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Incorporating nonlocal anyonic statistics into a graph decomposition
by Matthias Mühlhauser, Viktor Kott, Kai Phillip Schmidt
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Authors (as registered SciPost users):  Kai Phillip Schmidt 
Submission information  

Preprint Link:  scipost_202401_00019v2 (pdf) 
Date accepted:  20240503 
Date submitted:  20240424 11:07 
Submitted by:  Schmidt, Kai Phillip 
Submitted to:  SciPost Physics Core 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In this work we describe how to systematically implement a full graph decomposition to set up a linkedcluster expansion for the topological phase of Kitaev's toric code in a field. This demands to include the nonlocal effects mediated by the mutual anyonic statistics of elementary charge and flux excitations. Technically, we describe how to consistently integrate such nonlocal effects into a hypergraph decomposition for single excitations. The approach is demonstrated for the groundstate energy and the elementary excitation energies of charges and fluxes in the perturbed topological phase.
Author comments upon resubmission
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Response to the first referee

We thank the referee for the examination of our work.
Referee: This article is mathematically/algorithmically involved and results from several previous articles using the same technic (which is not a criticism in itself) and the same vocabulary. However, despite its length, the authors fail to make it selfconsistent and easy to follow. I am convinced the calculations are correct and involved, but the authors should give more detailed explanation of the articulations between technical parts, and more careful definitions of the objects they use, with a thought for those who have not read all the previous articles. I list questions and remarks containing examples of this below.
On another side, the novelty of the calculations is not clearly stated. What was the initial question that has motivated this work (except applying a wellmastered technique to a new model). The phase diagram of this model is already known (ref 25). What information can be obtained from the series ? Do the gap close at some magnetic field, or does the series diverge before ?
Is is the first time such a method is applied to a model with anyons ? If I understand correctly, the way fractionalized excitations are taken into account when (hyper ?)graphs contain loops is inherent to the linkedcluster decomposition, and is not realized through a special adaptation of the method, but the authors elegantly evidence how it appears through the spin background ?
Our response: We thank the referee for these questions which are very happy to address. The physical problem of the toric code in a magnetic field on the square lattice has been addressed by various techniques in the last 15 years (as cited in the article) so that many aspects of the rich groundstate phase diagram including first and secondorder quantum phase transitions have been quantitatively understood (although not all). For the secondorder transitions out of the topological phase it is indeed the charge and/or flux gap which closes at the quantum critical point. Among other numerical techniques, this in particular also includes highorder series expansions using the pCUT method which was a breakthrough at that time due to exotic anyon statistics in the topological phase (the closing of the charge and flux gap can be quantitatively located by extrapolation techniques of the obtained gap series). However, in these works the pCUT method has been set up using an expansion in large clusters (so that all fluctuations are contained in a single cluster or a decomposition in medium size rectangular clusters based on Enting's method) as outlined in section 3. The latter is not equivalent to a full graphdecomposition in minimal clusters and the question how to treat the (nonlocal) anyonic statistics on graphs has remained open. This is exactly what we address and solve in our current work via the hypergraph decomposition for the onecharge and oneflux sectors as detailed in section 4. It is therefore the combination of hypergraph expansions and anyonic statistics which is the key aspect of our work. This is the reason why we have not included a physical analysis of the obtained series. At the same time we benefitted from the known series expansions from the above mentioned works (except the highest order which we added in this work) which allowed us to validate our approach in terms of hypergraphs.
Below we address the specific points raised by the referee
 Referee: Sec. 2.1: the ground state energy E_O^{tc} is said to be N/2, but from Eq. (2), I would have expected N, as N is the number of plaquettes and of stars.
Our response: We thank the referee for this comment. However, the groundstate energy given in the paper is actually correct. There are N/2 stars and N/2 plaquettes where N is the total number of spins. In addition, there is a 1/2 in the Hamiltonian so that the groundstate energy of the bare toric code is N/2 in these units.
 Referee: In Sec. 3, the matrix elements of T(m) of Eq. 15 are calculated. But I don't understand where it leads. Where are the C(m) evaluated ? And how are they related to H^{TCF} ?
Our response: The C(m) are model independent rational coefficients inherent to the pCUT method so that they do not need to be calculated again. These coefficients are calculated by solving the flow equation in the perturbative formulation order by order. This is explained in the literature about the pCUT method (e.g. Eur. Phys. J. B vol. 13, p. 209 (2000)). But we agree with the referee that this was not optimally phrased in the article. We therefore have added an additional sentence in the revised version on page 7 at the beginning of 3.1 which also includes a reference to the relevant paper.
 Referee: Is Sec. 3 only a preparation to Sec. 4, where the real calculations are done, and does it only explain the calculation of the T(m) matrix elements ? Then, the title of subsections 3.1 and 3.2 (GS and 1QP energy) are misleading. The sequence of steps leading to the expansions, and their logic, should be explained more carefully.
