This approach is in stark contrast to conventional Trotter-Suzuki-type methods which evolve ρ^ on a linear quasicontinuous grid in inverse temperature β ≡ 1/ T. We refer to this scheme of doubling β in each step of the imaginary time evolution as the exponential tensor renormalization group (XTRG). Here, we speed up thermal simulations of quantum many-body systems in both one- (1D) and two-dimensional (2D) models in an exponential way by iteratively projecting the thermal density matrix ρ^ = e –βH^ onto itself. In applications, they solely affect the tensors of reduced matrix elements and therefore, once tabulated, allow one to completely sidestep the explicit usage of CGTs, and thus to greatly increase numerical efficiency. Akin to 6 j -symbols, X-symbols are generally much smaller than their constituting CGTs. They can be computed deterministically once and for all, and hence they can also be tabulated. These deal with the pairwise contraction of the generalized underlying Clebsch-Gordan tensors (CGTs). As shown in this paper, however, the elementary step of a pairwise contraction of tensors of arbitrary rank can be tackled in a transparent and efficient manner by introducing so-called X-symbols. While well established for no or just Abelian symmetries, this can become quickly extremely involved and cumbersome more » for general non-Abelian symmetries. Physical expectation values based on TNSs require the full contraction of a given tensor network, with the elementary ingredient being a pairwise contraction. From a practical perspective, it allows one to push numerical efficiency by orders of magnitude. From a theoretical perspective, it can offer deep insights into the entanglement structure and quantum information content of strongly correlated quantum many-body states. The full exploitation of non-Abelian symmetries in tensor network states (TNSs) derived from a given lattice Hamiltonian is attractive in various aspects. The same system is analyzed using several alternative symmetry scenarios based on combinations of U(1) of the exact result in this basis. A detailed analysis is presented for a fully screened spin- 3/2 three-channel Anderson impurity model in the presence of conservation of total spin, particle-hole symmetry, and SU(3) channel symmetry. In this paper, the focus is on the application of the non-abelian framework within the NRG. The unifying tensor-representation for quantum symmetry spaces, dubbed QSpace, is particularly suitable to deal with standard renormalization group algorithms such as the numerical renormalization group (NRG), the density matrix renormalization group (DMRG), or also more general tensor networks such as the multi-scale entanglement renormalization ansatz (MERA). The two crucial ingredients, the Clebsch-Gordan algebra for multiplet spaces as well as the Wigner-Eckart theorem for operators, are accounted for in a natural, well-organized, and computationally straightforward way. In the Beta version, you can go to Preferences > Version > Uncheck "Join the Beta App Program" at any time, then click "Change to Release Version" to install the Release version.A general framework for non-abelian symmetries is presented for matrix-product and tensor-network states in the presence of well-defined orthonormal local as well as effective basis sets. How do I switch from the Beta version to the Release version? The Beta version and the Release version can be switched with each other, and they share the same account and settings. Go to Preferences > Version > Check "Join the Beta App Program" to receive beta app updates.ģ. QSpace Pro V2.9.8 or above is required.Ģ. The Beta version will contain more new features, and the Release version will be more stable.ġ. Simply put, the two are different in terms of new features and stability. What is the difference between Beta and Release? In order to make new features available to users as soon as possible, the QSpace R&D team decided to launch the QSpace Pro Beta program. Version updates for QSpace also take longer to complete. With the improvement of QSpace, the functions are more and more abundant, and the user's use environment is also more and more diverse.
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