A chain of strongly correlated SU(2)_4 anyons: Hamiltonian and Hilbert space of states

0.2 MB·English

by  L. Martina

SU
A chain of strongly correlated (2) anyons:
4
Hamiltonian and Hilbert space of states
0 L. Martina, A. Protogenov∗, V. Verbus
1
0
2
Dipartimento di Fisica, Universita del Salento, Lecce, Italy

n
Institute of Applied Physics, RAS, Nizhny Novgorod, Russia
a
J Institute for Physics of Microstructures, RAS, Nizhny Novgorod, Russia

7
2

Abstract
]
el One-dimensional lattice model of SU(2)4 anyons containing a transition into the
- topological ordered phase state is considered. An effective low-energy Hamiltonian is
r foundforhalf-integerandintegerindicesofthetypeofstronglycorrelatednon-Abelian
t
s anyons. The Hilbert state space properties in the considered modular tensor category
.
t are studied.
a
m PACS:05.30.Pr, 11.25.Hf, 05.50.+q, 03.65.Vf, 64.60.De.
-
d
n 1 Introduction
o
c
Inbasicmodelsofquantummagnets,thestatesofparticleswithhalf-integerandintegerval-
[
uesofspin,whicharedistributedoverlatticesites,correspondtothefundamentaloradjoint
1
irreducible representation of the SU(2) group and are classified according to odd or even
v
2 representations of the permutation group. In spatially one-dimensional or two-dimensional
3 systems,insteadofthepermutationgroup,wedealwiththebraidgroupanditsAbelianand
9 non-Abelian irreducible representations. Therefore, in the generalized Heisenberg or Ising
4
models, neither fermions, nor bosons are distributed over lattice sites, but the quantum
.
1 statesofparticlesrealizingirreduciblerepresentationsofthe braidgroupandcorresponding
0
to fractional statistics.
0
1 Thesetofquantumnumbersindicatingthetypeoftheseso-calledanyonstatescomprises
: integers specifying the structure of quantum groups, e.g., SU(2) . Here, k is the SU(2)
v k k
i leveloftheWess-Zumino-Witten-Novikovtheory. Thisnumberwhichcoincideswiththeco-
X
efficientintheChern-Simonsaction,determinesthebraidingdegreeofexcitationworldlines
r in the (2+1)D systems. When k = 1, fermion worldlines are parallel. If k = 2, world-
a
lines are linked twice and we deal with semi-fermion quasiparticles, the so-called semions.
Such anyons are primary fields of the SU(2) conformal field theory as well as that of the
2
transverse field Ising model at the critical point [1, 2, 3].
Anyon states with k = 3 form the Fibonacci family. Detailed analysis of these non-
Abelian states [1, 4] and generalization of the Heisenberg Hamiltonian to the case of inter-
actingFibonaccianyonsarepresentedinRefs. [5,6]. Paper[7]isdevotedtocollectivestates
∗email: [email protected]
1
for a higher level of k in the model of interacting SU(2) anyons belonging to the sector
5
with integer spins. A characteristic feature of anyon systems for k > 2 is the presence of a
termdescribingthethree-bodyinteractionintheHamiltonian. Anotherimportantproperty
is the fact that the squaredtotal quantum dimensionof states with k =1,2,4is aninteger.
In this case, the density of braid grouprepresentations is such that it does not provide uni-
versal quantum calculations [8, 9]. For k , the Heisenberg Hamiltonian is restored for
→∞
particleswiththespin1/2,whileforparticleswithanarbitraryspin,wehaveitspolynomial
variant[10]. To elucidatethe Hamiltonianstructureofthe theoryfor intermediatevaluesof
k, in this paper we focus on the theory of anyons in the case of k =4.
