Quantum dynamics in 1D lattice models with synthetic horizons

We thank the referees for their constructive feedback on our manuscript. We agree that the manuscript wasn’t explicit enough about its connection to Ref. [42], and have now made this connection entirely clear, both in the introduction and in the conclusion. We reply to the other comments in detail below.

First Referee

1- The authors might not be surprised to learn that velocity profiles for the Dirac equation have been extensively considered in the graphene literature. In particular, it would be worthwhile to compare the results of this paper with J. Phys. Condens. Matter 21 095501 (2009), where the gamma=1 case is solved in the continuum (sec. 3.4) and a slightly different tight binding model is presented (appendix A). In this reference, the analytical solution for gamma=1 predicts that wave-packets are always transmitted (when there is a regular lattice on both sides of the v(x) = x region). This appears inconsistent with the formation of a horizon described here. Is the horizon related to the presence of open boundary conditions more than to the power law hopping?

We thank the referee for pointing out this relevant reference to us. However, it appears to us that it considers a linear variation of the mass, not the tilting, as a function of x. Moreover, wave packets may be transmitted in the Graphene model by scattering to another band. Indeed, unlike the two-band model considered in this reference, our work considers a model a single band.

2- Given the several analytical solutions presented, in particular for p =pi/2, is there any physical insight to explain why gamma=1 is the critical case for the emergence of an horizon? This result does not appear characteristic of Dirac fermions, but rather of the metric considered, and its geodesics.

We agree with the referee that the specificity of gamma=1 originates from the metric. That this value is special can be seen on the one hand from the fact that the metric at gamma=1 case corresponds to Rindler spacetime or uniform acceleration in a general relativistic picture. On the other hand, as discussed in Ref. [42], gamma=1 is a critical value above which the density of states diverges at zero energy in the limit of an infinite lattice. Therefore, the specific value gamma=1 provides a direct relation between the GR and condensed matter perspectives of the model, by observing that the presence of a horizon defined by at least uniform acceleration is also associated with the presence of a sufficient (diverging) density of states available at zero energy.

3- Another interesting point is the analytical solution for the eigenstates for gamma=1 and 1/2. Given Ref. [42] presented the DOS for these cases, can one understand the exact form of the DOS peak in the gamma=1 case? Does this analytical solution provide insight on why gamma=1 develops a singularity?

We thank the referee for bringing up this point. In fact, the mathematical properties of DOS and specially the peaks have been explored in our separate work (Ref. 46). There, we thoroughly explored the DOS of lattice models with general position-dependent hopping, and we prove that the singularity of the DOS for gamma=1 is of a logarithmic form. The analytical solutions presented here for the eigenfunctions cannot be used directly establish the energy dependence of the DOS.

4- A minor point: the effective metric in the continuum Dirac equation in Ref. [42] is different from the one derived here. Footnote [19] explains the coordinate change needed to bring this metric to the diagonal form in the current work. Again, for the benefit of the reader comparing the two works it would be worth mentioning this coordinate change and the relation with the anti de Sitter metric here as well.

We agree with the referee and point this out in the revised version of our manuscript.

5- The authors state that the Dirac equation with v(x) represents a Dirac field in a background metric given by Eq. 7, without providing any derivation or citation. This is a subtle point, as Dirac fermions do not couple to the metric, but to the tetrads or vielbein (see standard book “Quantum fields in curved space” by Birrel and Davis for example). Since a lot of the motivation of this paper is the comparsion with gravitational physics, I think this derivation could be fleshed out in more detail.

We thank the referee for this suggestion. In the revision, we added an appendix for the standard derivation of the Dirac equation (Eq. 6) in the presence of the metric given by Eq. 7.

6- Why is particle-hole symmetry discussed in Eq. 2? It is not mentioned at all in the rest of the paper.

We thank the referee for pointing out our lack of explanation here. The presence of an explicit particle-hole symmetry in the lattice model plays a role in the success of the low-energy effective description in terms of a Dirac field subjected to a background metric. The relativistic Dirac field spectrum necessarily has a particle-hole symmetry and therefore, in lattice models which already possess either particle-hole or chiral symmetry, the low-energy physics is more likely of Dirac form. We added a few lines to the revised manuscript to better explain this connection.

Second Referee

1- In my opinion, the transformation (2) should not be called a particle-hole symmetry. Any particle-hole symmetry should swap particles and holes of a given vacuum and thus should be anti-unitary. In contrast, the transformation (2) is a unitary transformation, so I would not call it a particle-hole transformation. Also, following another referee, I am curious about the importance of the present symmetry in the manuscript.

We thank the referee for making this important point, and agree this was not sufficiently clear in the original manuscript. As the referee states, any proper particle-hole symmetry (PHS) operator is anti-unitary and is thus always accompanied by complex conjugation {\cal K}. Since the Hamiltonian considered here is real, the complex conjugation does not affect it, and we had represented the PHS by a unitary operator U in the original manuscript. In the revised version we introduce the more precise definition of the PHS operator {\cal P} = {\hat U} {\cal K}, which is anti-unitary and also anti-commutes with the Hamiltonian (because the U operator does). Acting on on wave functions with this form of the PHS operator further illuminate its action on electron and hole states in an intuitive manner.

Finally, as mentioned in response to the first referee, the significance of the PHS symmetry is that it guarantees the possibility of an effective low-energy description by the Dirac equation, which always has a particle-hole symmetric spectrum.