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according to the principle of emergence , the rich properties and the many different forms of materials originate from the different ways in which the atoms are ordered in the materials . landau symmetry - breaking theory provides a general understanding of those different orders and resulting rich states of matter.@xcite it points out that different orders really correspond to different symmetries in the organizations of the constituent atoms . as a material changes from one order to another order ( i.e. , as the material undergoes a phase transition ) , what happens is that the symmetry of the organization of the atoms changes .    for a long time , we believed that landau symmetry - breaking theory describes all possible orders in materials , and all possible ( continuous ) phase transitions . however , in last twenty years , it has become more and more clear that landau symmetry - breaking theory does not describe all possible orders . after the discovery of high @xmath0 superconductors in 1986,@xcite some theorists believed that quantum spin liquids play a key role in understanding high @xmath0 superconductors@xcite and started to introduce various spin liquids.@xcite despite the success of landau symmetry - breaking theory in describing all kinds of states , the theory can not explain and does not even allow the existence of spin liquids . this leads many theorists to doubt the very existence of spin liquids . in 1987 , in an attempt to explain high temperature superconductivity , an infrared stable spin liquid chiral spin state was discovered,@xcite which was shown to be perturbatively stable and exist as quantum phase of matter ( at least in a large @xmath1 limit ) . at first , not believing landau symmetry - breaking theory fails to describe spin liquids , people still wanted to use symmetry - breaking to describe the chiral spin state . they identified the chiral spin state as a state that breaks the time reversal and parity symmetries , but not the spin rotation symmetry.@xcite however , it was quickly realized that there are many different chiral spin states that have exactly the same symmetry , so symmetry alone is not enough to characterize different chiral spin states . this means that the chiral spin states contain a new kind of order that is beyond symmetry description.@xcite this new kind of order was named@xcite topological order . the key to identify ( and define ) new orders is to identify new universal properties that are beyond the local order parameters and long - range correlations used in the landau symmetry breaking theory . indeed , new quantum numbers , such as ground state degeneracy@xcite , the non - abelian berry s phase of degenerate ground states@xcite and edge excitations@xcite , were introduced to characterize ( and define ) the different topological orders in chiral spin states . recently , it was shown that topological orders can also be characterized by topological entanglement entropy.@xcite more importantly , those quantities were shown to be universal ( robust against any local perturbation of the hamiltonian ) for chiral spin states.@xcite the existence of those universal properties establishes the existence of topological order in chiral spin states . near the end of 1980 s , the existence of chiral spin states as a theoretical possibility , as well as their many amazing properties , such as fractional statistics,@xcite spin - charge separation,@xcite chiral gapless edge excitations,@xcite were established reliably , at least in the large @xmath1-limit introduced in . even non - abelian chiral spin states can be established reliably in the large @xmath1-limit.@xcite however , it took about 10 years to establish the existence of a chiral spin state reliably without using large @xmath1-limit ( based on an exactly soluble model on honeycomb lattice).@xcite    soon after the introduction of chiral spin states , experiments indicated that high - temperature superconductors do not break the time reversal and parity symmetries . so chiral spin states do not describe high - temperature superconductors . thus the theory of topological order became a theory with no experimental realization . however , the similarity between chiral spin states and fractional quantum hall ( fqh ) states allows one to use the theory of topological order to describe different fqh states.@xcite just like chiral spin states , different fqh states all have the same symmetry and are beyond the landau symmetry - breaking description . also like chiral spin states , fqh states have ground state degeneracies@xcite that depend on the topology of the space.@xcite those ground state degeneracies are shown to be robust against any perturbations . thus , the different orders in different quantum hall states can be described by topological orders , and the topological order does have experimental realizations .    the topology dependent ground state degeneracy , that signal the presence of topological order , is an amazing phenomenon . in fqh states , the correlation of any local operators are short ranged . this seems to imply that fqh states are `` short sighted '' and they can not know the topology of space which is a global and long - distance property . however , the fact that ground state degeneracy does depend on the topology of space implies that fqh states are not `` short sighted '' and they do find a way to know the global and long - distance structure of space . so , despite the short - range correlations of any local operators , the fqh states must contain certain hidden long - range correlation . but what is this hidden long - range correlation ? this will be one of the main topic of this paper . since high @xmath0 superconductors do not break the time reversal and parity symmetries , nor any other lattice symmetries , some people concentrated on finding spin liquids that respect all those symmetries and hoping one of those symmetric spin liquids hold the key to understand high @xmath0 superconductors . between 1987 and 1992 , many symmetric spin liquids were introduced and studied.@xcite the excitations in some of constructed spin liquids have a finite energy gap , while in others there is no energy gap . those symmetric spin liquids do not break any symmetry and , by definition , are beyond landau symmetry - breaking description .    by construction , topological order only describes the organization of particles or spins in a gapped quantum state . so the theory of topological order only applies to gapped spin liquids . indeed , we find that the gapped spin liquids do contain nontrivial topological orders@xcite ( as signified by their topology dependent and robust ground state degeneracies ) and are described by topological quantum field theory ( such as @xmath2 gauge theory ) at low energies . one of the simplest topologically ordered spin liquids is the @xmath2 spin liquid which was first introduced in 1991.@xcite the existence of @xmath2 spin liquid as a theoretical possibility , as well as its many amazing properties , such as spin - charge separation,@xcite fractional mutual statistics,@xcite topologically protected ground state degeneracy,@xcite were established reliably ( at least in the large @xmath1-limit introduced in ) . later , kitaev introduced the famous toric code model which establishes the existence of the @xmath2 spin liquid reliably without using large @xmath1-limit.@xcite the topologically protected degeneracy of the @xmath2 spin liquid was used to perform fault - tolerant quantum computation . the study of high @xmath0 superconductors also leads to many gapless spin liquids . the stability and the existence of those gapless spin liquid were in more doubts than their gapped counterparts . but careful analysis in certain large @xmath1 limit do suggest that stable gapless spin liquids can exist.@xcite if we do believe in the existence of gapless spin liquids , then the next question is how to describe the orders ( the organizations of spins ) in those gapless spin liquids . if gapped quantum state can contain new type of orders that are beyond landau s symmetry breaking description , it is natural to expect that gapless quantum states can also contain new type of orders . but how to show the existence of new orders in gapless states ? just like topological order , the key to identify new orders is to identify new universal properties that are beyond landau symmetry description . clearly we can no longer use the ground state degeneracy to establish the existence of new orders in gapless states . to show the existence of new orders in gapless states , a new universal quantity  projective symmetry group ( psg ) was introduced.@xcite it was argued that ( some ) psgs are robust against any local perturbations of the hamiltonian that do not change the symmetry of the hamiltonian.@xcite so through psg , we establish the existence of new orders even in gapless states . the new orders are called quantum order to indicate that the new orders are related to patterns of quantum entanglement in the many - body ground state.@xcite what is missed in landau theory so that it fails to describe those new orders ? what is the new feature in the organization of particles / spins so that the resulting order can not be described by symmetry ?    to answer those questions , let us consider a simple quantum system which can be described with landau theory  the transverse field ising model in two dimensions : @xmath3 , where @xmath4 , @xmath5 , and @xmath6 are the pauli matrices on site @xmath7 . in @xmath8 limit , the ground state of the system is an equal - weight superposition of all possible spin - up and spin - down states : @xmath9 , where @xmath10 label a particular spin - up ( @xmath11 ) and spin - down ( @xmath12 ) configuration . in the @xmath13 limit , the system has two degenerate ground states @xmath14 and @xmath15 . the transverse field ising model has a @xmath16 symmetry . the ground state @xmath17 respect such a symmetry while the ground state @xmath18 ( or @xmath19 ) break the symmetry . thus the small @xmath20 ground state @xmath17 and the small @xmath21 ground state @xmath18 describe different phases since they have different symmetries . we note that @xmath18 is the exact ground state of the transverse field ising model with @xmath22 . the state has no quantum entanglement since @xmath18 is a direct product of local states : @xmath23 where @xmath24 is an up - spin state at site @xmath7 . the state @xmath17 is the exact ground state of the transverse field ising model with @xmath25 . it is also a state with no quantum entanglement : @xmath26 where @xmath27 is a state with spin in @xmath28-direction at site @xmath7 . we see that the states ( or phases ) described by landau symmetry breaking theory has no quantum entanglement at least in the @xmath25 or @xmath22 limits . in the @xmath29 and @xmath30 limits , the two ground states of the two limits still represent two phases with different symmetries . however , in this case , the ground states are not unentangled . on the other hand since the ground states in the @xmath25 or @xmath22 limits have finite energy gaps and short - range correlations , a small @xmath20 or a small @xmath21 can only modify the states locally . thus we expect the ground states have only short - range entanglement . the above example , if generalized to other symmetry breaking states , suggests the following conjecture : _ _ if a gapped quantum ground state is described by landau symmetry breaking theory , then it has short - range quantum entanglement . _ _    the direct - product states and short - range entangled states only represent a small subset of all possible quantum many - body states . thus , according to the point of view of the above conjecture , we see that what is missed by landau symmetry breaking theory is long - range quantum entanglement . it is this long - range quantum entanglement that makes a state to have nontrivial topological / quantum order . however , mathematically speaking , the above conjecture is a null statement since the meaning of short - range quantum entanglement is not defined . in the following , we will try to find a more precise description ( or definition ) of short - range and long - range quantum entanglement . we will start with a careful discussion of quantum phases and quantum phase transitions . to give a precise definition of quantum phases , let us consider a local quantum system whose hamiltonian has a smooth dependence on a parameter @xmath31 : @xmath32 . the ground state average of a local operator @xmath33 of the system , @xmath34 , naturally also depend on @xmath31 . if the function @xmath34 , in the limit of infinite system size , has a singularity at @xmath35 for some local operators @xmath33 , then the system described by @xmath32 has a quantum phase transition at @xmath35 . after defining phase transition , we can define when two quantum ground states belong to the same phase : let @xmath36 be the ground state of @xmath37 and @xmath38 be the ground state of @xmath39 . if we can find a smooth path @xmath32 , @xmath40 that connect the two hamiltonian @xmath37 and @xmath39 such that there is no phase transition along the path , then the two quantum ground states @xmath36 and @xmath38 belong to the same phase . we note that `` connected by a smooth path '' define an equivalence relation between quantum states . a quantum phase is an equivalence class of such equivalence relation . such an equivalence class is called an universality class . if @xmath36 is the ground state of @xmath37 and in the limit of infinite system size all excitations above @xmath36 have a gap , then for small enough @xmath31 , we believe that the systems described by @xmath32 are also gapped.@xcite in this case , we can show that , the ground state of @xmath32 , @xmath41 , is in the same phase as @xmath36 , for small enough @xmath31 , the average @xmath34 is a smooth function of @xmath31 near @xmath42 for any local operator @xmath33.