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and the surprising structure of vortices


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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

(no Fermi surface)

Paradigmatic Dirac system in 2D: graphene


Brillouin zone:

Two inequivalent (Dirac) points at :





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!)


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):


and the Hermitian matrices satisfy:


Particle-hole ``symmetry” :

BdG Hamiltonian (by construction) anticommutes with an antilinear (!) operator

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!

(Ryu, Chamon, Hou, Mudry, PRB 2009)

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:

what are other masses that satisfy

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


Why? (Chekhov’s gun firing)


or any traceless matrix


which anticommutes with the Hamiltonian the expectation value comes entirely from zero-energy states: (IH, PRL 2007)

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:

given, 16 X 16 representation of

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):

(IH, PRB 2012, Okubo, JMP 1991, ABS 1964)


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


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” :

two ``isospins” - 1/2

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:

p-h symmetry is like time-reversal in If

is the eigenstate with energy


, then its time reversed copy

is the eigenstate with energy


, and thus orthogonal to it:

Product state!


Two possibilities:



The “state” of isospin:

(mixture X pure state)

Single finite isospin ½!


Example: U(1) superconducting vortex

(s-wave, singlet) (IH, PRL 2010)

: {CDW, Kekule BDW1, Kekule BDW2}

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

Lattice: 2K component

External staggered potential

Core is insulating !

(Ghaemi, Ryu, Lee, PRB 2010)


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


3 is the number operator


4 and


5 are superconducting.


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)



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 =>


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


zero-modes of Jackiw-Rossi-Dirac Hamiltonian in ``harmonic approximation”

(IH and C-K Lu, PRB 2011)


The vortex-core spectrum:

(IH and C-K Lu, PRB 2011)




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