SciPost Submission Page
Absence of topological order in the $U(1)$ checkerboard toric code
by Maximilian Vieweg , Viktor Kott, Lea Lenke, Andreas Schellenberger, Kai Phillip Schmidt
Submission summary
| Authors (as registered SciPost users): | Andreas Schellenberger · Kai Phillip Schmidt |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202511_00017v1 (pdf) |
| Date submitted: | Nov. 11, 2025, 12:25 p.m. |
| Submitted by: | Kai Phillip Schmidt |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
We investigate the $U(1)$ checkerboard toric code which corresponds to the $U(1)$-symmetry enriched toric code with two distinct star sublattices. One can therefore tune from the limit of isolated stars to the uniform system. The uniform system has been conjectured to possess non-Abelian topological order based on quantum Monte Carlo simulations suggesting a non-trivial ground-state degeneracy depending on the compactification of the finite clusters. Here we show that these non-trivial properties can be naturally explained in the perturbative limit of isolated stars. Indeed, the compactification dependence of the ground-state degeneracy can be traced back to geometric constraints stemming from the plaquette operators. Further, the ground-state degeneracy is fully lifted in fourth-order degenerate perturbation theory giving rise to a non-topological phase with confined fracton excitations. These fractons are confined for small perturbations so that they cannot exist as single low-energy excitation in the thermodynamic limit but only as topologically trivially composite particles. However, the confinement scale is shown to be surprisingly large so that finite-size gaps are extremely small on finite clusters up to the uniform limit which is calculated explicitly by high-order series expansions. Our findings suggest that these gaps were not distinguished from finite-size effects by the recent quantum Monte Carlo simulation in the uniform limit. All our results therefore point towards the absence of topological order in the $U(1)$ checkerboard toric code along the whole parameter axis.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
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Response to Referee 1 (Helene Spring):
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Requested changes:
I have some suggestions for improvement that I would like the authors to consider. * While the paper convincingly shows that the system is gapped and trivial, the magnitude of the gap remains small, even at λ = 1. It would be valuable to include a brief discussion of whether (and how) the gap could be made larger by adding symmetry-allowed perturbations. This might help distinguish the trivial phase more sharply from the finite-size effects that complicate numerical studies. This would also show that numerical studies alone could detect the triviality due to the presence of a gap. * The authors mention that their series expansion results agree with QMC gaps at λ = 1. Including a figure or table comparing these values explicitly could strengthen this point.
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Our response:
We thank the referee for the very positive examination of our work. In particular, for recommending acceptance for SciPost Physics.
Concerning the requested changes:
We thank the referee for the comments. Regarding the spectral gap, we have added a sentence discussing possible perturbations to enlarge the gap in Section 3.1 where we discuss the ground-state manifold and the compactification. Further, regarding the quantitative comparison of the gap calculated by series expansions and QMC, we think it is rather interesting that
the value for our energy gap (series expansion) is already quite close to the one calculated with QMC for the same system size. One does not expect that the value is quantitatively the same, e.g., we evaluated the bare series, which does not contain loop processes coupling different ground states present on finite systems. We have reformulated the corresponding sentence in our article.
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Response to Referee 2:
Report:
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I commend the authors on undertaking a critical analysis of claims made in the literature. I find their solvable lambda=0 limit quite insightful, and it nicely reproduces the observed near-ground-state-degeneracy in Ref 4 and explains that this does not require a non-trivial origin (i.e., it is consistent with a trivial phase of matter, since the authors explain that there is a nonzero gap between these ground states at fourth-order in lambda). While this is a nice consistent explanation, which is considerably simpler than the explanation offered in Ref 4, I found Section 4 (devoted to the lambda -> 1 case) a bit lacking to be entirely conclusive. In addition, while this is a nice careful analysis of a previously-introduced model, and while it is important to rectify unjustified claims in the literature, it is not clear to me that the particular insights gained in this work meet the 'groundbreaking' or 'opening new pathways' criteria listed for SciPost Physics. The manuscript probably does meet the criteria for SciPost Physics Core (although see the comments below). I do not directly see why it meets the criteria for SciPost Physics, but I remain open-minded for a potential rebuttal, based on the below comments. Overall, the content of the manuscript feels a bit minimal, and it would be nice if it went beyond merely rectifying previous work. The current take-away message I have as a reader is "the (checkerboard) U1TC is not so interesting, move on", which is of course solid work but does not evoke the feeling of a groundbreaking piece of work.
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Our response:
We are strongly convinced that our results meet the criteria for SciPost Physics which is also reflected by the very positive evaluation of the first referee.
We do not agree with the assessment that we have only shown the U1TC to be uninteresting.
