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Quantized hall effect in superfluid helium-3 film
Volume
128, number
PHYSICS
5
QUANTIZED
LETTERS
HALL EFFECT IN SUPERFLUID
4 April 1988
A
HELIUM-3
FILM
G.E. VOLOVIK L.D. Landau Institute for Theoretical Physics, Kosygin Street 2, I 17334 Moscow, USSR and NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Received 5 February 1988; accepted Communicated by A.A. Maradudin
for publication
5 February
1988
Supertluid 3He-A confined in a slab of small enough width exhibits an effect analogous to the quantization of the Hall conductance racy,.The factor cr_ in the supercurrent transverse to the applied gradient of the chemical potential,j=~.~.~,,IxVp, is a topological invariant; 0X.“= ( 1/2h)n, where the integer n corresponds to the number of transverse energy levels below the Fermi energy, and I is the direction of the orbital momentum of Cooper pairs which plays the role of magnetic field.
It is now understood (see e.g. ref. [ 1 ] ) that the exact quantization of the Hall conductance of a twodimensional electron system (quantized Hall effect, QHE) as well as the quantization of other physical parameters both in condensed matter and particle physics [ 2,3 ] is a consequence of general topological properties of the system. Here we consider a system whose topology results in an effect quite similar to the QHE: superfluid 3He-A confined in a thin slab. Though the experimental observation of this effect is essentially more complicated than in an electronic system, since the ‘He atoms are electrically neutral, the theoretical background for this phenomenon is very simple, since no magnetic field and impurities are needed to produce the QHE in a superfluid helium film: the QHE occurs in a spatially uniform system. Superfluid 3He-A apart from other spontaneously broken symmetries exhibits also breaking of the time inversion symmetry. This superfluid phase is the socalled orbital ferromagnetic one with the common direction of the orbital angular momenta of Cooper pairs, described by the unit vector I which is odd under time inversion: I( -t) = -Z(t). Due to this broken symmetry the applied gradient of the chemical potential, V,u, which is equivalent to the electric field for an electron system, should cause the transverse “Hall” supercurrent 0375-9601/88/S ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
j=aJxVp,
(1)
which for the bulk liquid was calculated in ref. [ 41. We show here that in the confined geometry the “Hall conductance” ox,, is quantized. Let us consider the uniform state of ‘He-A in the parallel-plane geometry with distance a between the plates. The orbital unit vector 1 is fixed to be normal to the plates: l=i or I= -2. The spin degrees of freedom are not important for the QHE due to its essentially orbital origin, therefore the spin part of the order parameter (i.e. of the gap function, d) is omitted. The orbital part of the gap function for the pwave state of 3He-A with Z=i is A(k)=
2
(k,+ik,)
.
(2)
F
To find the rr_, value, i.e. the response of the supercurrent to VP, one must consider the Bogolyubov-Nambu hamiltonian for the quasiparticles, which is a 2X2 matrix in the particle-hole space, A(P) P(r) -p2/2m
=r3($
B.V.
-p(r))+
2
>
(rlpx-rz2pY),
(3)
277
Volume 128, number
PHYSICS
5
where p= - iV and T, are the Pauli matrices. of this hamiltonian the supercurrent is j=Tr
In terms
CpG, (0
(4)
where the trace is over all the eigenstates G-‘=H-tiw
and
.
Since we choose ,u to depend only on the coordinates x and y in the plane the transverse motion in the z direction is not distorted and is quantized in the conventional way, 2
x2n2 PI e,,=2m+~-~(X’y)-P~-~,(x,y),
(5)
and the current may be calculated n independently, i= C j,, ,I
in= &
Performing
Tr(pG,
for each eigenstate
) .
