# Duality and Form Factors in the Thermally Deformed Two-Dimensional Tricritical Ising Model

### Submission summary

 As Contributors: Máté Lencsés · Gabor Takacs Arxiv Link: https://arxiv.org/abs/2109.09767v2 (pdf) Date submitted: 2022-01-21 08:33 Submitted by: Takacs, Gabor Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Mathematical Physics Approaches: Theoretical, Computational

### Abstract

The thermal deformation of the critical point action of the 2D tricritical Ising model gives rise to an exact scattering theory with seven massive excitations based on the exceptional $E_7$ Lie algebra. The high and low temperature phases of this model are related by duality. This duality guarantees that the leading and sub-leading magnetisation operators, $\sigma(x)$ and $\sigma'(x)$, in either phase are accompanied by associated disorder operators, $\mu(x)$ and $\mu'(x)$. Working specifically in the high temperature phase, we write down the sets of bootstrap equations for these four operators. For $\sigma(x)$ and $\sigma'(x)$, the equations are identical in form and are parameterised by the values of the one-particle form factors of the two lightest $\mathbb{Z}_2$ odd particles. Similarly, the equations for $\mu(x)$ and $\mu'(x)$ have identical form and are parameterised by two elementary form factors. Using the clustering property, we show that these four sets of solutions are eventually not independent; instead, the parameters of the solutions for $\sigma(x)/\sigma'(x)$ are fixed in terms of those for $\mu(x)/\mu'(x)$. We use the truncated conformal space approach to confirm numerically the derived expressions of the matrix elements as well as the validity of the $\Delta$-sum rule as applied to the off-critical correlators. We employ the derived form factors of the order and disorder operators to compute the exact dynamical structure factors of the theory, a set of quantities with a rich spectroscopy which may be directly tested in future inelastic neutron or Raman scattering experiments.

###### Current status:
Has been resubmitted

### Author comments upon resubmission

We are grateful to the referees for their constructive criticism, and especially to Gerard Watts to his very careful reading of the paper, and suggesting so many points of improvement. We give our detailed answers in the list of changes.

### List of changes

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\section{Answers to Referee 1}

We added the two references suggested by the referee.

\section{Answers to Gerard Watts}

\subsection{Comments in the Report section}
\begin{enumerate}
\item

The first is to which extent do the TCSA calculations confirm, agree with, or simply not disagree with the form-factor calculations? Some of the TCSA data appears in excellent agreement with the exact results for a large range of parameters - for example $\langle 0 | \sigma | \left\{n,-n\right\}\rangle_{1,2}$ in figure 6.1,
$\langle | \sigma' | \left\{n,-n\right\}\rangle_{1,2}$ for $n>0$
in figure 6.2, $\sqrt{\rho_{1,2}}|\langle 0 | \sigma | \left\{n,-n\right\}\rangle|$
for $n>1$
in figure 6.3,
$\sqrt{\rho_{1,1}}|\langle 0 | \sigma' | \left\{n,-n\right\}\rangle|$ for $n>1$
in figure 6.4,
$\sqrt{\rho_{1,1}}|\langle 0 | \sigma | \left\{n,-n\right\}\rangle|$
for $n>0$
in figure 6.6, $\sqrt{\rho_{1,1}}|\langle 0 | \sigma' | \left\{n,-n\right\}\rangle|$
for $n>0$
in figure 6.6, but many other comparisons are far worse. I could not find a commentary or critical discussion of the differences between TCSA and exact results when the agreement was poor, and why some results were better than others, other than that some poor agreement is down to truncation effects or small volumes leading to large exponential corrections, and so do not really know whether the TCSA results actually give support to the details of the form-factor expressions or not. Some discussion would be very helpful.

\textcolor{blue}{We added a discussion on different sources of errors when form factors to TCSA data, see Appendix C.2. We also updated the captions of the plots to point out the main features.}

\item
The authors also go to some lengths to explain that at a finite cutoff $\Lambda$, $\lambda_{\Lambda}$
should be non-zero to keep the $E_7$
symmetry. I would have thought that the same argument applied to the TCSA scheme as well with its cutoff $N$ and that a (small) non-zero coupling $\lambda_N$
would also be needed. Perhaps the authors could comment on this, and whether the effect is important or not?

