May 30, 2008

Fig. 1: DC Hall conductivity of the topological
insulator as a function of the Hamiltonian parameter
M. |

In this page we discuss a prototype model for a time-reversal symmetry breaking topological insulator. We will show that the model exhibits a quantum phase transition between a topologically trivial state and a non-trivial state characterized by a finite quantized Hall conductivity. After writing down the model we will discuss the generic insulating bulk bandstructure and the existence of chiral edge states in a strip geometry. Then we will show the existence of gap-closing phase transitions across which the Hall conductance jumps from zero to an integer value.

The prototype model considered here has two orbitals in (2+1)-d on a square lattice with nearest-neighbor hoppings. We can expand any two orbital model in the basis of Pauli matrices and the Hamiltonian is

where *m,n* are the lattice indices on the
*x,y* directions respectively, and *M* is a parameter. With
periodic boundary conditions we can diagonalize this model by Fourier
transformation to get the bulk bandstructure

where *a* is the lattice constant which will be
set to 1 from now on. To gain intuition for this model one can expand it
around *k _{x}=k_{y}=0* and

Fig. 2: Strip geometry i.e. a cylinder. Blue
and red circulating lines indicate chiral edge states. |

which is the dispersion for a massive Dirac fermion
in (2+1)-d. Thus at low energies our lattice model looks Dirac-like. At
high energies the dispersion is no longer linear in *k* and the
effects of the lattice regularization enter.

One can see that for values of *M=0,2,4* the
spectrum no longer has a gap. For example at *M=0* we find
E_{+}(k_{x}=0,k_{y}=0)=E_{-}(k_{x}=0,k_{y}=0)
so the gap closes at (k_{x},k_{y})=(0,0). At *M=2*
the gap closes at two places
(k_{x},k_{y})=(0,π),(π,0) and for *M=4* the
gap closes at (k_{x},k_{y})=(π ,π). In between
these critical points the spectrum is insulating. One can pose an
obvious question, is there any way to distinguish the insulating phases?
The answer is yes and we will illustrate two common ways to do this
which are intimately related.

One can calculate the off-diagonal response of this
model to an electric-field. The transport in this case is the Hall
conductivity σ_{xy}. For this prototype model the
calculation of the Hall conductivity is straightforward, but I will not
reproduce it here. See the Refs. [1,2] for two different accounts. If we
parameterize our Fourier transformed Hamiltonian via

then the DC Hall conductivity is given by

where *i,j=x,y* and *a,b,c=1,2,3,*
ε^{ij},ε^{abc} are the 2 and 3 index
totally-antisymmetric tensors respectively, and the integral is over the
first Brillouin zone. The value of σ_{xy} depends on the
parameter *M* and we plot its value vs. *M* in Fig. 1.

One can see that exactly at *M=0,2,4,*
σ_{xy} jumps. At the
first and last transition there is a jump of magnitude 1 while
at the middle transition the jump is by 2. This difference is due
to the fact that the amount of the jump is proportional to the
number of gap closings. Thus the middle transition has two points
where the gap closes and the Hall conductivity jumps by 2. This
transport coefficient already gives us a physical way to
distinguish the phases but we would like to illustrate a
complementary view point.

We want to study the Hamiltonian in a strip geometry
shown in Fig. 2, with periodic boundary conditions along the
x-direction, but open boundary conditions on the y-direction. This means
that we terminate the lattice at y=0 and y=L so that it is finite in
this direction. We can no longer diagonalize the Hamiltonian by Fourier
transformation because the y-direction is no longer homogeneous. There
is an easy way to solve this problem however. We simply Fourier
transform only on the x-direction and numerically diagonalize the
simple 1d model that is left over for each value of k_{x}. The
partially Fourier transformed Hamiltonian is

We can solve the 1d tightbinding model
for each value of k_{x} and plot the spectrum vs. k_{x} to get
the bandstructure. We have done this in Fig. 3 for several values of
M.

Fig. 4: Square of the right-moving (red) and
left-moving (blue) surface state wavefunctions for the case
of M=5. |

In Fig. 4 we plot the squared wavefunctions of
left-moving and right-moving surface states for the case of
*M=0.5*. Note that they lie on different edges. This gives the
system a chirality and indicates that time-reversal symmetry has been
broken. In fact, when the system is not gapless, time-reversal has been
broken. Typically an energy crossing like the edge state crossing when
*M=0.5* is not generic, and the states will backscatter to open a
gap. However, in this case, one state lies on one edge and the other on
the opposite edge. This means that for any backscattering is supressed
by the width of the sample. To open a gap the electron would have to hop from one edge to the other. Thus for edges which are far apart electrons
travelling along the edge cannot be backscattered which leads to some
interesting transport effects.

The source code for all of the calculations done in this work can be found here. We used the GNU scientific library for the matrix diagonalization routine and complex numbers. This library is standard in all GNU Linux distributions and can be used on any platform including Windows via the Cygwin emulator.

© 2008 T. L. Hughes. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] D. J. Thouless, M. Kohmoto, M.P. Nightingale, and
M. den Nijs, Phys. Rev. Lett., **49**,405(1982).

[2] X.-L. Qi, Y.-S. Wu, S.-C. Zhang, Phys. Rev. B
**74**,085308(2006).