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FIG. 20: Quantity X2/X1 obtained by taking the ratio of the contraction value of the double tensor in two different ways. X»2/Xy1 is invariant under change of scale, basis transforma- tion 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.

Figure 20 Quantity X2/X1 obtained by taking the ratio of the contraction value of the double tensor in two different ways. X»2/Xy1 is invariant under change of scale, basis transforma- tion 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.

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FIG. 1: (Color online) Energy spectrum of a gapped sys- tem as a function of a parameter s in the Hamiltonian. (a,b) For gapped system, a quantum phase transition can happen only when energy gap closes. (a) describes a first order quan- tum phase transition (caused by level crossing). (b) describes a continuous quantum phase transition which has a contin- uum of gapless excitations at the transition point. (c) and (d) cannot happen for generic states. A gapped system may have ground state degeneracy, where the energy splitting be- tween the ground states vanishes when system size L — co: limp—+oo € = 0. The energy gap A between ground and excited states on the other hand remains finite as LD > oo.
FIG. 1: (Color online) Energy spectrum of a gapped sys- tem as a function of a parameter s in the Hamiltonian. (a,b) For gapped system, a quantum phase transition can happen only when energy gap closes. (a) describes a first order quan- tum phase transition (caused by level crossing). (b) describes a continuous quantum phase transition which has a contin- uum of gapless excitations at the transition point. (c) and (d) cannot happen for generic states. A gapped system may have ground state degeneracy, where the energy splitting be- tween the ground states vanishes when system size L — co: limp—+oo € = 0. The energy gap A between ground and excited states on the other hand remains finite as LD > oo.
FIG. 2: (Color online) (a) A graphic representation of a quantum circuit, which is formed by (b) unitary operations on patches of finite size 1. The green shading represents a causal structure.
FIG. 2: (Color online) (a) A graphic representation of a quantum circuit, which is formed by (b) unitary operations on patches of finite size 1. The green shading represents a causal structure.
tanglement and simplify the wave function. In Ref. 58, the wave function renormalization for string-net states is generated by the following two basic moves
tanglement and simplify the wave function. In Ref. 58, the wave function renormalization for string-net states is generated by the following two basic moves
FIG. 6: (Color online) (a) A gLU transformation U acts in region A of a state |®), which reduces the degree freedom in region A to those contained only in the support space of |®) in region A. (b) U'U = P is a projector that does not change the state |®).
FIG. 6: (Color online) (a) A gLU transformation U acts in region A of a state |®), which reduces the degree freedom in region A to those contained only in the support space of |®) in region A. (b) U'U = P is a projector that does not change the state |®).
FIG. 7: A quantum state ona graph G. There are N+1 states on each edge which are labeled by | = 0,...,N. There are N, states on each vertex which are labeled by a = 1,..., Nu.
FIG. 7: A quantum state ona graph G. There are N+1 states on each edge which are labeled by | = 0,...,N. There are N, states on each vertex which are labeled by a = 1,..., Nu.
FIG. 9: A quantum state on a graph G. If the three edges of a vertex are in the states 2, 7, and k respectively, then the vertex has Nij~ states, labeled by a = 1,..., Nijz. Note the orientation of the edges are point towards to vertex. Also note that i + j > k runs anti-clockwise.
FIG. 9: A quantum state on a graph G. If the three edges of a vertex are in the states 2, 7, and k respectively, then the vertex has Nij~ states, labeled by a = 1,..., Nijz. Note the orientation of the edges are point towards to vertex. Also note that i + j > k runs anti-clockwise.
If we apply eqn. (18) twice, we reproduce eqn. (14). Thus eqn. (14) is not independent and can be dropped. Denm tho cranhin ranrodcantatinn (11) Wa nata that
If we apply eqn. (18) twice, we reproduce eqn. (14). Thus eqn. (14) is not independent and can be dropped. Denm tho cranhin ranrodcantatinn (11) Wa nata that
two different paths; one contains two steps of local trans- formations and the other contains three steps of local transformations as described by eqn. (11). The two paths lead to the following relations between the wave func- tions: The consistence of the above two relations leads a condi- tion on the F’ tensor. on other part of graph, we obtain different wave func- tions W(a,8,x,m,n) which form a dimension Dj;xip+ space. In other words, Djjx1p* is the support dimension of the state ®g, on the region a, 8, v,m,n with bound- ary state i,7,k,l,p fixed (see the discussion in section VIIB). Since the number of choices of a, 8,x,m,n is Nijktp* = mn Nyim= Nemn«Ninp*; we have Dijkip« < Nijkips. Here we require a similar saturation condition as in (8):
two different paths; one contains two steps of local trans- formations and the other contains three steps of local transformations as described by eqn. (11). The two paths lead to the following relations between the wave func- tions: The consistence of the above two relations leads a condi- tion on the F’ tensor. on other part of graph, we obtain different wave func- tions W(a,8,x,m,n) which form a dimension Dj;xip+ space. In other words, Djjx1p* is the support dimension of the state ®g, on the region a, 8, v,m,n with bound- ary state i,7,k,l,p fixed (see the discussion in section VIIB). Since the number of choices of a, 8,x,m,n is Nijktp* = mn Nyim= Nemn«Ninp*; we have Dijkip« < Nijkips. Here we require a similar saturation condition as in (8):
The simplified second type renormalization can now be written as (since Djj« = 1) via the first type of renormalization steps (see Fig. 10), which are local unitary transformations. In the simplified
The simplified second type renormalization can now be written as (since Djj« = 1) via the first type of renormalization steps (see Fig. 10), which are local unitary transformations. In the simplified
It is not clear if eqn. (48) and eqn. (49) will lead to new independent equations or not. In the following discus- sions, we will not include eqn. (48) and eqn. (49). We find that, at least for simple cases, the equations without eqn. (48) and eqn. (49) are enough to completely deter- mine the solutions.
