# Entanglement spectrum and symmetries in non-Hermitian fermionic non-interacting models

### Submission summary

 Authors (as Contributors): Jens H Bardarson · Loïc Herviou
Submission information
Date accepted: 2019-11-13
Date submitted: 2019-10-18 02:00
Submitted by: Herviou, Loïc
Submitted to: SciPost Physics
Ontological classification
Specialties:
• Condensed Matter Physics - Theory
• Condensed Matter Physics - Computational
Approaches: Theoretical, Computational

### Abstract

We study the properties of the entanglement spectrum in gapped non-interacting non-Hermitian systems, and its relation to the topological properties of the system Hamiltonian. Two different families of entanglement Hamiltonians can be defined in non-Hermitian systems, depending on whether we consider only right (or equivalently only left) eigenstates or a combination of both left and right eigenstates. We show that their entanglement spectra can still be computed efficiently, as in the Hermitian limit. We discuss how symmetries of the Hamiltonian map into symmetries of the entanglement spectrum depending on the choice of the many-body state. Through several examples in one and two dimensions, we show that the biorthogonal entanglement Hamiltonian directly inherits the topological properties of the Hamiltonian for line gapped phases, with characteristic singular and energy zero modes. The right (left) density matrix carries distinct information on the topological properties of the many-body right (left) eigenstates themselves. In purely point gapped phases, when the energy bands are not separable, the relation between the entanglement Hamiltonian and the system Hamiltonian breaks down.

Published as SciPost Phys. 7, 069 (2019)

Most changes follow from the referee reports

### List of changes

Changes to the main text
- Page 3: corrected wrong reference to Eq. (10)
- In Sec III: clarified discussion on Jordan blocks
"If the correlation matrix is diagonalizable, [...] will generically be different in the two matrices."
- In Sec III: added the sentence "Note that the formula (25) is the same as in the Hermitian case."
- In Sec IVB: clarified our assertion and replaced "In particular, the $T_-$ and $P_+$ symmetries[...] line gap classification."
by "In particular [...] its topological properties."
- In Sec VI, we clarified that singular values and eigenvalues undergoes transition at the same moment, for Fig.3, 6 and 8.
- Added Ref. 87 at the beginning of Sec VII A
- In Conclusions: developed the discussionon the bulk-boundary correspondence and replaced
""It appears [...] density matrix."
by
"It appears [...] bulk-boundary correspondence generally holds."

- Added the following sentence, referencing a related work published on arXiv shortly after ours.
"Following this work, a similar approach was developed in Ref. 92 to study the entanglement entropy and therefore the effective low-energy conformal field theory describing critical points in non-Hermitian models."

Update to the acknowledgements
- Added "N.R. was supported by [...]Memorial Foundation."

Change in Figures
- Updated Fig. 3, 6 and 8 and their caption to underline when the entanglement spectrum undergoes a phase transition

Change in Appendix
-Added the following two sentences to App. A
"Each eigenspace (corresponding to independent Jordan blocks) can be treated separately."
"Due to the matrix $P$, the basis in which $H_E$ takes a Jordan form is not the same as in $C$."

Formatting changes
- changed the spacing from \vec{c}^\dagger to \vec{c}^{\: \dagger}
- Appendix to App. when relevant

### Submission & Refereeing History

Resubmission scipost_201909_00002v3 on 18 October 2019
Submission scipost_201909_00002v1 on 4 September 2019

## Reports on this Submission

### Strengths

several, see previous report

### Weaknesses

no significant weaknesses, see previous report

### Report

The authors have addressed all of my criticisms very well, so I can warmly recommend the paper in its current form.

I also thank the authors for adding a reference to a related work published on this topic.

### Requested changes

None.

• validity: top
• significance: high
• originality: high
• clarity: high
• formatting: perfect
• grammar: excellent