Local Scale Transformations on the Lattice with Tensor Network Renormalization

Consider the partition function of a classical system in two spatial dimensions, or the Euclidean path integral of a quantum system in two space-time dimensions, both on a lattice. We show that the tensor network renormalization algorithm [G. Evenbly and G. Vidal Phys. Rev. Lett. 115, 180405 (2015)] can be used to implement local scale transformations on these objects, namely, a lattice version of conformal maps. Specifically, we explain how to implement the lattice equivalent of the logarithmic conformal map that transforms the Euclidean plane into a cylinder. As an application, and with the 2D critical Ising model as a concrete example, we use this map to build a lattice version of the scaling operators of the underlying conformal field theory, from which one can extract their scaling dimensions and operator product expansion coefficients.

Figure 1

(a) Graphical illustration of the tensor network, made of copies of a tensor $A$, that represents the partition function $Z$ of a lattice model. Tensor network renormalization (TNR) produces a coarser tensor network made of tensors $A^{\prime }$. (b) The plane $z=x+iy$ is mapped into itself by a global scale transformation $w=\lambda z$ with constant rescaling factor $\lambda$, or to an infinite cylinder of unit radius by a local scaling transformation $w=\log (z)$ with local rescaling factor $\lambda =1/|z|$. In this second case, a circle $|z{|}^2=x^2+y^2$ on the plane is mapped into a circle at constant $s={\log }_2|z|$ and $\theta \in [0,2\pi )$ on the cylinder, where $w=s+i\theta$, and a translation $s\to s+a$ on the cylinder can be interpreted as a change of scale $z\to e^{a}z$ in the plane. Alternatively, an infinite cylinder of finite radius $r$ can also be obtained from the plane $z=x+iy$ by, e.g., identifying the coordinate $x$ with $x+2\pi r$.

Figure 2

(a) Tensor network obtained by removing four tensors $A$ from the representation of the partition function $Z$ on the plane shown in Fig. 1. The puncture at the origin of the plane leaves eight open indices. (b)–(f) Applying TNR to implement a global scale transformation on the plane (see Ref. [17], Sec. A) but without touching the open indices results in a new square tensor network with a puncture, made of tensors $A^{\prime }$, together with two concentric rows of tensors around the origin. The inset relates these tensors to the auxiliary tensors that appear while coarse graining the bulk.

Figure 3

(a) Partition function $Z$ expressed in terms of the resulting cylindrical tensor network, Eq. (4). At the basis of the cylinder, a block of four tensors $A$, denoted $|\mathbb{I}⟩$, has been reinserted to remove the puncture previously introduced at origin of the plane. On a finite system, the cylinder is also finite and has a set of lines, denoted $⟨v_0|$, at the top. (b) Critical scaling map $\mathcal{R}$ in Eq. (5). (c) Eigenvector $|{\varphi }_{\alpha }⟩$ of the scaling map $\mathcal{R}$. (d) Transfer matrix $\mathcal{M}$ in Eq. (8).

Figure 4

(a) A local scale transformation maps the plane with two punctures into a pair of pants. On the lattice, from this pair of pants we can extract the OPE coefficient $C_{\alpha \beta \gamma }$ by inserting scaling operators $|{\varphi }_{\alpha }⟩$ and $|{\varphi }_{\beta }⟩$ on the punctures and evaluating the amplitude of obtaining scaling operator $|{\varphi }_{\gamma }⟩$ as a result of their fusion. (b) A logarithmic conformal transformation maps an $n$-sheeted Riemann surface, consisting of $n$ copies (here, $n=2$) of the plane periodically connected through semi-infinite branch cuts along the positive $x$ axis to a cylinder ($s$, $\theta$) with $s\in \mathbb{R}$ and $\theta \in [0,n2\pi )$.

Figure 5

First 101 scaling dimensions ${\mathrm{\Delta }}_{\alpha }$ of the critical Ising model, obtained from the eigenvalues $2^{-{\mathrm{\Delta }}_{\alpha }}$ of the scaling maps $\mathcal{R}$ (first and second columns) and ${\mathcal{R}}^{\prime }$ (third and fourth columns; see Ref. [17], Sec. B). Even and odd spin ${\mathbb{Z}}_2$ symmetry sectors are indicated by $p=+1$ and $p=-1$, respectively. The primary fields’ identity $\mathbb{I}$, spin $\sigma$, and energy density $\epsilon$, together with their descendants, are the local scaling operators; the disorder $\mu$ and the fermions $\psi$ and $\overline{\psi }$ and their descendants are nonlocal scaling operators.