SciPost Submission Page
Anderson Impurities In Edge States with Nonlinear and Dissipative Perturbations
by Vinayak M Kulkarni , N. S. Vidhyadhiraja
Submission summary
| Authors (as registered SciPost users): | Vinayak M. Kulkarni |
| Submission information | |
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| Preprint Link: | scipost_202502_00039v1 (pdf) |
| Date submitted: | 2025-02-18 14:40 |
| Submitted by: | Kulkarni, Vinayak M. |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
A nontrivial renormalization of impurity problems shown by anisotropic and pseudochiral (\(\mathcal{PC}\)) interactions as dissipative perturbations. Using renormalization group (RG) analysis, Fock-space diagonalization, and relaxation-time calculations, we explore how nonlinear dispersion and potential scattering terms lead to exceptional points (EPs) and Hermiticity breaking in weak to intermediate interaction regimes. Inspired by Shnirman et al.Phys. Rev. A {\bf 50}, 3453, showed that cubic nonlinearity induces non-Hermiticity(NH), and more recently Nakai et al.Phys. Rev. A {\bf 109}, 144203 show sensitivity to perturbation in non-Hermitian(NH) problems using condition number analysis and Fike Bauer Theorem. Here we derive an effective Kondo model with emergent anisotropic Dzyaloshinskii-Moriya (\(\mathcal{DM}\)) interactions leading to the breaking of Hermiticity under perturbative RG. The RG analysis identifies new dissipative fixed points with a scaling collapse in spin relaxation time. The model exhibits an extended Lie group structure due to \(\mathcal{PC}\)-symmetric degrees of freedom, which reveal dissipative fixed points without renormalizing the invariant. Extending to a two-impurity Kondo model, we confirm that anisotropy is necessary for \(\mathcal{PC}\) symmetry, and the RG equations show a "Sign Reversion" (SR) regime for anisotropic-Hermitian problems at critical nonlinear coupling \(J_{k^3}\). Additionally, local Hamiltonians in Fock space display topological features at these dissipative fixed points.
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
Current status:
Reports on this Submission
Report #1 by Pradip Kattel (Referee 1) on 2025-3-7 (Invited Report)
Strengths
1) The authors study a topological bath Hamiltonian that includes quadratic, cubic, and linear momentum–dependent terms. The study of higher-order effects is a timely and interesting topic.
2) The study of non-Hermitian phenomena, exceptional points, and emergent topological transitions in condensed matter systems is yet another timely subject.
3) The authors use a range of complementary techniques—such as renormalization group analysis, Fock-space diagonalization, and transport calculations—to study the impurity problem.
4) The prediction of emergent exceptional points and sign-reversion regimes in the RG flow are interesting.
Weaknesses
1) The paper is extremely heavy on formal derivations and lengthy algebraic manipulations. This way of presentation makes it extremely difficult to understand the main message of the paper.
2) The reliance on perturbative techniques in regimes that may be strongly interacting is not adequately justified. Specially since the authors study a Hermitian model and upon using perturbative method, end up with non-Hermitian Hamiltonian -- this begs the question whether the approach being used here is appropriate.
3) Citations are missing and appear as [?] in quite a few places such as "his non-interacting spectrum is similar to Rashba study [?] except
from nonlinear ..." and at the end of page 3 "we can also construct in operator form [?]"
Report
The paper needs some significant improvement in presentation styles. Right now, it is quite difficult to read through dense and detail derivations that are interleaved with physical discussion without a clear separation.
Even with that, there is more problem in the main motivation of the paper. The authors start by saying "Role [of Non-Hermiticity] in closed condensed matter systems remains less explored" while it is explored in open quantum systems in those described by Lindbladian and/or phenomenalogical non-Hermitian Hamiltonian. Then, they write down a Hermitian Hamiltnonian from which they some how obtain non-Hermitian model. For instance after Eq.7, the say "The above equation 7 satisfies the pseudo-chirality, as η Hη−1 = − H†, when the couplings are complex-valued." but it is hard to understand where these complex valued couplings microscopically arise from! It is hard to understand if the authors are saying that within their perturbative RG, the couplings are becoming complex valued or they are considering a non-Hermitian model with the parameters themselved being non-hermitian a priori.
