SciPost Submission Page
Entanglement spectrum and symmetries in non-Hermitian fermionic non-interacting models
by Loïc Herviou, Nicolas Regnault and Jens H. Bardarson
This is not the current version.
|As Contributors:||Jens H Bardarson · Loïc Herviou|
|Submitted by:||Herviou, Loïc|
|Submitted to:||SciPost Physics|
|Subject area:||Condensed Matter Physics - Theory|
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.
Submission & Refereeing History
Reports on this Submission
Report 1 by Gunnar Möller on 2019-10-9 Invited Report
1- good review of background information regarding the Hamiltonian and density matrix for both Hermitian and non-Hermitian systems
2- convincing extension of the definition of entanglement spectrum for non-Hermitian systems, for the two possible definitions of reduced density matrices
3- generalisation of Wick's theorem and the corresponding extension of how to calculate the entanglement of free fermions from the correlation matrix of non-Hermitian Hamiltonians
4- comprehensive classification of possible entanglement spectra as a function of the non-Hermitian symmetries present in the original Hamiltonian
5- range of specific examples, including both analytical and numerical studies
1- The paper does not offer many illustrations of the complex relations between the possible symmetries. Reading could be facilitated if the authors add an illustration of the symmetry and generic structure of the entanglement spectrum, at least for some of the examples discussed in Section IV.
This is an excellent paper, which introduces and studies two different notions of entanglement spectrum for non-Hermitian systems. The two possible definitions of ES arise naturally from the alternative use of left and/or right eigenvectors in constructing the system's density matrix.
The analytical results (see strenghts) are convincingly derived, and backed up by the study of a range of example systems. Thanks to their focus on non-interacting systems, the authors can study large system sizes, which display the features of the non-Hermitian ES. The authors believe that these will essentially carry over to interacting systems, which would imply that the results will enjoy significant importance in a range of different physical systems.
The scientific aspects are clearly sound, so this paper has my strong recommendation.
I have one minor comment about a loosely worded sentence in the conclusions, and a list of minor changes, given below.
1- In their conclusions, the authors should clarify the following sentence: "It appears then, that the bulk-boundary correspondence holds for the ES in line-gapped Hamiltonians, when considering the bi-orthogonal density matrix." How strong can this point be made, and if needed, what qualifications of this statement apply?
2- No comment is given to whether the locations of phase transitions seen for the singular values and eigenvalues agree in Figures 6 and 8. Given the scale, it is difficult to discern if they coincide, so could the authors add some vertical lines as visual guides to clarify the location of the other transition, respectively, in the different panels. Some discussion of this point would also be helpful.
3- In equation (3) and thereafter, the typesetting of $\vec c^\dagger$ requires some attention, as the vector and dagger signs overlap.
4- On page, three, below (12), the authors refer to equation (8), where I believe the reference should be to (10). This referencing error is repeated below (14).
5- On page 4, discussing the case of non-diagonalisable correlation matrices, the authors assert that the Jordan blocks of $H_E$ and $C$ are identical, while their form bases differ. Please expand on this assertion, if needed within an appendix.
6- The authors should point out explicitly that the resulting entanglement spectrum (25) follows the same relation to the correlation spectrum as in the Hermitian case.
7- On page 8, the sentence "[...] symmetries do not carry on to the right density matrix [...]" is unclear.