4D Quantum hall physics

Theorry:

A Four-Dimensional Generalization of the Quantum Hall Effect (2001)

Four-Dimensional Quantum Hall Effect with Ultracold Atoms  (2015)

Experiment:

Exploring 4D quantum Hall physics with a 2D topological charge pump  (2017)

Photonic topological boundary pumping as a probe of 4D quantum Hall physics

Magnon

Bose Einsrein condensate of Magnon

Electronic Tuning of Spin Waves

Topological Quantum Effects for Neutral Particles

Snell’s Law for Spin Waves

Observation of the Magnon Hall Effect

Theoretical Prediction of a Rotating Magnon Wave Packet in Ferromagnets

Hall effect in Phonon and Photon

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.100.145902

Phenomenological Evidence for the Phonon Hall Effect

Hall Effect of Light

Observation of magnetically induced transverse diffusion of light

Type II Dirac Fermion

The standard model of particle physics describes all the known elementary particles, like electrons and quarks. Many of these particles have analogs in condensed matter, where they arise as collective states, or quasiparticles. One example is an electronic state in graphene that behaves like a massless Dirac fermion—a spin-1/2 particle that is not its own antiparticle. But condensed-matter physics may offer a longer list of “elementary particles” than found in the standard model. This is due to the fact that—unlike fundamental particles—quasiparticles in solids are not constrained by so-called Lorentz invariance. A Lorentz-violating quasiparticle is one whose momentum-energy relation depends on the direction it travels. Three separate teams  have collected the first experimental evidence of quasiparticles called type-II Dirac fermions, which break Lorentz invariance. These electronic states, which have no counterpart in the standard model, could be associated with a new type of superconductivity, which has potential applications in thermoelectric devices and quantum computing.

For type-I Dirac [and Weyl semimetals  which obey Lorentz invariance, massless Dirac fermions with linear dispersions are expected at the Dirac or Weyl points. Type-II Dirac and Weyl fermions  emerge at the topologically protected touching points of electron and hole pockets, and they show highly tilted Dirac cones along certain momentum direction (see schematics bottom figure), thereby breaking the Lorentz invariance . The anisotropic electronic structure can also lead to anisotropic MR, and negative MR is expected only along directions where the cones are not tilted enough to break the Lorentz invariance . Type-II Dirac semimetal can be tuned to a Weyl semimetal or topological crystalline insulator when the crystal symmetry or time reversal symmetry is broken , and therefore they are ideal candidates for investigating topological phase transitions and potential device applications.

Materials: PtTe2, PtSe2,,PdTe2,


https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.016401

https://arxiv.org/abs/1607.03643


https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.041201

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.95.155112

website:
http://nccr-marvel.ch/fr/news/highlights/2017-07-typeII-dirac-fermions-now-experimetally-verified

Topological Magnon Insulator

The dispersion relations of magnons in ferromagnetic pyrochlores with Dzyaloshinskii-Moriya interaction are shown to possess all types of topologically protected states:.

           The breakthrough experiment that lead to such state is observation of Magnon hall effect .

Edge states in topological magnon insulators

Weyl magnons in breathing pyrochlore antiferromagnets

Topological magnon insulator in insulating ferromagnet

Magnon nodal-line semimetals and drumhead surface states in anisotropic pyrochlore ferromagnets

Weyl and Nodal Ring Magnons in Three-Dimensional Honeycomb Lattices 

Topological Superconductor

Particle hole symmetry is one of the remarkable conclusion of  BCS theory.Superconductor similar to insulator in the the sense  they have energy gap  between the quasiparticle excitation. Now for superconductor there is intrinsic symmetry that valence and conduction are exactly mirror image  of each other. But for insulator those are different. Adding an electron in conduction band is equivalent to electron from valence bond. Now for insulator these two case are independent as the valence and conduction band are independent of each other. In superconductor these states are actually same states !! In case of superconductor,  For superconductor creating an particle at +E and destroying a particle at -E are in same state. This known as particle hole symmetry. 

