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 $L=\{6,9,18,54,\infty \}$ 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.

Figure 1

Entanglement renormalization scheme for a square lattice. A block of $3\times 3$ sites of lattice ${\mathcal{L}}_{\tau -1}$ (i) is mapped onto one site of ${\mathcal{L}}_{\tau }$ (v). The RG transformation involves (ii) applying disentanglers $u$ between the corners of adjacent blocks followed by (iii) disentanglers $v$ which act across the sides of adjacent blocks and (iv) isometries $w$ which act within a block. Tensors $u$, $v$, and $w$ have a varying number of incoming and outgoing indices (vi) according to Eq. (1).

Figure 2

Spontaneous and transverse magnetizations $⟨{\sigma }_x⟩$ and $⟨{\sigma }_z⟩$ as a function of the applied magnetic field $\lambda$ and for different lattice sizes $L$. Results for small systems correspond to exact diagonalization while results for larger systems were obtained with a $\chi =6$ MERA. As $L$ increases, the magnetizations are seen to converge toward their thermodynamic limit values. Results for $L=54$ could not be visually distinguished from results for $L=18$ and have been omitted in the plot. As it is characteristic of a second order phase transition, for large $L$ both magnetizations develop a discontinuity in their derivative, with $⟨{\sigma }_x⟩$ (the order parameter) suddenly dropping to zero at the quantum critical point (see Fig. 3).

Figure 3

Magnetizations $⟨{\sigma }_x⟩$ and $⟨{\sigma }_z⟩$ as a function of the applied magnetic field $\lambda$ for different values of the refinement parameter $\chi$. Left: Spontaneous magnetization $⟨{\sigma }_x⟩$ for $L=54$. Data fits of the form $⟨{\sigma }_x⟩\sim (\lambda -{\lambda }_c{)}^{{\beta }_c}$ near the critical point give a critical magnetic field ${\lambda }_c=\{3.13,3.09,3.075\}$ and critical exponent ${\beta }_c=\{0.320,0.321,0.323\}$ for $\chi =\{2,4,6\}$. Current Monte Carlo estimates are ${\lambda }_c=3.044$ and ${\beta }_c=0.326$ [16]. Thus accuracy increases with $\chi$. Right: Transverse magnetization $⟨{\sigma }_z⟩$ for $L=6$. TTN results for large $\chi$ are taken as the exact solution (see Fig. 5). While a $\chi =2$ MERA produces significantly different values, results for $\chi =3$ are already very similar and those for $\chi =6$ MERA agree with the TTN solution on at least three significant digits.

Figure 4

Top: The energy gap as a function of the transverse magnetic field $\lambda$, computed by exact diagonalization for small system sizes $L=\{2,3,4\}$ and with a $\chi =6$ MERA for $L=\{6,9\}$. The gap scales as $1/L$ at the critical magnetic field. Bottom: Two-point correlators $⟨{\sigma }_x^{[r]}{\sigma }_x^{[r^{\prime }]}{⟩}_c$ at criticality and for different values of $\chi$. The scale invariant MERA produces correlators that decay polynomially with the distance $s\equiv |r-r^{\prime }|$. As $\chi$ increases their asymptotic scaling approaches $1/s^{1+\eta }$ with $\eta =0.03\pm 0.01$ [17]. Correlators have been computed at distances $s=3^k$ for $k=0,1,2,\dots$, where they can be evaluated with cost $O({\chi }^{16})$. For comparison, we have included correlators obtained with $D=2$ and $D=3$ infinite projected entangled-pair states (iPEPS) [18]. The latter are very accurate for $s=1,2$ but decay exponentially after a few sites.

Figure 5

Energy error as a function of the refinement parameter $\chi$ for finite systems of different sizes and for infinite systems. In absence of an exact solution for ground state energies, the errors are defined relative to the results obtained with (i) a $\chi =60$ TTN, (ii) a $\chi =9$ MERA, (iii),(iv) a $D=3$ iPEPS [18]. For finite systems (i),(ii), the MERA is compared against the TTN. The double $x$ axes for ${\chi }_{\mathrm{MERA}}$ and ${\chi }_{\mathrm{TTN}}$ have been adjusted so that they roughly correspond to the same computational cost. For $L=6$ the TTN is more efficient while for $L=9$ the MERA already gives significantly better results. Comparison between MERA and iPEPS results for (iii) an infinite system off criticality and (iv) an infinite system at criticality shows very similar accuracy between $\chi =3$ MERA and $D=2$ iPEPS, whereas $D=3$ iPEPS gives a lower (better) energy than $\chi =6$ MERA.