_{3,1}\times\bar{\phi}_{3,1}$ $\left(\frac{5}{7},\frac{5}{7}\right)$ $t$
Vacancy $1$ $\phi_{1,3}\times\bar{\phi}_{1,3}$
$\left(\frac{4}{3},\frac{4}{3}\right)$ $\mathbf{A}$ Irrelevant $2$
$\phi_{4,1}\times\bar{\phi}_{4,1}$ $\left(\frac{12}{7},\frac{12}{7}\right)$
$\mathbf{X}$ Irrelevant $1$ $\phi_{5,1}\times\bar{\phi}_{5,1}$
$\left(\frac{22}{7},\frac{22}{7}\right)$ $\mathbf{B}$ Irrelevant $1$
$\phi_{6,1}\times\bar{\phi}_{6,1}$ $\left(5,5\right)$ $\mathbf{Y}$ Irrelevant
$1$ $\phi_{4,1}\times\bar{\phi}_{3,1}$ $\left(\frac{12}{7},\frac{5}{7}\right)$
$\mathcal{F}$ Irrelevant, Spin 1 $1$ $\phi_{3,1}\times\bar{\phi}_{4,1}$
$\left(\frac{5}{7},\frac{12}{7}\right)$ $\bar{\mathcal{F}}$ Irrelevant, Spin
-1 $1$ $\phi_{4,1}\times\bar{\phi}_{2,1}$
$\left(\frac{12}{7},\frac{1}{7}\right)$ $\mathcal{U}$ Irrelevant, Spin 3 $1$
$\phi_{2,1}\times\bar{\phi}_{4,1}$ $\left(\frac{1}{7},\frac{12}{7}\right)$
$\bar{\mathcal{U}}$ Irrelevant, Spin -3 $1$ $\phi_{6,1}\times\bar{\phi}_{1,1}$
$\left(5,0\right)$ $\mathcal{W}$ Irrelevant, Spin 5 $1$
$\phi_{1,1}\times\bar{\phi}_{6,1}$ $\left(0,5\right)$ $\bar{\mathcal{W}}$
Irrelevant, Spin -5 $1$

Table 2: Operator content of the TPM at criticality.

$\mathbf{s}\times\mathbf{s}$ $=$
$\mathbbm{1}+\epsilon+t+\mathbf{X}+\mathbf{B}+\mathbf{Y}+\mathbf{s}+\mathbf{Z}+\mathbf{A}$
$\epsilon\times\epsilon$ $=$ $\mathbbm{1}+t$ $\mathbf{Z}\times\mathbf{Z}$ $=$
$\mathbbm{1}+\epsilon+t+\mathbf{X}+\mathbf{Y}+\mathbf{s}+\mathbf{A}$ $t\times
t$ $=$ $\mathbbm{1}+t+\mathbf{B}$ $\mathbf{A}\times\mathbf{A}$ $=$
$\mathbbm{1}+\mathbf{Y}+\mathbf{A}$ $\mathbf{X}\times\mathbf{X}$ $=$
$\mathbbm{1}+t+\mathbf{B}$ $\mathbf{B}\times\mathbf{B}$ $=$ $\mathbbm{1}+t$
$\mathbf{Y}\times\mathbf{Y}$ $=$ $\mathbbm{1}$ $\mathbf{s}\times\epsilon$ $=$
$\mathbf{s}+\mathbf{Z}$ $\mathbf{s}\times t$ $=$
$\mathbf{s}+\mathbf{Z}+\mathbf{A}$ $\mathbf{s}\times\mathbf{Z}$ $=$
$\epsilon+t+\mathbf{X}+\mathbf{B}+\mathbf{s}+\mathbf{Z}$
$\mathbf{s}\times\mathbf{A}$ $=$ $t+\mathbf{X}+\mathbf{s}$
$\mathbf{s}\times\mathbf{X}$ $=$ $\mathbf{s}+\mathbf{Z}+\mathbf{A}$
$\mathbf{s}\times\mathbf{B}$ $=$ $\mathbf{s}+\mathbf{Z}$
$\mathbf{s}\times\mathbf{Y}$ $=$ $\mathbf{s}$ $\epsilon\times\mathbf{Z}$ $=$
$\mathbf{s}+\mathbf{A}$ $\epsilon\times t$ $=$ $\epsilon+\mathbf{X}$
$\epsilon\times\mathbf{A}$ $=$ $\mathbf{Z}$ $\epsilon\times\mathbf{X}$ $=$
$t+\mathbf{Y}$ $\epsilon\times\mathbf{B}$ $=$ $\mathbf{X}+\mathbf{Y}$
$\epsilon\times\mathbf{Y}$ $=$ $\mathbf{B}$ $\mathbf{Z}\times t$ $=$
$\mathbf{s}+\mathbf{Z}$ $\mathbf{Z}\times\mathbf{A}$ $=$
$\epsilon+\mathbf{B}+\mathbf{Z}$ $\mathbf{Z}\times\mathbf{X}$ $=$
$\mathbf{s}+\mathbf{Z}$ $\mathbf{Z}\times\mathbf{B}$ $=$
$\mathbf{s}+\mathbf{A}$ $\mathbf{Z}\times\mathbf{Y}$ $=$ $\mathbf{Z}$
$t\times\mathbf{A}$ $=$ $\mathbf{s}$ $t\times\mathbf{X}$ $=$
$\epsilon+\mathbf{X}+\mathbf{Y}$ $t\times\mathbf{B}$ $=$ $t+\mathbf{B}$
$t\times\mathbf{Y}$ $=$ $\mathbf{X}$ $\mathbf{A}\times\mathbf{X}$ $=$
$\mathbf{s}$ $\mathbf{A}\times\mathbf{B}$ $=$ $\mathbf{Z}$
$\mathbf{A}\times\mathbf{Y}$ $=$ $\mathbf{A}$ $\mathbf{X}\times\mathbf{B}$ $=$
$\epsilon+\mathbf{X}$ $\mathbf{X}\times\mathbf{Y}$ $=$ $t$
$\mathbf{B}\times\mathbf{Y}$ $=$ $\epsilon$

Table 3: Fusion rules of the TPM at criticality.

Doublet operators like $\mathbf{s}$ and $\mathbf{Z}$ are charged under the
generator $\Omega$ of the subgroup $Z_{3}$. Let the components of the doublet
corresponding to the leading operator $\mathbf{s}$ be respectively $\sigma$
and $\tilde{\sigma}$. Similarly one defines the components of the subleading
order operator $\mathbf{Z}$ to be $Z$ and $\tilde{Z}$. As it happens in the
case of the Ising model Fradkin (2017), the tricritical Ising model Cubero et
al. (2022) and the three-state Potts model Capizzi et al. (2021); Caselle et
al. (2006), in view of the self-duality of the lattice model we can introduce
a leading disorder operator ${(\mu,\tilde{\mu})}$ associated to the leading
order operator ${(\sigma,\tilde{\sigma})}$. In complete analogy to the leading
counterpart, ${(\zeta,\tilde{\zeta})}$ is the subleading disorder operator
associated to the subleading order operator ${(Z,\tilde{Z})}$. The conformal
dimensions of these operators are:

$\displaystyle\left(\Delta_{\sigma},\bar{\Delta}_{\sigma}\right)=\left(\Delta_{\tilde{\sigma}},\bar{\Delta}_{\tilde{\sigma}}\right)=\left(\Delta_{\mu},\bar{\Delta}_{\mu}\right)=\left(\Delta_{\tilde{\mu}},\bar{\Delta}_{\tilde{\mu}}\right)=\left(\frac{1}{21},\frac{1}{21}\right),$
(7)
$\displaystyle\left(\Delta_{Z},\bar{\Delta}_{Z}\right)=\left(\Delta_{\tilde{Z}},\bar{\Delta}_{\tilde{Z}}\right)=\left(\Delta_{\zeta},\bar{\Delta}_{\zeta}\right)=\left(\Delta_{\tilde{\zeta}},\bar{\Delta}_{\tilde{\zeta}}\right)=\left(\frac{10}{21},\frac{10}{21}\right).$

The transformation properties under $S_{3}$ of the order and disorder
operators are as follows. Under the generator $\Omega$ of $Z_{3}$ the disorder
operators are invariant and the order operators acquire a phase:

$\displaystyle\Omega\sigma=\omega\sigma,$
$\displaystyle\Omega\tilde{\sigma}=\omega^{*}\tilde{\sigma},$
$\displaystyle\Omega Z=\omega^{2}Z,$
$\displaystyle\Omega\tilde{Z}=(\omega^{*})^{2}\tilde{Z},$ (8)
$\displaystyle\Omega\mu=\mu,$ $\displaystyle\Omega\tilde{\mu}=\tilde{\mu},$
$\displaystyle\Omega\zeta=\zeta,$
$\displaystyle\Omega\tilde{\zeta}=\tilde{\zeta},$

where $\omega=\exp(2i\pi/3)$. On the other hand, the generator $\mathcal{C}$
of $Z_{2}$ acts like a charge conjugation operator:

$\displaystyle\mathcal{C}\sigma$ $\displaystyle=\tilde{\sigma},$
$\displaystyle\mathcal{C}Z$ $\displaystyle=\tilde{Z},$ (9)
$\displaystyle\mathcal{C}\mu$ $\displaystyle=\tilde{\mu},$
$\displaystyle\mathcal{C}\zeta$ $\displaystyle=\tilde{\zeta}.$

The assumption that only order operators are charged with respect to $Z_{3}$
is consistent with an alternative definition of TPM, in terms of a
parafermionic algebra of central charge $6/7$ \- see Eq.(A.8) of Zamolodchikov
and Fateev (1985). The parafermionic theory is defined to be invariant under
the product group $Z_{3}\times\tilde{Z}_{3}$. The first $Z_{3}$ group acts
nontrivially only on order operators, while the second only on disorder ones.
However, the symmetry group of the TPM is $S_{3}$, which contains $Z_{3}$ and
$Z_{2}$ as subgroups. While $Z_{3}$ is the group of symmetry of the
parafermionic theory, the second group $Z_{2}$ acts as a charge conjugation
operator.

There are two important features of parafermionic theories. The first is that
they are self-dual by definition: order and disorder operators in the ordered
phase can be mapped into disorder and order operators of the disordered phase
respectively. As a consequence, we will only focus on the FFs in the
disordered phase. The second is that order and disorder operators are not
mutually local, in general. In particular, for the TPM the non-locality
factors are

$\displaystyle\gamma_{\sigma,\mu}=\gamma_{Z,\zeta}=-1/3,$
$\displaystyle\gamma_{\sigma,\zeta}=\gamma_{Z,\mu}=+1/3,$ (10)
$\displaystyle\gamma_{\tilde{\sigma},\mu}=\gamma_{\tilde{Z},\zeta}=1/3,$
$\displaystyle\gamma_{\tilde{\sigma},\zeta}=\gamma_{\tilde{Z},\mu}=-1/3.$

Identical results apply if we charge-conjugate both the first and second entry
in the $\gamma$’s.

##  III Thermal deformation of the TPM

Consider the fixed point action $\mathcal{A}_{0}$ of the TPM and define the
action $\mathcal{A}$ of its thermal deformation as

$\mathcal{A}=\mathcal{A}_{0}+\lambda\int\differential^{2}x\ \epsilon(x),$ (11)

where $\epsilon$ is the energy operator and $\lambda$ is its coupling
constant. In the scaling limit of the lattice version of the model, i.e. the
limit in which the temperature $T$ approaches its critical value $T_{c}$ and
the chemical potential is fixed at its critical value, we have
$\lambda\propto(T-T_{c})$. The action ${\mathcal{A}}$ corresponds to a massive
theory and there is a non-vanishing trace of the stress-energy tensor
$\Theta(z,\bar{z})$ given by

$\Theta(z,\bar{z})=2\pi\lambda(2-2\Delta_{\epsilon})\epsilon(z,\bar{z}).$ (12)

In the following we will often denote the quantum field theory associated to
the thermal deformation of the TPM as $\epsilon$-TPM. In addition to the
stress-energy tensor, in the $\epsilon$-TPM there are infinitely many local
conserved quantities which commute each other and therefore the theory is
integrable Fateev and Zamolodchikov (1990); Zamolodchikov (1989). The spins of
conserved quantities for this model are given by the infinite sequence of
residues

$\\{1,4,5,7,8,11\\}\hskip 8.53581pt{\rm mod\,\,h=12}\,\,\,.$ (13)

and they coincide, modulo the Coxeter number $h=12$, with the Coxeter
exponents of the exceptional algebras $E_{6}$.

The hidden $E_{6}$ structure of the model can be understood employing a
different CFT description of the TPM, i.e. the so-called coset construction Di
Francesco et al. (1997), which in this case consists of
$[(E_{6})_{1}\times(E_{6})_{1}]/(E_{6})_{2}$. Let’s discuss this point in more
detail. The affine algebra $(E_{6})_{k}$ (Fig. 1) has different integrable
highest weights depending on the level $k$. Calling $w_{i}$ ($i=0,1,\ldots,7$)
the fundamental weights of $(E_{6})_{k}$, which are the basis of the weight
space in the natural basis $w_{i}=\hat{e}_{i+1}$ (e.g.
$w_{1}=(0,1,0,0,0,0,0)$), the integrable highest weights at level $k=1$ are
$w_{0}$, $w_{1}$ and $w_{5}$, while at level $k=2$ we have $2w_{0}$, $2w_{1}$,
$2w_{5}$, $(w_{0}+w_{1})$, $(w_{0}+w_{5})$, $(w_{1}+w_{5})$, $w_{2}$ and
$w_{6}$. Through the coset fields, we can recover the operator content of the
TPM. As discussed for instance in Di Francesco et al. (1997), the procedure
consists of finding the allowed branchings, the fractional dimensions of the
integrable representations, conjugate representations (related to the symmetry
group of the non-affine Dynkin diagram) and coset fields, i.e. classes of
equivalent branchings related by automorphisms ($Z_{3}$ in the case of affine
$E_{6}$). The results are summarized in Table 4.

