Topological Properties of Tensor Network States From Their Local Gauge and Local Symmetry Structures

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by  Brian Swingle

Topological Properties of Tensor Network States
From Their Local Gauge and Local Symmetry Structures
Brian Swingle1 and Xiao-Gang Wen1
1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Tensor network states are capable of describing many-body systems with complex quantum en-
tanglement, including systems with non-trivial topological order. In this paper, we study methods
to calculate the topological properties of a tensor network state from the tensors that form the
state. Motivated by the concepts of gauge group and projective symmetry group in the slave-
particle/projective construction,andbythelow-dimensional gauge-likesymmetriesofsomeexactly
0 solvableHamiltonians,westudythed-dimensionalgaugestructureandthed-dimensionalsymmetry
1 structure of a tensor network state, where d ≤ dspace with dspace the dimension of space. The d-
0
dimensional gaugestructureandd-dimensional symmetrystructureallow ustocalculate thestring
2
operatorsandd-braneoperatorsofthetensornetworkstate. Thisinturnallowsustocalculatemany

n topologicalpropertiesofthetensornetworkstate,suchasgroundstatedegeneracyandquasiparticle
a statistics.
J

5
2 I. INTRODUCTION f g
e h
i j j
l] Topological order[1–3] is a new kind of ordering in m1 m2 m1 i m2 k
e
many-body quantum states. It represents new states of
r- quantum matter beyond the symmetry breaking states i j k t l l k q
t
s characterized by Landau symmetry breaking theory. m4 s m3 r
.
t This new kind of order and the new states of matter as- (a) (b)
a
sociated with it open up a whole new research direction
m
in condensed matter physics.[4–18] Intuitively, topologi-
d- calordercorrespondstoapatternoflongrangequantum FIG. 1: Tensor-network – a graphical representation of the
n entanglement in the ground state. tensor network wave function (1), (a) on a 1D chain or (b)
o Traditional mean-field approaches to calculating the on a 2D square lattice. The indices on thelinks are summed
c T = 0 phase diagram of a quantum system are based over.
[
on direct product states which have no long range quan-

1 tum entanglement. These approaches seem to exclude
ally, the d-brane operators) of the TNS, which in turn
v topological states from the very beginning. To study a
allows us to calculate the topological properties of the
7 phase diagram that may contain topological states, we
1 TNS.
have to find a way write down states with non-trivial
5
long range entanglement. The tensor network state
4
. (TNS)[19–22] method is one way to produce states with II. TNS, IDEAL WAVE FUNCTIONS, AND
1
longrangeentanglement.[23–25]Wecandevelopamean- IDEAL HAMILTONIANS
0
field approach for topologically ordered states based on
0
TNS.[26–31]
1 Whatis aTNS?Asanexample,letus consideraTNS
v: Physically, topological order is characterized and de- on a square lattice, where the physical states living on
i finedthroughmeasurablequantities,suchasgroundstate each vertex i are labeled by mi. The TNS is defined by
X degeneracy on a torus or other compact space[1, 32] and thefollowingexpressionforthemany-bodywavefunction
r fractionalized quantum numbers and statistics of quasi- Ψ({m }):
a i
particles [33, 34]. If we claim that TNS can describe
tlaotpeotlhogoisceamlsetaastuersa,bthleetnopitoilsogniactaulrqaulatnotuasmknhuomwbteorscafrlcoum- ΨT({mi})= X Tim,e1jifTjm,j2hkgTkm,l3qrkTlm,tl4si··· (1)
ijkl···
the TNS representation. In this paper, we will address
this question. We will showhow to calculate measurable Here Tmi is a complex tensor on vertex i with one
i,ejfi
topological properties of a TNS. physicalindexm andfourinnerindicesi,j,k,l,···. The
i
Ourapproachismotivatedbythegaugegroupandpro- physical index runs over the number of physical states
jective symmetry group in the slave-particle/projective D on each vertex, and the internal indices runs over D
p
construction [35, 36] and by the low-dimensional gauge- values. Note that tensors on different vertices can be
likesymmetriesofsomeexactlysolubleHamiltonians.[37] different. The TNS can be represented graphically as in
We will introduce the local gauge and local symmetry Fig. 1.
structures of a TNS from the tensors in the TNS. This We see that a TNS is characterized by tensors Tm
i,ijkl
allows us to obtain the string operators (or more gener- and by a network specifying the connectivity. Can we
2
A−1 A
i j
FIG. 2: (Color online) The patch P and the patch P . P FIG.3: Agraphicalrepresentationofagaugetransformation.
i j i
and P only differby a displacement.
j
calculate the topological properties of the many body
state Ψ ({m }) from its defining tensors Tm ? To an-
T i i,ijkl
swer this question, we need to define the problem more ξ
completely by introducing a Hamiltonian HT such that ξ
Ψ ({m }) is the exact ground state of H . We will call
T i T
H an “ideal” Hamiltonian and Ψ ({m }) an “ideal”
T T i
wave function.
FIG. 4: (Color online) A 1-brane and a 2-brane.
The Hamiltonian H has the following local form
T
HT =XOPi (2) is non-trivial in the large patch limit (i.e. 1−P 6= 0)
where OPi is an operator thait acts only on states in the because NPi ≤D2LPi <DpαL2Pi. If we choose Pi
patch P (see Fig. 2). How can we choose the operator
i
OPi sothatΨT({mi})is,hopefully,theonlygroundstate HT =X(1−PPi), (7)
of H ? In the following, we will describe a construction i
T
proposed in Ref. 38.
the TNS Ψ will be the exact ground state of H . We
Let ρ be the density matrix of the state Ψ ({m }) T T
Pi T i hope that in the large patch limit, the projection P is
onpatchPi obtainedby tracingoutthe physicaldegrees highlyrestrictiveandΨ is moreorless the onlygroPuind
of freedom outside P : T
i state.
ρPi =TrP¯i|ΨTihΨT| (3) sorNsoTwmthe,pwreobcalenmdceatenrmbeinbeeattner“iddeefianl”edH.aFmriolmtontihaentHen-
whereP¯ representtheregionoutsideofP andTr isthe i,ijkl T
i i A (7). Assuming H is gapped, the topological properties
T
trace over all the states in region A. From the structure
of the TNS are just those of the topological phase of
of the TNS, we can deduce that ρ must have the form
Pi HT. However, in this paper we will not try to calculate
the topological properties by directly diagonalizing H .
D2LPi T
This task is generically difficult because [P ,P ] 6= 0.
ρPi = X |ψaihφa|, (4) Instead, we will find a way to calculate thePitopPojlogical
a=1 properties from the structure of the tensors Tm di-
i,ijkl
where|ψ iand|φ iarestatesonthepatchP ,andL is rectly.
a a i Pi
the numberoflinksonthe boundaryofthe patchP (for
i
example L =10for the patchin Fig. 2). Equivalently,
Pi
ρPi must have the form III. LOCAL GAUGE STRUCTURE OF A TNS
NPi
ρPi =Xρa|ϕaihϕa|,. NPi ≤D2LPi, (5) ferOenntetoefntshoresimTpmortacnatnpgroivpeerrtiiseestoof athTeNsaSmisetphhaytsidciaf-l
a=1 i,ijkl
wave function Ψ ({m }). To be more precise, two sets
T i
where |ϕai are normalized states on the patch Pi. This oftensorsTm andT˜m relatedby a“gaugetransfor-
i,ijkl i,ijkl
means that ρ can be constructed from only N ≤
Pi Pi mation” (GT)
D2LPi states on patch Pi. Since the number of states
onPi growslikeDpαL2Pi, α=O(1)foralargepatch(here ΛiT˜im,l′r′d′u′ (8)
d =2), the following projection operator
space = X(Ai−x,i)l′l(Ai+x,i)r′r(Ai+y,i)u′u(Ai−y,i)d′dTim,lrdu,
NPi lrud
PPi =X|ϕaihϕa| (6) X(Aj,i)kj(Ai,j)ki =Λijδij (9)
a=1
k
3
giverisetothesamewavefunctionΨ ({m })uptosome The d-fSG is very similar to the “low-dimensional
T i
scaling factors (see Fig. 3). In the above Λ and Λ are gauge-like symmetries” introduced in Ref. 37. A differ-
i ij
scalingfactorsthatdonotdependontheindicesi,j,k,.... encebetweenthe twoconcepts isthatd-fSGisa symme-
Here A and A are D × D invertible matrices de- try of a ground state while the “low-dimensional gauge-
i,j j,i
finedonthe linkshi,ji. AGT is calleda “d-dimensional like symmetries” are symmetries of an exactly solvable
gauge transformation” (d-GT) if A 6= 1 only on links Hamiltonian. Aspointed inRef.37, the low-dimensional
j,i
withinwitha“d-brane”. By“d-brane”wemeanaregion gauge-like symmetries are useful tools to calculate the
near a d-dimensionalsubspace (i.e. a d-dimensionalsub- topological properties of a model Hamiltonian. Simi-
spacewithafinite “thickness”ξ asillustratedinFig. 4). larly, d-fSG should be a useful tool to calculate topo-
Notethatthe“thickness”ξ isfixedforaTNS,regardless logical properties from a ground state.
the size of the d-dimensional subspace and the value of We note that if d-fSG is non trivial, then we can use
d. The size L of the d-dimensional subspace is typically elements in d-fSG on different d-branes to construct a
much bigger than ξ: ξ/L→0. non-trivial d′-fSG for all d′ >d. Thus, to revealthe new
Using d-GT we can introduce the concept of a “d- structurethatappearsineachdimensiond,weintroduce
dimensional full invariant gauge group” (d-fIGG) as in the “d-dimensional symmetry group” d-SG. As in the
Ref. 35. d-fIGG is a property of the tensors Tm and a previous section, the d-fSG contains a normal subgroup
i,ijkl
fixed d-brane. d-fIGG is a groupformed by all the d-GT thatisformedbyelementsin(d−1)-fSG.Wedefined-SG
in the d-brane that leave Tm invariantup to a scaling as
i,ijkl
factor.
Wenotethatifd-fIGGisnontrivialthenwecanused- d-SG≡d-fSG/(d−1)-fSG (12)
GTind-fIGGwithdifferentd-branestoconstructanon-
trivial d′-fIGG for all d′ > d. Thus, to reveal the new We can make contact with the string-net framework
gauge structure that appears at every dimension d, we by noticing that the elements of 1-SG are actually the
willintroducethe“d-dimensionalinvariantgaugegroup” ideal string operators [12, 16, 39] that satisfy a “zero
(d-IGG). The d-fIGG containsa normalsubgroupwhose law”.[40] Similarly, the elements of d-SG are the ideal
elements are in (d−1)-fIGG. The d-IGG is defined as d-brane operators that satisfy a “zero law”.[41, 42] So
d-SG and “low-dimensional gauge-like symmetries” are
d-IGG≡d-fIGG/(d−1)-fIGG (10)
closely related to the string operators and the d-brane
The d-IGG for the tensors Tm defined this way may operators that play a key role in topologically ordered
i,ijkl
depend on the topology of the d-brane. states.[14–16]
As an example of this notation, we note that for a The above discussion of d-SG (or “low-dimensional
d -dimensional TNS, the d -IGG is like the group gauge-likesymmetries”)isforagenerictopologicalstate.
space space
of uniform global gauge transformations in a gauge the- Now the question is how to calculate d-SG for a TNS?
ory. Naively,wewouldliketosaythatd-SGareformedbylo-
calphysicaltransformationsthatleavethe tensorsTm
i,ijkl
invariant. But due to the gauge structure of the TNS,
IV. LOCAL SYMMETRY STRUCTURE OF A the invariance of the TNS and the invariance of the ten-
TNS sors Tm are not equivalent. Instead, d-SG are formed
i,ijkl
by local physical transformations that leave the tensors
Intheprevioussectionwediscussedgaugetransforma- Tim,ijkl invariant up to gauge transformations. Thus the
tionsthatleavethetensorsoftheTNSinvariant(uptoa gauge transformations that change the tensors play an
scaling factor). These gauge transformations define the important role in calculating d-SG for a TNS.
d-fIGG. In this section, we will discuss gauge transfor- In the above, we have defined a notion of d-IGG and
mations that do change the tensors Tm . d-SG for a TNS. We will see that both d-IGG and d-
i,ijkl
Letusfirstintroducesomeconcepts. A“localphysical SG play a very important role in characterizing a TNS.
transformation” of a TNS is generated by unitary D × Many (if not all) topologicalproperties of a TNS can be
p
DpmatricesSi,mm′ actingonthephysicalindicesm(here calculatedfromits d-IGGandd-SG.Inthe following,we
D is the range of the physical index m): willcarryoutthethisprogramforthesimplestTNSwith
p
a non-trivial topological order.
Tim,ijkl →XSi,mm′Tim,ij′kl (11)
m′
V. Z TOPOLOGICALLY ORDERED STATE
A“d-dimensionallocalphysicaltransformation”(d-lPT) 2
is a “local physical transformation” that is non-trivial
(i.e. Si,mm′ 6= δmm′) only on a d-brane. A “d- A. Z2 String Condensed State and its Ideal
dimensionalsymmetry transformation”(d-ST) of a TNS Hamiltonian
is a d-lPT that leave the TNS invariant. All the d-ST
on a fixed d-brane from a group which will be called the The simplest topological phase that we can consider
“d-dimensional full symmetry group” d-fSG. is the Z topological phase.[8, 9] Such a phase can
2
4
These operators can be taken to be
σz leg
I σx σx σx X(C)= Yσix (14)
σzσz p i∈C
edgeσx σx whereC isacurverunningalongtheedgesofthelattice.
i σx The operatorX(C) is a string creationoperatorbecause
σx acts as a spin flip operator for eigenstates of σz =m
sothatwearesimplyflipping the spins(adding astring)
alongthecurveC. NotethattheoperatorX(C)canalso
FIG.5: (Color on line) AKagome lattice viewed as thelinks remove a string because the string is its own anti-string
of a honeycomb lattice. meaning that two strings can annihilate each other.
Thegroundstate|Ωihastheremarkablefeaturethatit
is an eigenstate of X(C) for all C with eigenvalue 1. We
say that the strings created by X(C) have condensed in
the state |Ωi. Another useful point of view comes from
σx=−1 Open
Strings thinking about the Z2 condensed state in terms of Z2
gauge theory where the operator X(C) is nothing but a
σx=1 U Wilson-Wegnerloopoperator. Inthestringcondensedor
deconfinedphasethe Wilson-WegnerloopX(C)satisfies
Closed g a “zero law” as opposed to an area or perimeter law.[40]
Strings This “zero law” expresses the existence of strings at all
scales in the ground state.
FIG. 6: (Color online) A spin configuration of σz = 1 (filled For an open string operator that satisfies the “zero
dot) and σz = −1 (open dot) on the Kagome lattice can be law”, the action of the string operator will create a pair
viewed as a string configuration on the honeycomb lattice. of excitations at its two ends. Such excitations will in
The end of an open string costs an energy U. The closed-
generalhave fractionalstatistics andfractionalquantum
string sector has an energy gap g. The dashed lines describe
numbers. FortheX(C)string,theexcitationsatitsends
theduallatticeofthehoneycomblattice. Thethickbluelines
correspond to Z charge. Thus we will call the X(C)
arestringsinthehoneycomblattice. Thethickgreenlinesare 2
string an electric string.
strings in thedual lattice.
There is another string operator that we canconsider,
which we define as
blaettriceea:l[i1z2e,d1t6h]rough a spin-1/2 model on the Kagome Z(C∗)= Y σiz (15)
i∈C∗
H =UX(1−QI)+gX(1−Bp) whereC∗ isacurvealongthelinksoftheduallattice(see
Figure 6). It is characteristic of the Z string condensed
I p 2
phasethatthese stringsarealsocondensed: Z(C∗)|Ωi=
QI = Y σiz, Bp = Y σix. (13) |Ωi for any loop C∗.
legsofI edgesofp Forthe Z(C∗)string,the excitationsatitsends corre-
spond to Z vortices. Thus we will call the Z(C∗) string
2
Here we have viewed the Kagome lattice as the links of a magnetic string.
the honeycomb lattice (see Fig. 5). The vertices of the This was all for the case of a system on an infinite
honeycomb lattice are labeled by I, the links (which are plane where there is a unique ground state. Things
the sites of the Kagome lattice) by i, and the faces by changequalitativelywhenthesystemlivesonamanifold
p. PI sums overall verticesand Pp overall faces. The withnon-trivialtopology. Whenthemanifoldinquestion
Hamiltonian(13)indeed hasthe formofa summationof hasnon-trivial1-cycles(non-contractibleloops)thenthe
projectors as in eqn. (7). string operators corresponding to these non-contractible
Wecaninterpretagivenconfigurationofspinsinterms loops become interesting observables. We will return to
ofstringsbyviewingthe stateσz =1asthe absenceofa the case of non-trivial topology later.
string and the state σz = −1 as the presence of a string
(note the choiceofσz here). The aboveHamiltonianhas
thepropertythat,inthelargeU >0limit,thelowenergy B. Z2 Tensor Network Representation
Hilbert space is the space of all spin configurations that
containonly closedstrings (see Figure 6). Forg >0,the The Z condensed state, and many other topologi-
2
groundstate of H is the equal weighted superpositionof callyorderedstates,canbesystematicallyrepresentedin
allclosedstringstates|Ωi= |Xi. Itisusefulto termsofaTNS.[24,25]Fig. 7and8representthetensor
PXclosed
considerstringoperatorsthataddastringtothesystem. networkonthehoneycomblatticegraphically. Notethat
5
gauge transformations:
(A ) =λ δ , (A ) =λ δ . (17)
a iI ab iI ab Ii ab Ii ab
b
gm These transformations only change the tensors g, T and
T′ by a scaling factor.
Next, let us calculate 2-IGG. We consider the “gauge
T’
T transformations”thatleavegamb onlink-iinvariant,upto
scaling factor:
Λigam′b′ =X(AIi)a′a(AJi)b′bgamb, for m=±1. (18)
ab
Note that I and J label the vertices of the honeycomb
lattice and i is the link that connect the two vertices I
FIG. 7: A tensor network formed by vertices and links. The and J. The above equation requires that AIi) and AJi)
links that connect the dots carry indices a,b,.... Each triva- be diagonal. Using the “local gauge transformation”:
lent vertex represents a T-tensor. The vertices on the A-
tsiucbelsaottnictehe(oBp-esnubdloattstilcaeb(eslheaddbeydIdAot)srIeBpr)erseepnrteTseanbct.TTa′bhc.eTvehre- gamb →Xλ1δa′aλ2δb′bgamb, (19)
dots on the links represent the gm-tensor gamb. In the tensor ab
tracetTr⊗IAT⊗IBT′⊗igmi,thea,b,c,...indicesonthelinks we can fix (A ) =(A ) =1 and
thatconnectthedotsaresummedover. Thetensortracepro- Ii 00 Ji 00
duces a complex number Φ(m1,m2,...) that depends on mi
which can beviewed as a wave function. (AIi)11(AJi)11 =1. (20)
b From eqn. (9), we find that A must be also diagonal.
iI
Using the 0-GT in 0-IGG=0-fIGG we can transform the
c a a diagonal A into the following form
A B iI
m
a (AiI)00 =1, (AiI)11 =1/(AIi)11. (21)
(a) (b)
A must leave T invariant and thus
iI
aFnIGd.(b8): tThheeggmra-tpehniscorregpamrAesbBen.tation of (a) theT-tensor,Tabc, Ta′b′c′ =Xabc(AiI)a′a(AjI)b′b(AkI)c′cTabc, (22)
where i, j, and k are the three links that connect to the
the tensor network is formed by three tensors Tabc, Ta′bc, vertex I. The above gives us
and gm. To describe the Z condensed state, we choose
ab 2
D =2 and (A ) (A ) =(A ) (A ) =(A ) (A ) =1.
iI 11 jI 11 jI 11 kI 11 kI 11 iI 11
(23)
gm=1 =1, gm=−1 =1, other gm =0;
00 11 ab
′
T =T =1 if a+b+c=even, Such a non-linear equation gives two sets of solutions
abc abc
′
T =T =0 if a+b+c=odd. (16)
abc abc
A =A =1, (24)
iI Ii
where a,b,c=0,1.
In the following, we will calculate the string operators and
directly from the above tensors using the d-IGG and d-
SG of the tensor network. This allows us to understand A =A =σz. (25)
iI Ii
the topological properties of the TNS directly from its
defining tensors. wherewehaveusedeqn.(21). So2-IGGoftheZ tensor
2
network contains only two elements given by the above
two expressions. Such a 2-IGG is a Z group.
2
C. d-IGG of the Z2 Tensor Network Based on this discussion, we see that we can ob-
tain the 2-IGG from the 2-fIGG by removing the 0-GT,
What is the d-IGG of the above Z tensor network? i.e. 2-IGG = 2-fIGG/0-fIGG. This implies that the 1-
2
First, we find that the 0-IGG is formed by the following IGG is the trivial group containing only the identity.
6
is not an elements of 1-SG. Such an action will change
the state, but not the energy. Applying such a Z(C∗)
string operatorto the Z condensedstate ontorus along
2
the x- and y-directions will give us the four degenerate
ground state of the Z condensed state on a torus. This
2
is how do wecalculate the groundstate degeneracyfrom
the tensors in the TNS.
We would like to stress that the Z(C∗) string opera-
tor is directly obtained from the Z gauge transforma-
2
tion. The Z group structure of the gauge group also
2
determines the Z structure of the Z(C∗) string opera-
2
tor: [Z(C∗)]2 =1.
C* g’
g
E. Z d-SG and the Electric String Operator
2
FIG.9: (Coloronline)TheZ−2gaugetransformationA = We have seen that the d-IGG of a TNS can be calcu-
iI
A = σz acts only on the shaded region. The tensor g is lated by finding all the gauge transformations that leave
Ii
changed tog˜on theboundary of the stripe. the tensors invariant (up to a scaling factor). Similarly,
the d-SG of a TNS can be calculated by finding all the
combined local physical transformations and the gauge
D. Ground State Degeneracy from d-IGG
transformations that leave the tensors invariant.
For the case of the Z condensed state, we have
2
Using the d-IGG of the TNS, we can calculate some seen that the magnetic string operator Z(C∗) on a con-
topological properties of the TNS. In the Z2 condensed tractible loop C∗ is a 1-ST (an element of 1-SG). The
state we found that 2-IGG=Z2. Now we will show that tensors in the tensor network are invariant under Z(C∗)
the ground state degeneracy of the Z2 condensed state followed by a 2-GT on the disk D bounded by C∗. The
on a torus is the same as the ground state degeneracyof 1-SG generated by such a 1-ST, Z(C∗), will be called a
Z2 gauge theory on a torus. non-local 1-SG.
If we apply the Z gauge transformation A =A =
2 iI Ii Thereisanothertypeof1-ST,suchthatthetensorsin
σz everywhere,thetensorsT,T′ andg arenotchanged.
thetensornetworkareinvariantunderthe 1-STfollowed
Ifweapply the Z gaugetransformationonlyonastripe
2 by a 1-GT on the same curve on which the 1-ST is de-
(see Fig. 9), then the tensors on the boundary of the
fined. The 1-SG generated by this type of 1-ST will be
stripe will be changed. For the particular choice of the
called a local 1-SG.
stripeinFig. 9thetensorgischangedtog˜onthebound-
The Z condensed state also has a local 1-SG, which
ary of the stripe, where 2
is generated by the electric string operator X(C) (see
g˜m=1 =1, g˜m=−1 =−1, other g˜m =0. (26) eqn. (14)). One can directly check that the tensors g,
00 11 ab
T, and T′ on the tensor network are invariant under the
We note that the change g → g˜ along the boundary of
action of X(C) followed by a gauge transforation on the
the stripe is generatedbythe Z(C∗) stringoperator(see
loop C with
eqn. (15)). Therefore the Z gauge transformation on a
2
finite region defines a type of string operator along the
A =A =σx (27)
boundary. iI Ii
Since the application of the Z gauge transformation
2
does not changethe energyof the state, we find that the on the loop.
application of the Z(C∗) string operator also does not We have seen that the ground state degeneracy on a
change the energy of the state. We note that the action torus can be calculated from the string operators. We
oftwoZ(C∗)stringoperatorsalongthe topandthebot- can also calculate the quasiparticle statistics from the
tomboundaryofthestripeinFig. 9isequivalenttoaZ stringoperators. Thisisbecauseanopenstringoperator
2
gauge transformation on the stripe. In other words, the creates a pair of quasiparticles at its ends. The electric
tensors are invariant under the action of the two Z(C∗) string operator X(C) creates a pair of Z charges, and
2
stringoperators,uptoaZ gaugetransformationonthe the magnetic string operator Z(C∗) creates a pair of Z
2 2
stripe. Thus the product of the two Z(C∗) string opera- vortices. The open string operators can also be viewed
tors is an element in 1-SG. The action of the two Z(C∗) as hopping operators for the quasiparticles. The statis-
string operators does not change the state. tics ofthe quasiparticlescanthenbe calculatedfromthe
On the other hand the action of one Z(C∗) string op- algebra of these hopping operators (i.e. the open string
eratorisnotequivalenttoaZ gaugetransformationand operators).[14]
2
7
VI. DISCUSSION AND CONCLUSIONS TNS, such as ground state degeneracy and quasiparticle
statistics can be calculated from the resulting string op-
In this paper, we vieweda TNS as anidealwavefunc- erators. This allows us to identify the topological order
tion that is the exact ground state of a ideal Hamilto- of the TNS just from the tensors that form the TNS.
nian. When the ideal Hamiltonian has a finite energy
While completing this paper, we became aware
gap, the TNS can represent a quantum phase with non-
a preprint by Schuch, Cirac, and Perez-Garcia
trivial topologicalorder. We arguedthat the topological
(arXiv:1001.3807)whereaclassofTNSsatisfyingthe“G-
properties of such a phase can be calculated just from
injective” condition are studied (injectivity means that
the tensors that form the TNS.
one can achieve any action on the inner indices a,b,c,...
Motivated by the gauge group and projective symme-
by acting on the physical indices m). From the symme-
try group in the slave-particle/projective construction
try properties of the tensors (similar to our d -IGG),
space
[35] and by low-dimensional gauge-like symmetries of
theycanalsocalculategroundstatedegeneracyandother
model Hamiltonians[37], we introduced the d-IGG and
physical properties from the tensors of the tensor net-
the d-SG for a TNS. Using the d-IGG and the d-SG of
work.
a TNS, we cancalculate the stringoperators(ord-brane
operators)oftheTNS,aswedemonstratedinthesimple This research is supported by NSF Grant No. DMR-
Z condensed state. Many topological properties of the 0706078
2
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