It is not clear if eqn. (48) and eqn. (49) will lead to new independent equations or not. In the following discus- sions, we will not include eqn. (48) and eqn. (49). We find that, at least for simple cases, the equations without eqn. (48) and eqn. (49) are enough to completely deter- mine the solutions.
Due to the relation (18), the different components of the tensor F; am are not independent. There are only four independent potentially non-zero components which are denoted as fo,...,f3: We note that FyJ?” in eqn. (11) relates wave functions on two graphs. In the above we have drawn the two related graphs after the F' tensor, where the first graph following F' corresponds to the graph on the left-hand side of eqn. (11) and the second graph corresponds to the graph on the right-hand side of eqn. (11). The doted line corresponds to the |0)-state on the link and the solid line corresponds to the |1)-state on the link. There are four potentially non-zero components in Pr , which are denoted by pg,...,p3:
Due to the relation (18), the different components of the tensor F; am are not independent. There are only four independent potentially non-zero components which are denoted as fo,...,f3: We note that FyJ?” in eqn. (11) relates wave functions on two graphs. In the above we have drawn the two related graphs after the F' tensor, where the first graph following F' corresponds to the graph on the left-hand side of eqn. (11) and the second graph corresponds to the graph on the right-hand side of eqn. (11). The doted line corresponds to the |0)-state on the link and the solid line corresponds to the |1)-state on the link. There are four potentially non-zero components in Pr , which are denoted by pg,...,p3:
The 7 = 1 fixed-point state corresponds to the Zz loop condensed state whose low energy effective field theory is the Zz gauge theory.**©° We call such a state, simply, the Zo state. The 7 = —1 fixed-point state corresponds to the double-semion state whose low energy effective field theory is the U(1)xU(1) Chern-Simons gauge theory®?° where 7 = +1. The above simplified equations can be solved exactly. We find two solutions parameterized by n=Hl:
The 7 = 1 fixed-point state corresponds to the Zz loop condensed state whose low energy effective field theory is the Zz gauge theory.**©° We call such a state, simply, the Zo state. The 7 = —1 fixed-point state corresponds to the double-semion state whose low energy effective field theory is the U(1)xU(1) Chern-Simons gauge theory®?° where 7 = +1. The above simplified equations can be solved exactly. We find two solutions parameterized by n=Hl:
Note that F’s described by f; and fs are the only F’s that change the number of |1)-links and the number of |1)|1)|1)-vertices. So we can use the local unitary trans- formation ef (OM +oMii1) to make f; and fs to be positive real numbers. (Here Mj is the total number of |1)-links and M,1, is the total number of |1)|1)|1)-vertices.) We also use the freedom of adjusting the total sign of Fy7;” to make Re(fo) > 0. Again, due to the relation (18), the different compo- nents of the tensor F;j”" are not independent. Now there are seven independent potentially non-zero components which are denoted as fo,...,f¢:
Note that F’s described by f; and fs are the only F’s that change the number of |1)-links and the number of |1)|1)|1)-vertices. So we can use the local unitary trans- formation ef (OM +oMii1) to make f; and fs to be positive real numbers. (Here Mj is the total number of |1)-links and M,1, is the total number of |1)|1)|1)-vertices.) We also use the freedom of adjusting the total sign of Fy7;” to make Re(fo) > 0. Again, due to the relation (18), the different compo- nents of the tensor F;j”" are not independent. Now there are seven independent potentially non-zero components which are denoted as fo,...,f¢:
The fixed-point state corresponds to the N = 1 string-net condensed state°® whose low energy effective field theory is the doubled SO(3) Chern-Simons gauge theory.®°
The fixed-point state corresponds to the N = 1 string-net condensed state°® whose low energy effective field theory is the doubled SO(3) Chern-Simons gauge theory.®°
EES RENAE RENEE nls inveietenemeerineneeunarin: PR Ieee gS STIRS RAE 0 SAMIAM Eee eee eneeone enn VGeRWNe neg meena vwwmtanen emcees eters the tensor F; a are not independent. There are eight independent potentially non-zero components which are denoted as fo,...,f7: The above N;;; satisfies eqn. (9), eqn. (30), and eqn. (47). Te ee A bee: woxlenktons TG hw Jefe: amon: -
EES RENAE RENEE nls inveietenemeerineneeunarin: PR Ieee gS STIRS RAE 0 SAMIAM Eee eee eneeone enn VGeRWNe neg meena vwwmtanen emcees eters the tensor F; a are not independent. There are eight independent potentially non-zero components which are denoted as fo,...,f7: The above N;;; satisfies eqn. (9), eqn. (30), and eqn. (47). Te ee A bee: woxlenktons TG hw Jefe: amon: -
FIG. 11: (Color online) Left: Tensor T’ representing a 2D quantum state on hexagonal lattice. 7 is the physical index, a, 8,y 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.
