Holographic spin networks from tensor network states

In the holographic correspondence of quantum gravity, a global on-site symmetry at the boundary generally translates to a local gauge symmetry in the bulk. We describe one way how the global boundary on-site symmetries can be gauged within the formalism of the multiscale renormalization ansatz (MERA), in light of the ongoing discussion between tensor networks and holography. We describe how to “lift” the MERA representation of the ground state of a generic one dimensional (1D) local Hamiltonian, which has a global on-site symmetry, to a dual quantum state of a 2D “bulk” lattice on which the symmetry appears gauged. The 2D bulk state decomposes in terms of spin network states, which label a basis in the gauge-invariant sector of the bulk lattice. This decomposition is instrumental to obtain expectation values of gauge-invariant observables in the bulk, and also reveals that the bulk state is generally entangled between the gauge and the remaining (“gravitational”) bulk degrees of freedom that are not fixed by the symmetry. We present numerical results for ground states of several 1D critical spin chains to illustrate that the bulk entanglement potentially depends on the central charge of the underlying conformal field theory. We also discuss the possibility of emergent topological order in the bulk using a simple example, and also of emergent symmetries in the nongauge (gravitational) sector in the bulk. More broadly, our holographic model translates the MERA, a tensor network state, to a superposition of spin network states, as they appear in lattice gauge theories in one higher dimension.

Figure 1

(a) Graphical representation of a fragment of the infinite MERA tensor network representation of a quantum many-body state $|{\mathrm{\Psi }}^{\text{bound}}⟩$ of an infinite lattice $\mathcal{L}$. The thick arrows indicate that the tensor network extends infinitely in the top vertical and both horizontal directions. The vertical direction corresponds to length scale; ${\mathcal{L}}_0\to {\mathcal{L}}_1\to {\mathcal{L}}_2\to {\mathcal{L}}_3\to \cdots$ is a sequence of increasing coarse-grained lattices where ${\mathcal{L}}_0\cong \mathcal{L}$ is the ultraviolet lattice. The MERA may be viewed a tiling of the hyperbolic plane. In the graph metric, in which each edge has unit length, tiles that have the same shape have the same area. For example, the two blue tiles have the same shape but appear to have different areas because we have stretched out a tiling of hyperbolic plane on a flat plane. (b) Indices are decorated with arrows, as depicted in the box, which indicate how the symmetry acts on the tensors. Tensors $\widehat{u}$ and $\widehat{w}$ commute with the action of the symmetry as shown, see Eq. (4). (c) Graphical representation of equalities Eq. (5) fulfilled by the isometric tensors $\widehat{u}$ and $\widehat{w}$.

Figure 2

The lifted MERA tensor network obtained by inserting a 4-index tensor $(\widehat{c}{)}_{l^{\prime }{r}^{\prime }}^{lr}$ on every bond of the MERA representation of a 1D quantum many-body state. Each open index of the lifted MERA labels an orthonormal basis on a different site of the bulk lattice $\mathcal{M}$. Each bond of the MERA is associated with two bulk sites, corresponding to the open indices $l$ and $r$. These two sites carry the left and right gauge transformations, respectively. The lifted MERA represents a quantum state of $\mathcal{M}$, whose probability amplitudes are obtained by contracting all the tensors of the lifted tensor network. The tensor $\widehat{c}$ is not fixed and parametrizes our ansatz for the holographic dual of the 1D state.

Figure 3

(a) Graphical representation of the copy tensor $(\widehat{c}{)}_{l^{\prime }{r}^{\prime }}^{lr}$. The symmetry acts as ${\widehat{V}}_g$ (blue solid circle) on an incoming index and as ${\widehat{V}}_g^{†}$ (red solid circle) on an outgoing index for all $g\in \mathcal{G}$. (b,c) The action of a symmetry operator on index $l$ (index $r$) transfers to the index $l^{\prime }$ (index $r^{\prime }$). The left hand side of each equality depicts the action of the symmetry on the copy tensor according to index arrows, while the right hand side depicts an equivalent action of the symmetry on the tensor (not necessarily according to the arrows). (d) Tensor $\widehat{c}$ remains invariant under the action of the symmetry according to the index arrows.

