Entanglement renormalization in two spatial dimensions

We propose and test a scheme for entanglement renormalization capable of addressing large two-dimensional quantum lattice systems. In a translationally invariant system, the cost of simulations grows only as the logarithm of the lattice size; at a quantum critical point, the simulation cost becomes independent of the lattice size and infinite systems can be analysed. We demonstrate the performance of the scheme by investigating the low energy properties of the 2D quantum Ising model on a square lattice of linear size L={6,9,18,54,inf} with periodic boundary conditions. We compute the ground state and evaluate local observables and two-point correlators. We also produce accurate estimates of the critical magnetic field and critical exponent beta. A calculation of the energy gap shows that it scales as 1/L at the critical point.

Note:

4 pages, 5 figures, RevTeX 4. Revised for greater clarity

References(20)

Figures(5)

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The tensors of the MERA are called disentanglers u or isometries w depending on whether they are in charge of eliminating short-range entanglement or of mapping a block of sites into a single site [1, 4, 5]. This distinction is somewhat arbitrary: one can consider tensors that fulfill the two roles simultaneously, such as tensor v in Fig. 1, that we still call disentangler. All these tensors must be isometric, that is u† u = I, v† v = I, w† w = hermitian conjugation (†) in Eq. (1) appears for consistency with previous references

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Calculations for χ = 6 are achieved by using a disentangler u with χ = 4 on selected indices. The computation time is reduced to a few hours per point by re-using a MERA previously converged (for a similar magnetic field) as the starting point of a simulation

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