\textcolor{blue}{In the new Appendix C.2 we discuss the issue of running couplings.}

\item
Secondly, the form-factor equations have only been solved for low-particle number states, and not even, as far as I could tell, for all two-particle states, and so I would like to see some comments on whether there are any potential obstructions to finding form factors for general particle states. This does not affect the relevance of the results here which stand on their own.

\textcolor{blue}{All the Form Factors can be obtained in principle starting from Form Factors on generic $n$-multiparticle state containing only particles of $A_1$, which must be constructed as solutions to the Form Factor equations. All other Form Factors by looking at the residues on the various channels. There is no particular obstruction to doing so, we decided to do differently because we are more interested in using the lowest energy Form Factors relevant for experimental comparison.}

\item
Finally, the construction of the full CFT of the TIM including non-local fields is not completely straightforward and that without care, even the OPEs (2.15) and (3.12) are hard to understand, the OPEs with branch cuts cannot be defined (they are ambiguous up to a sign) and a full construction of all the correlation functions in the CFT is problematic. This does not affect any of the results in the paper as the [missing] details of the CFT construction are irrelevant, I think, to what is done here, and the extra details would again not, I think, affect the TCSA calculations which only need the action of the (local) perturbing field, so it is far from a criticism of the paper, but I would have hoped to see at least a mention of the ambiguity in the OPEs.

\textcolor{blue}{We agree with this comment. In both equations we have substituted $\sim$ with $=$, which means adopting a specific convention for the spin fields, as it was done, for instance, in one of the Appendices of the Belavin-Polyakov-Zamolodchikov paper on CFT. We also inserted comments regarding the branch cuts (see also below). In any case, this issue is not particular relevant for the rest of our paper.
}
\end{enumerate}

\subsection{Comments in the Requested changes section}
\begin{enumerate}
\item
On page 9, equation (2.15) has a couple of issues which mean it is hard to understand what it conveys. Most obviously, there are factors of $1/\sqrt{2}$ missing (see equation (2.14)), so is (2.15) only defined up to a constant?

\textcolor{blue}{We have corrected the OPE with the factor $\frac{1}{\sqrt{2}}$, and also added a comment on sigma/mu being defect operators changing the boundary conditions for the fermion field.
}

\item On page 9, equation (2.16) is technically wrong as

\langle 0 |\sigma(\infty) = \lim_{z\rightarrow\infty}\langle 0|\sigma(z) = 0.

It would be better to write $\sigma|$
instead of $\langle 0|\sigma(\infty)$
where

\langle \sigma | = \lim_{z\rightarrow\infty |z|^{1/4}}\langle 0| \sigma(z).

or define what $\sigma(\infty)$ is.

\textcolor{blue}{Corrected.}

\item On page 11, equation (2.32) looks wrong. It has the wrong dimension - is this instead
$F^2_0$
? I could not find the value of $\langle \mu \rangle$ in reference [26], instead that paper gives $\langle \sigma \rangle$
in the low temperature limit as $\langle \sigma \rangle=2^{1/12}e^{-1/8}A^{3/2}m^{1/8}$ which is not the square-root of $F_0$ as given in (2.32). I do not know what is going on. Is $F_0$
here found by applying duality to the result of [26]? Is the missing factor of $2^{1/6}$
because of a typo in [26] or a typo here or a different choice of ground state? I think this needs some explanation.

\textcolor{blue}{Updated. We meant to write $\langle\sigma\rangle$ as it is in [26].}

\item On page 11, in equation (2.13), what does $\simeq$
mean? Does this mean that $1/|x|^{1/4}$ is the leading behaviour as $x\rightarrow0$
? Why is it not "$=$"?

