Boundary Criticality of Complex Conformal Field Theory: A Case Study in the Non-Hermitian 5-State Potts Model

1-First numerical study of boundary conditions in a complex CFT

1-Does not connect to existing results about boundary conditions in non-unitary CFT.
2-Lacks precision about representation-theoretical aspects.

It was recently argued that conformally invariant critical behaviour
can exist in the Potts model with $Q > 4$ states, provided the
coupling constant of the field theory (Coulomb gas) is analytically
continued to complex values. This prediction was followed by two
numerical studies of specific models, where this scenario is
realised. The first one (ref. [25]) studied a loop model on the
triangular lattice, for which the loop fugacity can encode any complex
value of Q. The second one (ref. [26]) studied a non-hermitian 1+1D
spin chain, with integer $Q=5$. In both cases it was found that CFT
properties such as the central charge and bulk critical exponents are
in accord with the scenario of analytic continuation.

The present manuscript continues the study of the $Q=5$ spin chain,
but now focusing on boundary critical behaviour. The results indicate
that also the boundary-CFT properties are in agreement with analytic
continuation. Unfortunately, in a number of cases the authors
themselves seem unable to fully reach this conclusion, as they appear
to be ignorant of a large body of existing results for non-unitary
models that were previously studied for real values of $Q < 4$. This
flaws the manuscript in two ways. First, it leads in a number of cases
the authors to claim novelty for things which are in fact well
known. Second, it often prevents the authors from reaching clear
conclusions, even when they could have done so.

The problem becomes conspicuous already in the introduction, where the
authors state that "the investigation of irrational cases [of boundary
CFT] is difficult (...), [with results being] limited to numerical
calculations". Fortunately this is not so. The loop models of Potts
and $O(n)$ type, which beautifully realise the irrational CFT with
real $c < 1$, and which are closely related to the Coulomb-gas
approach in the bulk case, have been thoroughly studied also in the
boundary case. For instance, the study of a set of boundary conditions
that amount to allowing a lesser number of states, $Q_1 < Q$, for the
spins along one boundary was initiated by Jacobsen and Saleur in
Nucl. Phys. B 788, 137 (2008), then further studied and extended by
the same authors and their collaborators in a long series of papers
over the following years. These boundary conditions contain all those
being studied in this manuscript as special cases.

Section 2 contains a review of some well-known features of bulk and
boundary CFT. The focus is to a large extent on minimal models. The
authors present, misleadingly in view of the comments made above, the
complex CFT studied here as the first example of non-unitary CFT. But
in fact, most features of complex CFT are already present in the
non-unitary CFT with $Q < 4$. Other details need correction, too. After
eq. (5) it is stated that the "formalism was originally proposed for
integer $Q \le 4$", whereas it is in fact for integer $m$, viz. for
the minimal models. After eq. (6) the "higher represetations" alluded
to are more precisely representations of the global symmetry
$S_Q$. The discussion about Cardy conditions and Ishibashi states
applies, as stated, only to rational CFT, but this is not
mentioned. Towards the end of section 2, the statement that there are
"little [few] known results for irrational BCFT" is again despairing.

In section 3 the spin chain being studied is defined. It would have
been nice to know more about the boundary fields, in particular how
should they be chosen in order to respect quantum-group symmetry?
Shortly after eq. (25) the authors suddenly mention that the free and
fixed boundary conditions that they study are "known as blob
bcs". This terminology is derived from the literature related to the
Jacobsen-Saleur paper cited above, via an earlier paper by Martin and Saleur, Lett. Math. Phys. 30, 189 (1994), so it is surprising that this is
brought up without the proper context (only a number of much later and
tangentially related numerical papers are cited).

In section 4 the authors finally arrive at their numerical
results. The authors should have stated up to which size the
computations have been made (judging from the figures it is
$L=11$). Some of the representation-theoretical statements should be
made more precise, for instance in the caption to figure 3 the "lowest
$S_5$ vector operator" is presumably the one associated with the Young
diagram $(4,1)$, but it is not clear what is meant by "the lowest
higher-representation field".

In the discussion about free-free boundary conditions (section 4.2.1),
it is not clear what is meant by "the non-unitary nature of the hidden
fixed points". The first displayed formula in that section gives the
decomposition of the annulus partition function in terms of
characters, with coefficients $1, 4, 11$. It is a standard exercise of
Temperley-Lieb representation theory that these coefficients are the
even-order Chebyshev polynomials of the second kind, of which the
first are $1$, $Q-1 = 4$, $Q^2 - 3Q + 1 = 11$ indeed. This can be
found in Saleur and Bauer, Nucl. Phys. B 320, 591 (1989).

In the next section about fixed boundary conditions (section 4.2.2)
the results can also be explained in terms of existing analytical
results. For instance, the fact that the free/fixed boundary condition
changing operator is $\phi_{2,1}$ was established by Cardy in
J. Phys. A 25, L201 (1992) and put in a more general setting in
section 3.2 of the paper by Jacobsen and Saleur cited above. Moreover,
the expansion coefficients of $Z_{A,A}$ in eq. (31) can be inferred
from eq. (3.19) in Jacobsen and Saleur, J. Stat. Mech. (2008) P01021
(take the sum of the two equations). The first non-trivial coefficient
is $Q - 2 = 3$ as observed in eq. (31), and the next coefficient (not
provided there) should be $Q^2 - 4Q + 2 = 7$. The remaining results
can be established from the two-boundary Temperley-Lieb model, which
was further studied e.g. in Dubail, Jacobsen and Saleur, Nucl. Phys. B
813, 430 (2009).

