Topological Insulators and Superconductors - A Curved Space Approach
The method of the space dependent basis is applied to study electronic spinors in a crystal. The crystal in the momentum space is described by the Brillouine zone which might contains obstructions or degeneracies for which requires different gauges for different regions. The electronic bands are classified according to their topology. The connection and curvature determines the physical properties which are clasified according to the topological invariants. We apply this method to the Topological Insulators, Topological Superconductors, Persistent Currents in coupled rings and photoemission for a curved crystal-face boundary.
Keywords:Topological Insulators, connection,curvature, curved space, Topological Invariants
One of the important ideas in Condensed Matter Physics is the concept of topological order Volkov (); Haldane (); Golterman (); Kreutz (); Thouless (); Berry (); Mele (); Kane (); Bellissard (); davidSpinorbit (); More (); ZhangField (); Hasan (); Chao (); Zhangnew (); Simon (); Prodan (); rings (); genus (). Insulators with a single Dirac cone which lies in a gap such as , and Volkov (); Hasan () represent the experimental realization of Topological Insulators (). At the surface of the three dimensional Topological Insulator (), one obtains a two dimensional metallic surface characterized by an odd number of chiral excitations, due to Kramers theorem, electrons are protected against backscattering balatsky () and localization Hai (); davidT (). When time reversal is broken localization effects are observed Ando (). The surface physics has been realized in quantum wells. The quantized spin-Hall effect has been proposed Haldane () and observed by Zhangnew (); Wu () and recently the Anomalous Hall effect has been measured Takahashi (). The spin resolved photoemission Hasan (); Nature (); FanZ () has been used to identify the surface states. Topological superconductors and their identification through the Majorana Fermions have been observed Alicea (). In order to study physical properties in the Brilouine Zone () we need to use the concept of parallel transport since the spinors might rotate in The Brillouin zone contains obstructions, degeneracies and therefore the gauge transformation between different different regions is needed. This behavior is studied with the help of the connection (the vector potential in the momentum space) Nakahara () which is similar to the parallel transport on curved surfaces. The derivative (external)Nakahara () of the connection defines the curvature strength which measures the obstructions in the Brillouin Zone. According to the symmetry involved (time reversal symmetry, parity inversion, mirror symmetry or charge conjugation) the eigenvectors satisfy certain constraint equations. The solutions of the constraint symmetry gives rise to specific gauge symmetry for the connection in the momentum space. This gauge symmetry is used to compute the electromagnetic response with the coefficients which characterizes topological invariants . The electromagnetic (magnetoelectric) response is characterized by the second Chern number.. The interplay of Topological Insulators () and Superconductivity gives rise to Majorana Fermions Alicea ().
The plan of this paper is as folows: In Sec.2. the method of paralel transport in the momentum space is introduced. Sec.3. is devoted to the studies of topological invariants which can be obtained using external fields to measur the responce of the system. In Sec.4.1 we consider the topological invariants for superconductors. In Sec.4.2 we derive the topological response which is obtained from sound waves. In Sec.5 we show derive the equation of motion in the momentum space. Sec.6. is devoted to the computation of topological invariants in two space dimensions using the mapping to four space dimensions. In Sec.7. we study the topology of two coupled rings in the presence of Majorana fermion and compute the persistent current. In Sec.8 we study the effects of the topology of curved surfaces on the Photoemission. In Sec.9 we present our conclusion.
2. -The method of parallel transport in momentum space
In this section I will develop the method of Topological Insulators based on the ideas of parallel transport in a curved space. The space is represented by the The spinors in the presence of the spin-orbit interaction davidSpinorbit () wary from point to point in . This demands the use of parallel transport which is defined in terms of the spinors spinors ( is the band-spin index). The parallel transport is defined in terms of the connection and the curvature for the coordinates. This allows to introduce the first and second chern character , which are given in terms of the covariant coordinate (or polarization) ( is the coordinate in the momentum space ) . This tools are essential for studies of Topological Insulators and Superconductors which are gaped phases of Fermionic systems which exhibit topological protected boundary modes to arbitrary deformation as long discrete symmetry such as time reversal, particle hole and chiral symmetry are respected. Due to the symmetries the Hamiltonian is invariant at the time invariant point which obey (time reversal with ) or charge conjugation (). The product of the charge conjugation (particle-hole) with the time reversal allows to define the unitary chiral symmetry which holds in the entire . As a result we have symmetry classes. For the case that the inversion symmetry and the time reversal symmetry hold, it has been proposed Kane (); Fu () that computing determine the topological invariant index , . The real challenge is to relate the index to the quantized electromagnetic response ZhangField (); Zhangnew () .