Our response: We thank the referee for these questions. Section 3 explains how one can derive perturbative series in the thermodynamic limit from calculations on appropriately chosen finite (large) clusters for the toric code in the lowfield limit in the topological phase. While this has already been described in a compact form in Phys. Rev. B 79, 033109 (2009), we here present a detailed explanation how to calculate onequasiparticle energies (dispersion of a single flux or charge) and groundstate energy (0QP sector) of the effective pCUT Hamiltonian for the perturbed toric code. The title of the subsections 3.1 (Set up), 3.2 (Groundstate energy), and 3.3 (Onequasiparticle energies) are appropriate.
As the modelindependent coefficients C(m) (see above), also the operator sequences T(m) of the effective pCUT Hamiltonian are modelindependent. However, these operator products are not normalordered so that physical properties can not be read off directly. The normal ordering of the T(m) is therefore the essential step which is modeldependent. This is done typically via the calculation of matrix elements of the effective Hamiltonian on large clusters (like in section 3) or graphs (like in section 4), which boils down essentially to determine matrix elements of T(m) operator sequences with appropriate unperturbed basis states. In contrast to many applications of the pCUT method in the literature, for the perturbed toric code and its anyonic excitations one has to take care of the nonlocal mutual statistics. While for (large) clusters this has been done before (detailed in section 3), a full hypergraph expansion as done section 4 has never been realized before. As a consequence, section 3 contains all basic and general information while in section 4 we fully concentrate on the specific aspects arising in the new hypergraph expansion. In the revised version we have further clarified these points.
 Referee: The last paragraph of p.7 is hard to folllow with the introduction of bonds b^(\alpha)}. Would the notation b_i^(\alpha)} not be more adapted, with a sum over i and α replacing the sums over b(α) ? (this notation with sites in index is already used in the caption of Fig 4, but not in the figure, and should be used in Table 1 for example)
It seems that "bond" designs two different objects when applied to a site i and to an operator b(α). b(α) is a composite object containing a site i and 0 or 2 vortice, and 0 or 2 plaquettes, also called 'bond'.
Our response: We thank the referee for this point. In the revised version we have tried to make this part more comprehensive. A bond specifies where the respective tauoperator acts. However, as this operator does not act equivalently on all sites within the bond, the notion of a set is not enough. Instead, there needs to be more structure distinguishing the different roles of the sites within the bond. We therefore stick to the notation b^alpha which is now used in the whole article.
 Referee: What means 'ordered set' for such a bbond ? Is it not enough to precise that τ first acts on site, then on flux sites and finally on charge sites ?
Our response: We thank the referee for this point. We have avoided this notion in the revised article.
 Referee: p. 14: between Eqs 50 and 51, "note that the sum runs over all subclusters c of C and not over all distinct subclusters of C". What is the difference between the two expressions ?
Our response: In the sum of Eq. 50, the symbol means "included in and different from", as C is included in itself. We intended that \subset means proper subcluster and \subseteq means subcluster. We have adapted the text to clarify this point.
 Referee: p. 14: a definition of a cluster is required, and/or of H_{eff}.
Does a cluster include sites ? bbonds ? plaquettes ? Same question for H^c_{eff}. Do a "cluster" with no plaquette and no star has H_{eff}=0 ? These notations are used all along p. 14, and only explained p. 17, with many references to other articles. An hypergraph and a König representation of an hypergraph could be better defined.
Our response: We thank the referee for these questions. We have added a short paragraph about notion "cluster" at the beginning of section 4 as well as a description of what a "hypergraph" is. This should then also allow to access the restriction of H_eff to a specific cluster.
For the König representation we wrote in the paper: "The two parts of the bipartite König graph represent the edges (edgepart) and the vertices (vertexpart) of the hypergraph.
Two vertices in the König representation are adjacent if and only if they correspond to an incident edgevertex pair within the hypergraph." We are convinced that this definition is sufficient in the current context.
 Referee: After Eq. 56, what does a "local state" represent ? A cluster is thus not just a collection of bonds, but includes a spinconfiguration ? or is it simply the x,y or z from the bbonds ?
Our response: The proper definition of clusters (see last point) should also help here. Consider that a cluster is a set of charge, flux, and spin sites, together with a set of bonds  these are subsets of the sites and bonds of the entire system. The state on the cluster is just the restriction of the product state in full (infinite system) Hilbert space to the (finite) Hilbert space of the cluster.
 Referee: Could the authors precise in which sense the bonds joins multiple sites not necessarily in a symmetric way ? (p.15)
Our response: The different charge, flux, and spin sites play different roles in the bonds, and the tauoperators may act differently on these sites.