The following section on free/fixed-mixed boundary conditions (section
4.2.3) contains a number of vague (and hence rather incomprehensible)
statements. For instance, what is meant by "it is difficult to align
each multiplet with corresponding Virasoro characters"? Same thing
about the mentioning of the Kac indices being allowed to take
fractional values: it is not written what those fractional values
are. It seems that the authors' findings about fixed/mixed boundary
conditions are inconclusive, but results are available in the papers
on the two-boundary TL model cited above. A final vague statement is
that "the symmetry of most configurations is much smaller than...".

In section 4.3 about duality, it should be mentioned that the sum over
fixed boundary conditions is usually called wired boundary
conditions. There is a global factor of $Q$ missing in the displayed
formula. It is wrong to state that there is a duality between free and
fixed boundary conditions: the correct statement is that there is a
duality between free and wired. And the final claim that "here for the
first time we consider such duality relations" is at best misleading:
all of these results are well understood in the context of non-unitary
models with real values of Q (see the papers cited above), and so the
extension to complex $Q$ is hardly surprising.

In the same vein, the concluding section 5 is too optimistic about
what the authors have achieved and oblivious about the existing
literature.

In conclusion, this manuscript most definitely cannot be accepted in
its present form, although it does open the interesting subject of
complex boundary CFT. It is possible that a thorough revision could be
reconsidered for publication in SciPost Core, but in my opinion not in
the flagship journal SciPost Phys.

See main report.

Ask for major revision

In this paper, the authors study the boundary critical phenomena in the quantum non-Hermitian 5-state Potts chain, which is described in terms of a complex Conformal Field Theory (CFT). The complex CFT is a novel area of study and of significant current interest especially in its relevance for the pseudo-critical behavior of the standard Hermitian 5-state Potts chain (equivalent to the standard 2-dimensional 5-state Potts model in classical statistical mechanics).
The boundary "complex CFT" is an unexplored area and this paper is a significant first step into that direction. The combination of numerical and theoretical analyses lead to beautiful results extending the known boundary (non-complex) CFTs.
This is an important piece of work which I can strongly recommend for publication in SciPost.
However, I would like to ask the authors to revise the paper on the following points before publication.
- In studying the non-free boundary conditions, the authors introduce the boundary fields $h_{L,R}$. However, the description "In all cases, we take $h_L$ and $h_R$ to be a large number to ensure the long-wavelength limit of this model could flow to corresponding boundary fixed points." in the paper is rather vague. For the reproducibility of the numerical results, the authors should provide more details (actual values of $h_{L,R}$ used in the actual calculations. You can send $h_{L,R}$ to (positive) infinity, so that the boundary spins are retricted to the subset of 5 states --- this is used in many other works and maybe also in this paper but please clarify as much as practically possible. Of course, changing the values of $h_{L,R}$ to observe the boundary Renormalization Group flow would be interesting, but perhaps it goes beyond the scope of the paper.
- In any case, if $h_{L,R}$ are negative you will see a different boundary condition, so perhaps it should be declared that $h_{L,R} \geq 0$.
- I find it unfortunate that the same symbol $h$ is used for the conformal dimension $h_{r,s}$, the bulk field $h$, and the boundary fields $h_{L,R}$. Most of the time it is obvious from the context, but it is still somewhat confusing (e.g. when you say $\mathrm{Re}{(h)} \leq 5$.) Maybe you can consider using different symbols.
- The analysis of the scaling dimensions through the rescaled gap as in Eq. (27) is reasonable. The "numerical" value of 2 in the second row is exact because of the way of the analysis; this is an obvious consequence of Eq. (27) but it would still be nice to comment that in Table 2. (also applies Table 3).
- Table 3: "the whole spectrum is shifted by the lowest scaling dimension within each b.c."--- it would be nice to describe more details explicitly (although I can guess roughly what was done).
- By comparing the actual finite-size gap and the expected conformal dimension, one can estimate the "spin-wave velocity" for the critical non-Hermitian 5-state Potts chain. While its value is not relevant for the main objective of the present study, it would be nice to provide the value as a reference.
- I agree with Referee 1 that the English of this paper should be improved (especially since the content is great!). Just for example, in page 3,
"the stability under RG flows is not aware" --- "aware" is used for example "he is not aware of the fact that..." and "stability" cannot be aware of anything. (Perhaps you can say "... is not known".)
"If exist,..." perhaps it should be "If they exist,..."
There are many more issues throughout the paper. While the paper is largely comprehensible, there is much room for improvement for better readability.

Ask for minor revision

This paper using exact diagonalization to study a Hamiltonian associated with the 5-state Potts model, which is non-Hermitian. The results for the spectrum of the theory match exact predictions for various boundary conditions, and confirm these previous predictions that the fixed point is a complex CFT.

The paper seems correct and a nice addition to the literature. My main issue is that there are many grammatical mistakes throughout the paper, which make it somewhat difficult to read. I would advice the authors to use ChatGPT or something similar to proofread the text for grammar, before resubmitting.

Ask for minor revision