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arXiv:1309.2255 · Physical Review B

Tree tensor networks and entanglement spectra

Iztok Pižorn, Frank Verstraete, Robert Konik

Published Nov 1, 2013• 23 citations
arXivHTMLPDF

Abstract

A tree tensor network variational method is proposed to simulate quantum many-body systems with global symmetries where the optimization is reduced to individual charge configurations. A computational scheme is presented, namely how to extract the entanglement spectra in a bipartite splitting of a loopless tensor network across multiple links of the network, by constructing a matrix product operator for the reduced density operator and simulating its eigenstates. The entanglement spectra of $2\ifmmode\times\else\texttimes\fi{}L$, $3\ifmmode\times\else\texttimes\fi{}L$, and $4\ifmmode\times\else\texttimes\fi{}L$ with either open or periodic boundary conditions on the rungs are studied using the presented methods, where it is found that the entanglement spectrum depends not only on the subsystem but also on the boundaries between the subsystems.

DOI
10.1103/physrevb.88.195102

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Tree tensor network states (TTNS) decompose the system wavefunction to the product of low-rank tensors based on the tree topology, serving as the foundation of the multi-layer multi-configuration time-dependent Hartree method. In this work, we present an algorithm that automatically constructs the optimal and exact tree tensor network operators (TTNO) for any sum-of-product symbolic quantum operator. The construction is based on the minimum vertex cover of a bipartite graph. With the optimal TTNO, we simulate open quantum systems, such as spin relaxation dynamics in the spin-boson model and charge transport in molecular junctions. In these simulations, the environment is treated as discrete modes and its wavefunction is evolved on equal footing with the system. We employ the Cole-Davidson spectral density to model the glassy phonon environment and incorporate temperature effects via thermo-field dynamics. Our results show that the computational cost scales linearly with the number of discretized modes, demonstrating the efficiency of our approach.
Abstract We propose the entanglement bipartitioning approach to design an optimal network structure of the tree tensor network (TTN) for quantum many-body systems. Given an exact ground-state wavefunction, we perform sequential bipartitioning of spin-cluster nodes so as to minimize the mutual information or the maximum loss of the entanglement entropy associated with the branch to be bipartitioned. We demonstrate that entanglement bipartitioning of up to 16 sites gives rise to nontrivial tree network structures for S = 1/2 Heisenberg models in one and two dimensions. The resulting TTNs enable us to obtain better variational energies, compared with standard TTNs such as the uniform matrix product state and perfect binary tree tensor network.
We present a numerical strategy to efficiently estimate bipartite entanglement measures, and in particular the entanglement of formation, for many-body quantum systems on a lattice. Our approach exploits the tree tensor operator tensor network Ansatz, a positive loopless representation for density matrices which, as we demonstrate, efficiently encodes information on bipartite entanglement, enabling the upscaling of entanglement estimation. Employing this technique, we observe a finite-size scaling law for the entanglement of formation in 1D critical lattice models at finite temperature for up to 128 spins, extending to mixed states the scaling law for the entanglement entropy.
Tensor network states are an indispensable tool for the simulation of strongly correlated quantum many-body systems. In recent years, tree tensor network states (TTNS) have been successfully used for two-dimensional systems and to benchmark quantum simulation approaches for condensed matter, nuclear, and particle physics. In comparison to the more traditional approach based on matrix product states (MPS), the graph distance of physical degrees of freedom can be drastically reduced in TTNS. Surprisingly, it turns out that, for large systems in D>1D>1D>1 spatial dimensions, MPS simulations of low-energy states are nevertheless more efficient than TTNS simulations. With a focus on D=2D=2D=2 and 3, the scaling of computational costs for different boundary conditions is determined under the assumption that the system obeys an entanglement (log-)area law, implying that bond dimensions scale exponentially in the surface area of the associated subsystems.
The tree tensor network (TTN) provides an essential theoretical framework for the practical simulation of quantum many-body systems, where the network structure defined by the connectivity of the isometry tensors plays a crucial role in improving its approximation accuracy. In this paper, we propose a TTN algorithm that enables us to automatically optimize the network structure by local reconnections of isometries to suppress the bipartite entanglement entropy on their legs. The algorithm can be seamlessly implemented to such a conventional TTN approach as the density-matrix renormalization group. We apply the algorithm to the inhomogeneous antiferromagnetic Heisenberg spin chain, having a hierarchical spatial distribution of the interactions. We then demonstrate that the entanglement structure embedded in the ground state of the system can be efficiently visualized as a perfect binary tree in the optimized TTN. Possible improvements and applications of the algorithm are also discussed.