$(3;3)$$(2;2)$$(1;1)$$(4;2)$$(5;1)$$(6;2)$$(0;1)$

Figure 1: Dynkin diagram of the affine $E_{6}$ algebra. The first term in
parenthesis labels the root, while the second its comark.

Cosets Operators
$\\{\left[w_{0},w_{0},2w_{0}\right]\sim\left[w_{1},w_{1},2w_{5}\right]\sim\left[w_{5},w_{5},2w_{5}\right]\\}$
$\mathbbm{1},\mathbf{Y}$
$\\{\left[w_{0},w_{1},w_{0}+w_{1}\right]\sim\left[w_{5},w_{0},w_{0}+w_{5}\right]\sim\left[w_{1},w_{5},w_{1}+w_{5}\right]\\}$
$\sigma$
$\\{\left[w_{1},w_{0},w_{0}+w_{1}\right]\sim\left[w_{0},w_{5},w_{0}+w_{5}\right]\sim\left[w_{5},w_{1},w_{1}+w_{5}\right]\\}$
$\tilde{\sigma}$
$\\{\left[w_{0},w_{0},w_{6}\right]\sim\left[w_{1},w_{1},w_{2}\right]\\}$
$\epsilon,\mathbf{B}$
$\\{\left[w_{1},w_{5},w_{6}\right]\sim\left[w_{5},w_{0},w_{2}\right]\\}$ $Z$
$\\{\left[w_{5},w_{0},w_{2}\right]\sim\left[w_{5},w_{1},w_{6}\right]\\}$
$\tilde{Z}$
$\\{\left[w_{0},w_{0},w_{1}+w_{5}\right]\sim\left[w_{1},w_{1},w_{0}+w_{5}\right]\sim\left[w_{5},w_{5},w_{0}+w_{1}\right]\\}$
$t,\mathbf{X}$
$\\{\left[w_{0},w_{1},2w_{5}\right]\sim\left[w_{5},w_{0},2w_{1}\right]\sim\left[w_{1},w_{5},2w_{0}\right]\\}$
$A$
$\\{\left[w_{1},w_{0},2w_{5}\right]\sim\left[w_{0},w_{5},2w_{1}\right]\sim\left[w_{5},w_{1},2w_{0}\right]\\}$
$\tilde{A}$

Table 4: Identification of coset fields of
$[(E_{6})_{1}\times(E_{6})_{1}]/(E_{6})_{2}$ with the operator content of the
TPM. An element of the coset $[\lambda,\mu,\nu]$ is such that it appears in
the branching $\lambda+\mu=\sum_{\nu}b_{\lambda,\mu}^{\nu}\nu$.

The coset construction of the TPM only holds for the critical theory. Away
from criticality there is however a correspondence between the thermal
deformation of the TPM and the exceptional algebra $E_{6}$ and this involves
the Toda field theories. These theories are built starting from a Lagrangian
associated to a Lie algebra of finite rank (see, e.g. Mussardo (2020); Braden
et al. (1990) and references therein). In Eguchi and Yang (1989) it was proven
that the deformation of the Toda field theory of $E_{6}$ perturbed with the
adjoint representation, i.e. the longest root, is integrable and has conserved
charges whose spins coincide with those of the thermal deformation of the TPM.
Thus the scattering matrix of this perturbed Toda field theory has the same
minimal scattering matrix as our model: its mass spectrum contains six
particles, two particle-antiparticle doublets, $(l,\bar{l})$ and
$(h,\bar{h})$, and two self-conjugated particles, $L$ and $H$, and the masses
of these particles can be expressed as

$\displaystyle m_{l}=m_{\bar{l}}=M(\lambda),$ $\displaystyle
m_{L}=2\cos\left(\pi/4\right)m_{l},$ (14) $\displaystyle
m_{h}=m_{\bar{h}}=2\cos\left(\pi/12\right)m_{l},$ $\displaystyle
m_{H}=4[\cos\left(\pi/4\right)]^{2}m_{l},$

where $M(\lambda)$ is the mass of the lightest kink of the theory and depends
on the coupling constant $\lambda$ in Eq. (11) as Fateev (1994)

$M(\lambda)=C_{\epsilon}|\lambda|^{\frac{7}{12}},$ (15)

where

$C_{\epsilon}=\left[4\pi^{2}\lambda^{2}\gamma\left(\frac{4}{7}\right)\gamma\left(\frac{9}{14}\right)\gamma\left(\frac{5}{7}\right)\gamma\left(\frac{11}{14}\right)\right]^{\frac{7}{24}}\frac{2\Gamma\left(\frac{1}{4}\right)\Gamma\left(\frac{13}{12}\right)}{\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{2}{3}\right)\Gamma\left(\frac{7}{6}\right)}\approx
3.74656.$ (16)

In the latter equation $\gamma(z)$ is a shorthand for $\Gamma(z)/\Gamma(1-z)$.

For the thermal deformation of the TPM, the excitations are ordinary particles
only in the disordered phase $\lambda>0$, while some of them become kinks in
the ordered phase $\lambda<0$. In a QFT with localized interactions, particles
are associated to asymptotic states which transform according to irreducible
representations of the symmetry group of the theory. For the thermal
deformation of the TPM, we have two $S_{3}$-doublets and two singlets. Since
the particles in the doublets $(l,\bar{l})$ and $(h,\bar{h})$ are generated by
order (magnetization-like) operators, we define the non-locality properties of
the particles with respect to disorder operators to be

$\displaystyle\gamma_{l,\mu}=\gamma_{l,\zeta}=1/3,$
$\displaystyle\gamma_{\bar{l},\mu}=\gamma_{\bar{l},\zeta}=-1/3.$ (17)

Identical relations hold if $l$ is replaced by $h$. This choice of non-local
factors is consistent with the scattering theory of the $\epsilon$-TPM, which
will now be outlined.

###  III.1 Scattering Theory

In ${(1+1)}$-dimensional theories, all particles scatter along a straight
line. It is useful to use the rapidity variable $\theta$ which parameterizes
the on-shell momentum of the particle of type $a$ and mass $m_{a}$ as

$p_{a}(\theta_{a})=m_{a}(\cosh\theta_{a},\sinh\theta_{a})$ (18)

and denote the one-particle state as $|A_{a}(\theta)\rangle$. For the
multiparticle state we suppose, as a convention, that all incoming particles
can be written from left to right, with decreasing rapidity. On the other
hand, outgoing states are written in the converse: particles with the highest
rapidity on the right and particles with lowest rapidity on the left. Examples
are provided in Table 5.

Picture State Convention
$\theta$$A_{a}(\theta_{1})$$A_{b}(\theta_{2})$$A_{c}(\theta_{3})$$\dots$
Incoming
$\displaystyle{|A_{a}(\theta_{1})A_{b}(\theta_{2})A_{c}(\theta_{3})\dots\rangle_{IN}}$
Outgoing $\displaystyle{|\dots
A_{c}(\theta_{3})A_{b}(\theta_{2})A_{a}(\theta_{1})\rangle_{OUT}}$

Table 5: Conventions for incoming and outgoing states, where
$\theta_{1}>\theta_{2}>\theta_{3}$.

Integrable models have commuting conserved quantities $Q_{s}$ that can be
diagonalized simultaneously with the Hamiltonian. This condition imposes
strong constraints on the scattering matrix. Apart from being unitary and
satisfying crossing symmetry, the $S$-matrix is elastic and all $n$-particle
processes can be factorized in terms of two-particle scatterings Zamolodchikov
(1989); Mussardo (2020). These two properties, along with the assumptions of
bootstrap, allow for the determination of two-body scattering amplitudes. In
the case of the $\epsilon$-TPM, the $S$-matrix Fateev and Zamolodchikov
(1990); Sotkov and Zhu (1989) is diagonal, i.e. reflection amplitudes vanish,
and the various amplitudes are reported in Table 6. They can be expressed in
terms of the building block functions

$[x]\equiv s_{x/12}(\theta)=\frac{\sinh[(\theta+i\pi
x/12)/2]}{\sinh[(\theta-i\pi x/12)/2]}\,\,\,.$ (19)

$a$ $b$ $S_{ab}$ $a$ $b$ $S_{ab}$ $l$ $l$
$\displaystyle{\overset{\bar{l}}{[8]}[6]\overset{\bar{h}}{[2]}}$ $L$ $L$
$\displaystyle{-[10]\overset{L}{[8]}[6]^{2}[4]\overset{H}{[2]}}$ $\bar{l}$
$\bar{l}$ $\displaystyle{\overset{l}{[8]}[6]\overset{h}{[2]}}$ $L$ $h$
$\displaystyle{\overset{l}{[10]}[8]^{2}[6]^{2}[4]^{2}[2]}$ $l$ $\bar{l}$
$\displaystyle{-[10]\overset{L}{[6]}[4]}$ $L$ $\bar{h}$
$\displaystyle{\overset{\bar{l}}{[10]}[8]^{2}[6]^{2}[4]^{2}[2]}$ $l$ $L$
$\displaystyle{\overset{l}{[9]}[7]\overset{h}{[5]}[3]}$ $L$ $H$
$\displaystyle{\overset{L}{[11]}[9]^{2}\overset{H}{[7]^{3}}[5]^{3}[3]^{2}[1]}$
$\bar{l}$ $L$
$\displaystyle{\overset{\bar{l}}{[9]}[7]\overset{\bar{h}}{[5]}[3]}$ $h$ $h$
$\displaystyle{\overset{\bar{l}}{[10]}\overset{\bar{h}}{[8]^{3}}[6]^{3}[4]^{2}[2]^{2}}$
$l$ $h$
$\displaystyle{[9]\overset{\bar{h}}{[7]}[5]^{2}[3]\overset{\bar{l}}{[11]}}$
$\bar{h}$ $\bar{h}$
$\displaystyle{\overset{l}{[10]}\overset{h}{[8]^{3}}[6]^{3}[4]^{2}[2]^{2}}$
$\bar{l}$ $\bar{h}$
$\displaystyle{[9]\overset{h}{[7]}[5]^{2}[3]\overset{l}{[11]}}$ $h$ $\bar{h}$
$\displaystyle{-[10]^{2}[8]^{2}\overset{H}{[6]^{3}}[4]^{3}[2]}$ $l$ $\bar{h}$
$\displaystyle{\overset{L}{[9]}[7]^{2}[5]\overset{H}{[3]}[1]}$ $h$ $H$
$\displaystyle{\overset{l}{[11]}\overset{h}{[9]^{3}}[7]^{4}[5]^{4}[3]^{3}[1]}$
$\bar{l}$ $h$ $\displaystyle{\overset{L}{[9]}[7]^{2}[5]\overset{H}{[3]}[1]}$
$\bar{h}$ $H$
$\displaystyle{\overset{\bar{l}}{[11]}\overset{\bar{h}}{[9]^{3}}[7]^{4}[5]^{4}[3]^{3}[1]}$
$l$ $H$ $\displaystyle{\overset{h}{[10]}[8]^{2}[6]^{2}[4]^{2}[2]}$ $H$ $H$
$\displaystyle{-\overset{L}{[10]^{3}}\overset{H}{[8]^{5}}[6]^{6}[4]^{5}[2]^{3}}$
$\bar{l}$ $H$ $\displaystyle{\overset{\bar{h}}{[10]}[8]^{2}[6]^{2}[4]^{2}[2]}$

Table 6: $S$-matrix of the $\epsilon$-TPM Fateev and Zamolodchikov (1990);
Sotkov and Zhu (1989). Here $[x]$ is a shorthand for $s_{x/12}(\theta)$ (see
eq.(19)). The superscript refers to the associated bound state. Notice that
$12$ is the Coxeter number of the exceptional algebra $E_{6}$.

By looking at bound states with positive residue of two-particle scattering
amplitudes it is possible to extract the mass spectrum of the model, which
also coincides with the one of the $E_{6}$ Toda field theory Eq. (14).

In the $\epsilon$-TPM the degeneracy on the particle-antiparticle states
$(l,\bar{l})$ and $(h,\bar{h})$ is lifted if we require that particle singlets
and doublets transform according to the singlet and doublet irreducible
representations of $S_{3}$ respectively. In particular, we suppose that
particles $l$ and $h$ have $Z_{3}$-charge $\omega$, $\bar{l}$ and $\bar{h}$
have $Z_{3}$-charge $\omega^{*}$, while $L$ and $H$ do not transform under the
generator $\Omega$ of $S_{3}$. On the other hand, $Z_{2}$ charge conjugation
maps particles to their respective antiparticles. In the specific case of the
two singlets $L$ and $H$, particle and antiparticle coincide.

##  IV Generalities on Form Factors

The aim of this section is outlining the main properties of FFs, in particular
the ones useful to address one and two-particle states. A further discussion
of the analytic properties of such objects can be found in Appendix A.

