and the surprising structure of vortices

Order parameters, real Clifford algebras, and the surprising structure of vortices

in Dirac materials

Igor Herbut (Simon Fraser University, Vancouver)

Chi-Ken Lu (Indiana) Bitan Roy (Maryland) Vladimir Juricic (Utrecht)

NTU, Taipei, October 25, 2013

Two triangular sublattices: A and B; one electron per site (half filling)

Tight-binding model ( t = 2.5 eV ):

(Wallace, PR 1947)

The sum is complex => two equations for two variables for zero energy =>

Dirac points

Paradigmatic Dirac system in 2D: graphene

Brillouin zone:

Two inequivalent (Dirac) points at :

+

K

-K

Dirac fermion: 4 components (no spin, 2^d with time-reversal, IH, PRB 2011)

“Low - energy” Hamiltonian: i=1,2

,

(isotropic, v = c/300 = 1, in our units). Neutrino-like in 2D!

,

and so map zero-energy modes, when they exist, into each other! Zero-energy subspace (when there!) is invariant under both commuting and anticommuting operators!!

= >

Chiral symmetry: anticommute with Dirac Hamiltonian

Anton P. Chekhov: If in the first act you hang the pistol on the wall, then in the following one it should

be fired!

“Masses” = chiral symmetries (rich in 2D!)

1)

Broken valley symmetry, preserved time reversal

+

2) Broken time-reversal symmetry, preserved valley

 +



In either case the spectrum becomes gapped:

= ,

,

On lattice?

1) m staggered density, or Neel (with spin); preserves translations (Semenoff, PRL 1984)

2) topological insulator (circulating currents, Haldane PRL 1988, Kane-Mele PRL 2005)

( Raghu et al, PRL 2008, generic phase diagram IH, PRL 2006 )

3) + Kekule bond-density-wave

(Hou,Chamon, Mudry, PRL 2007)

(Roy and IH, PRB 2010, Lieb and Frank, PRL 2011)

Real thing: ( + spin + Nambu )

Original Dirac Hamiltonian, with spin included, is 8 x 8:

Dirac-Nambu Hamiltonian is then 16 x 16 (16 = 2 x 2 x 2 x 2):

where

and the Hermitian matrices satisfy:

Particle-hole ``symmetry” :

In ``Majorana” (“real”) basis:

and the Hamiltonian becomes imaginary! So, we can distinguish between

- Imaginary (masses)

- Real (i=1,2, gamma matrices)

There are then 8 different types of masses:

1) 4 insulating masses (CDW, two BDWs, TI: singlet and triplet): 4 x 4 =16

2) 4 superconducting order parameters ( s-wave (singlet), f-wave (triplet), 2 Kekule (triplet)): 2 + 3 x ( 2 x 3) = 20 (Roy and IH, PRB 2010)

Altogether: 36 masses in 2D!

How many are mutually ``compatible” (i. e. anticommuting)?

Computer (list) : 5

Why? What does it mean?

Mass-vortex: (in real physical space)

with masses insulating and/or superconducting, but always anticommuting

imaginary, and, of course,

The problem:

and are imaginary? How many are mutually anticommuting? (5)

Why? (Chekhov’s gun firing)

F

M

Dirac-BdG Hamiltonian is 16 x 16, and therefore has four zero-modes! (Jackiw, Rossi, NPB 1981)

Internal structure !

(Not necessarily physical spin and electric charge)

Physics of anticommutation: Clifford algebra C(p,q):

p+q mutually anticommuting generators p of them square to +1

q of them square to -1

Vortex Hamiltonian:

2 real Gamma matrices

2 imaginary masses (when mutliplied by ``i” become real and square to -1)

The question: what is the maximal value of q for p=2 (or p>2) for which a

real 16X16 representation of C(p,q) exists?

Real representations of C(p,q):

So there exist three more mutually anticommuting masses (5 = 2 + 3):

and

form an irreducible real representation of the Clifford algebra

Quaternionic representation: there are three nontrivial real ``Casimirs”

Define instead the imaginary

We then find three more solutions (5 = 2 + 3’) :

which satisfy the desired relations

and commute with the old solutions:

In summary:

and true in d=1 (for domain wall) and d=3 (for hedgehog) (IH, PRB 2012)

Order in the defect’s ``core” :

In the four dimensional zero-energy subspace in some basis:

Perturbed (chem. potential, magnetic field, lattice…) Dirac-BdG Hamiltonian:

with small, and matrix also imaginary.

Splitting of the zero modes:

is the eigenstate with energy

+E

is the eigenstate with energy

–E

Product state!

Two possibilities:

E

0

The “state” of isospin:

(mixture X pure state)

Single finite isospin ½!

Example: U(1) superconducting vortex

: {CDW, Kekule BDW1, Kekule BDW2}

: {Haldane-Kane-Mele TI (triplet)}

Lattice: 2K component

External staggered potential

Core is insulating !

Example: insulating vortex (sharp particle number) (IH, PRL 2007)

1) Kekule BDW {Neelx, Neely, Neelz} (insulating, spin-1/2)

{CDW, sSC¹, sSC²} (mixed, spin-0) => meron charged 2) Neel, x-y {Neelz, KekuleBDW1, KekuleBDW2} (insulating)

{QSHz, fSCz1, fSCz2} (mixed) => meron charged

E

M’

M’

Some skyrmions are therefore electrically charged:

1) Neelx, Neely, QSHz => charge 2

2) QSHx, QSHy, QSHz => charge 2 (Grover and Senthil, PRL 2008) and six more!

Every texture in masses carries some generalized charge; if the Hamiltonian is

the conserved current is the topological current

with the matrix (IH, Lu, Roy, PRB 2012)

Summary:

1) Fundamental Clifford algebra for graphene-like systems:

C(2, 5)

2) Zero modes => defect’s cores in Dirac systems are never normal ; there is always some other (compatible) order inside =>

meron

3) Textures of masses carry a generalized ``charge”: in graphene, sometimes, the true electric charge! ( IH, Chi-Ken Lu, Bitan Roy, PRB 2012)

4) D-wave superconductors: without velocity anisotropy, the same algebra as graphene C(2,5); with anisotropy, the same as spinless graphene C(2,3)

Real representations of Clifford algebras; further consequences

1) Bilayers: quadratic dispersion, new symmetry => doubling of charge values (Chi-Ken Lu and IH, PRL 2012, Moon, PRB 2012)

2) Neutrino physics: time-reversal, Weyl fermions, and the dimension of space (IH, Phys. Rev. D 2013)

Introduce bosonic and fermionic operators

a la Dirac :

so that

Digression:

(IH and C-K Lu, PRB 2011)

The vortex-core spectrum: