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VOLUME 78, NUMBER 26 PHYSICAL REVIEW LETTERS 30 JUNE 1997 Simon and Lee Reply: Volovik and Kopnin [1] have raised two separate issues in the preceding Comment— (1) that the linearization of the quasiparticle spectrum around the gap node limits the maximum temperature for which the discussion in Ref. [2] is valid, and (2) that the use of the geometric average light cone velocity in Ref. [2] gives an incorrect estimate of the crossover scale. Both of these objections are valid and will be discussed below. We note that, with the exception of this incorrectly predicted crossover scale, the results of our paper remain unchanged. We first address the issue of linearization of the quasi- particle spectrum. As mentioned in [2], the validity of the linearized dispersion is expected to be restricted to a tem- perature range T ø D 2 yE F (with D , T c as the maxi- mum gap) where excitation energies are small enough such that the quadratic part of the Hamiltonian (the part repre- senting curvature of the Fermi surface) is much smaller than the leading linearized piece. On the other hand, the form of the quasiparticle spectrum described by Volovik and Kopnin [Eq. 1 of Ref. [1] ] accounts for the curvature of the Fermi surface and therefore can be used in describ- ing excitations at energy scales up to the order of D 2 yE F for quasiclassical calculations for which $ p is considered to be a good quantum number [3,4]. It should be noted, however, that as mentioned in [2], in practice, the Fermi surface in the high T c compounds can be quite flat at the gap nodes such that the Fermi surface curvature is smaller than one would expect for a model circular Fermi surface, and thus the range of validity of the linearization used in [2] may be somewhat larger than otherwise expected. It should also be noted that when curvature of the Fermi surface is important (such as for the calculation of the thermal Hall coefficient), it can be effectively treated using perturbation theory [2]. We now turn to the separate issue of the crossover scale. Volovik and Kopnin point out two crossover scales T 1 , T c p HyH c2 and T 2 ,sT 2 c yE F d p HyH c2 . In the ab- sence of magnetic field, the velocity in the direction per- pendicular to the Fermi surface is the Fermi velocity y F , whereas the velocity tangential to the Fermi surface is roughly ,y F T c yE F . In a magnetic field, the peri- odicity of the vortex lattice is given by the magnetic length l 0 , thus the Brillouin zone edge is at momentum k max , 1yl 0 . Neglecting the vector potential and assum- ing that momentum remains a good quantum number, we find that the energies of the states at the zone edge in the two different directions correspond to the two crossover energy scales discussed above. In Ref. [2] it was incor- rectly assumed that in a magnetic field, the semiclassi- cal states precess thus obtaining a single geometrically averaged velocity between the two different directions. However, as correctly treated in [4], the Eilenberger semi- classical approach leaves $ p a good quantum number even in a magnetic field so that the momentum states do not actually precess. This result can also be seen from the dynamics of the linearized Hamiltonian in [2]. In fact, however, the situation is somewhat more complicated than the above paragraph would lead us to believe. When we add a magnetic field [2], one component of the vector potential (times y F ) acts as a periodic scalar potential for delocalized quasiparticles, resulting in a gap at the zone edge of size T 1 , in both directions in the Brillioun zone. Whether a gap is actually observed in the density of states depends on the details of the band structure. Nonetheless, it is clear that this should be an important crossover scale being that this is also the typical energy of the periodic potential. The prediction of the additional crossover scale at energy T 2 is due to the somewhat different physics of the bound vortex core states. Volovik and Kopnin [1,3,4] have calculated that the spacing of the core states is approximately T 2 at low energy. In this low energy range, however, the major contribution to the density of states is from the delocalized quasiparticles [3], so we would probably see this discretization clearly only if a gap occurs in the spectrum of the extended states. Thus, an important direction for future research will be to attempt an exact quantum mechanical treatment of the spectrum at these low energies. This work was supported by NSF Grant No. DMR-95- 23361. Steven H. Simon and Patrick A. Lee Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Received 5 May 1997 [S0031-9007(97)03415-7] PACS numbers: 74.25.Fy, 74.25.Jb, 74.72. – h [1] G. E. Volovik and N. B. Kopnin, preceding Comment, Phys. 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)]. 0031-9007y 97y 78(26) y5029(1)$10.00 © 1997 The American Physical Society 5029

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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