Characteristic Properties of a Composite System of Topological Phases Separated by Gapped Domain Walls via an Exactly Solvable Hamiltonian Model

We would like to extend our deep gratitude to the referee for taking the time to carefully review our manuscript, "Characteristic Properties of a Composite System of Topological Phases Separated by Gapped Domain Walls via an Exactly Solvable Hamiltonian Model." We appreciate the detailed feedback and the valuable insights, especially the thought-provoking question about anyon condensations. We recognize the importance of these queries and are eager to address them in our revised manuscript.

(1) Is the intermediate theory (a unitary fusion category) describing the domain wall $\mathbb{Z}_2×\text{Ising}$?

Yes. This keen observation is correct. The algebra describing the domain wall is $\mathbb{Z}_2×\text{Ising}$, where $\mathbb{Z}_2$ is a group and $\text{Ising}$ is a fusion algebra. Based on the definitions of charge and flux observables presented in Ref.20[PRB 97.19 (2018): 195154], the $\text{Ising}$ component accounts for the "charge" of the DW quasiparticles, whereas the $\mathbb{Z}_2$ part characterizes their "flux":
$$\begin{array}{|cc|c|c|}\hline
& \text{Charge} & 1 & \sigma & \psi \\
\text{Flux} &&&\\ \hline
1 && 1 & \chi & e \\ \hline
m && m & \bar\chi & \epsilon \\ \hline
\end{array}$$
This phenomenon seems to recur in a broad class of gapped domain walls. For example, the boundary of a $\mathbb{Z}_2$ toric code model can be described by $\mathbb{Z}_2\times E$, where $E$ is the trivial fusion algebra/trivial group describing the vacuum. Nevertheless, the situation gets intricate when discussing the boundaries of the non-abelian quantum-doubled theory, e.g., $D(S_3 )$[Comm. in math. Phys. 306 (2011): 663-694, PRL 123.5 (2019): 051602.], suggesting the need for further study. Thus, we have chosen not to include this less-understood material in this article.

(2) About the ground-state degeneracy of the composite system on a torus, when using the vertical loop, one could think of having it entirely in DI or TC rather than in DW. And then one would naively expect that the groundstate degeneracy would not be the same. Could you explain this apparent paradox?

Thanks for the insightful question about the ground-state degeneracy of our composite system on a torus. The key is that GSD on a torus equals the number of globally invariant topological sectors of the system. In a single topological phase, the globally invariant topological sectors are the anyon species in this phase. Nevertheless, in our composite system, neither DI anyon species nor TC anyon species are globally invariant topological sectors.

As discussed in Section 6 of our paper, the ground-state Hilbert space can be generated by a complete set of vertical loop operators. Each vertical loop operator is labeled by a quasiparticle species and generates a ground state basis state. TC vertical loop operators are incomplete in generating all ground states because of the confinement of $\chi$ and $\bar\chi$. The vertical loop operators labeled by $\chi$ and $\bar\chi$ commute with all TC loop operators, thereby generating ground states orthogonal to all those generated by the TC loop operators. Meanwhile, DI loop operators are overcomplete because two DI loop operators labeled by distinct DI anyon species could be equal due to identification:
$$W_{V_\text{DI}}^{1\bar 1} = W_{V_\text{DW}}^{1} = W_{V_\text{DI}}^{\psi\bar\psi},\qquad\qquad
W_{V_\text{DI}}^{\psi\bar 1} = W_{V_\text{DW}}^{\epsilon} = W_{V_\text{DI}}^{1\bar\psi},\\
W_{V_\text{DI}}^{\sigma\bar 1} = W_{V_\text{DW}}^\chi = W_{V_\text{DI}}^{\sigma\bar\psi},\qquad\qquad
W_{V_\text{DI}}^{1\bar\sigma} = W_{V_\text{DW}}^{\bar\chi} = W_{V_\text{DI}}^{\psi\bar\sigma},$$
where $V_\text{DI}$ is a non-contractible vertical loop in DI domain, and $V_\text{DW}$ is the loop along the domain wall.

On the contrary, all quasiparticles on the torus can move into the gapped domain wall. When DW vertical loop operators deform and finally return to their original position on the domain wall, the DW quasiparticle species labeling the loop operators remain unchanged. Therefore, DW vertical loop operators exactly generate the six orthonormal basis of the ground-state Hilbert space without omission or duplication, and their count gives us the true GSD on the torus.

(3) In equation (3), the $G$ objects should be named $6j$ symbols or $F$-symbols, which is their usual names in the literature; Just before and in equation (11), $\delta$ should be replaced by the fusion matrix $N$.

Thanks for the keen attention to detail. We would like to clarify that our choice to use $G$-symbols and $\delta$-tensors aligns with conventions established in our primary reference Ref.20[PRB 97.19 (2018): 195154]. Although we acknowledge the value of using more widespread conventions, we have opted to maintain our current notation for consistency with this reference.

The $G$-symbols are essentially $6j$ symbols, which can have various symmetry conditions. The specific conditions we adopt are
$$G_{kln}^{ijm} = G_{ijn^*}^{klm^*} = G_{lkn^*}^{jim}= G_{nk^* l^*}^{mij},$$
where $m^*$ is the antipole of simple object $m$. When compared to the $F$-symbols in Eq. (9) of the original paper on the Levin-Wen model [PRB 71.4 (2005): 045110],
$$F_{kln}^{ijm}=\sqrt{d_m d_n}G_{kln}^{ijm},$$
where $d_k$ is the quantum dimension of simple object $k$. The $\delta$-tensor adheres to Eq. (10) of Levin and Wen’s paper [PRB 71.4 (2005): 045110]. Compared to the fusion tensor $N$,
$$N_{ij}^k =\delta_{ijk^*}.$$
It is particularly used when the fusion rules are multiplicity-free: $0\le N_{ij}^k\le 1$.

(4) Suggestions on other typos and citations.

Thanks to the referee for all these careful reviews and thoughtful comments. We'll be sure to address each point in our revised manuscript, including adding the necessary citations, removing redundant content, and giving a more accurate description of the spectrum. All the feedback has really helped improve our work.

Once again, we would like to extend our heartfelt thanks to both the editor and the referee for investing time and expertise in our manuscript. The insights have really helped elevate the quality of our research.

We thank the referee for the thorough review and valuable feedback on our manuscript, "Characteristic Properties of a Composite System of Topological Phases Separated by Gapped Domain Walls via an Exactly Solvable Hamiltonian Model". We appreciate the time and effort invested in providing us with insightful suggestions, and we will revise our manuscript in accordance with the referee's suggestions, particularly in addressing the question regarding the generalization of our work to all kinds of anyon condensations.

(1) Our Model can Describe Non-Simple Current Condensation.

Indeed, our model is applicable to non-simple current condensation. Moreover, it provides an intuitive representation of this type of anyon condensation.

During non-simple current anyon condensation, the condensed anyons are identified as trivial anyons in the resulting child topological phase, which necessitates that their quantum dimensions must be $1$. Therefore, these non-simple currents must possess internal charges, and anyon condensation only impacts those internal degrees of freedom with quantum dimensions of $1$. At the same time, the other components persist as nontrivial anyons or confined quasiparticles in the child phase. This process is described by the phenomenon of splitting.

Consider the rough gapped boundary of the doubled-Ising topological phase as a familiar example. The gapped boundary is a specific type of gapped domain wall separating the doubled Ising phase and the trivial topological phase. When anyons $1\bar{1}$, $\psi\bar{\psi},$ and $\sigma\bar{\sigma}$ enter the gapped boundary, they undergo condensation. Here, anyon $\sigma\bar{\sigma}$ is not a simple current.

But this description does not fully capture the process. As we've noted in our manuscript, the anyon $\sigma\bar{\sigma}$ consists of two components: $(\sigma\bar{\sigma}, 1)$ and $(\sigma\bar{\sigma}, \psi)$, both having a quantum dimension of $1$. In fact, the rough gapped boundary only condenses the component $(\sigma\bar{\sigma}, \psi)$, remaining the other component $(\sigma\bar{\sigma}, 1)$ confined on the boundary.

Correspondingly, in the trivial topological phase, the condensation term added to the Hamiltonian is
$$\Delta H = \sum_{E\in\text{Trivial Phase}}\Lambda(W_E^{\psi\bar{\psi}; 1,1} + 2W_E^{\sigma\bar{\sigma}; \psi,\psi} )$$,
where $\Lambda\to\infty$. Here the factors $2$ arise from the fractional fusion rules of internal charge $(\sigma\bar{\sigma}, \psi)$, see [ArXiv: 2304.08475]. The projector
$$P_\text{eff} = \sum_{E\in\text{Trivial Phase}}\frac{W_E^{1\bar{1}; 1,1} + W_E^{\psi\bar{\psi}; 1,1} + 2W_E^{\sigma\bar{\sigma}; \psi,\psi} }{4}$$
projects the original Hilbert space to the effective Hilbert space of the system with a rough gapped boundary.

(2) Slightly Small Audience and Appropriate for SciPost Physics Core.

We understand the referee's concerns about the technical nature of our work and its appeal to a potentially narrower audience. Nevertheless, we are confident that our work aligns well with the standards of SciPost Physics, which does not necessitate a very wide audience appeal for its publications.

In our pursuit to provide an intuitive understanding of the physics of anyon condensation and gapped domain walls, we found it more appropriate to delve into the detailed technical aspects of our work via constructing an exactly solvable Hamiltonian model. We acknowledge that this approach adds a layer of complexity and may limit the breadth of our work's appeal. Nevertheless, we believe that our detailed and explicit descriptions of the intricate aspects of composite topological ordered systems will still resonate with the readers of SciPost Physics who are engaged in this field. While our manuscript only discusses a special technical example of the exactly solvable model, our construction methodology and conclusions provide an accurate depiction of a wide range of anyon condensation gapped domain walls, including non-simple current condensation. This universality of our model allows for the natural extension of our conclusions to systems that do not have a lattice model description yet, such as chiral systems, inspiring further general studies of composite systems and anyon condensation among peers studying topological phases. For instance, in our recent work (ArXiv: 2304.08475, under review in PRL), we proposed the generalized Verlinde formulae that lead to the fractional and irrational fusion rules, which are related to the symmetry breaking and generalized Landau-Ginzburg paradigm of anyon condensation [JHEP 2022.3 (2022): 1-45].

We kindly hope that our perspective can be taken into account and that our manuscript can possess the opportunity for reconsideration for publication in SciPost Physics. We are committed to making the necessary revisions to meet the high standards of SciPost Physics and eagerly look forward to the opportunity to contribute to the scientific discourse within its pages.