Pis'ma v ZhETF, vol.94, iss.3, pp. 252-258
© 2011 August 10
Flat bands in topological media
T. T. Heikkila *1), N. B. Kopnin*+, G. E. Volovik*+ *Low Temperature Laboratory, Aalto University, School of Science and Technology, P.O. Box 15100, FI-00076 AALTO, Finland
+ Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia Submitted 22 June 2011
Topological media are systems whose properties are protected by topology and thus are robust to deformations of the system. In topological insulators and superconductors the bulk-surface and bulk-vortex correspondence gives rise to the gapless Weyl, Dirac or Majorana fermions on the surface of the system and inside vortex cores. Here we show that in gapless topological media, the bulk-surface and bulk-vortex correspondence is more effective: it produces topologically protected gapless fermions without dispersion - the flat band. Fermion zero modes forming the flat band are localized on the surface of topological media with protected nodal lines [1, 2] and in the vortex core in systems with topologically protected Fermi points (Weyl points) [3]. Flat band has an extremely singular density of states, and we show that this property may give rise in particular to surface superconductivity which could exist even at room temperature.
1. Introduction. Topological matter is characterized by a nontrivial topology in momentum space [4-7]. This topology may be represented by the momentum-space invariants as depicted in Fig. 1. They are in many respects similar to the real-space invariants which describe topological defects in condensed matter systems and in particle physics. In particular, the Fermi surface in metals is topologically stable, because it is analogous to the vortex loop in superfluids or superconductors [4]. In the same way, the Fermi point (the Weyl point) corresponds to the real-space point defects, such as hedgehog in ferromagnets or magnetic monopole in particle physics. The fully gapped topological matter, such as topological insulators and fully gapped topological superfluids represent skyrmions in momentum space: they have no nodes in their spectrum or any other singularities, and they correspond to non-singular objects in real space - textures or skyrmions (Fig. 1 top right).
Recently the interest to topological media has been mainly concentrated on the fully gapped topological media, such as topological insulators and superfluids or superconductors of the 3He-B-type. These systems contain topologically protected gapless fermions on the surface [6, 7], and in the core of topological objects [8-10], some of which have an exotic Majorana nature.
The gapless topological media also exhibit exotic fermion zero modes with interesting properties. In particular they may have Fermi arc (Fermi surface which terminates on monopole in Fig. 1 bottom left) [11-13], and a dispersionless branch of the spectrum with zero energy - the flat band [1-3,14,15]. Historically the flat
^e-mail: tero.heikkila0tkk.fi, kopnin0boojum.hut.fi, volovikOboojum.hut.fi
bands were first discussed in relation to Landau levels, but they may emerge even without magnetic fields. They were suggested in strongly interacting systems [16-19], in the core of quantized vortices [21], in 2+1-dimensional quantum field theory which is dual to a gravitational theory in the anti-de Sitter background [22], in rhombohedral graphite [23] and on graphene edge [24], and on the surface of superconductors with gap nodes in the bulk [24, 1, 14]. Flat band is the momentum-space analog of a domain wall (soliton) terminating on a half-quantum vortex [18] (Fig. 1 bottom right). The topologically protected flat bands, which we discuss here, as well as Fermi arc add a new twist in the investigation of the 3-dimensional topological matter, shifting the interest from the topological insulators and fully gapped superfluids/superconductors to their gapless 3-dimensional counterparts, such as superfluid 3He-A, graphite, topological semi-metals and gapless topological superconductors. Dispersionless bands may serve as a good starting point for obtaining interesting correlated and symmetry breaking states [25].
This paper is based on the earlier arXiv version [26].
2. Surface flat band in a semimetal. Consider the semimetal with topologically protected nodal line in the form of a spiral as in Fig. 2a and 3a. The topological invariant in the bulk, supporting the existence and topological stability of the nodal line and as a result of the surface flat band with respect to interactions, is the contour integral in momentum space
A"-. --¿r ir { din, IT 1 , (1)
47Ti Jc
where H(p) is the effective matrix Hamiltonian (inverse Green's function at zero energy), and az is pseudospin.
