Continuous N\'{e}el-VBS Quantum Phase Transition in Non-Local one-dimensional systems with SO(3) Symmetry

Dear editor,

We sincerely apologize for the delayed response. This year has been very stressful for many of us.

We want to thank both referees for their very careful reading of our paper, and their helpful questions/comments. Following is our response to the reports, and summary of changes according to each comment from the report.

Response to report 2:

1, We view our system as a two-layer system: layer-2 is an ordinary Wilson-Fisher critical point in the 2+1d bulk, and layer-1 is a SPT state in the 2+1d bulk with a nontrivial 1+1d boundary state. Then we couple the 1+1d boundary of layer-1 to layer-2. After introducing fractional dimensional epsilon, our system becomes 3-epsilon dimensional critical point (4-epsilon space-time dimension) from the bulk of layer-2, but still coupled to the 1+1d boundary of layer-1. We do not virtually generalize the spatial dimensionality of the layer-1 boundary, and we do not view the boundary of layer-1 as a 2-epsilon dimensional system. Hence the epsilon expansion is only for the layer-2, which is a standard Wilson-Fisher fixed point. And indeed for the Wilson-Fisher fixed point (and its boundary) the epsilon expansion is controlled. We have clarified this in our revised manuscript.

2, This boundary condition is for layer-2, which is always at the critical point in the bulk, hence there is no long range order in layer-2 at least in the bulk. Hence we believe it is fine to take the standard boundary condition of layer-2. The main physics we discuss is the novel CFT-like fixed point at the coupled 1+1d boundary, which also has no long range order. Then our discussion is based on the vicinity of the novel fixed point at the coupled 1+1d boundary. If the layer-1 forms a long range order at the 1+1d boundary due to the runaway flow from the fixed point, this long range order can in principle induce an order at the boundary of layer-2 through a “back-reaction”. But this back-reaction will involve higher order effects of the coupling between the two layers, which should correspond to higher-order in the epsilon expansion, because the coupling between the two layers is at the order of epsilon, if we focus on the vicinity of the fixed point at the coupled 1+1d boundary.

3, Lines 2 and 3 in Eq.5 are not Gaussian approximation of the bulk theory. The anomalous dimension from the Wilson-Fisher fixed point is already accounted for in Eq.5 (to the order of epsilon), which means that Feynman diagrams of self-interaction of the order parameter from layer-2 are already taken into account in Eq.5. So line-2 and 3 should be valid to the leading order of epsilon. Then we take Eq.5 as the starting point of studying the interaction between the layer-2 and layer-1. There could be further corrections to the “effective actions” of the Phi fields in line-2 and 3 in Eq.5, but these corrections should correspond to higher order expansion of epsilon, either from higher order epsilon-expansion of the standard boundary theory of the Wilson-Fisher fixed point, or from correction from the coupling between the two layers at the boundary. Let us again keep in mind that the couplings between the two layers (g_n and g_v in Eq.5) are also at order of epsilon in the vicinity of the main fixed point we are dicussing, so if we include the corrections from g_n and g_v to the effective action of \Phi fields in Eq.5, these corrections will be at least at the g_n^2 or g_v^2 order, which means epsilon^2 order.

4, Yes we want to explain the observation from numerics that the scaling at the AKLT boundary is not universal when the bulk is tuned to a critical point, i.e. the scaling at the boundary depends on the coupling constants at the boundary. Our interpretation is that the models used in the numerical references are in principle in the AFM ordered side of our RG flow diagram, but this long range order may emerge only at very large RG scale. So at different scale away from the fixed point (tunable by the boundary coupling constant) the system may appear to have different scaling behavior.

5, Indeed, more precisely we should have said that an instantaneous long range interaction between spins in the 1d spin-chain considered in Ref.8,9, are analogous to the effective 1d theory Eq.5 after taking the velocity of the effective action of \Phi to infinity. By the way here we view Eq.5 as an independent 1+1d theory, rather than a theory inferred from a bulk theory. We have clarified this in our revised manuscript. In our calculation we took the velocity of the effective action of \Phi to be larger than the velocity of the original SU(2)_1 CFT, and indeed the velocity of the CFT increases under RG, and flow to the velocity of the effective action of \Phi in Eq.5. This “acceleration” under RG will lead to dynamical exponent z < 1, which is qualitatively consistent with what was observed numerically.

