Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension

1. Excellently written, pedagogical and self contained.
2. Introduces an interesting lattice model that realises anyonic statistics in 1D in a 'natural' way, provides a very complete discussion of its basic properties, and also gives numerical evidence for some features.
3. Also derives the appropriate continuum limit model and shows that it is consistent with previous results for certain anyonic wavefunctions.
4. High quality, useful, figures.

1. The later subsections of section V are perhaps not quite as lucid as the earlier parts of the manuscript.
2. It may be that the model can't be realised experimentally (though it is still interesting theoretically).

In this excellently written, and highly readable manuscript the authors propose a lattice model that realises anyonic statistics of a certain type in 1D.
Models with unusual statistics are of fundamental interest, and this model is novel because it retains Galilean invariance, unlike previously introduced models of anyonic statistics in 1D. Furthermore, the mechanism behind the statistics is atypical because it invokes a three body hard core constraint.

The key results are the introduction of the lattice model, and its continuum counterpart, which open new research directions for those working on topological phases and many-body quantum physics in 1D.
The lattice model is also useful in that when written in terms of so-called traid-anyons it resembles the familiar bose-Hubbard model, which makes interpreting it somewhat more natural. The price of this simple form for the traid-anyons is that complicated non-local strings in the hopping terms for the underlying boson model become non-local commutation relations for the traid-anyons.
The introductory sections (I and II) are clear, pedagogical, well self-contained, and a model for other authors. The figures are of high standard and useful to the reader.

There are some queries I'd like the authors to address.

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?
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?
3. Two body and three body hard-core constraints and their implications are, understandably, discussed in detail. Is there any meaning to higher, n-body constraints? Might they lead to different physics, or possibly would they still lead to a similar continuum theory, in which there is an emergent two-body hard core constraint anyway?
4. The authors refer to pseudo-fermions. It might be useful to explain a bit more why the traid-anyons with the ----... signature are only pseudo-fermions, and what the origin of the difference between fermions and pseudo-fermions in fig 9 is.
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.
6. The third line of Eq 20 doesn't seem relevant - possibly I'm missing something, but does this definition for $N_{jk}$ get used anywhere, as $sgn(i-j)$ is zero for $i=j$ anyway?
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 two-body hardcore interactions". Please consider rephrasing this.
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 J-W 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 Neveu-Schwarz sectors. Is something similar happening for the traid-anyon field theory?

Typos:
8. Page 7, "In comparison neither $\mathcal{T}$ for $\mathcal{P}$...", for -> nor?
9. Page 13, just above the start of section V. Something has gone wrong with the definition of $t_\alpha$, presumably the text in red was supposed to be removed?

- The numerical indications of a possible fractional exclusion principle are intriguing.
- The paper is well written.

- Any experimental realization of the proposed model appears out of reach.

In the manuscript "Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension”, Nagies et al. propose a 1D lattice model for so-called "traid" anyons and study its properties including the continuum limit. The authors first review exchange statistics in continuum models, discussing conventional statistics, braid statistics, and traid statistics, a concept introduced by some of the authors. The possibility of traid statistics exists for particles with a three-body hard-core constraint in 1D. The authors construct a lattice model for abelian traid anyons. The starting point is a bosonic model with number dependent Peierls phases, which provide the required geometric phases. This construction is in some sense analogous to the description of Abelian anyons in terms of flux tube -particle composites. The model itself is interesting and appears sound. That it contains a non-local many-particle string operator, however, will make any realization, be it with ultracold atoms in optical lattices or any other platform, challenging if not unrealistic. The authors also investigate the ground-sate properties of the model and find tentative indications for fractional exclusions statistics, including fractional Friedel oscillations, a step-wise behavior of the chemical potential as a function of the total particle number and near-integer occupations of the natural orbitals. The signatures the authors report are not unambiguous, but intriguing and in need for further investigation.

The manuscript is an the long side, but well written. It will not be of interest to a wide readership, but to the best of my judgement valued by experts.

In summary, I recommend it for publication in its present form.

1) An extensive work of high professional quality.
2) Very well written.
3) Many useful figures.
4) Original ideas that may find future applications.

This article is about particle statistics in one dimension, in
particular what the authors call traid statistics. They cite two of
their own earlier articles, refs. (16,17), where they define this
concept and discuss it in more detail. They present here the
continuum theory of traid statistics as a background for their
construction of a lattice model, which is their main purpose.

It is my opinion that the work is of high quality, and I recommend
publication. This conclusion comes with the qualification that I have
read the article somewhat selectively, and mainly the parts
concerning the continuum theory.

I have especially one comment, as follows, which I would like to see answered.

a) Theory, earlier version

The case of one dimension was discussed by Leinaas and Myrheim, ref. 1
in this article. The relative configuration space, ignoring the
centre of mass coordinate, was there identified as a space with
boundaries. Taking three identical particles as an example, the
relative space is the wedge depicted in Fig. 15(b) in the present
article. The solutions of the Schr{\"o}dinger equation must satisfy
boundary conditions designed to make the probability current normal to
the boundary vanish. The formula for the probability current depends
on the Hamiltonian, in standard quantum mechanics it is quadratic in
the wave function $\psi$, and a suitable boundary condition is the
linear condition

$\psi_n = \eta\psi,$

where $\psi_n$ is the normal derivative of the wave function, and $\eta$
is a real parameter. Here $\eta=0$ means bosons, $1/\eta=0$ means
fermions, but any intermediate positive or negative value of $\eta$ is
theoretically possible.

