Adaptive-weighted tree tensor networks for disordered quantum many-body systems

We introduce an adaptive-weighted tree tensor network for the study of disordered and inhomogeneous quantum many-body systems. This Ansatz is assembled on the basis of the random couplings of the physical system with a procedure that considers a tunable weight parameter to prevent completely unbalanced trees. Using this approach, we compute the ground state of the two-dimensional quantum Ising model in the presence of quenched random disorder and frustration, with lattice size up to $32\times 32$. We compare the results with the ones obtained using the standard homogeneous tree tensor networks and the completely self-assembled tree tensor networks, demonstrating a clear improvement of numerical precision as a function of the weight parameter, especially for large system sizes.

Figure 1

(a) Representation of a homogeneous binary TTN for a one-dimensional spin chain. Yellow circles indicate the sites of the chain connected to the tree by the physical indices; the green circles are the tensors of the TTN. (b) Representation of a TTN completely self-assembled from the nearest-neighbor interactions between the spins of the chain: the strength of the couplings determines the structure of the tree.

Figure 2

(a) Two-dimensional homogeneous TTN Ansatz. Yellow circles indicate the physical sites of the lattice connected to the tree by physical indices; green circles are the tensors making up the TTN. (b) Projection of the TTN structure on the plane of the physical lattice. The three partitions used for the computation of the entanglement entropy are highlighted.

Figure 3

Behavior of the von Neumann entanglement entropy as a function of the effective area of the (a) partition $A^{\left(1\right)}$, (b) partition $A^{\left(2\right)}$, and (c) partition $A^{\left(3\right)}$. We consider the case with $L=8$ and 200 realizations of the random disorder.

Figure 4

Construction of the adaptive-weighted TTN for two-dimensional lattice and a specific realisation of the random disorder. (a) A tensor is placed on top of two spins that interact with the strongest coupling in absolute value: the two original spins are then considered as a single effective spin (dashed green line). (b) This procedure is iterated for the next strongest coupling and (c) so forth. (d) If two effective spins share a border larger than one unit, the couplings between them are summed in absolute value. (e) If the newly defined coupling is the strongest one at this stage, a tensor is placed on the two underlying effective spins. (f) The procedure is iterated for the remaining couplings of the lattice. In all the previous steps, the considered couplings for the construction of the adaptive-weighted TTN depend on a weight parameter $\alpha$ as defined in Eq. (5).

Figure 5

Average ground-state energy for $h=2$, for different values of the weight parameter, as a function of the bond dimension $m$, and (a) $L=8$, (b) $L=16$, and (c) $L=32$. Error bars refer to the standard deviation of the average over the different realizations of the disorder. Insets in (a) and (b): average energy, in logarithmic scale, obtained for the largest bond dimension as a function of $\alpha$; the dashed line represents the energy obtained by using the homogeneous binary TTN.

Figure 6

Average ground-state energy for (a) $h=0.61$ (close to the critical point) and (b) $h=0.0$ (spin-glass phase), for different values of the weight parameter, as a function of the bond dimension $m$, and $L=8$. Error bars refer to the standard deviation of the average over the different realizations of the disorder. Insets in (a) and (b): average energy obtained for the largest bond dimension as a function of $\alpha$; the dashed line represents the energy obtained by using the homogeneous binary TTN.