Studying of anyon systems for small k is important for several reasons. First of all,
due to their possible use for decoherence-free quantum computations based on the rigidity
of the wave-function phase of non-Abelian excitations [11]. For this reason, anyons of the
transverse field Ising model and the SU(2) theory with k = 2 differing by the sign of the
k
Frobenius-Schur indicator and Fibonacci anyons for k = 3 were analyzed [12] in detail as
probable candidates for the role of non-Abelian states in the fractional quantum Hall effect
withthefillingfactorν =2+ k . Phaseeffectsininterferometry[13,14]andtunneling[15]
k+2
ofnon-Abeliananyonstatesforfindingexperimentalconsequenceswerestudiedinanumber
ofrecentpapers(seealsoreview[1]). Discussionofthe criteriaofanyondeconfinement[16],
searchofcorrespondencesbetweenthe lattice theoryatthe criticalpoint andthe conformal
field theory determined on coset-spaces as the source of the nucleation effects in parent
two-dimensional spin liquid and edge modes at their boundaries [7, 17], as well as detailed
description of topology driven quantum phase transitions [18] in time-reversal invariant
systems (in quantum doubles) are subjects of recent papers in this challenging field.
In this paper, which employs the approach developed in Refs. [6, 7], we have found the
Hamiltonian and the Hilbert space of states in the SU(2) theory for chains of half-integer
4
and integer spins indicating sectors of interacting anyons. Since we are interested in the
low-energy states, the effective low-energy Hamiltonian has the form of projectors on the
groundstatewithafixedspin. Someaspectsoftheproblemassociatedwithrepresentationof
the Hamiltonian by means of projectors of the Temperley-Lieb algebra have been discussed
in Refs. [5, 18]. Maps of Hamiltonians [19, 20] of the exact solvable quantum models
[11, 21], study of correlation functions [22], and classification of the toloplogical order [23]
and quantum phase transitions are also considered.
We will pay attention to some universal features of externally different models and dis-
cuss a correspondence to the conformal field theory in description of the braiding effects.
The main employed tools belong to the theory of modular tensor categories, in terms of
which the anyon theory is formulated now. Therefore, in the next section we present some
technique from the theory of modular tensor categories for describing the Hamiltonian and
the eigenstate space considered in the third and the fourth sections. In conclusion, we dis-
cussthe unsolvedproblemsrelatedto theuniversalrepresentationofHamiltoniandynamics
of the exact integrable spin models describing the topological ordered phase states, which
are characterizedby non-Abelian anyon excitations.
2 Chain of SU(2) -anyons
4
Let us consider the Klein-type Hamiltinian family [24]
2
N
H = g P(j). (1)
− j i
i=1;j
X
Here, index i enumerates links of the chain, g are the coupling constants, and P(j) are
{ j} i
projectors on states with the spin j.
Insystemswith non-Abelianquasiparticles,stronglycorrelatedstates formsomepartof
the low-energyHilbert space. Since we are primarily interestedin the low-energylimit, this
property of non-Abelian anyons is an argument for representation of effective Hamiltonian
(1) by means of operators with two eigenvalues corresponding to the ground and excitated
