Pis'ma v ZhETF, vol.90, iss.5, pp.433-439
© 2009 September 10
Spectrum of bound fermion states on vortices in 3He-B
M. A. Silaev1)
Institute for Physics of Miciostiuctuies RAS, 603950 Nizhny Novgorod, Russia Low Temperature Laboratory, Helsinki University of Technology, 02150 Espoo, Finland
Submitted 31 Jule 2009
We study subgap spectra of fermions localized within vortex cores in 3He-B. We develop an analytical treatment of the low-energy states and consider the characteristic properties of fermion spectra for different types of vortices. Due to the removed spin degeneracy the spectra of all singly quantized vortices consist of two different anomalous branches crossing the Fermi level. For singular o and u vortices the anomalous branches are similar to the standard Caroli-de Gennes-Matricon ones and intersect the Fermi level at zero angular momentum yet with different slopes corresponding to different spin states. On the contrary the spectral branches of nonsingular vortices intersect the Fermi level at finite angular momenta which leads to the appearance of a large number of zero modes, i.e. energy states at the Fermi level. Considering the v, w and uvw vortices with superfluid cores we show that the number of zero modes is proportional to the size of the vortex core.
PACS: 74.25.—q, 74.78.Na
1. Introduction. Since the pioneering work of Car-oli, de Gennes and Matricon (CdGM) [1] it is well known that quantized vortices in superconductors and Fermi suprfluids have nontrivial internal electronic structure. It consists of low energy fermionic excitations localized within the vortex cores with characteristic interlevel spacing defined as Aq¡Ep -C A0, where A0 is the energy gap far from the vortex line and Ep is the Fermi energy. For conventional s-wave superconductors the excitation spectrum of each individual vortex E(Q) of a subgap state varies from —A0 to +A0 as one changes the angular momentum Q defined with respect to the vortex axis.
At small energies \E\ -C A0 the spectrum is a linear function of Q:
E(Q) ~ -Qw, (1)
where w k, A0/k. f, A0 is the superconducting gap value far from the vortex axis, k± = yk],-k?, kp is the Fermi momentum, kz is the momentum projection on the vortex axis, £ = HVp/A0 is the coherence length, Vp is the Fermi velocity, and Q is half an odd integer. Under some exotic conditions [2] several vortices can merge and then one obtains a multiquantum vortex with a certain winding number M. The number of anomalous branches per spin projection [3] is equal to the vorticity M. For the states with an even vorticity all the anom-
^ e-mail: msilaeveipm.sci-nnov.ru
alous branches cross the Fermi level at nonzero angular momentum Qf.
E(Q)^^(Q±Qj)A0/k±^ (2)
where j = 1...M/2, Qm/2 ~ k. For a vortex with an odd winding number there appears a branch crossing the Fermi level at zero impact parameter.
The quantized vortices in 3He-B have much in common with vortices in ordinary s-wave superconductors. However in multi-component superfluid system 3He axial symmetry allows the nucleation of additional order parameter components inside vortex core. Thus vortices in this system are in general nonsingular, i.e. may have a superfluid core unlike singular vortices in s-wave superconductors which always have a normal core. There exist five types of vortices with different internal core structures in 3He-B: o, u, v, w and uvw vortices [4-6]. The o vortex is the most symmetric one, it has no super-fluid core and consists of almost pure B-phase without inclusions of other phases. Other vortices break some of the discrete symmetries existing for the most symmetric o vortex. Among them the u vortex is singular while the remain v, w and uvw vortices have superfluid cores.
According to the analysis in the framework of Ginzburg-Landau theory [5 - 7] near the critical temperature only v vortex is stable. The cores of such vortices are occupied by an A phase and a ferromagnetic ¡3 phase [4-6]. These additional phases correspond to a nonzero total angular momentum projection on the vortex axis and a zero vorticity in the real space. Therefore nuclea of additional phases remain finite at the vortex center. Nucleation of ferromagnetic ¡3 phase inside vortex cores
explains a large spontaneous magnetic moment of vortices revealed in the NMR experiments in rotating 3He-B [8]. The first order phase transition seen in the NMR experiments was associated with the change of the symmetry of the internal core structure [8, 9].
