Reply to "Comment on 'High-Spin Polaron in Lightly Doped CuO 2 Planes'"
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Lau, Berciu and Sawatzky reply:
In the precedingComment[1], Lee and Lee bring to attention their inter-esting variational calculation for the single band Hubbardmodel further reduced to a t - t ′ - t ′′ - J model [2]. Its mainresult was to reveal new low-energy one-hole states calledspin-bags (SB), possibly forming a continuum. SBs con-sist of a quasiparticle (QP) plus a spin-wave excited in theAFM background, and are found to cross below the QPband in some regions of the Brillouin zone (BZ). Basedon this, Lee and Lee claim that the SBs explain, withina one-band model, the spin- polaron that we recentlyfound in a three-band model [3]. They conclude that ourclaim that this model reveals physics that cannot be de-scribed within one-band models, i.e. in the framework ofZhang-Rice singlets (ZRS), is not justified.While superficial similarities exist between the SB andthe spin- polaron, we disagree that they describe thesame physics, on several grounds:(i) In the Supplemental Material of Ref. [3], we ruledout the possibility that the spin- polaron is a spin- polaron plus a free magnon, because its band lies belowthe continuum describing such states. It can be roughlythought of as a bound-state of a spin- polaron and amagnon, with a very distinct local spin structure aroundthe charge. The existence of such bound states, whichmight be a better analog of our spin- polaron, is notanalyzed for the one-band model, in Refs. [1, 2];(ii) As shown in our Fig. 2, the spin − polaron’sband has significant dispersion, comparable to that ofthe spin − polaron [3]. In contrast, the low-energy edgeof the SB continuum is rather flat throughout the BZ,see Fig. 1(b) of Ref. [2]. This striking difference in theirspectra is likely an indication of a very different nature ofthe two types of low-energy states. There is currently noevidence that the two models have comparable dispersionfor spin excitations, regardless of their nature.(iii) While the spin- polaron band crosses below thespin- polaron band in certain regions, just like the SBcontinuum is below the QP band in certain regions ofthe BZ, a careful comparison shows yet more differences.In our model, this happens in two separate regions, cen-tered at (0 ,
0) and ( π, π ). In the variational solution forthe one-band model, this happens in one larger regioncentered at k = ( π, π ) which, coincidentally, is the AFMorder vector. The difference is most clearly visible alongthe ( π, − (0 , π ) cut, where we find no crossing whereasthe variational calculation predicts the QP as the low-energy state only near ( π , π ). If bound-states were foundin the one-band model, the comparison would be worsesince this would further increase the crossing region.Such differences result in very different physics, eg. atthe nodal point. While the vanishing quasiparticle weightat (0 , π ) is explained as being due to the SB state inthe one-band model, we find Z = 0 here because of theorthogonal reflection parity between the lowest electron-removal state and the lowest spin- eigenstate [4]. A second point raised in the Comment is that if aZR-like state is built from a superposition of configu-rations like that of Fig. 3a, AFM correlations on the eand d bonds are similar to those calculated in Ref. [2].This is taken as proof that the spin- polaron is simi-lar to the ZR-based QP state, as well. First, Fig. 3ais for a state of momentum ( π , π ), so naive π rotationslead to a state with an ill defined momentum. In fact,even though these bonds are related by the exact ˆ P x ↔ y symmetry of our Hamiltonian, the quoted values are notequal; this is wrong. In any case, the fact that bondsrather far from the hole show robust AFM correlationsis hardly surprising. The key observation in our model isthe strong FM correlation between the spins neighboringthe hole, which points to the three-spin polaron (3SP) asthe proper framework to understand the spin- polaronand the inner core of the spin- polaron. Since the 3SPcan be written as the sum of singlets between the holeand each of its neighboring spins [3], it does have a finiteoverlap with a ZR state [5]. Its additional degrees of free-dom, however, allow it to describe correlations beyondthose possible in a ZR-based model. This invalidates theComment’s claim that a low-energy non-bonding state isthe only signature of breakdown for a one-band model. Itis very important, in this context, to also point out thatthe model used in Ref. [2] is a further simplification ofthe ZR scenario – the O states are no longer present anda discussion of the spin correlations around an O holebecomes meaningless.For all these reason we remain convinced that boththe spin- and spin- polarons in our 3 band model arequite different objects from the QP and SB obtained froma single band description However, caution is necessarysince comparisons between a variational solution basedon mean-field and our exact diagonalization (ED) for afinite cluster may be misleading. If ED results for a one-band model revealed similar low-energy spin- states,and FM correlations between the spins sandwiching thehole, our position would have to be reconsidered. To ourknowledge, the former is not the case and the latter isnot possible for a one-band model.Bayo Lau, Mona Berciu and George A. Sawatzky Department of Physics and AstronomyUniversity of British ColumbiaVancouver, British Columbia, Canada V6T 1Z1[1] W.-C. Lee and T. K. Lee, preceding Comment(arXiv:1108.5413v1).[2] W.-C. Lee, T. K. Lee, C.-M. Ho, and P. W. Leung. Phys.Rev. Lett. , 057001 (2003).[3] B. Lau, M. Berciu, and G. A. Sawatzky. Phys. Rev. Lett.106