We have provided compelling arguments that the U1TC is not topologically ordered and have offered an interpretation of its low-energy excitations as confined fractons.
Beyond topological order, the weak Hilbert-space fragmentation observed in the U1TC remains an intriguing feature.
Since the U1TC in the (+,+) sector maps exactly to the Ising model in a small transverse field, our results, and the interpretation of the low-energy excitations
as confined fractons, also apply to that model (see, for example, Refs. [6, 7]). We have expanded the Introduction and Conclusion to clarify this connection.
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The ED discussion is quite brief (roughly a paragraph long) and is restricted to a 4x4 geometry (presumably 442 qubits, which is then reduced using quantum numbers such as U(1) conservation etc). I found certain claims a bit confusing such as "No higher-energy states for small λ = 0 approach the ground state on the full parameter line". If we are only given one fixed system size, it is nearly impossible to tell whether there is a trend of states approaching the ground state. Moreover, this section would benefit from a more detailed comparison to the numerical claims made in Ref 4. For instance, that work commented on the phase being gapped due to having a large gap. Can the authors of the present work compare their gap to the one reported in Ref 4?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our response:
We agree that our ED analysis, being restricted to 4x4 systems, yields limited predictions concerning the low-energy states approaching the ground state.
The purpose of this section is to provide a global picture of the system and to show that, for the 4x4 system size, the conclusion that no phase transition occurs is consistent and in agreement with our other analyses.
Let us stress that Ref. 4 also did ED on 4x4 systems for lambda=1 so that a quantitative comparison between ED and QMC offers no real insight for the isotropic systems. More relevant is the comparison of our series expansion for the gap and the value of QMC in Ref. 4. Here it is interesting that the value for our energy gap (series expansion - obtained for a system of size L) is already quite close to the one calculated with QMC for the same system size. Note that one does not expect that the value is quantitatively the same, e.g., we evaluated the bare series, which does not contain loop processes coupling different ground states present on finite systems. We have reformulated the corresponding sentence in our article.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
On a related note: is there a reason the authors do not use a duality transformation to reduce the number of qubits? By applying the usual Kramers-Wannier transformation, one would obtain a model of qubits on the vertices of the square lattice, where the Hamiltonian would contain terms of the form "X", "ZXZ" and "ZZXZZ" (like the square lattice cluster model), and the U(1) symmetry is now generated by \sum Z_i Z_j. In this representation, the "4x4" lattice would only have 16 qubits, which is easily simulable even without quantum numbers. It thus seems likely one can push to larger system sizes. This might allow the authors to consider more than one system size, such that one can get a sense of the behavior of the gap in the thermodynamic limit et cetera.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our response:
Reducing the Hilbert-space dimension via the Kramers–Wannier transformation may indeed be possible, but it is effectively equivalent to the reduction obtained by exploiting the local symmetries of the B_p operators. For the full 4x4 system the dimension is 2^32. Applying the local symmetries yields 2^17, and incorporating the two independent Wilson loops reduces this further to 2^15,
without invoking additional symmetries (U(1)-symmetry, translation symmetry).
The bottleneck of the ED calculation is the diagonalization of the large matrix rather than the construction of the basis. Consequently, we do not expect that employing the Kramers–Wannier transformation would render ED feasible for larger system sizes. The next interesting size, 6x6, has a Hilbert-space dimension of 2^35 after using all local symmetries and Wilson-loop operators
(or 2^36 if one uses the Kramers–Wannier transformation).
One can further reduce the dimension using translation, mirror, and global U(1) symmetries, but this is still insufficient to bring the problem to a manageable scale (for 6x6 or larger). Moreover, Ref. 4 did not reach larger system sizes using ED, and Ref. 7 explicitly states that their exact enumeration algorithm is applicable to 6x6 systems because they only construct the matrix and do not attempt to diagonalize it.
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The authors perhaps over-emphasize the non-abelian claims of Ref 4. If I take a cursory glance at Ref 4, it does not even mention "non-abelian" in the abstract, and the word only appears a total of three times in their manuscript. The first mention is on p6 of their published manuscript: "These arguments suggest a logical possibility that the U(1) toric code may realize non-Abelian topological order". This feels like a pretty mild claim, and I do not think that the non-abelian nature is really an explicit or main conjecture, it is merely offered as a logical explanation. More importantly, Ref 4 does clearly claim topological ground state degeneracy, so that feels like a safer point to argue against.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our response:
We agree that the authors of Ref. 4 only claim to present strong evidence for topological order and discussed the possibility of non-Abelian anyons to account for the three-fold ground-state degeneracy.