As was shown by Volovik [10,11] in vortices with dissolved core singularity the spectrum of bound fermion states can be substantially modified in contrast to ordinary CdGM spectrum of singular vortices. In particular the presence of other superfluid phases inside vortex core leads to the appearance of large number of zero modes, i.e. the spectral branches crossing the Fermi level. The number of these zero modes can be as high as Ep/Ao ~ fcfC I- Thus a minigap in spectrum of bound fermions, which is a characteristic feature of CdGM spectrum [1] is absent for nonsingular vortices in 3He-B. Zero modes also exist even for a singular and most symmetric o vortex. Although the number of them is much smaller than for nonsingular vortices it can be effectively controlled by external magnetic field [12, 13]. Even in zero magnetic field due to a broken relative spin-orbital symmetry in B-phase of 3He the spin degeneration of the energy spectrum is removed [12]. As a result the CdGM spectral branches acquire spin dependent shift which closes the minigap. Localized fermions which occupy the negative energy states on the spectral branches intersecting zero energy level form a one dimensional Fermi liquid inside vortex core, which can lead to the instability of the vortex core structure [13].
In this Letter we develop a generalization of CdGM theory for the case of vortices in B phase of superfluid 3He. We derive a general expression for spectrum of vortex core quasiparticles in the presence of multiple order parameter components inside vortex core and analyze the spectra of several particular vortex types.
The method that we use is based on the approximate analytical solution of quasiclassical Andreev equation describing the motion of quasiparticles along the trajectories inside vortex core. Earlier this method was applied to study the spectrum of quasiparticles localized within the cores of multiquanta vortices [3, 14]. Generally the Andreev equation for two component wave function \j) = (U, V) along the quasiclassical trajectory has the form
dtf!
ds
Tt<'-' - T2 A/,„(.' = Elf),
(3)
where 71,2,3 are Pauli matrices of Bogolubov-Nambu spin, | is a length scale of the order of coherence length, AR = (A + A+)/2 and iAIm = (A - A+)/2 are the hermitian and anti-hermitian parts of normalized gap operator, E is energy normalized to the bulk value of gap function Aq.
Here we should take into account the spinor structure of quasiparticle wave functions which is essential in 3He. In this case the coefficients A#,im in Andreev equation (3) are 2x2 matrices in spinor space. Then the matrix Eq. (3) is a system of 4 scalar equations. If matrices Ar and Aim commute [A#, Aim] = 0 the fourth order Andreev equation can be reduced to 2 equations of the second order. However this can not always be the case. To develop a general perturbation theory we note that if Aim = 0 the exact solution of the Andreev Eq. (3) corresponding to E = 0 can be obtained in a spinor basis diagonalizing the matrix Ar = diag(A#i, Ar2). Then we obtain two degenerate solutions \j) 1,2 = (1, —¿)T/i,2(s) corresponding to
the zero energy, where fj(s) = Aj exp Jo ^Rids^j
and A j is an eigen spinor of matrix Ar. As we will see below in case of a single-quantum vortex the functions A#i,2(s) have asymptotics of different signs Affi(+oo)Affi(^oo) < 0. We assume that A#j(+oo) > > 0 therefore the solutions /1,2(5) decay at s = ±00. Using this localized solution as a zero-order approximation for the wave function the spectrum can be found within the first order perturbation theory assuming that \E\ -C 1 and |AIm(s)| -C 1. In general /1,2(5) are not the eigen spinors of the operator Aim which therefore couples the and ^2 states. Then the standard perturbation theory yields the secular equation
det
S
11 -
Si
■E
S
12
S2
•Et
= 0,
(4)
where the matrix elements are
¿>11(22) = 2{/i(2)|AIm|/1(2)) and S12 = 2(/i|AIm|/2).
In general the accuracy of the first order perturbation correction should be determined by the factor 0(Ajm), where |Aim| C 1 is a small parameter. However in a particular case of Eq.(3) the second order correction to the zero energy level is exactly zero and therefore the accuracy of Eq.(4) is much better: 0(A3m). To prove this result we assume for simplicity that [A#, Aim] = 0 so that S12 = 0. Then if the eigen function \j) = (U, V)T of Eq.(3) with Aim = 0 corresponds to the energy en the other function ^ = (—V,U)T corresponds to the energy —en. Therefore it is easy to check that the contribution from negative energy levels to the second order perturbation of the energy level E = 0 exactly compensates the contribution from the positive levels. The proof modification to the general case [A#, Aim] ^ 0 is straightforward.
2. Basic formulas. Our further consideration is based on the Bogoulubov-Nambu equation for the quasiparticles near the Fermi level. From the beginning we
assume the system to be homogeneous in г direction which coincides with the vortex axis. Then we obtain two-dimensional Bogoulubov-Nambu equations with the effective Fermi energy E± = Ep — H2k2/2m and the Fermi momentum in xy plane k± = \fkj,-Щ-
/То*.- - г,&nv - fL>Ai,r,i.' Ev, (5)
where H0 = f3(p2 - %2k2L)/2m, and p = -iHW. Further we will assume that the gap function and energy are normalized to the bulk value of the energy gap До-
Generally the gap function in 3He-B can be parameterized as follows: A = —iay(a ■ d), where d is a vector in 3D spac
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