Pis'ma v ZhETF, vol.93, iss.2, pp.69-72 ©2011 January 25
Flat band in the core of topological defects: bulk-vortex correspondence in topological superfluids with Fermi points
G. E. Volovik1^
Low Temperature Laboratory, Aalto University, School of Science and Technology, FI-00076 AALTO, Finland Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia Submitted 25 November, 2010
We discuss the dispersionless spectrum with zero energy in the linear topological defects - vortices. The flat band emerges inside the vortex living in the bulk medium containing topologically stable Fermi points in momentum space. The boundaries of the flat band in the vortex are determined by projections of the Fermi points in bulk to the vortex axis. This bulk-vortex correspondence for flat band is similar to the bulk-surface correspondence discussed earlier in the media with topologically protected lines of zeroes. In the latter case the flat band emerges on the surface of the system, and its boundary is determined by projection of the bulk nodal line on the surface.
1. Introduction. When the fermion zero modes localized on the surface or on the topological defects are studied in topological media, the investigation is mainly concentrated on the fully gapped topological media, such as topological insulators and superflu-ids/superconductors of the 3He-B type [1-3]. However, the gapless topological media may also have fermion zero modes with interesting properties, in particular they may have the dispersionless branch of spectrum with zero energy - the flat band [4, 5].
The dispersionless bands, where the energy vanishes in a finite region of the momentum space, have been discussed in different systems. Originally the flat band has been discussed in the fermionic condensate - the Khodel state [6-9], and for fermion zero modes localized in the core of vortices in superfluid 3He-A [10-12]. The flat band has also been discussed on the surface of the multi-layered graphene (see [13, 14] and references therein). In particle physics, the Fermi band (called the Fermi ball) appears in a 2+1 dimensional nonrelativis-tic quantum field theory which is dual to a gravitational theory in the anti-de Sitter background with a charged black hole [15].
Recently it was realized that the flat band can be topologically protected in gapless topological matter. It appears in the 3D systems which contain the nodal lines in the form of closed loops [4] or in the form of spirals [5]. In these systems the surface flat band emerges on the surface of topological matter. The boundary of the surface flat band is bounded by the projection of the nodal loop or nodal spiral onto the corresponding surface. Here we extend this bulk-surface correspondence
to the bulk-vortex correspondence, which relates the flat band of fermion zero modes in the vortex core to the topology of the point nodes (Dirac or Fermi points) in the bulk 3D topological superfluids.
2. Vortex-disgyration. As generic example we consider topological defect in 3D spinless chiral super-fluid/superconductor of the 3He-A type, which contains two Fermi points (Dirac points). Fermions in this chiral superfluid are described by Hamiltonian
H = T3e(p) + c (up • ei + r2p • e2), c(p) =
P2 ~ Pf
2 to
(1)
where 71,2,3 are Pauli matrices in the Bogoliubov-Nambu space, and in bulk liquid the vectors ej and e2 are unit orthogonal vectors. There is only one topologically stable defect in such superfluid/superconductor, since the homotopy group m(G/H) = m(SOs) = Z2- We choose the following order parameter in the topologically non-trivial configuration (in cylindrical coordinates r = (p,(t>,z)):
ei(r) = fi{p)4>, e2(r) = zsinA - /2(p)pcosA, (2)
with /1,2(0) = 0, /1,2(00) = 1. The unit vector I, which shows the direction of the Dirac points in momentum space, p± = ±PfI, is
i(r) =
ei x e2
/2 (p) z cos A + p sin A leiXe2l \J/|(p) cos2 A + sin2 A
• (3)
Asymptotically at large distance from the vortex core one has
^e-mail: volovik0boojum.hut.fi
ei (p = 00) = (j), e2 (p = 00) = z sin A — p cos A, I(p = 00) = z cos A + p sin A,
70
G. E. Volovîk
which means that changing the parameter A one makes the continuous deformation of the pure phase vortex at A = 0 to the disgyration in the 1 vector without vorticity at A = 7r/2, and then to the pure vortex with opposite circulation at A = 7r (circulation of the superfluid velocity around the vortex core is j> ds ■ vs = k cos A, where k = irh/m.). We consider how the flat band in the core of the defect evolves when this parameter A changes. In bulk, i.e. far from the vortex core, the Dirac points are
P± = ±PfUp = oo) = ±pF (z cos A + p sin A). (5)
Due to the bulk-vortex correspondence, which we shall discuss in the next section, the projection of these two points on the vortex axis gives the boundary of the flat band in the core of the topological defect:
E(p:) = 0, pi < p2p cos2 X.
(6)
This is the central result of the paper: in general the boundaries of the flat band in the core of the linear topological defect (a vortex) are determined by the projections on the vortex axis of the topologically protected point nodes in bulk. In the next section we consider the topological origin of the flat band and geometrical derivation of its boundaries. In Sec.4, the boundaries of the flat band (6) are obtained analytically.
