Spectral Flow, the Magnus Force, an the arxiv:con-mat/9602094v1 16 Feb 1996 Josephson-Anerson Relation P. Ao Department of Theoretical Physics Umeå University, S-901 87, Umeå, SWEDEN October 18, 2018 Abstract We show that the spectral flow ue to a moving vortex is ientical to the phase slippage process, an conclue that its evaluation confirms the results by the Berry s phase calculation of the Magnus force. 1
Consier a rectilinear vortex moving in a superconuctor with a small velocity v v relative to the backgroun of the crystal lattice, both the Berry s phase[1] an the irect total transverse force[2] calculations lea to the Magnus force: F M = L q v h ρ s 2 v v ẑ. (1) Here ρ s is the superflui electron number ensity, q v = ±1 is the vorticity, an the vortex line is along the z-irection with the length L. The crystal lattice backgroun of the superconuctor is hel at rest in the laboratory frame to avoi further complications. Looking from the point of view of the electron flui, a Fermi liqui, moving vortices cause phase slippages, generating a potential ifference. Therefore the electron flui also feels a force, corresponing to the measure electric fiel. This process has been analyze by Josephson[3] an Anerson[4]. Recently, it has been reanalyze by the calculation of the spectral or momentum flow in the electron flui[5, 6]. Since this spectral flow has a sign opposite to that of the Magnus force, one might be tempte to conclue that it cancels the Magnus force. The purpose of the present letter is to show that those two views of looking at the consequences of a moving vortex are equivalent as the action-reaction forces. This equivalence must have been implicitly containe in the literature, but it has not been explicitly state yet. Instea, some confusion still exists.[7, 8] We start by stuying the momentum flow in the electron flui ue to a moving vortex trappe by a potential locate at r 0. The trapping potential can be arbitrarily weak, an just efines the vortex position. The electron flui system is homogeneous other than this trapping potential, an the system of vortex-electron flui is translational invariant. The momentum flow in the electron flui ue to the moving of the trapping potential, the moving of the vortex, is t P(t) = Tr[ ˆρ(t)ˆP]. (2) 2
Here the ensity matrix for the whole electron flui is governe by the equation i h ˆρ(t) = [Ĥ, ˆρ(t)], with ˆP the total momentum operator an Ĥ the truncate Hamiltonian of the system: the lattice phonon inuce effective electron-electron interaction is treate as an attractive one, which gives arise to the superconuctivity. One may inclue the lattice ynamics into the formulation. In this case, because the electron-lattice system consists of a clean superconuctor, our following conclusion will remain unmoifie. We first evaluate the momentum flow from a safe istance far away from the vortex core. Later we point out that it can be evaluate near the vortex core by the spectral flow metho an the results of the two calculations are equal as guarantee by the momentum conservation law. We note that for a slow moving vortex, the aiabatic conition hols: At a given time the ensity matrix ˆρ of the electron flui system can be approximate by its instantaneous equilibrium ensity matrix ˆρ 0 an the eviation ˆρ 1 : ˆρ = ˆρ 0 + ˆρ 1. Here ˆρ 1 is etermine by the equation [Ĥ, ˆρ 1] = i h ˆρ 0, an ˆρ 0 (t) = n f n Ψ n >< Ψ n, (3) with f n = e En/k BT / ne En/k BT the normalize Boltzmann factor an Ψ n the n-th eigenstate of the Hamiltonian. The eigenenergies {E n } are inepenent of time. The time epenence of the ensity matrix is through the vortex position epenence of the wavefunction. Then we have, to the lowest orer in the vortex velocity, { } P(t) = Tr f n v v [ r0 Ψ n >< Ψ n + Ψ n >< r0 Ψ n ]ˆP t n. (4) Here the operator r0 is the graient with respect to the vortex position, an v v = ṙ 0. Since the vortex position, or the position of the trapping potential, is the only reference point in the electron flui system, we have the ientity r0 Ψ n ({r j },r 0 ) = j rj Ψ n ({r j },r 0 ), (5) 3
