Frustrated antiferromagnets with entanglement renormalization: ground state of the spin-1/2 Heisenberg model on a kagome lattice

Entanglement renormalization techniques are applied to numerically investigate the ground state of the spin-1/2 Heisenberg model on a kagome lattice. Lattices of N={36,144,inf} sites with periodic boundary conditions are considered. For the infinite lattice, the best approximation to the ground state is found to be a valence bond crystal (VBC) with a 36-site unit cell, compatible with a previous proposal. Its energy per site, E=-0.43221, is an exact upper bound and is lower than the energy of any previous (gapped or algebraic) spin liquid candidate for the ground state.Comment: 6 pages, 7 figures, RevTeX 4. Revised version with improved numerical results

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 analyzed. 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,∞} 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 β. A calculation of the energy gap shows that it scales as 1/L at the critical point

Entanglement renormalization in fermionic systems

We demonstrate, in the context of quadratic fermion lattice models in one and two spatial dimensions, the potential of entanglement renormalization (ER) to define a proper real-space renormalization group transformation. Our results show, for the first time, the validity of the multi-scale entanglement renormalization ansatz (MERA) to describe ground states in two dimensions, even at a quantum critical point. They also unveil a connection between the performance of ER and the logarithmic violations of the boundary law for entanglement in systems with a one-dimensional Fermi surface. ER is recast in the language of creation/annihilation operators and correlation matrices.Comment: 5 pages, 4 figures Second appendix adde

Entanglement renormalization in free bosonic systems: real-space versus momentum-space renormalization group transforms

The ability of entanglement renormalization (ER) to generate a proper real-space renormalization group (RG) flow in extended quantum systems is analysed in the setting of harmonic lattice systems in D=1 and D=2 spatial dimensions. A conceptual overview of the steps involved in momentum-space RG is provided and contrasted against the equivalent steps in the real-space setting. The real-space RG flow, as generated by ER, is compared against the exact results from momentum-space RG, including an investigation of a critical fixed point and the effect of relevant and irrelevant perturbations.Comment: 14 pages, 11 figures, substantial revision

Non-local scaling operators with entanglement renormalization

The multi-scale entanglement renormalization ansatz (MERA) can be used, in its scale invariant version, to describe the ground state of a lattice system at a quantum critical point. From the scale invariant MERA one can determine the local scaling operators of the model. Here we show that, in the presence of a global symmetry $\mathcal{G}$, it is also possible to determine a class of non-local scaling operators. Each operator consist, for a given group element $g\in\mathcal{G}$, of a semi-infinite string \tGamma_g with a local operator $\phi$ attached to its open end. In the case of the quantum Ising model, $\mathcal{G}= \mathbb{Z}_2$, they correspond to the disorder operator $\mu$, the fermionic operators $\psi$ and $\bar{\psi}$, and all their descendants. Together with the local scaling operators identity $\mathbb{I}$, spin $\sigma$ and energy $\epsilon$, the fermionic and disorder scaling operators $\psi$, $\bar{\psi}$ and $\mu$ are the complete list of primary fields of the Ising CFT. Thefore the scale invariant MERA allows us to characterize all the conformal towers of this CFT.Comment: 4 pages, 4 figures. Revised versio

Algorithms for entanglement renormalization

We describe an iterative method to optimize the multi-scale entanglement renormalization ansatz (MERA) for the low-energy subspace of local Hamiltonians on a D-dimensional lattice. For translation invariant systems the cost of this optimization is logarithmic in the linear system size. Specialized algorithms for the treatment of infinite systems are also described. Benchmark simulation results are presented for a variety of 1D systems, namely Ising, Potts, XX and Heisenberg models. The potential to compute expected values of local observables, energy gaps and correlators is investigated.Comment: 23 pages, 28 figure

Low energy excitations of the kagome antiferromagnet and the spin gap issue

In this paper we report the latest results of exact diagonalizations of SU(2) invariant models on various lattices (square, triangular, hexagonal, checkerboard and kagome lattices). We focus on the low lying levels in each S sector. The differences in behavior between gapless systems and gapped ones are exhibited. The plausibility of a gapless spin liquid in the Heisenberg model on the kagome lattice is discussed. A rough estimate of the spin susceptibility in such an hypothesis is given.The evolution of the intra-S channel spectra under the effect of a small perturbation is consistent with the proximity of a quantum critical point. We emphasize that the very small intra-S channel energy scale observed in exact spectra is a very interesting information to understand the low T dynamics of this model.Comment: 6 pages, 5 figures, revised version with a more extended discussion on the issue of a possible proximity with a quantum critical point, a few more details and references, a modified Fig