We have accordingly revised our statements in the Abstract, Introduction, and Chapter 2.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Response to Referee 1 (Helene Spring):
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Requested changes:
I have some suggestions for improvement that I would like the authors to consider. * While the paper convincingly shows that the system is gapped and trivial, the magnitude of the gap remains small, even at λ = 1. It would be valuable to include a brief discussion of whether (and how) the gap could be made larger by adding symmetry-allowed perturbations. This might help distinguish the trivial phase more sharply from the finite-size effects that complicate numerical studies. This would also show that numerical studies alone could detect the triviality due to the presence of a gap. * The authors mention that their series expansion results agree with QMC gaps at λ = 1. Including a figure or table comparing these values explicitly could strengthen this point.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our response:
We thank the referee for the very positive examination of our work. In particular, for recommending acceptance for SciPost Physics.
Concerning the requested changes:
We thank the referee for the comments. Regarding the spectral gap, we have added a sentence discussing possible perturbations to enlarge the gap in Section 3.1 where we discuss the ground-state manifold and the compactification. Further, regarding the quantitative comparison of the gap calculated by series expansions and QMC, we think it is rather interesting that
the value for our energy gap (series expansion) is already quite close to the one calculated with QMC for the same system size. One does not expect that the value is quantitatively the same, e.g., we evaluated the bare series, which does not contain loop processes coupling different ground states present on finite systems. We have reformulated the corresponding sentence in our article.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Response to Referee 2:
Report:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I commend the authors on undertaking a critical analysis of claims made in the literature. I find their solvable lambda=0 limit quite insightful, and it nicely reproduces the observed near-ground-state-degeneracy in Ref 4 and explains that this does not require a non-trivial origin (i.e., it is consistent with a trivial phase of matter, since the authors explain that there is a nonzero gap between these ground states at fourth-order in lambda). While this is a nice consistent explanation, which is considerably simpler than the explanation offered in Ref 4, I found Section 4 (devoted to the lambda -> 1 case) a bit lacking to be entirely conclusive. In addition, while this is a nice careful analysis of a previously-introduced model, and while it is important to rectify unjustified claims in the literature, it is not clear to me that the particular insights gained in this work meet the 'groundbreaking' or 'opening new pathways' criteria listed for SciPost Physics. The manuscript probably does meet the criteria for SciPost Physics Core (although see the comments below). I do not directly see why it meets the criteria for SciPost Physics, but I remain open-minded for a potential rebuttal, based on the below comments. Overall, the content of the manuscript feels a bit minimal, and it would be nice if it went beyond merely rectifying previous work. The current take-away message I have as a reader is "the (checkerboard) U1TC is not so interesting, move on", which is of course solid work but does not evoke the feeling of a groundbreaking piece of work.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our response:
We are strongly convinced that our results meet the criteria for SciPost Physics which is also reflected by the very positive evaluation of the first referee.
We do not agree with the assessment that we have only shown the U1TC to be uninteresting.
We have provided compelling arguments that the U1TC is not topologically ordered and have offered an interpretation of its low-energy excitations as confined fractons.
Beyond topological order, the weak Hilbert-space fragmentation observed in the U1TC remains an intriguing feature.
Since the U1TC in the (+,+) sector maps exactly to the Ising model in a small transverse field, our results, and the interpretation of the low-energy excitations
as confined fractons, also apply to that model (see, for example, Refs. [6, 7]). We have expanded the Introduction and Conclusion to clarify this connection.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The ED discussion is quite brief (roughly a paragraph long) and is restricted to a 4x4 geometry (presumably 442 qubits, which is then reduced using quantum numbers such as U(1) conservation etc). I found certain claims a bit confusing such as "No higher-energy states for small λ = 0 approach the ground state on the full parameter line". If we are only given one fixed system size, it is nearly impossible to tell whether there is a trend of states approaching the ground state. Moreover, this section would benefit from a more detailed comparison to the numerical claims made in Ref 4. For instance, that work commented on the phase being gapped due to having a large gap. Can the authors of the present work compare their gap to the one reported in Ref 4?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our response:
We agree that our ED analysis, being restricted to 4x4 systems, yields limited predictions concerning the low-energy states approaching the ground state.
The purpose of this section is to provide a global picture of the system and to show that, for the 4x4 system size, the conclusion that no phase transition occurs is consistent and in agreement with our other analyses.