states. Besides, when the considered particles are spatially separated, the low-energy space
proves to be degenerate; constrains are imposed on the degree of degeneracy, which are
characterized by topological quantum numbers.
Oneofthemethodsfordescribingthetopologyeffectsrelatedtobraidingofquasiparticle
worldlines is based on the use of Temperley-Lieb algebra operators. The Temperley-Lib
algebra operators e satisfy the commutation relations [25]
i
e2 =de ,
i i
e e e =e ,e e =e e , (j =i 1).
i i+1 i i i j j i
6 ±
This implies that the operators P = e /d are projectors: P2 = P . Here, the isotopic
i i i i
parameter is d=2cos( π ).
k+2
The Hamiltonian for a small anisotropy parameter x can be obtained using the transfer
matrix T =τ1τ3 τ2N−1I0τ2τ4 τ2N−2I2N with τi =Ii+xei and the identity matrix Ii.
··· ···
The Temperley-Lieb algebra operators e are defined here [26, 27, 28] as
i
j′
e[i]|ji−1jiji+1i = Xj′i (cid:16)e[i]jjii−+11(cid:17)jii|ji−1j′iji+1i
e[i]ji+1 j′i = δ Sj0iSj0′i , (2)
(cid:16) ji−1(cid:17)ji ji−1,ji+1 vuuSj0i−1Sj0i+1
t
where Sj′ = 2 sin π(2j+1)(2j′+1) is the modular S-matrix SU(2) of the conformal
j k+2 k+2 k
field theory. q (cid:16) (cid:17)
It follows from expressions (2) that in the considered weight representation of the
operators e , anyon degrees of freedom are distributed over the three neighboring links
i
(i 1,i,i+1) of the chain. Their filling by some value of the spin a from the anyon-type
−
set j = 0,1/2,1,...,k/2 will be used below as the collective label of the quantum states
ji−1,ji,ji+1 >. The relationship between the admissible values from the set a,b,c of the
| { }
angular momentum values is determined by the fusion rules (tensor product) of the two
anyon states in the SU(2) theory:
k
j j = j j +(j j +1)+...+min(j +j ,k j j ). (3)
1 2 1 2 1 2 1 2 1 2
× | − | | − | − −
Composition of the fields truncated by the level k after fusion is such that instead of the
standard result in the case of the SU(2) tensor product, we have a smaller number of fields
at the output.
3
The fusion rules a b= Nc c of primary fields for possible values of the index j for
× c ab
k = 4, the quantum dimensions d = [2j +1] = q(j+1/2)−q−(j+1/2) = sin[π(2j+1)]/sin[ π ]
P j q q1/2−q−1/2 k+2 k+2
(forq =e2πi/(k+2))beingthelargesteigenvaluesofthematrix(N )c withthetotalquantum
a b
dimension D= d2 = √k+22 =2√3, and the conformalscale dimensions h = j(j+1)
a a sin( π ) j k+2
k+2
determining theptoPpologicalspin θj =e2πihj are summarized in the table
j =0 d =1 h =0
0 0
j =1/2 d =√3 h = 1
1/2 1/2 8
j =1 d =2 h = 1
1 1 3
j =3/2 d =√3 h = 5
3/2 3/2 8
j =2 d =1 h =1
2 2
1/2 0=1/2
×
1/2 1/2=0+1 1 0=1
× ×
1/2 1=1/2+3/2 1 1=0+1+2 3/2 0=3/2
× × ×
1/2 3/2=1+2 1 3/2=1/2+3/2 3/2 3/2=0+1 2 0=2
× × × ×
1/2 2=3/2 1 2=1 3/2 2=1/2 2 2=0
× × × ×
Table 1: Anyontypesnumbered byvaluesoftheindexj,quantum andconformaldimensions,andfusion
rulesfortheSU(2)4 theory.
When two fields fuse, the vectorspace Vc ofthe dimension dimVc equalto Nc occurs.
ab ab ab
In our case, Nc = 0,1. For the non-Abelian states indexed in the table by the spin values
ab
j, d > 1. Non-Abelian anyons are also characterized by a multiplicity of allowed channels
j
of fusion (splitting) ( Nc 2 for some value of the spin b).
c ab ≥
Splitting of the anyon b into anyon states with the spins a,b,c corresponds to the space
P
