## Quantum Hall effect, graphene, topological insulators

### Kaleidoscope of topological phases with multiple Majorana species

A slice of the phase diagram of the square-octagon variant of the Kitaev honeycomb model. The dark lines indicate phase transitions where the higher negative energy band and the lower positive energy band touch. Inside each phase the Chern numbers of the two negative bands are also indicated. The top number indicates the higher negative energy band, the one lying closest to the zero of energy. The dotted lines indicate where these negative energy bands touch. In this case the internal transfer of Chern integers does not change the total Chern invariant. All phases with total Chern number 3 and 4 have topological orders for which we do not know of previous lattice realizations.

The Kitaev honeycomb model is an exactly solvable spin model that has
attracted a lot of attention because of its topological properties. In
particular, there can be a non-abelian phase that maps to the spinless
p-wave BCS state and hosts as such the much discussed Majorana
fermions. We are working on finding a variant of the Kitaev model that
would map to spinful p-wave paired fermions, among other with the rich
topological phases of superfluid He-3 in mind. In this endeavor we
have looked at the square-octagon variant of the Kitaev model, which
because of having twice as many sites per unit cell maps naturally to
a spinful Bogoliubov-de Gennes hamiltonian (however, with a mixture of
both spin-singlet and spin-triplet channels). The phase diagram is
indeed startingly rich. The phases can be characterized by a
topological invariant called the Chern number. Phases with odd Chern
number can host topologically protected Majorana fermions. In systems
with Majorana fermions one has so far for phases with odd Chern
number only unveiled integers *±1*. We give the first example of
Majorana fermions related to a higher integers, namely *±3*, which can
be shown to be a different species than those related to *±1*.

References

“Kaleidoscope of topological phases with multiple Majorana
species”, G. Kell, J. Kailasvuori, J. Slingerland and J. Vala
[arXiv:1011.4323].

### One-dimensional view on the quantum Hall system

Relation between the circumference of torus

*L*and the distance of guiding center coordinates,

_{1}*2 \pi/L*. As

_{1}*L*is decreased, the mutual support of neighboring Gaussian-weighted wave functions becomes small, one-dimensional solvable models can be constructed. Also the limit of thick and short cylinder (not shown), where the spatial positions of all single particle states coincide, is amenable to a detailed analysis.

_{1}Recent work has established that there is a simple one-dimensional limit of the (fractional) quantum Hall system that is exactly solvable, but still exhibits the rich structure of, and is adiabatically connected to the bulk 2D system. In this limit, the fractionally charged quasiparticles appear as domain walls and reveals close connections both to the concept of charge fractionalization in one-dimension and to the mathematical structures of the conformal field theory description of the quantum Hall effect. Recent work at MPI-PKS work includes studies of the physics beyond the solvable limit by means of studying possible instabilities, and by exploring its connection to spin chains and integrable models.

References

[1] E. J. Bergholtz, A. Karlhede, Phys. Rev. B **77**, 155308 (2008); J. Stat. Mech. (2009) P04015.

[2] M. Nakamura, E. J. Bergholtz, J. Suorsa, Phys. Rev. B **81**, 165102 (2010).

[3] E. Wikberg, E. J. Bergholtz, A. Karlhede, J. Stat. Mech. (2009) P07038.

[4] E. J. Bergholtz, A. Karlhede, Phys. Rev. Lett. **94**, 026802
(2005); J. Stat. Mech. (2006) L04001; Phys. Rev. B **77**, 155308
(2008).

[5] E. J. Bergholtz, J. Kailasvuori, E. Wikberg, T. H. Hansson, A. Karlhede, Phys. Rev. B **74**, 081308(R) (2006).

[6] E. J. Bergholtz, M. Hermanns, T. H. Hansson, A. Karlhede, Phys. Rev. Lett. **99**, 256803 (2007).

### Entanglement Skyrmions in multi-component quantum Hall systems

Electrons moving in two dimensions, subject to a strong perpendicular magnetic field, exhibit a wide variety of unusual many-body phenomena. The most celebrated of these are the quantum Hall effects, after which the field of their study is named. These effects appear when the number of flux quanta is commensurate with the number of electrons in the system. Near commensurability, the elementary excitations can be non-trivial charged spin textures known as Skyrmions. We have studied the properties of Skyrmions in systems with several internal degrees of freedom (such as graphene or bilayer systems). We have found that, even in the presence of anisotropies, there are families of degenerate Skyrmions which differ in the degree to which the different internal degrees of freedom are entangled. We discovered that there exists a unique ‘most entangled’ topological excitation, which we call the ‘entanglement Skyrmion’.

References

B. Doucot, M. O. Goerbig, P. Lederer, and R. Moessner, Entanglement Skyrmions
in multicomponent systems, Phys. Rev. B **78**, 195327 (2008).

### Fundamental aspects and applications of spin-coherent Boltzmann equations in graphene and spintronics

A sketch showing a simplified picture of the conductivity in monolayer graphene with screened charged impurities. Far away from the Dirac point the conductivity follows Drude result with a linear behavior as a function of carrier density. However, because of electron-hole coherent quantum effects there can be a small shift to the Drude result.

Graphene is in its simplicity a fascinating system both for applications and for addressing fundmental issues. In our case graphene prompted a re-examination of several common approaches for deriving Boltzmann type semiclassical kinetic equations from nonequilibrium quantum many-body theory. Such semiclassical equations provides also in quantum systems a powerful and intuitively appealing tool for studying transport. The massless Dirac electrons in graphene offer in the most distilled form the feature - the simple coupling between momentum and a spin-type degree of freedom (in this case the pseudospin) - that we needed for unveiling a hitherto unknown inconsistency of various existing approaches [1]. Although they agree in ordinary applications, they are in general not equivalent when some degree of freedom — typically a spin or band index — is not treated semiclassically. Furthermore, we have shown how one solves analytically the resulting spin-coherent Boltzmann equations for Dirac electrons in randomized impurities. The thereby derived quantum corrections to the Drude conductivity offer one possible contribution to an observed and so far unexplained shift in the conductivity of graphene. We have also unveiled a special case where these rather difficult equations become relatively simple, namely in bilayer graphene with point-like impurities. There we can take the electron-hole coherent quantum corrections into account to all order, which results in a conductivity minimum [2]. We have also worked on theoretical aspects of spin-coherent Boltzmann equations related to semiconductor systems with a small Rashba spin-orbit coupling [3]. On the application side we have derived spin-coherent two-body collision integrals and addressed the problem of interplay between electron-electron interaction and different types of spin-orbit coupling [4]. In particular we can explain the finite life-time observed in experiments on the Persistent Spin Helix. We have also within a band-coherent phenomenlogical Boltzmann approach addressed thermoelectric properties like the Nernst effect in strongly correlated systems to show that electronic correlations can lead to an increase of the Nernst effect beyond the semiclassical Sondheimer result [5].

References

[1] “Quantum corrections in the Boltzmann conductivity of graphene
and their sensitivity to the choice of formalism”,
J. Kailasvuori, M. Lüffe, J. Stat. Mech. P06024 (2010).

[2] “Finite Conductivity Minimum in Bilayer Graphene without Charge
Inhomogeneities”, M. Trushin, J. Kailasvuori, J. Schliemann,
A. H. MacDonald, Phys. Rev. B **82**, 155308 (2010).

[3] “Boltzmann approach to the spin Hall effect revisited and
electric field modified collision integrals”, J. Kailasvuori,
J. Stat. Mech. P08004 (2009).

[4] “Relaxation mechanisms of the persistent spin helix”,
J. Kailasvuori, M. C. Lüffe, T. Nunner [arXiv:1103.0773].

[5] “A Nernst effect beyond the Sondheimer formula”, J. Kailasvuori [arXiv:1011.4323].

### Topological zero-modes and supersymmetry in multilayer graphene with a random vector potential

An imagined typical spectrum for electrons in graphene with a gap and a random magnetic field due to ripples. (The horizontal distribution carries no meaning.) The two valleys

*K*and

*K'*are time-reversed copies in absence of a real magnetic field. For ripples without any spatial symmetries the spectrum of each valley will in general be non-degenerate and of random spacing, except for the threshold modes at

*|E| = m*. Their degeneracy depends only on the total effective flux

*Φ = 4.1 φ*. The rest of the spectrum is symmetric around

_{0}*E = 0*due to supersymmetry.

An index theorem tells that 2*d* Dirac electrons as found in single-layer
graphene will have exact zero-energy states in a random magnetic
field. The number of such modes is given by the number of flux quanta
penetrating the system. We have shown how this argument can be
generalized to multilayer graphene with the vector potential given by
an external magnetic field or by an internal effective field due to
ripples. We have also shown that just like for the case of Dirac
electrons one can discuss properties of the multilayer graphene
hamiltonians in terms of supersymmetric quantum mechanics concepts. In
particular, even when the vector potential is fully random and chiral
symmetry is absent because of the presence of a mass term, the
spectrum will still be fully symmetric, apart from the modes that were
zero-modes in the massless case [1].

References

[1] “Pedestrian index theorem a la Aharonov-Casher for bulk
threshold modes in corrugated multilayer graphene”,
J. Kailasvuori, Europhys. Lett. **87**, 47008 (2009).

### Dirac fermions in solids

Graphene, a single sheet of carbon atoms in a honeycomb lattice, has attracted enormous interest recently. Its quasiparticles obey a two-dimensional Dirac equation, whose speed of light is replaced by the Fermi velocity (being 1/300th the speed of light). This open up the possibility of observing relativistic effects in a condensed matter experiment. We have analyzed the properties of disordered graphene, with special attention to the density of states. We have also shown that graphene can be used to simulate the basic model of quantum optics, the Jaynes Cummings Hamiltonian and that Rabi oscillations are observable in its optical response.

References

[1] B. Dora, K. Ziegler, P. Thalmeier, Phys. Rev. B **77**, 115422 (2008).

[2] B. Dora, K. Ziegler, T. Thalmeier, M. Nakamura, Phys. Rev. Lett. **102**, 036803 (2009).

### Transport phenomena in multilayer and bilayer graphenes

Recently, graphene, a single layer graphite was extracted and exotic physical properties are observed such as universal minimum conductivity and anomalous integer quantum Hall effect. These phenomena are essentially explained by the Dirac-type equation since the dispersion relation of the honeycomb lattice is linear in the vicinity of the Fermi energy. We have studied multilayer systems which consist of few layer graphenes based on the matrix decomposition method which reduces the Hamiltonian into those of bilayer and monolayer systems. Then we found that conductivity and Hall conductivity behave differently depending on the stacking structures and number of layers.

References

M. Nakamura and L. Hirasawa, Phys. Rev. B **77**, 045429 (2008).