Figure 4

Local gauge symmetry of the bulk state $|{\mathrm{\Psi }}^{\text{bulk}}⟩$. Here we illustrate that the lifted MERA, and thus $|{\mathrm{\Psi }}^{\text{bulk}}⟩$, remains invariant under the action of two elementary gauge transformations on the bulk lattice $\mathcal{M}$, corresponding to two different group elements $g,g^{\prime }\in \mathcal{G}$ respectively. One gauge transformation acts on the 4 bulk sites (counting clockwise starting at the top left in the graphical representation) immediately surrounding tensor $\widehat{w}$ (highlighted green) as ${\widehat{V}}_g\otimes {\widehat{V}}_g^{†}\otimes {\widehat{V}}_g^{†}\otimes {\widehat{V}}_g^{†}$, and the other acts on the 4 bulk sites immediately surrounding tensor $\widehat{u}$ (highlighted yellow) as ${\widehat{V}}_{g^{\prime }}\otimes {\widehat{V}}_{g^{\prime }}\otimes {\widehat{V}}_{g^{\prime }}^{†}\otimes {\widehat{V}}_{g^{\prime }}^{†}$. This is shown by means of two equalities: $(\mathrm{a})=(\mathrm{b})$, which results from the symmetry properties of the copy tensor [Fig. 3], and $(\mathrm{b})=(\mathrm{c})$, which results from the fact that the tensors $\widehat{u}$ and $\widehat{w}$ are $\mathcal{G}$-symmetric [Fig. 1].

Figure 5

(a,b) Wigner-Eckart decomposition of tensors $\widehat{u}$ and $\widehat{w}$ into degeneracy tensors and intertwiners of the symmetry group $\mathcal{G}$, Eq. (10). (c,d) Intertwiners ${\widehat{\mathcal{I}}}_e$ and ${\widehat{\mathcal{J}}}_e$ expressed in terms of two Clebsch-Gordan coefficients (solid black circles), Eqs. (11) and (12). The red lines carry the intermediate intertwining charges $e$. (e) Wigner-Eckart decomposition of the copy tensor $\widehat{c}$ according to Eq. (13). The symmetry properties depicted in Fig. 3 imply that the intermediate charge $e$ is trivial, $e=0$ (depicted by the absence of any red line).

Figure 6

The lifted MERA decomposes as a sum of tensor product of two parts: (left) a tensor network composed of degeneracy tensors, and (right) a tensor network composed of intertwiners of the symmetry group, namely, a spin network. The sum is over tuples of symmetry charges $b$ and $e$. Here $b\equiv (b_1,b_2,\cdots )$ is the tuple of symmetry charges carried by all bond and open indices of the lifted MERA, and $e\equiv (e_1,e_2,\cdots )$ is the tuple of intertwining symmetry charges associated with all the intertwiners in the spin network, see Fig. 5. The decomposition separates out the gauge degrees of freedom, dual to the global symmetry at boundary, from the remaining bulk degrees of freedom. The spin network states span the gauge-invariant support of the bulk state, while we view the remaining degrees of freedom to possibly include “gravitational” degrees of freedom in a holographic interpretation of the MERA.

Figure 7

An illustration of the tensor network contraction equating to the expectation value of a gauge-invariant loop operator in the bulk that acts nontrivially on the gauge degrees of freedom (the spin networks) and as the identity on the remaining degrees of freedom. For $\mathcal{G}=Z_2$ and $\widehat{X}$ defined according to Eq. (a8), the loop operator can be understood as a Wilson loop in a $Z_2$ lattice gauge theory (here defined on a hyperbolic lattice).

Figure 8

Useful equalities satisfied by the $\mathcal{G}$-symmetric copy tensor $\widehat{c}$. (a) Contraction of the identity on the open indices of $\widehat{c}$ results in an identity. (b) Tensor $\widehat{c}$ is an isometry.

Figure 9

The bulk sites (red squares) located in the region highlighted yellow were considered to obtain the plots shown in Fig. 10 and Fig. 11. These sites are located along an infinitely long tower of the $\widehat{w}$ tensors. The bulk sites located in the region highlighted blue were considered to obtain the entanglement negativities listed in Table 4. These consist of sites located around a loop of tensors and two sites located at the bottom of the loop.

