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Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension
by Sebastian Nagies, Botao Wang, A. C. Knapp, André Eckardt, N. L. Harshman
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Submission summary
Authors (as registered SciPost users):  André Eckardt · Nathan Harshman · Sebastian Nagies 
Submission information  

Preprint Link:  https://arxiv.org/abs/2309.04358v3 (pdf) 
Date accepted:  20240307 
Date submitted:  20240117 08:00 
Submitted by:  Harshman, Nathan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Anyons obeying fractional exchange statistics arise naturally in two dimensions: hardcore twobody constraints make the configuration space of particles not simplyconnected. The braid group describes how topologicallyinequivalent exchange paths can be associated to nontrivial geometric phases for abelian anyons. Braidanyon exchange statistics can also be found in one dimension (1D), but this requires broken Galilean invariance to distinguish different ways for two anyons to exchange. However, recently it was shown that an alternative form of exchange statistics can occur in 1D because hardcore threebody constraints also make the configuration space not simplyconnected. Instead of the braid group, the topology of exchange paths and their associated nontrivial geometric phases are described by the traid group. In this article we propose a first concrete model realizing this alternative form of anyonic exchange statistics. Starting from a bosonic lattice model that implements the desired geometric phases with numberdependent Peierls phases, we then define anyonic operators so that the kinetic energy term in the Hamiltonian becomes local and quadratic with respect to them. The groundstate of this traidanyonHubbard model exhibits several indications of exchange statistics intermediate between bosons and fermions, as well as signs of emergent approximate Haldane exclusion statistics. The continuum limit results in a Galilean invariant Hamiltonian with eigenstates that correspond to previously constructed continuum wave functions for traid anyons. This provides not only an aposteriori justification of our lattice model, but also shows that our construction serves as an intuitive approach to traid anyons, i.e. anyons intrinsic to 1D.
Author comments upon resubmission
List of changes
REFEREE 1:
We thank the Referee 1 for evaluating our submission to SciPost Physics and we are happy about the positive feedback "The reasoning, although far from trivial, is explained very clearly and pedagogically for readers who are not experts on this topic, making the paper selfcontained. [...] In general, the lattice approach and the pedagogical style retained throughout the manuscript make the paper a great resource for other physicists who might want to approach this novel topic." We appreciate that the referee believes that (pending revisions) our manuscript is on track for publication. We also thank the Referee for their constructive critical remarks, regarding especially the numerical simulation of the traid anyon lattice model, which we address in detail below and according to which we made revised the manuscript. (Please note, all references now refer to the new arXiv version.)
Request 1: If the authors have in mind a specific setup in which Hamiltonian (14) can be efficiently realized, it would be very valuable if they could discuss it explicitly. Otherwise, I think the authors should be more open about this limitation and discuss it explicitly in the introduction and the conclusions.
Response 1: Referee 1 observes correctly that we do not formulate an experimental implementation for our model in this article. As we emphasize in the text, the exchange phase depends on ordinality (i.e., the number of particles to the left or right of a given exchange). Ordinality is a nonlocal notion that depends on the $N$body configuration space, although perhaps calling it an $N$body interaction obscures its relative simplicity. We think there there may be several possible approaches to implementing ordinalitydependent phases in the lab, but it is too premature to include these speculations. So we have added new sentences to the manuscript to clarify that although the experimental implementation of traid anyons is a motivation, we do not actually propose an implementation in this article and that such an implementation would be more difficult than the recent implementation of the (braid)anyonHubbard model.
In the introduction, we now write: "While we do not propose a specific implementation for our model, we do identify the necessary structure of the numberdependent tunneling phases than an implementation would require, and that due to the nonlocal multibody interactions required, it will be more complicated than in the recentlyrealized anyonHubbard model [14]."
Also later, in the conclusions, we have added a comment: "It is an interesting task for future research to design experimentally accessible schemes for the implementation of this (or related) traidanyon models, though such an implementation will have to address the challenge of effectively realizing the required nonlocal interactions."
Request 2: One quantity studied by the authors is the chemical potential.