(6)
the gradient expansion
(see e.g. ref. [ 3 ] ),
(7) one obtains the contribution current from the nth band, j = fl(pL,) n 2hlXVK
to the “Hall”
(8)
(12)
where p=k’,/37c’ is the bulk density of the liquid. The change in r? takes place when one of the pL, crosses zero, i.e. the eigenvalue of the transverse motion crosses the Fermi energy. At that moment the gap in the quasiparticle energy spectrum
E,(k,)=(c:+A;k:/k:)“2
(13)
disappears. Thus the conductance a,, as a function of the width a or of the Fermi energy p has plateaus of quantized values in the regions where the quasiparticle spectrum has a gap, while the jump in ox,, occurs at points where the gap disappears. This just illustrates the general theory of the quantization of physical parameters [ l-3 1. The topological reason of the quantization of s,, comes from the structure of the order parameter (2 ). On the Fermi-surface kFn= (2m,u,) ‘12, corresponding to the nth occupied level, the order parameter phase @,
(14)
has nonzero winding number, p= 1, when going around the Fermi-surface (in the two-dimensional case the Fermi-surface is a circle). This winding number is related with the Chern number (see ref. [ 1 ] ) and produces the quantized contribution to the Hall conductance from a single filled level; in the general case of winding number p, which corresponds to the Cooper pairing with orbital momentum projection p, the contribution from a single filled band is
(10)
(11)
where E is the largest IZsatisfying the condition (9). For large width a, when n’- kFa/x >>1, the bulk 278
j= :lXVp,
to the expression
(15)
and the total cupercurrent is given by eq. (1) with quantized value of the oXxv, fi s\-,. = s >
transforms
F
(9)
1 2 )
“Hall” supercurrent found in ref. [ 41,
A(k)=Ao~ei”:
and for these n the supercurrent j, does not depend on n. Thus the contribution to the Hall conductance from a single filled band is o:,,=
4 April 1988
A
super-
where &x)=1 for x>O and 19(x)=0 for x
LETTERS
In conclusion QHE should exist in thin films of superfluid 3He-A. This together with the fractional statistics of the topological objects in the film [ 51 makes an experimental investigation of superfluid 3He films necessary. I thank A. Luther, A. Nersesyan for useful discussions.
and A. Talapov
Volume
128, number
5
References [ 1] M. Kohmoto,
Ann. Phys. 160 ( 1985) 343.
PHYSICS
LETTERS [ 21 [3] [4] [5 ]
A
4 April 1988
R. Jackiw, Comm. Nucl. Part. Phys. 13 ( 1984) 141. A.J. Niemi and G.W. Semenoff, Phys. Rep. 135 (1986) 99. N.D. Mermin and P. Muzikar, Phys. Rev. B 21 ( 1980) 980. G.E. Volovik, submitted to Phys. Ser.
279
128, number
PHYSICS
5
QUANTIZED
LETTERS
HALL EFFECT IN SUPERFLUID
4 April 1988
A
HELIUM-3
FILM
G.E. VOLOVIK L.D. Landau Institute for Theoretical Physics, Kosygin Street 2, I 17334 Moscow, USSR and NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Received 5 February 1988; accepted Communicated by A.A. Maradudin
for publication
5 February
1988
Supertluid 3He-A confined in a slab of small enough width exhibits an effect analogous to the quantization of the Hall conductance racy,.The factor cr_ in the supercurrent transverse to the applied gradient of the chemical potential,j=~.~.~,,IxVp, is a topological invariant; 0X.“= ( 1/2h)n, where the integer n corresponds to the number of transverse energy levels below the Fermi energy, and I is the direction of the orbital momentum of Cooper pairs which plays the role of magnetic field.
It is now understood (see e.g. ref. [ 1 ] ) that the exact quantization of the Hall conductance of a twodimensional electron system (quantized Hall effect, QHE) as well as the quantization of other physical parameters both in condensed matter and particle physics [ 2,3 ] is a consequence of general topological properties of the system. Here we consider a system whose topology results in an effect quite similar to the QHE: superfluid 3He-A confined in a thin slab. Though the experimental observation of this effect is essentially more complicated than in an electronic system, since the ‘He atoms are electrically neutral, the theoretical background for this phenomenon is very simple, since no magnetic field and impurities are needed to produce the QHE in a superfluid helium film: the QHE occurs in a spatially uniform system. Superfluid 3He-A apart from other spontaneously broken symmetries exhibits also breaking of the time inversion symmetry. This superfluid phase is the socalled orbital ferromagnetic one with the common direction of the orbital angular momenta of Cooper pairs, described by the unit vector I which is odd under time inversion: I( -t) = -Z(t). Due to this broken symmetry the applied gradient of the chemical potential, V,u, which is equivalent to the electric field for an electron system, should cause the transverse “Hall” supercurrent 0375-9601/88/S ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
j=aJxVp,
(1)
which for the bulk liquid was calculated in ref. [ 41. We show here that in the confined geometry the “Hall conductance” ox,, is quantized. Let us consider the uniform state of ‘He-A in the parallel-plane geometry with distance a between the plates. The orbital unit vector 1 is fixed to be normal to the plates: l=i or I= -2. The spin degrees of freedom are not important for the QHE due to its essentially orbital origin, therefore the spin part of the order parameter (i.e. of the gap function, d) is omitted. The orbital part of the gap function for the pwave state of 3He-A with Z=i is A(k)=
2
(k,+ik,)
.