with r j the position of the j-th electron in the system. Note that the total momentum operator is ˆP = i h j rj i h R, Eq.(4) gives us { } P(t) = i h f n [ < R Ψ n v v R Ψ n > + < v v R Ψ n R Ψ n >] t n = i hv v n f n j 3 r j [ R R Ψ n ({r j})ψ n({r j })] {r j }={r j }. (6) Following the same proceure as in Ref.[2], we first reuce the N-boy ensity to the oneboy ensity matrix ρ 1 : t P(t) = i hv v 3 r 1 [ r1 r 1 ρ 1 (r 1 ;r 1)] r 1 =r 1, (7) with ρ 1 (r 1;r 1 ) N n f n j 1 3 r j Ψ n (r 1,{r j })Ψ n(r 1,{r j }), (8) an N the total electron number. Then we use the Stokes theorem to evaluate the integral far away from the vortex core, an obtain t P(t) = L hv v ẑ r i 2 [( r r )ρ 1 (r ;r)] r =r = L q v h ρ s 2 v v ẑ, (9) which shows that the force felt by the electron flui has the same magnitue as the Magnus force, but with the opposite sign. We will return to this point below. In the integration leaing to Eq.(9) we have use the fact that the integran is the momentum ensity. The two-flui moel has been employe to account for the fact that the momentum generate by the vortex correspons to a supercurrent. The momentum flow calculation can also be performe near the vortex core by counting the flow of the energy spectrum, the so-calle spectral flow metho, anhave been oneinref.[5]. The mainiea isthat, by linearizing the time-epenent Bogoliubov-e Gennes equation near the Fermi surface, a pseuorelativistic Dirac equation will be obtaine. This linearization reuces the original 3+1 imensional problem to an effective 1+1 imensional one. As a vortex moves, there is a continuous flow 4
of the energy spectrum, emerging from(or sinking into) the Fermi sea. The spectral flow rate is proportional to the vortex velocity. By counting their contributions to the momentum, the same result as Eq.(9) has been arrive at in Ref.[5]. This proceure has been further verifie by a moel calculation.[8] The agreement between two seemingly totally ifferent ways, lookingfromasafeistance away fromthecoreanwatching thespectral flownear the core, of calculating the momentum flow may first appear surprising. It has been, however, explicitly emonstrate in Ref.[9] in a slightly ifferent context in the 3 He-A phase that they are inee the complementary two ways of keeping track of the same physics, an they are equal as a result of the momentum conservation in the electron flui system. In the present situation the link between those two ways has been iscusse in Ref.[6]. It is the ifferential form of the momentum flow: t (m v s )+ µ = q v hv v ẑ δ 2 (r r 0 ), (10) with the electrochemical potential µ as the sum of the chemical potential µ 0, the flui kinetic energyantheelectricpotential µ = µ 0 + m 2 v2 s+e (φ+ 1 r r A(r,t)), anthesuperflui c t velocity istribution m v s = h θ e c A. Here m an e are effective mass an charge of a Cooper pair, respectively, an θ is the phase of the superconucting conensate wavefunction. Eq.(10) is gauge invariant, an is a precise statement about the phase slippage process ue to a moving vortex. Its various consequences have been explore by Josephson[3] an Anerson[4]. An ientical equation for the special case of v v = v s in the neutral superflui has been stuie by Anerson[4]. We may call Eq.(10) the Josephson-Anerson relation. It isclear now that a moving vortex feels a transverse force, the Magnus force; if one looks from the point of view of the electron flui, the flui also feels a force with the magnitue equal to but thesign oppositeto themagnus force. The two forces areacting ontwo ifferent objects, the vortex an the electron flui, an are the action-reaction forces. In the following 5