FIG. 11: (Color online) Left: Tensor T’ representing a 2D quantum state on hexagonal lattice. 7 is the physical index, a, 8,y 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.
FIG. 13: (Color online) F-move in the renormalization proce- dure: (1) combining double tensors T; and Tz on neighboring sites into a single double tensor T (2)Splitting double tensor T into tensor T (3) SVD decomposition of tensor T into tensors T, and Ty. FIG. 12: (Color online) Double tensor T represented as two layers of tensor T’ with the physical indices contracted. The gray layer is the lower layer.
FIG. 13: (Color online) F-move in the renormalization proce- dure: (1) combining double tensors T; and Tz on neighboring sites into a single double tensor T (2)Splitting double tensor T into tensor T (3) SVD decomposition of tensor T into tensors T, and Ty. FIG. 12: (Color online) Double tensor T represented as two layers of tensor T’ with the physical indices contracted. The gray layer is the lower layer.
FIG. 15: (Color online) P-move in the renormalization pro- cedure: (1) contracting three tensors that meet at a triangle Ta, I>, Tc to form a new tensor T™ on one site of the renor- malized hexagonal lattice. (2)constructing the double tensor T® from T™ so that we can start to do F-moves again. FIG. 14: Original hexagonal lattice (gray line) and renormal- ized lattice (black line) after F-move has been applied to the neighboring pairs of sites circled by dash line.
FIG. 15: (Color online) P-move in the renormalization pro- cedure: (1) contracting three tensors that meet at a triangle Ta, I>, Tc to form a new tensor T™ on one site of the renor- malized hexagonal lattice. (2)constructing the double tensor T® from T™ so that we can start to do F-moves again. FIG. 14: Original hexagonal lattice (gray line) and renormal- ized lattice (black line) after F-move has been applied to the neighboring pairs of sites circled by dash line.
FIG. 17: (Color online) Renormalization procedure on square lattice part 1: (1) decomposing double tensor T into tensor T (2) SVD decomposition of T in two different directions, resulting in tensors T,, Ty, and T., Ta respectively.
FIG. 17: (Color online) Renormalization procedure on square lattice part 1: (1) decomposing double tensor T into tensor T (2) SVD decomposition of T in two different directions, resulting in tensors T,, Ty, and T., Ta respectively.
FIG. 19: (Color online) Renormalization procedure on square lattice part 2: (1) combining four tensors that meet at a square into a single tensor (2) constructing a double tensor from it. FIG. 18: Original square lattice (gray line) and renormalized lattice (black line) after the operations in Fig. 17 has been applied. Physical indices of the tensors are not drawn here.
FIG. 19: (Color online) Renormalization procedure on square lattice part 2: (1) combining four tensors that meet at a square into a single tensor (2) constructing a double tensor from it. FIG. 18: Original square lattice (gray line) and renormalized lattice (black line) after the operations in Fig. 17 has been applied. Physical indices of the tensors are not drawn here.
[see eqns. (9, 30, 47 12, 18, 23, 33, 36)]. From the data (Nizks Fim 8 | pk-8) we can further find out the fixed- kln,x6 9~ 4 point wave function by solving the following equations for A‘,i =0,..., N: First, we find that the fixed-point local uni- tary transformations are described by the data (Nizrs Fim pliant that satisfy a
[see eqns. (9, 30, 47 12, 18, 23, 33, 36)]. From the data (Nizks Fim 8 | pk-8) we can further find out the fixed- kln,x6 9~ 4 point wave function by solving the following equations for A‘,i =0,..., N: First, we find that the fixed-point local uni- tary transformations are described by the data (Nizrs Fim pliant that satisfy a
Related topics:
Tensor product semigroupsFixed Point TheoryQuantum Phase Transition
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