@xcite after scaling @xmath31 to @xmath43 , we find that : @xmath44 $ , then there is no phase transition along the path $ g$. } } \ ] ] in other words , for gapped system , a quantum phase transition can happen only when energy gap closes@xcite ( see fig . [ trans ] ) . is finite for all @xmath31 in the segment @xmath45 $ ] , show that the algebra of local operators for different @xmath31 s are isomorphic to each other at low energies , using a quasi - adiabatic continuation of quantum states . ] here , we would like to assume that the reverse is also true : a closing of the energy gap for a gapped state always induces a phase transition . or more precisely @xmath46 $ , and $ |\phi(0)\>$ and $ |\phi(1)\>$ are ground states of $ h(0)$ and $ h(1)$ respectively . } } \ ] ] the above two boxed statements imply that two gapped quantum states are in the same phase @xmath47 if and only if they can be connected by an adiabatic evolution that does not close the energy gap . given two states , @xmath36 and @xmath38 , determining the existence of such a gapped adiabatic connection can be hard . we would like to have a more operationally practical equivalence relation between states in the same phase . here we would like to show that _ two gapped states @xmath36 and @xmath38 are in the same phase , if and only if they are related by a local unitary ( lu ) evolution . _ we define a local unitary(lu ) evolution as an unitary operation generated by time evolution of a local hamiltonian for a finite time . that is , @xmath48 |\phi(0)\>\end{aligned}\ ] ] where @xmath49 is the path - ordering operator and @xmath50 is a sum of local hermitian operators . note that @xmath51 is in general different from the adiabatic path @xmath32 that connects the two states . first , assume that two states @xmath36 and @xmath38 are in the same phase , therefore we can find a gapped adiabatic path @xmath32 between the states . the existence of a gap prevents the system to be excited to higher energy levels and leads to a local unitary evolution , the quasi - adiabatic continuation as defined in , that maps from one state to the other . that is , @xmath52\ ] ] the exact form of @xmath51 is given in and will be discussed in more detail in appendix .     in the hamiltonian . ( a , b ) for gapped system , a quantum phase transition can happen only when energy gap closes . ( a ) describes a first order quantum phase transition ( caused by level crossing ) . ( b ) describes a continuous quantum phase transition which has a continuum of gapless excitations at the transition point . ( c ) and ( d ) can not happen for generic states . a gapped system may have ground state degeneracy , where the energy splitting between the ground states vanishes when system size @xmath53 : @xmath54 . the energy gap @xmath55 between ground and excited states on the other hand remains finite as @xmath53 . ]    on the other hand , the reverse is also true : _ if two gapped states @xmath36 and @xmath38 are related by a local unitary evolution , then they are in the same phase . _ since @xmath36 and @xmath38 are related by a local unitary evolution , we have @xmath56 |\phi(0)\ > $ ] . let us introduce @xmath57 .\end{aligned}\ ] ] assume @xmath36 is a ground state of @xmath37 , then @xmath58 is a ground state of @xmath59 . if @xmath60 remains local and gapped for all @xmath61 $ ] , then we have found an adiabatic connection between @xmath36 and @xmath38 . first , let us show that @xmath60 is a local hamiltonian . since @xmath62 is a local hamiltonian , it has a form @xmath63 where @xmath64 only acts on a cluster whose size is @xmath65 . @xmath65 is called the range of interaction of @xmath62 . we see that @xmath60 has a form @xmath66 , where @xmath67 . to show that @xmath68 only acts on a cluster of a finite size , we note that for a local system described by @xmath51 , the propagation velocities of its excitations have a maximum value @xmath69 . since @xmath70 can be viewed as the time evolution of @xmath64 by @xmath71 from @xmath72 to @xmath73 , we find that @xmath68 only acts on a cluster of size @xmath74,@xcite where @xmath75 is the range of interaction of @xmath76 . thus @xmath60 are indeed local hamiltonian . if @xmath62 has a finite energy gap , then @xmath60 also have a finite energy gap for any @xmath77 . as @xmath77 goes for @xmath78 to @xmath43 , the ground state of the local hamiltonians , @xmath60 , goes from @xmath36 to @xmath38 . thus the two states @xmath36 and @xmath38 belong to the same phase . this completes our argument that states related by a local unitary evolution belong to the same phase . thus through the above discussion , we show that @xmath79 a more detailed and more rigorous discussion of this equivalence relation is given in appendix a , where exact bounds on locality and transformation error is given . the relation defines an equivalence relation between @xmath36 and @xmath38 . the equivalence classes of such an equivalence relation represent different quantum phases . so the above result implies that the equivalence classes of the lu evolutions are the universality classes of quantum phases for gapped states . using the lu evolution , we can obtain a more precise description ( or definition ) of short - range entanglement : @xmath80 if a state can not be transformed into an unentangled state through a lu evolution , then the state has long - range entanglement . we also see that @xmath81 thus all states with short - range entanglement belong to the same phase . the local unitary evolutions we consider here do not have any symmetry . if we require certain symmetry of the local unitary evolutions , states with short - range entanglement may belong to different symmetry breaking phases , which will be discussed in section [ symtop ] .    since a direct - product state is a state with trivial topological order , we see that a state with a short - range entanglement also has a trivial topological order . this leads us to conclude that a non - trivial topological order is related to long - range entanglement . since two gapped states related by a lu evolution belong to the same phase , thus two gapped states related by a local unitary evolution have the same topological order . in other words , @xmath82 or more pictorially , topological order is a pattern of long - range entanglement . in , it was shown that the `` topologically non - trivial '' ground states , such as the toric code,@xcite can not be changed into a `` topologically trivial '' state such as a product state by any unitary locality - preserving operator . in other words , those `` topologically non - trivial '' ground states have long - range entanglement .    . the green shading represents a causal structure . ] the lu evolutions introduced here is closely related to _ quantum circuits with finite depth_. to define quantum circuits , let us introduce piece - wise local unitary operators . a piece - wise local unitary operator has a form @xmath83 where @xmath84 is a set of unitary operators that act on non overlapping regions . the size of each region is less than some finite number @xmath85 . the unitary operator @xmath86 defined in this way is called a piece - wise local unitary operator with range @xmath85 . a quantum circuit with depth @xmath87 is given by the product of @xmath87 piece - wise local unitary operators ( see fig . [ qc ] ) : @xmath88 . in quantum information theory , it is known that finite time unitary evolution with local hamiltonian ( lu evolution defined before ) can be simulated with constant depth quantum circuit and vice - verse . therefore , the equivalence relation can be equivalently stated in terms of constant depth quantum circuits : @xmath89 where @xmath87 is a constant independent of system size . because of their equivalence , we will use the term `` local unitary transformation '' to refer to both local unitary evolution and constant depth quantum circuit in general . the lu transformation defined through lu evolution is more general . it can be easily generalized to study topological orders and quantum phases with symmetries ( see section [ symtop]).@xcite the quantum circuit has a more clear and simple causal structure . however , the quantum circuit approach breaks the translation symmetry . so it is more suitable for studying quantum phases that do not have translation symmetry . in fact , people have been using quantum circuits to classify many - body quantum states which correspond to quantum phases of matter . in , the local unitary transformations described by quantum circuits was used to define a renormalization group transformations for states and establish an equivalence relation in which states are equivalent if they are connected by a local unitary transformation . such an approach was used to classify 1d matrix product states . in , the local unitary transformations with disentanglers was used to perform a renormalization group transformations for states , which give rise to the multi - scale entanglement renormalization ansatz ( mera ) in one and higher dimensions . the disentanglers and the isometries in mera can be used to study quantum phases and quantum phase transitions in one and higher dimensions . later in this paper , we will use the quantum circuit description of lu transformations to classify 2d topological orders through classifying the fixed - point lu transformations . in the above discussions , we have defined phases without any symmetry consideration . the @xmath51 or @xmath86 in the lu transformation does not need to have any symmetry and can be sum / product of any local operators . in this case , two hamiltonians with an adiabatic connection are in the same phase even if they may have different symmetries . also , all states with short - range entanglement belong to the same phase ( under the lu transformations that do not have any symmetry ) .    on the other hand , we can consider only hamiltonians @xmath62 with certain symmetries and define phases as the equivalent classes of symmetric local unitary transformations : @xmath90 where @xmath51 or @xmath91 has the same symmetries as @xmath62 . we note that the symmetric local unitary transformation in the form @xmath92 always connect to the identity transformation continuously . this may not be the case for the transformation in the form @xmath91 . to rule out that possibility , we define symmetric local unitary transformations as those that connect to the identity transformation continuously . the equivalent classes of the symmetric lu transformations have very different structures compared to those of lu transformations without symmetry . each equivalent class of the symmetric lu transformations is smaller and there are more kinds of classes , in general .     without any symmetry . ( b ) the possible phases for a hamiltonian @xmath93 with some symmetries . the shaded regions in ( a ) and ( b ) represent the phases with short range entanglement ( those ground states can be transformed into a direct product state via a generic lu transformations that do not have any symmetry . ) ]    fig . [ topsymm ] compares the structure of phases for a system without any symmetry and a system with some symmetry in more detail . for a system without any symmetry , all the short - range - entangled ( sre ) states ( those ground states can be transformed into a direct product state via a generic lu transformations that do not have any symmetry ) are in the same phase ( sre in fig . [ topsymm](a ) ) . on the other hand , long range entanglement ( lre ) can have many different patterns that give rise to different topological phases ( lre 1 and lre 2 in fig . [ topsymm](a ) ) . the different topological orders usually give rise to quasiparticles with different fractional statistics and fractional charges .    for a system with some symmetries , the phase structure can be much more complicated . the short - range - entangled states no longer belong to the same phase , since the equivalence relation is described by more special symmetric lu transformations : + ( a ) states with short range entanglement belong to different equivalent classes of the symmetric lu transformations if they have different broken symmetries . they correspond to the symmetry - breaking ( sb ) short - range - entanglement phases sb - sre 1 and sb - sre 2 in fig . [ topsymm](b ) . they are landau s symmetry breaking states . + ( b ) states with short range entanglement can belong to different equivalent classes of the symmetric lu transformations even if they do not break any symmetry of the system . ( in this case , they have the same symmetry . ) they correspond to the symmetric ( sy ) short - range - entangled phases sy - sre 1 and sy - sre 2 in fig . [ topsymm](b ) . we say those states have symmetry protected topological orders . haldane phase@xcite and @xmath94 phase of spin-1 chain are examples of states with the same symmetry which belong to two different equivalent classes of symmetric lu transformations ( with parity symmetry).@xcite band and topological insulators@xcite are other examples of states that have the same symmetry and at the same time belong to two different equivalent classes of symmetric lu transformations ( with time reversal symmetry ) . also , for a system with some symmetries , the long - range - entangled states are divided into more classes ( more phases ) : + ( c ) symmetry breaking and long range entanglement can appear together in a state , such as sb - lre 1 , sb - lre 2 , in fig . [ topsymm](b ) . the topological superconducting states are examples of such phases.@xcite + ( d ) long - range - entangled states that do not break any symmetry can also belong to different phases such as the symmetric long - range - entanglement phases sy - lre 1 , sy - lre 2 , in fig . [ topsymm](b ) . the many different @xmath2 symmetric spin liquids with spin rotation , translation , and time - reversal symmetries are examples of those phases.@xcite some time - reversal symmetric topological orders were also called topological mott - insulators or fractionalized topological insulators.@xcite after defining topological order as the equivalent classes of many - body wave functions under lu transformations , we like to ask : how to describe ( or label ) the different equivalent classes ( the different topological orders or patterns of long - range entanglement ) ?    one simple way to do so is to use the full wave function which completely describe the different topological orders . but the full wave functions contain a lot of non - universal short - range entanglement . as a result , such a labeling scheme is a very inefficient many - to - one labeling scheme of topological orders . to find a more efficient or even one - to - one labeling scheme , we need to remove the non - universal short - range entanglement .     into a direct - product state , if and only if the state @xmath95 has no long - range quantum entanglement . here , a dot represents a site with physical degrees of freedom . a vertical line carries an index that label the different physical states on a site . the presence of horizontal lines between dots represents quantum entanglement . ]     into a direct product . removing / adding the degrees of freedom in the form of direct product defines an additional equivalence relation that defines the topological order ( or classes of long - range entanglement ) . ]    as the first application of the notion of lu transformation , we would like to describe a wave function renormalization group flow introduced in , . the wave function renormalization can remove the short - range entanglement and simplify the wave function . in , the wave function renormalization for string - net states is generated by the following two basic moves @xmath96{xikljo } \epm   = & \delta_{ij } \phi \bpm \includegraphics[height=0.3in]{xi0 } \epm \\ \label{fmv }   \phi \bpm \includegraphics[height=0.3in]{xijklmxo } \epm   = & \sum_{n } f^{jim^*}_{lk^*n^ * } \phi \bpm \includegraphics[height=0.3in]{xijklnxo } \epm\end{aligned}\ ] ] ( note that the definition of the f - tensor in is slightly different from the definition in this paper . ) the two basic moves can generate a generic wave function renormalization which can reduce the string - net wave functions to very simple forms.