Vabc, which can be represented in the form of tensor products of two splitted anyon spaces
d
by matching appropriate pairs of indices. This can be done by two isomorphic ways:
Vabc = Vab Vec = Vaf Vbc, (4)
d ∼ e ⊗ d ∼ d ⊗ f
e f
M M
To introduce the notion of associativity at the level of splitting spaces, it is necessary
to describe the set of unitary isomorphismsbetween different decompositions, whichcan be
considered as a change of the basis. These isomorphic transformations can be written as a
diagram using the so-called F-matrix:
(5)
S
F
The F-matrix satisfies the pentagonal equation
FmlqFjip Fjsn =FjipFriq , (6)
kpn mns lkr qkr mls
n
X
4
which has the solution [29] proportional to the q-deformed analog of the 6j-symbols:
j j j
Fj1j2j3 =( 1)j1+j2+j3+j [2j +1] [2j +1] 1 2 12 , (7)
jj12j23 − q 12 q 23 q(cid:26) j3 j j23 (cid:27)q
j j j
1 2 12 =∆(j ,j ,j )∆(j ,j ,j)∆(j ,j ,j )∆(j ,j ,j)
j j j 1 2 12 12 3 2 3 23 1 23
(cid:26) 3 23 (cid:27)q
( 1)n[n+1] !
− q
× n ([n−j1−j2−j12]q![n−j12−j3−j]q![n−j2−j3−j23]q![n−j1−j23−j]q!
X
1
,
× [j1+j2+j3+j−n]q![j1+j12+j3+j23−n]q![j2+j12+j+j23−n]q!)
[ j +j +j ] ![j j +j ] ![j +j j ] ! n
∆(j ,j ,j )= − 1 2 3 q 1− 2 3 q 1 2− 3 q , [n] ! [m] .
1 2 3 s [j1+j2+j3+1]q! q ≡m=1 q
Y
In the following two sections,we will calculate the F-matrixdifferent from zeroor unity
by using Eq. (7). We have not presented here the expressions for the braiding Rc -matrix
ab
of two anyons and the hexagon equation, because the latter is a consequence [30] of the
pentagon equation.
3 The Hamiltonian and the Hilbert space of states
When constructing effective Hamiltonian (1), we employ the representation[5]
ha′,f′,d′|P(j)|a,e,di=Fda′′jbfc′Fdaebjcδa,a′δd,d′ (8)
for projectors in the channel with the total spin j, using the F-matrix. In Eq. (8), the
summation over repetitive indices is absent. In the case of the Heisenberg Hamiltonian
H = (S S ) for the spin equal to unity, we have (S S )=3P(2)+P(1) 2. For the
i i i+1 i i+1 i i −
Affleck-Kennedy-Lieb-Tasakimodel H = P(2). In our case, the superscripts (b,c) in (8)
P i i
will acquire the value 1/2. In the sector with an integer value of the indices, these labels of
P
the anyon species will be as follows: b=c=1.
Let us consider first the half-integer values b = c = 1/2 of the anyon index in the
F-matrix. In this case, the admissible state space a,f,d is fourteen-dimensional and
{| i}
represents the following state set shown in the right-hand side of Eq. (5):
0,0,0 , 0,1,1 , 1/2,0,1/2 , 1/2,1,1/2 , 1/2,1,3/2 , 1,1,0 , 1,0,1 , (9)
{| i | i | i | i | i | i | i
1,1,1 , 1,1,2 , 3/2,1,1/2 , 3/2,0,3/2 , 3/2,1,3/2 , 2,1,1 , 2,0,2 .
| i | i | i | i | i | i | i}
TheHilbertspace a,e,d belongingtotheleft-handsideofEq. (5)consistsofthefollowing
{| i}
states:
5
0,1/2,0 , 0,1/2,1 , 1/2,0,1/2 , 1/2,1,1/2 , 1/2,1,3/2 , 1,1/2,0 , 1,1/2,1 , (10)
{| i | i | i | i | i | i | i
1,3/2,1 , 1,3/2,2 , 3/2,1,1/2 , 3/2,1,3/2 , 3/2,2,3/2 , 2,3/2,1 , 2,3/2,2 .
| i | i | i | i | i | i | i}
Let us write out the F-matrix components equal to unity:
011 011 111 111
F 22 =F 22 =F222 =F 22 = (11)
010 111 311 011
2 2 2 2
111 311 211 211
F 22 =F222 =F 22 =F 22 =1.
231 111 131 230
2 2 2 2
Inthecaseofk =4andb=c=1/2,anontrivialcontributiontotheF-matrixstructure
is made by three two-dimensionalmatrices M :
j
1 1
M12 =F1221e12f21 = FF2111220100 FF2111220111 = √13(cid:18) √−12 √12 (cid:19) , (12)
2 2
 
