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Non-Abelian adiabatic phases in vortex dynamics. Magnus force in Fermi superfluids
s __
30 September
__ Biz
1996
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 22 1 (1996) 277-28 1
Non-Abelian adiabatic phases in vortex dynamics. Magnus force in Fermi superfluids E. hnhek Department Received
ofPhysics. University ofCalifornia, Riverside, CA 92521. USA 14 February 1996; accepted for publication Communicated by A. Lagendijk
27 June 1996
Abstract The Magnus force acting on a vortex in a two-dimensional neutral Fermi superfluid is derived using the method of adiabatic phases. The large core contribution to this force is generated by the non-Abelian structure of the gauge potentials.
The Abelian approximation
yields a small hydrodynamic
force sensitive
to the bulk band structure.
PACS: 74.20. - z; 74.60.Ge; 62.20.D~; 62.2O.Fe
The problem of the Magnus force acting on a vortex in Fermi super-fluids received a renewed interest in recent years. The method of the adiabatic (geometric) phases [ 11 has become a powerful tool in such studies, as it yields the dynamics of the slow variable, the vortex center. Haldane and Wu [2] calculated the geometric phase for a vortex in a Bose superfluid transported adiabatically around a closed contour. The effective Lagrangian associated with this phase can be used to deduce the Magnus force. More recently, this approach was applied to a vortex in a Fermi superfluid by Ao and Thouless [3] and Gaitan [4]. In these papers, the superconducting phase is coupled to the superfluid density and the resulting Magnus force is predicted to be a robust quantity that is insensitive to disorder (except for a possible reduction of the bulk superfluid density). As pointed out by Feigel’man et al. [5], this result contrasts with previous microscopic calculations [6,7] 0375-%01/96/$12.00 Copyright PII SO375-9601(96)00539-7
in which the Magnus force is strongly reduced by a relatively weak disorder. This is because, in these calculations, the Magnus force is found to originate predominately from excitations involving the core bound states that are sensitive to disorder. This point of view was substantiated in the recently reported derivation of the effective action of a vortex in a two-dimensional superfluid [8]. Applying the Stratonovich transformation to the attractive fermion-fermion interaction [9], the effective action of Ref. [8] is obtained as a functional of the vortex displacement U. Expanding the action to second order in U, the Magnus force is found to contain two distinct contributions: the hydrodynamic term that is first order in U, and the core contribution that is second order in U. Actually, the hydrodynamic term was discarded in Ref. [8]. This is justified as long as particle-hole symmetry is imposed [9]. Recent work by van Otterlo et al. [lo] considers
0 1996 Elsevier Science B.V. All rights reserved.
278
E. &mcinek/Physics
Letters A 221 (1996) 277-281
the possibility of a nonvanishing hydrodynamic term due to the breaking of particle-hole symmetry. The competition between this term and the core contribution provides a novel interpretation of the sign change in the Hall effect in the superconducting state observed for temperatures just below T, [ 11,121. There are major differences between the results of Ref. [lo] and the adiabatic phase approaches [3,4] to the Magnus force. The latter yields a full-sized hydrodynamic term, even in the presence of particlehole symmetry. Moreover, the core contribution is entirely missing in the derived Magnus force. In view of this controversy, our microscopic understanding of the vortex dynamics in fermion superfluids is incomplete. This theoretical ambiguity has unsettling upshots for the interpretation of experimental data. For instance, recent measurements of the terahertz impedance in YBa,Cu,O,_ s thin films yield a Magnus parameter that is less than 0.07 of the clean limit [13]. Nevertheless, the authors, clinging to the concept of a robust Magnus force, attribute this reduction to the effect of the anisotropic gap on vortex dynamics [ 131. This situation prompts us to take another look at the microscopic basis of the Magnus force in Fermi superfluids. In the present Letter, we attempt to reconcile the results of Ref. [lo] with the method of adiabatic phases. We find that non-Abelian phase factors are needed to generate the core contribution