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Fermionization and boundary states in 1+1 dimensions

by Yoshiki Fukusumi, Yuji Tachikawa, Yunqin Zheng

This is not the current version.

Submission summary

As Contributors: Yunqin Zheng
Arxiv Link: (pdf)
Date submitted: 2021-03-19 13:16
Submitted by: Zheng, Yunqin
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


In the last few years it was realized that every fermionic theory in 1+1 dimensions is a generalized Jordan-Wigner transform of a bosonic theory with a non-anomalous $\mathbb{Z}_2$ symmetry. In this note we determine how the boundary states are mapped under this correspondence. We also interpret this mapping as the fusion of the original boundary with the fermionization interface.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2021-6-11 (Invited Report)


1- The authors explain how to use a Z2-orbifolding procedure to move between different bosonic and fermionic models, and they apply these ideas to boundary conditions

2- The paper makes clear that the four models called A,D,F,F' there are closely related also on the level boundary conditions by including twisted sectors in the bosonic models.


1- In giving the fermionic boundary states, it could be made clearer what constitutes a boundary condition, and why some solutions to consistency conditions are preferred over others.


Overall, I think this is a helpful paper with new and relevant results, which will give rise to further research into fermionic conformal field theories.

Since when it came to writing this report, the first reviewer had already submitted a very detailed report, I can be brief here: I agree with and support the comments made there. Accordingly, I will recommend publication, once the points raised in that report have been addressed.

Requested changes

In addition to the first report, I would like to comment on the sentence "It should be noted that the only difference ..." below (1.7). I find this sentence slightly misleading: the statement is true on the level of partition functions, but not on the level of structure constants, which will be different (by signs which are not normalisations) between F and F'. If the authors agree, maybe the formulation can be changed.

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

Author:  Yunqin Zheng  on 2021-07-14

(in reply to Report 2 on 2021-06-11)
answer to question

Dear referee,

Thank you very much for the comments and suggestions. We will address the questions, and we have also revised the manuscript accordingly. Please find our responses below.

In the introduction section, we added more discussions of the definition of boundary conditions and boundary states for the fermionic theory. We also addressed the comments from the first report.

Summary of changes

  1. We deleted this sentence in the revised draft.

Sincerely, Yoshiki Fukusumi, Yuji Tachikawa, Yunqin Zheng

Report 1 by Gerard Watts on 2021-5-2 (Invited Report)


1- This paper gives a clear summary of relations between bulk theories and results for their boundary conditions which were only partially known in the literature

2- There are ample examples provided


1- There are at least two ways to define a boundary condition for a fermionic theory, this needs to be clarified.

2- The main result seems likely to be correct (for one interpretation of what a boundary condition is) but the argument is not clear.

2- There are some technical details which should be cleared up


I think that this paper certainly meets the principal criterion 3, that it is very likely to provoke new research on conformal field theories and their boundary conditions through its investigation of the boundary conditions of related theories, and the posing of possible new avenues to pursue.

However, I have several questions, of which the first is just that the authors do not give a definition of a boundary condition. One might think this is unnecessary, but since there are (I think) at least two different results for the boundary conditions of a free fermion, I think it would be helpful.

Cardy's constraint is definitely a constraint on allowed boundary conditions, but (to my mind) the principal condition is that a boundary condition enables one to calculate all correlation functions of bulk fields. For that, one needs at least all bulk field 1-point functions, or equivalently, the couplings of bulk spinless fields to weight zero boundary fields. These are constrained by sewing equations, and can be reformulated as a "Classifying algebra" that determines the one-point functions of bulk fields and hence the boundary conditions. As a case in point, take the free fermion. This has bulk fields 1, $\psi$, $\bar\psi$, $\epsilon$ in the NS sector and $\sigma^\pm$ in the R sector. Consequently there are three bulk, spinless, bosonic fields, with the same bulk-boundary OPEs as in the Ising model, and so there are three boundary conditions for the free fermion, two of which differ only by the coupling of the bosonic spin field to the identity field on the boundary and hence only differ by the sign of the Ramond sector boundary state. If one considers only the sewing relations for the fermion field, these two appear identical, leading to the statement that there are "two boundary conditions for the Majorana fermion". This is true, if one only considers the fermion field; including the other sectors leads to three boundary conditions as we found in reference [9].

If in the context of this paper you include the option of both signs in the bosonic R-sector then the $F'$ theory has exactly the same number and labelling of boundary states as the $A$ theory, and the $F$ theory as the $D$ theory. This is to be expected since the bosonic subsector of the $F'$ theory, $S+T$, is the untwisted sector of the $A$ theory, and $S+U$ is the bosonic subsector of $F$ and the untwisted sector of $D$, and I think the boundary conditions are classified just by these parts of the respective theories (since it is only fields that can couple to the identity on the boundary that play a role).

So my question is: what do the authors think a boundary condition is, and do they think the one-point functions of bulk bosonic Ramond sector fields are a property of a boundary condition or not. If not, then their answers look plausible; if not, then they are not finding all boundary conditions but only up to equivalence, and this explains the disparity between the number of boundary conditions of the bosonic and fermionic theories.

There is also, it seems, a problem, either with the argument in the paper or in its presentation, which also seems to me due to the reliance on Cardy's constraint. The main result, the identification of the boundary states of the F and F' theories, but moving from the relations (2.32) to (2.41), requires a choice for the normalisation constants $P_i$ for which the only stated constraint is Cardy's condition. While the choice (2.40) works, so, it seems to me, does $P_1=P_3=\sqrt 2, P_2=P_4=1/\sqrt 2$, which would lead to the boundary states in (2.42) [which naturally also satisfy Cardy's condition]. As such, I do not see that the authors can say that they have found (2.41) to be true. Did I miss something, can one really decide between (2.41) and (2.42) as the correct boundary states for $F$ based on some other information in the paper? I am sorry if it is there, I could not see it and it is not pointed out in the text. I would have thought that consideration of the bosonic subsectors of $F$ and $F'$ and their implications for the boundary conditions would make an identification possible, even easy, but it would also make it likely that the $|i\rangle^F$ and $|a\rangle^{F'}$ boundary conditions would be doubled (with the boundary states differing just by signs in the R sector) thereby returning the boundary conditions of the $F$ theory to be in 1-1 correspondence with those of the $D$ theory and those of the $F'$ theory to be in 1-1 correspondence with those of the $A$ theory. I think this problem, whether the $P_i$ are actually determined or not needs to be fixed before I can recommend publication.

I should say that the discussion at the start of 2.3 is very helpful and the explanation of the "cond-mat" convention is good to see.

I found the discussion of the free-fermion had some problems. The boundary states in equation (3.21) looked very odd as I would have thought that the states $|1/16\rangle$ were in the R sector and $|0\rangle, |1/2\rangle$ in the NS. This makes the labelling in (3.21) look wrong to me. The constants also look wrong, I would have expected $2^(1/4)$, but maybe I just misunderstood completely what is going on.

On page 3, the authors say "It should be noted that the only difference of the theories $F$ and$ F'$ are in the assignment of the fermionic parity on the R-sector." This is not true, the theories are intrinsically different - they will typically different structure constants for example - it is just that the bulk field content when viewed forgetting fermion parity is the same.

At the top of page 4, it is very confusing when the authors refer to "the resulting boundary states as $|i\rangle^F$.." and straight after say "The corresponding boundary states.. $|i\rangle^F_{NS}$ etc". I think the first statement should refer to the "boundary conditions" not the "boundary states". I think they mean that the boundary conditions are [my notation!] $(i)^F$, $(a\psi)^F$ and each of these boundary conditions has an NS and R boundary state. If they instead mean $|i\rangle^F = |i\rangle^F_{NS} + |i\rangle^F_R$, or something else, please can they make it clear?

Finally, I find it unhelpful that the results in (1.9) and (1.10) are not strictly true, or at least they are not the results this paper finds. The $F$ and $F'$ theories involve a tensor product with "Kitaev" which encodes the fermionic nature, and so (1.9) and (1.10) are wrong while (2.41) and (2.42) are [very probably] right, up to the question of R-sector signs. I think it is important to get this right, the bulk Hilbert space of $F$ is not that of $A$, it differs in some important respects.

Requested changes

1- Please make clear what distinguishes different boundary conditions.

2- Please make clear how the choice $P_i=1$ is made. Can it be decided or is it a guess? If there is a good reason, this needs to be clear to be able to recommend publication. If not, then I suspect the rest of the paper would need to be substantially re-written to accommodate this ambiguity.

3- I think there is a mistake in the text before eqn (2.29) - rather than "the only solution.. to (2.31) is..", should it be "the only solution.. to (2.28) is.."

4- Make clear on page 4 the difference between boundary conditions and boundary states

5- Either change (1.9) and (1.10) to be (2.41) and (2.42), or make it clear that the dependence on Kitaev is suppressed and the correct formulae are actually (2.41) and (2.42)

6- Please check R- and NS- assignments and the constants in the free-fermion discussion.