Fermi surface: vortex ring in p-space Fully gapped topological matter:
Metals, normal 3He C ro Py (Pz) ^ p> Pz skyrmion in p-space
rt h=+ca • p 3He-B, topological insulators, 3He-A film, vacuum of Standard Model
\ PF = 2p
Weyl point - hedgehog in p-space 3He-A, vacuum of SM, topological semimetals
Fig. 1. Topological matter, represented in terms of topological objects in momentum space. (top left) - Fermi surface is the momentum-space analog of the vortex line: the phase of the Green's function changes by 2ttNi around the element of the line in (w, p)-space. (top middle) - Fermi point (Weyl point) is the counterpart of a hedgehog and a magnetic monopole. The hedgehog in this figure has integer topological charge N3 = +1, and close to this Fermi point the fermionic quasiparticles behave as Weyl fermions. Nontrivial topological charges N1 and N3 in terms of Green's functions support the stability of the Fermi surfaces and Weyl points with respect to perturbations including interactions [4, 27]. (top right) - Topological insulators and fully gapped topological superfluids/superconductors are textures in momentum space: they have no singularities in the Green's function and thus no nodes in the energy spectrum in the bulk. This figure shows a skyrmion in the two-dimensional momentum space, which characterizes two-dimensional topological insulators exhibiting intrinsic quantum Hall or spin-Hall effect. (bottom left) - Flat band emerging in strongly interacting systems [17]. This dispersionless Fermi band is analogous to a soliton terminated by half-quantum vortices: the phase of the Green's function changes by tt around the edge of the flat band [18]. (bottom right) - Fermi arc on the surface of 3He-A [11] and of topological semi-metals with Weyl points [12, 13] serves as momentum-space analog of Dirac string terminating on monopole. The Fermi surface formed by the surface bound states terminates on the points where the spectrum of zero energy states merge with the continuous spectrum in bulk, i.e. with Weyl points
Flat band (Khodel state): m-vortex in p-space
The nodal line is stable if the integral over the contour C around the line is nonzero. On the other hand one may choose the contour C as a straight line along the direction p- normal to the surface, i.e. at fixed momentum p|| = (px,py) along the surface. Due to periodic boundary conditions, the points p- = ±7r/a are equivalent and the contour of integration C forms a closed loop, giving integer values to the integral iV2(p||) if the integration path does not cross the point in the bulk where the energy is zero. This integral iV2(P||) = 1 for any point p|| within the projection Sp of the spiral on
the surface, and iV2(p||) = 0 outside this region. The states with momentum py € Sp cannot be adiabatically transformed into states in topologically trivial (N2 = 0) media, and therefore a surface state with zero energy is formed for all momenta within Sp [2, 1, 24]. This is in contrast with topological insulators, where this type of momentum-space invariants can be defined only for some particular values of momenta (but in principle, the insulators with topologically protected surface flat bands are possible, we are indebted to A. Kitaev for this comment).
Fig. 2. Topological media which exhibit topologically protected dispersionless fermionic spectrum with exactly zero energy -the flat band, (a) - Three-dimensional topological semi-metal. Flat band can be obtained by stacking of graphene layers, each alone represented by Dirac points. In case of proper stacking, represented by the coupling of the pseudospins in each layer, the nodal line is formed in the bulk in the form of a spiral which ends up on the faces of the Brillouin zone[2, 29]. This nodal line has a non-zero topological charge N2 = 1. This nontrivial charge protects the surface states with zero energy e(px,py) = 0, in the whole region within the projection of the spiral on the surface, see Fig. 3a. (b) - Dirac points in graphene (top: real-space crystal lattice; bottom: reciprocal space). They lead to the flat band on the zig-zag edge, see Fig. 3b. (c) -Cuprate superconductors have also topologically protected nodal lines [19, 31]. They lead to the flat band on a side surface, see Fig. 3c. (d) - Superfluid 3He-A has two three-dimensional Weyl points, with N3 = +1 and N3 = —1. The projections of the Weyl points on the direction of the vortex line determine the boundaries of the region where the spectrum of fermions bound to the vortex core is exactly zero, e(pz) = 0, see Fig. 3d and Fig. 5
Flat band: half-quantum vortex in p-space
Flat band in the vortex core
Dirac point
t
Ns B-l
(d)
Pz Dirac
point
Pl
n3 = +1
Px
O (L
n = 1
Fig. 3. Flat band formation, (a, top) - Two-dimensional flat band appears on the surface of gapless systems with a topologically protected nodal line [1, 2]. (a, bottom) - The nodal spiral in a semi-metal. This nodal line has a non-zero topological charge in momentum space, N2 = 1, and this charge protects the surface states with zero energy e(px,py) = 0, in the whole region within the projection of the spiral on the surface, (b) - One-dimensional topologically protected flat band
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