to the Magnus force. Though, in the Bose case [2], the Abelian phase is sufficient, the gauge fields associated with the adiabatic transport of a vortex in a Fermi superfluid acquire a non-Abelian structure. Wilczek and Zee [ 141 showed that non-Abelian gauge potentials arise when degenerate states are subjected to adiabatically changing external parameters. In our problem of the Magnus force, the relevant external parameter is the coordinate of the vortex center R. The matrix elements of V, between the electronic states act as effective gauge potentials for the vortex motion. We note that the spurious infinite degeneracy of the BCS ground state does not play a role in the formation of the gauge potentials. Hence, if transitions to excited states are neglected, the potentials for the vortex motion remain Abelian as assumed in Refs. [3] and [4]. However, as shown in the present work, there is an important matrix element of V, between the ground state and an excited state
corresponding to the breaking of a pair formed from the core states next to the Fermi level [ 151. This off-diagonal transition cannot be disregarded, even in the adiabatic limit, as it leads to a contribution to the Magnus force that is of the same magnitude as the hydrodynamic term derived in Refs. [3] and [4]. We note that the concept of non-Abelian gauge potentials arising from nondegenerate electronic states is not new. Zygelman [ 161 discussed their appearance in connection with the method of the perturbed stationary states applied to atomic collision problems. We consider a two-dimensional neutral Fermi superfluid with s-wave pairing containing a vortex at R(r). Following the molecular analogy, we adopt the Born-Oppenheimer method [16,17] by decomposing the system wave function into electronic and vortex center components, q(R,
r> = ;@,(R)&L m
r),
(1)
where r stands for the totality of the fermion coordinates. The wave function (1) is an eigenfunction of the full Hamiltonian
(2) where m, is the effective vortex mass and Xe is the fermion Hamiltonian for fixed vortex center. At any instant, for R = R(t), the instantaneous fermion eigenstates ‘p, satisfy ze(R,
~)co~(R, r) =E,cp,(R,
r).
(3)
Introducing Eqs. (1) and (2) into the Schrodinger equation for TP, multiplying this equation on the left by q”* and integrating over the set of fermion coordinates r, we obtain the Schriidinger equation for the amplitudes @,,, [16] -&[V”_‘L(R)]‘@+V^(R)@=iz, ”
(4)
where @ is an N-dimensional column vector with the components Qm. The quantities A^ and v^ are N X N Hermitian matrices given by Lij = i/ d*r p; (R, r)V,qj(
R, r)
(5)
E. &nrinek/Physics
and pij= SijEj(R).
(6)
Applying the generalized Ehrenfest theorem [ 161, we calculate the acceleration of the vortex center d2RC” dt2
1
=im,
where fiR is the vortex-center Hamiltonian operating on @ (see Eq. (4)). Paying attention to the non-commutativity of the spatial components of i(R), Eq. (7) yields dR d2R cL 112-__=F”pLyY dt ’ ” dt2
(8)
where
ak
F^PY=__-_ aRp
atip 3R”
In terms of these amplitudes, the ground state is given by I&>=
1
(9)
These equations are Lorentz-type equations describing the motion of a particle in a non-Abelian gauge field [18]. The @h component of the Magnus force is obtained from Eq. (8) by taking the expectation value for a solution of Eq. (4). In the adiabatic limit, at T = 0, the column vector @ has only the ground state component that we denote as @, . Then the expectation value of Eq. (9) becomes it
[ Al”j, A;,].
( 10)
j=2
We now proceed to the evaluation of this expression for a vortex in a clean BCS-superfluid. The electronic ground state is constructed by pairing the states u, and Us*which are solutions of Bogoliubov’s equations [ 191
(11)
n[~,,,(r,-R) m
where 10) is the zero-particle state and b: = is an operator that creates a pair of 4T L fermions with angular momenta CL,,,- 3 and - p,,, f . The product in Eq. (14) contains both the core bound states with pLm< 0 only, and the scattering states with both signs of p,,, (see Ref. [20]). The excited state I (p2) which couples to I ‘p, > via the operator V, is obtained by breaking the occupied pair with pm = - 3. This is accomplished by defining the Bogoliubov-Valatin operators [21] =
U&
rl,,
A=
IA(r-R)lexp[-iB(r-R)].
(12)
Following Caroli et al. [ 191, the Bogoliubov amplitudes are expressed (in two dimensions) as u”(r) =f!“‘( r) exp[i( pn - $)0],
( 134
r) exp[i( pn + +)0].