 Referee: Is the cluster expansion realised on the initial Hamiltonian or on the pCUT Hamiltonian H_{eff} (which seems to be the case according to Eq. 5153). Then again, the C(m) coefficients are required, but their obtention not detailed.
Our response: We thank the referee for this point. The cluster expansion is done for the effective Hamiltonian. We are convinced that this is now clearer in the revised version after clarifying the meaning of the C(m), T(m), and of the clusters and bonds.
Further changes:
 Fixed some inconsistencies in the formulation of the perturbation theory

Response to the first referee

We thank the referee for the examination of our work.
Referee: This article is mathematically/algorithmically involved and results from several previous articles using the same technic (which is not a criticism in itself) and the same vocabulary. However, despite its length, the authors fail to make it selfconsistent and easy to follow. I am convinced the calculations are correct and involved, but the authors should give more detailed explanation of the articulations between technical parts, and more careful definitions of the objects they use, with a thought for those who have not read all the previous articles. I list questions and remarks containing examples of this below.
On another side, the novelty of the calculations is not clearly stated. What was the initial question that has motivated this work (except applying a wellmastered technique to a new model). The phase diagram of this model is already known (ref 25). What information can be obtained from the series ? Do the gap close at some magnetic field, or does the series diverge before ?
Is is the first time such a method is applied to a model with anyons ? If I understand correctly, the way fractionalized excitations are taken into account when (hyper ?)graphs contain loops is inherent to the linkedcluster decomposition, and is not realized through a special adaptation of the method, but the authors elegantly evidence how it appears through the spin background ?
Our response: We thank the referee for these questions which are very happy to address. The physical problem of the toric code in a magnetic field on the square lattice has been addressed by various techniques in the last 15 years (as cited in the article) so that many aspects of the rich groundstate phase diagram including first and secondorder quantum phase transitions have been quantitatively understood (although not all). For the secondorder transitions out of the topological phase it is indeed the charge and/or flux gap which closes at the quantum critical point. Among other numerical techniques, this in particular also includes highorder series expansions using the pCUT method which was a breakthrough at that time due to exotic anyon statistics in the topological phase (the closing of the charge and flux gap can be quantitatively located by extrapolation techniques of the obtained gap series). However, in these works the pCUT method has been set up using an expansion in large clusters (so that all fluctuations are contained in a single cluster or a decomposition in medium size rectangular clusters based on Enting's method) as outlined in section 3. The latter is not equivalent to a full graphdecomposition in minimal clusters and the question how to treat the (nonlocal) anyonic statistics on graphs has remained open. This is exactly what we address and solve in our current work via the hypergraph decomposition for the onecharge and oneflux sectors as detailed in section 4. It is therefore the combination of hypergraph expansions and anyonic statistics which is the key aspect of our work. This is the reason why we have not included a physical analysis of the obtained series. At the same time we benefitted from the known series expansions from the above mentioned works (except the highest order which we added in this work) which allowed us to validate our approach in terms of hypergraphs.
Below we address the specific points raised by the referee
 Referee: Sec. 2.1: the ground state energy E_O^{tc} is said to be N/2, but from Eq. (2), I would have expected N, as N is the number of plaquettes and of stars.
Our response: We thank the referee for this comment. However, the groundstate energy given in the paper is actually correct. There are N/2 stars and N/2 plaquettes where N is the total number of spins. In addition, there is a 1/2 in the Hamiltonian so that the groundstate energy of the bare toric code is N/2 in these units.
 Referee: In Sec. 3, the matrix elements of T(m) of Eq. 15 are calculated. But I don't understand where it leads. Where are the C(m) evaluated ? And how are they related to H^{TCF} ?
Our response: The C(m) are model independent rational coefficients inherent to the pCUT method so that they do not need to be calculated again. These coefficients are calculated by solving the flow equation in the perturbative formulation order by order. This is explained in the literature about the pCUT method (e.g. Eur. Phys. J. B vol. 13, p. 209 (2000)). But we agree with the referee that this was not optimally phrased in the article. We therefore have added an additional sentence in the revised version on page 7 at the beginning of 3.1 which also includes a reference to the relevant paper.
 Referee: Is Sec. 3 only a preparation to Sec. 4, where the real calculations are done, and does it only explain the calculation of the T(m) matrix elements ? Then, the title of subsections 3.1 and 3.2 (GS and 1QP energy) are misleading. The sequence of steps leading to the expansions, and their logic, should be explained more carefully.
Our response: We thank the referee for these questions. Section 3 explains how one can derive perturbative series in the thermodynamic limit from calculations on appropriately chosen finite (large) clusters for the toric code in the lowfield limit in the topological phase. While this has already been described in a compact form in Phys. Rev. B 79, 033109 (2009), we here present a detailed explanation how to calculate onequasiparticle energies (dispersion of a single flux or charge) and groundstate energy (0QP sector) of the effective pCUT Hamiltonian for the perturbed toric code. The title of the subsections 3.1 (Set up), 3.2 (Groundstate energy), and 3.3 (Onequasiparticle energies) are appropriate.
