Unconstrained tree tensor network: An adaptive gauge picture for enhanced performance

We introduce a variational algorithm to simulate quantum many-body states based on a tree tensor network ansatz which releases the isometry constraint usually imposed by the real-space renormalization coarse graining. This additional numerical freedom, combined with the loop-free topology of the tree network, allows one to maximally exploit the internal gauge invariance of tensor networks, ultimately leading to a computationally flexible and efficient algorithm able to treat open and periodic boundary conditions on the same footing. We benchmark the novel approach against the 1D Ising model in transverse field with periodic boundary conditions and discuss the strategy to cope with the broken translational invariance generated by the network structure. We then perform investigations on a state-of-the-art problem, namely, the bilinear-biquadratic model in the transition between dimer and ferromagnetic phases. Our results clearly display an exponentially diverging correlation length and thus support the most recent guesses on the peculiarity of the transition.

Figure 1

(a) Structure of the binary TTN with $L$ layers and $N$ sites. The maximal bond dimension is $m$. (b) The same TTN displayed in a PBC configuration, showing its natural capability of treating OBC and PBC on the same footing.

Figure 2

Isometrization of a three-legged tensor $\Lambda$. The arrow indicates the leg with respect to which the tensor is isometric. In the picture, $Q^{*}$ is the element-wise complex complex conjugate of $Q$, and it is also drawn as vertically reflected.

Figure 3

TTN isometrization with respect to the tensor highlighted in red, i.e., all isometrization arrows point towards this tensor. To obtain this gauge, $QR$-decompose all tensors in order of decreasing distance (indicated by red numbers).

Figure 4

Mapping operation induced by the tensor ${\Lambda }^{[l,n]}$ for the case of upward-directed isometrization arrow. The operation maps the effective Hamiltonian terms at sites $[l+1,2n-1]$ and $[l+1,2n]$ to the new terms at site $[l,n]$. For the two other possible isometrization directions similar mapping operations hold. The index $\alpha$ has been dropped for simplicity.

Figure 5

Definition of the action of the effective Hamiltonian $H_{\text{eff}}^{[l,n]}$ on the degrees of freedom of the tensor ${\Lambda }^{[l,n]}$. (Again, the index $\alpha$ has been dropped for simplicity.)

Figure 6

Generic optimization move: (a) after optimizing ${\Lambda }^{[l,n]}$, the tensor ${\Lambda }^{[l^{\prime },n^{\prime }]}$ is targeted for optimization. (b) Only tensors and effective Hamiltonian terms located on the path connecting ${\Lambda }^{[l,n]}$ and ${\Lambda }^{[l^{\prime },n^{\prime }]}$ (colored in blue) need to be updated in order to enable the optimization of ${\Lambda }^{[l^{\prime },n^{\prime }]}$. (c) Targeting each tensor in the tree multiple times results in a sweeping pattern. After completing a sweep, resume at the top (as indicated by the encircled marks).

Figure 7

Expectation value of a local observable $O^{\left[n\right]}$. Due to the isometry condition, the calculation always reduces to a contraction involving only three tensors.

Figure 8

Location of the central site $s_c$ in a 1D binary TTN.

Figure 9

Ising model: Error of the ground-state energy per site $\Delta E/N$ as a function of the number of optimization sweeps for two different system sizes $N=32$ and $N=1024$ (left). Error as a function of the system size (right upper). The same quantity as a function of the bond dimension $m$. The data for $N=32$ is fitted by $a{m}^{-b}exp(-cm)$, with $a=12.8$, $b=5.5$, $c=0.11$. Extrapolation to the thermodynamic limit suggests a polynomial decay of the form $\Delta E\propto m^{-3.3}$ (right-lower).

Figure 10

Error of the ground-state energy per site $\Delta E/N$ as a function of the number of optimization sweeps. Comparison between a constrained and an adaptive TTN. The generalized eigenvalue problems present in the constrained TTN algorithm cause strong instabilities in the energy minimization, hindering a reliable convergence behavior.

Figure 11

(Top) Spin-correlation functions and fitted critical exponents for a PBC quantum Ising chain of $N=128$ sites, bond dimension $m=100$. The exact exponents are ${\eta }_x=0.25$, ${\eta }_y=2.25$, ${\eta }_z=2$. In order to account for the TTN-induced translational symmetry breaking, the correlations of distance $r$ are obtained by averaging over lattice locations within branches wide enough to span $r$ (see text). (Bottom) Von Neumann entropy of a $N=256$ ($m=100$) chain, for partitions of length $\ell$ and with fitted central charge. The exact central charge is $c=0.5$.

Figure 12

Error of the local energy $E_n$ as a function of the lattice site $n$ for different bond dimensions $m$, in the critical quantum Ising model with PBC defined in Eq. (6). The solid lines are the errors of the system-wide averages. The vertical dashed line indicates the location of the center site $s_c$ (left). Error of the spin two-point correlator $C_{n,n+r}^y$ for bond dimension $m=100$ (right); the two insets show the data at fixed distance $r$ together with system-wide averages for $r=8$ (top) and $r=-32$ (bottom).

Figure 13

Phase diagram of the bilinear-biquadratic model of Eq. (7), with four firmly established phases: ferromagnetic, trimerized [44], Haldane antiferromagnet [45, 46], and dimerized [47, 48]. According to the most recent studies and the data obtained here with adaptive TTN, no intermediate nematic phase (originally hypothesized location indicated by red hatched region [49, 50]) is emerging around ${\theta }_{\mathrm{F}}$ (black dot). The white dot identifies the purely biquadratic point [47, 51, 52] whose Bethe ansatz solution in Eq. (11) we use to benchmark the numerical precision of our algorithm.

Figure 14

Extrapolation to the thermodynamic limit of the ground-state energy at the purely biquadratic point $\theta =-\pi /2$. The inset shows the extrapolation in the bond dimension for $N=128$ using a fit similar to the one indicated in Fig. 9.

Figure 15

(Top) Dimerization parameter $D$ for various system sizes. Each data point is the result of an extrapolation $m\to \infty$ in the bond dimension. The dashed lines shown in the inset are linear fits. (Bottom) Quadrupolar correlation length ${\xi }_{\mathrm{Q}}$, obtained from fitting the correlation function Eq. (12). Each $C^{\mathrm{Q}}\left(r\right)$ has been extrapolated to $m\to \infty$ (for every $r$) before the correlation length was extracted. Reported error bars are fit errors. The large error bars for $N=32$ are caused by the fact that the chain is too small to contain meaningful information on the long-range property ${\xi }_{\mathrm{Q}}$.