Magnetic impurities in the honeycomb Kitaev modelcmworksh/LDQS/talks/Kusum.pdf · Magnetic impurities in the honeycomb Kitaev model Kusum Dhochak, TIFR Low Dimensional Quantum Systems

Magnetic impurities in the honeycomb Kitaevmodel

Kusum Dhochak, TIFR

Low Dimensional Quantum SystemsHRI, Allahabad

I Collaborators:Vikram Tripathi, TIFR, MumbaiR. Shankar, IMSc Chennai

I Discussions:G. Baskaran, IMSc ChennaiKedar Damle, TIFRD. Dhar, TIFRKrishnendu Sengupta, IACS Kolkata

I Ref: Phys. Rev. Lett. 105, 117201 (2010)

Outline

I Introduction, MotivationI Brief overview of the honeycomb Kitaev ModelI Kondo Effect in the Kitaev modelI Topological nature of the Kondo transition and non abelian anyonI Inter-impurity coupling

Introduction

I Kitaev model: spin-1/2 model on honeycomb latticeI Interesting properties: exactly solvable, gapless Majorana

fermionic excitations.I Spin liquid ground state with very short range correlations.I Anyonic excitations, relevant to quantum computation.

I Magnetic/non-magnetic impurities effectsI probe of electronic state of the host system (gapless/gapped,

nature of quasiparticle excitations etc).I gives rise to interesting effects: unusual Kondo effect in gapless

bosonic spin liquids (Florens et. al. PRL 2006), renormalized Curiesusceptibility (Sachdev et. al. PRB 2006), creation andmanipulation of non-abelian anyons.

Introduction

Magnetic impurity effects in the Kitaev modelI Probe the quantum spin liquid state.I Intersting Kondo effect due to Majorana fermion nature of the

quasiparticles.I Unstable fixed point in the Kondo coupling scaling.I Change of flux sector/topology of the ground state.I Localized fluxes with zero energy majorana fermion at its core: non

abelian anyonic statistics.I Inter-impurity interactions mediated by dispersing Majorana

modes: long range interaction present despite very short rangecorrelations in ground state.

I Non-zero only if the impurities couples to more that oneneighbouring Kitaev spins.

I Very different from the RKKY interaction mediated by electrons.

Kitaev Model: brief description

A honeycomb lattice of spin-1/2 with nearest neighbour directiondependent interactions.

H0 = −Jx

∑x-links

σxj σ

xk − Jy

∑y -links

σyj σ

yk − Jz

∑z-links

σzj σ

zk ,

(b)

q2 q1

kF

z

3

6

(a)

5

4y 2

1B

A

!n1

W2W1

x

!n2

W3

Conserved flux operators on each plaquett:

Wp = σx1σ

y2σ

z3σ

x4σ

y5σ

z6 , eigenvalues ±1


The Hamiltonian can be solved exactly by representing spins in termsof Majorana fermions σαi = ibαi ciThe ground state manifold corresponds to a vortex free state whereall Wi are equal.Gauge condition: Di = ibx

i byi bz

i ci = 1.

uαij = ibαi bαj , [uij ,H] = 0, [uij ,ukl ] = 0

Wp =∏〈ij〉,i∈A,j∈B uij ; In the vortex free state, fix uij = 1

H0 =i4

∑jk

Ajk cjck ,

Ajk = 2Jαjkif j , k are neighboring sites on an α−bond and zero

otherwise.


I Model of free majorana fermions on honeycomb lattice.I Excitation spectrum: εα(q) = ±2|(Jxeiq·n1 + Jy eiq·n2 + Jz)|I q varies over only half of the BZI Gapless phase when J ’s satisfy triangle inequality|Jz | ≤ |Jx |+ |Jy | etc.

I Linear dispersion near the Fermi point.

Kondo coupling to external magnetic spin

I Kondo effect in metals:H =

∑k,α εk c†k,αck,α + K

2

∑i,α,β

~Si · c†i,α~σ ci,β

d lnKd lnD ∝ −Kρ(εF )

Flows to infinity for antiferromagnetic K and to 0 forferromagnetic.

I Possibility of a new type of Kondo effect in Kitaev model due toMajorana nature of excitations


I Kondo coupling term in the Hamiltonian:

VK =∑α

KαSασα(0) = i∑

q∈HBZ ,α

Kα

√2N

Sαbα(cq + c†q)

Kα

bα

c

Bare T matrix


Poor man’s scaling for K .I perturbation expansion T = T (1) + T (2) + · · · in increasing

powers of K , for the T−matrix element: 〈bβ |iK βSβbβca,A|(q, α)〉.I follow its variation as a function of the decrease of the bandwidth

(−D,D).

I Diagrams that contribute to 3rd order term:

(a)

b! q!, !q, !b!

S!, iS!, i S!, i

(b)q, ! b!

q!, !

S!, i

b!

S!, i S!, i


I Adding the two contributions (taking E ' 0),

T (3) ∼ − iK βSβ√2

ρ(D)δDJD

∑β

(K β)2(Sβ)2.

I If either the impurity is a S = 12 spin, or the Kondo interaction is

rotationally symmetric, the above contribution renormalizes theKondo coupling constant.

I For S 6= 12 with anisotropic coupling, new terms are generated

,need to go to higher order diagrams to obtain the scaling ofthese new coupling terms.


I Linear dispersion gives further contribution to the scaling of K (D.Withoff and E. Fradkin, PRL 64, 1835 (1990)).

I For ρ(ε) = (1/2πv2F )|ε| ≡ C|ε|r , r = 1

K → K (D′/D)r , (D′ = D − |δD|).