@xcite later in , the wave function renormalization for generic states was discussed in a more general setting , and was called mera . the two basic string - net moves and correspond to the isometry and the disentangler in mera respectively . in the mera approach , the isometries and the disentanglers are applied in a layered fashion , while in the string - net approach , the two basic moves can be applied arbitrarily . in this section , we will follow the mera setup to describe the wave function renormalization . later in this paper , we will follow the string - net setup to study the fixed - point wave functions . note that we can use a lu transformation @xmath97 to transform some degrees of freedom in a state into direct product ( see fig . [ entre ] ) . we then remove those degrees of freedom in the form of direct product . such a procedure does not change the topological order . the reverse process of adding degrees of freedom in the form of direct product also does not change the topological order . we call the local transformation in fig . [ entre ] that changes the degrees of freedom a generalized local unitary ( glu ) transformation . it is clear that a generalized local unitary transformation inside a region @xmath98 does not change the reduced density matrix @xmath99 for the region @xmath98 . this is the reason why we say that ( generalized ) local unitary transformations can not change long - range entanglement and topological order . similarly , the addition or removal of decoupled degrees of freedom to or from the hamiltonian , @xmath100 , will not change the phase of the hamiltonian ( the ground states of @xmath62 and @xmath101 are in the same phase ) , if those degrees of freedom form a direct product state ( the ground state of @xmath102 is a direct product state ) . let us define the glu transformation @xmath97 more carefully and in a more general setting . consider a state @xmath95 . let @xmath99 be the reduced density matrix of @xmath95 in region @xmath98 . @xmath99 may act in a subspace of the total hilbert space @xmath103 in region a , which is called the support space @xmath104 of region @xmath98 . the dimension @xmath105 of @xmath104 is called support dimension of region @xmath98 . now the hilbert space @xmath103 in region a can be written as @xmath106 . let @xmath107 , @xmath108 be a basis of this support space @xmath104 , @xmath107 , @xmath109 be a basis of @xmath110 , where @xmath111 is the dimension of @xmath103 , and @xmath112 , @xmath113 be a basis of @xmath103 . we can introduce a lu transformation @xmath114 which rotates the basis @xmath112 to @xmath107 . we note that in the new basis , the wave function only has non - zero amplitudes on the first @xmath105 basis vectors . thus , in the new basis @xmath107 , we can reduce the range of the label @xmath115 from @xmath116 $ ] to @xmath117 $ ] without losing any information . this motivates us to introduce the glu transformation as a rotation from @xmath112 , @xmath113 to @xmath118 , @xmath108 . the rectangular matrix @xmath97 is given by @xmath119 . we also regard the inverse of @xmath97 , @xmath120 , as a glu transformation . a lu transformation is viewed as a special case of glu transformation where the degrees of freedom are not changed . clearly @xmath121 and @xmath122 are two projectors . the action of @xmath123 does not change the state @xmath95 ( see fig . [ glut](b ) ) . we note that despite the reduction of the degrees of freedom , a glu transformation defines an equivalent relation . two states related by a glu transformation belong to the same phase . the renormalization flow induced by the glu transformations always flows within the same phase . acts in region a of a state @xmath95 , which reduces the degree freedom in region a to those contained only in the support space of @xmath95 in region a. ( b ) @xmath121 is a projector that does not change the state @xmath95 . ] as an application of the wave function renormalization , in this section , we will study the structure of fixed - point wave functions under the wave function renormalization , which will lead to a classification of topological order ( without any symmetry ) . we note that as wave functions flow to a fixed point , the glu transformations in each step of the renormalization also flow to a fixed point . so instead of studying fixed - point wave functions , here , we will study the fixed - point glu transformations . for this purpose , we need to fix the renormalization scheme . in the following , we will discuss a renormalization scheme motivated by the string - net wave function.@xcite after we specify a proper wave function renormalization scheme , then the fixed - point wave function is simply the wave function whose `` form '' does not change under the wave function renormalization .    since those fixed - point glu transformations do not change the fixed - point wave function , their actions on the fixed - point wave function do not depend on the order of the actions . this allows us to obtain many conditions that glu transformations must satisfy . from those conditions , we can determine the forms of allowed fixed - point glu transformations . this leads to a general description and a classification of topological orders and their corresponding fixed - point wave functions . the renormalization scheme that we will discuss was first used in to characterize the scale invariant string - net wave function . it is also used in to simplify the string - net state in a region , which allows us to calculate the entanglement entropy of the string - net state exactly . a similar approach was used in to show quantum - double / string - net wave function to be a fixed - point wave function and its connection to 2d mera.@xcite in the following , we will generalize those discussions by not starting with string - net wave functions . we just try to construct local unitary transformations at a fixed point . we will see that the fixed - point conditions on the glu transformations lead to a mathematical structure that is similar to the tensor category theory  the mathematical framework behind the string - net states . since the wave function renormalization may change the lattice structure , we will consider quantum states defined on a generic trivalence graph @xmath124 : each edge has @xmath125 states , labeled by @xmath126 ( see fig . [ nijkv ] ) . we assume that the index @xmath115 on the edge admits an one - to - one mapping @xmath127 : @xmath128 that satisfies @xmath129 . as a result , the edges of the graph are oriented . the mapping @xmath128 corresponds to the reverse of the orientation of the edge ( see fig . [ llstar ] ) . each vertex also has physical states , labeled by @xmath130 ( see fig . [ nijkv ] ) .    each labeled graph ( see fig . [ nijkv ] ) corresponds to a state and all the labeled graphs form an orthonormal basis . our fixed - point state is a superposition of those basis states : @xmath131{strnet}\emm \right ) \left |\bmm \includegraphics[scale=0.2]{strnet}\emm\right \>$ ] . here we will make an important assumption about the fixed - point wave function . we will assume that the fixed - point wave function is `` topological '' : two labeled graphs have the same amplitude if the two labeled graphs can be deformed into each other continuously on the plane without the vertices crossing the links . for example @xmath132{gs1}\epm = \psi_\text{fix}\bpm \includegraphics[scale=0.25]{gs2}\epm $ ] . due to such an assumption , the topological orders studied in this paper may not be most general . we also assume that all the fixed - point states on each different graphs to have the same `` form '' . this assumption is motivated by the fact that during wave function renormalization , we transform a state on one graph to a state on a different graph . the `` fixed - point '' means that the wave functions on those different graphs are all determined by the same collection of the rules , which defines the meaning of having the same `` form '' . however , the wave function for a given graph can have different total phases if the wave function is calculated by applying the rules in different orders . those rules are noting but the fixed - point glu transformations .    . there are @xmath125 states on each edge which are labeled by @xmath133 . there are @xmath134 states on each vertex which are labeled by @xmath130 . ] corresponds to the reverse of the orientation of the edge ]      . if the three edges of a vertex are in the states @xmath115 , @xmath135 , and @xmath136 respectively , then the vertex has @xmath137 states , labeled by @xmath138 . note the orientation of the edges are point towards to vertex . also note that @xmath139 runs anti - clockwise . ]    before describing the wave function renormalization , let us examine the structure of entanglement of a fixed - point wave function . first , let us consider a fixed - point wave function @xmath140 on a graph . we examine the wave function on a patch of the graph , for example , @xmath141{f1g}\emm$ ] . the fixed - point wave function @xmath142{f1g}\epm$ ] ( only the relevant part of the graph is drawn ) can be viewed as a function of @xmath143 : @xmath144{f1 g } \epm$ ] if we fix @xmath145 and the indices on other part of the graph . ( here the indices on other part of the graph is summarized by @xmath146 . ) as we vary the indices @xmath146 on other part of graph ( still keeping @xmath115 , @xmath135 , @xmath136 , and @xmath85 fixed ) , the wave function of @xmath143 , @xmath147 , may change . all those @xmath147 form a linear space of dimension @xmath148 . @xmath149 is an important concept that will appear later . we note that the two vertices @xmath150 and @xmath151 and the link @xmath152 form a region surrounded by the links @xmath145 . so we will call the dimension-@xmath148 space the support space @xmath153 and @xmath148 the support dimension for the state @xmath140 on the region surrounded by the fixed boundary state @xmath145 . similarly , we can define @xmath154 as the support dimension of the @xmath142{ijkal}\epm$ ] on a region bounded by links @xmath155 . since the region contains only a single vertex @xmath150 with @xmath134 physics states , we have @xmath156 . we can use a local unitary transformation on the vertex to reduce the range of @xmath150 to @xmath157 where @xmath158 . in the rest of this paper , we will implement such a reduction . so , the number of physical states on a vertex depend on the physical states of the edges that connect to the vertex . if the three edges of a vertex are in the states @xmath115 , @xmath135 , and @xmath136 respectively , then the vertex has @xmath137 states , labeled by @xmath138 ( see fig . [ nijk ] ) . here we assume that @xmath159    we note that in the fixed - point wave function @xmath142{f1g}\epm$ ] , the number of choices of @xmath143 is @xmath160 . thus the support dimension @xmath148 satisfies @xmath161 . here we will make an important assumption  the saturation assumption : _ the fixed - point wave function saturate the inequality : _ @xmath162 we will see that the entanglement structure described by such a saturation assumption is invariant under the wave function renormalization . our wave function renormalization scheme contains two types of renormalization . the first type of renormalization does not change the degrees of freedom and corresponds to a local unitary transformation . it corresponds to locally deform the graph @xmath163{f1 g } \emm$ ] to @xmath163{f2 g } \emm$ ] . ( the parts that are not drawn are the same . ) the fixed - point wave function on the new graph is given by @xmath142{f2g}\epm$ ] . again , such a wave function can be viewed as a function of @xmath164 : @xmath165{f2 g } \epm$ ] if we fix @xmath145 and the indices on other part of the graph . the support dimension of the state @xmath142{f2g}\epm$ ] on the region surrounded by @xmath145 is @xmath166 . again @xmath167 , where @xmath168 is number of choices of @xmath164 . the saturation assumption implies that @xmath169 . the two fixed - point wave functions @xmath142{f1g}\epm$ ] and @xmath142{f2g}\epm$ ] are related via a local unitary transformation . thus @xmath170 , which implies @xmath171 we express such an unitary transformation as @xmath172 or graphically as @xmath173{f1 g } \epm    \simeq \sum_{n=0}^n \sum_{\chi=1}^{n_{kjn^ * } } \sum_{\del=1}^{n_{nil^ * } }   f^{ijm,\al\bt}_{kln,\chi\del } \phi_\text{fix } \bpm \includegraphics[scale=.40]{f2 g } \epm .\end{aligned}\ ] ] where @xmath174 means equal up to a constant phase factor . ( note that the total phase of the wave function is unphysical . ) we will call such a wave function renormalization step a f - move . we would like to remark that relates two wave functions on two graphs @xmath175 and @xmath176 which only differ by a local reconnection . we can choose the phase of @xmath177 to make @xmath174 into @xmath178 : @xmath179{f1 g } \epm    = \sum_{n=0}^n \sum_{\chi=1}^{n_{kjn^ * } } \sum_{\del=1}^{n_{nil^ * } }   f^{ijm,\al\bt}_{kln,\chi\del } \phi_\text{fix } \bpm \includegraphics[scale=.40]{f2 g } \epm .\end{aligned}\ ] ] but such choice of phase only works for a particular pair of graphs @xmath175 and @xmath176 . to use @xmath177 to relate all pair of states that only differ by a local reconnection , in general , we may have a phase ambiguity , with the value of phase depend on the pair of graphs . so can only be a relation up to a total phase factor .    for fixed @xmath115 , @xmath135 , @xmath136 , and @xmath85 , the matrix @xmath180 with matrix elements @xmath181 is a @xmath182 dimensional matrix ( see ) . the mapping @xmath183 generated by the matrix @xmath180 is unitary . since , as we change @xmath146 , @xmath184 and @xmath147 span two @xmath182 dimensional spaces . thus @xmath180 is a @xmath185 unitary matrix @xmath186 where @xmath187 when @xmath188 , @xmath189 , @xmath190 , and @xmath191 otherwise .    we can deform @xmath192{f1 g } \emm$ ] to @xmath192{f1ga } \emm$ ] , and @xmath192{f2 g } \emm$ ] to @xmath192{f2ga } \emm$ ] . we see that @xmath193{f1ga } \epm \simeq \sum_{n=0}^n \sum_{\chi,\del }   f^{kl^*m^*,\bt\al}_{ij^*n^*,\del\chi } \phi_\text{fix } \bpm \includegraphics[scale=.40]{f2ga } \epm .\end{aligned}\ ] ] eqn . ( [ ihwave ] ) and eqn . ( [ ihwavea ] ) relate the same pair of graphs , and thus @xmath194 ( where we have used the condition @xmath195 . )    using the relation , we can rewrite as @xmath196{f2 g } \epm \simeq \sum_{m=0}^n \sum_{\al,\bt }   ( f^{ijm,\al\bt}_{kln,\chi\del})^ * \phi_\text{fix } \bpm \includegraphics[scale=.40]{f1 g } \epm   .