For a relativistic theory, a generic matrix element of an Hermitian operator
$\mathcal{O}$ between an incoming and outgoing state is given by

$\begin{split}\prescript{}{OUT}{\langle}A_{{b}_{1}}(q_{1})\dots&A_{{b}_{m}}(q_{m})|\mathcal{O}(x)|A_{{a}_{1}}(p_{1})\dots
A_{{a}_{n}}(p_{n})\rangle_{IN}=\exp{i\left(\sum_{j=1}^{m}(q_{j})_{\mu}-\sum_{i=1}^{n}(p_{i})_{\mu}\right)x^{\mu}}\times\\\
&\times\prescript{}{OUT}{\langle}A_{{b}_{1}}(q_{1})\dots
A_{{b}_{m}}(q_{m})|\mathcal{O}(0)|A_{{a}_{1}}(p_{1})\dots
A_{{a}_{n}}(p_{n})\rangle_{IN}.\end{split}$ (20)

FFs are nothing more than matrix elements between the vacuum of the theory and
incoming multi-particle states, as pictorially described in Fig. 2. They are
related to the previous equation by crossing, are analytic functions of the
rapidity difference ${\theta_{ij}=\theta_{i}-\theta_{j}}$ and are defined as

$F_{{a}_{1}\dots a_{n}}^{\mathcal{O}}(\theta_{1},\dots,\theta_{n})=\langle
0|\mathcal{O}(0)|A_{{a}_{1}}(\theta_{1})\dots A_{{a}_{n}}(\theta_{n})\rangle.$
(21)

For a scalar operator, knowledge of FFs allows us to write the spectral series
for the correlation functions which, for the two-point correlators, reads
Zamolodchikov (1991)

$\langle\mathcal{O}(x)\mathcal{O}(0)\rangle=\sum_{n=0}^{\infty}\int\frac{\differential\theta_{1}\cdots\differential\theta_{n}}{(2\pi)^{n}n!}\absolutevalue{F_{a_{1}\dots
a_{n}}^{\mathcal{O}}(\theta_{1},\dots,\theta_{n})}^{2}\exp{-\absolutevalue{x}\sum_{k=1}^{n}m_{k}\cosh\theta_{k}}.$
(22)

An important feature of these spectral series is their rapid convergence Cardy
and Mussardo (1993); Mussardo (2020). Practically, lowest mass particle FFs
contribute more than higher mass particle ones, as it will be evidenced later
by applications of the $\Delta$-theorem Delfino et al. (1996) which is a sum
rule for the conformal weight of an operator $\Phi$

$\Delta_{\Phi}=-\frac{1}{4\pi\expectationvalue{\Phi}}\int\differential^{2}x\
\expectationvalue{\Theta(x)\Phi(0)}.$ (23)

This sum rule is similar to Zamolodchikov’s $c$-theorem sum rule for the UV
central charge Zamolodchikov (1986)

$c=\frac{3}{2}\int\differential r\ r^{3}\langle\Theta(r)\Theta(0)\rangle.$
(24)

$\mathcal{O}$$\theta_{1}$$\theta_{2}$$\theta_{n-1}$$\theta_{n}$…

Figure 2: Form factor of the operator $\mathcal{O}$.

For integrable theories, FFs of scalar operators can be parameterized as

$F_{a_{1}\dots
a_{n}}^{\mathcal{O}}(\theta_{1},\dots,\theta_{n})=\frac{Q_{a_{1}\dots
a_{n}}^{\mathcal{O}}(\theta_{1},\dots,\theta_{n})}{D_{a_{1}\dots
a_{n}}(\theta_{1},\dots,\theta_{n})}\prod_{i<j}F_{a_{i}a_{j}}^{min}(\theta_{ij}),$
(25)

where ${D_{a_{1}\dots a_{n}}(\theta_{1},\dots,\theta_{n})}$ is a $2\pi
i$-periodic even function of the rapidity differences, ${Q_{a_{1}\dots
a_{n}}^{\mathcal{O}}(\theta_{1},\dots,\theta_{n})}$ encodes the non-locality
properties of $\mathcal{O}$ with respect to the operator generating particles,
and $F_{a_{i}a_{j}}^{min}(\theta_{ij})$ are the so-called ‘minimal’ FFs, which
satisfy

$\begin{split}&F^{min}_{ij}(\theta)=-S_{ij}(\theta)F^{min}_{ij}(-\theta),\\\
&F^{min}_{ij}(\theta+2\pi i)=F^{min}_{ij}(-\theta).\end{split}$ (26)

Being relevant to Section VI, the non-locality equations satisfied by FFs in
the two-particle case read

$F_{a_{1}a_{2}}^{\mathcal{O}}(\theta)=S_{a_{1}a_{2}}(\theta)e^{i2\pi\gamma_{a_{2},\mathcal{O}}}F_{a_{1}a_{2}}^{\mathcal{O}}(\theta+2\pi
i),$ (27)

with $\theta=\theta_{1}-\theta_{2}$.

FFs can be found by solving a set of linear equations in correspondence of
kinematic and dynamical poles, for a suitable parametrization of
${Q_{a_{1}\dots a_{n}}^{\mathcal{O}}(\theta_{1},\dots,\theta_{n})}$.
Concerning bound state poles, e.g. $A_{i}(\theta)A_{j}(\theta+iu_{ij}^{k})\to
A_{k}(iu_{ij}^{k})$, a two-particle FF obeys

$-i\underset{\theta\to iu_{ij}^{k}}{\lim}(\theta-
iu_{ij}^{k})F^{\mathcal{O}}_{a_{i},a_{j}}(\theta)=\Gamma_{ij}^{k}F^{\mathcal{O}}_{a_{k}},$
(28)

where $\Gamma_{ij}^{k}$ is the real three-point vertex

$\Gamma_{ij}^{k}=\left[-i\underset{\theta\to iu_{ij}^{k}}{\lim}(\theta-
iu_{ij}^{k})S_{ij}(\theta)\right]^{1/2}.$ (29)

There are also residue equations coming from double poles of the $S$-matrix.
The equation for a pole located at ${i\phi}$ is reported in Acerbi et al.
(1996) and reads

$-i\underset{\theta\to
i\varphi}{\lim}(\theta-i\varphi)F_{ij}(\theta)=\Gamma_{id}^{c}\Gamma_{\bar{d}j}^{e}F_{ce}(i\gamma),$
(30)

where $\gamma=\pi-u_{\bar{c}d}^{\bar{i}}-u_{d\bar{e}}^{\bar{j}}$ ($\bar{j}$
denoting the antiparticle of $j$). Kinematic poles are due to particle-
antiparticle annihilation and in these cases two-particle FFs obey Yurov and
Zamolodchikov (1991)

$-i\underset{\theta\to
i\pi}{\lim}(\theta-i\pi)F^{\mathcal{O}}_{a_{i},a_{\bar{i}}}(\theta)=\left[1-e^{i2\pi\gamma_{a_{i},\mathcal{O}}}\right]\langle\mathcal{O}\rangle,$
(31)

where $\langle\mathcal{O}\rangle$ is the vacuum expectation value (VEV) of the
operator $\mathcal{O}$.

In the two-particle case, ${Q_{a_{i}a_{j}}^{\mathcal{O}}(\theta)}$ is a
polynomial in $e^{\theta}$, which satisfies an upper bound on its maximal
degree Delfino and Mussardo (1995). Define $y_{\mathcal{O}}$ to be the
coefficient multiplying $\theta_{i}$ as it approaches infinity, i.e.

$F_{a_{i}a_{j}}^{\mathcal{O}}(\theta_{i},\theta_{j})\underset{|\theta_{i}|\to\infty}{\sim}\exp{y_{\mathcal{O}}|\theta_{i}|}.$
(32)

The upper bound on the degree of the FF reads

$y_{\mathcal{O}}\leq\Delta_{\mathcal{O}},$ (33)

where $\Delta_{\mathcal{O}}$ is the conformal weight of the operator
$\mathcal{O}$ in the UV CFT.

The latter bound and the aforementioned linear equations are in most cases
sufficient to fix the FFs as functions of the VEV of some operator. However,
it may happen that they are not enough. In computing the FFs of order and
disorder, leading and subleading, operators of the thermally deformed TPM, we
also made use of second order equations coming from the asymptotic
factorization of FFs Delfino et al. (1996).

##  V Form Factors of the Stress-Energy Tensor

This section outlines the computation of the FFs of the stress-energy tensor
$\Theta(x)$, as originally obtained in Acerbi et al. (1996).

Since the FFs of the stress-energy tensor corresponding to the same particle
states have all the same pole structure, the two-particle FFs of the trace
$\Theta$ can be parametrized by the following polynomial:

$Q^{\Theta}_{ab}(\theta)=\left(\cosh\theta+\frac{m_{a}^{2}+m_{b}^{2}}{2m_{a}m_{b}}\right)^{1-\delta_{m_{a},m_{b}}}\sum_{i=0}^{\deg
Q^{\Theta}_{ab}}c_{ab}^{i}(\cosh\theta)^{i},$ (34)

where the degree is determined by the asymptotic behavior Eq. (33). Notice
that when particles have the same mass, i.e. in the case of a particle-
antiparticle doublet or a singlet, the prefactor in the latter equation is
completely absorbed in the unknown coefficients $c_{ab}^{(i)}$. Being a
conserved quantity of the theory, the FFs of $\Theta$ don’t obey the usual
residue equations for particle-antiparticle annihilation, rather:

$F_{a\bar{a}}^{\Theta}(i\pi)=2\pi m_{a}^{2}.$ (35)

Apart from these properties, $\Theta$ is a $Z_{3}$-invariant operator, like
$\mu$ and $\zeta$. The problem of finding the one and two-particle FFs is
therefore similar for those three operators. In the following we’ll present
the parametrization and the equations satisfied by the one-particle and first
two-particle FFs (Table 7).

state | $E_{CM}/m_{l}$
---|---
$L$ | $1.41421\dots$
$l\bar{l},\bar{l}l$ | $\geq 2$
$H$ | $2.73205\dots$
$LL$ | $\geq 2.82843\dots$
$l\bar{h},\bar{l}h,\bar{h}l,h\bar{l}$ | $\geq 2.93185\dots$
$lll,\bar{l}\bar{l}\bar{l}$ | $\geq 3$
Table 7: Multi-particle states with lowest center of mass energy and
nontrivial FFs with respect to the $Z_{3}$-invariant operators.

The first two-particle FFs of $\Theta$ are parametrized as follows

$\begin{split}F_{l\bar{l}}^{\Theta}(\theta)&=c_{l\bar{l}}^{0}\frac{F_{l\bar{l}}^{min}(\theta)}{D_{l\bar{l}}(\theta)},\\\
F_{LL}^{\Theta}(\theta)&=\left(c_{LL}^{0}+c_{LL}^{1}\cosh\theta\right)\frac{F_{LL}^{min}(\theta)}{D_{LL}(\theta)},\\\
F_{l\bar{h}}^{\Theta}(\theta)&=c_{l\bar{h}}^{0}\left(\cosh\theta+\frac{m_{l}^{2}+m_{h}^{2}}{2m_{l}m_{h}}\right)\frac{F_{l\bar{h}}^{min}(\theta)}{D_{l\bar{h}}(\theta)},\end{split}$
(36)

and satisfy the equations

$\displaystyle F_{l\bar{l}}^{\Theta}(i\pi)=2\pi m_{l}^{2},$ (37)
$\displaystyle-i\underset{\theta\to iu_{l\bar{l}}^{L}}{\lim}(\theta-
iu_{l\bar{l}}^{L})F_{l\bar{l}}^{\Theta}(\theta)=\Gamma_{l\bar{l}}^{L}F_{L}^{\Theta},$
(38) $\displaystyle F_{LL}^{\Theta}(i\pi)=2\pi m_{L}^{2},$ (39)
$\displaystyle-i\underset{\theta\to iu_{LL}^{L}}{\lim}(\theta-
iu_{LL}^{L})F_{LL}^{\Theta}(\theta)=\Gamma_{LL}^{L}F_{L}^{\Theta},$ (40)
$\displaystyle-i\underset{\theta\to iu_{LL}^{H}}{\lim}(\theta-
iu_{LL}^{H})F_{LL}^{\Theta}(\theta)=\Gamma_{LL}^{H}F_{H}^{\Theta},$ (41)
$\displaystyle-i\underset{\theta\to
i\pi/2}{\lim}(\theta-i\pi/2)F_{LL}^{\Theta}(\theta)=\Gamma_{Ll}^{l}\Gamma_{L\bar{l}}^{\bar{l}}F_{l\bar{l}}^{\Theta}(0),$
(42) $\displaystyle-i\underset{\theta\to iu_{l\bar{h}}^{L}}{\lim}(\theta-
iu_{l\bar{h}}^{L})F_{l\bar{h}}^{\Theta}(\theta)=\Gamma_{l\bar{h}}^{L}F_{L}^{\Theta},$
(43) $\displaystyle-i\underset{\theta\to iu_{l\bar{h}}^{H}}{\lim}(\theta-
iu_{l\bar{h}}^{H})F_{l\bar{h}}^{\Theta}(\theta)=\Gamma_{l\bar{h}}^{H}F_{H}^{\Theta},$
(44) $\displaystyle-i\underset{\theta\to i\pi 7/12}{\lim}(\theta-i\pi
7/12)F_{l\bar{h}}^{\Theta}(\theta)=\Gamma_{ll}^{\bar{l}}\Gamma_{\bar{h}\bar{l}}^{l}F_{\bar{l}l}^{\Theta}(i\pi
2/12).$ (45)

Solutions of this system of equations are provided in Table 8, and reproduce
the ones of Acerbi et al. (1996). As a further check of consistency, we
computed the contributions to the $c$-theorem Eq. (24) (Table 9).