(13b)
un( r) =fl”‘(
-
=u;ctpl.
$C-,I ,
( 154
+u;cpr.
(15b)
Similar to the vortex-free case, we then write using Eqs. (14) and (15) I(P2)=L,2,Y:,2J = - n’(u,
m
Id
( 16)
+ umbi)
x(uf,,,--uf,,,b:,,z
) IO>,
(17) where the prime denotes a product with the bound state, pm = - 3, missing. Owing to the angular momentum conservation, I cp2> is the only relevant excited state. Hence, the gauge fields defined in (5) are 2 x 2 Hermitian matrices. Performing the integration over the fermion coordinates, the diagonal matrix element obtained by using Eq. (14) in (5) is A^,, =i(cp, IV,Icp,> =
/
d2XS(r-R)VRO(r-R),
(18)
where s= c[
where so is the free-electron Hamiltonian, and the gap function is of the form
+U,,,(rm--R)b~]lO), (14)
Yi PT i 1 AaM, A*’ .
279
Letters A 221 (1996) 277-281
IUJ*(-&+;)+
lu,l’(-j&-g].
m (19)
This result agrees with the Abelian adiabatic phase obtained recently by Gaitan [4]. A similar calculation, done with the use of Eq. (171, yields
E. .%nbnek/Physics
280
Letrem A 221 (1996) 277-281
We see from this equation that, due to the core excitation, the geometric phase of the excited state is diminished compared to that of the ground state. Using Eq. (18), the Abelian part of the curvature (10) is written as
(21) For a point vortex in two dimensions,
(FXY&~berian=
we have
V,xV,O=2mY(r-R)z,,
(22)
where z0 is the unit vector perpendicular to the superfluid sheet. Thus the first term of Eq. (21) gives us 2nS(O). Evaluating the second term in polar coordinates, we have
&dp P/ 2ndv(
V,SX
which a large hydrodynamic Magnus force is obtained by dropping the I p,,, I 2 terms from S (see Eq. (19)). On the other hand, Eq. (25) gives a Magnus force which agrees (in magnitude) with the result of Ref. [lo] (applied to the case of an uncharged superfluid). According to Eq. (lo), the non-Abelian part of the adiabatic curvature is given by -i(A:,&
2TZ,
/
a;;
=
-_1 .
3:
-
ve)
dp=2n[S(x)
au-
u_,,~----
so that aA:,
(F1Xy)*belian=aX-aY=2~~(tC)-
(24
Due to the exponential decay of the core bound states, the expression (19) for S(m) contains only scattering states. Since in the bulk of the superfluid, the quantities 1u, I 2 and 1u, I 2 are independent of the sign of p., [20], the terms in Eq. (19) that are proportional to CL,,cancel out and we are left with S(a)
=+c(
n
Iu,12-
1/2
ax
i k,/2.
au - l/2 ax
~ I/*
(27)
-S(O>]z,,
(23) aA:,
(26)
Using Eqs. (14) and (17), we obtain from Eq. (5) the off-diagonal matrix element
0
=
-Ay2A&).