\end{aligned}\ ] ] we can also express @xmath197{f2 g } \epm $ ] as @xmath198{f2 g } \epm \simeq \sum_{m=0}^n \sum_{\al,\bt }   f^{jkn,\chi\del}_{l^*i^*m^*,\bt\al } \phi_\text{fix } \bpm \includegraphics[scale=.40]{f1 g } \epm\end{aligned}\ ] ] using the relabeled . so we have @xmath199 since the total phase of the wave function is unphysical , the total phase of @xmath177 can be chosen arbitrarily . we can choose the total phase of @xmath177 to make @xmath200 if we apply twice , we reproduce . thus is not independent and can be dropped . from the graphic representation , we note that @xmath201 when @xmath202 or @xmath203 , the left - hand - side of is always zero . thus @xmath204 when @xmath202 or @xmath205 . when @xmath206 or @xmath207 , wave function on the right - hand - side of is always zero . so we can choose @xmath204 when @xmath206 or @xmath207 . the f - move maps the wave functions on two different graphs through a local unitary transformation . since we can locally transform one graph to another graph through different paths , the f - move must satisfy certain self consistent condition . for example the graph @xmath208{pent1 g } \emm $ ] can be transformed to @xmath209{pent3 g } \emm $ ] through two different paths ; one contains two steps of local transformations and the other contains three steps of local transformations as described by . the two paths lead to the following relations between the wave functions :    @xmath210{pent1 g } \epm & \simeq \sum_{q,\del,\eps } f^{mkn,\bt\chi}_{lpq,\del\eps } \phi_\text{fix } \bpm \includegraphics[scale=.40]{pent2 g } \epm   \simeq \sum_{q,\del,\eps;s,\phi,\ga } f^{mkn,\bt\chi}_{lpq,\del\eps } f^{ijm,\al\eps}_{qps,\phi\ga } \phi_\text{fix } \bpm \includegraphics[scale=.40]{pent3 g } \epm , \end{aligned}\ ] ]    @xmath211{pent1 g } \epm & \simeq \sum_{t,\eta,\vphi } f^{ijm,\al\bt}_{knt,\eta\vphi } \phi_\text{fix } \bpm \includegraphics[scale=.40]{pent4 g } \epm   \simeq \sum_{t,\eta,\vphi;s,\ka,\ga } f^{ijm,\al\bt}_{knt,\eta\vphi } f^{itn,\vphi\chi}_{lps,\ka\ga } \phi_\text{fix } \bpm \includegraphics[scale=.40]{pent5 g } \epm \nonumber\\ & \simeq \sum_{t,\eta,\ka;\vphi;s,\ka,\ga;q,\del,\phi } f^{ijm,\al\bt}_{knt,\eta\vphi } f^{itn,\vphi\chi}_{lps,\ka\ga } f^{jkt,\eta\ka}_{lsq,\del\phi } \phi_\text{fix } \bpm \includegraphics[scale=.40]{pent3 g } \epm .\end{aligned}\ ] ]    the consistence of the above two relations leads a condition on the @xmath212 tensor .    to obtain such a condition , let us fix @xmath115 , @xmath135 , @xmath136 , @xmath85 , @xmath213 , and view @xmath214{pent1 g } \epm$ ] as a function of @xmath215 : @xmath216{pent1 g } \epm$ ] . as we vary indices on other part of graph , we obtain different wave functions @xmath217 which form a dimension @xmath218 space . in other words , @xmath218 is the support dimension of the state @xmath140 on the region @xmath215 with boundary state @xmath219 fixed ( see the discussion in section [ entstru ] ) . since the number of choices of @xmath215 is @xmath220 , we have @xmath221 . here we require a similar saturation condition as in : @xmath222 similarly , the number of choices of @xmath223 in @xmath214{pent3 g } \epm $ ] is also @xmath224 . here we again assume @xmath225 , where @xmath226 is the support dimension of @xmath214{pent3 g } \epm $ ] on the region bounded by @xmath219 . so the two relations and can be viewed as two relations between a pair of vectors in the two @xmath218 dimensional vector spaces . as we vary indices on other part of graph ( still keeping @xmath219 fixed ) , each vector in the pair can span the full @xmath218 dimensional vector space . so the validity of the two relations and implies that @xmath227 which is a generalization of the famous pentagon identity ( due to the extra constant phase factor @xmath228 ) . we will call such a relation projective pentagon identity . the projective pentagon identity is a set of nonlinear equations satisfied by the rank-10 tensor @xmath177 and @xmath229 . the above consistency relation is equivalent to the requirement that the local unitary transformations described by on different paths all commute with each other up to a total phase factor . the second type of wave function renormalization does change the degrees of freedom and corresponds to a generalized local unitary transformation . one way to implement the second type renormalization is to reduce @xmath208{4to1a } \emm $ ] to @xmath192{4to1b } \emm $ ] ( the part of the graph that is not drawn is unchanged ) : @xmath230{4to1a } \epm \simeq \sum_{\la=1}^{n_{ijk } } f ^{abc,\al\bt\ga } _ { ijk,\la } \phi_\text{fix } \bpm \includegraphics[scale=.40]{4to1b } \epm .\end{aligned}\ ] ]        but we can define a simpler second type renormalization , by noting that @xmath163{4to1a } \emm$ ] can be reduced to @xmath163{tpolij } \emm$ ] via the first type of renormalization steps ( see fig . [ tritpol ] ) , which are local unitary transformations . in the simplified second type renormalization , we want to reduce @xmath163{tpolij } \emm$ ] to @xmath163{iline } \emm$ ] , so that we still have a trivalence graph . this requires that the support dimension @xmath231 of the fixed - point wave function @xmath232{tpolij } \epm $ ] is given by @xmath233 this implies that @xmath234{tpolij } \epm \simeq \del_{ii ' } \phi_\text{fix } \bpm \includegraphics[scale=.40]{tpol } \epm   .\end{aligned}\ ] ] the simplified second type renormalization can now be written as ( since @xmath235 ) @xmath236{tpol } \epm \simeq   p_i^{kj,\al\bt } \phi_\text{fix } \bpm \includegraphics[scale=.40]{iline } \epm   .\end{aligned}\ ] ] we will call such a wave function renormalization step a p - move.@xcite here @xmath237 satisfies @xmath238 and @xmath239 the condition ensures that the two wave functions on the two sides of have the same normalization . we note that the number of choices for the four indices @xmath240 in @xmath241 must be equal or greater than @xmath43 : @xmath242    notice that @xmath243{tpol } \epm \simeq \sum_{m,\la,\ga } f^{jj^*k,\bt\al}_{i^*i^*m^*,\la\ga } \phi_\text{fix } \bpm \includegraphics[scale=.40]{buble } \epm \nonumber\\ & \simeq \sum_{m,\la,\ga , l,\nu,\mu } f^{jj^*k,\bt\al}_{i^*i^*m^*,\la\ga } f^{i^*mj,\la\ga}_{m^*i^*l,\nu\mu } \phi_\text{fix } \bpm \includegraphics[scale=.40]{tpolud } \epm\end{aligned}\ ] ] using and its variation @xmath244{tpolud } \epm \simeq p_{i^*}^{lm,\mu\nu } \phi_\text{fix } \bpm \includegraphics[scale=.40]{iline } \epm   .\end{aligned}\ ] ] we can rewrite as @xmath245 which is a condition on @xmath241 . more conditions on @xmath177 and @xmath241 can be obtained by noticing that @xmath234{f1gp } \epm   \simeq \sum_{n=0}^n \sum_{\chi,\del }   f^{ijm,\al\bt}_{kln,\chi\del } \phi_\text{fix } \bpm \includegraphics[scale=.40]{f2gp } \epm , \end{aligned}\ ] ] which implies that @xmath246{ikl1 } \epm \nonumber\\ & \simeq \sum_{n,\chi,\del }   f^{ijm,\al\bt}_{kln,\chi\del } p^{jp,\chi\eta}_{k^ * } \del_{kn } \phi_\text{fix } \bpm \includegraphics[scale=.40]{ikl2 } \epm .\end{aligned}\ ] ] we find @xmath247      in the last section , we discussed the conditions that a fixed - point glu transformation @xmath248 must satisfy . after finding a fixed - point glu transformation @xmath248 , in this section , we are going to discuss how to calculate the corresponding fixed - point wave function @xmath140 from the solved fixed - point glu transformation @xmath248 . first we note that , using the two types of wave function renormalization introduced above , we can reduce any graph to @xmath192{oi } \emm$ ] . so , once we know @xmath214{oi } \epm$ ] , we can reconstruct the full fixed - point wave function @xmath140 on any connected graph . let us assume that @xmath234{oi } \epm = a^i = a^{i^*}\end{aligned}\ ] ] here @xmath249 satisfy @xmath250 the condition @xmath251 is simply the normalization condition of the wave function . the condition @xmath252 come from the fact that the graph @xmath163{oi } \emm$ ] can be deformed into the graph @xmath192{oia } \emm$ ] on a sphere .    to find the conditions that determine @xmath249 , let us first consider the fixed - point wave function where the index on a link is @xmath115 : @xmath253{iga1 } \epm$ ] , where @xmath146 are indices on other part of graph . we note that the graph @xmath163{iga1 } \emm$ ] can be deformed into the graph @xmath192{iga2 } \emm$ ] on a sphere . thus @xmath197{iga1 } \epm= \phi_\text{fix}\bpm \includegraphics[scale=.35]{iga2 } \epm$ ] . using the f - moves and the p - moves , we can reduce @xmath192{iga2 } \emm$ ] to @xmath192{oi } \emm$ ] : @xmath254{iga1 } \epm= \phi_\text{fix}\bpm \includegraphics[scale=.40]{iga2 } \epm \simeq f(i,\ga ) \phi_\text{fix } \bpm \includegraphics[scale=.40]{oi } \epm\end{aligned}\ ] ] we see that @xmath255{oi } \epm = a^i\neq 0\end{aligned}\ ] ] for all @xmath115 . otherwise , any wave function with @xmath115-link will be zero .    to find more conditions on @xmath249 , we note that @xmath234{ooijm } \epm \simeq   p^{mj,\ga\la}_i \phi_\text{fix } \bpm \includegraphics[scale=.40]{oi } \epm =   p^{mj,\ga\la}_ia^i .\end{aligned}\ ] ] by rotating @xmath192{ooijm } \emm$ ] by 180@xmath256 , we can show that @xmath257 or @xmath258 we also note that @xmath234{thetaijk } \epm & \simeq \sum_{m,\la,\ga }   f^{ijk^*,\al\bt}_{j^*im,\la\ga } \phi_\text{fix } \bpm \includegraphics[scale=.40]{ooijm } \epm .\end{aligned}\ ] ] this allows us to show @xmath259 where @xmath260{thetaijk } \epm \nonumber\\ &   \phi^\th_{ikj,\al\bt } = \e^{\imth \th_{a2 } } \phi^\th_{kji,\al\bt } , \nonumber\\ & \phi^\th_{ikj,\al\bt}=0 , \text { if } n_{ikj}=0.\end{aligned}\ ] ] the condition @xmath261 comes from the fact that the graph @xmath192{thetaijk } \emm$ ] and the graph @xmath192{thetakij } \emm$ ] can be deformed into each other on a sphere . also , for any given @xmath262 that satisfy @xmath263 , the wave function @xmath264{ijkga } \epm$ ] must be non - zero for some @xmath146 , where @xmath146 represents indices on other part of the graph . then after some f - moves and p - moves , we can reduce @xmath192{ijkga } \emm$ ] to @xmath192{thetakij } \emm$ ] . so , for any given @xmath262 that satisfy @xmath263 , @xmath214{thetakij } \epm$ ] is non - zero for some @xmath151 . since such a statement is true for any choices of basis on the vertex @xmath150 , we find that for any given @xmath262 that satisfy @xmath263 and for any non - zero vector @xmath265 , @xmath266{thetakij } \epm$ ] is non - zero for some @xmath151 . this means that the matrix @xmath267 is invertible , where @xmath267 is a matrix whose elements are given by @xmath268{thetakij } \epm $ ] . let us define @xmath269{thetakij } \epm \big ] = \det(m_{kji})$ ] , we find that @xmath270{thetakij } \epm \big ]   = \det [ \phi^\th_{kji,\al\bt } ] \neq 0.\end{aligned}\ ] ] the above also implies that @xmath271 the conditions eqns . ( [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] ) allow us to determine @xmath249 ( and @xmath272 ) .    from , we see that relation @xmath197{thetaijk } \epm = \phi_\text{fix}\bpm \includegraphics[scale=.35]{thetakij } \epm$ ] leads to some equations for @xmath273 , @xmath241 , and @xmath249 . more equations for @xmath273 , @xmath241 , and @xmath249 , can be obtained by using the relations @xmath274{tetra1 } \epm & = \phi_\text{fix } \bpm \includegraphics[scale=.40]{tetra2 } \epm \nonumber\\ & = \phi_\text{fix } \bpm \includegraphics[scale=.40]{tetra3 } \epm , \end{aligned}\ ] ] from the tetrahedron rotation symmetry@xcite and @xmath275{tetra1 } \epm   \simeq \sum_{\ga\la } f^{ijm^*,\al\bt}_{kln,\ga\la } \phi_\text{fix } \bpm \includegraphics[scale=.40]{sq00 } \epm \nonumber\\ & \simeq \sum_{\ga\la , p\si\eps } f^{ijm^*,\al\bt}_{kln,\ga\la } f^{ln^*i^*,\del\la}_{nlp^*,\si\eps } \phi_\text{fix } \bpm \includegraphics[scale=.40]{osqo } \epm \nonumber\\ & \simeq \sum_{\ga\la , p\si\eps } f^{ijm^*,\al\bt}_{kln,\ga\la } f^{ln^*i^*,\del\la}_{nlp^*,\si\eps } p^{pl^*,\si\eps}_{n } \phi_\text{fix } \bpm \includegraphics[scale=.40]{sqo } \epm \nonumber\\ & = \sum_{\ga\la , p\si\eps } f^{ijm^*,\al\bt}_{kln,\ga\la } f^{ln^*i^*,\del\la}_{nlp^*,\si\eps } p^{pl^*,\si\eps}_{n } \phi^\th_{kjn^*,\ga\chi } .\end{aligned}\ ] ] it is not clear if and will lead to new independent equations or not . in the following discussions , we will not include and . we find that , at least for simple cases , the equations without and are enough to completely determine the solutions . to summarize , the conditions ( [ nnnn ] , [ di ] , [ nnstar ] , [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] , [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) form a set of non - linear equations whose variables are @xmath137 , @xmath273 , @xmath241 , @xmath249 , and @xmath276 . finding @xmath137 , @xmath273 , @xmath241 , and @xmath249 that satisfy such a set of non - linear equations corresponds to finding a fixed - point glu transformation that has a non - trivial fixed - point wave function . so the solutions @xmath277 give us a characterization of topological orders . this may lead to a classification of topological order from the local unitary transformation point of view . in this section , let us find some simple solutions of the fixed - point conditions ( [ nnnn ] , [ di ] , [ nnstar ] , [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] , [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) for the fixed - point glu transformations @xmath278 and the fixed - point wave function @xmath249 .      formally , the solutions of the fixed - point conditions are not isolated . they are parameterized by several continuous phase factors . in this section , we will discuss the origin of those phase factors . we will see that those different phase factors do not correspond to different states of matter ( different equivalence classes of glu transformations ) . so after removing those unimportant phase factors , the solutions of the fixed - point conditions are isolated ( at least for the simple examples studied here ) . we notice that , apart from two normalization conditions , all of the fixed - point conditions are linear in @xmath241 and @xmath249 . thus if @xmath279 is a solution , then @xmath280 is also a solution . however , the two phase factors @xmath281 do not lead to different fixed - point wave functions , since they only affect the total phase of the wave function and are unphysical . thus the total phases of @xmath282 and @xmath249 can be adjusted . we can use this degree of freedom to set , say , @xmath283 and @xmath284 . similarly the total phase of @xmath273 is also unphysical and can be adjusted . we have used this degree of freedom to reduce to . but this does not totally fix the total phase of @xmath273 . the transformation @xmath285 does not affect . we can use such a transformation to set the real part of a non - zero component of @xmath273 to be positive . the above three phase factors are unphysical . however , the fixed - point solutions may also contain phase factors that do correspond to different fixed - point wave functions . for example , the local unitary transformation @xmath286 does not affect the fusion rule @xmath137 , where @xmath287 is the number of links with @xmath288-state and @xmath289-state . such a local unitary transformation changes @xmath279 and generates a continuous family of the fixed - point wave functions parameterized by @xmath290 . those wave functions are related by local unitary transformations that continuously connect to identity . thus , those fixed - point wave functions all belong to the same phase . similarly , we can consider the following local unitary transformation @xmath291 that acts on each vertex with states @xmath292 on the three edges connecting to the vertex . such a local unitary transformation also does not affect the fusion rule @xmath137 . the new local unitary transformation changes @xmath279 and generates a continuous family of the fixed - point wave functions parameterized by the unitary matrix @xmath293 . again , those fixed - point wave functions all belong to the same phase .    