Entanglement renormalization in two spatial dimensions

TL;DR

A calculation of the energy gap shows that it scales as 1/L at the critical point, and a scheme for entanglement renormalization capable of addressing large two-dimensional quantum lattice systems is proposed and tested.

Abstract

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 analysed. 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,inf} 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.

Figures (5)

• Figure 1: (Color online) 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. \ref{['eq:tensors']}.
• Figure 2: (Color online) Spontaneous and transverse magnetizations $\left\langle {\sigma _x } \right\rangle$ and $\left\langle {\sigma _z } \right\rangle$ as a function of the applied magnetic field $\lambda$ and for different lattice sizes $L$. Results for small systems correspond to exact diagonalization whilst 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 $\left\langle {\sigma _x } \right\rangle$ (the order parameter) suddenly dropping to zero at the quantum critical point (see Fig. \ref{['fig:MagRefine']}).
• Figure 3: (Color online) Magnetizations $\left\langle {\sigma _x } \right\rangle$ and $\left\langle {\sigma _z } \right\rangle$ as a function of the applied magnetic field $\lambda$ for different values of the refinement parameter $\chi$. Left: Spontaneous magnetization $\left\langle {\sigma _x } \right\rangle$ for $L=54$. Data fits of the form $\left\langle {\sigma _x } \right\rangle \sim \left( {\lambda - \lambda _c } \right)^{\beta_c }$ near the critical point give a critical magnetic field $\lambda _c = \left\{ {3.13,3.09,3.075} \right\}$ and critical exponent $\beta _c = \{ 0.320,0.321,0.323\}$ for $\chi = \left\{ {2,4,6} \right\}$. Current Monte Carlo estimates are $\lambda _c = 3.044$ and $\beta _c = 0.326$MC. Thus accuracy increases with $\chi$. Right: Transverse magnetization $\left\langle {\sigma _z } \right\rangle$ for $L=6$. TTN results for large $\chi$ are taken as the exact solution (see Fig. \ref{['fig:2DEnergyErrorNew']}). Whilst 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 3 significant digits.
• Figure 4: (Color online) Top: The energy gap as a function of the transverse magnetic field $\lambda$, computed by exact diagonalization for small system sizes $L=\left\{2,3,4\right\}$ and with a $\chi=6$ MERA for $L=\left\{6,9\right\}$. The gap scales as $1/L$ at the critical magnetic field. Bottom: Two-point correlators $\langle {\sigma_x^{[r]} \sigma_x^{[r']} } \rangle_{c}$ at criticality and for different values of $\chi$. The scale invariant MERA produces correlators that decay polynomially with the distance $s \equiv |r-r'|$. As $\chi$ increases their asymptotic scaling approaches $1/s^{ 1 + \eta }$ with $\eta = 0.03 \pm 0.01$critphenom. Correlators have been computed at distances $s=3^k$ for $k=0,1,2,\ldots$, where they can be evaluated with cost $O(\chi^{16})$. For comparison, we have included correlators obtained with a $D=2$ and $D=3$ iPEPS iPEPS. The latter are very accurate for $s=1,2$ but decay exponentially after a few sites.
• Figure 5: (Color online) 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 iPEPS. For finite systems (i,ii), the MERA is compared against the TTN. The double $x$-axes for $\chi_{\hbox{\tiny MERA}}$ and $\chi_{\hbox{\tiny TTN}}$ have been adjusted so that they roughly correspond to the same computational cost. For $L=6$ the TTN is more efficient whilst 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.