научная статья по теме DIMENSIONAL CROSSOVER IN TOPOLOGICAL MATTER: EVOLUTION OF THE MULTIPLE DIRAC POINT IN THE LAYERED SYSTEM TO THE FLAT BAND ON THE SURFACE Физика

Текст научной статьи на тему «DIMENSIONAL CROSSOVER IN TOPOLOGICAL MATTER: EVOLUTION OF THE MULTIPLE DIRAC POINT IN THE LAYERED SYSTEM TO THE FLAT BAND ON THE SURFACE»

Pis'ma v ZhETF, vol.93, iss.2, pp.63-68

© 2011 January 25

Dimensional crossover in topological matter: Evolution of the multiple Dirac point in the layered system to the flat band on the surface

T. T. Heikkila*1^, G. E. Volovik *+ *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 consider the dimensional crossover in the topological matter, which involves the transformation of different types of topologically protected zeroes in the fermionic spectrum. In the considered case, the multiple Dirac (Fermi) point in quasi 2-dimensional system evolves into the flat band on the surface of the 3-dimensional system when the number of atomic layers increases. This is accompanied by formation of the spiral nodal lines in the bulk. We also discuss the topological quantum phase transition at which the surface flat band shrinks and changes its chirality, while the nodal spiral changes its helicity.

1. Introduction. Topological matter is characterized by nontrivial topology of the Green's function in momentum space [1, 2]. The topological objects in momentum space (zeroes in the spectrum of fermionic quasipar-ticles) in many respects are similar to the topological defects in real space, and are also described by different ho-motopy groups including the relative homotopy groups. In particular, the Fermi surface is the momentum-space analog of the vortex loop in superfluids/superconduc-tors; the Fermi point (or Dirac point) corresponds to the real-space point defects, such as hedgehog (monopole) in ferromagnets; the fully gapped topological matter is characterized by skyrmions in momentum space, which are analogs of non-singular objects - textures; etc.

Here we discuss the transformations of the topologically protected zeroes, which occur during the dimensional crossover from a 2-dimensional to a 3-dimensional system. We consider the dimensional crossover which involves such topological objects as a nodal line in a 3D system; the flat band, which is an analog of a domain wall terminating into a half-quantum vortex; and the topologically protected Dirac points with multiple topological charge \N\ > 1 in quasi 2-dimensional substance, which are analogous to the multiply quantized vortex.

The Fermi bands, where the energy vanishes in a finite region of the momentum space, and thus zeroes in the fermionic spectrum have co-dimension 0, have been discussed in different systems. The flat band appears in the so-called fermionic condensate [3 - 6]. Topologically protected flat band exists in the spectrum of fermion zero modes localized in the core of some vortices [7-9]. In particle physics, the Fermi band (called the Fermi ball) appears in a 2+1 dimensional nonrelativistic quan-

^e-mail: tero.heikkila0tkk.fi, volovik0boojum.hut.fi

turn field theory which is dual to a gravitational theory in the anti-de Sitter background with a charged black hole [10]. The flat band has also been discussed on the surface of the multi-layered graphene [11] and on the surface of superconductors without inversion symmetry [12].

The topologically protected 2-dimensional and 3-dimensional Dirac points with multiple topological charge N were considered both in condensed matter [1,5,13-20] and for relativistic quantum vacua [21, 1, 22, 5]. In the vicinity of the multiple Dirac point with topological charge N the spectrum may have the form E2 oc p2N. We consider the special model of the multi-layered system discussed in [20], where the topological charge N of the Dirac point coincides with the number of layers. In this model, when N oo, the multiple Dirac point transforms to the flat band in the finite region of the two-dimensional momentum on the surface of the sample. The interior layers in the limit N oo transform to the bulk state, which represents a semi-metal in which the nodal line (line of zeroes) forms a spiral. The projection of this spiral onto the edge layer produces the boundary of the flat band. The latter is similar to what occurs in superconductors without inversion symmetry, where the region of the flat band on the surface is also determined by the projection of the topological nodal line in the bulk on the corresponding surface [12].

2. Flat band and spiral nodal line. Let us first consider the model in Ref. [20], specified in Sec.3 below, in the limit N oo. The effective Hamiltonian in the 3-dimensional bulk system which emerges in the continuous limit N oo is the 2x2 matrix

H= (J. ty,f = P*-iPv~ t+^iaP* ~ f1)

Here t+ = |i+|e!^+ and = |i_|e!<^ are the hopping matrix elements between the layers and a is the interlayer distance. The hopping matrix element proportional to is i+(_). The energy spectrum of the bulk system

E2 = \px - |i+| cos(ap; - (/>+) - \t-\cos(apz + \py + |i+| sin( ap~ - (l>+) - \t-\sm(apz - 4>

has zeroes on the line (see Fig.l):

(2)

-1.0 -2

Fig.l. Fermi line for the case t = \t+\ = 10|i_| (with circles) and t = \t- \ = 10|i+| along with their projection to the pz = 0 plane (dashed). This projection represents the boundary of the dispersionless flat band on the surface. In both cases <p+ = <j>- = 0. Note that the helicity of the two lines is opposite, which gives the opposite signs of the invariant iVi(pj_) = ±1 in (5), and the opposite chiralities of the flat band

px = |i+| cos(apz - <j>+) + |i_| cos(apz -py = |i_| sin(ap- - (/>-) - |i+| sin( ap~ - <j>+).

(3)

These zeroes are topologically protected by the topological invariant [5]

iVi =

ini

T tr <p dl <jzH ViH,

(4)

where the integral is along the loop around the nodal line in momentum space. The winding number around the element of the nodal line is N\ = 1. For the interacting system the Hamiltonian matrix must be substituted by the inverse Green's function at zero frequency, H G^1(w = 0,p), which plays the role of effective Hamiltonian, see also [23].

The same invariant can be written if the contour of integration is chosen parallel top-, i.e. at fixed p i . Due to periodic boundary conditions, the points pz = ±7r/a

are equivalent and the contour of integrations forms the closed loop.

Nlip±) = -—tr

/+ix/a

1

-7T/a

dpz a.H-'V^H. (5)

For interacting systems, this invariant can be represented in terms of the Green's function expressed via the 3D vector g(p-,w) [5]:

G 1(u),pz) = igz(w,pz) - gx(w,pz)ax + gy(w,pz)ay

In our model the components g(p-,w) are:

9x(Pz,u)=Px~\t+\cos(apz - (/>+) - \t- \ cos(ap-9y(Pz,^)=Py+\t+\s'm(aPz - </>+) - |i_|sin(ap- -9z{Vz,w) = u-

Then the invariant (5) becomes [5]

(6)

(7)

iVi

(8)

where g = g/|g|. It describes the topological properties of the fully gapped ID system, with px and py being the parameters of the system.

Let us first consider the case i_ = 0. For i_ = 0, the nodal line in (3) forms a spiral, the projection of this spiral on the plane p- = const being the circle

= pi + Py = \t+\2 (the spiral survives for i_ ^ 0, but circle transforms to an ellipse, see Fig.l for the case |i_| < |i+|). The topological charge in (5) isiVi(pj_) = 1 for momenta |pj_| < |i+|. If the momentum pj_ is considered as a parameter of the ID system, then for |p_i_| < |i+| the system represents the ID topological insulators. For |pj_| > |i+| one has iVi(pj_) = 0 and thus the non-topological ID insulator. The line |pj_| = |i+| marks the topological quantum phase transition between the topological and non-topological ID insulators.

Topological invariant iVi(pj_) in (5) determines also the property of the surface bound states of the ID system: the topological insulator must have the surface states with exactly zero energy. These states exist for any parameter within the circle |pj_| = |i+|. This means that there is a flat band of states with exactly zero energy, £J(|pj_| < \t+\) = 0, which is protected by topology. The bound states on the surface of the system can be obtained directly from the Hamiltonian:

H = ax(px - |i+| cos(ap-)) + Vy(py + |t+| sin(ap-)), pz = -idz

z < 0.

(9)

IlHCbMa B ?K3T<1> TOM 93 Bbin.1-2 2011

We assumed that the system occupies the half-space z < 0 with the boundary at z = 0, and made rotation in (px,py) plane to remove the phase <f>+ of the hopping element t+. This Hamiltonian has the bound state with exactly zero energy, E(p±) = 0, for any |pj_| < |i+|, with the eigenfunction concentrated near the surface:

Ip-lI < IM-

The normalizable wave functions with zero energy exist only for pj_ within the circle |pj_| < |i+|, i.e. the surface flat band is bounded by the projection of the nodal spiral onto the surface. Such correspondence between the flat band on the surface and lines of zeroes in the bulk has been also found in Ref. [12].

Restoring the non-zero hopping element i_, we find that for |i_| < |i+| there is still the region of the momentum pj_ for which the topological charge in (5) is Ni(p±) = 1. However, the area of the projection of the nodal line on the surface (and thus the area of the flat band) is reduced. Finally at |i_| = |i+|, the nodal line becomes flat, its projection on the surface shrinks to the line segment py/px = tan(^>+ — <f>-)/2, and the flat band disappears (Fig.2). For |i_| > |i+|, the spiral

pydt

Fig.2. The Fermi line (nodal line) in the bulk at the topological quantum phase transition, which occurs at t = |i+| = |i_|, for two values of the phase shift <j>- — <j>+. In this case the projection of the nodal line to the pz = 0 plane shrinks to the line segment with zero area because the Fermi line is flat: it is within the corresponding plane drawn on the figure. As a result the dispersionless flat band on the surface is absent at the transition, and has opposite chiralities on two sides of the transition

the topological quantum phase transition, at which the flat band changes its orientation (or actually its chiral-ity). At the transition line |i_| = |i+| the flat band on the surface does not exist. The non-zero

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