Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

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.

Figure 1

(Color online) Energy spectrum of a gapped system as a function of a parameter $s$ in the Hamiltonian. [(a) and (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) cannot happen for generic states. A gapped system may have ground-state degeneracy, where the energy splitting between the ground states vanishes when system size $L\to \infty$: ${\text{lim}}_{L\to \infty }ϵ=0$. The energy gap $\Delta$ between ground and excited states on the other hand remains finite as $L\to \infty$.

Figure 2

(Color online) (a) A graphic representation of a quantum circuit, which is formed by (b) unitary operations on patches of finite size $l$. The dark shading represents a causal structure.

Figure 3

(Color online) (a) The possible phases for a Hamiltonian $H\left(g_1,g_2\right)$ without any symmetry. (b) The possible phases for a Hamiltonian $H_{\text{symm}}\left(g_1,g_2\right)$ with some symmetries. The shaded regions in (a) and (b) represent the phases with short-range entanglement (i.e., those ground states can be transformed into a direct product state via a generic LU transformations that do not have any symmetry).

Figure 4

(Color online) A finite depth quantum circuit can transform a state $|\Phi ⟩$ into a direct-product state, if and only if the state $|\Phi ⟩$ 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.

Figure 5

(Color online) A piecewise local unitary transformation can transform some degrees of freedom in a state $\Phi$ 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).

Figure 6

(Color online) (a) A gLU transformation $U$ acts in region A of a state $|\Phi ⟩$, which reduces the degree of freedom in region A to those contained only in the support space of $|\Phi ⟩$ in region A. (b) $U^{†}U=P$ is a projector that does not change the state $|\Phi ⟩$.

Figure 7

A quantum state on a graph $G$. There are $N+1$ states on each edge which are labeled by $l=0,\dots ,N$. There are $N_v$ states on each vertex which are labeled by $\alpha =1,\dots ,N_v$.

Figure 8

The mapping $i\to i^{\ast }$ corresponds to the reverse of the orientation of the edge.

Figure 9

A quantum state on a graph $G$. If the three edges of a vertex are in the states $i$, $j$, and $k$, respectively, then the vertex has $N_{ijk}$ states, labeled by $\alpha =1,\dots ,N_{ijk}$. Note the orientation of the edges are point toward to vertex. Also note that $i\to j\to k$ runs anticlockwise.

Figure 10

A “triangle” graph can be transformed into a “tadpole” via two steps of the first type of wave function renormalization (i.e., two steps of local unitary transformations).

Figure 11

(Color online) Left: tensor $T$ representing a 2D quantum state on hexagonal lattice. $i$ is the physical index and $\alpha ,\beta ,\gamma$ are the 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.

Figure 12

(Color online) Double tensor $\mathbb{T}$ represented as two layers of tensor $T$ with the physical indices contracted. The gray layer is the lower layer.

Figure 13

(Color online) F-move in the renormalization procedure: (1) combining double tensors ${\mathbb{T}}_1$ and ${\mathbb{T}}_2$ on neighboring sites into a single double tensor $\mathbb{T}$, (2) splitting double tensor $\mathbb{T}$ into tensor $\tilde{T}$, and (3) SVD decomposition of tensor $\tilde{T}$ into tensors $T_a$ and $T_b$.

Figure 14

Original hexagonal lattice (gray line) and renormalized lattice (black line) after F-move has been applied to the neighboring pairs of sites circled by dash line.

Figure 15

(Color online) P-move in the renormalization procedure: (1) contracting three tensors that meet at a triangle $T_a,T_b,T_c$ to form a new tensor $T^{\left(1\right)}$ on one site of the renormalized hexagonal lattice. (2) Constructing the double tensor ${\mathbb{T}}^{\left(1\right)}$ from $T^{\left(1\right)}$ so that we can start to do F-moves again.

Figure 16

(Color online) The corner double line tensor which is a fixed point of the renormalization algorithm. The three groups of indices $\left\{\alpha ,\beta ,i_1\right\}$, $\left\{\gamma ,\delta ,i_2\right\}$, and $\left\{ϵ,\lambda ,i_3\right\}$ are entangled within each group but not between the groups.

Figure 17

(Color online) Renormalization procedure on square lattice part 1: (1) decomposing double tensor $\mathbb{T}$ into tensor $\tilde{T}$ and (2) SVD decomposition of $\tilde{T}$ in two different directions, resulting in tensors $T_a,T_b$ and $T_c,T_d$, respectively.

Figure 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.

Figure 19

(Color online) Renormalization procedure on square lattice part 2: (1) combining four tensors that meet at a square into a single tensor and (2) constructing a double tensor from it.

Figure 20

Quantity $X_2/X_1$ obtained by taking the ratio of the contraction value of the double tensor in two different ways. $X_2/X_1$ is invariant under change in 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.

Figure 21

(Color online) $X_2/X_1$ for tensors [Eq. (79)] under the renormalization flow. As the number of RG steps increases, the transition in $X_2/X_1$ becomes sharper and finally approaches a step function at fixed point. The critical point is at ${\lambda }_c=0.358$.

Figure 22

(Color online) $X_2/X_1$ for tensors [Eq. (80)] under the renormalization flow. As the number of RG steps increases, the transition in $X_2/X_1$ becomes sharper and finally approaches a step function at fixed point. The critical point is at $g_c=0.804$.