Exact results for spin dynamics and fractionization in the Kitaev Model

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by  G. Baskaran

Exact results for spin dynamics and fractionization in the Kitaev Model
G. Baskaran, Saptarshi Mandal and R. Shankar
The Institute of Mathematical Sciences, CIT Campus, Chennai 600 113, India.
Wepresentcertainexactanalytical resultsfordynamicalspincorrelation functionsintheKitaev
Model. It is the first result of its kind in non-trivial quantum spin models. The result is also
novel: in spite of presence of gapless propagating Majorana fermion excitations, dynamical two
7 spin correlation functions are identically zero beyond nearest neighbor separation. This shows
0 existenceof agapless butshort range spin liquid. Anunusual, all energy scale fractionization of a
0 spin -flip quanta, into two infinitely massive π-fluxesand a dynamical Majorana fermion, is shown
2 to occur. As the Kitaev Model exemplifies topological quantum computation, our result presents
new insights into qubitdynamics and generation of topological excitations.
n
a
PACS numbers: 75.10.jm, 03.67.-a, 03.67.Lx, 71.10.Pm
J

5

In the field of quantum computers and quantum com- states. In actual quantum computations, key manipula-
]
l munications, practical realizations of qubits that are ro- tions such as braiding involve parametric change of the
e
- bust and escape decoherence is a foremost challenge[1]. Hamiltonian and adiabatic transport of topological de-
r InthiscontextKitaevproposed[2]certainemergenttopo- greesoffreedom[7]. Inprinciple,someoftheneeded‘non
t
s logical excitations in strongly correlated quantum many equilibrium’ dynamical correlationfunctions may be ob-
.
t body systems as robust qubits. In a fault tolerant quan- tained by convolution of our results with suitable Berry
a
m tum computation scheme[2, 3, 4], Kitaev constructed phase factors.
a non-trivial and exactly solvable 2-dimensional spin In our work we follow Kitaev[2] and use the Majo-
-
d model[2]andillustratedbasicideas. Insomelimititalso rana fermion representation of spin-half operators and
n
becomes the celebrated ‘toric code’ Hamiltonian. The an enlarged Hilbert space. What is remarkable is that,
o
Kitaevmodelhascomeclosertoreality,afterrecentpro- because of the presence of certain local conserved quan-
c
[ posalsforexperimentalrealizations[5,6]andschemesfor tities in the Kitaev Model, Hilbert space enlargement

manipulation and detection[7]. In initialisation, error only produces ‘gauge copies’, without altering the en-
2
correction and read out operations, it is ‘spins’ rather ergy spectrum. This luxury is absent for standard 2D
v
7 than emergent topological degrees of freedom that are Heisenbergmodelswhenstudiedusingenlargedfermionic
4 directly accessed from outside. Thus an understanding Hilbert space [9, 10].
5 ofdynamicspincorrelationsisofparamountimportance. The Kitaev Hamiltonian is
1
We present certain exact analytical results for time
1 H = J σxσx J σyσy J σzσz (1)
6 dependent spin correlation functions, in arbitrary eigen- − x X i j − y X i j − z X i j
0 states of the Kitaev Model. Our results are non-trivial hijix hijiy hijiz
/ and novel, with possible implications for new quantum wherei,j labelthesitesofahexagonallattice, ij , a=
at computational schemes. Further our result is unique in x,y,z denotes the nearestneighbor bonds in thhe aia’th di-
m
the sense that it is the first exact result for equilibrium rection. Themodelhasnocontinuousglobalspinsymme-
- dynamical spin correlation functions in a non trivial 2D try. AllbondinteractionsareIsinglike,albeitindifferent
d
quantum spin model. quantisationdirectionsx,yandz,inthreedifferentbond
n
o Weshowthatdynamicaltwospincorrelationfunctions types, making the model quantum mechanical. Further,
c are short ranged and vanish identically beyond nearest it renders a high degree of frustration; that is, even at
v: neighborsitesfor alltime t, forallvaluesofthe coupling a classical level a given spin can not satisfy conflicting
i constants Jx,Jy and Jz, evenin the domain of J’s where demands, from 3 neighbors, of orientations in mutually
X
the model is gapless. Our result shows rigorouslythat it orthogonal directions. The model has a rich local sym-
ar is a shortrangequantumspin liquidandlongrangespin metry. A specific product of 6 spin components in every
order is absent. We obtain a compact form for the time elementaryhexagon,σyσzσxσyσzσx (figure1)commutes
1 2 3 4 5 6
dependence, which makes the physics transparent. with the full Hamiltonian. Thus there is one conserved
Kitaev Model is known to support dynamical Majo- Z charge 1, at every dual lattice site of the hexago-
2
±
rana fermions and static π-flux eigen-excitations. We nal lattice. The model is exactly solvable and becomes
showhowfractionization[8,9]ofalocalspin–flipquanta non interacting Majorana fermions, propagating in the
into a bound pair of static π-flux excitations and a free background of static Z gauge fields. Different possible
2
Majorana fermion occurs. Z chargesseparatethe Hilbertspaceintosuperselected
2
In the present paper we have restricted our calcula- sectors. ThegroundstatecorrespondstoallZ charges=
2
tiontodynamicalcorrelationfunctionsfortimeindepen- +1. Inthissector,forarangeofJ’s,Majoranafermions
dentHamiltonians,inarbitraryeigen-statesandthermal are gapless, including the special point J =J =J .
