Entanglement renormalization in noninteracting 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 the validity of the multiscale entanglement renormalization ansatz to describe certain ground states in two dimensions, including quantum critical states. 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.

Figure 1

(Color online) 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. (7).

Figure 2

(Color online) Scaling of the entanglement entropy $S_L$ (Ref. 10) in 1D systems. Left: quantum Ising model, $\gamma =1$. Bold (solid and dotted) lines represent entanglement at criticality, $\lambda =1$. The system is an entangled fixed point of our RG transformation: the correlation matrices $\left\{{\Gamma }^{\left(1\right)},{\Gamma }^{\left(2\right)},\dots \right\}$ quickly converge to a fixed ${\Gamma }_{\text{Ising}}^{\ast }$. In particular, the renormalized entanglement of a block is constant. Thin lines correspond to a noncritical system, $\lambda =1.001$, which the RG flow, as generated by ER, eventually brings a product (unentangled) ground state. Right: quantum $XX$ model, $\gamma =0$. Bold and thin lines represent two critical cases, $\lambda =0$ and $\lambda =\text{cos}\left(15\pi /16\right)$. They belong to the same universality class and are found to indeed converge to the same correlation matrix ${\Gamma }_{XX}^{\ast }$, (with ${\Gamma }_{XX}^{\ast }\ne {\Gamma }_{\text{Ising}}^{\ast }$) and in particular to the same renormalized entropy.

Figure 3

(Color online) Dispersion relation of Hamiltonian (1) in 1D with $\gamma =0$, the quantum spin $XX$ model, under successive RG transformations. Shading indicates the Fermi sea. A sequence of local, coarse-grained Hamiltonians is obtained $\left\{H^{\left(1\right)},H^{\left(2\right)},\dots \right\}$ with their corresponding dispersion relations $\left\{{\nu }_1,{\nu }_2,\dots \right\}$ converging to a straight line, a fixed point of the RG flow. Convergence is achieved very quickly at half filling $\left(\lambda =0\right)$ and slower for $\lambda =\text{cos}\left(3\pi /4\right)$. These results have been obtained by minimizing the energy while keeping eight modes in each effective site.

Figure 4

(Color online) Entanglement entropy $S_L$ of a block of $L\times L$ modes in 2D models. Left: in the critical phase II and the noncritical phase III (bold and 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 noncritical model. We have considered $\gamma =1$ and $\lambda =2,\lambda =2.05$ for the critical and noncritical cases. Right: the critical phase II system $\left(\gamma ,\lambda \right)=\left(1,2\right)$ is replotted for comparison against critical phase I, $\left(\gamma ,\lambda \right)=\left(0,0\right)$, where the system has a 1D Fermi surface and the entanglement entropy has a logarithmic correction, $S_L\sim L{\text{log}}_{2}L$. Here disentanglers are not able to reduce the renormalized entanglement down to a constant.