However, the very notion of chemical potential appears to be illdefined for these systems. Calling E0(N), the ground state energy of a system with
N particles, the authors define the chemical potential as E0(N)−E0(N−1). However, the model is not fully specified by N, since every new particle introduces a sign choice in the traid group representation. In what sense is the choice made by the author meaningful? Is there a way of relating the notion of chemical potential to physically measurable properties here to resolve this ambiguity? To remove this ambiguity, could the behavior of twopoint functions be a better quantity to investigate?
Response 2: As Referee 1 notes, for each particle that is added, one has a choice between adding it "like a boson" symmetrically or "like a fermion" antisymmetrically, and so in that sense there is not a unique thermodynamic limit. However, when we restrict ourselves to the two alternating representations $(++\cdots)$ and $(++\cdots)$ (or more generally, to any representation with a repeating pattern of $\tau_i$, including also the case of bosons $\tau_i=1$, and fermions, $\tau_i=1$), then we can speak of a welldefined thermodynamic limit.
In the revised manuscript, this issue is now pointed out more clearly. Moreover, we also discuss the reason for choosing this quantity in more detail. We have added the text:
"Note that this quantity is well defined only for a specific rule for how the sign $\tau_N$ is chosen, when the $N+1$st particle is added to the system. It is of interest, since it directly provides a perspective on the relation between exchange statistics and exclusion statistics (see Appendix~\ref{app:exclusion} below). Free bosons and free fermions (where by ``free'' we mean noninteracting) are the extreme cases. Any number of bosons can occupy the ground state, so each additional particle requires the same amount of energy, whereas each additional fermion must occupy the next higher singleparticle energy eigenstate. Free particles with fractional exclusions statistics, where singleparticle states can be occupied by a welldefined finite number of particles, would, in a similar fashion, give rise to a clear fingerprint in the behaviour of the chemical potential as a function of the total particle number. Below, we will, indeed, see traces of such behaviour, but also deviations from it, which will be attributed to the fact that, as a result of the numberdependent tunneling phases, even for $U=0$ the traidanyon Hubbard model does not describe free particles."
Request 3: I would expect the dependency with N not to be determined entirely by topological effects, but by interactions as well. Could the author comment on these effects?
The Referee is absolutely right. A clean fingerprint of exclusion statistics is visible in the groundstate chemical potential only for free (i.e. noninteracting particles). It turns out, however, that even for $U=0$ the particles of the proposed traidanyon Hubbard model are not free. This is also known from the braidanyonHubbard model, where the effective interactions between the bosons have been analyzed; c.f. Ref.[30]. Nevertheless, we still see signatures in the dependence of the chemical potential with respect to the total particle number that reflect the expected behaviour, but slightly modified by interactions. In response to the Referee, we now discuss this point in more detail. See the new text cited in our response to point 2 above as well as the new sentence "This hints at an effectively induced attractive interaction, something previously noted for the braidanyonHubbard model [30].", which we included later in the same section IV B.
Request 4: Could the authors add an explanation of why the fermions and pseudofermions results are different? Naively, I would have thought that the two should coincide.
Reply 4: We have added a sentence explaining this discrepancy, which is evident for higher densities. This occurs because pseudofermions act like bosons on the same site, and this effectively reduces the energy cost of adding pseudofermions to the system. In the revised manuscript, we now write in chapter IV C about pseudofermions (in view of the discrepancy with real fermions in Fig. 9):
"Their chemical potential agrees nicely with that of free fermions in the dilute regime (roughly below quarter filling or $N=7$), but as the density increases one can see a deviation because multiplyoccupied sites are allowed for pseudofermions, unlike for true fermions."
Request 5: Secondly, the authors observe some approximate quantization of the eigenvalues of the oneparticle density matrix, which the authors refer to as (near) integervalued occupation numbers''. However, numerically, this seems to happen only for some choices of N and tau_i. In the current form, it is not clear if this "(near) integervalued occupation" is a coincidence that happens sometimes for very small system sizes or something physically meaningful.
What does "near" mean in this context? Is it intended that in some appropriate limit, the occupation numbers will tend to integer values? Otherwise, is there a sense for which the (near) quantization can be expected in many instances of N and statistics tau_i?