x y z
2
z c2z In the gauge field sector we have gauge invariant Z2
vortexcharges 1(0andπ-fluxes),definedasproductof
y 1 2 3 x c2x c2 cy2 u ij a around e±ach elementary hexagonal plaqauette.
2 cy cx hEiquation 6, with conserved uˆ is the Hamiltonian
3 3 hijia
x 6 5 4 y 3 of free Majorana fermions in the background of frozen
c Z vortices or π-fluxes. Since Z gauge fields have no
σ2y=ic2y(χ/\23χ/\+/\23χ/\ =+/\23(/\)cy2+; iσc3yy3)= c3(χ/\23/\_χ+/\23/\c)3z 3 dsrvtaay2anrntiaaaembfielnriecmsts.h,ioWeanl2els12ewaNingildeldnrtisemhtfeaeertn(e2tsso)io32ctnNahaneldfbFiomeormcewk2nersirsitpoatanesncamelaasosptftpaetcrrhoeesdoecufcictZotM2sralaoijnfnodka-
thelatterasgaugefieldsector. Gaugecopies(eigen-states
FIG.1: Elementaryhexagonand‘bondfermion’construction.
A spin is replaced with 4 majorana fermions (c,cx,cy,cz). with same energy eigen-values) spanning corresponding
Bond fermion χh23i for the bond joining site 2 and site 3 is extendedHilbertspaceareobtainedbylocalgaugetrans-
shown . Spin operators are also defined. formations u τ u τ .
ij a i ij a j
h i → h i
It turns out that if we attempt to calculate spin-spin
correlationfunctionswiththeuseofabovefreeMajorana
Following Kitaev, we represent the spins in terms of
Hamiltonian and the Z fields u ’s, it is difficult to
Majorana fermions. At each site, we define 4 Majorana proceed further. 2 hijia
fermions, cα, α=0,x,y,z,
It is here we have invented a simple but key trans-
formation that facilitates exact computation of all spin
cα,cβ =2δ (2)
{ } αβ correlationfunctions. We call this as ‘bond fermion’ for-
mation. Inthe processwe alsodiscovera ‘quantumfrac-
Four Majorana (real) fermions make two complex
tionization’ phenomenon in the Kitaev Model, that has
fermions, making the Hilbert space 4 dimensional. No-
an unusual validity at all energy scales.
tionally, Hilbert space dimension of a Majorana fermion
Hereinafter,wefollowtheconventionthatiinthebond
is√2,anirrationalnumber,remindingusthatMajorana
ij , belongs to A and j to B sub-lattice. We define
fermionshavetooccurinpairs(leadingtoa√2 √2=2 h ia
× complex fermions on each link as,
dimensional Fock space) in physical problems.