\textcolor{blue}{Yes, it means that it is just the leading term and specifies the operator normalisation.}

\item On page 17, the introduction of superfields (with no reference to superspace) seems very odd, especially as the superspace formalism is only applied to the NS sector, and there is no corresponding discussion of the R sector as "spin fields". The fields in this expansion also have the wrong normalisation - if $t=G_{-1/2}\bar{G}_{-1/2}\epsilon$ then $t(z)t(w)\sim (-1/25) |z-w|^{-12/5}$

\textcolor{blue}{We only intended to use the superfields for the NS sector, where we think it provides the most transparent to summarize the sectors operator content. Indeed the normalisation was not treated carefully, and corrected the text accordingly. For the R sector, we rather concentrated on an exposition that emphasizes the structures underlying KW duality.}

\item On page 17, there is no need to introduce $(-1)^F$ to deduce there is a two-dimensional space of $R$
highest weight states. The fact that $\{G_0,\bar{G}_0\}=0$
is sufficient.

\textcolor{blue}{That is true, but we decided to keep the fermionic number operator as well for full analogy with the Ising model.}

\item On page 18, the structure constants are missing from (3.12). This could be very confusing. There are same issues about the choice of sign in the square-root branch cut. [This could be solved by defining fields $G(z)$ and $\mu(z)$ as defect-creating fields for the $Z_2$ defect and a convention for the placement of the corresponding defect lines as apply in the free-fermion case - not that this is needed, but some observation would help]

\textcolor{blue}{We have corrected the OPE taking care of the proper normalization of the zero-mode of the field $G$.}

\item On page 21, the fact that $\Delta_{\epsilon}$ is small means that $y_{\epsilon} =1-\Delta_{\epsilon}$ is large and perturbation theory is unreliable. I don't think this affects the qualitative nature of the results, but I think some comment would be helpful.

\textcolor{blue}{We agree, but we are not doing any perturbation theory in our paper. The first terms of the beta functions are correct as they are, independently of any further use of perturbation theory.}

\item On page 24, I think it is relevant to the discussion of fine-tuning of parameters in expriments to recall that there are four relevant fields and so four parameters need fine-tuning. The magnetic field parameters would be set to zero, but since one of these is the most relevant field, it would need to very accurately set to zero not to trigger a flow to a different fixed point.

\textcolor{blue}{We agree, of course. We have added an extra sentence concerning the $Z_2$ odd sector of the theory which simply requires to properly control the absence of any external magnetic field.}

\item On page 26, in equation (5.5), are the coefficients of $a_{i,j}^k$ known exactly or only numerically?

\textcolor{blue}{They are known exactly as integrals given in the Appendix. Now this is indicated in the text, see page 27.}

\item On page 33, table 6.1, there is no indication of the accuracy or reliability of the TCSA data. If all the digits can be relied upon, then this table shows disagreement. There are errors of various sorts here - truncation and exponential correction - and an estimate of the error would be helpful, rather than just reporting large numbers of decimal places which are, presumably, "wrong".

\textcolor{blue}{A more careful exponential fit was carried out using different volume windows for the fit, see also the discussion in Appendix C.2. Now we only show the digits which are stable during the fit, except the last one. We indicate the variation of the last digit using different windows.}

\item On page 5, it says "the different Verma modules of the conformal field theory". The Verma module is not actually a physically relevant space, since it includes all Virasoro descendants, and states which are identically zero. It would be better to find a re-formulation of this sentence which avoids the use of the words "Verma" and instead referred to "highest-weight modules" or something similar.

\textcolor{blue}{We have changed the terminology to "highest-weight modules" to avoid confusion, as suggested.}

\item In figures 6.1-6.6, it appears that there is excellent agreement only for $F^i_{12}$. For $F^i_{14}$ and $F^i_{23}$
, the fit is substantially worse, to my eyes. I wonder if the authors could comment on this? What would a "wrong" form-factor look like? I cannot tell from these figures how much of the behaviour is dictated simply by the two-particle S-matrix, and how much from the detail of the form-factor, and whether the figures really indicate confirmation of any form-factors apart from $F^i_{12}$.
For example, in table 6.2, the TCSA and exact results for $F^{\sigma'}_{34}(i\pi)$
differ by close to $10\%$. Is this a sign of agreement or disagreement?

