Tensor Network Renormalization

We introduce a coarse-graining transformation for tensor networks that can be applied to study both the partition function of a classical statistical system and the Euclidean path integral of a quantum many-body system. The scheme is based upon the insertion of optimized unitary and isometric tensors (disentanglers and isometries) into the tensor network and has, as its key feature, the ability to remove short-range entanglement or correlations at each coarse-graining step. Removal of short-range entanglement results in scale invariance being explicitly recovered at criticality. In this way we obtain a proper renormalization group flow (in the space of tensors), one that in particular (i) is computationally sustainable, even for critical systems, and (ii) has the correct structure of fixed points, both at criticality and away from it. We demonstrate the proposed approach in the context of the 2D classical Ising model.

Figure 1

(a) As an example, we consider a square lattice (slanted 45°) of classical spins, where ${\sigma }_k\in \{+1,-1\}$ is an Ising spin on site $k$. (b) Graphical representation of a part of the tensor network, where each circle denotes a tensor $A$, for the partition function $Z$ of the classical spin model, see Eq. (1). Here, tensor $A_{ijkl}$ encodes the Boltzmann weights of the spins $\{{\sigma }_i,{\sigma }_j,{\sigma }_k,{\sigma }_l\}$ according to the Hamiltonian function $H$; see Eq. (5). (c) Insertion of a pair of disentanglers $u{u}^{†}$ between four tensors, where tensors $\tilde{A}$ are obtained from tensors $A$ through a gauge transformation on their horizontal indices [23], followed by an insertion of four projectors of the form $v{v}^{†}$ (or $w{w}^{†}$). These projectors introduce a truncation error. (d) Tensor $\delta$, whose norm $\parallel \delta \parallel$ measures the truncation error introduced by the isometries $v$ and $w$. Disentanglers and isometries are chosen so as to minimize $\parallel \delta \parallel$.

Figure 2

(a)–(d) Steps of a TNR transformation to produce tensor $A^{(s+1)}$ from tensor $A^{(s)}$. In (a), the insertion of disentanglers $u$ and isometries $v$ and $w$ is made according to Fig. 1. Insets (e)–(g) contain the definition of the auxiliary tensors $B^{(s)}$ and $C^{(s)}$ and the coarse-grained tensor $A^{(s+1)}$.

Figure 3

Benchmark results for the square lattice Ising model on a lattice with $2^{39}$ spins. (a) Relative error in the free energy per site $\delta f$ at the critical temperature $T_c$, comparing the TRG and TNR over a range of bond dimensions $\chi$. The TRG errors fit $\delta f\propto {\chi }^{-3.02}$ (the inset displays them using log-log axes), while TNR errors fit $\delta f\propto \exp (-0.305\chi )$. Extrapolation suggests that the TRG would require bond dimension $\chi \approx 750$ to match the accuracy of the $\chi =42$ TNR result. (b) Spontaneous magnetization $M(T)$ obtained with TNR with $\chi =6$ [30]. Even very close to the critical temperature, $T=0.9994T_c$, the magnetization $M\approx 0.48$ is reproduced to within 1% accuracy.

Figure 4

(a) Singular values ${\lambda }_{\alpha }$ of the matrix $[A^{(s)}{]}_{(ij)(kl)}$ obtained after $s$ RG steps [31] using TNR (the filled circles) or the TRG (the empty circles) for the 2D Ising model at critical temperature $T_C$. (b) Singular values for $T=1.1T_C$. (c) Plot of the von Neumann entropy $-\sum _{\alpha }{\tilde{\lambda }}_{\alpha }\log ({\tilde{\lambda }}_{\alpha })$ of the (normalized) singular values of tensors $[A^{(s)}{]}_{(ij)(kl)}$ obtained with the TRG (the empty circles) or TNR (the filled circles).