Continuum Limits of Homogeneous Binary Trees and the Thompson Group

Tree tensor network descriptions of critical quantum spin chains are empirically known to reproduce correlation functions matching conformal field theory (CFT) predictions in the continuum limit. It is natural to seek a more complete correspondence, additionally incorporating dynamics. On the CFT side, this is determined by a representation of the diffeomorphism group of the circle. In a remarkable series of papers, Jones outlined a research program where the Thompson group $\mathsf{T}$ takes the role of the latter in the discrete setting, and representations of $\mathsf{T}$ are constructed from certain elements of a subfactor planar algebra. He also showed that, for a particular example of such a construction, this approach only yields—in the continuum limit—a representation which is highly discontinuous and hence unphysical. Here we show that the same issue arises generically when considering tree tensor networks: the set of coarse-graining maps yielding discontinuous representations has full measure in the set of all isometries. This extends Jones’s no-go example to typical elements of the so-called tensor planar algebra. We also identify an easily verified necessary condition for a continuous limit to exist. This singles out a particular class of tree tensor networks. Our considerations apply to recent approaches for introducing dynamics in holographic codes.

Figure 1

In addition to the isometry $V$: ${\mathbb{C}}^d\to {\mathbb{C}}^d\otimes {\mathbb{C}}^d$ (green triangles), a tensor at the bottom representing a suitable state $|\psi ⟩\in {\mathbb{C}}^d$ fully determine the state under consideration.

Figure 2

The inner product $⟨\mathrm{\Phi },{\rho }^V(f_3)\mathrm{\Psi }⟩$. Here we assume periodic boundary conditions. After three layers of application of the fine-graining isometry $V$, each state is represented by an eight-qudit state. The inner product is obtained by translating the second state $\mathrm{\Psi }$ by $1/2^3$ (which corresponds to one lattice spacing if the qudits are arranged equidistantly on the unit circle), then contracting the tensor network obtained by stacking the adjoint of $\mathrm{\Phi }$ on top. In Fig. 2, appearances of the tensor $x$ [see Fig. 3] are indicated by dotted boxes. In Fig. 2 the same inner product is shown with $x$ inserted, and with dotted boxes indicating the position of $\mathcal{R}(x)$ [see Fig. 3]. Figure 2 shows the same inner product with $\mathcal{R}(x)$ inserted; now, the dotted boxes indicate the position of ${\mathcal{R}}^{\circ 2}(x)$. Finally, Fig. 2 shows the final form of the inner product which corresponds to the right hand side of Eq. (3).

Figure 3

The element $x\in \mathcal{B}({\mathbb{C}}^d\otimes {\mathbb{C}}^d)$ [cf. (4)] and the definition of $\mathcal{R}(z)\in \mathcal{B}({\mathbb{C}}^d\otimes {\mathbb{C}}^d)$ [cf. (5)] used in the proof of Lemma 1.