IuJ2)
(25) where wu is the Debye frequency, [ is the singleparticle energy measured from the chemical potential II, and d, is the energy gap in the bulk. Eqs. (8), (24) and (25) define the hydrodynamic contribution to the Magnus force. Its magnitude and sign are determined by the derivative of the density of states, N( 5 + p), at the Fermi level [lo]. In BCS theory, this quantity is very small, so that S(m) is negligible compared to the electron density. This result contrasts with the conclusion of Ref. [4] in
This approximate result becomes exact for k, ,$’* 1, in which case the two-dimensional integration in Eq. (27) is dominated by terms involving the derivatives of the radial parts of the amplitudes (13). In a similar way, we obtain A:, = - fkF_ With use of these results, Eq. (26) yields (FXY)non_Abeiian~ -+ki=
-2nn,,
(28)
where n, is the sheet superfluid density. Introducing the results (24) and (28) into Eq. (lo), we obtain from Eq. (8) the net Magnus force F,v,=2~[n,-S(cc)](zoXV),
(29)
where V = d is the velocity of the vortex center. This result should be compared with Eq. (2) of van Otterlo et al. [ 101 in the clean limit. We see that there is an agreement except for the sign of the hydrodynamic term. Specifically, if we expand N( [ + p) in Eq. (25) about the Fermi level, the hydrodynamic term in Eq. (29) has a negative sign if the derivative, N’(p), is positive. This finding is of relevance for the interpretation of the sign changes of the Hall angle in terms of broken particle-hole symmetry
DOI. In the presence of disorder, the core contribution to F, is modified as first shown in Ref. [6]. In the present formulation, the corresponding reduction factor can be obtained by converting the curvature (26)
E. .%ma’nek/Physics
Letters A 221 (1996) 277-281
to a Kubo formula, and evaluating the latter within the relaxation-time approximation (see Ref. [15] for details). The disorder-induced reduction of the core part of the Magnus force can be simply understood as a breakdown of the adiabatic approximation: in the presence of level broadening, there is no gap in the excitation spectrum so that the vortex cannot be moved slowly enough to prevent real excitations which tend to wash out the topological effects [22]. To summarize, we use the method of the geometric phase to derive an expression for the Magnus force on a vortex in an uncharged BCS superfluid at T = 0. The gauge potentials associated with vortex motion are shown to acquire a non-Abelian structure. The equation of motion for the vortex yields a hydrodynamic part determined by the electron density of states in the bulk, and a core part generated by the non-Abelian potentials. Except for the relative sign of these two parts, our results agree with the recent work by van Otterlo et al. [ 101 based on a path integration method of Refs. [81 and [91. I would like to thank Dardo Piriz for a helpful suggestion.
References [ I] A. Shapere and F. Wilczek, eds., Geometric phases of physics (World Scientific,
Singapore,
1989).
l21 F.D.
281
Haldane and Y.-S. Wu, Phys. Rev. Lett. 55 (1985) 2887. [31 P. Ao and D.J. Thouless, Phys. Rev. Lett. 70 (1993) 2158. [41 F. Gaitan, Phys. Rev. B 51 (1995) 9061. V.B. Geshkenbein, A.I. Larkin and S. [51 M.V. Feigel’man, Levit, JETP Lett. 57 (1993) 711. b1 N.B. Kopnin and V.E. Kravtsov, JETP Lett. 23 (1976) 578. Phys. Rev. B 44 [71 N.B. Kopnin and M.M. Salomaa, (1991) 9667. b31E. Sim&rek, Phys. Lett. A 194 (1994) 323. l91 V. Ambegaokar, U. Eckem and G. Schon, Phys. Rev. Lett. 48 (1982) 1745. [lOI A. van Otterlo, M. Feigel’man, V. Geshkenbein and G. Blatter, Phys. Rev. Lett. 75 (1995) 3736. llll S.J. Hagen, C.J. Lobb, R.L. Green, M.G. Forrester and J.H. Kang, Phys. Rev. B 41 (1990) 11630. [121 J.M. Harris, N.P. Gng and Y.F. Yan, Phys. Rev. Lett. 71 (1993) 1455. D.T. Nemeth, F. [131 B. Parks, S. Speilman, J. Orenstein, Ludwig, J. Clarke, P. Merchant and D.J. Lew, Phys. Rev. Lett. 74 (1995) 3265. [141 F. Wilczek and A. Zee, Phys. Rev. Lett. 52 (1984) 2111. I151 E. Simanek, Phys. Rev. B 52 (1995) 10336. l161 B. Zygelman, Phys. Lett. A 125 (1987) 476. 1171 J. Moody, A. Shapere and F. Wilczek, Phys. Rev. Lett. 56 (1986) 893. 1181 S.K. Wong, Nuovo Cimento A 65 (1970) 689. 1191 C. Caroli, P.G. de Gennes and J. Matricon, Phys. Lett. 9 ( 1964) 307. [201 J. Bardeen, R. Kiimmel, A.E. Jacobs and L. Tewordt, Phys. Rev. 187 (1969) 556. Theory of superconductivity (Benjamin, 1211 J.R. Schrieffer, New York, 1964). l221 M. Stone, Phys. Rev. D 33 (1986) 1191.