Let us stress that Ref. 4 also did ED on 4x4 systems for lambda=1 so that a quantitative comparison between ED and QMC offers no real insight for the isotropic systems. More relevant is the comparison of our series expansion for the gap and the value of QMC in Ref. 4. Here it is interesting that the value for our energy gap (series expansion - obtained for a system of size L) is already quite close to the one calculated with QMC for the same system size. Note that one does not expect that the value is quantitatively the same, e.g., we evaluated the bare series, which does not contain loop processes coupling different ground states present on finite systems. We have reformulated the corresponding sentence in our article.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
On a related note: is there a reason the authors do not use a duality transformation to reduce the number of qubits? By applying the usual Kramers-Wannier transformation, one would obtain a model of qubits on the vertices of the square lattice, where the Hamiltonian would contain terms of the form "X", "ZXZ" and "ZZXZZ" (like the square lattice cluster model), and the U(1) symmetry is now generated by \sum Z_i Z_j. In this representation, the "4x4" lattice would only have 16 qubits, which is easily simulable even without quantum numbers. It thus seems likely one can push to larger system sizes. This might allow the authors to consider more than one system size, such that one can get a sense of the behavior of the gap in the thermodynamic limit et cetera.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our response:
Reducing the Hilbert-space dimension via the Kramers–Wannier transformation may indeed be possible, but it is effectively equivalent to the reduction obtained by exploiting the local symmetries of the B_p operators. For the full 4x4 system the dimension is 2^32. Applying the local symmetries yields 2^17, and incorporating the two independent Wilson loops reduces this further to 2^15,
without invoking additional symmetries (U(1)-symmetry, translation symmetry).
The bottleneck of the ED calculation is the diagonalization of the large matrix rather than the construction of the basis. Consequently, we do not expect that employing the Kramers–Wannier transformation would render ED feasible for larger system sizes. The next interesting size, 6x6, has a Hilbert-space dimension of 2^35 after using all local symmetries and Wilson-loop operators
(or 2^36 if one uses the Kramers–Wannier transformation).
One can further reduce the dimension using translation, mirror, and global U(1) symmetries, but this is still insufficient to bring the problem to a manageable scale (for 6x6 or larger). Moreover, Ref. 4 did not reach larger system sizes using ED, and Ref. 7 explicitly states that their exact enumeration algorithm is applicable to 6x6 systems because they only construct the matrix and do not attempt to diagonalize it.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The authors perhaps over-emphasize the non-abelian claims of Ref 4. If I take a cursory glance at Ref 4, it does not even mention "non-abelian" in the abstract, and the word only appears a total of three times in their manuscript. The first mention is on p6 of their published manuscript: "These arguments suggest a logical possibility that the U(1) toric code may realize non-Abelian topological order". This feels like a pretty mild claim, and I do not think that the non-abelian nature is really an explicit or main conjecture, it is merely offered as a logical explanation. More importantly, Ref 4 does clearly claim topological ground state degeneracy, so that feels like a safer point to argue against.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our response:
We agree that the authors of Ref. 4 only claim to present strong evidence for topological order and discussed the possibility of non-Abelian anyons to account for the three-fold ground-state degeneracy.
We have accordingly revised our statements in the Abstract, Introduction, and Chapter 2.
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List of changes
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List of Changes
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Abstract (Page 1): Removed the reference to non-Abelian topological order.
Introduction (Page 2, Paragraph 2): Rephrased the statement regarding topological order and briefly expanded the discussion of Hilbert-space fragmentation.
Chapter 2 (Page 4, Paragraph 3): Removed the 'non-Abelian' from 'non-Abelian topologically ordered ground state'.
Section 3.1 (Page 7, Paragraph 1): Added a note on possible perturbations to enlarge the gap.
Chapter 4 (Page 10, Paragraph 3): Expanded the discussion of the perturbative results and their comparison with quantum Monte Carlo (QMC) data.
Conclusion (Page 11, Paragraph 2): Extended the discussion of the relevance of the results to other models.
List of Changes
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Abstract (Page 1): Removed the reference to non-Abelian topological order.
Introduction (Page 2, Paragraph 2): Rephrased the statement regarding topological order and briefly expanded the discussion of Hilbert-space fragmentation.
Chapter 2 (Page 4, Paragraph 3): Removed the 'non-Abelian' from 'non-Abelian topologically ordered ground state'.
Section 3.1 (Page 7, Paragraph 1): Added a note on possible perturbations to enlarge the gap.
Chapter 4 (Page 10, Paragraph 3): Expanded the discussion of the perturbative results and their comparison with quantum Monte Carlo (QMC) data.
Conclusion (Page 11, Paragraph 2): Extended the discussion of the relevance of the results to other models.
Current status:
In refereeing
Reports on this Submission
Report #1 by Helene Spring (Referee 1) on 2025-11-11 (Invited Report)
Report
I thank the authors for addressing my suggestions and I am satisfied with the outcome. I also thank the second referee for raising interesting points which I had not considered. Following the ensuing author response and after reviewing the revised manuscript, I stand by my original appraisal and recommend the manuscript for publication in Scipost Physics.
Recommendation
Publish (meets expectations and criteria for this Journal)