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

by A. Cortés Cubero, R. M. Konik, M. Lencsés, G. Mussardo, G. Takács

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Submission summary

Authors (as registered SciPost users): Máté Lencsés · Gabor Takacs
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-09-28 14:28
Submitted by: Takacs, Gabor
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
Approaches: Theoretical, Computational


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

Submission & Refereeing History

Resubmission 2109.09767v3 on 11 April 2022

Reports on this Submission

Report 2 by Gerard Watts on 2021-12-28 (Invited Report)

  • Cite as: Gerard Watts, Report on arXiv:2109.09767v1, delivered 2021-12-28, doi: 10.21468/SciPost.Report.4097


1- This paper gives the first calculation of the form factors of the magnetic fields and their duals in the tri-critical Ising model.

2- The paper presents the calculation of potentially experimentally observable signatures

3- The form-factor calculations are checked using the truncated conformal space approach, giving an independent check of the results

4- There is a discussion of the simpler Ising model results and of the space of RG flows of the tri-critical Ising model which help make the paper accessible.


1- There are various technical issues with the paper (see report) which need addressing. (These are not critical)

2- The truncated conformal space results do not agree with the exact results to the number of digits reported, and some of the graphs are very hard to read, so that the degree of agreement is not clear.

3- The form factor equations are not solved in general, the results are only presented for some low-particle number states.


The core of the paper is the demonstration that the form-factor equations for the leading and sub-leading magnetisation operators $\sigma$ and $\sigma'$ in the thermally-perturbed tri-critical Ising model, and their dual disorder operators $\mu$ and $\mu'$, can be solved in terms of the high-temperature vacuum expectation values $\langle\mu\rangle$, $\langle\mu'\rangle$. This is supplemented by a check on various properties of the form-factor solutions using the truncated conformal space approach and a demonstration that the results give potentially experimentally verifiable signatures.

Since the tri-critical Ising model is one of the few experimentally realisable models, this seems a very worthwhile result and very deserving of publication and entirely suitable for publication in SciPost Physics.

The paper is generally very well written and easy to follow.

Having said that, I think the paper needs some minor revisions - there are places where it is unclear, and there are also some minor errors and typos that I have listed in the "requested changes" section.

Apart from the detailed changes, there are a few areas where some more commentary would be very helpful.

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 $\langle 0|\sigma|\{n,-n\}\rangle_{1,2}$ in figure 6.1, $\langle 0|\sigma'|\{n,-n\}\rangle_{1,2}$ for $n>0$ in figure 6.2, $\sqrt{\rho_{12}}|\langle 0|\sigma|\{n,-n\}\rangle|$ for $n>1$ in figure 6.3, $\sqrt{\rho_{12}}|\langle 0|\sigma|\{n,-n\}\rangle|$ for $n>1$ in figure 6.4, $\sqrt{\rho_{11}}|\langle 0|\sigma|\{n,-n\}\rangle|$ for $n>0$ in figure 6.6, $\sqrt{\rho_{11}}|\langle 0|\sigma'|\{n,-n\}\rangle|$ 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.

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?

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.

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.

(As an aside, I believe the only full discussions in which all structure constants can be calculated and consistency shown are in the work of Fröhlich et al which gives a construction from TFT data (in this approach, the fields $\psi(z)$, $\mu(z)$ etc, arise as defect creation fields at the ends of $Z_2$ defect lines, which is a "bosonic" theory in that there is no coupling to the spin-structure of the underlying space) and in the subsequent work of Runkel et al [see arXiv:1506.07547 for the Ising model] of a "fermionic" version which does couple to the spin structure. Whether these are in some way equivalent on the plane or on the cylinder, or indeed in general, has not yet been checked, I think.)

I would like to apologise to the authors for the time taken to submit this report - the pandemic has meant that my administrative load has been much higher than normal and it has taken until the Christmas vacation period for me to find time to write a considered response.

Requested changes

- 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?

-On page 9, equation (2.16) is technically wrong as
\lim_{z\to\infty}\langle0|\sigma(z) =0\;.\]
It would be better to write $\langle\sigma|$ instead of $\langle0|\sigma(\infty)$ where
\[ \langle\sigma| = \lim_{z\to\infty} |z|^{1/4} \langle0|\sigma(z)\;.\]
or define what $\sigma(\infty)$ is.

- On page 11, equation (2.32) looks wrong. It has the wrong dimension - is this instead $F_0^2$? 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.

- On page 11, in equation (2.13), what does $\simeq$ mean? Does this mean that $1/|x|^{1/4}$ is the leading behaviour as $x\to 0$? Why is it not "="?

- 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 $t(z) t(w) \sim (-1/25)|z-w|^{-12/5}$

- 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.

- 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]

- 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.

- 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.

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

- 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".

- 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.

- 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?

- 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).

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

- 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.

- 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 $\bar\psi_0$.

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

- 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.

- 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?

- 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.

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

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2021-11-12 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2109.09767v1, delivered 2021-11-12, doi: 10.21468/SciPost.Report.3837


The authors consider the problem of the determination of form factors in the thermal (E7) integrable deformation of the tricritical Ising model in two dimensions. This model possesses two magnetisation operators (leading and subleading) which are not distinguished by a symmetry and for this reason satisfy the same form factor equations. From this point of view the problem is similar to that solved by Delfino and Simonetti in Phys. Lett. B 383, 450 (1996) for the magnetic (E8) integrable deformation of the Ising model. In that case the magnetisation and energy operators satisfy the same form factor equations because the magnetic field leaves no symmetry. An additional feature of the E7 case is that high-low temperature duality provides another sector containing a pair of disorder operators (duals of the magnetisation operators). The authors then exploit the fact that the clustering property provides a relation between the form factors of magnetisation and disorder operators, as observed by Cardy and Delfino in Nucl. Phys. B 519, 551 (1998) for the Potts model. Having determined the form factors, the authors compare them with the numerical results obtained by the truncated conformal space approach and observe an excellent agreement. Equally successful is the check performed using the sum rule for the conformal weights (Delta theorem). The form factors are then used for the calculation of the dynamical structure factors. The paper also contains introductory sections in which the authors review properties of the tricritical Ising model such as duality, Landau-Ginzburg description, supersymmetry and renormalisation group flows. The paper provides a very interesting application of the form factor bootstrap and I recommend publication in SciPost after inclusion of the two references mentioned above.

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