\textcolor{blue}{Possible sources of discrepancies comparing to TCSA now are carefully discussed in Appendix C.2. Regarding $F_{14}$ and $F_{23}$, we were able to find only the the lowest Bethe--Yang state, therefore the finite size effects are more pronounced for larger rapidities. Due to the coupled nature of the form factor equations, the different quantities are eventually very much related and so the agreement is very tight, which may be obscured somewhat by the fact that for some rapidity values the finite size effects are large due to the given data being extracted from relatively small volume. We also commented on the issue of $F^{\sigma'}_{34}(i\pi)$ in the caption of Table 6.2.}

\item On page 54, there is an ansatz for the form factor in (A.8) which disagrees with the choices in (5.15) for non-locality of the particles with respect to the fields. If appendix A is meant to explain the form-factor ansatz, then I think the choice of the location of the factors $\cosh(\theta/2)$ and $1/\cosh(\theta/2)$ needs to be mentioned - as it is, appendix A does not do the job of explaining (5.15).

\textcolor{blue}{We have slightly extended the discussion in the Appendix to give reason of the appeareance of this term $\cosh\theta/2$ (either in the numerator or denominator of the form factors) according to the non-locality of the operator wrt the excitations.}

\item On page 58, I think the conformal transformation should be $z=\exp(2\pi w/L)$ so that $z$ is invariant under $w\rightarrow w + iL$.

\textcolor{blue}{We have corrected it, thanks!}

\item On page 11, it is said that "$\mu$ has a semi-local index equal to $1/2$ with respect to the operator $\sigma$
" but in the appendix this is referred to as a "mutual locality index". I think it would be better to use the same name in both places.

\textcolor{blue}{We have made the terminology uniform.}

\item On page 9, the choice of representation (2.14) is not very symmetric, requiring $\bar{\psi}_0$ to be given by $\pm\sigma_2$
. There is nothing wrong with this choice, but it might be helpful to give the representation of $\bar{\psi}_0$.

\textcolor{blue}{We have not introduced any representation for $\bar\psi$, simply because we do not need it and we did not want to further complicate the text.}

\item On page 3, it says "The TIM is the second conformal minimal model", when it is the second *unitary* conformal minimal model

\textcolor{blue}{We have included "unitary".}

\item On page 11, at the top, the authors say "we consider local operator fields" but then very soon say that $\mu$ is not local. I do wonder why the top sentence is there - it might be better to change it.

\textcolor{blue}{We have changed it.}

\item On page 29, the authors say they introduce an extra $\cosh(\theta/2)$ in $F^{\tilde{\Phi}}_{13}(\theta)$ "because of the kink nature", but earlier on page 25 they say they are dealing with the high temperature phase in which the particles are just regular particle, the kinks only appearing in the low temperature phase (according to page 19). Perhaps this could be clarified?

\textcolor{blue}{What really matters is the non-locality of the operator wrt the excitations. The simplest way of determining this fact is to consider the low-temperature phase, where the excitations which are non-local wrt the order operator $\sigma$ are the kinks. Going to high-temperature phase, this converts to the non-locality of the {\em now} particles BUT with respect to the disorder operator $\mu$. We have added a few extra words in the text in this respect.}

\item On page 8, the introduction of "P" and "A" seems unnecessary and indeed confusing. These are often used to describe the periodicity on the cylinder/torus and have exactly the opposite identification (P=R, A=NS). Since "P" and "A" are not used outside this subsection, as far as I can tell, maybe it would be better not to introduce them here? I also found it confusing to say that the branch cut for the free fermion starts from the origin - there must be branch cuts terminating at each insertion of a spin or disorder field, it is just that in the case of the plane with the only insertion at the origin and infinity then the branch cut must run from 0 to infinity. The description here will not confuse anybody who already knows what is going on, but I do wonder if it would help someone who did not understand the system and was reading it for the first time.

