Topological superfluid 3He-B in magnetic field and Ising variable

0.14 MB·English

by  G. E. Volovik

Pis’ma v ZhETF
Topological superfluid 3He-B in magnetic field and Ising variable
G.E. Volovik 1)
LowTemperatureLaboratory,AaltoUniversity,SchoolofScienceandTechnology, P.O.Box15100,FI-00076AALTO,Finland
LandauInstitute forTheoretical PhysicsRAS,Kosygina2,119334Moscow,Russia
SubmittedJanuary16,2010
The topological superfluid 3He-B provides many examples of the interplay of symmetry and topology.
Here we consider the effect of magnetic field on topological properties of 3He-B. Magnetic field violates the
0 time reversal symmetry. As a result, the topological invariant supported by this symmetry ceases to exist;
1 and thusthe gapless fermions on the surface of 3He-B are not protected any more bytopology: they become
0
fullygapped. Nevertheless,ifperturbationofsymmetryissmall,thesurfacefermionsremain relativisticwith
2
mass proportional to symmetry violating perturbation – magnetic field. The 3He-B symmetry gives rise to

n the Ising variable I = ±1, which emerges in magnetic field and which characterizes the states of the surface
a
of 3He-B. This variable also determines the sign of the mass term of surface fermions and the topological
J
invariantdescribingtheireffectiveHamiltonian. Thelineonthesurface, whichseparatesthesurfacedomains
9
with different I, contains 1+1 gapless fermions, which are protected by combined action of symmetry and
1
topology.

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r
e
PACS:67.30.ht, 71.90.+q, 04.50.-h
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o
1. INTRODUCTION in 2+1 momentum space. The value of this invariant
.
at is determined by the Ising type variable I, which char-
m Recently topological insulators, semimetals, super- acterizes the surface of 3He-B in the presence of mag-
-conductors,superfluidsandothertopologicallynontriv- netic field. The surface of 3He-B can be in two states,
dial gapless and gapped phases of matter have attracted with I = +1 and I = −1, and both states of the sur-
na lot of attention, see e.g. [1, 2, 3, 4]. Superfluid 3He-B
face have been recently probed by NMR experiments
o
crepresentsthe fully gapped topologicalsuperfluids with with 3He-B confined in a single micron-size slab [15].
[time reversalsymmetry[3,4,5]. This3+1systemcon- These experiments also detected the coexistence of the

5 tains 2+1gaplessAndreev boundstates onthe surface domains with different I on the surface of 3He-B; these
voftheliquidorattheinterfacesbetweenthebulkstates domains are separated by the one-dimemsional domain
4with different topological invariants;these fermion zero
walls(lines)onthesurface. Sincethedomainsarechar-
1modes resemble 2+1 Majorana fermions in relativistic
acterizedbydifferentvaluesofthetopologicalinvariant,
5
quantum field theories [6, 7, 8]. Andreev bound states
1 the boundary line between the domains contains 1+1
.onthesurfaceof3He-Bwerediscussedtheoretically(see gapless fermions with “relativistic” spectrum.
1
e.g. [9, 10, 11]) and probed experimentally [12, 13, 14].
0
0However,theMajoranasignatureofthesestateshasnot
1yet been observed. One of the possible tools for obser-
:vationofthe“relativistic”spectrumoftheboundstates 2. DISTORTION OF ORDER PARAMETER
v
iisNMR,whichrequiresanexternalmagneticfield. The
X Magnetic field destroys the rotational symmetry
influence of magnetic field on Majorana fermions has
r of 3He-B, leading to the anisotropic gap. The gap
abeen recently discussed in [11]. Here we consider the
anisotropy is also induced by the walls. The order pa-
effect of magnetic field in more detail.
