Tensor network renormalization yields the multi-scale entanglement renormalization ansatz

TL;DR

This work shows how to build a multiscale entanglement renormalization ansatz (MERA) representation of the ground state of a many-body Hamiltonian H by applying the Euclidean time evolution operator e(-βH) for infinite β and extends the MERA formalism to classical statistical systems.

Abstract

We show how to build a multi-scale entanglement renormalization ansatz (MERA) representation of the ground state of a many-body Hamiltonian $H$ by applying the recently proposed \textit{tensor network renormalization} (TNR) [G. Evenbly and G. Vidal, arXiv:1412.0732] to the Euclidean time evolution operator $e^{-βH}$ for infinite $β$. This approach bypasses the costly energy minimization of previous MERA algorithms and, when applied to finite inverse temperature $β$, produces a MERA representation of a thermal Gibbs state. Our construction endows TNR with a renormalization group flow in the space of wave-functions and Hamiltonians (and not just in the more abstract space of tensors) and extends the MERA formalism to classical statistical systems.

Figures (13)

• Figure 1: (a) Tensor network the ground state $\hbox{$| \Psi \rangle$}$ of $H$ on an infinite lattice. It is made of copies of tensor $A$ and restricted to the upper half plane $(x,\tau^{+})$, with a row of open indices at $\tau = 0$. (b) By coarse-graining the tensor network while leaving the open indices untouched, we obtain a new tensor network with tensors $A'$ together with one row of disentaglers and isometries. (c) Further coarse-graining of the tensor network produces new coarse-grained tensors $A"$ and a second layer of disentanglers and isometries. (d) By iteration we obtain a full MERA approximation for state $\hbox{$| \Psi \rangle$}$.
• Figure 2: (a) Tensor network on an infinite strip of finite width $\beta$, with two rows of open indices. It is proportional to the thermal state $e^{-\beta H}/Z$. (b) By coarse-graining the tensor network while leaving the open indices untouched, we obtain a new tensor network with tensors $A'$ together with an upper and lower row of disentaglers and isometries. (c) Futher coarse-graining produces a thermal MERA.
• Figure 3: (a) Tensor network on a semi-infinite vertical cylinder of finite width $L$ and with a row of open indices, proportional to the ground state of $H$ on a periodic chain made of $L$ sites. (b) Result of coarse-graining the initial tensor network while not touching its open indices. (c) MERA connected to a semi-infinite vertical cylinder of $O(1)$ width. Inset: transfer matrix $T$ of this cylinder. The eigenvectors of $T$ with largest eigenvalues correspond to the low energy eigenstates of $H$. (d) MERA for the ground state/low energy excited states of $H$, where the top tensor is an eigenvector of the transfer matrix $T$.
• Figure 4: (a) Thermal energy per site (above the ground state energy) as a function of the inverse temperature $\beta$, for the quantum Ising model $H= \sum_i X_i X_{i+1} + \lambda \sum_i Z_i$ in an infinite chain, for different values of magnetic field $\lambda$. Continuous lines correspond to the exact solution. (b) Connected two-point correlators at the critical magnetic field $\lambda = 1$, as a function of the distance $d$, for several values of $\beta$. Continuous lines correspond again to the exact solution. (c) Low energy eigenvalues of $H$ for critical $\lambda=1$ as a function of $1/L$. (d) Low energy spectrum of $H$ for critical $\lambda = 1$ and corresponding momentum (in unites of $2\pi/L$) for $L=1024$ sites, which appear organized according to the conformal towers of the identity $\mathbb{I}$ (red), spin $\sigma$ (green), and energy density $\epsilon$ (blue) primary fields of the Ising CFT, Henkel. Discontinuous lines in (c) and (d) correspond to the finite-size CFT prediction, which ignores corrections of order $L^{-2}$.
• Figure 5: (a) Tensor network for the Euclidean path integral obtain by multiplying small (Euclidean time) two-site gates of the form $e^{-\epsilon h_{i,i+1}}$. (b) Same tensor network after decomposing each two-site gate using a singular value decomposition, according to the inset. (c) Final tensor network for the Euclidean path integral in terms of copies of a single tensor $A$.