Tensor Network Renormalization Yields the Multiscale Entanglement Renormalization Ansatz

We show how to build a multiscale entanglement renormalization ansatz (MERA) representation of the ground state of a many-body Hamiltonian $H$ by applying the recently proposed tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)] to the Euclidean time evolution operator $e^{-\beta H}$ for infinite $\beta$. This approach bypasses the costly energy minimization of previous MERA algorithms and, when applied to finite inverse temperature $\beta$, produces a MERA representation of a thermal Gibbs state. Our construction endows tensor network renormalization with a renormalization group flow in the space of wave functions and Hamiltonians (and not merely in the more abstract space of tensors) and extends the MERA formalism to classical statistical systems.

Figure 1

(a) Tensor network, the ground state $|\mathrm{\Psi }⟩$ 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 $A^{\prime }$ together with one row of disentanglers and isometries. (c) Further coarse graining of the tensor network produces new coarse-grained tensors $A^{\prime \prime }$ and a second layer of disentanglers and isometries. (d) By iteration we obtain a full MERA approximation for state $|\mathrm{\Psi }⟩$.

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 $A^{\prime }$ together with an upper and lower row of disentanglers and isometries. (c) Further 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 the largest eigenvalues correspond to the low energy eigenstates of $H$. (d) MERA for the ground state or 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 units 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 conformal field theory, [30]. Discontinuous lines in (c) and (d) correspond to the finite-size conformal field theory prediction, which ignores corrections of order $L^{-2}$.