30 September
__ Biz
1996
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 22 1 (1996) 277-28 1
Non-Abelian adiabatic phases in vortex dynamics. Magnus force in Fermi superfluids E. hnhek Department Received
ofPhysics. University ofCalifornia, Riverside, CA 92521. USA 14 February 1996; accepted for publication Communicated by A. Lagendijk
27 June 1996
Abstract The Magnus force acting on a vortex in a two-dimensional neutral Fermi superfluid is derived using the method of adiabatic phases. The large core contribution to this force is generated by the non-Abelian structure of the gauge potentials.
The Abelian approximation
yields a small hydrodynamic
force sensitive
to the bulk band structure.
PACS: 74.20. - z; 74.60.Ge; 62.20.D~; 62.2O.Fe
The problem of the Magnus force acting on a vortex in Fermi super-fluids received a renewed interest in recent years. The method of the adiabatic (geometric) phases [ 11 has become a powerful tool in such studies, as it yields the dynamics of the slow variable, the vortex center. Haldane and Wu [2] calculated the geometric phase for a vortex in a Bose superfluid transported adiabatically around a closed contour. The effective Lagrangian associated with this phase can be used to deduce the Magnus force. More recently, this approach was applied to a vortex in a Fermi superfluid by Ao and Thouless [3] and Gaitan [4]. In these papers, the superconducting phase is coupled to the superfluid density and the resulting Magnus force is predicted to be a robust quantity that is insensitive to disorder (except for a possible reduction of the bulk superfluid density). As pointed out by Feigel’man et al. [5], this result contrasts with previous microscopic calculations [6,7] 0375-%01/96/$12.00 Copyright PII SO375-9601(96)00539-7
in which the Magnus force is strongly reduced by a relatively weak disorder. This is because, in these calculations, the Magnus force is found to originate predominately from excitations involving the core bound states that are sensitive to disorder. This point of view was substantiated in the recently reported derivation of the effective action of a vortex in a two-dimensional superfluid [8]. Applying the Stratonovich transformation to the attractive fermion-fermion interaction [9], the effective action of Ref. [8] is obtained as a functional of the vortex displacement U. Expanding the action to second order in U, the Magnus force is found to contain two distinct contributions: the hydrodynamic term that is first order in U, and the core contribution that is second order in U. Actually, the hydrodynamic term was discarded in Ref. [8]. This is justified as long as particle-hole symmetry is imposed [9]. Recent work by van Otterlo et al. [lo] considers
0 1996 Elsevier Science B.V. All rights reserved.
278
E. &mcinek/Physics
Letters A 221 (1996) 277-281
the possibility of a nonvanishing hydrodynamic term due to the breaking of particle-hole symmetry. The competition between this term and the core contribution provides a novel interpretation of the sign change in the Hall effect in the superconducting state observed for temperatures just below T, [ 11,121. There are major differences between the results of Ref. [lo] and the adiabatic phase approaches [3,4] to the Magnus force. The latter yields a full-sized hydrodynamic term, even in the presence of particlehole symmetry. Moreover, the core contribution is entirely missing in the derived Magnus force. In view of this controversy, our microscopic understanding of the vortex dynamics in fermion superfluids is incomplete. This theoretical ambiguity has unsettling upshots for the interpretation of experimental data. For instance, recent measurements of the terahertz impedance in YBa,Cu,O,_ s thin films yield a Magnus parameter that is less than 0.07 of the clean limit [13]. Nevertheless, the authors, clinging to the concept of a robust Magnus force, attribute this reduction to the effect of the anisotropic gap on vortex dynamics [ 131. This situation prompts us to take another look at the microscopic basis of the Magnus force in Fermi superfluids. In the present Letter, we attempt to reconcile the results of Ref. [lo] with the method of adiabatic phases. We find that non-Abelian phase factors are needed to generate the core contribution to the Magnus force. Though, in the Bose case [2], the Abelian