## 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 : **

** +**

**K **

**and**

**-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” :**

^{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) **

**F**

**or any traceless matrix**

**M **

**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): **

**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” : **

**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 **

**+E**

**, then its time reversed copy**

**is the eigenstate with energy **

**–E**

**, and thus orthogonal to it:**

**Product state! **

**Two possibilities: **

** E **

**0 **

**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 {Neel****x****, Neel****y****, Neel****z****} (insulating, spin-1/2) **

** {CDW, sSC**^{1}**, sSC**^{2}**} (mixed, spin-0) => meron charged **
**2) Neel, x-y {Neel****z****, KekuleBDW****1****, KekuleBDW****2****} (insulating) **

** **

** **

** {QSH****z****, fSC****z1****, fSC****z2****} (mixed) => meron charged **
** **

**E**

^{3}**is the number operator**

** M’**

^{4}

^{ and }**M’**

^{5}**are superconducting.**

** **

**Some skyrmions are therefore electrically charged: **

**1) Neel****x****, Neel****y****, QSH****z**** **=> **charge 2 **

**2) QSH****x****, QSH****y****, QSH****z** => **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: **

**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)**