in the following , we will study some simple solutions of the fixed - point conditions . we find that , for those examples , the solutions have no addition continuous parameter apart from the phase factors discussed above . this suggests that the solutions of the fixed - point conditions correspond to isolated zero - temperature phases . let us first consider a system where there are only two states @xmath295 and @xmath296 on each link of the graph . we choose @xmath297 and the simplest fusion rule that satisfies , , is @xmath298 since @xmath299 , there is no states on the vertices . so the indices @xmath300 labeling the states on a vertex can be suppressed . the above fusion rule corresponds to the fusion rule for the @xmath294 loop state discussed in . so we will call the corresponding graphic state @xmath294 loop state . due to the relation , the different components of the tensor @xmath301 are not independent . there are only four independent potentially non - zero components which are denoted as @xmath302, ... ,@xmath303 : @xmath304 \fbox \ilnkaa \jlnkaa \klnkaa \llnkaa \mlnkaa \nlnkaa \end{tikzpicture}\emm & = f_{0 } \nonumber\\ f^{000}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkaa \klnkbb \llnkbb \mlnkaa \nlnkbb \end{tikzpicture}\emm & = ( f^{011}_{100 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbb \klnkbb \llnkaa \mlnkbb \nlnkaa \end{tikzpicture}\emm ) ^*=(f^{101}_{010 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkaa \klnkaa \llnkbb \mlnkbb \nlnkaa \end{tikzpicture}\emm ) ^ * \nonumber\\ & = f^{110}_{001 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkaa \llnkaa \mlnkaa \nlnkbb \end{tikzpicture}\emm = f_{1 } \nonumber\\ f^{011}_{011 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbb \klnkaa \llnkbb \mlnkbb \nlnkbb \end{tikzpicture}\emm & = ( f^{101}_{101 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkaa \klnkbb \llnkaa \mlnkbb \nlnkbb \end{tikzpicture}\emm ) ^*=f_{2 } \nonumber\\ f^{110}_{110 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkbb \llnkbb \mlnkaa \nlnkaa \end{tikzpicture}\emm & = f_{3}\end{aligned}\ ] ] we note that @xmath301 in relates wave functions on two graphs . in the above we have drawn the two related graphs after the @xmath212 tensor , where the first graph following @xmath212 corresponds to the graph on the left - hand side of and the second graph corresponds to the graph on the right - hand side of . the doted line corresponds to the @xmath295-state on the link and the solid line corresponds to the @xmath296-state on the link . there are four potentially non - zero components in @xmath282 , which are denoted by @xmath305, ... ,@xmath306 : @xmath307    we can adjust the total phases of @xmath308 and @xmath249 to make @xmath309 and @xmath310 . we can also use the local unitary transformation @xmath311 with @xmath312 to make @xmath313 , since the @xmath212 s described by @xmath314 in are the only @xmath212 s that change the number of @xmath296-links . the fixed - point conditions ( [ nnnn ] , [ di ] , [ nnstar ] , [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] , [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) form a set of non - linear equations on the ten variables @xmath315 , @xmath308 , and @xmath249 . many of the non - linear equations are dependent or even equivalent . using a computer algebraic system , we simplify the set of non - linear equations . the simplified equations are ( after making the phase choice described above ) @xmath316 where @xmath317 . the above simplified equations can be solved exactly . we find two solutions parameterized by @xmath317 : @xmath318 we also find @xmath319    the @xmath320 fixed - point state corresponds to the @xmath2 loop condensed state whose low energy effective field theory is the @xmath2 gauge theory.@xcite we call such a state , simply , the @xmath2 state . the @xmath321 fixed - point state corresponds to the double - semion state whose low energy effective field theory is the @xmath322 chern - simons gauge theory@xcite @xmath323      to obtain another class of simple solutions , we modify the fusion rule to @xmath324 while keeping everything the same . the above @xmath137 also satisfies , , and . the new fusion rule corresponds to the fusion rule for the @xmath294 string - net state discussed in . so we will call the corresponding graphic state @xmath294 string - net state .    again , due to the relation , the different components of the tensor @xmath301 are not independent . now there are seven independent potentially non - zero components which are denoted as @xmath302, ... ,@xmath325 : @xmath326 \fbox \ilnkaa \jlnkaa \klnkaa \llnkaa \mlnkaa \nlnkaa \end{tikzpicture}\emm & = f_{0 } \nonumber\\ f^{000}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkaa \klnkbb \llnkbb \mlnkaa \nlnkbb \end{tikzpicture}\emm & = ( f^{011}_{100 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbb \klnkbb \llnkaa \mlnkbb \nlnkaa \end{tikzpicture}\emm ) ^*=(f^{101}_{010 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkaa \klnkaa \llnkbb \mlnkbb \nlnkaa \end{tikzpicture}\emm ) ^ * \nonumber\\ & = f^{110}_{001 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkaa \llnkaa \mlnkaa \nlnkbb \end{tikzpicture}\emm = f_{1 } \nonumber\\ f^{011}_{011 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbb \klnkaa \llnkbb \mlnkbb \nlnkbb \end{tikzpicture}\emm & = ( f^{101}_{101 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkaa \klnkbb \llnkaa \mlnkbb \nlnkbb \end{tikzpicture}\emm ) ^*=f_{2 } \nonumber\\ f^{011}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbb \klnkbb \llnkbb \mlnkbb \nlnkbb \end{tikzpicture}\emm & = ( f^{101}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkaa \klnkbb \llnkbb \mlnkbb \nlnkbb \end{tikzpicture}\emm ) ^*=f^{111}_{011 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkaa \llnkbb \mlnkbb \nlnkbb \end{tikzpicture}\emm \nonumber\\ & = ( f^{111}_{101 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkbb \llnkaa \mlnkbb \nlnkbb \end{tikzpicture}\emm ) ^*=f_{3 } \nonumber\\ f^{110}_{110 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkbb \llnkbb \mlnkaa \nlnkaa \end{tikzpicture}\emm & = f_{4 } \nonumber\\ f^{110}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkbb \llnkbb \mlnkaa \nlnkbb \end{tikzpicture}\emm & = ( f^{111}_{110 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkbb \llnkbb \mlnkbb \nlnkaa \end{tikzpicture}\emm ) ^*=f_{5 } \nonumber\\ f^{111}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkbb \llnkbb \mlnkbb \nlnkbb \end{tikzpicture}\emm & = f_{6}\end{aligned}\ ] ] note that @xmath212 s described by @xmath314 and @xmath327 are the only @xmath212 s that change the number of @xmath296-links and the number of @xmath328-vertices . so we can use the local unitary transformation @xmath329 to make @xmath314 and @xmath327 to be positive real numbers . ( here @xmath330 is the total number of @xmath296-links and @xmath331 is the total number of @xmath328-vertices . ) we also use the freedom of adjusting the total sign of @xmath301 to make re@xmath332 . there are five potentially non - zero components in @xmath282 , which are denoted by @xmath305, ... ,@xmath333 : @xmath334 we use the freedom of adjusting the total phase of @xmath282 to make @xmath305 to be a positive number . we can also use the freedom of adjusting the total phase of @xmath249 to make @xmath335 to be a positive number . the fixed - point conditions ( [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] , [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) form a set of non - linear equation on the variables @xmath315 , @xmath308 , and @xmath249 , which can be simplified . the simplified equations have the following form @xmath336 let @xmath337 be the positive solution of @xmath338 : @xmath339 . we see that @xmath340 . the above can be written as @xmath341 we also find @xmath319    the fixed - point state corresponds to the @xmath294 string - net condensed state@xcite whose low energy effective field theory is the doubled @xmath342 chern - simons gauge theory.@xcite      the above simple examples correspond to non - orientable string - net states . here we will give an example of orientable string - net state . we choose @xmath343 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 the above @xmath137 satisfies , , and .    due to the relation , the different components of the tensor @xmath301 are not independent . there are eight independent potentially non - zero components which are denoted as @xmath302, ... ,@xmath349 : @xmath326 \fbox \ilnkaa \jlnkaa \klnkaa \llnkaa \mlnkaa \nlnkaa \end{tikzpicture}\emm & = f_{0 } \nonumber\\ f^{000}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkaa \klnkbc \llnkbc \mlnkaa \nlnkbc \end{tikzpicture}\emm & = ( f^{011}_{200 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbc \klnkcb \llnkaa \mlnkbc \nlnkaa \end{tikzpicture}\emm ) ^*=f^{120}_{002 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkcb \klnkaa \llnkaa \mlnkaa \nlnkcb \end{tikzpicture}\emm \nonumber\\ & = ( f^{202}_{020 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkaa \klnkaa \llnkcb \mlnkcb \nlnkaa \end{tikzpicture}\emm ) ^*=f_{1 } \nonumber\\ f^{000}_{222 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkaa \klnkcb \llnkcb \mlnkaa \nlnkcb \end{tikzpicture}\emm & = ( f^{022}_{100 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkcb \klnkbc \llnkaa \mlnkcb \nlnkaa \end{tikzpicture}\emm ) ^*=(f^{101}_{010 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkaa \klnkaa \llnkbc \mlnkbc \nlnkaa \end{tikzpicture}\emm ) ^*\nonumber\\ & = f^{210}_{001 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkbc \klnkaa \llnkaa \mlnkaa \nlnkbc \end{tikzpicture}\emm = f_{2 } \nonumber\\ f^{011}_{011 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbc \klnkaa \llnkbc \mlnkbc \nlnkbc \end{tikzpicture}\emm & = f^{022}_{022 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkcb \klnkaa \llnkcb \mlnkcb \nlnkcb \end{tikzpicture}\emm = ( f^{101}_{202 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkaa \klnkcb \llnkaa \mlnkbc \nlnkcb \end{tikzpicture}\emm ) ^*\nonumber\\ & = ( f^{202}_{101 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkaa \klnkbc \llnkaa \mlnkcb \nlnkbc \end{tikzpicture}\emm ) ^*=f_{3 } \nonumber\\ f^{011}_{122 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbc \klnkbc \llnkcb \mlnkbc \nlnkcb \end{tikzpicture}\emm & = ( f^{101}_{121 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkaa \klnkbc \llnkcb \mlnkbc \nlnkbc \end{tikzpicture}\emm ) ^*=f^{112}_{021 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkbc \klnkaa \llnkcb \mlnkcb \nlnkbc \end{tikzpicture}\emm \nonumber\\ & = ( f^{112}_{102 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkbc \klnkbc \llnkaa \mlnkcb \nlnkcb \end{tikzpicture}\emm ) ^*=f_{4 } \nonumber\\ f^{022}_{211 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkcb \klnkcb \llnkbc \mlnkcb \nlnkbc \end{tikzpicture}\emm & = ( f^{202}_{212 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkaa \klnkcb \llnkbc \mlnkcb \nlnkcb \end{tikzpicture}\emm ) ^*=f^{221}_{012 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkcb \klnkaa \llnkbc \mlnkbc \nlnkcb \end{tikzpicture}\emm \nonumber\\ & = ( f^{221}_{201 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkcb \klnkcb \llnkaa \mlnkbc \nlnkbc \end{tikzpicture}\emm ) ^*=f_{5 } \nonumber\\ f^{112}_{210 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkbc \klnkcb \llnkbc \mlnkcb \nlnkaa \end{tikzpicture}\emm & = ( f^{120}_{221 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkcb \klnkcb \llnkcb \mlnkaa \nlnkbc \end{tikzpicture}\emm ) ^*=(f^{210}_{112 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkbc \klnkbc \llnkbc \mlnkaa \nlnkcb \end{tikzpicture}\emm ) ^ * \nonumber\\ & = f^{221}_{120 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkcb \klnkbc \llnkcb \mlnkbc \nlnkaa \end{tikzpicture}\emm = f_{6 } \nonumber\\ f^{120}_{110 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkcb \klnkbc \llnkbc \mlnkaa \nlnkaa \end{tikzpicture}\emm & = ( f^{210}_{220 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkbc \klnkcb \llnkcb \mlnkaa \nlnkaa \end{tikzpicture}\emm ) ^*=f_{7}\end{aligned}\ ] ] there are nine potentially non - zero components in @xmath282 , which are denoted by @xmath305, ... ,@xmath350 : @xmath351 using the transformations discussed in section [ fixphase ] , we can fix the phases of @xmath314 , @xmath352 , @xmath325 , and @xmath305 to make them positive .    the fixed - point conditions ( [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] , [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) form a set of non - linear equation on the variables @xmath315 , @xmath308 , and @xmath249 , which can be solved exactly . after fixing the phases using the transformations discussed in section [ fixphase ] , we find only one solution @xmath353 we also find @xmath319    the fixed - point state corresponds to the @xmath344 string - net condensed state@xcite whose low energy effective field theory is the @xmath322 chern - simons gauge theory@xcite @xmath354 which is the @xmath344 gauge theory . we note that all the above simple solutions also satisfy the standard pentagon identity , although we solved the weaker projective pentagon identity . it is not clear if we can find non - trivial solutions that do not satisfy the standard pentagon identity . there are several ways to define time reversal operation for the graphic states . the simplest one is given by @xmath355 where @xmath146 represents the labels on the vertices and links which are not changed under @xmath356 . ( this corresponds to the situation where the different states on the links and the vertices are realized by different occupations of scalar bosons . ) for such a time reversal transformation , @xmath357 and the real solutions of the fixed - point conditions ( [ nnnn ] , [ di ] , [ nnstar ] , [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] , [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) give us a classification of time reversal invariant topological orders in local spin systems . note that the time reversal invariant topological orders are equivalent class of local orthogonal transformations that connect to the identity transformation continuously . different real solutions @xmath358 of the fixed - point conditions do not always correspond to different time reversal invariant topological orders . the solutions differ by some unimportant phase factors ( which are @xmath359 signs ) correspond to the same topological order .    