phase is sufficient, the gauge fields associated with the adiabatic transport of a vortex in a Fermi superfluid acquire a non-Abelian structure. Wilczek and Zee [ 141 showed that non-Abelian gauge potentials arise when degenerate states are subjected to adiabatically changing external parameters. In our problem of the Magnus force, the relevant external parameter is the coordinate of the vortex center R. The matrix elements of V, between the electronic states act as effective gauge potentials for the vortex motion. We note that the spurious infinite degeneracy of the BCS ground state does not play a role in the formation of the gauge potentials. Hence, if transitions to excited states are neglected, the potentials for the vortex motion remain Abelian as assumed in Refs. [3] and [4]. However, as shown in the present work, there is an important matrix element of V, between the ground state and an excited state
corresponding to the breaking of a pair formed from the core states next to the Fermi level [ 151. This off-diagonal transition cannot be disregarded, even in the adiabatic limit, as it leads to a contribution to the Magnus force that is of the same magnitude as the hydrodynamic term derived in Refs. [3] and [4]. We note that the concept of non-Abelian gauge potentials arising from nondegenerate electronic states is not new. Zygelman [ 161 discussed their appearance in connection with the method of the perturbed stationary states applied to atomic collision problems. We consider a two-dimensional neutral Fermi superfluid with s-wave pairing containing a vortex at R(r). Following the molecular analogy, we adopt the Born-Oppenheimer method [16,17] by decomposing the system wave function into electronic and vortex center components, q(R,
r> = ;@,(R)&L m
r),
(1)
where r stands for the totality of the fermion coordinates. The wave function (1) is an eigenfunction of the full Hamiltonian
(2) where m, is the effective vortex mass and Xe is the fermion Hamiltonian for fixed vortex center. At any instant, for R = R(t), the instantaneous fermion eigenstates ‘p, satisfy ze(R,
~)co~(R, r) =E,cp,(R,
r).
(3)
Introducing Eqs. (1) and (2) into the Schrodinger equation for TP, multiplying this equation on the left by q”* and integrating over the set of fermion coordinates r, we obtain the Schriidinger equation for the amplitudes @,,, [16] -&[V”_‘L(R)]‘@+V^(R)@=iz, ”
(4)
where @ is an N-dimensional column vector with the components Qm. The quantities A^ and v^ are N X N Hermitian matrices given by Lij = i/ d*r p; (R, r)V,qj(
R, r)
(5)
E. &nrinek/Physics
and pij= SijEj(R).
(6)
Applying the generalized Ehrenfest theorem [ 161, we calculate the acceleration of the vortex center d2RC” dt2
1
=im,
where fiR is the vortex-center Hamiltonian operating on @ (see Eq. (4)). Paying attention to the non-commutativity of the spatial components of i(R), Eq. (7) yields dR d2R cL 112-__=F”pLyY dt ’ ” dt2
(8)
where
ak
F^PY=__-_ aRp
atip 3R”
In terms of these amplitudes, the ground state is given by I&>=
1
(9)
These equations are Lorentz-type equations describing the motion of a particle in a non-Abelian gauge field [18]. The @h component of the Magnus force is obtained from Eq. (8) by taking the expectation value for a solution of Eq. (4). In the adiabatic limit, at T = 0, the column vector @ has only the ground state component that we denote as @, . Then the expectation value of Eq. (9) becomes it
[ Al”j, A;,].
( 10)
j=2
We now proceed to the evaluation of this expression for a vortex in a clean BCS-superfluid. The electronic ground state is constructed by pairing the states u, and Us*which are solutions of Bogoliubov’s equations [ 191
(11)
n[~,,,(r,-R) m
where 10) is the zero-particle state and b: = is an operator that creates a pair of 4T L fermions with angular momenta CL,,,- 3 and - p,,, f . The product in Eq. (14) contains both the core bound states with pLm< 0 only, and the scattering states with both signs of p,,, (see Ref. [20]). The excited state I (p2) which couples to I ‘p, > via the operator V, is obtained by breaking the occupied pair with pm = - 3. This is accomplished by defining the Bogoliubov-Valatin operators [21] =
U&
rl,,
A=
IA(r-R)lexp[-iB(r-R)].
(12)
Following Caroli et al. [ 191, the Bogoliubov amplitudes are expressed (in two dimensions) as u”(r) =f!“‘( r) exp[i( pn - $)0],
( 134
r) exp[i( pn + +)0].
(13b)
un( r) =fl”‘(
-
=u;ctpl.
$C-,I ,
( 154
+u;cpr.
(15b)
Similar to the vortex-free case, we then write using Eqs. (14) and (15) I(P2)=L,2,Y:,2J = - n’(u,
m
Id
( 16)
+ umbi)
x(uf,,,--uf,,,b:,,z
) IO>,
(17) where the prime denotes a product with the bound state, pm = - 3, missing. Owing to the angular momentum conservation, I cp2> is the only relevant excited state. Hence, the gauge fields defined in (5) are 2 x 2 Hermitian matrices. Performing the integration over the fermion coordinates, the diagonal matrix element obtained by using Eq. (14) in (5) is A^,, =i(cp, IV,Icp,> =
/
d2XS(r-R)VRO(r-R),
(18)
where s= c[
where so is the free-electron Hamiltonian, and the gap function is of the form
+U,,,(rm--R)b~]lO), (14)
Yi PT i 1 AaM, A*’ .
279
Letters A 221 (1996) 277-281
IUJ*(-&+;)+
lu,l’(-j&-g].
m (19)
This result agrees with the Abelian adiabatic phase obtained recently by Gaitan [4]. A similar calculation, done with the use of Eq. (171, yields
E. .%nbnek/Physics
280
Letrem A 221 (1996) 277-281
We see from this equation that, due to the core excitation, the geometric phase of the excited state is diminished compared to that of the ground state. Using Eq. (18), the Abelian part of the curvature (10) is written as
(21) For a point vortex in two dimensions,
(FXY&~berian=
we have
V,xV,O=2mY(r-R)z,,
(22)
where z0 is the unit vector perpendicular to the superfluid sheet. Thus the first term of Eq. (21) gives us 2nS(O). Evaluating the second term in polar coordinates, we have
&dp P/ 2ndv(
V,SX
which a large hydrodynamic Magnus force is obtained by dropping the I p,,, I 2 terms from S (see Eq. (19)). On the other hand, Eq. (25) gives a Magnus force which agrees (in magnitude) with the result of Ref. [lo] (applied to the case of an uncharged superfluid). According to Eq. (lo), the non-Abelian part of the adiabatic curvature is given by -i(A:,&
2TZ,
/
a;;
=
-_1 .
3:
-
ve)
dp=2n[S(x)
au-
u_,,~----
so that aA:,
(F1Xy)*belian=aX-aY=2~~(tC)-
(24
Due to the exponential decay of the core bound states, the expression (19) for S(m) contains only scattering states. Since in the bulk of the superfluid, the quantities 1u, I 2 and 1u, I 2 are independent of the sign of p., [20], the terms in Eq. (19) that are proportional to CL,,cancel out and we are left with S(a)
=+c(
n
Iu,12-
1/2
ax
i k,/2.
au - l/2 ax
~ I/*
(27)
-S(O>]z,,
(23) aA:,
(26)
Using Eqs. (14) and (17), we obtain from Eq. (5) the off-diagonal matrix element
0
=
-Ay2A&).