Param.$/m_{l}^{2}$ |
---|---
$F_{L}^{\Theta}$ | $1.2613539473\dots$
$c_{l\bar{l}}^{0}$ | $6.2831853071\dots$
$F_{H}^{\Theta}$ | $0.2920374047\dots$
$c_{LL}^{0}$ | $21.765592370\dots$
$c_{LL}^{1}$ | $9.1992217564\dots$
$c_{l\bar{h}}^{0}$ | $25.226482640\dots$

Param./$\langle\epsilon\rangle$ |
---|---
$F_{L}^{\epsilon}$ | $-0.9499625864\dots$
$d_{l\bar{l}}^{0}$ | $-4.7320508076\dots$
$F_{H}^{\epsilon}$ | $-0.2199419194\dots$
$d_{LL}^{0}$ | $-16.392304845\dots$
$d_{LL}^{1}$ | $-6.9282032303\dots$
$d_{l\bar{h}}^{0}$ | $-18.998802632\dots$

Table 8: Parameters of the first FFs of the trace of the stress-energy tensor $\Theta$ and the energy operator $\epsilon$. State |  $c$-series
---|---
$A_{L}$ | $0.7596528$
$A_{l\bar{l}}$ | $0.0853913$
$A_{H}$ | $0.0029236$
$A_{LL}$ | $0.0024957$
$A_{l\bar{h}}$ | $0.0049010$
Total | 0.85536…
Relative error | $0.207\%$
Table 9: $c$-theorem contributions of the first five particle states
($c=6/7\approx 0.85714$).

Since the fundamental mass $m_{l}$ obeys Eq. (35), it is possible to compute
$\langle\Theta\rangle$ from the cluster equation Koubek and Mussardo (1993)

$\underset{\theta\to\infty}{\lim}F_{LL}^{\Theta}(\theta)=\frac{w_{LL}}{\langle\Theta\rangle}F_{L}^{\Theta}F_{L}^{\Theta},$
(46)

where $w_{LL}$ is a phase, which can be either $1$ or $-1$. In order to make
the $\Delta$-theorem contributions for the energy operator $\epsilon$
positive, we choose the phase $w_{LL}$ such that $\langle\Theta\rangle$ is
negative, yielding

$\langle\Theta\rangle=-1.3277932893\dots m_{l}^{2}.$ (47)

Combining Eqs. (12) and (15) we get

$\langle\epsilon\rangle=-1.7303450451\dots\lambda^{1/6},$ (48)

which coincides with the VEV of the energy operator in the thermal
perturbation of the whole minimal model $\mathcal{M}_{6,7}$ in the conformal
normalization Fateev et al. (1998).

The FFs of $\epsilon$ retain the same pole structure and polynomial dependence
of the ones of $\Theta$, with modified coefficients; in particular, the FFs
are parametrized as follows

$\begin{split}F_{l\bar{l}}^{\epsilon}(\theta)&=d_{l\bar{l}}^{0}\frac{F_{l\bar{l}}^{min}(\theta)}{D_{l\bar{l}}(\theta)},\\\
F_{LL}^{\epsilon}(\theta)&=\left(d_{LL}^{0}+d_{LL}^{1}\cosh\theta\right)\frac{F_{LL}^{min}(\theta)}{D_{LL}(\theta)},\\\
F_{l\bar{h}}^{\epsilon}(\theta)&=d_{l\bar{h}}^{0}\left(\cosh\theta+\frac{m_{l}^{2}+m_{h}^{2}}{2m_{l}m_{h}}\right)\frac{F_{l\bar{h}}^{min}(\theta)}{D_{l\bar{h}}(\theta)},\end{split}$
(49)

with the unknown coefficients satisfying

$d_{ab}^{i}=\frac{c_{ab}^{i}}{\langle\Theta\rangle}\langle\epsilon\rangle.$
(50)

For the sake of completeness, their values are summarized in Table 8 and the
contributions of the respective FFs in the $\Delta$ sum rule are shown in
Table 10.

State |  $\Delta_{\epsilon}$-series
---|---
$A_{L}$ | $0.0953527$
$A_{l\bar{l}}$ | $0.0313356$
$A_{H}$ | $0.0013696$
$A_{LL}$ | $0.0026516$
$A_{l\bar{h}}$ | $0.0046187$
Total | 0.135328
Relative error | $5.27\%$
Table 10: $\Delta$-theorem contributions for the energy operator $\epsilon$ of
the first five particle states ($\Delta_{\epsilon}=1/7\approx 0.14286$).

##  VI Form Factors of Order and Disorder Leading and Subleading Operators

This section is dedicated to the derivation of the first FFs of magnetic
operators in the thermally deformed TPM. Let us start from the leading
disorder operator.

###  VI.1 Leading Disorder Operator $\mu$

Because of $\mathcal{C}$-invariance, $\langle\tilde{\mu}\rangle$ and
$\langle\mu\rangle$ coincide. In Table 7 are presented the first states with
lowest mass and nontrivial FFs. Since the computation of these FFs involves
non-locality factors, the derivation will be outlined step-by-step.

Let us start from ${F_{l\bar{l}}^{\mu}(\theta)}$. It is possible to show that

$F_{l\bar{l}}^{\mu}(\theta)=F_{\bar{l}l}^{\mu}(-\theta)=F_{\bar{l}l}^{\tilde{\mu}}(\theta)=F_{l\bar{l}}^{\tilde{\mu}}(-\theta).$
(51)

Since the disorder operator has nonzero non-locality factor with respect to
particles $l$ and $\bar{l}$, the FF will also possess an additional term which
takes into account particle-antiparticle annihilation. Because of the
asymptotic behaviors of ${F_{l\bar{l}}^{min}(\theta)}$ and
${D_{l\bar{l}}(\theta)}$, $Q_{l\bar{l}}^{\mu}(\theta)$ satisfies the following
inequality on its degree:

$(\deg Q_{l\bar{l}}^{\mu})\theta-\frac{5}{6}|\theta|\leq\frac{1}{21}|\theta|.$
(52)

Notice the absence of the absolute value on the first term in the l.h.s.. The
polynomial ${Q_{l\bar{l}}^{\mu}(\theta)}$ is not symmetric in the exchange
${\theta\to-\theta}$, since the non-locality factors of $\mu$ with respect to
particles $l$ and $\bar{l}$ are nonzero. The non-locality equation that must
be satisfied is

$F_{l\bar{l}}^{\mu}(\theta)=S_{l\bar{l}}(\theta)e^{i2\pi\gamma_{\bar{l},\mu}}F_{l\bar{l}}^{\mu}(\theta+2\pi
i),$ (53)

with ${\gamma_{\bar{l},\mu}=-1/3}$. Solutions are sought of the form
$\exp(\alpha\theta)$, provided that they satisfy

$e^{\alpha\theta}=-e^{-i2\pi/3}e^{\alpha(\theta+2\pi i)}.$ (54)

Therefore the parametrization for this FFs is

$F_{l\bar{l}}^{\mu}(\theta)=\left(a_{l\bar{l}}^{0}\
e^{\frac{5}{6}\theta}+a_{l\bar{l}}^{1}\
e^{-\frac{1}{6}\theta}\right)\frac{F_{l\bar{l}}^{min}(\theta)}{\cosh\left(\frac{\theta}{2}\right)D_{l\bar{l}}(\theta)}.$
(55)

Pole equations

$\begin{split}&-i\underset{\theta\to
i\pi}{\lim}(\theta-i\pi)F_{l\bar{l}}^{\mu}(\theta)=\left[-i\underset{\theta\to
i\pi}{\lim}(\theta-i\pi)F_{\bar{l}l}^{\mu}(\theta)\right]^{*}=\left(1-e^{i2\pi/3}\right)\langle\mu\rangle,\\\
&-i\underset{\theta\to iu_{l\bar{l}}^{L}}{\lim}(\theta-
iu_{l\bar{l}}^{L})F_{l\bar{l}}^{\mu}(\theta)=-i\underset{\theta\to
iu_{l\bar{l}}^{L}}{\lim}(\theta-
iu_{l\bar{l}}^{L})F_{\bar{l}l}^{\mu}(\theta)=\Gamma_{l\bar{l}}^{L}F_{L}^{\mu},\end{split}$
(56)

allow us fixing all three unknowns - $a_{l\bar{l}}^{0}$, $a_{l\bar{l}}^{1}$
and $F_{L}^{\mu}$ \- as functions of the VEV of the leading disorder operator
$\mu$ (Table 11).

As far as ${F_{LL}^{\mu}(\theta)}$ is concerned, the absence of non-locality
phases simplifies the parametrization of the FF:

$F_{LL}^{\mu}(\theta)=\left(a_{LL}^{0}+a_{LL}^{1}\cosh\theta\right)\frac{F_{LL}^{min}(\theta)}{D_{LL}(\theta)}.$
(57)

However, the corresponding $S$-matrix has a second-order pole; therefore the
set of pole equations is

$\begin{split}&-i\underset{\theta\to iu_{LL}^{L}}{\lim}(\theta-
iu_{LL}^{L})F_{LL}^{\mu}(\theta)=\Gamma_{LL}^{L}F_{L}^{\mu},\\\
&-i\underset{\theta\to iu_{LL}^{H}}{\lim}(\theta-
iu_{LL}^{H})F_{LL}^{\mu}(\theta)=\Gamma_{LL}^{H}F_{H}^{\mu},\\\
&-i\underset{\theta\to
i\pi/2}{\lim}(\theta-i\pi/2)F_{LL}^{\mu}(\theta)=\Gamma_{L\bar{l}}^{\bar{l}}\Gamma_{Ll}^{l}F_{\bar{l}l}^{\mu}(0)=\Gamma_{Ll}^{l}\Gamma_{L\bar{l}}^{\bar{l}}F_{l\bar{l}}^{\mu}(0).\end{split}$
(58)

Again, all three unknowns ($F_{H}^{\mu}$, $a_{LL}^{0}$ and $a_{LL}^{1}$) can
be fixed by simply solving these linear equations (Table 11). The first
nontrivial check of the consistency of these solutions is given by the second
order cluster equation

$\underset{\theta\to\infty}{\lim}F_{LL}^{\mu}(\theta)=\frac{w_{LL}}{\langle\mu\rangle}F_{L}^{\mu}F_{L}^{\mu},$
(59)

satisfied when the phase $w_{LL}=1$.

The FF $F_{l\bar{h}}^{\mu}(\theta)$ is obtained along the lines of
${F_{l\bar{l}}^{\mu}(\theta)}$. In particular, its parametrization reads

$F_{l\bar{h}}^{\mu}(\theta)=\left(a_{l\bar{h}}^{0}\
e^{\frac{4}{3}\theta}+a_{l\bar{h}}^{1}\
e^{\frac{1}{3}\theta}+a_{l\bar{h}}^{2}\
e^{-\frac{2}{3}\theta}\right)\frac{F_{l\bar{h}}^{min}(\theta)}{D_{l\bar{h}}(\theta)},$
(60)

and satisfies the pole equations

$\begin{split}&-i\underset{\theta\to iu_{l\bar{h}}^{L}}{\lim}(\theta-
iu_{l\bar{h}}^{L})F_{l\bar{h}}^{\mu}(\theta)=\Gamma_{l\bar{h}}^{L}F_{L}^{\mu},\\\
&-i\underset{\theta\to iu_{l\bar{h}}^{H}}{\lim}(\theta-
iu_{l\bar{h}}^{H})F_{\bar{h}l}^{\mu}(\theta)=\Gamma_{l\bar{h}}^{H}F_{H}^{\mu},\\\
&-i\underset{\theta\to i\pi 7/12}{\lim}(\theta-i\pi
7/12)F_{l\bar{h}}^{\mu}(\theta)=\Gamma_{ll}^{\bar{l}}\Gamma_{\bar{l}\bar{l}}^{l}F_{\bar{l}l}^{\mu}(i\pi
2/12),\\\ &-i\underset{\theta\to i\pi 7/12}{\lim}(\theta-i\pi
7/12)F_{\bar{h}l}^{\mu}(\theta)=\Gamma_{ll}^{\bar{l}}\Gamma_{\bar{l}\bar{l}}^{l}F_{l\bar{l}}^{\mu}(i\pi
2/12).\end{split}$ (61)

Notice that the latter two pole equations have a peculiar dependence on the
two-particle FFs ${F_{\bar{l}l}^{\mu}(\theta)}$ and
${F_{l\bar{l}}^{\mu}(\theta)}$. Solutions are reported in Table 11. We also
computed the first contributions to the $\Delta$-theorem (Table 12).

Param.$/\langle\mu\rangle$ |
---|---
$F_{L}^{\mu}$ | $-0.3477104392\dots$
$a_{l\bar{l}}^{0}$ | $-0.3169872981\dots$
$a_{l\bar{l}}^{1}$ | $-1.1830127018\dots$
$F_{H}^{\mu}$ | $-0.0589332596\dots$
$a_{LL}^{0}$ | $-5.1961524220\dots$
$a_{LL}^{1}$ | $-0.9282032290\dots$
$a_{l\bar{h}}^{0}$ | $-0.6587868452\dots$
$a_{l\bar{h}}^{1}$ | $-6.0223791136\dots$
$a_{l\bar{h}}^{2}$ | $-4.2584651101\dots$

Param.$/\langle\zeta\rangle$ |
---|---
$F_{L}^{\zeta}$ | $-2.2476356112\dots$
$b_{l\bar{l}}^{0}$ | $-4.4150635077\dots$
$b_{l\bar{l}}^{1}$ | $-7.6471143143\dots$
$b_{l\bar{l}}^{2}$ | $-2.3660254027\dots$
$F_{H}^{\zeta}$ | $-0.8408344177\dots$
$b_{LL}^{0}$ | $-49.9807620886\dots$
$b_{LL}^{1}$ | $-38.7646096627\dots$
$b_{l\bar{h}}^{0}$ | $-22.6098992318\dots$
$b_{l\bar{h}}^{1}$ | $-74.3815188798\dots$
$b_{l\bar{h}}^{2}$ | $-59.3127674978\dots$
$b_{l\bar{h}}^{3}$ | $-9.4994013120\dots$

Table 11: Parameters of the first FFs of the leading disorder operator $\mu$ and subleading disorder operator $\zeta$. State |  $\Delta_{\mu}$-series |  $\Delta_{\zeta}$-series
---|---|---
$A_{L}$ | $0.0349015$ | $0.2256069$
$A_{l\bar{l}}$ | $0.0095534$ | $0.1266482$
$A_{H}$ | $0.0003670$ | $0.0051113$
$A_{LL}$ | $0.0005463$ | $0.0121824$
$A_{l\bar{h}}$ | $0.0010622$ | $0.0257981$
Total | 0.0464304 | 0.395347
Relative error | $2.50\%$ | $16.98\%$
Table 12: $\Delta$-theorem contributions for the leading disorder operator
$\mu$ ($\Delta_{\mu}=1/21\approx 0.047619$) and the subleading disorder
operator $\zeta$ of the first five particle states
($\Delta_{\zeta}=10/21\approx 0.47619$).