M1 =F11e12f21 = FF111112300 FF111121311 != √12(cid:18) −11 11 (cid:19) , (13)
2 2
3 3
M32 =F3223e12f21 = FF2333221200 FF2333221211 =r23 3−12/14 3112/4 ! . (14)
2 2
 
For simplicity of the notation, in the three latter formulas we have omitted fixed indices
of the elements of the F-matrices. In calculation of the elements of the matrices in these
formulas, we used explicit expression (7) for the q-deformed values of the 6j-symbols.
Let us employ the given values of the F-matrix to calculate the projectors by means of
the F-matrix. In the case of b = c = 1/2 and the level k = 4, the following two projectors
prove to be nonzero:
P(0) =diag (0,0,A,0,0,0,0,0,0,0,0,0,0), (15)
P(1) =diag (0,0,B,1,0,0,0,0,1,C,0,0), (16)
where the 2 2 matrices A,B,C are present in diagonals and have the forms
×
1 2 1 1
F2 F2 F2 1 1 √2
A= 2100 2100 2101 = − , (17)
 F(cid:16)12110F1(cid:17)2100 F11201F11210  3(cid:18) −√2 2 (cid:19)
2 2 2 2
 
1 1 1 1
F1210F1201 F1210F1211 1 2 √2
B = 21 21 2 1 2 2 = 3 √2 1 , (18)
F1211F1210 F1211 (cid:18) (cid:19)
2 2 2
 (cid:16) (cid:17) 
3 3 3 3
F3210F3211 F3210F3221 1 √2 2
C = 2 3 2 2 23 23 = 3 −1 √−2 . (19)
F3211 F3211F3221 (cid:18) (cid:19)
2 2 2
 (cid:16) (cid:17) 
6
The enumerationin Eqs. (15), (16) correspondsto listing ofstates in Eq. (10). As a result,
the Hamiltonian of the chain of the SU(2) anyons is as follows:
4
N
H = g P(0)+g P(1) (20)
− 0 i 1 i
Xi=1(cid:16) (cid:17)
with projectors from Eqs. (15), (16).
4 Integer anyon index
In this section, we shall consider integer values of the index c = b = 1 of anyon types. In
this case, the admissible state space a,f,d is nineteen-dimensional and represents the
{| i}
state set shown in the right-hand side of Eq. (5):
0,0,0 , 0,1,1 , 0,2,2 , 1/2,0,1/2 , 1/2,1,1/2 , 1/2,1,3/2 , (21)
{| i | i | i | i | i | i
1/2,2,3/2 , 1,0,1 , 1,1,1 , 1,2,1 , 1,1,0 , 1,1,2 ,
| i | i | i | i | i | i
3/2,1,1/2 , 3/2,2,1/2 , 3/2,0,3/2 , 3/2,1,3/2 ,
| i | i | i | i
2,2,0 , 2,1,1 , 2,0,2 .
| i | i | i}
The Hilbert state space a,e,d shown in the left-hand side of Eq. (5) consists of the
{| i}
following states:
0,1,0 , 0,1,1 , 0,1,2 , 1/2,1/2,1/2 , 1/2,3/2,1/2 , 1/2,1/2,3/2 , (22)
{| i | i | i | i | i | i
1/2,3/2,3/2 , 1,0,1 , 1,1,1 , 1,2,1 , 1,1,0 , 1,1,2 ,
| i | i | i | i | i | i
3/2,1/2,1/2 , 3/2,3/2,1/2 , 3/2,1/2,3/2 , 3/2,3/2,3/2 ,
| i | i | i | i
2,1,0 , 2,1,1 , 2,1,2 .
| i | i | i}
The one-dimensional F-matrices equal to unity correspond to the following sequence of
indices:
F011 =F02121 =F211221 =F12121 = (23)
010 111 311 011
2 2 2
111 311 211 211
F 22 =F222 =F 22 =F 22 =1.
231 111 131 230
2 2 2 2
The four two-dimensional F-matrices Mˆ are equal to
j
1 1
Mˆ12 =F2121e1f1 = FF21122113200 FF21122113211 = √12(cid:18) −11 11 (cid:19) , (24)
22 22
 
1 1
Mˆ′ =F2111 = F232121 F232122 = 1 −1 1 , (25)
12 23ef  F32131 F32132  √2(cid:18) 1 1 (cid:19)
22 22
 
3 3
Mˆ32 =F2123e1f1 = FF21122313211 FF21122313222 = √12(cid:18) −11 11 (cid:19) , (26)
22 22
 
7
3 3
Mˆ′ =F2311 = F232210 F232211 = 1 1 −1 . (27)
23 32ef  F33230 F33231  √2(cid:18) −1 −1 (cid:19)
22 22
 
The F-symbol also contains a three-dimensional matrix having the form
F1 F1 F1 1 √2 1
100 101 102 1 −
Mˆ =F111 = F1 F1 F1 = √2 0 √2 . (28)
1 1ef  110 111 112  2 − 
F1 F1 F1 1 √2 1
120 121 122
   
The projectorsPˆ(j) with j =0,1,2ande=0,1,2,f =0,1,2determined by Eq. (28) in
the three-dimensional subspace are equal to
F F F F F F 1 √2 1
00 00 00 01 00 02 1 −
Pˆ(0) =F111F111 = F F F F F F = √2 2 √2 , (29)
1e0 10f  10 00 10 01 10 02  4 − − 
F F F F F F 1 √2 1
20 00 20 01 20 02
−
   
F F F F F F 1 0 1
01 10 01 11 01 12 1 −
Pˆ(1) =F111F111 = F F F F F F = 0 0 0 , (30)
1e1 11f  11 10 11 11 11 12  2 
F F F F F F 1 0 1
21 10 21 11 21 12
−
   
F F F F F F 1 √2 1
02 20 02 21 02 22 1
Pˆ(2) =F111F111 = F F F F F F = √2 2 √2 . (31)
1e2 12f  12 20 12 21 12 22  4 
F F F F F F 1 √2 1
22 20 22 21 22 22
   