This mathematical approach is called in the present article the
manifold approach, because it is based on the configuration space
which is a manifold with a boundary, in one-to-one correspondence with
the physical configurations.

b) Theory, version presented here

An alternative approach is taken in the present article. The wave
functions are defined on $R^3$, the configuration space describing a
system of identical particles carrying arbitrary labels 1,2,3, and the
indistinguishability is manifested by identifications between
different sectors where the particles 1,2,3 appear in different orders
along the line, as illustrated in Fig. 13. A boundary between two
sectors is singular, because the identification implies that it has
only one side.

They call this the orbifold approach, as opposed to the manifold
approach, because the space where the wave functions are defined is an
orbifold and not a manifold, it contains singularities which are
manifolds of lower dimensions.

c) Difference between the two versions

It is surprising to me that the possibility of statistics intermediate
between bosons and fermions, as argued in ref. 1, is not recognized in
the orbifold approach. One would think that this is a case where
conclusions should not depend on the mathematical approach.

It is not immediately clear, however, how to translate from the
manifold to the orbifold formulation of the quantum theory. One
problem is that it has to be defined by convention what should happen
to the wave function $\psi$ along a curve crossing the boundary. The
necessary conditions are that the probability density $|\psi|^2$ has to
be symmetric across the boundary, and the probability current must
have a component normal to the boundary which is antisymmetric.

In the bosonic case, the obvious convention is that the wave function
is symmetric across the boundary, implying that the normal derivative
is zero on the boundary. Similarly in the fermionic case, the wave
function is naturally assumed to be antisymmetric across the boundary,
vanishing on the boundary in order to be continuous. In both cases,
the wave function and its derivative will be continuous along a
differentiable curve crossing the boundary. But it should be realized
that these rules for crossing the boundary are strictly speaking pure
conventions. It is quite possible to describe bosons by antisymmetric
wave functions and fermions by symmetric wave functions.

The question then is how to formulate an orbifold version of the
quantum theory with "$\eta$-statistics". It must be possible, but I do
not know how (or am too lazy) to do it.

d) Two more comments

They argue in addition that a strong three-body repulsion excluding
from the configuration space points where three or more particles
collide makes the one-dimensional traid anyons more similar to
two-dimensional braid anyons. It is clear that this similarity has
its limits. As seen in Fig. 13, a loop encircling the excluded point
in the three-particle case will have to pass through six boundaries
where two particles cross, and each crossing can at most result in a
change of sign of a one-component wave function. Therefore the loop
can not result in phase factors more general than +-1.

I wonder, by the way as a wild idea, whether it is possible to assume
that the crossing of the boundary results in a complex conjugation of
the wave function (possibly accompanied by the multiplication with a
fixed phase factor).

1- The paper is very well written and clear.
2- The paper introduces traid-anyon statistics and many related concepts in a very pedagogical fashion.

1- The proposed lattice Hamiltonian features all-to-all N-body couplings and is therefore quite unrealistic.
2- Numerical results are very qualitative. In particular, it is unclear which aspects are inherently topological (i.e. can be achieved only through traid-anyon statistics and not through local interactions in a bosonic Hamiltonian).

After summarizing the conceptual points that lead to anyon braid statistics in 2+1D, the authors introduce anyon traid statistics: a relatively novel type of statistics that can be defined in 1+1D. The reasoning, although far from trivial, is explained very clearly and pedagogically for readers who are not experts on this topic, making the paper self-contained. Throughout the text, the authors introduce a lattice model for traid anyons and analyze it numerically. Finally, they study the continuum limit of the model and show that they recover results that had been previously derived in different ways.

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.

I think the manuscript is a valuable contribution that could foster new work on the topic from a broader community of condensed matter physicists. However, the numerical simulations and their interpretation seem rather confusing and it is often unclear what rationale led the authors to certain interpretations.
Therefore I would recommend its publication only if the numerical study of the model is substantially improved.

In Sec. III, the authors summarize the construction of 1D lattice models designed to mimic braid anyon statistics. Proceeding along similar lines the authors are able to identify a lattice model with non-trivial traid statistics.
Having a lattice Hamiltonian displaying this type of statics is potentially useful to both theoretical and experimental physicists, as the authors explain in their motivation (i)-(iii) on page 2.

However, I think that the model constructed here addresses motivation (i) and (iii) only partially: the proposed lattice models have non-local and even $N$-body interactions, where $N$ is the total number of particles (used in the hopping operator of the rightmost pair of particles). In the current implementation, this is needed to correctly reproduce the traid phases on the lattice. In this sense, the model does not seem realistic even for an artificial quantum system.

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.

In Sec. IV, the authors then proceed to numerically investigate the properties of traid anyons. This could be, in principle a strong point of the paper, since it is enabled by the lattice construction here presented for the first time. However, in its current form, this might be the weakest part of the paper. I am sympathetic towards the fact that the physics of these systems is new and good paradigms to interpret numerical data might be missing, but the current discussion does not provide much value.

One quantity studied by the authors is the chemical potential.
However, the very notion of chemical potential appears to be ill-defined for these systems. Calling $E_0(N)$, the ground state energy of a system with $N$ particles, the authors define the chemical potential as $E_0(N)-E_0(N-1)$. However, the model is not fully specified by N, since every new particle introduces a sign choice in the traid group representation.

2- 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 two-point functions be a better quantity to investigate?

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?

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.

Secondly, the authors observe some approximate quantization of the eigenvalues of the one-particle density matrix, which the authors refer to as ``(near) integer-valued 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) integer-valued occupation" is a coincidence that happens sometimes for very small system sizes or something physically meaningful.

5- 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 $\tau_i$?

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?

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.

Overall, I want to emphasize the manuscript is a valuable contribution. Its weakest point is the numerical section, where it is unclear how the results should be interpreted. Provided that this is substantially improved, I would recommend its publication in SciPost Physics.

I would ask the authors to address points 1-7 above and change the manuscript accordingly.