we strengthen this point by a straightforwar proof. We write Eq.(2) in its equivalent form of the Ehrenfest theorem: Using R Ĥ = r0 Ĥ, we obtain P(t) = Tr[ˆρ(t) RĤ]. (11) t t P(t) = Tr[ˆρ(t) r 0 Ĥ] = F M. (12) This shows that the two forces uner iscussion are inee the action-reaction forces. In the last equality we have use the formal efinition of the Magnus force[2]. The present result, the equivalence between the Magnus force an the spectral flow as action-reaction forces, may seem obvious. Nevertheless it has never been explicitly spelle out in the literature. Instea, some recent work have treate the spectral flow as a way to cancel the Magnus force[5, 7, 8], which is incorrect accoring to the present emonstration. It shoul be pointe out that the spectral flow is a counting of contributions from extene states. There is no involvement of the localize core states. This can be explicitly checke by expressing the one-boy ensity matrix in Eq.(9) in terms of extene an localize states, an latter gives zero contribution, as having been note in Ref.[2] in the evaluation of the Magnus force. Incientally, in Ref.[5] the Wess-Zumino term for a moving vortex has been ientifie as the same force as the one ue to the spectral flow. Recent as well as earlier work have shown that the Wess-Zumino term gives exactly the Magnus force[10, 11, 12, 6], not the spectral flow. Comparing the Berry s phase calculation away from the vortex core with the spectral or momentum flow counting near the vortex core, we fin that the former only epens on a few global properties of a superconuctor, namely the topology of a vortex, an the latter is a rather etaile calculation. The topological constraints behave like conservations laws. 6
Results obtaine uner them shoul, an have to, be borne out by etaile calculations, which are concrete realizations. For the Magnus force, it is inee the case. Acknowlegements: The author thanks Davi Thouless an Qian Niu for numerous iscussions, an Mike Stone an Frank Gaitan for informative corresponences. The paper was initiate at the Institute of Scientific Information at Turin in the fall of 1993, an was shape into the present form at the Aspen Center for Physics in the summer of 1995. Their hospitalities are gratefully acknowlege. The work was supporte in part by Sweish Natural Science Research Council an by US NSF Grant No. DMR-9220733. Present aress: Department of Physics, Box 351560, University of Washington, Seattle, WA 98195, USA. References [1] P. Ao an D.J. Thouless, Phys. Rev. Lett. 70, 2158 (1993); P. Ao, Q. Niu, an D.J. Thouless, Physica B194-196, 1453 (1994). [2] D.J. Thouless, P. Ao, an Q. Niu, Transverse force on a quantize vortex in a superflui, preprint, High-Tc Upate, Nov. 15, 1995. [3] B.D. Josephson, Phys. Lett. 1, 251 (1962); ibi. 16, 242 (1965). [4] P.W. Anerson, Rev. Mo. Phys. 38, 298 (1966). [5] G.E. Volovik, JETP Lett. 57, 244 (1993); JETP 77, 435 (1993). [6] F. Gaitan, J. Phys. Con. Matt. 7, L165 (1995); Phys. Rev. B51, 9061 (1995). [7] G.E. Volovik, JETP Lett. 62, 65 (1995). 7
[8] Y.G. Makhlin an T.S. Misirpashaev, JETP Lett. 62, 83 (1995). [9] M. Stone an F. Gaitan, Ann. Phys.(N.Y.) 178, 89 (1987). [10] M. Hatsua, S. Yahikozawa, P. Ao, an D.J. Thouless, Phys. Rev. B49, 15870 (1994); an references therein. [11] P. Ao, D.J. Thouless, an X.-M. Zhu, Mo. Phys. Lett. B9, 755 (1995); I.J.R. Aitchison, P. Ao, D.J. Thouless, an X.-M. Zhu, Phys, Rev, B51, 6531 (1995). [12] M. Stone, Int. J. Mo. Phys. B9, 1359 (1995). 8