Entanglement renormalization

TL;DR

Numerical simulations with the ground state of a 1D lattice at criticality show that the resulting coarse-grained sites require a Hilbert space dimension that does not grow with successive RG transformations, and calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales.

Abstract

In the context of real-space renormalization group methods, we propose a novel scheme for quantum systems defined on a D-dimensional lattice. It is based on a coarse-graining transformation that attempts to reduce the amount of entanglement of a block of lattice sites before truncating its Hilbert space. Numerical simulations involving the ground state of a 1D system at criticality show that the resulting coarse-grained site requires a Hilbert space dimension that does not grow with successive rescaling transformations. As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with a computational effort that scales logarithmically in the system's size. The calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales. At a quantum critical point, each rellevant length scale makes an equivalent contribution to the entanglement of a block with the rest of the system.

Figures (4)

• Figure 1: Isometries and disentanglers. Left: a standard numerical RG transformation builds a coarse-grained site $s'$, with Hilbert space dimension $m$, from a block of two sites $s_1$ and $s_2$ through the isometry $w$ of Eq. (\ref{['eq:isometry']}). Right: by using the disentanglers $u_1$ and $u_2$ of Eq. (\ref{['eq:disentanglers']}), short-range entanglement residing near the boundary of the block is eliminated before the coarse-graining step. As a result, the coarse-grained site $\tilde{s}$ requires a smaller Hilbert space dimension $\tilde{m}$, $\tilde{m} < m$.
• Figure 2: Left: MERA for a 1D lattice with periodic boundary conditions. Notice the fractal nature of the tensor network. Translational symmetry and scale invariance can be naturally incorporated, substantially reducing the computational complexity of the numerical simulations. Right: building block of a MERA for a 2D lattice. Disentanglers and isometries address one of the $x$ and $y$ spatial directions at a time.
• Figure 3: Scaling of the entropy of entanglement in 1D quantum Ising model with transverse magnetic field. Up: in a critical lattice [$h=1$ in Eq. (\ref{['eq:ising_ham']})], the unrenormalized entanglement of the block scales with the block size $L$ according to Eq. (\ref{['eq:critical']}). Instead, renormalized entanglement remains constant along successive RG transformations, as a clear manifestation of scale invariance. Line (i) corresponds to using disentanglers only in the first RG transformation. Line (ii) corresponds to using disentanglers only in the first and second RG transformation. Down: in a noncritical lattice [$h=1.001$ in Eq. (\ref{['eq:ising_ham']})], the unrenormalized entanglement scales roughly as in the critical case until it saturates (a) for block sizes comparable to the correlation length. Beyond that length scale, the renormalized entanglement vanishes (b) and the system becomes effectively unentangled.
• Figure 4: Spectrum of the reduced density matrix of a spin block. Up: as the size $L$ of the spin increases, the number $m$ of eigenvalues $\{p_i\}$ required to achieve a given accuracy $\epsilon$, see Eq. (\ref{['eq:truncation']}), also increases. In particular, $m$ grows roughly exponentially in the number $\tau=\log_2 L$ of RG transformations. The spectrum resulting from applying disentanglers leads to a significantly smaller $\tilde{m}$ invariant along successive RG transformations. Down: spectrum of the reduced density matrix of $2^\tau$ spins immediately before and after using the disentanglers at the $\tau^{th}$ coarse-graining step. These spectra are essentially independent of the value of $\tau=1,\cdots, 14$.