Exact physical quantities of a competing spin chain in the thermodynamic limit

The problem is interesting, see attached pdf.

The applied methods are cumbersome and the results are not as explicit as they can be, see attached pdf.

See attached pdf.

Many calculations can be extremely shortened.

1- The authors compute various properties of an integrable spin chain with several couplings within an approach based on the roots of the corresponding transfer matrix eigenvalues.

1- The implicit assumptions used in the analysis of the Bethe equations restrict their use to the computation of properties in the thermodynamical limit (and therefore, in the case of the isotropic model, effectively parallel boundary fields)

2- Results are not discussed in the context of previous ones obtained using conventional Bethe ansatz methods.

The authors apply the Bethe ansatz scheme introduced in Refs. [22,23] to study bulk and boundary properties of the Heisenberg spin chain with competing interactions constructed from an inhomogeneous six vertex model with open boundary conditions. The scheme is based on the discrete set of discrete inversion relations (2.17). In this formulation boundary conditions breaking the $U(1)$ symmetry of the bulk system appear not to lead to the complications when dealing with unparallel boundary fields in the conventional TBA approach.

For the derivation of the integral form (4.2) of the Bethe equations, however, the authors have implicitely assumed that the inversion relations hold for a continuous variable $u$ -- not just at the special points $u=\theta_j$. This is not correct: for example, in this approach the spectrum would depend only on the effective parameters $p$ and $\bar{q}$ appearing in the eigenvalues of the boundary matrices (which determine the absolute value of the boundary fields) but not on the relative orientation of the boundary fields leading to the breaking of the $U(1)$ symmetry! In fact, it is well known that equations (2.17) only hold up to terms vanishing as powers of $(u-\theta_j)$.

Ignoring this fact (2.17) turns into a functional equation for the transfer matrix eigenvalues. Imposing constraints on their analytical properties (e.g. by considering a particular root pattern $\{z_j\}$) one selects an eigenstate and the functional equation can be solved directly by Fourier methods. As has been observed in previous works this yields the correct results for the bulk and boundary contributions to the corresponding energy in the thermodynamic limit (i.e. with boundaries being infinitely separated). Corrections of order $1/L$ can not be obtained in this way.

Therefore the results obtained in Section 4 based on the configurations of roots of the transfer matrix eigenvalues are correct although straightforward generalizations of what is known for the homogeneous Heisenberg model ($a=0$), see e.g. Grisaru et al., J. Phys. A28 (1995) 1027-1046 and Kapustin & Skorik, J. Phys. A29 (1996) 1629-1638.

Similarly, the spectrum of bulk elementary excitations (Sect. 5) has been studied for the periodic staggered ($a\neq 0$) spin chain before in Frahm & Rödenbeck, Europhys. Lett. 33 (1996) 47-52.

In Sect. 6 the authors study the boundary excitations and observe a different behaviour of their energy around $p=0$ for the homogeneous ($a=0$) and the staggered ($a\neq0$) spin chains. Given the $p$-dependence of the boundary term (2.3) this not really surprising.

In summary, most of the physical quantities considered in the manuscript are either known or straightforward extensions of known results which could have been obtained without using the 'novel Bethe ansatz scheme'. Moreover, the tacit assumption underlying the integral Eqs. (4.2) rules out an application of the proposed scheme to studies of the finite size spectrum. This would be necessary to address the particularly interesting case of unparallel fields advertised in the abstract and the introduction. Given these limitations I see little potential for the proposed scheme beyond what has already been done using established Bethe ansatz methods.

Therefore, I can not recommend to accept this manuscript publication in SciPost Physics.

1- The authors should extend their discussion the effect of the inhomogeneities $\theta_j$ at the end of Section 2. Also, only in Fig.2 their choice underlying the numerical data is stated. I assume that $\theta_j\equiv 0$ in Figs. 3, 6 and 7 (i.e. alternating inhomogeneities $\pm a$) but that should be clearly stated in the figure captions.

2- Summation indices in (4.1) should be $l$ and $k$, not $j$.

3- Where in the derivation of (4.3) has $\sigma(\theta)=\delta(\theta)$ been used?

4- The boundary elementary excitation (Sect. 6) should described more clearly: it is unclear how the root configuration of the excitation displayed in Fig. 7(a) for parameters from regime III can be obtained from the ground state one similar to that in Fig. 3(a) by just changing the boundary string (i.e. what happens to the roots at $\pm \alpha$ and $\pm i\beta$)?