The topologicalinvariantfor 3He-B which gives rise rameterin3He-Bis3×3matrixAαi [16]. Ittransforms
as vector under spin rotation(the Greek index), and as
to the 2+1 gapless fermion zero modes on its surface
vectorunder orbitalrotation(the Latinindex). For the
is supported by the time reversal symmetry [5]. Mag-
distorted 3He-B it has the form
netic field destroys this symmetry, as a result the An-
dreev bound states acquire gap (mass) proportional to
∆ 0 0
the magnitude of magnetic field. The resulting massive k
∆
2+1 Andreev-Majorana fermions also have nontrivial Aαi =RαiA(ij0) , A(ij0) = 0 ∆k 0  , q = ∆⊥ ,
k
topology,describedbythenonzerotopologicalinvariant 0 0 ∆
 ⊥
1)[email protected].fi   (1)
1
2 G.E.Volovik
where q is the distortion parameter with q = 1 in the lows that in the geometry of experiment, the distor-
non-deformed B-phase and q = 0 in the planar state; tion parameter q at temperature T = 0.54T is about
c
∆ isthegapforquasiparticlespropagatingindirection q ∼ 0.6−0.7. From this figure it also follows that in
⊥
along the normal to the wall; ∆ is the gap for quasi- coolingoneobtainsthreedifferentstates: thestatewith
k
particles propagating in directions parallel to the wall; I = +1 characterized by small positive frequency shift;
R is rotation matrix, which connects spin and orbital the state with I = −1 characterized by large negative
αi
spaces. Assaid,thedistortionofthegapiscausedboth frequencyshift;andthestatewithcoexistingdomainsof
by magnetic field and by boundaries. Let us introduce both orientations of the Ising variable, which is charac-
the unit vectorhˆ =H/H in directionofmagnetic field, terized by two peaks in the NMR spectrum. The latter
and the unit vector ˆl = hˆ R in the orbital space. statecontainsdomainwall(s)–theline(s)onthesurface
i α αi
The orbital vector ˆl shows the direction of the orbital separatingsurfacedomainswithoppositeorientationsof
angular momentum of Cooper pairs (see e.g. [17, 18]) I.
andplaystheroleoftheanisotropyaxisfortheuniaxial Letusalsoforcompletenessconsiderthefieldparal-
deformation of the gap. lel to the plates. In this case theˆl-vector is perpendic-
ular to the field and I =ˆl·hˆ = 0. The frequency shift
is positive, ω(I =0)−ω =−(ω(I =−1)−ω )>0.
L L
3. ISING VARIABLE PROBED BY NMR
We have seen that the orientation of the magnetic
field with respect to the Ising spin is crucial for the
Let us consider NMR experiments with superfluid
NMRexperiments. Letusconsidertheeffectoforienta-
3He-B confined in a single micron size slab [15], where
the field is normal to the plates: H = Hˆz. Typical tion of the magnetic field on the spectrum of Majorana
fermions.
fields used for NMR exceed the dipole field (spin-orbit
coupling). As a result the boundary conditions for ˆl-
vector is that ˆl is normal to the walls, ˆl׈z = 0 [19]. 4. MAJORANA FERMIONS IN DISTORTED
This boundary condition ensures that the anisotropy B-PHASE
axis ˆz caused by the walls and anisotropy axisˆl caused
by magnetic field coincide. Fermions in magnetic field are described by the fol-
In a given geometry with the field being normal to lowing Hamiltonian:
theplates,theprojectionoftheˆl-vectoronthemagnetic
k2−k2 1
field direction H= Fτ − γH·σ+
I =ˆl·hˆ, (2) 2m∗ 3 2
k k k
x y z
τ ∆ (z)σ˜ +∆ (z)σ˜ +∆ (z)σ˜ , (4)
represents the Ising variable. Thus is because only two 1 k xk k yk ⊥ zk
orientationsofˆl-vectorwithrespecttomaneticfieldare (cid:18) F F F(cid:19)
allowed by boundary conditions: I = +1, and I = −1. whereτi arePaulimatricesofBogolyubov-Nambuspin;
σ arePaulimatricesof3Henuclearspin; γ isthegyro-
The degeneracy is lifted by spin-orbit interaction. The α
magnetic ratio ofthe 3He atom; andfinally σ˜ =σ R
latter is small compared with all other energies but is i α αi
are the Pauli matrices obtained from matrices σ by ro-
important for the measured NMR frequency shift from
tation. In the absence of magnetic field there is a topo-
the Larmorvalue ω , whichisjust causedbyspin-orbit
L
logical invariant for 3He-B [5]:
interaction. Themeasurementsofthefrequencyshiftal-
lows to determine experimentally the value of the Ising
e
spin I on the surface. N = 24ijπk2 tr τ2 d3k H−1∂kiHH−1∂kjHH−1∂kkH .
For the distorted 3He-B, the corresponding fre- (cid:20) Z (cid:21)
(5)
quency shifts can be found from equations (4.5) and At H=0, when the time reversal symmetry is pre-
(4.10) of Ref. [20]. The frequency shift is positive when
served, the integral (5) is invariant under deformations
ˆl-vectoris parallelto the field, i.e. for I =+1,andneg-
because the matrixτ anti-commuteswithHamiltonian
ative for ˆl-vector antiparallel to field, i.e. for I = −1. H. The nonzero val2ue of this invariant supports gap-
The ratio of the frequency shifts is
less fermions at the interface between bulk states with
ω(I =+1)−ω 1−q2 different values of N [6] and on the surface of 3He-B
L
ω −ω(I =−1) = 1+2q2 . (3) [8]. These fermions contribute to the ground-state spin
L
supercurrent [6].
