Universal non-Hermitian flow in one-dimensional PT-symmetric quantum criticalities

Analysis of a timely problem

Lack of clarity, lack of a complete bibliography, unclear statements on boundary conditions (see the requested changes)

This paper investigates the finite-size behavior of the ground state energy in the non-Hermitian Su-Schrieffer-Heeger (SSH) model. This system exhibits several phases distinguished by their topological properties, making it a popular model for understanding topological phases and entanglement in the absence of Hermiticity. The different phases of the model are separated by critical points described by the $bc$-ghost conformal field theory (CFT) with a central charge $c=-2$.

The authors first numerically verify the finite-size scaling behavior of the ground state energy at these critical points, as predicted by CFT. The main goal of the paper is to examine how this scaling behavior changes as the imaginary chemical potential, which renders the model non-Hermitian, is varied. When this parameter is zero, the system reduces to the Hermitian tight-binding model, described by the massless Dirac fermion with a central charge $c=1$.

Therefore, the paper analyzes the crossover in the finite-size scaling of the ground state energy between a non-unitary CFT with $c=-2$ and a unitary CFT with $c=1$. By taking the continuum limit of the non-Hermitian SSH model, the authors derive an analytical scaling function that describes this crossover and verify it against exact numerical results on the lattice.

The paper might be suitable for being published in SciPost Core after a major revision that should take into account the following points. The bibliography should be expanded to include all the mentioned references.

1. The finite-size scaling behavior of the ground state energy in non-unitary CFTs is a well-established result that has been previously verified. The expressions in Eq. (4) are generally valid only for unitary CFTs. As shown in EPL 2, 91 (1986), for non-unitary CFTs, these expressions must be modified by replacing the central charge $c$ with the effective central charge $c_{\text{eff}} = c - 24 h_0$, where $h_0$ is the conformal dimension of the lowest primary field, which can be negative. In the case of the $bc$-ghost theory, the lowest primary field has a conformal dimension $h_0 = 0$, so $c_{\text{eff}} = c = -2$, and the expressions in Eq. (4) remain valid. The same modification applies to the entanglement entropy, where the formula $c/3 \log l$ valid for unitary CFTs should be replaced by $c_{\text{eff}}/3 \log l$ in non-unitary CFTs, as found in Ref. [65]. The discussion on the finite-size scaling of the ground state energy in non-unitary CFTs should be amended to account for the effective central charge, which is also relevant when different boundary conditions are considered.

2. At the end of the conclusions, the authors highlight that in Ref. [66], it is found by studying the entanglement entropy that the effective central charge of the model is $c_{\text{eff}} = -2$ for periodic boundary conditions (PBC) and $c_{\text{eff}} = 1$ for antiperiodic or open boundary conditions (OBC). As discussed in 2304.08609, the reason why $c_{\text{eff}} = 1$ in OBC is that the lowest primary in that case has a conformal dimension $h_0 = -1/8$ and, therefore, $c_{\text{eff}} = c - 24 h_0 = 1$. In the case of PBC, the lowest conformal weight is $h_0 = 0$ and $c_{\text{eff}} = c = -2$. The fact that the authors obtain $c_{\text{eff}} = -2$ from the finite size scaling for OBC would mean that either they are actually considering PBC or they are not choosing the true ground state but the conformal vacuum. The authors should revise this point as it is crucial in their calculations in Sec. 4.

3. It is not clear why the limit $|\Lambda|\to\infty$ corresponds to taking OBC for the CFT and the point $\Lambda=0$ to PBC. If one fixes $|t_2-t_1|=\gamma$ and varies $\gamma$, the boundary conditions of the Hamiltonian (1) do not change (in particular at the Hermitian point $\gamma=0$), unless the authors are doing something else that they do not explicitly specify. This should be clarified.

4. The cross-over between a non-unitary CFT with central charge $c=-2$ and a unitary CFT with $c=1$ has been also considered in J. Phys. A 41, 295206 (2008) in a lattice model very similar to the non-Hermitian SSH chain: the XX spin chain with imaginary staggered magnetic field. The authors should mention it and perhaps compare their findings with those of that reference. Other example of flow from a unitary to a non-unitary CFT is the Ising->Lee-Yang flow studied in J Stat Phys, 110, 3 (2003). A version of the c-theorem for the effective central charge $c_eff$ has been proposed in J. Phys. A 50, 424002 (2017), which is also valid for some unitary to non-unitary flows as the Ising->Lee-Yang case.

5. In the introduction, it seems that the entanglement entropy in non-unitary CFTs has been generically studied in Ref. [66]. However, Ref. [66] only analyses the entanglement entropy of the non-Hermitian SSH model. The entanglement entropy of generic non-unitary CFTs has been examined in Ref. [65], Nucl. Phys. B 896, 835 (2015), J. Phys. A: Math. Theor. 49 (2016) 154005, Phys. Rev. Lett. 119, 040601 (2017), and SciPost Phys. 4, 031 (2018). The entanglement in the non-Hermitian SSH model has been further studied in Phys. Rev. B 107, 205153 (2023), J. Stat. Mech. (2024) 063102, 2304.08609, and Ref. [63].

6. Before Eq. (10), what is $\psi_A$ and $\psi_B$?

7. At the end of Sec. 2, "while keeps" => "while keeping"

8. At the beginning of Sec. 3, "Any states in CFT" => "Any state in CFT"

9. After Eq. (4), "resultin" => "resulting"

10. In the second paragraph of Sec. 3.2.2, "continnuously" => "continuously"

Ask for major revision

Not sure.

Seems to re-invent well known facts, peppering them with a bit of new language, and betrays embarrassing ignorance of the literature on the side of the authors.

This paper is almost exclusively formulated in the language of topological phase transitions, but from a technical point of view, it simply revisits well known facts about spin chains, non-unitarity and quantum criticality. The total lack of reference to the relevant literature in this area is embarrassing to say the least, and prevents one from appreciating what might be new in the paper, if anything. I think a complete rewriting is necessary before publication can be considered. It may then appear that the authors have discovered something interesting, but as of now, I cannot judge.

The free fermion model considered by the authors becomes an XX chain with boundary terms after a Jordan-Wigner transformation. The correspondence should be discussed, and the results compared with what is known in this field - see e.g. the paper by Hinrichsen Rittenberg Physics Letters B 275 (1992) 350-354 and the many references therein. See papers by Pasquier Saleur about quantum groups, see papers by Nichols Rittenberg and others about Temperley and two-boundary Temperley-Lieb algebras.

The emergence of c=-2 with open BC and c=1 with periodic BC is well known in the context of fermionic CFTs. See papers by Kausch, the whole literature about dimers, papers by Ruelle and coworkers about sand piles etc. The author's discussion of this point in the paper is particularly naive. The behavior of levels of Hamiltonians for non-unitary CFTs is studied in an old paper by Itzykson Saleur and Zuber. The behavior of entanglement in non-unitary systems is studied in a paper by Couvreur Jacobsen Saleur, and discussed in great detail in papers by Castro-Alvaredo, Doyon and Ravanini.

The study of scaling functions of central charge or effective central charge in non-unitary CFTs is familiar, and so is the zig zag behavior observed by the authors. See papers by A. Zamolodchikov, on his own or with Fendley Saleur.

I think the authors should do the job of situating their paper in the vast literature on the subject, and explain clearly what it is they think they have discovered that is new and interesting.

Ask for major revision