The dimension of Hilbert space of N spins is 2N. The 1
e√n2larg√ed2)HNil.beSrttastpeavceechtoarssaodfitmheenpshioynsi4cNal=Hi(l√be2rt×s√pa2c×e χhijia = 21(cid:0)cai +icaj(cid:1) (7)
satis×fy the condition, χ†hijia = 2(cid:0)cai −icaj(cid:1) (8)
Di Ψ phys = Ψ phys (3) The link variables are related to the number operator of
| i | i
Di ≡ ci cxicyiczi (4) tehigeesnestfeartmesiocnasn, tuˆhheijrieafo≡re ibceaiccajho=sen2χt†hoijhiaaχvheijaiad−efi1n.iteAχll
The spin operators can then be represented by, fermion occupation number. The Hamiltonian is then
blockdiagonal,eachblockcorrespondingtoadistinctset
σia =icicai, a=x,y,z (5) of χ fermion occupation numbers. Thus all eigenstates
in the extended Hilbert space take the factorized form,
When projected into the physical Hilbert space, the op-
erators defined above satisfy the algebra of spin 1/2 op- Ψ˜ = ; (9)
erators, [σa,σb] = iǫ σcδ . The Hamiltonian written | i |MG Gi ≡ |MGi|Gi
in terms ofithej Majoraabcna feijrmions is, and χ†hijiaχhijia|Gi = nhijia|Gi (10)
H =− X Ja X iciuˆhijiacj, (6) swthaetereinnhtijhiea m=authtiejir2as+e1ctaonrd, c|MorrGeispiosnadminagntyobaodgyiveeingeZn-
a=x,y,z hijia field of . In terms of bond fermions, spin operator2s
|Gi
with uˆ icaca. Kitaev showed that [H,uˆ ] = 0 become,
hijia ≡ i j hijia
and u become constants of motion with eigen-values
ij a
(uIhsiijniag)h=Zi±1g.auTgheefivealdrisaobnlesthuehibjioanadrse.iKdeintateifiveHdawmitihltsotnaitainc σia = ici(cid:16)χhijia +χ†hijia(cid:17) (11)
2
(teeqnudaedtioHnil6b)erhtasspaacleo.caFloZr2pgraacutgiceailnpvaurripaonscees,inthteheloecxa-l σja = cj(cid:16)χhijia −χ†hijia(cid:17) (12)
Z gauge transformation amounts to u τ u τ , Three components of a spin operator at a site, gets
2 ij a i ij a j
with τ 1. Equation(3) is the Gauss lhawi a→ndthehpihys- connected to three different Majorana fermions defined
i
±
ical subspace is the gauge invariant sector. on the three different bonds ! Written in the above
3
form, the effect of σa on any eigen-state, which we re- 2(J sink +J sink ), k =k.n , k =k.n and n =
i x 1 y 2 1 1 2 2 1,2
fer to as a ”spin flip”, becomes clear. In addition to 1e √3e are unit vectors along x and y type bonds.
adding a Majoranafermionatsite i, it changesthe bond A2txth±e p2oinyt, J =J =J , we get Saa (0)= 0.52.
fermionnumber from0 to 1andvice versa(equivalently, x y z hijia −
To compute g (t) we substitute for the σ’s from
u ij a u ij a), at the bond ij a. The end result equation (7) andhi(j8i)a. We choose a gauge where u =
ishthiat→on−e π hfluix each is added tho tiwo plaquettes that hijia
are shared by the bond hijia (figure 2). We denote this t−h1e iambpolvyeincgonχd†hiltjiiobn|sGiim=poχs†heidkibat|Gti==00w. iWll econnotitneutehatot
symbolically as
be true at all times since the bond fermion numbers are
σia =ici(cid:16)χhijia +χ†hijia(cid:17) → ici πˆ1hijia πˆ2hijia (13) conserved. We then have,
with πˆ1 ij a and πˆ2 ij a defined as operators that add ghijia(t) = hMG|hG| ici(t)χ†hijia(t)χhijia(0) cj(0)|Gi|MGi
π fluxeshtoi plaquetthesi1 and 2 shared by a bond ij a (18)
h i
(figure 2). Further πˆ2 =1, since adding two π fluxes
1 ij a Thetimedependenceevolutioncanbeexpressedinterms
h i
is equivalent to adding (modulo 2π) zero flux.
ofthehamiltonianandnotingitisdiagonalinthenumber
Now we wish to calculate spin-spin correlation func-
operators χ χ, we get,
tions in physical subspace. Since the spin operators are †
gauge invariant, we can compute the correlation in any g ij a(t)= eiH[Gia]tici(0)e−iH[Gia]t( 1)cj(0)
gauge fixed sector and the answer will be the same as in h i hMG| − |M(1G9i)
the physical gauge invariant subspace. (We have con-
where H[ ia] is the tight binding hamiltonian in the
firmed this by a calculation in the projected physical backgrounGd of the static gauge field configuration ia.
subspace.) So we consider the 2-spin dynamical corre- G
The ( 1) factor is u ij . This expression can be writ-
a
lationfunctions,inanarbitraryeigen-stateoftheKitaev − h i
tenin terms ofthe time evolutionunder H[ ] asfollows,
Hamiltonian in some fixed gauge field configuration , G
G
Siajb(t)=hMG|hG|σia(t)σjb(0)|Gi|MGi (14) ghijia(t) = hMG|ici(t)T (cid:16)e−2JaR0tuhijiaci(τ)cj(τ)dτ(cid:17)
u c (0) (20)
HereA(t) eiHtAe iHt istheHeisenbergrepresentation hijia j |MGi
−
≡
of an operator A, keeping h¯ =1. As discussed above, The above equation is written in an arbitrary gauge.