2.1. The space dependent basis in the B.Z.
The seminal work of Bellissard () on the Integer Quantum Hall opened the door to study disorder as a problem in a curved space, this work has been further developed in the Mathematical literature by Prodan () and his collaborators. In an early paper we have realized that due to the spin-orbit interaction davidSpinorbit () the spinors wary from point to point in . Using the formal language of space dependent basis we introduce the tangent vector. In the local frame we have a set of vectors which are related to the cartesian coordinates, ,
For translational invariant systems we have in the B.Z. the Bloch spinors which are a momentum dependent basis (when orbitals are included ).
In the presence of a non translational invariant potential we replace by, . Any Bloch Spinor can be represented in terms of the momentum dependent basis (or ),
In Quantum Mechanics we have the following matrix element for the momentum derivative . In real space- , is the coordinate operator. In momentum space we have , -
We borrow the following concepts from Differential Geometry Nakahara () is the exterior derivative which allow to introduce . is the spin connection and is the curvature operator.
The spin connection operator: with the matrix element:
The curvature operator : , with the matrix elements:
The derivative of the operator
The covariant derivative for the spinors
The physics of electrons in a periodic crystal is determined by the eigenvectors (spinors) ( is the band-spin index) behavior in the Brillouin Zone (torus in a dimensional momentum space). This behavior is similar to the parallel transport of a vector around a curve. We need to find the way the eigenvectors change under transport in the Brillouin Zone davidSpinorbit (); Blount (); Zak (). The topological properties are encoded into the connection (the vector potential in the momentum space) which measures the changes of when it is transported in the Brillouin Zone. The changes are given by:
(an index which appears twice implies a summation ). The matrix is given by where is the connection. Applying twice the (exterior) derivative we define the curvature ( see eqs. , Nakahara (2008) page 285 Nakahara ()) and find :
; ( the symbol represents the wedge product )
where is the matrix curvature with the matrix elements given in terms of the commutator of the covariant derivative , ; .
2.2 Observing the topology using external sources or disorder potential
The Hamiltonian can be express in terms of the eigenvalues. To obtain information about the topology, we have to transport the spinor around the B.Z. Alternatively we can include space dependent scalar and vector potentials , and probe the response.
The Hamiltonian with spin half and two orbitals in the presence of the external vector potential and scalar potential is given by: The four component spinors for the Hamiltonian :
; . and are four component spinors for particles and antiparticles.
The surface Due to T.R.S. invariance with and finite chemical potential the integrated Fermi Surface curvature is .
The situation is similar to spin-orbit scattering giving rise to This can be demonstrated using a diagrammatic or a Non linear sigma approach Due to the spin connections we find that the Cooperon changes sign! As a result the conductivity increases. The surface Hamiltonian is given by : . For a finite chemical potential we have the eigen spinors
The effect of the random potential:
As a result the multiple scattering matrix obeys:
This result is the reason for anti-localization which we have obtained in ref. davidT ().
3.Topological invariants from response theory
In order to demonstrate the emergent of the topological invariant we will consider a typical Hamiltonian for the materials , , . We introduce the tensor product ( stands for the and stands for the orbitals). A four band model is obtained Chao () which can be written in the chiral form. .
