Simulation of interacting fermions with entanglement renormalization

Rapid Communication

Simulation of interacting fermions with entanglement renormalization

Philippe Corboz, Glen Evenbly, Frank Verstraete, and Guifré Vidal

Phys. Rev. A 81, 010303(R) – Published 21 January 2010

Abstract

We propose an algorithm to simulate interacting fermions on a two-dimensional lattice. The approach is an extension of the entanglement renormalization technique [Phys. Rev. Lett. 99, 220405 (2007)] and the related multiscale entanglement renormalization ansatz. Benchmark calculations for free and interacting fermions on lattices ranging from $6 \times 6$ to $162 \times 162$ sites with periodic boundary conditions confirm the validity of this proposal.

Received 7 May 2009

DOI:https://doi.org/10.1103/PhysRevA.81.010303

Authors & Affiliations

Philippe Corboz¹, Glen Evenbly¹, Frank Verstraete², and Guifré Vidal¹

¹School of Mathematics and Physics, The University of Queensland, Brisbane, Queensland 4072, Australia
²Fakultät für Physik, Universität Wien, Boltzmanngasse 3, A-1090 Wien, Austria

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Vol. 81, Iss. 1 — January 2010

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Images

Figure 1
(i) A $4 \times 4$ lattice $L$ is coarse-grained by applying, first, (ii) disentanglers $u$ on blocks of four sites and, then, (iii) isometries $w$ , also on blocks of four sites, producing (iv) a $2 \times 2$ lattice $L^{'}$ . Note that the Jordan-Wigner order for the sites is such that each isometry $w$ acts on four consecutive sites.Reuse & Permissions
Figure 2
(i) TTN to approximate the ground state $| Ψ 〉$ of $H$ obtained by adding a top tensor (representing the state of $L^{'}$ ) to the isometries $w$ in Fig. 1. Note that the sites (vertical lines) are arranged in one dimension according to the Jordan-Wigner order. Isometries $w$ are local even when written in terms of spin operators. (ii) MERA obtained by adding disentanglers $u$ to the TTN according to Fig. 1. Disentanglers $u$ are delocalized and decompose into sums of terms that contain strings of $Z$ s (represented by ribbons).Reuse & Permissions
Figure 3
(i) The fermionic isometry $w$ preserves parity; cf. Eq. (6). (ii) By construction, an isometry $w$ is such that the pair $w$ $w^{†}$ annihilates into the identity $I$ . (iii) Coarse-graining of the operator $c_{2}^{†} c_{12}$ into $A_{1^{'}} Z_{2^{'}} B_{3^{'}}$ [cf. Eq. (7)], by using properties (i) and (ii). (iv) The operator $c_{2}^{†} c_{12}$ also commutes with a disentangler $u$ if $c_{2}^{†} c_{12}$ and $u$ have disjoint fermionic support.Reuse & Permissions
Figure 4
Left panel: Phase diagram of the free fermion model, Eq. (8). Right panels: Error in the ground-state energy obtained with TTN and MERA simulations of a $6 \times 6$ lattice with PBC. The lines correspond to different values of the refinement parameter $χ$ , as indicated in the legend to the left panel. The entanglement entropy $S_{1 / 2}$ for one-half of the lattice is larger for values of $λ$ and $γ$ that lead to larger errors.Reuse & Permissions
Figure 5
Convergence of the ground-state energy of interacting spinless fermions, Eq. (9), on a $6 \times 6$ lattice with PBC for $γ = 1, λ = 2$ , and varying interaction strength $V$ . The plot shows $Δ E \equiv E_{χ} - E_{χ_{\max}}$ , the difference between the energy as a function of $χ$ (squares, TTN; circles, MERA) and our best MERA result $E_{χ_{\max}}$ , where $χ_{\max} = 120$ . Again, the entanglement entropy $S_{1 / 2}$ shows a strong correlation between ground-state entanglement and convergence in $χ$ .Reuse & Permissions
Figure 6
The error in the ground-state energy as a function of the linear size $L$ of a square lattice, obtained with the fermionic MERA algorithm with $χ = 4$ (noninteracting case; $γ = 1$ ), is of the same order of magnitude for small and large systems, even in the critical regime $λ < 2$ . Simulations with interacting fermions (not plotted) show an analogous pattern of energies, but there are no exact results for comparison.Reuse & Permissions

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