Reply to Comment on Perturbative calculations of quantum spin tunneling in effective spin systems with a transversal magnetic field and transversal anisotropy

the by Garg on our manuscript ‘ Perturbative calculations of quantum spin tunneling in effective spin systems with a transversal magnetic ﬁ eld and transversal anisotropy ’ . Within our work [ 1 ] we presented a perturbative solution for the resonant tunnel splitting energy E D of an arbitrary effective single spin system described by the following Hamiltonian: two formulas for the ground doublet energy splitting, one for the integer spins and one for the half-integer. For the integer spin case the formula has the following structure:

Further, we investigated the influence of the transversal magnetic field on the energy splitting for higher integer quantum spins and we introduced an exact formula, obtained by the ratio lim 1 Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
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The comment by Garg on our manuscript 'Perturbative calculations of quantum spin tunneling in effective spin systems with a transversal magnetic field and transversal anisotropy' contains two points. First, the author of the comment introduces a more compact version of our formula equations (2) and (3) for the tunnel splitting energy, Which agrees very well with our formula in equation (2) (see table 1).
We disagree with the implication by the author of the comment that his formula in equation (5) should replace ours (equations (2) and (3)), because our intention was not to generate a compact formula, but rather to derive formulas which can distinguish between mixed ( B K x D ) and pure ( B K , x D D ) tunneling paths. We have chosen this approach in order to obtain a mathematical structure which enables us to make relations between the tunneling paths, without any further transformations.
To obtain a similar separated expression of the tunneling paths (like in our formalism) based on equation (5) by the author of the comment, one should have to do some transformations, which would make the formula less compact.
Second, the author of the comment disagrees with our interpretation of equation (4) regarding the energy splitting values. The author of the comment shows that the values of E D , generated by the parameters which fulfills our equation (4), leads to the energy splitting E 0 D = .
Here we agree with the author of the comment that our interpretation of equation (4) is not exact, because in our technical numerical procedure the tunneling terms do not cancel each other completely out as predicted by equation (4). The conclusion by the author of the comment is correct that these special values of B x lead to quenched tunneling points [2], where the energy splitting is exactly zero. This means that our formula in equation (4) generates parameters (B K K , and  figure 1(a) we show a spin s = 5 system, which is the smallest possible system where this effect appears. We see that under the value of B x = 0.021(arb.units), which depends on the parameters of K z and K, the mixed B K x D paths are vanishing. The spin s = 13 system in figure 1(b) demonstrates that the number of these critical values of B x , where the mixed B K x D paths are vanishing, depends on the spin quantum number. It is important to mention that the number of these B x values has the tendency to increase with increasing spin quantum number. Moreover, this increase seems not to be monotonous: for spin s = 7 there are three critical values of B x , but for spin s = 8 only two.
These values differ for each spin number. They are also different from the values of the quenched tunnel splitting, and lead to a linear combination of two pure paths B x D and K D (see equation (2)) for integer spins and to the single contribution B x D (see equation (3)) for the half integer spins. This leads to the conclusion that in contrast to the quenched tunnel splitting, where the energy splitting is vanishing and so the quantum spin tunneling, our formalism reveals a situation where quantum spin tunneling occurs but the influence of the transversal magnetic field is drastically reduced, both for integer and half-integer spins. We can interpret this Table 1. Comparison between the energy splitting E D obtained by the formula of the author of the comment ( E comment D ) and by our formulas from the original publication ( E ours D ). Shared parameter K z = 1.0 (meV). Parameters for spin s = 3: B x = 0.05 (meV), K = 0, 002 (meV). Parameters for spin s = 4: B x = 0.002 (meV), K = 0, 01 (meV). Parameters for spin s = 5: B x = 0.01 (meV), K = 0, 06 (meV).  Here we show a spin s = 13 system to illustrate the spin quantum number dependence for the number of the vanishing points. We see that the spin s = 13 system contains, in contrast to the spin s = 5 system, five vanishing points for B x . Parameters for spin s = 13: K z = 1 (arb.units), K = 0.001 (arb.units).