###  VI.2 Subleading Disorder Operator $\zeta$

The computation of FFs for the subleading disorder operator $\zeta$ presents
little differences if compared to the case of $\mu$. First of all, since the
$\zeta$ has the same transformation properties as $\mu$, the non-vanishing FFs
of $\zeta$ are the same as the ones of $\mu$ (Table 7). The only difference in
computing the FFs between the leading and subleading disorder operator lies in
their different anomalous dimensions, which in turn affects the upper bound on
the FFs degree. On the other hand non-locality coefficients remain unchanged
in passing from the leading to the subleading case. These considerations are
accounted for in the parametrization of some of the $Q$ polynomials:

$\begin{split}F_{l\bar{l}}^{\zeta}(\theta)&=\left(b_{l\bar{l}}^{0}\
e^{\frac{5}{6}\theta}+b_{l\bar{l}}^{1}\
e^{-\frac{1}{6}\theta}+b_{l\bar{l}}^{2}\
e^{-\frac{7}{6}\theta}\right)\frac{F_{l\bar{l}}^{min}(\theta)}{\cosh\left(\frac{\theta}{2}\right)D_{l\bar{l}}(\theta)},\\\
F_{LL}^{\zeta}(\theta)&=\left(b_{LL}^{0}+b_{LL}^{1}\cosh\theta\right)\frac{F_{LL}^{min}(\theta)}{D_{LL}(\theta)},\\\
F_{l\bar{h}}^{\zeta}(\theta)&=\left(b_{l\bar{h}}^{0}\
e^{\frac{4}{3}\theta}+b_{l\bar{h}}^{1}\
e^{\frac{1}{3}\theta}+b_{l\bar{h}}^{2}\
e^{-\frac{2}{3}\theta}+b_{l\bar{h}}^{3}\
e^{-\frac{5}{3}\theta}\right)\frac{F_{l\bar{h}}^{min}(\theta)}{D_{l\bar{h}}(\theta)}.\end{split}$
(62)

The set of linear equations for $\zeta$ remains the same as the ones for
$\mu$, except for a change in label $\mu\to\zeta$ in all of them. However, it
turns out that there is one linearly dependent equation, which doesn’t allow
fixing one of the unknowns, e.g. $F_{L}^{\zeta}$. Studying the cluster
equation Koubek and Mussardo (1993)

$\underset{\theta\to\infty}{\lim}F_{LL}^{\zeta}(\theta)=\frac{w_{LL}}{\langle\zeta\rangle}F_{L}^{\zeta}F_{L}^{\zeta},$
(63)

provides a solution to the problem. If ${w_{LL}=1}$, we have two nonphysical
solutions

$F_{L}^{\zeta}=(-0.2725122015\dots,2.8678582521\dots)\langle\zeta\rangle,$

since the first value gives a contribution to the $\Delta$-theorem which is
positive but too small, while the second gives a negative contribution. If
${w_{LL}=-1}$, one finds

$F_{L}^{\zeta}=(-2.2476356112\dots,-0.3477104393\dots)\langle\zeta\rangle.$

While the second solution is associated to the disorder operator $\mu$, the
first one is acceptable: its contribution to the $\Delta$-theorem is given in
Table 12 and it is compatible with the conformal weight of the operator.
Notice that convergence is slower than the leading case, as it generally
happens when one considers less relevant operators. Solving the system of
equations, together with Eq. (63), yields the results of Table 11.

###  VI.3 Leading Order Operator $\sigma$ and Subleading Order Operator $Z$

The FFs of the leading and subleading order operators are simpler to compute
if compared to their disorder counterparts, because there are no non-locality
factors involved. The first nontrivial FFs are summarized in Table 13. It
turns out that the solution of their pole equations require the knowledge of
the FFs of disorder operators.

state | $E_{CM}/m_{l}$
---|---
$\bar{l}$ | $1$
$\bar{h}$ | $1.93185\dots$
$ll$ | $\geq 2$
$\bar{l}L$ | $\geq 2.41421\dots$
$lh$ | $\geq 2.93185\dots$
$\bar{l}l\bar{l}$ | $\geq 3$
$\bar{h}L$ | $\geq 3.34606\dots$
Table 13: Multi-particle states with lowest center of mass energy and
nontrivial FF with respect to the order operator $\sigma$. The one of the
subleading order operator $Z$ are retrieved by charge conjugating all
particles.

The parametrizations of the first FFs of $\sigma$ are

$\begin{split}F_{ll}^{\sigma}(\theta)&=a_{ll}^{0}\frac{F_{ll}^{min}(\theta)}{D_{ll}(\theta)},\\\
F_{\bar{l}L}^{\sigma}(\theta)&=\left(a_{\bar{l}L}^{0}+a_{\bar{l}L}^{1}\cosh\theta\right)\frac{F_{\bar{l}L}^{min}(\theta)}{D_{\bar{l}L}(\theta)},\\\
F_{lh}^{\sigma}(\theta)&=\left(a_{lh}^{0}+a_{lh}^{1}\cosh\theta\right)\frac{F_{lh}^{min}(\theta)}{D_{lh}(\theta)},\\\
F_{\bar{h}L}^{\sigma}(\theta)&=\left(a_{\bar{h}L}^{0}+a_{\bar{h}L}^{1}\cosh\theta+a_{\bar{h}L}^{2}(\cosh\theta)^{2}\right)\frac{F_{\bar{h}L}^{min}(\theta)}{D_{\bar{h}L}(\theta)}.\end{split}$
(64)

and the set of pole equations is

$\displaystyle-i\underset{\theta\to iu_{ll}^{\bar{l}}}{\lim}(\theta-
iu_{ll}^{\bar{l}})F_{ll}^{\sigma}(\theta)=\Gamma_{ll}^{\bar{l}}F_{\bar{l}}^{\sigma},$
(65) $\displaystyle-i\underset{\theta\to iu_{ll}^{\bar{h}}}{\lim}(\theta-
iu_{ll}^{\bar{h}})F_{ll}^{\sigma}(\theta)=\Gamma_{ll}^{\bar{h}}F_{\bar{h}}^{\sigma},$
(66) $\displaystyle-i\underset{\theta\to
iu_{\bar{l}L}^{\bar{l}}}{\lim}(\theta-
iu_{\bar{l}L}^{\bar{l}})F_{\bar{l}L}^{\sigma}(\theta)=\Gamma_{\bar{l}L}^{\bar{l}}F_{\bar{l}}^{\sigma},$
(67) $\displaystyle-i\underset{\theta\to
iu_{\bar{l}L}^{\bar{h}}}{\lim}(\theta-
iu_{\bar{l}L}^{\bar{h}})F_{\bar{l}L}^{\sigma}(\theta)=\Gamma_{\bar{l}L}^{\bar{h}}F_{\bar{h}}^{\sigma},$
(68) $\displaystyle-i\underset{\theta\to iu_{lh}^{\bar{l}}}{\lim}(\theta-
iu_{lh}^{\bar{l}})F_{lh}^{\sigma}(\theta)=\Gamma_{lh}^{\bar{l}}F_{\bar{l}}^{\sigma},$
(69) $\displaystyle-i\underset{\theta\to iu_{lh}^{\bar{h}}}{\lim}(\theta-
iu_{lh}^{\bar{h}})F_{lh}^{\sigma}(\theta)=\Gamma_{lh}^{\bar{h}}F_{\bar{h}}^{\sigma},$
(70) $\displaystyle-i\underset{\theta\to i\pi 5/12}{\lim}(\theta-i\pi
5/12)F_{lh}^{\sigma}(\theta)=\Gamma_{lh}^{\bar{l}}\Gamma_{l\bar{l}}^{L}F_{\bar{l}L}^{\sigma}(i\pi/12),$
(71) $\displaystyle-i\underset{\theta\to
iu_{\bar{h}L}^{\bar{l}}}{\lim}(\theta-
iu_{lh}^{\bar{l}})F_{\bar{h}L}^{\sigma}(\theta)=\Gamma_{\bar{h}L}^{\bar{l}}F_{\bar{l}}^{\sigma},$
(72) $\displaystyle-i\underset{\theta\to i\pi 8/12}{\lim}(\theta-i\pi
8/12)F_{\bar{h}L}^{\sigma}(\theta)=\Gamma_{\bar{l}\bar{h}}^{l}\Gamma_{lL}^{l}F_{ll}^{\sigma}(i\pi
4/12),$ (73) $\displaystyle-i\underset{\theta\to i\pi 4/12}{\lim}(\theta-i\pi
4/12)F_{\bar{h}L}^{\sigma}(\theta)=\Gamma_{\bar{l}\bar{h}}^{l}\Gamma_{lL}^{h}F_{lh}^{\sigma}(i\pi/12),$
(74) $\displaystyle-i\underset{\theta\to i\pi 6/12}{\lim}(\theta-i\pi
6/12)F_{\bar{h}L}^{\sigma}(\theta)=\Gamma_{\bar{h}L}^{\bar{l}}\Gamma_{l\bar{l}}^{L}F_{\bar{l}L}^{\sigma}(i\pi/12).$
(75)

These equations are linearly dependent and $F_{\bar{l}}^{\sigma}$ cannot be
fixed in this way. This value, however, can be obtained employing the cluster
equation

$\underset{\theta\to+\infty}{\lim}F_{l\bar{l}}^{\mu}(\theta)=\frac{1}{\langle\mu\rangle}F_{l}^{\bar{\sigma}}F_{\bar{l}}^{\sigma}=\frac{1}{\langle\mu\rangle}\left(F_{\bar{l}}^{\sigma}\right)^{2}\,\,\,.$
(76)

This equation allows us to fix $F_{\bar{l}}^{\sigma}$ up to a sign, which
however is not relevant to the computation of two-point correlation functions.
The parameters are summarized in Table 14.

Param.$/\langle\mu\rangle$ |
---|---
$F_{\bar{l}}^{\sigma}$ | $0.6143373207\dots$
$F_{\bar{h}}^{\sigma}$ | $-0.1788394265\dots$
$a_{ll}^{0}$ | $3.2187030369\dots$
$a_{\bar{l}L}^{0}$ | $5.8427632193\dots$
$a_{\bar{l}L}^{1}$ | $1.7461593992\dots$
$a_{lh}^{0}$ | $66.6327984972\dots$
$a_{lh}^{1}$ | $61.9658050968\dots$
$a_{\bar{h}L}^{0}$ | $45.6244273941\dots$
$a_{\bar{h}L}^{1}$ | $45.6244273770\dots$
$a_{\bar{h}L}^{2}$ | $5.4770385951\dots$

Param.$/\langle\zeta\rangle$ |
---|---
$F_{l}^{Z}$ | $2.2927380937\dots$
$F_{h}^{Z}$ | $-1.6446346242\dots$
$b_{\bar{l}\bar{l}}^{0}$ | $18.4497693307\dots$
$b_{\bar{l}\bar{l}}^{1}$ | $12.8748121373\dots$
$b_{lL}^{0}$ | $46.9842992716\dots$
$b_{lL}^{1}$ | $42.1249702982\dots$
$b_{\bar{l}\bar{h}}^{0}$ | $695.4260997639\dots$
$b_{\bar{l}\bar{h}}^{1}$ | $960.8141050048\dots$
$b_{\bar{l}\bar{h}}^{2}$ | $276.4554994639\dots$
$b_{hL}^{0}$ | $531.5286291647\dots$
$b_{hL}^{1}$ | $851.3434042817\dots$
$b_{hL}^{2}$ | $325.5818137048\dots$

Table 14: Parameters of the first FFs of the leading order operator $\sigma$
and subleading order operator $Z$.