Both positive and negative frequency shifts have been In the presence of magnetic field, the integral (5)
observed in [15]. From Fig. 1 of [15] and Eq.(3) it fol- doesnotrepresentthetopologicalinvariantbecausethe
Topological superfluid 3He-B in magnetic field and Ising variable 3
Paulitermdoesnotanti-commutewithτ ,andthegap- appears due to violation of the time reversal symme-
2
lessness ofthe Andreev bound states is notguaranteed. try by magnetic field. The mass (12) was obtained in
ThePaulitermcanbeexpressedintermsoftherotated Ref. [11] only for the case of magnetic field normal to
spin σ˜: the wall. However, as distinct from Ref. [11], in our
case the equation (12) is valid for any orientation of
H·σ =H σ =H R σ˜ =H ˆl·σ˜ =(ˆl·zˆ)Hσ˜ . (6)
α α α αi i z magnetic field. This is because we consider relatively
large magnetic fields used in NMR experiments, when
Equation (6) demonstrates that the Hamiltonian (4)
thespin-orbitinteractionisrelativelysmall: thoughthe
does not depend on the orientation of magnetic field.
spin-orbit interaction is responsible for the frequency
It depends only on the Ising variableˆl·ˆz = ±1, which
shift, it does not influence the orientation of the vector
describestwopossibleorientationsofthevectorˆlonthe
ˆlonthesurface. AsaresultthePauliterm(6)expressed
surface. This is because in large fields used in experi-
via the rotated spin does not depend on orientation of
ments, the spin-orbit interaction can be neglected and
magnetic field.
the boundary conditions for the vectorˆl on the surface
This is contrary to the effect of magnetic field dis-
are fully determined by the orbital effects, which are
cussed in Refs. [7, 11], where the magnetic field is as-
invariant under rotation of the magnetic field together
sumed to be so small that the boundary conditions are
with the spin subsystem.
solely determined by spin-orbit interaction. This leads
Let us, however, consider the orientation of mag-
to anisotropy of magnetic susceptibility [7, 11], which
netic field along the axis z. This allows us to express
the HamiltonianintermsofthevariableI =ˆl·hˆ in(2), however disappears in large fields. Note also that in
our case the spin which enters the Hamiltonian (9) for
which is regulated in NMR experiments in Ref. [15]
Majorana fermions, is obtained from the real spin by
where the orientation of the field normal to the plates
rotations provided by the order parameter matrix R .
is used: αi
This means that in large fields, when the spin-orbit in-
k2−k2 1 teractioncanbeneglected,thequantizationaxisforthe
H= Fτ − γHIσ˜ +
2m∗ 3 2 z real spin of Majorana fermions is not fixed.
kx ky kz The effective Hamiltonian (9) has nontrivial topol-
τ ∆ (z)σ˜ +∆ (z)σ˜ +∆ (z)σ˜ . (7)
1(cid:18) k xkF k ykF ⊥ zkF(cid:19) ogy described by the topological invariant
The second order secular equation (10) in [8] pro-
1
duces the following 2×2 Hamiltonian for fermion zero N = tr d2kdω G∂ G−1G∂ G−1G∂ G−1 ,
2+1 4π2 kx ky ω
modes: (cid:20)Z (cid:21)
(13)
H+(1+) H+(1−) =c −21γHI ky+ikx , (8) where G = 1/(iω − Hzm) is the Green’s function of
H−(1+) H−(1−)! ky−ikx 21γHI ! fdeurcmeidoninzreerloatmivoidsteisc.2T+he1itnhveaorriaienst[(2123,)2w3,a2s4fi].rsIttiinstrroe--
which can be written as sponsible for quantization of Hall conductivity induced
bytopologicalstructureofenergy-momentumspace,the
γHI
H =cˆz·(σ˜ ×k)− ˆz·σ˜ . (9) so-called intrinsic quantum Hall effect. It was later in-
zm
2
dependently introducedforthe filmofsuperfluid3He-A
This the so-called helical (see e.g. [21] and references in condensed matter [25, 26]. Under experimental con-
therein)Hamiltonianproducestherelativisticspectrum ditions in Ref. [15], the value of this invariant is deter-
of massive particles: mined by the Ising variable I in (2):
E2 =c2k2+M2, (10)
I
N = . (14)
2+1
2
where the ‘speed of light’ [8]
∞dz∆k(z)exp − 2 zdz′∆ (z′) NMR experiments in Ref. [15] detected coexistence of
c= 0 kF vF 0 ⊥ . (11) the surface domains with different I, which are thus
R ∞dzexp −(cid:16)2 zRdz′∆ (z′) (cid:17) separated by the one-dimensional domain wall on the
0 vF 0 ⊥
surface. Sincethedomainshavedifferentvaluesoftopo-
R (cid:16) R (cid:17)
The mass of the surface fermions logicalinvariantN , then accordingto the index the-
2+1
orem (see e.g. [27]), the line separating two domains
1
M = γH, (12) contains gapless 1+1 fermion zero modes.