Request 6: The authors further write "We again emphasize that this is a feature of the traid anyons, and it is strikingly different from bosons or fermions." While I agree that free bosons will have different eigenvalues, I expect that these values will depend on the details of the interactions as well. Is there a sense in which some features of the eigenvalue are purely "topological" and cannot be mimicked by a local interacting Hamiltonian for bosons?
Combined reply to 5 and 6: The motivation for this subsection on natural orbitals is again to try and probe the interplay among exchange statistics and exclusion statistics, by comparing the traidanyonHubbard model to bosons and fermions. For $N$ free bosons, we would expect one SPDM eigenvector (i.e., the ground state) to have occupation $N$ and the rest to be $0$, whereas for free fermions there would be $N$ eigenvectors (i.e., natural orbitals) with occupation $1$, corresponding to the filled Fermi sea. For free bosons and fermions, the natural orbitals would be the singleparticle states. In this context, integer values for occupation numbers are taken a signature of a system of "free" particles in its ground state.
For anyons obeying fractional exclusion statistics, we expect that singleparticle orbitals can be occupied by several particles up to a maximum occupation (being 2 for the example of semions). For a system of free anyons with fractional exclusion statistics in its ground state, each orbital would be occupied by an integer number of particles, not exceeding the maximum possible occupation.
There are however discrepencies. We tested whether this was a finitesize effect (the answer was no), and as we describe in the text our best understanding is that this is due to the interaction effectively induced by the numberdependent phases on the lattice. But we admit that our results are not yet conclusive.
We have extensively modified the text of Sec. VI C, to address these questions and make it more clear why nearinteger values for the expectations is physically motivated.
Request 7: I also wanted to ask why numerical DMRG results are limited to rather small sizes. I don't know if studying larger sizes would be beneficial here, but if there are issues with this numerical technique, it would be useful if the authors could comment on them.
Reply 7: The lowparticle number, lowdensity regime is where we believe the connection to topological exchange statistics will be most evident. In this regime, exact diagonalization is sufficient to achieve numerical results. We added a clarifying note that only for the higherparticle numbers of Fig. 9 did we apply DMRG to calculate ground state energies.
A further motivation for studying this low fillingfactor regime is the recent implementation of the braidanyonHubbard model, which currently has two particles on eight sites. However, other experimental groups are seeking to look at the higherdensity regime, and future studies with DMRG will be useful there.
Summary: We made substantial changes to Section IV, where the numerical results are presented, to address the Referee's concerns and demonstrate that the numerical results we present are critical to addressing the research goals of this article.
Further, we made clarifications in the introduction and conclusion addressing the challenging possibility of implementation.
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REFEREE 2
We thank Referee 2 for the positive report. Further, we appreciate the question about comparing the LeinaasMyrheim $\eta$statistics for 1D anyons to the orbifold approach. One of the goals of creating and exploring the traidanyonHubbard model is to understand the interplay of exchange statistics, exclusion statistics, and interactions, so the referee's question is most welcome, and we have several comments in response.
In the LeinaasMyrheim approach, the $\eta$parameter describes the boundary condition at the twobody coincidences. The parameter is a degree of freedom that remains after the constraint is applied that there is no probability current through the boundary. Alternatively, in the bosonic case, this parameter can be identified (up to a constant) with the twobody coupling constant for the LiebLiniger gas, and when LeinaasMyrheim anyons are symmetrically extended to the configuration space $\mathbb{R}^N$, they have been called LiebLiniger anyons; c.f. Ref. [45]. The limiting case corresponds to the TonksGirardeau limit, c.f. Ref. [39], and the bosons are said to have been `fermionized', meaning that their probability density and energies are equivalent to fermions, although their momentum distribution is not.
The point of this digression is that the LeinaasMyrheim approach is equivalent to imposing twobody interactions on bosons (or alternatively, a more complicated derivative deltainteraction on fermions, c.f. [40, 41]). This interaction has consequences for exclusion statistics (a la Haldane), but seems to be largely independent of exchange statistics. In fact in the Leinaas and Myrheim approach, fermions are equivalent to bosons, they just satisfy different boundary conditions.