Figure 10

A probability distribution of the Renyi entanglement entropy $R^{\mathrm{tower}}$ per site [Eq. (16)], computed from randomly sampled bulk states dual to the ground state of the critical Ising model. We sampled $10^5$ bulk states and sorted the corresponding entropy densities into 100 equally spaced bins.

Figure 11

The maximum (square, star), mode (triangle), and minimum (circle) of the probability distribution of the Renyi entanglement entropy $R^{\mathrm{tower}}$ per site [Eq. (16)] per site, obtained from randomly sampled $10^5$ bulk states, dual to the ground state of each of the critical models listed along the $x$-axis. The various $Z_2$ data for XXZ models correspond to $\mathrm{\Delta }\in \{0,0.71,0.81,0.87,1\}$. Explanation in Sec. 5c.

Figure 12

The spectrum of a reduced density matrix obtained from a randomly selected bulk state, dual to the ground state of each of the critical models listed in Eq. (15). The reduced density matrix corresponds to one bulk site, obtained by tracing out all remaining bulk sites and also tracing out the gauge degrees of freedom of that site.

Figure 13

(a) The $Z_2$-symmetric copy tensor ${\widehat{c}}^{[Z_2]}$, Eq. (a7) is invariant under the action of $\widehat{Z}$ on any two of its indices. (b,c) The lifted MERA ${\mathcal{T}}^{\prime }$, and thus the bulk state it represents, is invariant under the action of the loop (dashed red contour) of $\widehat{Z}$’s shown here. Namely, the contraction depicted in (b) simply recovers the lifted MERA (c).

Figure 14

(a) $\widehat{X}$ operators applied on the two open indices of the $Z_2$ copy tensor ${\widehat{c}}^{[Z_2]}$ [Eq. (a7)] transfers to the two bond indices. (b) A diagonal bond transformation ${\widehat{M}}_k^{[Z_2]}$ applied on any of the two bond indices of the copy tensor ${\widehat{c}}^{[Z_2]}$ transfers to the closer open index. Here illustrated for one of the bond indices only. (c,d) Equalities illustrating that tensors ${\widehat{u}}^{\text{topo}}$ and ${\widehat{w}}^{\text{topo}}$ remain invariant under the action of the $\widehat{X}$, Eq. (a8), applied on any two indices.

Figure 15

The bulk state $|{\mathrm{\Phi }}_{\text{topo}}^{\text{bulk}}⟩$ remains invariant under the action of $\widehat{X}$ operators applied on bulk sites located around a plaquette on the bulk lattice, here illustrated for the action of $\widehat{X}$’s on the highlighted plaquette (red). This is shown by means of two equalities. The first equality results from applying the equality depicted in Fig. 14 to all the copy tensors located around the plaquette. The second equality results from using the equalities depicted in Fig. 14, which eliminates the $\widehat{X}$ operators.

Figure 16

(a) Tensor network contraction equating to the bulk expectation value of a loop of $\widehat{X}$’s. (b) The tensor network contraction resulting from simplifying the tensors located below the loop. First, the tensors located at the bottom are contracted with their adjoints and thus cancel out. Consequently, pairs of copy tensors are seen to be contracted together and also cancel out, thanks to the equality depicted Fig. 8. (c) Zoom in to the contraction around tensor $v_4$. Tensor $\widehat{u}$ cancels with its adjoint, resulting in two separate contractions. Each of these is identically zero since the trace of $\widehat{X}$ is 0.

Figure 17

(a) The graphical representation of the decomposition the $\mathcal{G}$-symmetric copy tensor in terms of a trivalent tensor $\widehat{x}$, Eq. (b3), exposing an intermediate bond index $\gamma$ (red) that carries the trivial symmetry charge. (b) A bipartition of the lifted MERA into parts $\mathcal{P}$ and $\mathcal{R}$ by a path (dashed contour) that only intersects (the red bonds of) $N$ copy tensors. $\{p_1,p_2,\dots ,p_N\}$ and $\{r_1,r_2,\dots ,r_N\}$ denote the open indices of the $N$ intersected copy tensors.

Figure 18

The bulk sites (red squares) acted on by the vertex operator ${\widehat{h}}^{[v]}$ and the plaquette operator ${\widehat{h}}^{[p]}$, which appear in Eq. (c1), are highlighted yellow and green respectively.