Now a question is , Does a Hamiltonian (antiunitary) that satisfy particle hole symmetry have a topological class. If yes then that would be a topological superconductor !! 

 In 1D there is possibility of Z2 topological superconductor. These Topological superconductor end host Majorana fermions at the edge with zero mode.

In 2D, we will have chiral majorana edge states which are protected as integer quantum hall state. But here because of particle hole symmetry the chiral state with positive and negative are redundant

here time reversal symmetry is broken and it ins Z(n) topological insulator as any integer number (n) of chiral state are possible not Z2  as in case of 1D .It is possible to observe majorana zero mode at the vortex of 2D topological superconductor.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.100.096407

https://www.nature.com/articles/nphys3036

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.103.107002

https://www.nature.com/articles/nmat3255

 possible topological superconductivity in ${\mathrm{Sn}}_{1-x}{\mathrm{In}}_x\mathrm{Te}$

${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$

$$ {\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$$

Now what about 3D. from the following periodic table

By Kiataev and in PRB 

Classification of topological insulators and superconductors in three spatial dimensions

Classification of topological quantum matter with symmetries

The DIII could host a 3D topological superconductor.The quasiparticles on the surface of a 3D topological insulator are massless Dirac fermions, familiar from graphene. The superconducting counterpart has massless Majorana fermions on its surface

The Majorana Fermions

Story start with discovery of antiparticle (Positron) by Dirac by an attempt to unified  Theory of relativity and Quantum mechanics. 1932 positron was discovered in 192 in cosmic ray. Dirac equation describe charged spin 1/2 Fermions.
In 1937 Majorana predicted one new type of perticle by smartly choosing  gamma matrices of Diraction so that the wave function of the particle is real and then this particle will be self conjugate. But don't forgate it is still a spin 1/2 Fermion !! Waoo.. that.s amazing a charge neutral, spin1/2, self conjugate (a particle with it's own anti particle) Fermion.

In condensed matter there is findings of Majorana Bound state on the boundary of a 1D topological supercoductor  (0+1) that are localize not propagating

Edge of a 1D superconductor :

https://arxiv.org/abs/cond-mat/0010440

If in a 1D wire,in the middle there is no state but there is at the end. Now if it has a state in positive side then it should have it’s partner in negative side. But this not true for  the is zero energy state at the end and it  don’t have any partner. Now this state we can’t get rid of it.It is topologically protected. If we try to move it away then the partner could be nowhere to be found. Now adding a particle in this zero energy state is equivalent to removing the same from this zero energy state!! So this zero energy particle is it’s antiparticle means it’s half of it’s particle. Called Majorana particle.

http://iopscience.iop.org/article/10.1070/1063-7869/44/10S/S29/meta
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.177002
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.077001
Expt. from Yazdani group in Prince ton:
http://science.sciencemag.org/content/346/6209/602
http://science.sciencemag.org/content/336/6084/1003

Majorana Bound states in vortex of a 2D topological superconductor:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.104.040502
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.100.096407
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.81.125318
expt:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.257003

Latter found propagating Majorana Fermions in chiral topological super conductor by sandwich of quantum anomalous hall insualtor and a superconductor.

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.82.184516
 proposal of detecting majorana by transport measurement: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.83.100512
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.064520

A majorana fermion can be imgined as half of Dirac electron. So it will show hall conductance  will be half of normal electron quantum hall conductance !!
Expt. Observation by S.C. Zhang idea:
http://science.sciencemag.org/content/357/6348/294

Well the experiment has been objected by many group and claimed the observed effect could be because of quantum decoherence.

https://www.condmatjclub.org/uploads/2017/09/JCCM_September_2017_01.pdf
https://arxiv.org/abs/1708.06214
https://arxiv.org/abs/1708.06752

Here is the reply that the observed effect is real by  .C.Zhang Stanford
https://arxiv.org/abs/1709.05558

There is alternate suggestion of expt.:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.102.216403
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.102.216404