to understand the above result , we notice that , from the structure of the fixed - point conditions , if @xmath279 is a solution , then @xmath360 is also a solution , where @xmath361 , @xmath362 , and @xmath363 . however , the three phase factors @xmath364 , @xmath365 , and @xmath366 do not lead to different fixed - point wave functions , since they only affect the total phase of the wave function and are unphysical . on the other hand , the fixed - point solutions may also contain phase factors that do correspond to different fixed - point wave functions . for example , the local orthogonal transformation @xmath367 does not affect the fusion rule @xmath137 , where @xmath287 is the number of links with @xmath288-state and @xmath289-state . such a local orthogonal transformation changes @xmath279 and generates a discrete family of the fixed - point wave functions . similarly , we can consider the following local orthogonal transformation @xmath368 that acts on each vertex with states @xmath292 on the three edges connecting to the vertex . such a local orthogonal transformation also does not affect the fusion rule @xmath137 . the new local orthogonal transformation changes @xmath279 and generates a family of the fixed - point wave functions parameterized by the orthogonal matrix @xmath369 . now the question is that do those solutions related by local orthogonal transformations have the same time reversal invariant topological order or not . we know that two gapped wave functions have the same time reversal invariant topological order if and only if they can be connected by local orthogonal transformation that _ connects to identity continuously_. it is well known that an orthogonal matrix whose determinant is @xmath370 does not connect to identity . thus it appears that local orthogonal transformations some times can generate different time reversal invariant topological orders . however , when we use the equivalent classes of local orthogonal transformations to define time reversal invariant topological orders , we not only assume the local orthogonal transformations to connect to identity continuously , we also assume that we can expand the local hilbert spaces ( say by increasing the range of the indices @xmath115 and @xmath150 that label the states on the edges and the vertices ) . the local orthogonal transformations can act on those enlarged hilbert spaces and can connect to identity in those enlarged hilbert spaces . even when a local orthogonal transformation can not be deformed into identity in the original hilbert space , it can always be deformed into identity continuously in an enlarged hilbert space . thus two real wave functions related by a local orthogonal transformation always have the same time reversal invariant topological order . for example , an orthogonal matrix @xmath371 that acts on states @xmath295 and @xmath296 does not connect to identity within the space of two by two orthogonal matrices . however , we can embed the above orthogonal matrix into a three by three orthogonal matrix that acts on @xmath295 , @xmath296 , and @xmath372 : @xmath373 . such a three by three orthogonal matrix does connect to identity within the space of three by three orthogonal matrices . this completes our argument that all local orthogonal transformations can connect to identity continuously at least in an enlarged hilbert space . so , after factoring out the unimportant phase factors discussed above , the real solutions of the fixed - point conditions may uniquely correspond to time reversal invariant topological orders . the four types of real solutions discussed in the last section are examples of four different time reversal invariant topological orders .    for the @xmath294 loop states and the @xmath294 string - net state , we only have two states on each link . in this case , we can treat the two states as the two states of an electron spin . the time reversal transformation now becomes @xmath374 on each link . for such a time reversal transformation @xmath375 . the two @xmath294 loop states and the @xmath294 string - net state are not invariant under such an time - reversal transformation . once the fixed - point states have been identified and the labeling of topological orders has being found , we then face the next important issue : given a generic ground state wave function of a system , how to identify the topological orders in the state ? in other words , how to calculate the data @xmath376 that characterize the topological orders from a generic wave function ?    one way to address the above issue is to have a general renormalization procedure which flows other states in the same phase to the simple fixed - point state so that we can identify topological order from the resulting fixed - point state . that is , we want to find a local unitary transformation which removes local entanglement and gets rid of unnecessary degrees of freedom from the state . how to find the appropriate unitary transformation for a specific state is then the central problem in this renormalization procedure . such a procedure for one dimensional tensor product states(tps ) ( also called matrix product states ) has been given in . here we will propose a method to renormalize two dimensional tps , where nontrivial topological orders emerge . the basic idea is to use the glu transformation discussed in section [ wvrg ] . note that , through the glu transformation , we can reduce the number of labels in a region a to the minimal value without loosing any quantum information ( see fig . [ glut ] ) . this is because the glu transformation is a lossless projection into the support space of the state in the region a. by performing such glu transformations on _ overlapping _ regions repeatedly ( see fig . [ qc]a ) , we can reduce a generic wave function to the simple fixed - point form discussed above . it should be noted that any state reducible in this way can be represented as a mera @xcite .    in the following , we will present this renormalization procedure for two dimensional tps where we find a method to calculate the proper glu transformations . the tensor product states are many - body entangled quantum states described with local tensors . by making use of the entanglement information contained in the local tensors , we are able to come up with an efficient algorithm to renormalize two dimensional tps .    this algorithm can be very useful in the study of quantum phases . due to the efficiency in representation , tps has found wide application as variational ansatz states in the studies of quantum many - body systems @xcite . suppose that in a variational study we have found a set of tensors which describe the ground state of a two dimensional many - body hamiltonian and want to determine the phase this state belongs to . we can apply our renormalization algorithm to this tensor product state , which removes local entanglement and flows the state to its fixed point . by identifying the kind of order present in the fixed - point state , we can obtain the phase information for the original state .    in this section , we will give a detailed description of the algorithm and in the next section we will present its application to some simple ( but nontrivial ) cases . the states we are concerned with have translational symmetry and can be described with a translational invariant tensor network . to be specific , we discuss states on a hexagonal lattice . generalization to other regular lattices is straight - forward . consider a two - dimensional spin model on a hexagonal lattice with one spin ( or one qudit ) living at each vertex . the hilbert space of each spin is @xmath377-dimensional . the state can be represented by assigning to every vertex a set of tensors @xmath378 , where @xmath115 labels the local physical dimension and takes value from @xmath43 to @xmath377 . @xmath379 are inner indices along the three directions in the hexagonal lattice respectively . the dimension of the inner indices is @xmath380 . note that the figures in this note are all sideviews with inner indices in the horizontal plane and the physical indices pointing in the vertical direction , if not specified otherwise .     representing a 2d quantum state on hexagonal lattice . @xmath115 is the physical index , @xmath379 are inner indices . right : a tensor product state where each vertex is associated with a tensor . the inner indices of the neighboring tensors connect according to the underlying hexagonal lattice.,width=336 ]    the wave function is given in terms of these tensors by @xmath381 where @xmath382 denotes tensor contraction of all the connected inner indices on the links of the hexagonal lattice . a renormalization procedure of quantum states is composed of local unitary transformations and isometry maps such that the state flows along the path @xmath383 , @xmath384 , @xmath385, ... and finally towards a fixed point @xmath386 . with the tensor product representation , flow of states corresponds to a flow of tensors @xmath387 , @xmath388 , @xmath389 ... we will give the detailed procedure of how the tensors are mapped from one step to the next in the following section .      in one round of renormalization , we start from tensor @xmath390 , do some operation to it which corresponds to local unitary transformations on the state , and map @xmath390 to @xmath391 . the whole procedure can be broken into two parts : the f - move and the p move , in accordance with the two steps introduced in the previous section .      in the f - move , we take a @xmath392{f1}\emm$ ] configuration in the tensor network and map it to a @xmath393{f2}\emm$ ] configuration by doing a local unitary operation . we will see that the tensor product representation of a state leads to a natural way of choosing an appropriate unitary operation for the renormalization of the state .    in order to do so , first we define the double tensor @xmath394 of tensor @xmath395 as @xmath396 graphically the double tensor @xmath394 is represented by two layers of tensor @xmath395 with the physical indices connected . represented as two layers of tensor @xmath395 with the physical indices contracted . the gray layer is the lower layer.,width=144 ]    the tensor @xmath395 giving rise to the same double tensor @xmath394 is not unique . any tensor @xmath397 which differs from @xmath395 by an unitary transformation @xmath97 on physical index @xmath115 gives the same @xmath394 as @xmath97 and @xmath398 cancels out in the contraction of @xmath115 . on the other hand , an unitary transformation on @xmath115 is the only degree of freedom possible , i.e. any @xmath397 which gives rise to the same @xmath394 as @xmath395 differs from @xmath395 by a unitary on @xmath115 . therefore , in the process of turning a tensor @xmath395 into a double tensor @xmath394 and then split it again into a different tensor @xmath397 , we apply a non - trivial local unitary operation on the corresponding state . a well designed way of splitting the double tensor will give us the appropriate unitary transformation we need , as we show below .     and @xmath399 on neighboring sites into a single double tensor @xmath394 ( 2)splitting double tensor @xmath394 into tensor @xmath400 ( 3 ) svd decomposition of tensor @xmath400 into tensors @xmath401 and @xmath402.,width=336 ]    f - move has the following steps . first , construct double tensors for two neighboring sites on the lattice and combine them into a single double tensor with @xmath403 inner indices . @xmath404    note that with respect to the bipartition of indices @xmath405 and @xmath406 , @xmath394 is hermitian @xmath407 and positive semidefinite . therefore it has a spectral decomposition with positive eigenvalues @xmath408 . the corresponding eigenvectors are @xmath409 @xmath410 this spectral decomposition lead to a special way of decomposing double tensor @xmath394 into tensors . define a rank @xmath403 tensor @xmath400(as shown in fig . [ fig : f - move ] after step 2 ) as follows : @xmath411 @xmath400 has four inner indices @xmath412 of dimension @xmath380 and four physical indices @xmath413 also of dimension @xmath380 which are in the direction of @xmath412 respectively . as @xmath409 form an orthonormal set , it is easy to check that @xmath400 gives rise to double tensor @xmath394 . going from @xmath414 and @xmath415 to @xmath400 , we have implemented a local unitary transformation on the physical degrees of freedom on the two sites , so that in @xmath400 the physical indices and the inner indices represent the same configuration . in some sense , we are keeping only the physical degrees of freedom necessary for entanglement with the rest of the system while getting rid of those that are only entangled within this local region . now we do a singular value decomposition of tensor @xmath400 in the direction orthogonal to the link between @xmath414 and @xmath415 and @xmath400 is decomposed into tensors @xmath401 and @xmath402 . @xmath416 this step completes the f - move . ideally , this step should be done exactly so we are only applying local unitary operations to the state . numerically , we keep some large but finite cutoff dimension for the svd step , so this step is approximate .    on a hexagonal lattice , we do f - move on the chosen neighboring pairs of sites ( dash - circled in fig . [ fig : rg_hex ] ) , so that the tensor network is changed into a configuration shown by thick dark lines in fig . [ fig : rg_hex ] . physical indices are omitted from this figure . now by grouping together the three tensors that meet at a triangle , we can map the tensor network back into a hexagonal lattice , with @xmath417 the number of sites in the original lattice . this is achieved by the p - move introduced in the next section .           to form a new tensor @xmath388 on one site of the renormalized hexagonal lattice . ( 2)constructing the double tensor @xmath418 from @xmath388 so that we can start to do f - moves again.,width=336 ]    now we contract the three tensors that meet at a triangle together to form a new tensor in the renormalized lattice as shown in the first step in fig . [ fig : p - move ] @xmath419 where @xmath420 is the physical index of the new tensor which includes all the physical indices of @xmath421 : @xmath422 . note that in the contraction , only inner indices are contracted and the physical indices are simply group together . constructing the double tensor @xmath394 from @xmath395 , we get the renormalized double tensor on the new hexagonal lattice which is in the same form as @xmath423,@xmath399 and we can go back again and do the f - move . one problem with the above renormalization algorithm is that , instead of having one isolated fixed - point tensor for each phase , the algorithm has a continuous family of fixed points which all correspond to the same phase . consider a tensor with structure shown in fig . [ fig : cdl ] .     