For the subleading order operator, the parametrization of the FFs is

$\begin{split}F_{\bar{l}\bar{l}}^{Z}(\theta)&=\left(b_{\bar{l}\bar{l}}^{0}+b_{\bar{l}\bar{l}}^{1}\cosh\theta\right)\frac{F_{\bar{l}\bar{l}}^{min}(\theta)}{D_{\bar{l}\bar{l}}(\theta)},\\\
F_{lL}^{Z}(\theta)&=\left(b_{lL}^{0}+b_{lL}^{1}\cosh\theta\right)\frac{F_{lL}^{min}(\theta)}{D_{lL}(\theta)},\\\
F_{\bar{l}\bar{h}}^{Z}(\theta)&=\left(b_{\bar{l}\bar{h}}^{0}+b_{\bar{l}\bar{h}}^{1}\cosh\theta+b_{\bar{l}\bar{h}}^{2}(\cosh\theta)^{2}\right)\frac{F_{\bar{l}\bar{h}}^{min}(\theta)}{D_{\bar{l}\bar{h}}(\theta)},\\\
F_{hL}^{Z}(\theta)&=\left(b_{hL}^{0}+b_{hL}^{1}\cosh\theta+b_{hL}^{2}(\cosh\theta)^{2}\right)\frac{F_{hL}^{min}(\theta)}{D_{hL}(\theta)}\end{split}$
(77)

and the set of pole equations is the same as the one of $\sigma$, provided
that we substitute $\sigma\to Z$ and charge conjugate all particles, since
$\sigma$ and $Z$ have complex conjugate $Z_{3}$ phases. It turns out that, in
solving the set of linear equations, $F_{l}^{Z}$ and $F_{h}^{Z}$ remain
unknown. Conjointly solving

$\begin{split}\underset{\theta\to\infty}{\lim}F_{lL}^{Z}(\theta)&=\frac{w_{lL}}{\langle\zeta\rangle}F_{l}^{Z}F_{L}^{\zeta},\\\
\underset{\theta\to\infty}{\lim}F_{hL}^{Z}(\theta)&=\frac{w_{hL}}{\langle\zeta\rangle}F_{h}^{Z}F_{L}^{\zeta},\\\
\underset{\theta\to+\infty}{\lim}F_{l\bar{l}}^{\zeta}(\theta)&=\frac{w_{l\bar{l}}}{\langle\zeta\rangle}\left(F_{l}^{Z}\right)^{2},\end{split}$
(78)

allows us finding $F_{l}^{Z}$ and $F_{h}^{Z}$ without any phase ambiguity
between them - $F_{l}^{Z}$ is fixed up to an overall sign, still. The
parameters of the FFs of $Z$ are summarized in Table 14.

##  VII Universal Ratios of the Renormalization Group

In the following we are going to consider some universal ratios of the
Renormalization Group associated to the thermal deformation of TPM, i.e. to
the action Eq. (11). Consider an operator $\phi_{j}$ of the theory which has
at criticality the anomalous dimension $\mathcal{X}_{j}=2\Delta_{j}$ and
therefore renormalization group eigenvalue $y_{j}=2-2\Delta_{j}$. The coupling
$\lambda$ of the theory is a dimensional quantity related to the lightest
excitation of the theory $m_{l}$ as in Eq.(15). Denoting by $K_{j}$ the metric
factor associated to the operator $\phi_{j}$, on general ground the scaling
form of the VEV $\langle\phi_{j}\rangle$ can be written as (see Fioravanti et
al. (2000) for a more detailed discussion)

$\langle\phi_{j}\rangle=B_{j\epsilon}\lambda^{\mathcal{X}_{j}/y_{\epsilon}},$
(79)

with $B_{j\epsilon}:=K_{j}K_{\epsilon}^{\mathcal{X}_{j}/y_{\epsilon}}$.
Generalized susceptibilities

$\hat{\Gamma}_{jk}^{\epsilon}=\Gamma_{jk}^{\epsilon}\lambda_{\epsilon}^{(2-y_{j}-y_{k})/y_{\epsilon}},$
(80)

with $\Gamma_{jk}^{i}:=K_{j}K_{k}K_{\epsilon}^{(2-y_{j}-y_{k})/y_{\epsilon}}$,
are related to two-point connected off-critical correlators through the
fluctuation-dissipation theorem

$\hat{\Gamma}_{jk}^{\epsilon}=\int\differential^{2}x\langle\phi_{j}(x)\phi_{k}(0)\rangle^{\text{C}}_{\epsilon}.$
(81)

Universal ratios are defined as ratios involving the correlation length $\xi$,
the expectation values $\langle\phi_{j}\rangle$ and generalized
susceptibilities $\hat{\Gamma}_{jk}^{\epsilon}$ such that the dependence on
the metric factors $K_{j}$ cancels out Fioravanti et al. (2000). These
quantities are associated to the class of universality and therefore are the
same for different microscopic realizations of the model. The first trivial
universal ratio is the ratio between the correlation lengths in the disordered
and ordered phase:

$\frac{\xi_{+}}{\xi_{-}}=\frac{m_{L}}{m_{l}}=2\cos(\pi/4)=\sqrt{2}\approx
1.41421.$ (82)

In order to find nontrivial universal ratios starting from the FFs computed in
previous sections, Eq. (81) must be rewritten as to make the dependence on
$\lambda$ manifest. Notice however that some susceptibilities are identically
zero, because of $Z_{3}$-symmetry as, for instance,
$\hat{\Gamma}^{\epsilon}_{\sigma\mu}$, $\hat{\Gamma}^{\epsilon}_{\sigma Z}$
and $\hat{\Gamma}^{\epsilon}_{\sigma\tilde{\sigma}}$. Charge conjugation also
reduces the number of quantities to compute, e.g.
$\hat{\Gamma}^{\epsilon}_{\tilde{\mu}\tilde{\mu}}=\hat{\Gamma}^{\epsilon}_{\mu\mu}$.
In Table 15 are presented the generalized susceptibilities involving
$Z_{3}$-invariant operators ($\mu$, $\epsilon$ and $\zeta$), while Table 16 is
dedicated to magnetization ones ($\sigma$ and $Z$).

State | ${\hat{\Gamma}^{\epsilon}_{\mu\mu}\frac{m_{l}^{2}}{\langle\mu\rangle^{2}}}$ | ${\hat{\Gamma}^{\epsilon}_{\epsilon\epsilon}\frac{m_{l}^{2}}{\langle\epsilon\rangle^{2}}}$ | ${\hat{\Gamma}^{\epsilon}_{\zeta\zeta}\frac{m_{l}^{2}}{\langle\zeta\rangle^{2}}}$ | ${\hat{\Gamma}^{\epsilon}_{\mu\epsilon}\frac{m_{l}^{2}}{\langle\mu\rangle\langle\epsilon\rangle}}$ | ${\hat{\Gamma}^{\epsilon}_{\mu\zeta}\frac{m_{l}^{2}}{\langle\mu\rangle\langle\zeta\rangle}}$ | ${\hat{\Gamma}^{\epsilon}_{\epsilon\zeta}\frac{m_{l}^{2}}{\langle\epsilon\rangle\langle\zeta\rangle}}$
---|---|---|---|---|---|---
$A_{L}$ | $0.12090$ | $0.90243$ | $5.05186$ | $0.33031$ | $0.78153$ | $2.13517$
$A_{l\bar{l}}$ | $0.02860$ | $0.29656$ | $6.90261$ | $0.09041$ | $0.35195$ | $1.19861$
$A_{H}$ | $0.00093$ | $0.01296$ | $0.18053$ | $0.00347$ | $0.01296$ | $0.04837$
$A_{LL}$ | $0.00110$ | $0.02509$ | $0.53618$ | $0.02329$ | $0.02329$ | $0.11530$
$A_{l\bar{h}}$ | $0.00237$ | $0.04371$ | $1.90493$ | $0.01005$ | $0.05368$ | $0.24416$
Sum | $0.15390$ | $1.28076$ | $14.5761$ | $0.45754$ | $1.22341$ | $3.74160$
Table 15: First contributions for the generalized susceptibilities of $Z_{3}$-invariant operators. State | ${\hat{\Gamma}^{\epsilon}_{\sigma\sigma}\frac{m_{l}^{2}}{\langle\mu\rangle^{2}}}$ | ${\hat{\Gamma}^{\epsilon}_{\tilde{Z}\tilde{Z}}\frac{m_{l}^{2}}{\langle\zeta\rangle^{2}}}$ | ${\hat{\Gamma}^{\epsilon}_{\sigma\tilde{Z}}\frac{m_{l}^{2}}{\langle\mu\rangle\langle\zeta\rangle}}$
---|---|---|---
$A_{\bar{l}}$ | $0.75482$ | $10.5133$ | $2.81702$
$A_{\bar{h}}$ | $0.01714$ | $1.44951$ | $0.15762$
$A_{ll}$ | $0.00838$ | $4.49913$ | $0.13430$
$A_{\bar{l}L}$ | $0.00919$ | $2.17994$ | $0.13840$
$A_{lh}$ | $0.00120$ | $1.33517$ | $0.02814$
$A_{\bar{h}L}$ | $0.00041$ | $0.33581$ | $0.01113$
Sum | $0.79113$ | $20.3128$ | $3.28662$
Table 16: First contributions for the generalized susceptibilities of
magnetization operators.

In general, FFs depend naturally on the VEV of some operator

$F^{\phi_{j}}_{a_{1},\dots,a_{n}}(\theta_{1},\dots,\theta_{n})=f^{\phi_{j}}_{a_{1},\dots,a_{n}}(\theta_{1},\dots,\theta_{n})\langle\phi_{j}^{\prime}\rangle_{i},$
(83)

where $\phi_{j}^{\prime}$ is often coincident with $\phi_{j}$, like in the
case of $\mu$, $\epsilon$ and $\zeta$, but not always, as it happens for
$\sigma$ and $Z$. Manipulations of Eq. (22) yield

$\Gamma_{jk}^{\epsilon}=2\pi
C_{\epsilon}^{-2}(B_{j\epsilon}^{\prime})^{*}B_{k\epsilon}^{\prime}\mathcal{F}_{jk}^{\epsilon},$
(84)

where

$\mathcal{F}_{jk}^{i}=\int_{0}^{\infty}\differential\tau\
\tau\Biggl{\\{}\sum_{\\{n\\}}\int_{-\infty}^{\infty}\frac{\differential\theta_{1}\dots\differential\theta_{n}}{n!(2\pi)^{n}}\exp\left(-\tau\sum\beta_{a_{i}}\cosh\theta_{i}\right)\left[f^{\phi_{j}}_{a_{1},\dots,a_{n}}(\theta_{1},\dots,\theta_{n})\right]^{*}f^{\phi_{k}}_{a_{1},\dots,a_{n}}(\theta_{1},\dots,\theta_{n})\Biggr{\\}}.$
(85)

Here $\beta_{a_{j}}=m_{a_{j}}/m_{l}$. By approximating the two-point
correlation function with the sum of the contributions associated to the one
and two-particle states with lowest center-of-mass energy, we present
hereafter some universal ratios of the thermal deformation of the TPM:

$\displaystyle(R_{c})^{\epsilon}_{\sigma\sigma}=\frac{\Gamma_{\epsilon\epsilon}^{\epsilon}\Gamma_{\sigma\sigma}^{\epsilon}}{B_{\mu\epsilon}B_{\mu\epsilon}}$
$\displaystyle\approx 0.0153975135,$ (86)
$\displaystyle(R_{c})^{\epsilon}_{\sigma
Z}=\frac{\Gamma_{\epsilon\epsilon}^{\epsilon}\Gamma_{\sigma
Z}^{\epsilon}}{B_{\mu\epsilon}B_{\tilde{\zeta}\epsilon}}$
$\displaystyle\approx 0.0639662952,$ (87)
$\displaystyle(R_{c})^{\epsilon}_{\mu\mu}=\frac{\Gamma_{\epsilon\epsilon}^{\epsilon}\Gamma_{\mu\mu}^{\epsilon}}{B_{\mu\epsilon}B_{\mu\epsilon}}$
$\displaystyle\approx 0.0029953833,$ (88)
$\displaystyle(R_{c})^{\epsilon}_{\mu\epsilon}=\frac{\Gamma_{\epsilon\epsilon}^{\epsilon}\Gamma_{\mu\epsilon}^{\epsilon}}{B_{\mu\epsilon}B_{\epsilon\epsilon}}$
$\displaystyle\approx 0.0089049910,$ (89)
$\displaystyle(R_{c})^{\epsilon}_{\mu\zeta}=\frac{\Gamma_{\epsilon\epsilon}^{\epsilon}\Gamma_{\mu\zeta}^{\epsilon}}{B_{\mu\epsilon}B_{\zeta\epsilon}}$
$\displaystyle\approx 0.0238108180,$ (90)
$\displaystyle(R_{c})^{\epsilon}_{\epsilon\epsilon}=\frac{\Gamma_{\epsilon\epsilon}^{\epsilon}\Gamma_{\epsilon\epsilon}^{\epsilon}}{B_{\epsilon\epsilon}B_{\epsilon\epsilon}}$
$\displaystyle\approx 0.0249269560,$ (91)
$\displaystyle(R_{c})^{\epsilon}_{\zeta\epsilon}=\frac{\Gamma_{\epsilon\epsilon}^{\epsilon}\Gamma_{\zeta\epsilon}^{\epsilon}}{B_{\zeta\epsilon}B_{\epsilon\epsilon}}$
$\displaystyle\approx 0.0728215391,$ (92)
$\displaystyle(R_{c})^{\epsilon}_{ZZ}=\frac{\Gamma_{\epsilon\epsilon}^{\epsilon}\Gamma_{ZZ}^{\epsilon}}{B_{\tilde{\zeta}\epsilon}B_{\tilde{\zeta}\epsilon}}$
$\displaystyle\approx 0.3953414813,$ (93)
$\displaystyle(R_{c})^{\epsilon}_{\zeta\zeta}=\frac{\Gamma_{\epsilon\epsilon}^{\epsilon}\Gamma_{\zeta\zeta}^{\epsilon}}{B_{\zeta\epsilon}B_{\zeta\epsilon}}$
$\displaystyle\approx 0.2836900244.$ (94)

It should be noted that the universal ratios involving the subleading order
and disorder operators are subject to more uncertainty than those containing
more relevant operators. Indeed for less relevant operators, the spectral
series converges more slowly, as evidenced by the $\Delta$-theorem
contributions.