The projectors Pˆ(j) dependent on the matrix Mˆ in two-dimensional subspaces with half-
j
integer indices a and d are zero: Pˆ(1/2) =Pˆ′(1/2) =Pˆ(3/2) =Pˆ′(3/2) =0.
Asaresult,fork =4anda=b=1wehaveP¯(j) =0forhalf-integervaluesoftheanyon
type and the following nonzero projectors in the integer-valued sector
P¯(0) =diag 0,0,0,0,0,0,0,Pˆ(0),0,0,0,0,0,0,0,0,0 , (32)
(cid:16) (cid:17)
P¯(1) =diag 0,1,0,0,0,0,0,Pˆ(1),1,1,0,0,0,0,0,1,0 , (33)
(cid:16) (cid:17)
P¯(2) =diag 0,0,0,0,0,0,0,Pˆ(2),0,0,0,0,0,0,0,0,0 . (34)
(cid:16) (cid:17)
where the three-dimensional matrices Pˆ(j) are located on diagonals.
All aforesaid means that the Hamiltonian of the chain of the SU(2) -anyons for the
4
ladder indices b=c=1 has the form
N
H = g P¯(0)+g P¯(1)+g P¯(2) (35)
− 0 i 1 i 2 i
Xi=1(cid:16) (cid:17)
with projectors from Eqs. (32), (33), and (34).
8
5 Conclusion
The features of the Hamiltonian dynamics in the case of the SU(2) theory can be revealed
4
by comparing it with the SU(2) theory [5] of Fibonacci anyons and the SU(2) theory
3 5
[6]. In the latter case for b = c = 1, of interest is the absence into total projectors (32)-
(34) of the contribution made by projectors in the two-dimensional subspace. Incontrast
with the Hamiltonian of Fibonacci anyons, the one-dimensional projector is absent in the
Hamiltonianofthe SU(2) theory,whenthe zeroenergyis assignedto the state with j =1.
4
For the coinciding coupling constants in the Hamiltonian (35), we have the model de-
scribing the critical state; in the continuous limit, it coincides with the rational conformal
field theory with the central charge c = 4/5. For the opposite sign of the coupling con-
stants, the considered case k = 4 in the continuous limit corresponds to the Z theory of
4
parafermions with the central charge c = 1. If the coupling constants do not coincide, we
can analyzethe type of gapped phase states by assigningthe zero energyto the phase state
with the spin j =1 in (20), i.e., by setting g =0 in (20) and g =0 with the spin j =0 in
1 0
(35). Thiscanbedonebychangingtheangleθforparametrizationofthecouplingconstants
g =sinθ,g =cosθ in(35)asitwasdoneinRefs. [7,6]. We planto considerthis problem
1 2
in a separatepaper. In this paper,we focus on the study of the Hamiltonianrepresentation
of exact integrable anyon systems in the form of projectors in a more general formulation.
When discussing weight representation (2) of the Temperley-Lieb algebra, we have al-
ready mentioned that the Hamiltonian in the extremely anisotropic case x 0 at the
→
boundary of some regimes of the exact integrable RSOS-model [31, 32] can be obtained
using the R-matrix. This matrix satisfies the Yang-Baxter equation
R R R =R R R (36)
12 13 23 23 13 12
In this equation, we employed standardnotations R =R R 1, etc. with the R matrix
→ 12 ⊗ ⊗
realizing self-mapping of the space V V ֒ in itself.
⊗
In its turn, the R-matrix of the models describing topological ordered phase states can
be constructed using the solution [33]
R = Ft4 −1F Ft F−1 t2 (37)
12,34 14 13 24 23
of the pentagon equation (cid:0) (cid:1) (cid:0) (cid:1)
F F F =F F . (38)
12 13 23 23 12
Here, R W W, where W = V V⋆, and the indices t and t mean, respectively,
12,34 i
∈ ⊗ ⊗
complete and partial transpositions in the i-th space. The indices of the F-matrices in
Eqs. (37) and (38) belong to the four faces of tetrahedron. In the general case, the F-
matrix also depends on the spectral parameter x [33, 34], the small value of which yields
the Hamiltonian. For example, the case [34]
R(x)=[x 1] +[x] e (39)
q q 1
−
yields the known answers [18, 35].
Let us consider the (2+1)D models and the tensor T which depends on the indices
ijkl
(i,j,k,l) belonging to the tetrahedron faces. Each of these indices is the collective label
comprising variables from the sets (a,b,c,d,e,f), (i ,j ,k ,l ) with s = 1,2,3,4 of the
s s s s
9
indicesbelongingrespectivelytoedgesandverticesofeachtetrahedronface. Employingthe
T -tensors, we can find the partition sum [36, 37]
ijkl
Z =Tr e−βH = T T T T =tTr T (40)
jfei hgjk qklr lits i
··· ⊗
ijkl
X
by computing the tensor trace tTr T. To carry out the calulations, anzats for the T -
i ijkl
⊗
tensor can be written as
T =Fabc λ(i ,j ,k ) λ(i ,j ,l ) λ(i ,k ,l ) λ(j ,k ,l ). (41)
[i][j][k][l] def · 1 1 1 · 2 2 2 · 3 3 3 · 4 4 4
Since we are usually interested in the system behavior at the distances much larger
than the lattice constant, now we consider also the final stage of the renormalizationgroup
flow, which does not contain data about the distribution of the degrees of freedom at small
distances. The fixed surface at this final stage of the renormalization group flow in the
terms of the functions from Eq. (41) corresponds to the condition that all functions of
local variables tend to be constant. In other words, all the functions are λ 1 up to
→
normalization. This conjecture means that topological universality of the behavior at large
distancesoccursintheinfraredlimitduetospecialcontributiontothepartitionsum,which
depends only on the F-matrix. Practicalimplementationofthe computationprogramofthe
resultof suchnormalizationgroupflow is relatedto the solutionof the problemoffinding a
convenient relation between the variables belonging to tetrahedron faces and the variables
belonging to tetrahedron edges and vertices. We plan to consider this problem in future.
6 Acknowledgement
We are grateful to R.M. Kashaev and V.G. Turaev for useful discussions. The authors are
also grateful to INFN for partial support in the framework of the LE41 project, to the
AbdusSalamInternationalCentreforTheoreticalPhysicsinTriesteandtheGalileoGaliley
Institute for Theoretical Physics in Florence for hospitality. This work was supported in
part by a grantof the E.I.N.S.T.E.IN consortium(L.M. and A.P.), the Russian Foundation
for Basic Research (Grants No. 09-01-92426 and 09-01-00268, V.V. and А.P.), and the
program"Basicproblems of nonlineardynamics" ofthe Presidiumof the RussianAcademy
of Sciences.
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England, 1999.
[4] Preskill, J.: Topological quantum computation, lecture notes (2004).
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