Entanglement renormalization and symmetry fractionalization

It is well known that the matrix product state (MPS) description of a gapped ground state with a global on-site symmetry can exhibit “symmetry fractionalization.” Namely, even though the symmetry acts as a linear representation on the physical degrees of freedom, the MPS matrices—which act on some virtual degrees of freedom—can transform under a projective representation. This was instrumental in classifying gapped symmetry-protected phases that manifest in one-dimensional (1D) quantum many-body systems. Here we consider the multiscale entanglement renormalization ansatz (MERA) description of 1D ground states that have global on-site symmetries. We show that, in contrast to the MPS, the symmetry does not fractionalize in the MERA description if the ground state is gapped, assuming that the MERA preserves the symmetry at all length scales. However, it is still possible that the symmetry can fractionalize in the MERA if the ground state is critical, which may be relevant for characterizing critical symmetry-protected phases. Our results also motivate the presumed use of symmetric tensors to implement global on-site symmetries in MERA algorithms.

Figure 1

(a) A tensor network made of three-index tensors arranged on a circle, which can be contracted to obtain the probability amplitudes of a matrix product state (see also Appendix pp1). Open indices correspond to sites of a lattice $\mathcal{L}$. We will consider only translation-invariant MPS, which consists of copies of the same tensor $B$ everywhere. (b) An MPS with a global on-site symmetry $\mathcal{G}$ fulfills the top equality. Here, $U_g$ is a unitary linear representation of the symmetry on each lattice site. Namely, the representation fulfills the group product exactly, $U_g{U}_h=U_{gh}$ for all group elements $g,h$ of $\mathcal{G}$. The local constraint equation (1) implies global symmetry, as illustrated by the bottom equality ($Y_g$ is unitary, $Y_g^{†}{Y}_g=I$).

Figure 2

(a) An entanglement renormalization transformation $W$ as a tensor network that implements a linear map from a lattice $\mathcal{L}$ to a coarse-grained lattice ${\mathcal{L}}^{\prime }$. For simplicity, we assume that copies of the same two tensors $u,w$ appear everywhere. (b) Tensors $u$ and $w$ are isometries; thus, they fulfill these equalities. (c) Tensor $A$ is obtained by contracting $u$ and $w$. (d) Entanglement renormalization transformation made instead from copies of tensor $A$. (e) MERA tensor network for a quantum many-body state of a lattice of 16 sites, obtained by composing several $W$ transformations; different tensors may appear in each $W$.

Figure 3

(a) Our main working assumption: a global on-site symmetry remains global and on site after coarse graining. This follows if tensor $A$ is symmetric [Eq. (3)] by the sequence of equalities (a) = (b) = (c) = (d). In (a) we introduced a resolution of identity $U_g^{\prime }{U_g^{\prime }}^{†}$ on the top open indices. In (b) we applied Eq. (3). In (c) we used $X_g{X}_g^{†}=I$ and the fact that $u,w$ are isometries [Fig. 2].

Figure 4

Proof of the reverse of implication (4). (a) We demand that on-site symmetry remains on site under RG for all group elements $g\in G$. $U_g$ and $U_g^{\prime }$ are the (linear) representations of the symmetry on each site of lattices $\mathcal{L}$ and ${\mathcal{L}}^{\prime }$, respectively. (b) Reorganizing the previous equality by moving the symmetry operators on the right to the left-hand side. (c) Since both $\tilde{W}$ and $W^{†}$ are isometries, the equality shown in (b) can be true only if $\tilde{W}$ is the adjoint of $W^{†}$, that is, $\tilde{W}=W$, as depicted here. (d) $W$ expressed as a translation-invariant matrix product state $|W\rangle$ by bending the top indices. The equation depicted in (c) translates to the MPS $|W\rangle$ having a global symmetry $\mathcal{G}$. (e) The local constraint fulfilled by each MPS tensor as implied by the global on-site symmetry.

Figure 5

The matrix $E_A\equiv {\sum }_i{A}_i\otimes A_i^{*}$ is a rank-1 projector $|\rho \rangle \langle I|$ since the isometry $u$ cancels out with its adjoint $u^{†}$. Thus, MPS $|W\rangle$ is injective. $|I\rangle$ and $|\rho \rangle$ are the left and right eigenvectors of $E_A$, respectively. $|I\rangle$ is the (vectorized) identity matrix. $|\rho \rangle$ is obtained by contracting isometries $w$ and $w^{†}$, as shown; this contraction can be viewed as acting with a completely positive map (whose Kraus operators are given by $w$) on the identity, which results in a positive semidefinite matrix $|\rho \rangle$. Thus, MPS $|W\rangle$ is also in the canonical form.

Figure 6

Coarse graining a normal MPS (top left) by means of entanglement renormalization, applied as a sequence of elementary operations. The resulting coarse-grained MPS is shown on the top right. Red arrows track the representations of the symmetry that appear on the bond indices of the intermediate MPS after each operation. The crucial decomposition step is elaborated in the box: A symmetric matrix $M$ can always be decomposed as a product of two symmetric matrices $P$ and $Q$, $M=PQ$. To see this, we note that $M$ is block diagonal in the symmetry basis (Schur's lemma). By applying a standard matrix decomposition such as eigenvalue decomposition to each block of $M$ separately, we can obtain factor matrices $P$ and $Q$, which are also block diagonal in the symmetry basis. Thus, $P$ and $Q$ are also symmetric.

Figure 7

A patch of the MERA composed of copies of tensors shown in Eq. (11). This MERA represents a state that exhibits both symmetry fractionalization and long-range entanglement. The latter can be deduced by considering blocks of sites (open indices at the bottom) on the 1D lattice with an increasing number of sites; for example, illustrated here are blocks of $\ell =4,8$, and 16 sites indicated in red, blue, and green, respectively. The entanglement of each block with the rest of the lattice is proportional to the number of singlets shared between the block and the remaining lattice; these are the singlets that are intersected by the geodesic (shown in the respective colors) that extends between the end points of the block through the tensor network. For the three blocks illustrated here the respective geodesic intersects four, six, and eight sites. Generally, a block of $\ell$ sites shares approximately ${log}_2\ell$ singlets with the remaining lattice.