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Exactly Solvable Models of Interacting Chiral Bosons and Fermions on a Lattice
by Manuel Valiente
Submission summary
| Authors (as registered SciPost users): | Manuel Valiente |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2501.16140v1 (pdf) |
| Date submitted: | 2025-01-29 14:58 |
| Submitted by: | Valiente, Manuel |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider one-dimensional theories of chiral fermions and bosons on a lattice, which arise as edge states of two-dimensional topological matter breaking time-reversal invariance. We show that hard core bosons or their spin chain equivalent exhibit properties that are similar to free fermions, solving the many-body problem exactly. For fermions, we study the effect of a static impurity exactly and show the orthogonality catastrophe in the continuum limit via bosonization. The interacting many-fermion problem in the continuum limit is solved exactly using simple momentum conservation arguments.
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
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Report
The idea to consider particles on a lattice with strictly linear dispersion analytically in momentum space (both fermions and bosons) is an interesting idea. It maybe useful for understanding the fundamental differences (similarities) between continuous and discrete theories, and give more benchmarks to test against and even help the numerical techniques by giving more insights about the structure of the wave functions.
While I do find this work suitable for SciPost Physics, there are issues that I would hope the authors can address.
1. It may be obvious to some, but it is not explained why the total momentum K=2k_*. And since the whole derivation of the following relies on it, I think it should be explained properly.
2. What is actually the form of the interaction that is considered for the interacting fermions? Later in the section they calculate matrix elements of operator V. It’s impossible to check the answers because it’s not very clear what is the form of the operator V that is being substituted there.
3. Before (31) a_q^\dagger was never defined. The transition in the second equality of (31) is not clear, I do not understand it, it should be explained.
4. “To prove the statement above, it suffices to apply second-order perturbation theory” - Why is it possible to use perturbation theory in 1d interacting fermionic case? It counters the fact that all the interaction effects for fermions in 1d are non-perturbative.
5. The form of states (33) and (34) is not really clear. What is l? What is j? A better explanation is needed for these notations.
6. The (36)-(37) transition I do not understand, should there be at least a reference to something that shows how to do such derivations?
7. As far as I understand the whole calculation of 32-38 was supposed to show that there is no difference in ground state energy and therefore no difference in the ground state. Is it a clear implication? Moreover later in 39-42 the Hartree shift, which is a shift in energy due to interaction (assuming that the ground state is the same), is calculated again, and here it is not equal to zero in the continuum limit. Does it mean that the whole derivation of 32-38 was wrong? In short, I do not find results (38) and (40),(42) consistent. One says that there is no change in energy in the continuum limit, the other gives the finite value of the shift.
8. The whole derivation of (42) is not clear. There is not even a single reference to all these seemingly very sophisticated techniques and facts mentioned in the paragraph above (42). Or do they mean that everything is explained in [19]?
9. What is \epsilon(q) in (52), it was not defined and calculated before, wasn’t it? So what does this comparison of \epsilon(q) and \epsilon_B(q) mean then?
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