(2)
F
To find the rr_, value, i.e. the response of the supercurrent to VP, one must consider the Bogolyubov-Nambu hamiltonian for the quasiparticles, which is a 2X2 matrix in the particle-hole space, A(P) P(r) -p2/2m
=r3($
B.V.
-p(r))+
2
>
(rlpx-rz2pY),
(3)
277
Volume 128, number
PHYSICS
5
where p= - iV and T, are the Pauli matrices. of this hamiltonian the supercurrent is j=Tr
In terms
CpG, (0
(4)
where the trace is over all the eigenstates G-‘=H-tiw
and
.
Since we choose ,u to depend only on the coordinates x and y in the plane the transverse motion in the z direction is not distorted and is quantized in the conventional way, 2
x2n2 PI e,,=2m+~-~(X’y)-P~-~,(x,y),
(5)
and the current may be calculated n independently, i= C j,, ,I
in= &
Performing
Tr(pG,
for each eigenstate
) .
(6)
the gradient expansion
(see e.g. ref. [ 3 ] ),
(7) one obtains the contribution current from the nth band, j = fl(pL,) n 2hlXVK
to the “Hall”
(8)
(12)
where p=k’,/37c’ is the bulk density of the liquid. The change in r? takes place when one of the pL, crosses zero, i.e. the eigenvalue of the transverse motion crosses the Fermi energy. At that moment the gap in the quasiparticle energy spectrum
E,(k,)=(c:+A;k:/k:)“2
(13)
disappears. Thus the conductance a,, as a function of the width a or of the Fermi energy p has plateaus of quantized values in the regions where the quasiparticle spectrum has a gap, while the jump in ox,, occurs at points where the gap disappears. This just illustrates the general theory of the quantization of physical parameters [ l-3 1. The topological reason of the quantization of s,, comes from the structure of the order parameter (2 ). On the Fermi-surface kFn= (2m,u,) ‘12, corresponding to the nth occupied level, the order parameter phase @,
(14)
has nonzero winding number, p= 1, when going around the Fermi-surface (in the two-dimensional case the Fermi-surface is a circle). This winding number is related with the Chern number (see ref. [ 1 ] ) and produces the quantized contribution to the Hall conductance from a single filled level; in the general case of winding number p, which corresponds to the Cooper pairing with orbital momentum projection p, the contribution from a single filled band is
(10)
(11)
where E is the largest IZsatisfying the condition (9). For large width a, when n’- kFa/x >>1, the bulk 278
j= :lXVp,
to the expression
(15)
and the total cupercurrent is given by eq. (1) with quantized value of the oXxv, fi s\-,. = s >
transforms
F
(9)
1 2 )
“Hall” supercurrent found in ref. [ 41,
A(k)=Ao~ei”:
and for these n the supercurrent j, does not depend on n. Thus the contribution to the Hall conductance from a single filled band is o:,,=
4 April 1988
A
super-
where &x)=1 for x>O and 19(x)=0 for x
LETTERS
In conclusion QHE should exist in thin films of superfluid 3He-A. This together with the fractional statistics of the topological objects in the film [ 51 makes an experimental investigation of superfluid 3He films necessary. I thank A. Luther, A. Nersesyan for useful discussions.
and A. Talapov
Volume
128, number
5
References [ 1] M. Kohmoto,
Ann. Phys. 160 ( 1985) 343.
PHYSICS
LETTERS [ 21 [3] [4] [5 ]
A
4 April 1988
R. Jackiw, Comm. Nucl. Part. Phys. 13 ( 1984) 141. A.J. Niemi and G.W. Semenoff, Phys. Rep. 135 (1986) 99. N.D. Mermin and P. Muzikar, Phys. Rev. B 21 ( 1980) 980. G.E. Volovik, submitted to Phys. Ser.
279