##  VIII Comparison with a Monte Carlo study

In this section we present some results coming from the numerical simulations
of the thermal deformation of the tricritical three-state Potts model on
isotropic square lattices of spacing $a$. Hereafter we denote the sizes of the
lattice in the two directions as $L_{t}=aN_{t}$ and $L_{x}=aN_{x}$; for all of
the simulations described here, we used $L_{t}=L_{x}\equiv L$. Periodic
boundary conditions are imposed in both directions; this choice, which
preserves translational symmetry under translations by integer multiples of
the lattice spacing along the $t$ and $x$ axes, minimizes the impact of
artefacts due to the finiteness of the system. The reduced Hamiltonian of the
lattice model is defined by Eq. (1), which here we re-write in the following
form,

$\displaystyle\frac{H}{k_{\mbox{\tiny{B}}}T}=-K\sum_{\langle
i,j\rangle}\delta_{s_{i},s_{j}}(1-\delta_{s_{i},0})-V\sum_{\langle
i,j\rangle}\delta_{s_{i},0}\delta_{s_{j},0}-D\sum_{i}\delta_{s_{i},0},$ (95)

to account for the implementation used in our simulation code, where the spin
and vacancy variables are simultaneously encoded into a single variable
$s_{i}\in\left\\{0,1,2,3\right\\}$ defined on each lattice site $i$, with the
value $0$ corresponding to a vacancy, while the other three values correspond
to the site being occupied by a spin. As in Eq. (1), the $\langle i,j\rangle$
notation denotes pairs of nearest-neighbor lattice sites. To simplify our
numerical study, we restricted our attention to simulations for the $V=0$ case
only – that is, we neglected the possibility of a vacancy-vacancy coupling –
so that the dynamics of the model depends only on the two parameters $K$ (the
coupling between nearest-neighbor spins) and $D$ (which acts as a chemical
potential for the vacancies).

For our Monte Carlo study, we created a dedicated C++ simulation algorithm,
which generates the configurations of $s_{i}$ variables combining Swendsen-
Wang cluster updates Swendsen and Wang (1987) (for the spins) with local
Metropolis updates Metropolis et al. (1953) (for both the spins and the
vacancies). One complete update consists of five Metropolis updates of the
entire lattice alternated with five cluster updates of the whole system. While
the non-local cluster updates dramatically reduce the critical slowing down
close to critical points, this numerical problem is not completely solved, due
to the persistence of non-negligible autocorrelations for the vacancies, whose
evolution in Monte Carlo time is affected only indirectly by the cluster
updates.

We studied the model for a range of $K$ values in the proximity of the
tricritical value $K_{\mbox{\tiny{tc}}}=1.649903(5)$, holding $D$ fixed to its
tricritical value $D_{\mbox{\tiny{tc}}}=3.152152(2)$. These values for both
$K_{\mbox{\tiny{tc}}}$ and $D_{\mbox{\tiny{tc}}}$ coincide with those reported
in Ref. Qian et al. (2005) where they were estimated by means of a different
method, i.e. the transfer-matrix technique. We performed simulations at $81$
equally spaced values of $K$ in the
$[K_{\mbox{\tiny{tc}}}-0.04,K_{\mbox{\tiny{tc}}}+0.04]$ interval; note that
values of $K$ larger than $K_{\mbox{\tiny{tc}}}$ correspond to thermal
deformations driving the system into a broken-symmetry phase, in which the
magnetization is expected to be non-zero in the thermodynamic limit. The
square lattices that we simulated had sizes $(L/a)^{2}=64^{2}$, $128^{2}$,
$256^{2}$, and $512^{2}$. For each value of $K$ and each lattice size, after
at least $10^{4}$ complete updates for reaching Monte Carlo thermalization, we
collected $8\cdot 10^{5}$ configurations for the lattices of sizes
$(L/a)^{2}=64^{2}$ and $128^{2}$, $4\cdot 10^{5}$ configurations for the
lattices of size $(L/a)^{2}=256^{2}$, and a number of configurations between
$387610$ and $751375$ for the lattices of size $(L/a)^{2}=512^{2}$. We used
these ensembles of configurations to compute the expectation values of all
observables described below, with the exception of the spin-spin correlation
function, which was estimated from ensembles consisting of $4\cdot 10^{5}$
configurations for each value of $K$ on the lattices of sizes
$(L/a)^{2}=64^{2}$, $128^{2}$, and $(L/a)^{2}=256^{2}$, and a number of
configurations between $201835$ and $386155$ for each $K$ value on the
lattices of size $(L/a)^{2}=512^{2}$.

The observables that we evaluated in our Monte Carlo simulations include:

  * •

The density of vacancies, which is the fraction of sites in the $s_{i}=0$
state

$\displaystyle\rho=\frac{1}{(L/a)^{2}}\sum_{i}\delta_{s_{i},0}.$ (96)

  * •

The contribution to the energy (in units of $k_{\mbox{\tiny{B}}}T$) coming
from spin-spin interactions, divided by the number of lattice sites, which is
denoted as

$\displaystyle\epsilon=\frac{K}{(L/a)^{2}k_{\mbox{\tiny{B}}}T}\sum_{\langle
i,j\rangle}\delta_{s_{i},s_{j}}(1-\delta_{s_{i},0}).$ (97)

In the following (with a slight abuse of terminology), we will also refer to
this quantity simply as ‘energy’; in addition, we considered the associated
susceptibility, too.

  * •

The magnetization in each configuration, which is defined as

$\displaystyle\sigma=(1-\rho)\frac{qf_{\mbox{\tiny{max}}}-1}{q-1},$ (98)

where $f_{\mbox{\tiny{max}}}$ denotes the fraction of spins in the majority
state in the given configuration,

$\displaystyle
f_{\mbox{\tiny{max}}}=\mathrm{max}_{k\in\left\\{1,2,3\right\\}}\left(\frac{1}{(L/a)^{2}}\sum_{i}\delta_{s_{i},k}\right).$
(99)

  * •

Finally, the two-point spin-spin correlation function, which is defined as

$\displaystyle
G_{ss}(r)=\frac{1}{2(L/a)^{2}}\sum_{i}\sum_{k=0}^{1}\left(1-\delta_{s_{i},0}\right)\delta_{s_{i},s_{i+r\hat{u}_{k}}},$
(100)

where $\hat{u}_{0}$ and $\hat{u}_{1}$ respectively denote the positively
oriented unit vectors in the $t$ and $x$ directions of the lattice.

For all of these variables, we computed the integrated autocorrelation time
(for a discussion about the numerical estimate of this quantity in Monte Carlo
simulations, see, e.g., Ref. Wolff (2004)), and grouped the whole set of
‘measurements’ of each observable obtained in every simulation into bins of
length equal to the integrated autocorrelation time. We verified that, as
expected, the average over each bin produced data with negligible residual
autocorrelation.

Figure 3 shows how the distribution of vacancies extracted from our
simulations varies as a function of $K_{\mbox{\tiny{tc}}}-K$. The first
observation is that the results from simulations on lattices of the different
sizes that we considered exhibit a remarkable collapse of data, indicating
that finite-volume corrections are under control. Furthermore, the data are
consistent with an inflection point at $K=K_{\mbox{\tiny{tc}}}$, and the value
of $\rho$ at that point is in full agreement with the result (denoted by the
black cross) obtained through the transfer-matrix technique in Ref. Qian et
al. (2005).

Figure 3: Average density of vacancies, displayed as a function of
$K_{\mbox{\tiny{tc}}}-K$, obtained from our simulations on square lattices of
different sizes (denoted by symbols of different shapes and colors). The plot
also shows the average density of vacancies at $K_{\mbox{\tiny{tc}}}$ (shown
by the black cross) that was computed in Ref. Qian et al. (2005) using a
different numerical technique.

Next, we consider the average value of $\epsilon$, which is plotted in Fig. 4,
together with the results for the associated susceptibility (displayed in the
inset plot, with a logarithmic scale on the vertical axis). As for the vacancy
density, also for $\langle\epsilon\rangle$ we observe a clear collapse of the
data obtained from simulations on lattices of different area, and an
inflection point at a location consistent with $K=K_{\mbox{\tiny{tc}}}$. For
this observable, the Monte Carlo results are expected to be modelled by a
function of the form:

$\displaystyle\langle\epsilon\rangle=e_{0}\left(K_{\mbox{\tiny{tc}}}-K\right)^{1/6}+e_{1}.$
(101)

Fitting the data in the range $0.005\leq K_{\mbox{\tiny{tc}}}-K\leq 0.04$ to
Eq. (101), one obtains $e_{0}=2.036(4)$ and $e_{1}=-1.6124(23)$. The curve
obtained from the fit is shown by the black line in Fig. 4. The good quality
of this fit, in which the exponent of $(K_{\mbox{\tiny{tc}}}-K)$ is fixed to
the value predicted analytically, confirms the validity of the conformal field
theory description of the results obtained from Monte Carlo simulations, at
least for the range of $K$ values and for the lattice sizes under
consideration in this study.

Figure 4: Average contribution to the energy density from spin-spin
interactions, plotted against $K_{\mbox{\tiny{tc}}}-K$, from Monte Carlo
calculations on square lattices of linear size ranging from $L/a=64$ to
$L/a=512$; the curve obtained by fitting the data for $0.005\leq
K_{\mbox{\tiny{tc}}}-K\leq 0.04$ to Eq. (101) is also shown. The inset plot
shows the results for the susceptibility associated with $\epsilon$ (rescaled
by the number of lattice sites), which are displayed with a logarithmic scale
for the vertical axis.

Our Monte Carlo results for the average magnetization are displayed in Fig. 5;
note that, in the broken-symmetry phase, the results obtained from simulations
on lattices of different area are consistent with each other. On the other
hand, in the symmetric phase, the results obtained from smaller lattices are
systematically larger than those from larger lattices: this is an artefact of
the definition of the magnetization in Eq. (98), which is based on the
fraction of spins in the majority state in each configuration: by definition,
for a system with a finite number of degrees of freedom, in the symmetric
phase $f_{\mbox{\tiny{max}}}$ is always larger than $1/3$ – except in the
exceedingly rare configurations in which the total number of variables in the
spin states $1$, $2$, and $3$ are exactly equal to each other. Nevertheless,
our Monte Carlo results show that the magnetization in the symmetric phase
vanishes when it is extrapolated to the thermodynamic limit.

In the broken-symmetry phase, fitting the results for the magnetization in the
range $-0.04\leq K_{\mbox{\tiny{tc}}}-K\leq-0.005$ to

$\displaystyle\langle\sigma\rangle=d_{0}\left|K_{\mbox{\tiny{tc}}}-K\right|^{1/18}+d_{1},$
(102)

one obtains $d_{0}=0.9526(15)$ and $d_{1}=0.1185(13)$. The result of this fit
is shown by the black line in Fig. 5. As for $\langle\epsilon\rangle$, the
fact that one obtains a good modelling of Monte Carlo results using this
simple functional form, with the exponent of
$\left|K_{\mbox{\tiny{tc}}}-K\right|$ fixed to the value that is predicted by
conformal field theory, confirms the validity of this description of Monte
Carlo results.

Figure 5: Average magnetization, plotted against $K_{\mbox{\tiny{tc}}}-K$,
from our Monte Carlo simulations on lattices with the same linear sizes as in
Fig. 3 and in Fig. 4. The curve obtained fitting our data in the broken-
symmetry phase to Eq. (102) is also shown (black line).

Next, we show examples of our results for the two-point spin-spin correlation
function in Fig. 6. In particular, it is interesting to study the universal
ratio between the correlation lengths in the symmetric and in the symmetry-
breaking phases. Sufficiently close to the tricritical point, it is expected
that the ratio of correlation lengths obtained at values of $K$ that are
symmetric with respect to $K_{\mbox{\tiny{tc}}}$ should become consistent with
the theoretical prediction $\xi_{+}/\xi_{-}=\sqrt{2}=1.41421...$.

As an example to illustrate this type of study, the data displayed in Fig. 6
are obtained from two test ensembles of Monte Carlo simulations on a lattice
of size $(L/a)^{2}=256^{2}$ at $K=K_{\mbox{\tiny{tc}}}\pm 0.01$, i.e., close
to the tricritical point.444Each of these two additional ensembles consisted
of $4\cdot 10^{5}$ configurations, that were produced after $10^{4}$ complete
updates for thermalization, like the other simulations on the lattices of size
$(L/a)^{2}=256^{2}$. Note that, in order to (slightly) reduce the computing
times, for these two test ensembles the normalization used in the Monte Carlo
calculation of the correlators explicitly dismissed an additive contribution
to $G_{ss}(r)$; this additive contribution does not depend on $r$, and, hence,
has no effect on the evaluation of the correlation lengths. Expressing all
distances in units of the lattice spacing, the correlators can be fitted to

$\displaystyle G_{ss}(r)=a_{0}\frac{\exp(-r/\xi)}{r^{4/21}}+a_{2},$ (103)

yielding $\xi=2.672(82)$ for $K=1.639903$ and $\xi=1.800(45)$ for
$K=1.659903$, whose ratio

$\displaystyle\frac{\xi_{K=1.639903}}{\xi_{K=1.659903}}=1.484(59)$ (104)

is within less than $1.2$ standard deviations from the analytically predicted
value.

Figure 6: Two-point spin correlation functions computed from Monte Carlo
simulations on a lattice of size $(L/a)^{2}=256^{2}$ at two different $K$
values, shown as a function of the spatial separation $r$ in units of the
lattice spacing $a$. The dashed lines are the curves obtained from the fits to
Eq. (103).