Before the authors clarify this point and explain how they obtained this non-Hermitian model (i.e. whether they are claiming that the microscopically and systematically projecting the degrees of freedom they obtained NH model or they are considering the NH model from scratch), my questions would be different so as the first step I suggest the authors to write about this part in more details and explain the origin of non-Hermiticity. It is puzzling specially because they take this old Bethe Ansatz paper as the motivation where they applied Bethe Ansatz to some non-integrable model and saw that in three particle sector the eigen-values of the Bethe Ansatz wavefunction was complex.
On the typesetting side, there are usages of too many inconsistent punctuations throughout the paper. One of the most repetitive issue is no spacing after the full stop in many sentence.
Due to this concern, I suggest the authors to make the paper more lucid by clearly explaining the motivation and supplementing the derivation of the non-Hermitian model in detail.
Requested changes
1) Explain how the model with complex valued coupling arise from Eq.1 or Eq.2.
2) Explain why should a low-energy of a completly hermitian model exhibit non-Hermitcity?
Recommendation
Ask for major revision
Author: Vinayak M. Kulkarni on 2025-03-11 [id 5281]
(in reply to Report 1 by Pradip Kattel on 2025-03-07)Dear Editor,
We sincerely thank you for considering our manuscript and for arranging the referee report. We are also grateful to the referee for their time, effort, and valuable feedback.
On behalf of the authors,
Vinayak
Below are our responses to the referee’s comments:
We appreciate the referee’s key question regarding the emergence of a non-Hermitian model from a Hermitian one. To clarify, the model remains Hermitian after derivation. However, the renormalization group (RG) flow induces non-Hermitian features due to the anisotropic DM interactions: the (x,y)-plane interaction J_k and the Z-direction cubic interaction J_k3 renormalize according to the relation ( J_k = 1/2 \pm \sqrt(1 + m J_k3) . Depending on the sign of J_k3 +ve (ferromagnetic) or -ve (antiferromagnetic) exceptional points emerge. We will clarify this point in the revised manuscript.
The appearance of exceptional points is tied to the RG invariants, which are specific to these types of anisotropic interactions. As in Ref. [5], the impurity-relevant regime is analyzed by constructing the Hilbert space of the RG-flowed model using these invariants. We followed a similar procedure, substituting our RG invariants into the model. For m > 0 and J_ k3 < 0 , and vice versa, diagonalization reveals exceptional points — still without any intrinsic non-Hermiticity.
To further understand this unexpected result, we performed additional loop corrections, including potential scatterings. Interestingly, the RG invariant structure remains robust, even when non-Hermitian terms are introduced via potential scattering. This prompted us to examine possible symmetries of the model, leading to the discovery of pseudochiral (PC) symmetry in the presence of anisotropy.
Motivated by this, we intuitively explored a three-particle sector picture, suspecting that interactions between two chiral particles in the bath and the impurity might explain the observed behavior. We found no prior studies on three-particle Bethe ansatz approaches in the context of the Kondo problem.So we mentioned about work Shnirman et al.Phys. Rev. A {\bf 50}, 3453. We give our best to present better motivation for the manuscript.
Regarding the validity of perturbative RG and strong coupling limit we completely agree with referee. For example the local moment and Kondo singlet regimes can not be clearly distinguished however we believe in the regime J_0 > J_k, J_k3 this works where we find fixed points. Definitely there could be exact linearised bethe ansatz approach capturing all the phases including the FP's. It is also interesting to look at 3-particle sector in anisotropic DM model.
In response to the referee’s feedback, we will improve the presentation by clearly distinguishing the Hermitian nature of the model, except for the non-Hermitian contributions from potential scattering. We will also explicitly discuss how potential scattering may introduce non-Hermiticity.
We noticed many formatting issues after the referees comments and these will be corrected in the revised version.
Lastly, we acknowledge the citation error pointed out by the referee and will correct it with appropriate explanations.
We greatly value the referee’s insightful comments and are confident that the revisions will strengthen the clarity and impact of our work.
Sincerely,
Vinayak