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

Author:  Yunqin Zheng  on 2021-07-14

(in reply to Report 1 by Gerard Watts on 2021-05-02)
answer to question

Dear Prof. Watts,

Thank you very much for the insightful comments. We will address the questions, and we have also revised the manuscript accordingly. Please find our responses below.

Response to the questions from the 5th paragraph:
We did not use the one-point function as the defining property of the boundary condition. Our fermionization map only produces 2 Cardy states rather than 3, as expected from Referee's comments above. However, in our construction, we have 2 more twisted states, while in the construction in ref [9], there is 1 twisted states. The total number of boundary states (untwisted $+$ twisted) is $3+1=2+2=4$. This difference comes from the fact that in our work, we demand that the fermionic boundary states are either in the R sector or NS sector, and we don't allow the superposition between the two, because they live in different Hilbert spaces. However, the boundary states in ref [9] are superpositions of the NS sector and R sector Ishibashi states, hence they don't live in a definite NS or R Hilbert space. As the referee may be considering, there exists the modern formulation of BCFT by considering correlation functions and OPE as the building block of the theory. This formulation may enable one to calculate more general objects in BCFT. However, our formulation is sufficient to construct the fermionic NS and R boundary states and their partition functions on a cyclinder geometry. We have added more discussions on the definition of boundary conditions and boundary states in the introduction.

Response to the questions from the 6th paragraph:
We added another supporting argument to rule out $P_1=P_3=\sqrt{2}, P_2=P_4=1/\sqrt{2}$ in theory $\mathsf{F}$, please see the discussion below (2.40) and (2.41) in the revised manuscript. The idea is to consider a completely trivial theory for $\mathsf{A}$. It should also be noted that our construction is consistent with appearance of paired or unpaired Majorana fermion at the interface in section 4 and reproduces triviality or nontriviality of the theories.

Response to the questions from the 8th paragraph:
It is related to the first response above. The factor $2^{1/4}$ appears if we allow superposition of boundary states with NS and R spin structures in the closed spatial cycle. However, in the present work, we think it is more appropriate to keep NS and R states separate, and don't allow the superposition because NS and R states live in different Hilbert spaces.

Response to the questions from the 9th paragraph:
We deleted this sentence.

Response to the questions from the 10th paragraph:
The boundary conditions are $(i)^\mathsf{F}$ and $(a\psi)^\mathsf{F}$. For each boundary condition, we can have either NS or R spin structure along the time cycle. After rotating the spacetime by 90 degree, we denote the boundary states as $|i\rangle_{NS}^F$, and $|a\psi\rangle_{NS}^{\mathsf{F}}$ if the original boundary conditions are equipped with NS spin structure, and $|i\rangle_{R}^F$ and $|a\psi\rangle_{R}^{\mathsf{F}}$ if the original boundary conditions are equipped with R spin structure.

Response to the questions from the 11th paragraph:
The tensor product factor is simply to emphasize the fermion parity of the state. (This tensor product factor does not have to be identified with the "Kitaev" chain.) Below (1.11) and (1.12) we have clarified the fermion parity of the states, hence there should be no ambiguity. We suppressed the tensor product factor in the introduction because we want to present (1.10), (1.11), (1.12) in the same fasion, where (1.10) does not contain the "tensor product factor".

Summary of Changes:

1.We distinguish the boundary conditions by their symmetry quantum numbers. We summarized this in the introduction. For bosonic theory with $\Z_2$ global symmetry, we classify the boundary condition by $\Z_2$ invariant and $\Z_2$ breaking ones. In addition, we also have $\Z_2$ twisted boundary condition for those are $\Z_2$ invariant. For the fermionic theory, we classify the boundary condition by their fermion parity $(-1)^F$, and for each fermion parity. In addition, we also have NS and R boundary conditions specified by whether fermions along the time cycle is antiperiodic (NS) or periodic (R).

2. we have added a paragraph below (2.40) and (2.41), to justify why $P_i=1$ is made, and not the other choice.

3.fixed, thank you for pointing out.

4.  We have added a related discussion in the introduction section, and emphasized the \emph{spin Cardy condition} for the fermionic theories vs the Cardy condition for bosonic theories.

5. The $\ket{\pm}$ should not be regarded as the Kitaev chain, which seems to confuse the reader and the referee. We added footnote 5 to clarify this.

6. This requirement is related to the previous comments. We have revised the presentation to make the point clearer.

Sincerely, Yoshiki Fukusumi, Yuji Tachikawa, Yunqin Zheng



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