are entangled within each group but not between the groups.,width=115 ]    the tensor is a tensor product of three parts which include indices @xmath424 respectively . it can be shown that this structure remains invariant under the renormalization flow . therefore , any tensor of this structure is a fixed point of our renormalization flow . however , it is easy to see that the state it represents is a tensor product of loops around each plaquette , which can be disentangled locally into a trivial product state . therefore , the states all have only short - range entanglement and correspond to the topologically trivial phase . the trivial phase has then a continuous family of fixed - point tensors . this situation is very similar to that discussed in . we will keep the terminology and call such a tensor a corner double line tensor . not only does corner double line tensor complicate the situation in the trivial phase , it leads to a continuous family of fixed points in every phase . it can be checked that the tensor product of a corner double line with any other fixed - point tensor is still a fixed - point tensor . the states they correspond to differ only by small loops around each plaquette and represent the same topological order . therefore any single fixed - point tensor gets complicated into a continuous class of fixed - point tensors . in practical application of the renormalization algorithm , in order to identify the topological order of the fixed - point tensor , we need to get rid of such corner double line structures . due to their simple structure , this can always be done , as discussed in the next section . now we present some examples where our algorithm is used to determine the phase of a tensor product state . the algorithm can be applied both to symmetry breaking phases and topological ordered phases . in the study of symmetry breaking / topological ordered phases , suppose that we have obtained some tensor product description of the ground state of the system hamiltonian . we can then apply our algorithm to the tensors , flow them to the fixed point , and see whether they represent a state in the symmetry breaking phase / topological ordered phase or a trivial phase .    for system with symmetry / topological order related to gauge symmetry , it is very important to keep the symmetry / gauge symmetry in the variational approach to ground state and search within the set of tensors that have this symmetry / gauge symmetry@xcite . the resulting tensor will be invariant under such symmetries / gauge symmetries , but the state they correspond to may have different orders . in the symmetry breaking case , the state could have this symmetry or could spontaneous break it . in the topological ordered phase , the state could have nontrivial topological order or be just trivial . our algorithm can then be applied to decide which is the case . in order to correctly determine the phase for such symmetric tensors , it is crucial that we maintain the symmetry / gauge symmetry of the tensor throughout our renormalization process . we will discuss in detail two cases : the ising symmetry breaking phase and the @xmath2 topological ordered phase . for simplicity of discussion and to demonstrate the generality of our renormalization scheme , we will first introduce the square lattice version of the algorithm . all subsequent applications are carried out on square lattice . ( algorithm on a hexagonal lattice would give qualitatively similar result , though quantitatively they might differ , e.g. on the position of critical point . )      tensor product states on a square lattice are represented with one tensor @xmath425 on each vertex , where @xmath115 is the physical index and @xmath426 are the four inner indices in the up , down , left , right direction respectively . we will assume translational invariance and require the tensor to be the same on every vertex . the renormalization procedure is be a local unitary transformation on the state which flows the form of the tensor until it reaches the fixed point . it is implemented in the following steps . first , we form the double tensor @xmath394 from tensor @xmath395 @xmath427     into tensor @xmath428 ( 2 ) svd decomposition of @xmath428 in two different directions , resulting in tensors @xmath401 , @xmath402 and @xmath0 , @xmath429 respectively.,width=240 ]    then do the spectral decomposition of positive operator @xmath394 into @xmath430 and form a new tensor @xmath431 @xmath432 @xmath85,@xmath152,@xmath433,@xmath434 are physical indices in the up , down , left , right directions respectively . this is illustrated in step 1 of fig . [ fig : rg_sq_1 ] . this step is very similar to the second step in the f - move on hexagonal lattice . next we do svd decomposition of tensor @xmath428 . for vertices in sublattice @xmath98 we decompose between the up - right and down - left direction as shown in step 2 of fig . [ fig : rg_sq_1 ] . for vertices in sublattice @xmath21 we decompose between the up - left and down - right direction as shown in step 3 of fig . [ fig : rg_sq_1 ] . @xmath435     has been applied . physical indices of the tensors are not drawn here.,width=115 ]    after the decomposition , the original lattice ( gray lines in fig . [ fig : rg_sq_2 ] ) is transformed into the configuration shown by thick dark lines in fig . [ fig : rg_sq_2 ] . physical indices are omitted from this figure . if we now shrink the small squares , we get a tensor product state on a renormalized square lattice . fig . [ fig : rg_sq_3 ] shows how this is done .        in step 1 , we contract the four tensors that meet at a small square @xmath436 where @xmath420 stands for the combination of all physical indices @xmath437 , @xmath438 . in step 2 , we construct a double tensor @xmath394 form tensor @xmath395 and completes one round of renormalization . now we can go back to step 1 in fig . [ fig : rg_sq_1 ] and flow the tensor further . now we are ready to discuss two particular examples , the ising symmetry breaking phase and the @xmath2 topological ordered phase , to demonstrate how our algorithm can be used to determine the phase of a tensor product state . a typical example for symmetry breaking phase transition is the transverse field ising model . consider a square lattice with one spin @xmath439 on each site . the transverse field ising model is @xmath440 where @xmath441 are nearest neighbor sites . the hamiltonian is invariant under spin flip transformation @xmath442 for any @xmath443 . when @xmath444 , the ground state spontaneously breaks this symmetry into either the all spin up state @xmath445 or the all spin down state @xmath446 . in this case any global superposition @xmath447 , the ground state has all spin polarized in the @xmath448 direction ( @xmath449 ) and does not break this symmetry .    in the variational study of this system , we can require that the variational ground state always have this symmetry , regardless if the system is in the symmetry breaking phase or not . then we will find for @xmath444 the ground state to be @xmath450 . such a global superposition represents the spontaneous symmetry breaking . for @xmath451 , we will find the ground state to be @xmath449 and does not break the symmetry . for @xmath452 , we will need to decide which of the previous two cases it belongs to . we can first find a tensor product representation of an approximate ground state which is symmetric under the spin flip transformation , then apply the renormalization algorithm to find the fixed point and decide which phase the state belongs to . below we will assume a simple form of tensor and demonstrate how the algorithm works . suppose that the tensors obtained from the variational study @xmath425 , where @xmath115,@xmath453,@xmath454,@xmath455,@xmath456 can be @xmath78 or @xmath43 , takes the following form @xmath457 @xmath458 is a parameter between @xmath78 and @xmath43 . under an @xmath448 operation to the physical index , the tensor is changed to @xmath400 @xmath459 @xmath400 can be mapped back to @xmath395 by switching the @xmath78,@xmath43 label for the four inner indices @xmath426 . such a change of basis for the inner indices does not change the contraction result of the tensor and hence the state that is represented . therefore , the state is invariant under the spin flip transformation @xmath460 and we will say that the tensor has this symmetry also .    when @xmath461 , the tensor represents state @xmath450 , which corresponds to the spontaneous symmetry breaking phase . we note that the @xmath461 tensor is a direct sum of dimension-1 tensors . such a direct - sum structure corresponds to spontaneous symmetry breaking , as discussed in detail in . when @xmath462 , the tensor represents state @xmath449 which corresponds to the symmetric phase . when @xmath463 , there must be a phase transition between the two phases however , as @xmath458 goes from @xmath78 to @xmath43 , the tensor varies smoothly with well defined symmetry . it is hard to identify the phase transition point . now we can apply our algorithm to the tensor . first , we notice that at @xmath461 or @xmath43 , the tensor is a fixed point for our algorithm . next , we find that for @xmath464 , the tensor flows to the form with @xmath461 , while for @xmath465 , it flows to the form with @xmath462 . therefore , we can clearly identify the phase a state belongs to using this algorithm and find the phase transition point . note that in our algorithm , we explicitly keep the spin flip symmetry in the tensor . that is , after each renormalization step , we make sure that the renormalized tensor is invariant under spin flip operations up to change of basis for the inner indices . if the symmetry is not carefully preserved , we will not be able to tell the two phases apart . obtained by taking the ratio of the contraction value of the double tensor in two different ways . @xmath466 is invariant under change of scale , basis transformation and corner double line structures of the double tensor and can be used to distinguish different fixed point tensors . for clarity , only one layer of the double tensor is shown . the other layer connects in exactly the same way.,width=240 ]    we also need to mention that for arbitrary @xmath458 , the fixed point that the tensor flows to can be different from the tensor at @xmath461 or @xmath43 by a corner double line structure . we need to get rid of the corner double line structure in the result to identify the real fixed point . this is possible by carefully examining the fixed point structure . another way to distinguish the different fixed points without worrying about corner double lines is to calculate some quantities from the fixed - point tensors that are invariant with the addition of corner double lines . we also want the quantity to be invariant under some trivial changes to the fixed point , such as a change in scale @xmath467 or the change of basis for physical and inner indices . one such quantity is given by the ratio of @xmath468 and @xmath469 defined as @xmath470 , and @xmath471 . [ fig : x2x1 ] gives a graphical representation of these two quantities . in this figure , only one layer of the double tensor is shown . the other layer connects in the exactly the same way . it is easy to verify that @xmath466 is invariant under the change of scale , basis transformation and corner double lines .     for tensors under the renormalization flow . as the number of rg steps increases , the transition in @xmath466 becomes sharper and finally approaches a step function at fixed point . the critical point is at @xmath472.,width=355 ]    we calculate @xmath466 along the renormalization flow . the result is shown in fig . [ fig : x1_ising ] . at the @xmath461 fixed point , @xmath473 while at @xmath462 , @xmath474 . as we increase the number of renormalization steps , the transition between the two fixed points becomes sharper and finally approaches a step function with critical point at @xmath472 . tensors with @xmath475 belongs to the symmetry breaking phase while tensors with @xmath476 belongs to the symmetric phase . the algorithm can also be used to study topological order of quantum states . in this section , we will demonstrate how the algorithm works with @xmath2 topological order . consider again a square lattice but now with one spin @xmath439 per each link . a simple hamiltonian on this lattice with @xmath2 topological order can be defined as @xmath477 where @xmath213 means plaquettes and @xmath478 is all the spin @xmath439s around the plaquette and @xmath479 means vertices and @xmath480 is all the spin @xmath439s connected to the vertex . the ground state wave function of this hamiltonian is a fixed - point wave function and corresponds to the @xmath294 loop state with @xmath320 as discussed in the previous section . the ground state wave function has a simple tensor product representation . for simplicity of discussion we split every spin @xmath439 into two and associate every vertex with four spins . the tensor @xmath481 has four physical indices @xmath482 and three inner indices @xmath483 . @xmath484 it can be checked that @xmath485 is a fixed - point tensor of our algorithm . this tensor has a @xmath2 gauge symmetry . if we apply @xmath486 operation to all the inner indices , where @xmath486 maps @xmath78 to @xmath78 and @xmath43 to @xmath370 , the tensor remains invariant as only even configurations of the inner indices are nonzero in the tensor . consider then the following set of tensor parameterized by @xmath31 @xmath487 at @xmath488 , this is exactly @xmath485 and the corresponding state has topological order . at @xmath42 , the tensor represents a product state of all @xmath78 and we denote the tensor as @xmath489 . at some critical point in @xmath31 , the state must go through a phase transition . this set of tensors are all invariant under gauge transformation @xmath490 on their inner indices and the tensor seems to vary smoothly with @xmath31 . one way to detect the phase transition is to apply our algorithm . we find that , at @xmath491 , the tensors flow to @xmath485 , while at @xmath492 , the tensors flow to @xmath489 . we determine @xmath35 to be between @xmath493 . as this model is mathematically equivalent to two dimensional classical ising model where the transition point is known to great accuracy , we compare our result to that result and find our result to be within @xmath494 accuracy ( @xmath495 ) . again in the renormalization algorithm , we need to carefully preserve the @xmath2 gauge symmetry of the tensor so that we can correctly determine the phase of the states . the fixed - point tensor structure might also be complicated by corner double line structures , but it is always possible to identify and get rid of them . similarly , we can calculate the invariance quantity @xmath466 to distinguish the two fixed points . @xmath474 for @xmath485 while @xmath473 for @xmath489 . the result is plotted in fig . [ fig : x1_z2 ] and we can see that the transition in @xmath466 approaches a step function after a large number of steps of rg , i.e. at the fixed point . the critical point is at @xmath496 . for @xmath492 , the tensor belongs to the trivial phase , while for @xmath491 , the tensor belongs to the @xmath2 topological ordered phase .     