3. Bulk-vortex correspondence. Let us first give the topological arguments, which support the existence of the flat band inside the vortex-disgyration line. Let us consider the Hamiltonian (1) in bulk (i.e. far from the vortex core) treating the projection p~ as parameter of the 2D system. At each p~ except for two values p~ = ±pfcosA corresponding to two Fermi points (see Figure), the Hamiltonian has fully gapped spectrum and thus describes the effective 2D insulator. One can check that this 2D insulator is topological for |p~| < pp\ cos A| and is topologically trivial for \pz\ > pp\ cos A|. For that one considers the following invariant describing the 2D topological insulators or fully gapped 2D supefluids [16]:
"4tt2
tr
N3(Pz) = dpxdpvdLjGdp.G-1GdpG-1GduG
(7)
where G is the Green's function matrix, which for non-interacting case has the form G= iu> — H. This invariant, which is applicable both to interacting and non-interacting systems, gives
N3(pz) = 1, \Pz\ <Pf|cosA|, N3(Pz) = 0 , \pz\ >pf|cosA|.
(8) (9)
N3(pz)=l
Fermi point
N3(pz) = 0
flat band
j Fermi point
-pF cosA,
Projections of Dirac (Fermi) points on the direction of the vortex axis (the z-axis) determine the boundaries of the flat band in the vortex core. Fermi point in 3D systems represents the hedgehog (monopole) in momentum space [16]. For each plane pz = const one has the effective 2D system with the fully gapped energy spectrum EPl (px,py), except for the planes with pz± = ±pF cos A, where the energy EPl{pX)py) has a node due to the presence of the hedgehogs in these planes. Topological invariant Ns(pz) in (7) is non-zero for < pf| cosA|, which means that for any value of the parameter pz in this interval the system behaves as a 2D topological insulator or 2D fully gapped topological superfluid. Point vortex in such 2D superfluids has fermionic state with exactly zero energy. For the vortex line in the original 3D system with Fermi points this corresponds to the dispersionless spectrum of fermion zero modes in the whole interval < pf| cos A|
At pz = ±pf|cosA|, there is the topological quantum phase transition between the topological 2D "insulator" and the non-topological one. The difference of 2D topological charges on two sides of the transition, N3(pz = pf cos A + 0) - N3(pz = pF cos A - 0) = N3, represents the topological charge of the Dirac point in the 3D system - hedgehog in momentum space [16]. As we know, the topological quantum phase transitions are accompanied by reconstruction of the spectrum of fermions bound to the topological defect: fermion zero modes appear or disappear after topological transition in bulk [2,17-19]. For the pure vortex, i.e. at A = 0 or A = 7r, we know from [10] that the vortex contains the fermionic level with exactly zero energy for any p~ in the region |P;| < Pf, i-e. in the region of parameters where the 2D medium has non-trivial topological charge, N3 = 1.
Flat band in the core of topological defects
71
On the other hand no such levels are present after the topological transition to the state of matter with N3 = 0.
The similar reconstruction of the spectrum at the topological quantum phase transition takes place for any parameter A ^ 7r/2 of the considered defect. This can be understood using the topology in the mixed real and momentum space [20, 21]. To study fermions with zero energy (Majorana fermions) in the core of a point vortex in a 2D topological superconductor, the Pontrya-gin invariant for mixed space has been exploited in Ref. [1]. The Pontryagin invariant describes classes of mappings S2 x S1 S2. Here the mixed space S2 x S1 is the space (px,py,(/>), where 4> is the coordinate around the vortex-disgyration far from the vortex core. This space is mapped to the sphere S2 of unit vector g(Px,Py,<t>) = g(Px,Py,<t>)/\g(Px,Py,<t>)\ describing the 2D Hamiltonian. In our case it is the Hamiltonian (1) outside the vortex core:
Hpz,\(Px,Py,<t>) = Ti9'{Px,Py, 0; Pz, A),
(10)
g3 _
Pi
2 to
MPz), KPz) =
Pf - P2z.
2 to '
91 = c(Py COS 4> ^ px sin (/>), g2 = c(pz sin A — cos A(px cos <f> + py sin (/>)), (11)
with pz and A being the parameters of this effective 2D Hamiltonian. The Pontryagin Z2 invariant is non-trivial and thus the zero energy state exists in the core of the defect in the effective 2D superconductor, if the parameters pz and A of the 2D Hamiltonian (10) satisfy condition \Pz\ <Pf|cosA|.
For the considered linear topological defect (vortex-disgyration) in the 3D system this implies that the core of this defect contains the dispersionless band in the interval of momentum \pz\ < pp| cos
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