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The negativity contour: a quasi-local measure of entanglement for mixed states
by Jonah Kudler-Flam, Hassan Shapourian, Shinsei Ryu
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Submission summary
| Authors (as registered SciPost users): | Jonah Kudler-Flam |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/1908.07540v1 (pdf) |
| Date submitted: | 2019-09-04 02:00 |
| Submitted by: | Kudler-Flam, Jonah |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In this paper, we study the entanglement structure of mixed states in quantum many-body systems using the $\textit{negativity contour}$, a local measure of entanglement that determines which real-space degrees of freedom in a subregion are contributing to the logarithmic negativity and with what magnitude. We construct an explicit contour function for Gaussian states using the fermionic partial-transpose. We generalize this contour function to generic many-body systems using a natural combination of derivatives of the logarithmic negativity. Though the latter negativity contour function is not strictly positive for all quantum systems, it is simple to compute and produces reasonable and interesting results. In particular, it rigorously satisfies the positivity condition for all holographic states and those obeying the quasi-particle picture. We apply this formalism to quantum field theories with a Fermi surface, contrasting the entanglement structure of Fermi liquids and holographic (hyperscale violating) non-Fermi liquids. The analysis of non-Fermi liquids show anomalous temperature dependence of the negativity depending on the dynamical critical exponent. We further compute the negativity contour following a quantum quench and discuss how this may clarify certain aspects of thermalization.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 2) on 2020-2-5 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1908.07540v1, delivered 2020-02-05, doi: 10.21468/SciPost.Report.1480
Report
In this manuscript, Kudler-Flam and company investigate the negativity contour for several models, including ones which have a Fermi surface. The negativity contour can be considered a generalization of the entanglement contour, which has been used to investigate which degrees of freedom contribute to the entanglement entropy. This generalization is particularly useful given the rise of interest in mixed states and thermalization of quantum many-body systems. I am confident this work is solid and timely and will be of interest to the community. I recommend publication after a few minor issues, listed below, have been addressed.
Requested changes
1) The authors should add a sentence about negativity and advantages of using logarithmic negativity over it.
2) The authors should include a Figure for Eq. 14.
3) The authors should consider citing Phys. Rev. B 100, 241108(R) 2019, which investigates entanglement contours after a quench in momentum-space instead of real-space and show only momentum-space degrees of freedom near the Fermi surface contribute.
4) How does logarithmic negativity decay with time (for fixed position in Fig. 4)? It would be nice for the authors to another plot at fixed position as a function of time.
5) f and f^dagger do not appear to be defined near eq. 42. The authors should define these operators.