2
4 G.E.Volovik
5. DISCUSSION smoothly violated, etc. The Green’s function matrices
with spin and band indices must be used for this clas-
The symmetry of superfluid 3He-B is important sification, since they take into account interaction. In
for its topological properties. Magnetic field violates this way we may finally obtain the full classification of
isotropy of 3He-B: the gap become anisotropic with topological matter, including insulators, superconduc-
∆ −∆ ∝ H2. This distortion of the gap does not tors,magnets,liquids,andvacuaofrelativisticquantum
k ⊥
change the topological invariant introduced in [5], and fields.
thus cannot destroy the massless relativistic fermions The problem of the interplay of topology and sym-
on the surface of 3He-B. However, the violation of the metry exists not only for the momentum-space topol-
other symmetry of 3He-B by magnetic field – the time ogy,butalsoforthereal-spacetopology,whichconsiders
reversal symmetry – is crucial. The topological invari- classification of topological defects and textures in sys-
antsupportedbythissymmetryceasestoexistandthus tems with broken symmetry. For example, some values
thegaplessfermionsonthesurfaceof3He-Barenotpro- ofthe windingnumberdescribingthe topologicaldefect
tectedanymorebytopology: theybecomefullygapped. are notcompatible with symmetry ofthis defect, which
Nevertheless, if perturbation of symmetry is small, the connectsforexamplethepointsrand−r. Thishappens
surface fermions remain relativistic with mass propor- with monopoles [34]; disclinations – topological defects
tional to symmetry violating perturbation – magnetic inliquidcrystals[35];vorticesinsuperfluids[17];cosmic
field. strings; etc. One of the tasks there is the classification
This is notthe whole story. The spontaneouslybro- oftopologicalobjectsinterms oftheirsymmetry. Simi-
ken relative spin-orbit symmetry of 3He-B [28] – sym- larly we need the symmetry classificationof topological
metry under relative rotations of spin subsystem with objects in momentum space, where the symmetry con-
the respect to the orbital one [29] – gives rise to the nects different points in momentum space, say k and
vector ˆl of orbital anisotropy in the presence of mag- −k [1].
netic field. This in turn leads to Ising variable I = ±1 The special role in classification of topological sys-
whichcharacterizesthesurfaceof3He-B.TheIsingvari- tems is played by dimensional reduction. See for ex-
abledeterminesthesignofthe masstermofthe surface ample Chapters 11 and 21 in [27] where the topological
fermions and the topological invariant (14) describing invariantsfor the 2Dfully gappedsystemsareobtained
their effectiveHamiltonian(9). Theline onthe surface, by dimensional reduction of the topological invariants
whichseparatesdomains withdifferentI, containsgap- for the 3D gapless system; the dimensional reduction
less fermions, which are protected by combined action hasbeenrecentlydiscussedin[36]. Thedimensionalre-
of symmetry and topology. So, the superfluid 3He-B duction allows us to use for classificationof the gapped
provides many examples of the interplay of symmetry systemsthescheme,whichwassuggestedbyHoravafor
and topology. the classification of the topologically nontrivial nodes
The other examples of the interplay of symmetry in spectrum [37]. But this also must be supplemented
and topology in topological matter and in vacua of rel- by the symmetry analysis, and also by analysis of the
ativistic quantum fields can be found in [27, 32, 33]. types ofthe emergentsymmetries whichnecessarilyap-
Some topological invariants exists only due to symme- pear within some topological classes.
try, while some values of the topological invariant are IowespecialthankstoJohnSaundersfordiscussion
not compatible with symmetry. It is right time for the onthe experiments [15]and amgratefulto Shinsei Ryu
general classification of topological matter in terms of and Liang Fu for discussion on symmetry and topol-
its symmetry andtopology. The attemptofsuchclassi- ogy. This work is supported in part by the Academy of
fication was made in [1, 2, 3, 4]. However, only non- Finland,CentersofExcellenceProgram2006–2011,and
interacting systems were considered, and not all the the Khalatnikov–Starobinsky leading scientific school
symmetries were exploited. Examples are again pro- (grant 4899.2008.2).
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