Response to report 1:

Response to general comments:

Yes we are only generalizing the bulk dimensionality of layer-2 to 3-epsilon, the boundary dimension of layer-1 (AKLT layer) is always kept at 1d.

Although the flow of velocity in our work was done perturbatively, the “acceleration” of the CFT modes, and the fact that z < 1, is indeed (and should be) qualitatively consistent with the 1d spin-chain with nonlocal interaction observed numerically. Also, the fact that we found the Neel order parameter has a larger scaling dimension than the VBS order parameter at the deconfined critical point is also consistent with what was observed numerically. Hence we believe at least at the qualitative level the comparison/connection between our theory and numerics is worth making.

Response to other points:

1, We have added more clarification about the difference between the SU(2) SPT state and AKLT state.

2, Yes we believe that the SPT-to-order state transition should be generally the same as the Wilson-Fisher transition. The SPT states can in many cases be described by a nonlinear-sigma model with a topological Theta term at Theta = 2Pi, and at Theta = 2Pi, the Theta term is reduced to boundary terms. We do not expect the Theta = 2Pi topological term to modify the universality class in the bulk.

3, We have added brief discussions about other choices of boundary conditions.

4, This is a very good question. Yes the situation we are discussing here is different from the extraordinary transition in the standard boundary criticality literature. The boundary of layer-1 cannot be a trivial disordered phase in principle, protected by topology; while the ordinary boundary system can enter a trivial disordered phase. Hence in our system, when the bulk is tuned closer and closer to a VBS (Ising) transition, the boundary will go through a transition between the “gapless spin chain phase” with SU(2)_1 CFT, to a VBS phase, before the bulk hits criticality. This boundary transition should be in the same universality class as the transition from an SU(2)_1 CFT to a VBS phase in a purely one-dimensional spin-1/2 chain with both nearest and next nearest neighbor Heisenberg interactions. This transition is not an ordinary 1+1d Ising transition and, hence, is different from the ``extraordinary transition" studied in the standard boundary criticality. We have explained this in the revised manuscript.

1, We further clarified our epsilon expansion. As was noted by referee-1, we are only generalizing the dimensionality of "layer-1", i.e. the layer with Wilson-Fisher criticality in the bulk. The boundary of the SPT phase is fixed at one dimension.

2, We added explanation about choosing the ordinary boundary condition of "layer-2", and briefly discussed other possible boundary conditions in the standard boundary criticality literature (in the footnote).

The reason for choosing this boundary condition, and the nature of the effective action of \Phi and \vec{\Phi} in Eq.5 were also further explained in the revised manuscript. The effective action is not a Gaussian approximation, it already contained self-interaction between the critical modes from the bulk, at least to the leading order of epsilon expansion. In principle it will receive further corrections, for example from the coupling to \vec{n}, but that will be higher order effect in the epsilon expansion, because this correction involves higher order effect of the coupling constant g_n. In our calculation, g_n, and g_v will be at order of epsilon, if we focus on the physics around the fixed point we found.

3, We have clarified our understanding of the velocity flow. We view Eq.2,4, and 5 together as an effective 1d theory, and what we meant was that, the instantaneous spatial interaction introduced in long range spin-chain, is equivalent to taking the velocity of the action of \vec{\Phi} to infinity in Eq.5. Although the flow of velocity in our work was done perturbatively, the “acceleration” of the CFT modes, and the fact that z < 1, is indeed (and should be) qualitatively consistent with the 1d spin-chain with nonlocal interaction observed numerically.

Also, the fact that we found the Neel order parameter has a larger scaling dimension than the VBS order parameter at the deconfined critical point is also consistent with what was observed numerically. Hence we believe at least at the qualitative level the comparison/connection between our theory and numerics is worth making.

4, We have added more clarification about the difference between the SU(2) SPT state and AKLT state.

5, As was suggested by referee-1, we added text and a footnote to explain that we believe the SPT-to-ordered state phase transition, is in the same universality class as the ordinary disorder-to-order phase transition.

6, We have explained that, indeed, when the boundary forms VBS order before the bulk in our case, the transition at the boundary is different from the standard "extra-ordinary transition" in boundary criticality literature.

Other typos and broken references have been fixed.