Topological Crystalline Insulator (TCI)

Unlike Z2 class of topological insulator where the states are protected by TR and SI symmetry. InTCI it is the crystal symmetry that protect it. Time-reversal symmetry protects strong topological insulators of the Z2 class, which possess an odd number of metallic surface states with dispersion of a Dirac cone. Topological crystalline insulators are merely protected by individual crystal symmetries and exist for an even number of Dirac cones. Here we mirror chern number.

SnTe is an example where the band inversion is protected by mirror symmetry but it is not the case for PbTe

In thus case te surface states Dirac cone neither is at corner of Brilouin Zone (Z2 TI) or in the symetry axis (Dirac semimetal) rather it is situated on mirror inversion point. inset of bottom of figure.

Unlike Graphene these Dirac point can"t annihilate. So what is the difference between TI and TCI ?

In TI there are odd no of Dirac cone situated on time reversal point in momentum space but for TCI there are even no of Dirac cone which are situated  any where except the time reversal point.

Now one interesting point is if we break the mirror inversion symmetry then a gap will appear in Dirac cone which will lead to appearance of mass. This symmetry can be broken by structural distortion upon lowering the temperature. There is a report here

http://science.sciencemag.org/content/341/6153/1496

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Prediction by a  toy model (2010)  https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.106.106802

https://www.nature.com/articles/ncomms1969

SnTe (2012): https://www.nature.com/articles/nphys2442

https://www.nature.com/articles/nmat3449

https://www.nature.com/articles/ncomms2191

Observation of ferroelectricity and proposal of ferroelectric tunneling random access memory.!! 

http://science.sciencemag.org/content/353/6296/274

A complete review by Liang Fu;

http://www.annualreviews.org/doi/10.1146/annurev-conmatphys-031214-014501

This paper demonstrate between Z2 to TCI: https://www.nature.com/articles/s41467-017-01204-0

Further Idea:

Hourglass fermions :https://www.nature.com/articles/nature17410

Non symorphic TI: https://journals.aps.org/prb/references/10.1103/PhysRevB.96.165143

One point to remember upto now we have not consider any interaction in TCI !!

Dirac Semimetal

Dirac semimetal are 3D versions of 2D graphene. In case of Graphene the Dirac point are only in the corner of hexagonal Briloune Zone which are time and space reversely protected. In Dirac semimetal, these Dirac points are now inside the Brilouine Zone along the symmetry axis.
                                                                         

Grphene Dirac cone

Dirac semimetal Dirac cone

This happen by considering the point group symmetry along with Time reversal and space inversion symmetry..PHYSICAL REVIEW B 85, 195320 a material prediction in Na3Bi in the year 2012.

In Na3Bi there is band crossing because of spin orbit interaction at K_D and there is other one also at -K_D. Now such crossing are protected against gap formation by point group symmetry as they they different representation they can"t never hybridize. 

Na3Bi has C3 point group symmetry (http://science.sciencemag.org/content/343/6173/864/tab-pdf)

Cd3As2 has C4 symmetry (https://www.nature.com/articles/nmat3990

                                             https://www.nature.com/articles/ncomms4786

                                             https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.113.027603

                                            

)

As already discussed if we broke either time reversal (by applying magnetic field or having a spontaneous magnetization ) or space inversion the we will get Wyel semimetal. As theoriticaly predicted in 2011 by Vishwanath

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.83.205101

Weyl found an important conclusion of Dirac equation:

1>Start with Durac equtaion and set the mass zero

2?Do not include any electromagnetic interaction

Then we have  a mass less Dirac Fermion with two different population of opposite chirality and they never mixed up. When there is mass term then this two chirality will mixed up.

In 2D this equivalent to parity. So let's talk about chirality in more details:

Weyl and Dirac Semimetal

What happen if we close the gal of a 3D topological insulator (TI)?

Then both bulk and surafce behave as a metal !!

Ok. there is something important about TIs is bulk bundary correspondence. Because of the bulk gap the 2D surface states with Dirac cone like dispersion are protected.  So in Z2 TIs the the time and space reversal symmetry  are conserved as roposed by Kane and Male (PRL 2005). Once we close the gap in bulk the these symmetry are broken !! Cane and male theory beak down and they can't be applied anymore. Now the question is , are the topological surface states still exist? 

Yes they do exist !! Now they are protected by crystal symmetry rather than the band topology.

So for trivial insulator the  highest occupied band and lowest empty band are separated by a gap. If SOC coupling is there then  there will be band inversion along with surface states, is a topological insulator. Now these are the two extreme cases. There is another case where, for a Ti the bulk band are just touched each other as similar in graphene. Then these staes is called dirac semimetal where the bulk as well as surface states are Dirac cone like.

Now in the previous case the bulk band touched in sigle point what if, these band crossed each other (Fig. B  above) and there will be two Dirac points along with the surface states , these are called Weyl semimetal. 

So if we measure the dispersion relation (Kz component: momentum perpendicular to surface) of a Dirac or weyl semimetal then both bulk and surface will show Dirac cone. NOw for topological insulator the Dirac cone was only for the suraface states not for the Bulk. So fora weyl semimetal  such measurement will show a single Dirac point at the surface but as soon as we go to the bulk this single point will disperse into two points.

So for topological insulator we have helical Dirac fermion ar the surface which is time and space reversely protected. Now by braking this symmetry we split the  the helical fermion (Fig d bottom) at the surface into  pair of  weyl (Fig.c bottom).

https://www.nature.com/articles/ncomms8373

Hence now the Fermi surface is fractionalised. Now we can"t do that, there should be some way out to complete the loop  with the other channel via bulk (fig.c top) and that's where the bulk come and make it semi-metal. 