The first term affects only the eigenvalues and not the eigenvectors , . The matrices are given as a tensor product :, , ,. The mass (gap) obeys and has points in the Brillouin where it vanishes. (On a lattice with the lattice constant we define the Cartesian component of the momentum .) The Hamiltonian is diagonalized using the four eigenvectors , are the spin helicity operator and represents the particles-antiparticles energies, with the mass which vanishes at and :
The Green’s function operator in the basis is given by: . In the eigen vector basis the Green’s function takes the form:
The transformation from the basis to the eigenvector basis replaces the coordinate with the covariant coordinate davidSpinorbit (). In the second quantized form the spinor operator is given by: , , . (It is important to stress that this representation is valid for momentum , for the region we need to choose a different representation.)The coupling of the to the electromagnetic field and is given by the action :
We compute the partition function integrating over the Grassman fields for four space dimensions. We find that the effective action for the electromagnetic fields obtained by Golterman () is given by,
Using the totally antisymmetric tensor for four space dimensions we find the electromagnetic response , polarization energy (given in terms of the electric and magnetic field ) :
with the quantized coefficient which dose not break the time reversal symmetry Golterman (). The response is given for space time dimensions. Golterman (); Weinberg (); Nakahara (); davidtop (); ZhangField (); Zhangnew (),
In the presence of additional interactions the Green’s function might have zeroe’s Gourarie (); Zhong (), for such a case the system seizes to be topological. If the renormalized Green’s function has no zeroe’s the topology is preserved. The renormalized Green’s function is given in terms of the wave function renormalization , where is the wave function renormalization. When the wave function renormalization is finite at we take the limit and obtain the second Chern character only for four space dimensions Zhong (). cancels and we find:
The trace operator acts only on the occupied bands . The Green’s function in the eigen vector basis representation replaces the calculation with a multiplicative covariant matrix coordinates . The Chern character is given by a matrix multiplication. The commutator gives the curvature in terms of the connection . The second Chern number is given by
which is either zero (exact form) or non-zero (non exact form). Therefore
dwhere is the Chern-Simons three form Nakahara () . can be found with the help of the gauge symmetry imposed by Nakahara (). The second Chern number in four dimensional space is given by: which has a winding number. is but not Nakahara (). This means that in some restricted regions of the Brillouin zone the integral is given by a Chern-Simons contour integral Nakahara ():
This result does not hold in the entire . We can identify two regions which are related by a transition function (a gauge transformation), a transformation matrix between states for the region which belongs to half of the positive sphere and for which belongs to second half, the negative shere . The matrix which transforms between the two regions is the Pfaffian matrix defined in terms of the Kramers pair. We use the eigenvectors to compute the matrix , the relation between the connections in the different regions and :
and the curvature transform like : . Applying Stokes theorem for the two different regions we obtain a boundary integral over the difference of the Chern-Simons terms defined for each region .The difference between the two Chern-Simons terms can be understood as a polarization difference between the two regions. The boundary is a surface perpendicular to the fourth direction (physically we can introduce the concept of polarization which is the difference of the electric flux on the two boundary surfaces perpendicular to the direction.)
We have ( are points in the Brillouin Zone where the Pfaffian matrix vanishes) . Due to the lattice periodicity Bloch theory allows us to define the polarization as modulo an integer. We recover the result for and for Kane (). represents the topological invariant for space obtained from the response theory.
This method is also be applicable to Topological Crystals where a mirror reflection invariant with the property replaces the invariant.
Inspired by Fu () the authors in ref. Bansil () proposed that has a mirror plane perpendicular to the direction. A band inversion at the four points in the Brillouin zone between and can be achieved for the mixed crystal . The model near an point Bansil () is given by:
where is the inverted mass (has zeros in the Brillouin zone), corresponds to the states with total angular momentum , and corresponds to the orbitals of the cation (Sn or Pb) and anion .
The mirror invariant with the property can be found such that the condition for the polarization is different from the one given by the time reversal invariant points.
The reflection symmetry from the plane perpendicular to is given by the transformation . The operator of reflection which acts on the states is given by a rotation of an angle around the axes and is accompanied by an inversion trough the origin. A simple calculation shows that the mirror operator is given by . As a result the state is transformed to and the Hamiltonian obeys the symmetry:
Next we include the anti unitary conjugation operator K and define the operator which obeys (this is similar to the time reversal operator ). We obtain the invariance transformation:
At the invariance points one obtains the conditions: