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VOLUME 78, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 30 JUNE 1997
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Simon and Lee Reply: Volovik and Kopnin [1] haveraised two separate issues in the preceding Commen(1) that the linearization of the quasiparticle spectrumaround the gap node limits the maximum temperatufor which the discussion in Ref. [2] is valid, and (2)that the use of the geometric average light cone velocin Ref. [2] gives an incorrect estimate of the crossovescale. Both of these objections are valid and will bdiscussed below. We note that, with the exception of thincorrectly predicted crossover scale, the results of opaper remain unchanged.
We first address the issue of linearization of the quasparticle spectrum. As mentioned in [2], the validity of thelinearized dispersion is expected to be restricted to a teperature rangeT ø D2yEF (with D , Tc as the maxi-mum gap) where excitation energies are small enough suthat the quadratic part of the Hamiltonian (the part reprsenting curvature of the Fermi surface) is much smallthan the leading linearized piece. On the other hand, tform of the quasiparticle spectrum described by Voloviand Kopnin [Eq. 1 of Ref. [1] ] accounts for the curvatureof the Fermi surface and therefore can be used in descring excitations at energy scales up to the order ofD2yEF
for quasiclassical calculations for which$p is considered tobe a good quantum number [3,4].
It should be noted, however, that as mentioned in [2],practice, the Fermi surface in the highTc compounds canbe quite flat at the gap nodes such that the Fermi surfacurvature is smaller than one would expect for a modcircular Fermi surface, and thus the range of validity othe linearization used in [2] may be somewhat larger thaotherwise expected. It should also be noted that whcurvature of the Fermi surface is important (such as fthe calculation of the thermal Hall coefficient), it can beeffectively treated using perturbation theory [2].
We now turn to the separate issue of the crossovscale. Volovik and Kopnin point out two crossover scaleT1 , Tc
pHyHc2 andT2 , sT2
c yEFdp
HyHc2. In the ab-sence of magnetic field, the velocity in the direction pependicular to the Fermi surface is the Fermi velocityF , whereas the velocity tangential to the Fermi surfacis roughly ,yFTcyEF . In a magnetic field, the peri-odicity of the vortex lattice is given by the magneticlength l0, thus the Brillouin zone edge is at momentumkmax , 1yl0. Neglecting the vector potential and assuming that momentum remains a good quantum number, wfind that the energies of the states at the zone edge intwo different directions correspond to the two crossoveenergy scales discussed above. In Ref. [2] it was incorectly assumed that in a magnetic field, the semiclas
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cal states precess thus obtaining a single geometricaaveraged velocity between the two different directionHowever, as correctly treated in [4], the Eilenberger semclassical approach leaves$p a good quantum number evenin a magnetic field so that the momentum states do nactually precess. This result can also be seen fromdynamics of the linearized Hamiltonian in [2].
In fact, however, the situation is somewhat morcomplicated than the above paragraph would leadto believe. When we add a magnetic field [2], oncomponent of the vector potential (timesyF) acts asa periodic scalar potential for delocalized quasiparticleresulting in a gap at the zone edge of sizeT1, in bothdirections in the Brillioun zone. Whether a gap is actualobserved in the density of states depends on the detailthe band structure. Nonetheless, it is clear that this shobe an important crossover scale being that this is alsotypical energy of the periodic potential.
The prediction of the additional crossover scaleenergy T2 is due to the somewhat different physics othe bound vortex core states. Volovik and Kopnin [1,3,have calculated that the spacing of the core statesapproximatelyT2 at low energy. In this low energy rangehowever, the major contribution to the density of statis from the delocalized quasiparticles [3], so we wouprobably see this discretization clearly only if a gap occuin the spectrum of the extended states. Thus, an importdirection for future research will be to attempt an exaquantum mechanical treatment of the spectrum at thelow energies.
This work was supported by NSF Grant No. DMR-9523361.
Steven H. Simon and Patrick A. LeeDepartment of PhysicsMassachusetts Institute of TechnologyCambridge, Massachusetts 02139
Received 5 May 1997 [S0031-9007(97)03415-7PACS numbers: 74.25.Fy, 74.25.Jb, 74.72.–h
[1] G. E. Volovik and N. B. Kopnin, preceding CommentPhys. Rev. Lett.78, 5028 (1997).
[2] S. H. Simon and P. A. Lee, Phys. Rev. Lett.78, 1548(1997).
[3] G. E. Volovik, Pis’ma Zh. Eksp. Theor. Fiz.58, 457 (1993)[JETP Lett.58, 469 (1993) ].
[4] N. B. Kopnin and G. E. Volovik, Pis’ma Zh. Eksp. Teor.Fiz. 64, 641 (1996) [JETP Lett.64, 690 (1996)].
© 1997 The American Physical Society 5029