\textcolor{blue}{We agree and therefore we have simplified the text accordingly.}

\item Some simple typos - I hope the authors do not mind me listing these, in case they missed them -
\begin{enumerate}
\item page 3 "orginary" should be "ordinary"? \textcolor{blue}{Corrected.}
\item page 8 "close contour" should be "closed contour" \textcolor{blue}{Corrected.}
\item page 11 "corresponds the form" should be "corresponds to the form" \textcolor{blue}{Corrected.}
\item page 20 "cooples"should be "couples" \textcolor{blue}{Corrected.}
\item page 26 "set of equation" should be "set of equations" \textcolor{blue}{Corrected.}
\item page 30 "consider set.. which relating.." could be "consider a set..which relate"? \textcolor{blue}{Corrected.}
\item pages 32 and 56 "factor ised" should be "factorised" \textcolor{blue}{Corrected.}
\item page 33 "and exponential" should be "an exponential" \textcolor{blue}{Corrected.}
\item page 35 "curves small" could be "curves for small"? \textcolor{blue}{Corrected.}
\item page 40 "to the for" could be "to that for"? \textcolor{blue}{Corrected.}
\item page 41 "just few" should be "just a few" \textcolor{blue}{Corrected.}
\item pages 54 and 56 "factor isation" should be "factorisation" \textcolor{blue}{Corrected.}
\item page 55 "$G_{\alpha}$" should be "$g_{\alpha}$" \textcolor{blue}{Corrected.}
\item page 61 "the the" should be "the" \textcolor{blue}{Corrected.}
\end{enumerate}

\end{enumerate}

\end{document}

### Submission & Refereeing History

Resubmission 2109.09767v3 on 11 April 2022

Resubmission 2109.09767v2 on 21 January 2022
Submission 2109.09767v1 on 28 September 2021

## Reports on this Submission

### Report 2 by Gerard Watts on 2022-4-3 (Invited Report)

• Cite as: Gerard Watts, Report on arXiv:2109.09767v2, delivered 2022-04-03, doi: 10.21468/SciPost.Report.4850

### Report

I would like to thank the authors for the changes they have made which I think answer my previous queries very satisfactorily. I would also like to apologise for the time taken to write this report, for which I can only say that pressure of work in the past semester has been rather high.

I have, I should say, found a few different points which I think the authors may want to clarify, but which do not detract from the main results, which I think are now perfectly acceptable.

The first point is that of the "$Z_2$" symmetry in the TIM. On page 12, this is said to be $\sigma \to - \sigma, \mu \to -\mu$ at the conformal point, but on page 11 it is said to be $\sigma\to-\sigma, \mu\to\mu$ in the high-temperature phase. I find this very confusing. If it is the same symmetry, then the symmetry ought to act in the massless limit in the same way it does in the massive theory. This led me to wonder what the symmetry of the CFT is - and given the full set of fields (local and non-local) and the non-zero OPE coefficients, I think it looks like $Z_2\times Z_2$, with two generators, the first acting as $\{\sigma\to-\sigma,\mu\to\mu,\psi\to-\psi,\bar\psi\to-\bar\psi\}$ and the second as $\{\sigma\to\sigma,\mu\to-\mu,\psi\to-\psi,\bar\psi\to-\bar\psi\}$, with their product being $\{\sigma\to-\sigma,\mu\to-\mu,\psi\to\psi,\bar\psi\to\bar\psi\}$. Anyway, I think the comments on page 10 and page 11 appear contradictory and it would be good to clarify what is going on.

Secondly, on page 20, I think it is not the currents that have spins $\{1,5,7..\}$, but the charges. In the simplest case, $T_{\mu\nu}$ has spin 2 (or $T(z)$ has spin 2, $\bar T(\bar z)$ has spin -2) but the charges $P_\mu$ have spin 1 (or $P$ spin 1 and $\bar P$ spin -1)

Finally, in equation (A.22), which is $\langle\Phi\rangle$? In the references, in [25] the equation relates the form factors of $\Phi_a$, $\Phi_b$ and $\Phi_c$ but does not divide by any expectation value, whereas [57] and [58] only deal with the case of a single field $\Phi$. At the most basic level, I think the authors should consider clarifying what the general formula (A.22) means, as well as the explicit case (A.23) and (A.24).

There are also some typos but none of these at all get in the way of intelligibility.

• validity: -
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### Report

The paper can be published.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
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