  Hence for a Weyl semi-metal we will have a co propagating Fermi arcs of the two separated Weyl points. So if we considr dispersion relation of a weyl semimetal then we will have a copropagating fermi arcs at the surface and well points at the bulk. 

So if we measure bulk Fermi surface we will see a dot instead of a Fermi surface which is common fro insulators. So in ARPES experiment we will see a non closed co-propagating Fermi arcs/ FRACTIONAL FERMI SURFACE (at boundary , low photon energy) and they will terminate at bulk in weyl points (high photon energy measurement).

Experiment by Hasan group from Princeton:

http://science.sciencemag.org/content/349/6248/613

Ok now the idea of Weyl semimetal is established. Let's push ourself  a bit more: For 3D topological insulator we have 2D surface metallic and bulk insulator. Now  Weyl semimetal is a 3D metllic topologically where is the bulk insulated part? We can imagine from the analogy of 3D TIs that as if weyl semimetal is 3D metalic state of a 4D TIs. Ahh this is exciting we are going of higher dimension like string theory.    

 

Different types of Available topological insulator till date

Upto now there are there types of available Topological insulator :

1> The chern Insulator : 2D free electron gas under magnetic field. The integar and fractional quantum hall effct. The chen number is basically the TKNN invariant of occupied band.

2> Z2 Type: As predicted by Cane ad Mele the time reversal invariant topological insulator.

3> Topological crystalline insulator:  proposed by Liang Fu PRL 106, 106802 (2011) here the richness and beauty of the crystal symmetry drives the topological nature

Experiment: http://science.sciencemag.org/content/341/6153/1496

One of the common thing among this three class is that they all have non vanishing Berry Curvature !

Now there is this paper predicting a NEW TOPOLOGICAL class without Bery Curvature but Berry connection.

3D (Topological Insulator) version of 2D fractional quantum hall effect

The fractional quantum hall effect observed in 2D system is one of the remarkable result of topological nature of band. Now one much more awaiting phenomenon is to observation 3D version of 2D fractional Quantum hall effect. For integer quantum hall effect (observed in 2D electron gas under magnetic field) the time reversal protected chiral edge state are in 1D. Now a 3D topological insulator where we have chiral edge states in 2D (as similar Graphene, but in Graphene everything occur from band theory whereas in topological insulator it's the symmetry protected surface topological states ). In this analogy, 3D topological insulator can be thought as a dimensional extension of 2D integar quantum hall effect.

   Similarly, one would expect to a 3D (Topological Insulator) version of 2D fractional quantum hall effect !! Waooo that would much more interesting from because the excitation in that system will be just extraordinary !! As in 2D (http://www.tandfonline.com/doi/abs/10.1080/00018739500101566) fractional quantum hall effect electron electron correlation played a crucial role (which not the case for integer quantum hall effect) , on have consider similar interaction in 3D topological insulator.

There is a proposal of this here
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.085422

Realization of quantum anomalous Hall effect from a magnetic Weyl semimetal

The following article predicted observation of quantum anomalous hall effect in Wyel semimetal. So far, there are already many candidates for the magnetic WSMs from theoretical proposals, and some of them host strong AHE. Now quantum anomalous hall effect is observed in 2D insulating system having chiral edeg state whereas wyel semimetal is 3D topological system (sort of metalic).  They proposed that Co3Sn2S2 is a promising candidate by breaking translational symmetry along one direction of low carrier density Wyel semimetal.

https://arxiv.org/abs/1712.08115

3D topological insulator

If we stack quantum hall insulator on upon other then we will get current only in the edge but this does not give rise a edge current on to and bottom of this state. Now for 3D topological hall insulator what we want is edge state driven current through the 2D surface state of a 3D bulk. So what we want is following:

For 2D topological insulator, edge state formed a cross sign "X"by up and down spin, for 3D topological insulator we want them to form cone, the 3D vrsion of the "Cross X " . So i am talking is a Dirac cone.

Bi based materials:
Bi has strong spin orbit coupling. If we take Bi-n, Bi-As then because of Rashba coupling give a pair of Fermi surface with up down spin at the surface. These are pair of Rashba states by breaking the time reversal symmetry at surface. 
 Now for  Bi2Se3, Bi-Sb the situtaion is different, we have only one Fermi surface and in momentum space the  K+ ,K- has opposite spin. As shown in following figure z axis is energy and x,y axis are momentum. If we cut the circle formed by the top of cone, the spin momentum has always opposite sign.So the spin form a clockwise spin texture at the top of the cone. Now if go towards center point of the cone the this opposite nature of spin is maintained. It has half the degree of freedom  compare to Rashba system. So at he end the surface is gapless 

Now this type band dispersion is exist only in the surface not for the bulk. Because such dispersion of energy(E) moment(k_x, k_y) is studied by doing ARPES at the surface of Bi2Se3. In order to make sure that it exist only at the surface not in the bulk, a K_z dependent study is needed. Now a K_z study revealed that the surface edge states , lates call them now Dirac band are non dispersive as function of K_z where as the bulk band is dispersive.

So the surface state contain, K_+ has spin polarization has up and K_ - has spin down. So this surface state has half the degree of freedom than the Rashba system. Remember the whole system is nonmagnetic. Now the Dirac cone make sure that back scattering is not allowed. The surface electron has spin momentum lock. These are called helical Fermion.

Now what is the fundamental difference between the normal insulator and such surface states with metallic state and bulk being insulator.

In the above figure one of Selenium site has been substituted by lower Z value atom salphar to decrease spin orbit coupling. The extreme the bulk and surface is also insulator. So this is  spin orbit coupled Bloch band insulator. In the right side there is formation of surface state provided the bulk band has closed the gap at 0.6. So there is phase transition , only odd number of band inversion has happen in he bulk then only surface states appear.