However, for a manifold with boundary, as in the Leinaas and Myrheim approach, the fundamental group is trivial, there is no notion of exchange by a continuous adabiatic path, and there is no possibility for topological exchange statistics. For a much longer discussion, see Ref. [17]. In Section V.E. and Appendix C, we have sketched how, even in the absence of twobody interactions/boundary conditions, there can still be nontrivial statistics. As the referee notes, this does not mean that one cannot additionally impose $\eta$type exclusion/interaction statistics on top of the topological structure of the orbifold. An earlier version of this paper included an appendix on this subject, but it was incomplete and far from the mainline of the article; we intend to perform just that extension in a future work. The exciting possibility that opens up in the orbifold approach is that there is a different $\eta$parameter for each sequential pair, and the required representations of the traid group are nonabelian.
As for the intriguing suggestions in the final comments of the referee about possible extensions, we have thought a little about that. As Valiente points out in Ref. 24 (see comments after e.q. (105)), one challenge is making sure that generalizations preserve the gaugeindependence of the interactions from the labels of the particles. If this accidentally introduces distinguishability to the particles, it can lead to spurious results. However, the possibility of looking for projective representations of the traid group (or the traid group combined with time reversal) would certainly be interesting to explore in more detail.
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REFEREE 3
We thank Referee 3 for reviewing our work and are gratified by the recommendation in current form. However, we have made changes in response to other referees that we think address two of the criticisms in the first report.
First, in response to Referee 1 we have added a sentence to the introduction about possible realizations, pointing out that the nonlocal dependence of the phase will challenge an implementation. However, we think that their might ways of effectively realizing the required nonlocal coupling, for instance on the basis of correlated quantum states that obey nonlocal string order. Second, we made other revisions in response to Referee 1 that clarify the significance of our numerical results.
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REFEREE 4
We thank Referee 4 for their comments and encouragement toward publication and the helpful comments and questions. We have responded to the referee's comments below and made the necessary changes to the paper:
Request 1: Unless I misunderstand, as written the manuscript seems to consider particles on a line, not on a ring. The results in figs 7, 8 seem to confirm this. Are there any complications for periodic boundary conditions? If so, are there any topological implications for the thermodynamic limit?
Response: The referee is correct; both the braidanyonmodel and our new traidanyonmodel are defined on a finite lattice with open boundary conditions. There exists work on the braidanyonHubbard on a ring, with both twisted and periodic boundary conditions, including for the thermodynamic limit. However, the traid group on a ring does not have abelian representations besides bosons and fermions. This is explained in Ref. [17], but the point is that even if the YangBaxter relation is broken, the fact that particles can `go around the back' enforces that all particles have the same statistics in abelian representations. On a line (or interval), one has a welldefined notion of ordinality, which our model depends on to define the numberdependent phase, but this is lost on the ring. Therefore, we did not explore periodic or twisted boundary conditions for our model of abelian anyons.
Request 2: The authors refer to their model as quadratic. I wouldn't consider the Hubbard model quadratic due to the U interaction term which is quartic in creation/annihilation operators. Can the authors please clarify this?
Response: Thanks for catching this. In most cases we were intending to refer to the kinetic term of the Hamiltonian, or referring to the case when there was no onsite interaction, but the language was incorrect. We have gone through and fixed this.
Request 3: Two body and three body hardcore constraints and their implications are, understandably, discussed in detail. Is there any meaning to higher, nbody constraints? Might they lead to different physics, or possibly would they still lead to a similar continuum theory, in which there is an emergent twobody hard core constraint anyway?
Response: For an $N$body constraint to have topological consequences, the defect it creates in configuration space must either make the space not pathconnected or not simplyconnected. The only $N$body interaction that breaks the pathconnected of space is twobody interactions in 1D. This is called a codimension 1 defect (like a line in a plane, or a plane in a 3D volume), because each twobody coincidence $x_i x_j =0$ is onedimensional. To make the configuration space not simply connected, the defect must have codimension 2, like a point in a plane or a line in a volume. There are only three $N$body interactions that do this (c.f. Refs. [23,24]): twobody in 2D with constraint defects described by $x_i  x_j =0$ and $y_i  y_j =0$; threebody in 1D with constraints like $x_i  x_j =0$ and $x_j  x_k =0$; and a nonlocal 4body interaction in 1D with constraints like $x_i  x_j =0$ and $x_k  x_l =0$. Dimensional analysis shows that this list really is exhaustive.
On a separate note, a twobody hardcore constraint in any dimension implies a threebody constrain, but only if all pairs of particles feel the constraint. In the case of the alternating representations of the traid group, the twobody constraint between sequential antisymmetric pairs is also enough to force the threebody constraint for the equivalent continuum bosonic model. However, for traid anyon representations with sequences of multiple consecutive $\tau_i = +1$, then the twobody constraints implied by the $\tau_i = 1$ for certain pairs are not sufficient to guaranteed threebody exclusion. This is mentioned in the text after Eq. 44.
Request 4: The authors refer to pseudofermions. It might be useful to explain a bit more why the traidanyons with the ... signature are only pseudofermions, and what the origin of the difference between fermions and pseudofermions in fig 9 is.
Referee 1 also commented on this, and We have added a comment to clarify.
Request 5: I found the discussion of Fig 6. a bit confusing. The text describes processes that aren't shown in the figure, e.g. for 6a the text describes hopping to the right to create. a double occupied site, and then hopping off again. But the latter process isn't shown. In fact none of 6 a,b,c seem to show a particle swap, so I think either a new figure or new description is needed.
Response: We have revised that paragraph in section III B to make it more clear.
Request 6: The third line of Eq 20 doesn't seem relevant  possibly I'm missing something, but does this definition for Njk get used anywhere, as sgn(i−j) is zero for i=j anyway?
Response: We thank the referee for pointing that out. We have changed Eq. (20) accordingly.
Request 7: I found the following sentence fragment on page 5 a bit hard to parse: "as they correspond to continuous transformations of strands that are not possible also in the presence of twobody hardcore interactions". Please consider rephrasing this.
Response: We thank the Referee for pointing that out. We have fixed the problem.
Request 8: The difference between the behaviour of even and odd numbers of particles is reminiscent of the behaviour of the quantum Ising model in the fermionic field theory limit after JW transformation. In that case the difference between even and odd numbers of particles means that both symmetric and antisymmetric boundary conditions have to be considered, and the spectrum separates into Ramond and NeveuSchwarz sectors. Is something similar happening for the traidanyon field theory?
Response: This is a fascinating comment, and worthy of further investigation. We were previously unaware of the RNS construction, but it may have relevance, at least as an analogy. Like the twobody hardcore contact interaction in 2D, the threebody hardcore contact interaction in 1D is conformally invariant, which seems to be an ingredient necessary for the RNS construction to be relevant. Further (as emphasized by Referee 2 and alluded to somewhat in Sect. V.E.), the continuum version of traid anyons can be mapped to bosons or fermions with boundary conditions. We thank the referee for this idea, and will follow up, but do not think it is necessary to include in this already long paper.
Finally, thanks for catching a few typos, which have now been corrected.
Published as SciPost Phys. 16, 086 (2024)
Reports on this Submission
Report
I am generally happy with the author's reply to my comments.
The discussion of the numerical results has greatly improved, especially in Subsection 4.3.
While the definition of the chemical potential still looks very arbitrary to me and the reasoning provided by the authors does not fully lift my objections, this is potentially the only weak point of an otherwise wellthoughtout and carefully written paper.
I would therefore recommend publication of the manuscript in its current form.
Report
I'm very happy with the authors' responses to my queries and the (minor) alterations to the manuscript.
I believe the manuscript meets criterion 3 of the journal expectations: by developing a Hubbard like 1D model of anyonic statistics that doesn't break Galilean invariance, the authors have opened up a significant new research pathway.
It also addresses a long standing stumbling block (criterion 2), by providing a useful model of anyonic statistics in 1D.
All the general acceptance criteria are met: this is an extremely wellwritten and complete manuscript.