I δK = −K δDD

(2K 2a2CDS(S + 1)/J − 1

).

I The effective coupling K has an unstable fixed point atKc =

√J/[2a2ρ(D)S(S + 1)] ∼

√J/S2a2CD ∼ J/S.

I For K > Kc , coupling flows to infinity independent of its sign(ferromagnetic or antiferromagnetic).

I Same scaling observed for bosonic Z2 bosonic spin-liquid withpseudogap DOS (Florens et al).

Localized Majorana, finite flux and change of topology

I In the strong coupling limit, the impurity spin forms a bound statewith the Kitaev spin (say at the origin)

I Majorana fermions associated with the Kitaev spin at the originno longer available for pairing with the Majoranas at the threeneighboring sites.

I We can equivalently study the Kitaev model with the site at originremoved.


I W1, W2 and W3 do not commute with H = H0 + VK .

I W0 = W1W2W3 is still conserved, W0 = 1 in the ground state ofunperturbed Kitaev model.

A

B

W3

W2

y

W1z

x

I Define composite operators τ x = W2W3Sx , τ y = W3W1Sy andτ z = W1W2Sz ,

I τα’s obey the SU(2) algebra, [τα, τβ] = 2iεαβγτγ .I The SU(2) symmetry is exact for all couplings, is realized in the

spin-1/2 representation(

(τα)2 = 1)

.


I All eigenstates, including the ground state are doubly degenerate(τ z = ±1).

I In strong coupling limit, double degeneracy arises from theKitaev model with one spin removed.

I For strong (antiferromagnetic) coupling limit JK →∞, the spin atthe origin forms a singlet |0〉 with the impurity spin,

|ψ〉 = |ψK−〉 ⊗ |0〉.

I Action of the SU(2) symmetry generators on these states:

τα = Wα ⊗ σα0 ⊗ Sα.

I τα|ψ〉 = −(Wα|ψK−〉)⊗ |0〉.Double degeneracy coming from Wα|ψK−〉.


I Three unpaired b−Majorana Fermions: bz3 , bx

2 and by1

1 2

3

bz3

by1

bx2

I ibx2by

1 : Conserved; Commutes with all the conserved Wi ’s.I Does not commute with iby

1bz3 and ibz

3bx2

I Choose a gauge s.t. i〈bx2by

1〉 = 1I One unpaired b−Majorana with dimension

√2; =⇒ unpaired

Majorana mode in the c sector (together give the full doublydegenerate zero energy mode).

Non-abelian anyons

I The bz3 mode is sharply localized, the wave function of the c

mode can be spread out in the lattice.I An unpaired zero energy Majorana Fermion is associated with

finite flux (Kitaev 06).I Also shown by Willans et al that in presence of a vacancy, the

state with finite flux pinned to vacancy has lower ground stateenergy.

I Unit flux with a localized zero energy b-Majorana fermion at itscore: Non-abelian statistics (similar to quantum half vortices inp-wave superconductors).

Interaction between external spins

I In bare Kitaev model ground state, spin correlations very shortranged (only nearest neighbour) due to b−Majoranas beinglocalized.

I Possibility of interaction between distant spins mediated bydispersing c−Majorana Fermions.

I Two distant impurities coupled to a single Kitaev site each do notinteract via dispersing Majoranas.

I Impurities can interact only if they are coupled more than oneneighbouring Kitaev spins: bonds.

(b) q,!

q!, !

S!, i S!, j

b!(a) q,!

b!b!

S!, jS!, i


I The Kondo coupling term is

VK = i∑

j∈hex1,α KαSα1 bαj cj + i∑

j∈hex2,α KαSα2 bαj cj .

I Effective interaction generated involving c−type Majoranafermions at two neighboring sites (i ∈ A, j ∈ B) :

Veff =∑ν

〈Ωb|V |ν〉1

E0 − Eν〈ν|V |Ωb〉

≈ 2J

∑a,<ij>α

(Kαij )2(Sαija )2cicj ,

|Ωb〉 refers to the ground state configuration of b− Majoranas i.e. all uij = ibibj = 1.

•

j, B•S2

j!, Bk!, !!

k, "!!i, Ai!, AS1


I Interaction between the two impurity spins is given by the secondorder term in Veff

1J2 〈(Kαij )2(Sαij

1 )2(K βi′ j′ )2(Sβi′ j′

2 )2cicjci′cj′〉.

I Performing the fermionic averaging,

J ij,i′ j′

12 ∼ −(Kαij )2(Sαij1 )2(K βi′ j′ )2(S

βi′ j′

2 )2

× a4

4vF J2π31 + cos(2α(kF ))− 2 cos(2kF · R12)

R312

.

I For spin-1/2 impurities, (Sα)2 = 1/4 and for isotropic couplingwhere

∑bond pairs(Sαij

1 )2(Sβi′ j′

2 )2 = const., no long-rangedinteraction is generated.

I Non-dipolar interaction unlike dipolar Si · Sj interaction in metals.

I Kondo effect in Kitaev model associated with an unstable fixedpoint separating sectors of different topology.

I Strong coupling limit corresponds to non-Abelian anyon with azero energy unpaired Majorana fermion.

I RKKY interaction is non-dipolar due to Majorana fermionicexcitations.

I Change of topology and nondipolar interaction make the KitaevKondo effect different from earlier studied Kondo effects inspin-liquids.

Thank You

Documents

Magnetic impurities in the honeycomb Kitaev modelcmworksh/LDQS/talks/Kusum.pdf · Magnetic impurities in the honeycomb Kitaev model Kusum Dhochak, TIFR Low Dimensional Quantum Systems