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arXiv:2111.12398 · Physical review. B./Physical review. B

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

Giovanni Ferrari, Giuseppe Magnifico, Simone Montangero

Published Jun 6, 2022• 22 citations
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Abstract

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 $32\ifmmode\times\else\texttimes\fi{}32$. 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.

DOI
10.1103/physrevb.105.214201

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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\ifmmode×\else\texttimes\fi3232\ifmmode\times\else\texttimes\fi{}3232\ifmmode×\else\texttimes\fi32. 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.
Novel randomness-induced disordered ground states in two-dimensional (2D) quantum spin systems have been attracting much interest. For quantitative analysis of such random quantum spin systems, one of the most promising numerical approaches is the tensor-network strong-disorder renormalization group (tSDRG), which was basically established for one-dimensional (1D) systems. In this paper, we propose a possible improvement of its algorithm toward 2D random spin systems, focusing on a generating process of the tree network structure of tensors, and precisely examine their performances for the random antiferromagnetic Heisenberg model not only on the 1D chain but also on the square- and triangular-lattices. On the basis of comparison with the exact numerical results up to 36 site systems, we demonstrate that accuracy of the optimal tSDRG algorithm is significantly improved for the 1D and 2D systems in the strong-randomness regime.
This work explores the use of a tree tensor network ansatz to simulate the ground state of a local Hamiltonian on a two-dimensional lattice. By exploiting the entropic area law, the tree tensor network ansatz seems to produce quasi-exact results in systems with sizes well beyond the reach of exact diagonalization techniques. We describe an optimization algorithm to accurately approximate ground states and apply it to the quantum Ising model on a torus of L X L sites for L= {4,6,8}. Approximate results for larger lattices are also obtained. As an application of the approach, we analyse the scaling of the ground state entropy at the quantum critical point of the model and confirm the presence of a logarithmic additive correction to the entropic area law.
The numerical simulation of two-dimensional quantum many-body systems away from equilibrium constitutes a major challenge for all known computational methods. We investigate the utility of tree tensor network (TTN) states to solve the dynamics of the quantum Ising model in two dimensions. Within the perturbative regime of small transverse fields, TTNs faithfully reproduce analytically known, but nontrivial and physically interesting dynamics of magnetic domains, for lattices up to 16 × 16 sites. Limitations of the method related to the rapid growth of entanglement entropy are explored within more general, paradigmatic quench settings. We provide and discuss comprehensive benchmarks regarding the benefit of GPU acceleration and the impact of using local operator sums on the performance.
We investigate the disordered spin-12\frac1221​Heisenberg model in two dimensions and employ tree tensor networks (TTNs) with a physics-informed structural optimization of the tree layout, to simulate dynamics in the many-body localization problem. We find that TTNs are able to capture two-dimensional entanglement patterns more effectively than matrix product states (MPS) while being more efficient to contract than projected entangled pair states (PEPS) to probe larger systems and longer times. Structural optimization of the trees based on time evolution of the entanglement in the system allows to keep the necessary bond dimensions low and to maximally exploit the increased expressiveness of TTNs over MPS. In this way, we achieve more accurate results in all considered parameter regimes both below and above the ergodicity-to-localization crossover at a comparable compute-time cost.
Tensor network states (TNSs) offer an efficient representation for the ground state of quantum many-body systems and play an important role in the simulations of them. Numerous TNSs have been proposed in the past few decades. However, due to the high cost of TNS for two-dimensional systems, a balance between the encoded entanglement and computational complexity of TNS is yet to be reached. In this work we introduce a new tree tensor network-based (TTN-based) TNS dubbed a ``fully augmented tree tensor network (FATTN) by releasing the constraint in augmented tree tensor network (ATTN). When disentanglers are augmented in the physical layer of TTN, FATTN can provide more entanglement than TTN and ATTN. At the same time, FATTN maintains the scaling of computational cost with bond dimension in TTN and ATTN. Benchmark results on the ground-state energy for the transverse Ising model are provided to demonstrate the improvement of accuracy of FATTN over TTN and ATTN. Moreover, FATTN is quite flexible, which can be constructed as an interpolation between TTN and the multiscale entanglement renormalization ansatz to reach a balance between the encoded entanglement and the computational cost.
We introduce a novel tensor network structure augmenting the well-established tree tensor network representation of a quantum many-body wave function. The new structure satisfies the area law in high dimensions remaining efficiently manipulatable and scalable. We benchmark this novel approach against paradigmatic two-dimensional spin models demonstrating unprecedented precision and system sizes. Finally, we compute the ground state phase diagram of two-dimensional lattice Rydberg atoms in optical tweezers observing nontrivial phases and quantum phase transitions, providing realistic benchmarks for current and future two-dimensional quantum simulations.