Wehavethusderivedasimplebutexactexpressionfor
σb(0) = c (0) ia (15)
j |Gi|MGi i |G i|MGi thespatialdependenceofthetwospindynamicalcorrela-
σa(t) = ei(H E)tc (0) jb (16) tionfunction. Wehavealsoobtainedanexactexpression
i |Gi|MGi − j |G i|MGi forthe time dependence intermsofthe correlationfunc-
where, ia(jb) denote the states with extra π fluxes tions of non-interacting Majorana fermions in the back-
|G i
added to on the two plaquettes adjoining the bond groundofstaticZ gaugefields. Equation(20)represents
G 2
ik a ( lj b )andE istheenergyeigenvalueoftheeigen- the propagation of a Majorana fermion in the presence
h i i h i i
state . Since the Z2 fluxes on each plaquette oftwoinjectedfluxes. ItcanbetreatedasanX-rayedge
|G|MGi
is a conserved quantity, it is clear that the correlation problemandcomputedin terms ofthe Toeplitz determi-
function in equation(14) which is the overlap of the two nant. We will not do this now but proceed to discuss
states in equations (15, 16) is zero unless the spins are some general features of our results.
on neighbouring sites. Namely, we have proved that the
dynamical spin-spin correlation has the form, k k
π Majorana π
Fermion
Sab(t) = g (t)δ , ij nearest neighbors (17) π π
ij hijia a,b i i
= 0 otherwise.
A (t)
_ = ij
iHt
Computation of gij(0) is straight forward in any eigen- e j
state . For the ground state where conserved Z
2
|MGi
chargesare unity at all plaquettes, the equal time 2-spin j j
correlation function for the bond ij a is given by the
h i
analytic expression: FIG. 2: Time evolution and fractionization of a spin flip at t
= 0 at site i, into a π-flux pair and a propagating Majorana
√3 fermion. ‘Shakeup’ of the Majorana fermion vacuum to an
σaσa Saa (0)= cosθ(k ,k )dk dk
h i ji≡ hijia 16π2 ZBZ 1 2 1 2 instantaneousadditionatt=0,ofaπ-fluxpairisnotshown.
Wherecosθ(k1,k2)= Eǫkk,Ek = (ǫ2k+∆2k),intheBril- Thenotionoffractionalizationofspin-flipquantaisthe
louin zone. ǫ = 2(J cosk +pJ cosk + J ), ∆ = naturalinterpretationofourresults[8,9]. Considertime
k x 1 y 2 z k
4
evolution of a single ‘spin-flip’ at site i given in equation our formalism. Further, quantum entanglement, a key
(16). Using the notation introduced in equation (13) we notion in quantum computation and quantum informa-
have, tion, is ultimately connected with some complicated
multi-spincorrelationfunction. Wehavecomputedsome
σia|Ψˆi≡ici(t)T(e2uhikiaJaR0tci(τ)ck(τ)dτ)πˆhikia1πˆhikia2 |Ψˆi entanglement measures, but do not discuss them in the
(21) present paper.
To summarise, this paper presents certain exact ana-
A spin-flip at site i at time t = 0 is a sudden perturba- lytical results for the spin dynamics and a spin-flip frac-
tion to the matter (Majorana fermion) sector, as it adds tionization scheme for the Kitaev Model. As this non-
two static π-fluxes to adjoining plaquettes. The time trivial spin model is also a model for topological quan-
ordered expression represents how a bond perturbation tum computation, our exact results should provide in-
term, i2u J c c evolvesthe Majoranafermion state, sights into qubit dynamics and possible ways of generat-
ik a a i k
in ‘interahctiion representation’. At long time scale the ing emergent topological qubits. Our formalism, which
resulting ‘shakeup’ is simple and represents a rearrange- usesthefactorizedcharacteroftheeigen-functionsinthe
ment (power law type for gapless case) of the Majorana extended Hilbert space, is easily adapted to the calcula-
fermion vacuum to added static π-flux pairs. The Ma- tion of multi-spin correlation functions, which is a key
jorana fermion, produced by a spin-flip, c (t) propagates step in the calculation and understanding of quantum
i
freely, as a function of time. entanglement properties.
As spin-flip at site i is a composite of a Majorana Acknowledgement
fermion and π-flux pair (equation 13), two spin corre- G.B. thanks Ashvin Vishwanath for bringing the Ki-
lation function defines the probability that we will de- taev Model to his attention and tutorials.
tect the added composite at site j after a time t. As the
added π-flux pair do not move, the above probability is
identicallyzero,unlesssites iandjarenearestneighbors
and spin components are a = b. This is why the spatial
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Multi spin correlation functions can be calculated in