Entanglement renormalization in fermionic systems

Abstract

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.

Figures (6)

• Figure 1: Top: A block of two sites (four modes) is coarse-grained into an effective site by first applying disentanglers $U$ across the boundary of the block and then using isometry $W$ to project out two modes. Bottom: Same RG transformation written in the language of correlation matrices, Eq. (\ref{['eq:truncated']}).
• Figure 2: Scaling of the entanglement entropy $S_L$logVidal in 1D systems. Left: Quantum Ising model, $\gamma=1$. Bold (solid/dotted) lines represent entanglement at criticality, $\lambda=1$. The system is an entangled fixed point of our RG transformation: the correlation matrices $\{\Gamma^{(1)}, \Gamma^{(2)}, \cdots \}$ quickly converge to a fixed $\Gamma_{\hbox{\tiny{Ising}}}^{*}$. In particular, the renormalized entanglement of a block is constant. Thin lines correspond to a non-critical system, $\lambda=1.001$, which the RG flow eventually brings a product (unentangled) ground state. Right: Quantum XX model, $\gamma=0$. Bold/thin lines represent two critical cases, $\lambda=0$ and $\lambda=\textrm{cos} (15\pi/16)$. They belong to the same universality class and are found to indeed converge to the same correlation matrix $\Gamma_{\hbox{\tiny{XX}}}^{*}$, (with $\Gamma_{\hbox{\tiny{XX}}}^{*} \neq \Gamma_{\hbox{\tiny{Ising}}}^{*}$) and in particular to the same renormalized entropy.
• Figure 3: Dispersion relation of Hamiltonian (\ref{['eq:Ham']}) in 1D with $\gamma=0$, quantum spin XX model, under successive RG transformations. Shading indicates the Fermi sea. A sequence of local, coarse-grained Hamiltonians is obtained $\{H^{(1)}, H^{(2)}, \cdots \}$ with their corresponding dispersion relations $\{\nu_1,\nu_2, \cdots\}$ converging to a straight line, a fixed point of the RG flow. Convergence is achieved very quickly at half filling ($\lambda=0$) and slower for $\lambda = \textrm{cos}(3\pi/4)$. These results have been obtained by minimizing the energy (Sect. IV of Ref. algorithms) while keeping $8$ modes in each effective site.
• Figure 4: Entanglement entropy $S_{L}$ of a block of $L\times L$ modes in 2D models. Left: in the critical phase II and the non-critical phase III (bold/fine lines respectively) the entanglement entropy grows linearly with the size $L$ of the boundary of the block, $S_L \sim L$ (boundary law). As in 1D, the renormalized entanglement is constant for the critical model and it eventually vanishes for the non-critical model. We have considered $\gamma=1$ and $\lambda=2, \lambda=2.05$ for the critical/non-critical case. Right: The critical phase II system $(\gamma,\lambda)=(1,2)$ is replotted for comparison against critical phase I, $(\gamma,\lambda)=(0,0)$, where the system has a 1D Fermi surface and the entanglement entropy has a logarithmic correction, $S_L\sim L\log L$. Here disentanglers are not able to reduce the renormalized entanglement down to a constant.
• Figure 5: 2D MERA on a square lattice as described in ERMERA.