Majorana Fermions -

in condensed matter systems?

Monday Morning Seminar, December 13, 2010

Dominik Zumbuhl, Physics Department, University of Basel

Frank Wilczek, “Majorana Returns”, Perspective, Nature Physics 5, 614 (2009)Ady Stern, “Non-Abelian States of Matter”, Review, Nature 464, 187 (2010)M. Franz, “Race for Majorana Fermions”, Viewpoint, Physics 3, 24 (2010)

Majorana Fermions - Particles and Antiparticles

particles which are their own antiparticles- neutral pions (spin 0) Klein-Gordon equation- photons (spin 1) Maxwell equations (EM)- gravitons (spin 2) Einstein Equations (GR)all neutral

particle created by operator / field particle = antiparticle: (real operator / field)

formulated usingreal numbers

neutron (spin ½) not it’s own antiparticle (but neutral)electrons, protons (spin ½) have distinct antiparticles

Dirac equation: complex numbers, complex fields, distinct antiparticles(explained why positrons, the antiparticles of electrons, exist)

Majorana (Nuovo Cimento 5, 171-184, 1937)simple, clever modification of Dirac equation using ONLY REAL numbersi.e. spin ½ particles which are their own antiparticlesconsistent with principles of relativity and quantum theoryEXISTENCE?

Dirac Equation

in covariant form

matrices obey Clifford Algebra

metric tensor of flat space

Dirac Equation

in covariant form

matrices obey Clifford Algebra

metric tensor of flat space

Majorana Equation

in covariant form

matrices obey Clifford Algebra

metric tensor of flat space

Motivation

new physicstopological state of matter, quasiparticle excitations not Fermions & not Bosons

but quasiparticles are NON-ABELIAN ANYONS...... WHAT?

quantum computation...topologically protected, i.e. DECOHERENCE FREE...!!...??

to date, non-Abelian states of matter have NOT been observed in nature

candidates:• FQHE = 5/2, 12/5, ...• ultracold atoms in optical lattices• josephson junction arrays• frustrated spin chains• ...

BUT: recently, theorists proposed clear experiments with smoking guns

Topology

topologically identical: continuously deformable

3D: two particle exchanges topologically equivalent to identity

... Fermions and Bosons

2D: two particle exchanges NOT topologically equivalent to identity: ANYONS can in principle by any phase

Non-Abelian Anyons

assume: degenerate ground state, represented by

braiding: represented by

exchange particles 1 and 2

exchange particles 2 and 3

generally....

non-Abelian, topological braiding statistics..... with deep, far reaching physical consequences... NEW PHYSICS

and better quantum computer?

Topological Quantum Computation...?

“conventional” quantum computer:

... or: error correction neededis a complication

idea/hope: protection from decoherence by TOPOLOGY

Topological Quantum Computation

topological QC: immune to decoherence

simplification: move only one particle

CNOT gate: braid upper two QP if control bit in state.... (entangling two-qubit gate, universal computation)

estimate of error of a topological QC:

sounds good?

estimate of error of superconducting (Josephson-Junctions etc) QC

obviously NAIVE...

AND: = 5/2 not sufficiently large Hilbert space for complete QC need unprotected operations (not topological)...

but: = 12/5 sufficient...

101212

Are Neutrinos Majorana Fermions?

short answer: maybe, not yet clear

Neutrinos: discovered 1956experiments show distinction between neutrinos and antineutrinos(lepton number conservation, separately for e, and )

but: recently: speculation about neutrino flavor oscillation(electron neutrino to muon or tau conversion and vice-versa, long distances)breaking lepton number conservation, only sum L + L + L conserved

currently: • experiments under way, to test neutrino and antineutrinos and Majorana’s hypothesis(neutrinoless double decay Ge76 -> Se76 + 2e, disproving lepton # conservation)

• leading ideas on neutrino masses predict: neutrinos are Majorana fermions

Supersymmetry and Dark Matter

supersymmetry- proposal to improve symmetry and coherence of equations- expansion of spacetime into new quantum dimensionpredicts:- every bosonic (integer spin) particle will have a heavier fermionic (half-integer spin) particle(unification of all forces)

- photon: photino (spin ½ particle, mirroring photon properties, including it is it’s own antiparticle)Majorana Fermion

- many other Majorana Fermions in Supersymmetry … LHC experiments

missing mass in the universe:WIMPs – weakly interacting massive particlescould be one of the massive supersymmetry partnersdominating the mass of the universecould be Majoranas

Condensed Matter Majorana Candidates

recently: exotic quasiparticle excitations in a variety of interesting condensedmatter systems might be Majorana fermions

second quantizationcreation operator creates electron, destroys a holeanihilation operator creates hoes, destroys an electron

Pauli exclusion principle

antisymmetry ofFermi-Diract statistics

completeness relation

charge conjugation / particle-hole interchange:

Excitons

bound electron – hole pair

created by

invariant under charge conjugation

i.e. excitons are their own antiparticles

BUT: excitons are always bosons (integer spin, photon absorption)so not Majoranas...