It should be noted that the extraction of the correlation length from the
Monte Carlo results for the correlators is non-trivial, since it is based on a
three-parameter, non-linear fit of cross-correlated data, and the fitted
parameters are quite strongly correlated with each other, too. This, in turn,
leads to a likely overestimate of the uncertainties affecting the fitted
parameters, including, in particular, the correlation length. However, we
followed a very conservative approach to the error budget, without attempting
to reduce these uncertainties. Figure 7 shows the correlation lengths computed
from Monte Carlo simulations on lattices of different sizes and at different
$K$ values. In particular, the main plot (in which a logarithmic scale is used
for both axes) displays the data in the symmetric phase and their fit to the
power-law dependence on $|K_{\mbox{\tiny{tc}}}-K|$ predicted from the theory:

$\displaystyle\xi=a_{0}|K_{\mbox{\tiny{tc}}}-K|^{-7/12}.$ (105)

The plot makes manifest that the simulation results precisely follow the
power-law behavior predicted by conformal field theory. The only free
parameter of the fit to Eq. (105) is the overall amplitude, for which we
obtain $a_{0}=0.356(1)$. Analogous results in the broken-symmetry phase are
shown in the inset plot (where a linear scale is used for both axes).

Figure 7: Correlation lengths computed from Monte Carlo simulations at
different $K$ values and different lattice sizes, and their fit to the power-
law behavior predicted from conformal field theory (black line), given by Eq.
(105). The main plot refers to simulations at $K<K_{\mbox{\tiny{tc}}}$,
whereas the inset plot shows analogous results in the broken-symmetry phase.

##  IX Discussion and Conclusions

In this work we have addressed the computation of the form factors of the
relevant operators of the TPM once this model is perturbed away from its
critical point by varying the temperature. The corresponding massive theory is
integrable and has a distinguished fingerprint associated to the $E_{6}$
exceptional Lie algebra. We have determined the FF associated to the leading
and subleading order and disorder operators and we have checked their correct
identification using the $\Delta$ sum rule. We have shown how to theoretically
compute several universal ratios of the Renormalization Group associated to
the class of universality of the model in any of its microscopic realizations.
We also presented some results coming from a Monte Carlo study of the lattice
model, which provided clear evidence that, in the range of parameters that we
considered, the results are in agreement with the expectations from CFT, and
that, in addition, the universal ratio of the masses in the high and low
temperature phases of the model is fully consistent with our theoretical
prediction.

## Acknowledgements

G.M. acknowledges the grants PNRR MUR Project PE0000023- NQSTI and PRO3
Quantum Pathfinder. The work of M.P. has been partially supported by the Spoke
1 “FutureHPC & BigData” of the Italian Research Center on High-Performance
Computing, Big Data and Quantum Computing (ICSC) funded by MUR (M4C2-19) –
Next Generation EU (NGEU), by the Italian PRIN “Progetti di Ricerca di
Rilevante Interesse Nazionale – Bando 2022”, prot. 2022TJFCYB, and by the
“Simons Collaboration on Confinement and QCD Strings” funded by the Simons
Foundation. The simulations were run on CINECA computers. We acknowledge
support from the SFT Scientific Initiative of the Italian Nuclear Physics
Institute (INFN).

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##  Appendix A Analytic Properties of Form Factors

In this Appendix, we will briefly touch upon some properties of FFs,
complementing the discussion of Section IV. Let us start with the connection
between the matrix element in the r.h.s. of Eq. (20) and Eq. (21), given by
crossing

$F^{\mathcal{O}}_{a_{1},\dots,a_{m},\bar{a}_{m+1},\dots,\bar{a}_{n}}(\theta_{1},\dots,\theta_{m},\theta_{m+1}-i\pi,\dots,\theta_{n}-i\pi)=F^{\mathcal{O}}_{a_{1},\dots,a_{n}}(\theta_{ij};\theta_{Ak};\theta_{BC}),$
(106)

where $1\leq i<j\leq m$, $m+1\leq k\leq m<A\leq n$ and $m+1\leq B<C\leq n$.

In analogy to the $S$-matrix, we expect FFs to be analytic functions in the
strip ${0<\imaginary\theta_{ij}<2\pi}$ and to have poles in correspondence of
bound states. To understand the analytic properties satisfied by FFs, it is
useful to define the ‘Faddeev-Zamolodchikov’ (FZ) algebra, which follows from
the definition of the scattering matrix. The FZ algebra is an associative
algebra of vertex operators which satisfies the following commutation
relations involving the $S$-matrix:

$\begin{split}A_{a_{i}}(\theta_{i})A_{a_{j}}(\theta_{j})&=S_{ij}(\theta)A_{a_{j}}(\theta_{j})A_{a_{i}}(\theta_{i}),\\\
A^{\dagger}_{a_{i}}(\theta_{i})A^{\dagger}_{a_{j}}(\theta_{j})&=S_{ij}(\theta)A^{\dagger}_{a_{j}}(\theta_{j})A^{\dagger}_{a_{i}}(\theta_{i}),\\\
A_{a_{i}}(\theta_{i})A^{\dagger}_{a_{j}}(\theta_{j})&=S_{ij}(\theta)A^{\dagger}_{a_{j}}(\theta_{j})A_{a_{i}}(\theta_{i})+2\pi\delta_{a_{i}a_{j}}\delta(\theta),\end{split}$
(107)

with $\theta=\theta_{i}-\theta_{j}$. Here $A_{a_{i}}(\theta_{i})$ is the
destruction operator and $A^{\dagger}_{a_{i}}(\theta_{i})$ its creation
counterpart. Also $a_{i}$ is a shorthand for the set of quantum numbers that
distinguishes a particle from the others in the spectrum.

It is straightforward to rephrase the definition of FFs Eq. (21) in the vertex
operator formalism. Suppose that the vertex operator $A^{\dagger}_{a_{i}}$ is
non-local with respect to the operator $\mathcal{O}$ Zamolodchikov and Fateev
[1985], which means that after a counterclockwise rotation of
$A^{\dagger}_{a_{i}}(\theta^{\prime})$ around $\mathcal{O}(\theta)$, the OPE
${A^{\dagger}_{a_{i}}(\theta^{\prime})\mathcal{O}(\theta)}$ acquires a phase
${\exp{i2\pi\gamma_{a_{i},\mathcal{O}}}}$. If $\gamma_{a_{i},\mathcal{O}}$ is
zero, then we have ‘mutual locality’. A graphical representation of this
process is presented in Figure 8.

$\mathcal{O}(\theta)$$\real\theta$$\imaginary\theta$$A^{\dagger}_{a_{i}}(\theta^{\prime})$

Figure 8: The analytic continuation that defines non-local fields.

Watson equations are direct consequences of the commutation relations Eq.
(107) and the non-locality properties of the vertex operators with respect to
$\mathcal{O}$:

$F_{\dots,a_{m},a_{m+1},\dots}^{\mathcal{O}}(\dots,\theta_{m},\theta_{m+1},\dots)=S_{a_{m},a_{m+1}}(\theta_{m}-\theta_{m+1})F_{\dots,a_{m+1},a_{m},\dots}^{\mathcal{O}}(\dots,\theta_{m+1},\theta_{m},\dots),$
(108) $F_{a_{1}\dots a_{n}}^{\mathcal{O}}(\theta_{1}+2\pi
i,\theta_{2},\dots,\theta_{n})=e^{i2\pi\gamma_{a_{1},\mathcal{O}}}F_{a_{2}\dots
a_{n},a_{1}}^{\mathcal{O}}(\theta_{2},\dots,\theta_{n},\theta_{1}).$ (109)

From these equations it is clear that minimal FFs introduced in Section IV are
nothing more than building-block solutions for the two-particle Watson
equation, if non-locality is not accounted for. For degenerate mass spectra
with reflectionless, i.e. diagonal, $S$-matrix, the general structure of a
two-particle amplitude is

$S_{ij}(\theta)=(-1)^{\delta_{i,\bar{j}}}\prod_{x\in\mathcal{A}_{ij}}s_{x}(\theta),$
(110)

where $s_{x}(\theta)$ is the building-block function Eq.(19) and
$\mathcal{A}_{ij}$ is the set of poles associated to the scattering amplitude
- see e.g. Table 6. Minimal form factors can be parametrized as

$F_{ij}^{min}(\theta)=\left[-i\sinh\left(\frac{\theta}{2}\right)\right]^{\delta_{ij}}\prod_{x\in\mathcal{A}_{ij}}\left[h_{x}(\theta)\right]^{p_{x}},$
(111)

where $p_{x}$ is the multiplicity associated to $x\in\mathcal{A}_{ij}$, which
coincides with the order of the pole in the corresponding $S$-matrix element.
The building-block functions $h_{x}(\theta)$ satisfy

$h_{x}(\theta)=-s_{x}(\theta)h_{x}(-\theta),\quad h_{x}(\theta+2\pi
i)=h_{x}(-\theta),$ (112)

and their minimal solution is

$h_{x}(\theta)=\exp{2\int_{0}^{\infty}\frac{\differential
t}{t}\frac{\sinh((1-x)t)}{\sinh^{2}(t)}\sinh^{2}\left(\frac{\hat{\theta}t}{2i\pi}\right)},$
(113)

with $\hat{\theta}=i\pi-\theta$. It should be noted that $h_{x}(\theta)$ has
poles on the imaginary axis of $\theta$, as it can be clearly see after
performing an analytic continuation. There are two equivalent ways: the first
is

$h_{x}(\theta)=\prod_{k=0}^{\infty}\frac{\left[\Gamma\left(k+\frac{1}{2}+\frac{x}{2}\right)\right]^{2}\Gamma\left(k+1-\frac{x}{2}-\frac{i\theta}{2\pi}\right)\Gamma\left(k+2-\frac{x}{2}+\frac{i\theta}{2\pi}\right)}{\left[\Gamma\left(k+\frac{3}{2}-\frac{x}{2}\right)\right]^{2}\Gamma\left(k+\frac{x}{2}-\frac{i\theta}{2\pi}\right)\Gamma\left(k+1+\frac{x}{2}+\frac{i\theta}{2\pi}\right)},$
(114)

while the second reads

$h_{x}(\theta)=\prod_{k=0}^{\infty}\left[\frac{1+\left(\frac{\hat{\theta}/2\pi}{k+\frac{1}{2}+\frac{x}{2}}\right)^{2}}{1+\left(\frac{\hat{\theta}/2\pi}{k+\frac{3}{2}-\frac{x}{2}}\right)^{2}}\right]^{k+1}.$
(115)

It is possible to find analytic continuations with a finite number of factors,
useful in numerical evaluations

$\begin{split}h_{x}(\theta)&=\prod_{k=0}^{N-1}\frac{\left[\Gamma\left(k+\frac{1}{2}+\frac{x}{2}\right)\right]^{2}\Gamma\left(k+1-\frac{x}{2}-\frac{i\theta}{2\pi}\right)\Gamma\left(k+2-\frac{x}{2}+\frac{i\theta}{2\pi}\right)}{\left[\Gamma\left(k+\frac{3}{2}-\frac{x}{2}\right)\right]^{2}\Gamma\left(k+\frac{x}{2}-\frac{i\theta}{2\pi}\right)\Gamma\left(k+1+\frac{x}{2}+\frac{i\theta}{2\pi}\right)}\times\\\
&\quad\times\exp{2\int_{0}^{\infty}\frac{\differential
t}{t}\frac{\sinh((1-x)t)}{\sinh^{2}(t)}e^{-2Nt}\sinh^{2}\left(\frac{\hat{\theta}t}{2i\pi}\right)},\end{split}$
(116)
$h_{x}(\theta)=\prod_{k=0}^{N+1}\left[\frac{1+\left(\frac{\hat{\theta}/2\pi}{k+\frac{1}{2}+\frac{x}{2}}\right)^{2}}{1+\left(\frac{\hat{\theta}/2\pi}{k+\frac{3}{2}-\frac{x}{2}}\right)^{2}}\right]^{k+1}\exp{2\int_{0}^{\infty}\frac{\differential
t}{t}\frac{\sinh((1-x)t)}{\sinh^{2}(t)}\left(1+N-Ne^{-2t}\right)e^{-2Nt}\sinh^{2}\left(\frac{\hat{\theta}t}{2i\pi}\right)}.$
(117)

The last property of $h_{x}(\theta)$ we mention is its asymptotic behavior
Acerbi et al. [1996], which finds a natural application in the asymptotic
factorization of FFs:

$h_{x}(\theta)\underset{|\theta|\to\infty}{\sim}\exp{\int_{0}^{\infty}\frac{\differential
t}{t}\left[\frac{\sinh((1-x)t)}{\sinh^{2}(t)}-\frac{1-x}{t}\right]}\exp{\frac{1-x}{2}(\absolutevalue{\theta}-i\pi)}.$
(118)

Concerning the parametrization of FFs Eq. (25), kinematic poles can be
implemented by adding a term ${\cosh(\theta_{i\bar{i}}/2)}$ at the denominator
of the FF, while for the poles coming from the $S$-matrix one can use:

$D_{ij}(\theta)=\prod_{x\in\mathcal{A}_{ij}}\left[P_{x}(\theta)\right]^{n_{x}},$
(119)

with

$P_{x}(\theta)=\frac{\cos(\pi x)-\cosh\theta}{2\cos^{2}(\pi x/2)},$ (120)

and

$n_{x}=\begin{cases}m+1,&\text{if }p_{x}=2m+1\text{ and the pole is in the
s-channel;}\\\ m,&\text{if }p_{x}=2m+1\text{ and the pole is in the
t-channel;}\\\ m,&\text{if }p_{x}=2m.\end{cases}$ (121)