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Entanglement Hamiltonian and effective temperature of non-Hermitian quantum spin ladders
by Pei-Yun Yang, Yu-Chin Tzeng
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Submission summary
| Authors (as registered SciPost users): | Yu-Chin Tzeng |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202409_00024v1 (pdf) |
| Date submitted: | 2024-09-20 13:54 |
| Submitted by: | Tzeng, Yu-Chin |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Quantum entanglement plays a crucial role not only in understanding Hermitian many-body systems but also in offering valuable insights into non-Hermitian quantum systems. In this paper, we analytically investigate the entanglement Hamiltonian and entanglement energy spectrum of a non-Hermitian spin ladder using perturbation theory in the biorthogonal basis. Specifically, we examine the entanglement properties between coupled non-Hermitian quantum spin chains. In the strong coupling limit ($J_\mathrm{rung}\gg1$), first-order perturbation theory reveals that the entanglement Hamiltonian closely resembles the single-chain Hamiltonian with renormalized coupling strengths, allowing for the definition of an \textit{ad hoc} temperature. Our findings provide new insights into quantum entanglement in non-Hermitian systems and offer a foundation for developing novel algorithms, such as applying finite-temperature Density Matrix Renormalization Group (DMRG) to non-Hermitian quantum systems.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2024-10-25 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202409_00024v1, delivered 2024-10-25, doi: 10.21468/SciPost.Report.9979
Strengths
The main calculation is correct and well explained, reporting all details necessary to reproduce it.
Report
The authors study a non-Hermitian Hamiltonian, a ladder of two coupled XXZ chains with non reciprocal interaction.
They consider the limit of strong coupling between the two chains (the "rungs" of the ladder), treating the intrachain interaction as a perturbation.
At first order in perturbation theory, the authors show that, tracing out one of the two chains, the entanglement Hamiltonian of the other one in the ground state of the ladder is in form analogous to the non-reciprocal XXZ Hamiltonian of the single chain, with a "renormalised" anisotropy parameter $\widetilde{\Delta} = \frac{1}{2}\left ( \Delta + \Delta^2 \right)$ and the same non-reciprocity $e^{\Psi}$.
The result of this paper is analogous to what happens in the Hermitian case in the reciprocal XXZ ladder at strong "rung" coupling.
The main computation is simple and well explained, presenting all the details necessary to reproduce the computation.
I recommend the publication in SciPost Physics Core.
Recommendation
Publish (meets expectations and criteria for this Journal)
Report #1 by Anonymous (Referee 1) on 2024-10-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202409_00024v1, delivered 2024-10-23, doi: 10.21468/SciPost.Report.9963
Report
The authors investigate the entanglement Hamiltonian of a non-Hermitian XXZ model on a ladder, focusing on the regime where the XXZ coupling between the two chains is much larger than the coupling within each chain. Using first-order perturbation theory, they demonstrate that the ground state entanglement Hamiltonian of a single chain is proportional to the chain's Hamiltonian, albeit with a modified XXZ anisotropy parameter.
While I don't find this result particularly surprising or groundbreaking, it is nonetheless a technical contribution that could merit publication in a specialized journal such as SciPost Physics Core.
I only have one comment that the author should address. Among the consequences of their result they say it "suggests that the ancilla trick applied for developing finite temperature Density Matrix Renormalization Group (DMRG) method, can be extended to non-Hermitian many-body systems as well". I am a bit confused by why they say so, since the "ancilla trick", as they call it, is a formal identity that enables the computation of thermal expectation values from the DMRG imaginary time evolution of pure states. It is widely used in finite-temperature calculations for 1-dimensional interacting systems and its applicability (in both Hermitian and non-Hermitian systems) does not rely on the fact that the ground state entanglement Hamiltonian of a ladder is proportional to the Hamiltonian. Therefore, I believe the authors should either elaborate more on this motivation or just cut it from the paper.
Recommendation
Ask for minor revision
Author: Yu-Chin Tzeng on 2024-10-27 [id 4904]
(in reply to Report 1 on 2024-10-23)We appreciate Referee 1's comments regarding the relevance of the ancilla trick in the context of our work. In response, we have removed the original discussion related to the finite-temperature DMRG method. Instead, we have reintroduced the ancilla trick as a well-established technique for studying finite-temperature properties, emphasizing its foundation in using maximally entangled states and imaginary time evolution. We now clarify that the proportionality of the entanglement Hamiltonian to the system Hamiltonian provides an alternative approach for exploring finite-temperature behavior. This alternative method could inspire experimentalists to design real systems that effectively simulate these finite-temperature properties. Our main work extends the knowledge from the Hermitian case to the non-Hermitian case, while the conclusion remains unchanged. We hope this addresses your concerns and strengthens the manuscript. Thank you once again for your valuable input.