IuJ2)
(25) where wu is the Debye frequency, [ is the singleparticle energy measured from the chemical potential II, and d, is the energy gap in the bulk. Eqs. (8), (24) and (25) define the hydrodynamic contribution to the Magnus force. Its magnitude and sign are determined by the derivative of the density of states, N( 5 + p), at the Fermi level [lo]. In BCS theory, this quantity is very small, so that S(m) is negligible compared to the electron density. This result contrasts with the conclusion of Ref. [4] in
This approximate result becomes exact for k, ,$’* 1, in which case the two-dimensional integration in Eq. (27) is dominated by terms involving the derivatives of the radial parts of the amplitudes (13). In a similar way, we obtain A:, = - fkF_ With use of these results, Eq. (26) yields (FXY)non_Abeiian~ -+ki=
-2nn,,
(28)
where n, is the sheet superfluid density. Introducing the results (24) and (28) into Eq. (lo), we obtain from Eq. (8) the net Magnus force F,v,=2~[n,-S(cc)](zoXV),
(29)
where V = d is the velocity of the vortex center. This result should be compared with Eq. (2) of van Otterlo et al. [ 101 in the clean limit. We see that there is an agreement except for the sign of the hydrodynamic term. Specifically, if we expand N( [ + p) in Eq. (25) about the Fermi level, the hydrodynamic term in Eq. (29) has a negative sign if the derivative, N’(p), is positive. This finding is of relevance for the interpretation of the sign changes of the Hall angle in terms of broken particle-hole symmetry
DOI. In the presence of disorder, the core contribution to F, is modified as first shown in Ref. [6]. In the present formulation, the corresponding reduction factor can be obtained by converting the curvature (26)
E. .%ma’nek/Physics
Letters A 221 (1996) 277-281
to a Kubo formula, and evaluating the latter within the relaxation-time approximation (see Ref. [15] for details). The disorder-induced reduction of the core part of the Magnus force can be simply understood as a breakdown of the adiabatic approximation: in the presence of level broadening, there is no gap in the excitation spectrum so that the vortex cannot be moved slowly enough to prevent real excitations which tend to wash out the topological effects [22]. To summarize, we use the method of the geometric phase to derive an expression for the Magnus force on a vortex in an uncharged BCS superfluid at T = 0. The gauge potentials associated with vortex motion are shown to acquire a non-Abelian structure. The equation of motion for the vortex yields a hydrodynamic part determined by the electron density of states in the bulk, and a core part generated by the non-Abelian potentials. Except for the relative sign of these two parts, our results agree with the recent work by van Otterlo et al. [ 101 based on a path integration method of Refs. [81 and [91. I would like to thank Dardo Piriz for a helpful suggestion.
References [ I] A. Shapere and F. Wilczek, eds., Geometric phases of physics (World Scientific,
Singapore,
1989).
l21 F.D.
281
Haldane and Y.-S. Wu, Phys. Rev. Lett. 55 (1985) 2887. [31 P. Ao and D.J. Thouless, Phys. Rev. Lett. 70 (1993) 2158. [41 F. Gaitan, Phys. Rev. B 51 (1995) 9061. V.B. Geshkenbein, A.I. Larkin and S. [51 M.V. Feigel’man, Levit, JETP Lett. 57 (1993) 711. b1 N.B. Kopnin and V.E. Kravtsov, JETP Lett. 23 (1976) 578. Phys. Rev. B 44 [71 N.B. Kopnin and M.M. Salomaa, (1991) 9667. b31E. Sim&rek, Phys. Lett. A 194 (1994) 323. l91 V. Ambegaokar, U. Eckem and G. Schon, Phys. Rev. Lett. 48 (1982) 1745. [lOI A. van Otterlo, M. Feigel’man, V. Geshkenbein and G. Blatter, Phys. Rev. Lett. 75 (1995) 3736. llll S.J. Hagen, C.J. Lobb, R.L. Green, M.G. Forrester and J.H. Kang, Phys. Rev. B 41 (1990) 11630. [121 J.M. Harris, N.P. Gng and Y.F. Yan, Phys. Rev. Lett. 71 (1993) 1455. D.T. Nemeth, F. [131 B. Parks, S. Speilman, J. Orenstein, Ludwig, J. Clarke, P. Merchant and D.J. Lew, Phys. Rev. Lett. 74 (1995) 3265. [141 F. Wilczek and A. Zee, Phys. Rev. Lett. 52 (1984) 2111. I151 E. Simanek, Phys. Rev. B 52 (1995) 10336. l161 B. Zygelman, Phys. Lett. A 125 (1987) 476. 1171 J. Moody, A. Shapere and F. Wilczek, Phys. Rev. Lett. 56 (1986) 893. 1181 S.K. Wong, Nuovo Cimento A 65 (1970) 689. 1191 C. Caroli, P.G. de Gennes and J. Matricon, Phys. Lett. 9 ( 1964) 307. [201 J. Bardeen, R. Kiimmel, A.E. Jacobs and L. Tewordt, Phys. Rev. 187 (1969) 556. Theory of superconductivity (Benjamin, 1211 J.R. Schrieffer, New York, 1964). l221 M. Stone, Phys. Rev. D 33 (1986) 1191.