Superconductors

How can one build Majorana Fermions from Electrons in solids?(electrons are charged, antiparticles are holes)

superconductor: Cooper pairs, bosons, condensateelectron number no longer conserved: two electrons can be added / removedfrom condensate without substantially changing its properties.charge no longer observable

existance zero (energy) modes: equal mixtures of particles and holes, spin ½

invariant under conjugation

Condensed Matter Majorana Candidates

not possible in ordinary superconductors, predicted in

(px + ipy) wave superconductors, angular momentum 1- fractional quantum Hall effect, =5/2 (Pfaffian / Moore-Read state)

Review: Ady Stern, Nature 464, 187 (March 2010)

- other exotic superconductors: strontium ruthenateDasSarma, Nayak, Tewari, PRB 73, 220502R (2006)

s-wave Cooper pairing if electrons in normal state obey Dirac-like equation- topological insulator surface with proximity effect to regular superconductor

Fu, Kane, PRL100, 096407 (2008);

- semiconductor SOC superconductorSau, Lutchyn, Tewari, DasSarma, PRL104, 040502 (2010);Alicea, PRB81, 125381 (2010)

- graphene with proximity effect to regular superconductorGhaemi, Wilczek, ArXive0709.2626 (2007).

- interacting quantum spins (toric codes)Kitaev, Ann. Phys. 321, 2 (2003).

- rotating bose einstein condensates (cold atoms)Cooper, Wiklin, Gunn, PRL87, 120405 (2001)

Ettore Majorana

Enrico Fermi apparently said:“There are many categories of scientists, people of second and third rank, who do their best, but do not go very far. There arealso people of first class, who make great discoveries, which are of capital importance for the development of science. But then there are the geniuses, like Galileo and Newton. Well, Ettore was one of these. “

Ettore Majorana

born in Catania, Sicily, 1906rose rapidly through academic ranksfriend and scientific collaborator of Fermi, Heisenberg etcstream of high quality papers

1933 problems… gastritis, reclusive, no publications for several years

1937 Fermi was allowed to write-up and submit underMajorana’s name his last and most profound paper whichMajorana had derived some years before.

At Fermi’s urging, Majorana applied and got Chair in Naples (1938)

March 1938: trip to Palermo, arrived, boarded a ship straight back to NapoliDISAPPEARED without a trace.

a) retired to monastery, to escape spiritual crisis and to embrace his deep Catholic faith

b) jumped overboard in suicide

Majorana had greater gifts than anyone else in the world; unfortunately he lacked one quality which other men generally have: plain common sense.

Ettore Majorana

Enrico Fermi apparently said:“There are many categories of scientists, people of second and third rank, who do their best, but do not go very far. There arealso people of first class, who make great discoveries, which are of capital importance for the development of science. But then there are the geniuses, like Galileo and Newton. Well, Ettore was one of these. “

Condensed Matter Majorana Candidates

not possible in ordinary superconductors, predicted in

(px + ipy) wave superconductors, angular momentum 1- fractional quantum Hall effect, =5/2 (Pfaffian / Moore-Read state)

Review: Ady Stern, Nature 464, 187 (March 2010)

- other exotic superconductors: strontium ruthenateDasSarma, Nayak, Tewari, PRB 73, 220502R (2006)

s-wave Cooper pairing if electrons in normal state obey Dirac-like equation- topological insulator surface with proximity effect to regular superconductor

Fu, Kane, PRL100, 096407 (2008);

- semiconductor SOC superconductorSau, Lutchyn, Tewari, DasSarma, PRL104, 040502 (2010);Alicea, PRB81, 125381 (2010)

- graphene with proximity effect to regular superconductorGhaemi, Wilczek, ArXive0709.2626 (2007).

- interacting quantum spins (toric codes)Kitaev, Ann. Phys. 321, 2 (2003).

- rotating bose einstein condensates (cold atoms)Cooper, Wiklin, Gunn, PRL87, 120405 (2001)