As the modelindependent coefficients C(m) (see above), also the operator sequences T(m) of the effective pCUT Hamiltonian are modelindependent. However, these operator products are not normalordered so that physical properties can not be read off directly. The normal ordering of the T(m) is therefore the essential step which is modeldependent. This is done typically via the calculation of matrix elements of the effective Hamiltonian on large clusters (like in section 3) or graphs (like in section 4), which boils down essentially to determine matrix elements of T(m) operator sequences with appropriate unperturbed basis states. In contrast to many applications of the pCUT method in the literature, for the perturbed toric code and its anyonic excitations one has to take care of the nonlocal mutual statistics. While for (large) clusters this has been done before (detailed in section 3), a full hypergraph expansion as done section 4 has never been realized before. As a consequence, section 3 contains all basic and general information while in section 4 we fully concentrate on the specific aspects arising in the new hypergraph expansion. In the revised version we have further clarified these points.
 Referee: The last paragraph of p.7 is hard to folllow with the introduction of bonds b^(\alpha)}. Would the notation b_i^(\alpha)} not be more adapted, with a sum over i and α replacing the sums over b(α) ? (this notation with sites in index is already used in the caption of Fig 4, but not in the figure, and should be used in Table 1 for example)
It seems that "bond" designs two different objects when applied to a site i and to an operator b(α). b(α) is a composite object containing a site i and 0 or 2 vortice, and 0 or 2 plaquettes, also called 'bond'.
Our response: We thank the referee for this point. In the revised version we have tried to make this part more comprehensive. A bond specifies where the respective tauoperator acts. However, as this operator does not act equivalently on all sites within the bond, the notion of a set is not enough. Instead, there needs to be more structure distinguishing the different roles of the sites within the bond. We therefore stick to the notation b^alpha which is now used in the whole article.
 Referee: What means 'ordered set' for such a bbond ? Is it not enough to precise that τ first acts on site, then on flux sites and finally on charge sites ?
Our response: We thank the referee for this point. We have avoided this notion in the revised article.
 Referee: p. 14: between Eqs 50 and 51, "note that the sum runs over all subclusters c of C and not over all distinct subclusters of C". What is the difference between the two expressions ?
Our response: In the sum of Eq. 50, the symbol means "included in and different from", as C is included in itself. We intended that \subset means proper subcluster and \subseteq means subcluster. We have adapted the text to clarify this point.
 Referee: p. 14: a definition of a cluster is required, and/or of H_{eff}.
Does a cluster include sites ? bbonds ? plaquettes ? Same question for H^c_{eff}. Do a "cluster" with no plaquette and no star has H_{eff}=0 ? These notations are used all along p. 14, and only explained p. 17, with many references to other articles. An hypergraph and a König representation of an hypergraph could be better defined.
Our response: We thank the referee for these questions. We have added a short paragraph about notion "cluster" at the beginning of section 4 as well as a description of what a "hypergraph" is. This should then also allow to access the restriction of H_eff to a specific cluster.
For the König representation we wrote in the paper: "The two parts of the bipartite König graph represent the edges (edgepart) and the vertices (vertexpart) of the hypergraph.
Two vertices in the König representation are adjacent if and only if they correspond to an incident edgevertex pair within the hypergraph." We are convinced that this definition is sufficient in the current context.
 Referee: After Eq. 56, what does a "local state" represent ? A cluster is thus not just a collection of bonds, but includes a spinconfiguration ? or is it simply the x,y or z from the bbonds ?
Our response: The proper definition of clusters (see last point) should also help here. Consider that a cluster is a set of charge, flux, and spin sites, together with a set of bonds  these are subsets of the sites and bonds of the entire system. The state on the cluster is just the restriction of the product state in full (infinite system) Hilbert space to the (finite) Hilbert space of the cluster.
 Referee: Could the authors precise in which sense the bonds joins multiple sites not necessarily in a symmetric way ? (p.15)
Our response: The different charge, flux, and spin sites play different roles in the bonds, and the tauoperators may act differently on these sites.
 Referee: Is the cluster expansion realised on the initial Hamiltonian or on the pCUT Hamiltonian H_{eff} (which seems to be the case according to Eq. 5153). Then again, the C(m) coefficients are required, but their obtention not detailed.
Our response: We thank the referee for this point. The cluster expansion is done for the effective Hamiltonian. We are convinced that this is now clearer in the revised version after clarifying the meaning of the C(m), T(m), and of the clusters and bonds.
Further changes:
 Fixed some inconsistencies in the formulation of the perturbation theory
List of changes
We have highlighted all changes in the revised version in red.
Published as SciPost Phys. Core 7, 031 (2024)