for tensors under the renormalization flow . as the number of rg steps increases , the transition in @xmath466 becomes sharper and finally approaches a step function at fixed point . the critical point is at @xmath496.,width=355 ]    our algorithm can also be used to demonstrate the stability of topological order against local perturbation . as is shown in ref . , local perturbations to the @xmath2 hamiltonian correspond to variations in tensor that do not break the @xmath2 gauge symmetry . we picked tensors in the neighborhood of @xmath485 which preserve this gauge symmetry randomly and applied our renormalization algorithm ( gauge symmetry is kept throughout the renormalization process ) . we find that as long as the variation is small enough , the tensor flows back to @xmath485 , up to a corner double line structure . this result shows that the @xmath2 topological ordered phase is stable against local perturbations . in this paper , we discuss a defining relation between local unitary transformation and quantum phases . we argue that two gapped states are related by a local unitary transformation if and only if the two states belong to the same quantum phase . we can use the equivalent classes of local unitary transformations to define `` patterns of long range entanglement '' . so the patterns of long range entanglement correspond to universality classes of quantum phases , and are the essence of topological orders.@xcite    as an application of this point of view of quantum phases and topological order , we use the generalized local unitary transformations to generate a wave function renormalization , where the wave functions flow within the same universality class of a quantum phase ( or the same equivalent class of the local unitary transformations ) . in other words , the renormalization flow of a wave function does not change its topological order . such a wave function renormalization allows us to classify topological orders , by classifying the fixed - point wave functions and the associated fixed - point local unitary transformations . first , we find that the fixed - point local unitary transformations are described by the data @xmath497 that satisfy @xmath498 @xmath499 @xmath500 [ see eqns . ( [ nnnn ] , [ di ] , [ nnstar ] [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] ) ] . from the data @xmath501 we can further find out the fixed - point wave function by solving the following equations for @xmath502 : @xmath503 [ see eqns . ( [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) ] .    the combined data @xmath504 that satisfy the conditions , , and classify a large class of topological orders . we see that the problem of classifying a large class of topological orders becomes the problem of solving a set of non - linear algebraic equations , , , and .    the combined data @xmath504 that satisfy the conditions , , , and also classify a large class of time reversal invariant topological orders , if we restrict ourselves to real solutions . the solutions related by local orthogonal transformations all belong to the same phase , since the local orthogonal transformations always connect to identity if we enlarge the hilbert space . we like to point out that we can not claim that the solutions @xmath505 classify all topological orders since we have assumed that the fixed - point local unitary transformations are described by tensors of finite dimensions . it appears that chiral topological orders , such as quantum hall states , are described by tensors of infinite dimensions . we note that the data @xmath504 just characterize different fixed - point wave functions . it is not guaranteed that the different data will represent different topological orders . however , for the simple solutions discussed in this paper , they all coincide with string - net states , where the topological properties , such as the ground state degeneracy , number of quasiparticle types , the quasiparticle statistics _ etc _ , were calculated from the data . from those topological properties , we know that those different simple solutions represent different topological orders . also @xmath506 are real for the simple solutions discussed here . thus they also represent topological orders with time reversal symmetry . ( certainly , at the same time , they represent stable topological orders even without time reversal symmetry . ) we also like to point out that our description of fixed - point local unitary transformations is very similar to the description of string - net states . however , the conditions , , , and on the data appear to be weaker than ( or equivalent to ) those@xcite on the string - net data . so the fixed - point wave functions discussed in this paper may include all the string - net states ( in 2d ) . last , we present a wave function renormalization scheme , based on the glu transformations for generic tps . such a wave function renormalization always flows within the same phase ( or within the same equivalence class of lu transformations ) . it allows us to determine which phase a generic tps belongs to by studying the resulting fixed - point wave functions . we demonstrated the effectiveness of our method for both symmetry breaking phases and topological ordered phases . we find that we can even use tensors that do not break symmetry to describe spontaneous symmetry breaking states : if a state described by a symmetric tensor has a spontaneous symmetry breaking , the symmetric tensor will flow to a fixed - point tensor that has a form of direct sum . we would like to thank i. chuang , m. hastings , m. levin , f. verstraete , z .- h . wang , y .- s . wu , s. bravyi for some very helpful discussions . xgw is supported by nsf grant no . zcg is supported in part by the nsf grant no . nsfphy05 - 51164 . in section [ qphucl ] and section [ tolren ] , we argued about the equivalence relation between gapped quantum ground states in the same phase . we concluded that two states are in the same phase if and only if they can be connected by local unitary evolution or constant depth quantum circuit : @xmath507 |\phi(0)\>\\   |\phi(1)\ > \sim |\phi(0)\ > & \text{\ iff\ } &   |\phi(1)\ > = u^m_{circ } |\phi(0)\>\end{aligned}\ ] ]    now we want to make these arguments more rigorous , by stating clearly what is proved and what is conjectured , and by giving precise definition of two states being the same , the locality of operators , etc . we will show the equivalence in the following steps :          ( all these discussions can be generalized to the case where the system has certain symmetries . @xmath51 and @xmath508 used in the equivalence relation will then have the same symmetry as the system hamiltonian @xmath62 . ) first , according to the definition in section [ qphucl ] , two states @xmath36 and @xmath38 are in the same phase if we can find a family of local hamiltonians @xmath32 , @xmath509 $ ] with @xmath41 being its ground state such that the ground state average of any local operator @xmath33 , @xmath510 changes smoothly from @xmath42 to @xmath488 . here we allow a more general notion of locality for the hamiltonian @xcite and require @xmath32 to be a sum of local operators @xmath511 : @xmath512 where @xmath511 is a hermitian operator defined on a compact region @xmath486 . @xmath513 sums over a set @xmath514 of regions . the set @xmath514 contains regions that differ by translations . the set @xmath514 also contains regions with different sizes . however , @xmath511 approaches zero exponentially as the size of the region @xmath486 approaches infinity . or more precisely , for all sites @xmath515 in the lattice @xmath516 for some positive constant @xmath517 . here sums over all regions in the set @xmath514 that cover the site @xmath515 , @xmath519 denotes operator norm , @xmath520 is the cardinality of @xmath486 , and @xmath521 is the diameter of @xmath486 . therefore , instead of being exactly zero outside of a finite region , the interaction terms can have an exponentially decaying tail . if @xmath36 and @xmath38 are gapped ground states of @xmath37 and @xmath39 , then we assume that for all @xmath510 to be smooth , @xmath32 must remain gapped for all @xmath31 . if @xmath32 closes gap for some @xmath35 , then there must exist a local operator @xmath33 such that @xmath510 has a singularity at @xmath35 . we call the gapped @xmath32 an adiabatic connection between two states in the same phase . let @xmath32 be a differentiable family of local hamiltonians and @xmath41 be its ground state . if the excitation gap above @xmath41 is larger than some finite value @xmath522 for all @xmath31 , then we can define @xmath523 , such that @xmath524 |\phi(0)\>$ ] .    where @xmath525 is a function which has the following properties : 1 . the fourier transform of @xmath525 , @xmath526 is equal to @xmath527 for @xmath528 . @xmath529 is infinitely differentiable . 3 . @xmath530 . under this definition , @xmath51 is local ( almost ) and satisfies @xmath531 \right \| \leq h'(\text{dist}(z , b ) ) \left | z \right \vert \left \vert \t h_z(g ) \right \| \left \vert o_b \right \| \label{local_th}\ ] ] where @xmath532 is any operator supported on site @xmath21 , @xmath533 is the smallest distance between @xmath21 and any site in @xmath486 , and @xmath534 is a function which decays faster than any negative power of @xmath85 . @xmath535 $ ] denotes commutator of two operators . this is called the quasi - adiabatic continuation of states . therefore , we can show that if @xmath36 and @xmath38 are in the same phase , then we can find a local ( as defined in ) hamiltonian @xmath536 such that @xmath524 |\phi(0)\>$ ] . in other words , states in the same phase are equivalent under local unitary evolution . note that here we map @xmath36 exactly to @xmath38 . there is another version of quasi - adiabatic continuation @xcite , where the mapping is approximate . in that case , @xmath536 can be defined to have only exponentially small tail outside of a finite region instead of a tail which decays faster than any negative power . @xmath537 |\phi(0)\>$ ] will not be exactly the same as @xmath38 , but any local measurement on them will give approximately the same result . next we want to show that the reverse is also true . suppose that @xmath36 is the gapped ground state of a local hamiltonian @xmath37 , @xmath538 and each @xmath539 is supported on a finite region @xmath486 . apply a local unitary evolution generated by @xmath540 to @xmath36 and take it to @xmath41 , @xmath541 , @xmath542 , where @xmath543 $ ] . @xmath41 is then ground state of @xmath544 . under unitary transformation @xmath545 the spectrum of @xmath37 does nt change , therefore @xmath32 remains gapped . to show that @xmath32 also remains local , we use the lieb - robinson bound derived in , which gives @xmath546 \right \| & = & \left \vert \left[u_gh_z(0)u_g^{\dagger } , o_b\right ] \right \|\\ & \leq & h(\text{dist}(z , b ) ) \left |z\right \vert \left \vert h_z(0 ) \right \| \left \vert o_b \right \|\end{aligned}\ ] ] where @xmath547 decays faster than any negative power of @xmath85 . therefore , @xmath511 remains local up to a tail which decays faster than any negative power . @xmath32 then forms a local gapped adiabatic connection between @xmath36 and @xmath38 .    to detect for phase transition , we must check whether the ground state average value of any local operator @xmath33 , @xmath510 , has a singularity or not . @xmath548 . using the lieb - robinson bound given in , we find @xmath549 remains local ( in the sense of ) and evolves smoothly with @xmath31 . therefore , @xmath510 changes smoothly . in fact , because @xmath32 is differentiable , the derivative of @xmath510 to any order always exists . therefore , there is no singularity in the ground state average value of any local operator @xmath33 and @xmath36 and @xmath38 are in the same phase . we have hence shown that states connected with local unitary evolution are in the same phase . this completes the equivalence relation stated in . the definition of locality is slightly different in different cases , so there is still some gap in the equivalence relation . however , we believe that the equivalence relation should be valid with a slight generalization of the definition of locality . lastly , we want to show that the equivalence relation is still valid if we use constant depth quantum circuit instead of local unitary evolution . this is true because we can always simulate a local unitary evolution using a constant depth quantum circuit and vice - verse .    to simulate a local unitary evolution of the form @xmath537 $ ] , first divide the total time into small segments @xmath550 such that @xmath551 \simeq e^{-i\delta t h(m\delta t)}$ ] . the set of local operators @xmath553 can always be divided into a finite number of subsets @xmath554 , @xmath555 ... such that elements in the same subset commute with each other . then we can do trotter expansion and approximate @xmath556 as @xmath557 , where @xmath558 , @xmath559 ... each term in @xmath560 , @xmath561 commute , therefore they can be implement as a piece - wise local unitary operator . putting these piece - wise local unitary operators together , we have a quantum circuit which simulates the local unitary evolution . the depth of circuit is proportional to @xmath562 . it can be shown that to achieve a simulation with constant error , a constant depth circuit would suffice@xcite .    on the other hand , to simulate a constant depth quantum circuit @xmath563 , where @xmath564 with a local unitary evolution , we can define the time dependent hamiltonian as @xmath565 , such that for @xmath566 , @xmath567 . it is easy to check that @xmath568=u_{circ}$ ] . the simulation time needed is @xmath569 . we can always choose a finite @xmath550 such that @xmath570 is finite and the evolution time is finite .
two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states . on the other hand , 
 gapped ground states remain within the same phase under local unitary transformations . 
 therefore , local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems . 
 since local unitary transformations can remove local entanglement , the above equivalence / universality classes correspond to pattern of long range entanglement , which is the essence of topological order . 
 the local unitary transformation also allows us to define a wave function renormalization scheme , under which a wave function can flow to a simpler one within the same equivalence / universality class . using such a setup , we find conditions on the possible fixed - point wave functions where the local unitary transformations have _ finite _ dimensions . 
 the solutions of the conditions allow us to classify this type of topological orders , which generalize the string - net classification of topological orders . 
 we also describe an algorithm of wave function renormalization induced by local unitary transformations . 
 the algorithm allows us to calculate the flow of tensor - product wave functions which are not at the fixed points . 
 this will allow us to calculate topological orders as well as symmetry breaking orders in a generic tensor - product state .