Abstract
Tensor network states are expected to be good representations of a large class of interesting quantum many-body wave functions. In higher dimensions, their utility is however severely limited by the difficulty of contracting the tensor network, an operation needed to calculate quantum expectation values. Here we introduce a method for the time-evolution of three-dimensional isometric tensor networks which respects the isometric structure and therefore renders contraction simple through a special canonical form. Our method involves a tetrahedral site-splitting which allows to move the orthogonality center of an embedded tree tensor network in a simple cubic lattice to any position. Using imaginary time-evolution to find an isometric tensor network representation of the ground state of the 3D transverse field Ising model across the entire phase diagram, we perform a systematic benchmark study of this method in comparison with exact Lanczos and quantum Monte Carlo results. We show that the obtained energy matches the exact groundstate result accurately deep in the ferromagnetic and polarized phases, while the regime close to the critical point requires larger bond dimensions. This behavior is in close analogy with the two-dimensional case, which we also discuss for comparison.
Highlights
The Hilbert space dimension of quantum many-body systems grows exponentially with the number of constituents, making the direct handling of many-body wave functions impractical for large systems
C indicates that all tensors are contracted, giving complex scalar coefficients, which is usually done by choosing the amount of virtual degrees of freedom per Tiσi equal to the lattice connectivity and contracting nearest neighbors
The accuracy is worst in the critical region, and here the performance is significantly improved upon increasing the bond dimensions, showing a clear trend toward the exact values both for the energy density and x magnetization
Summary
The Hilbert space dimension of quantum many-body systems grows exponentially with the number of constituents, making the direct handling of many-body wave functions impractical for large systems. Tensor networks are an attempt to tame the many-body wave function, by expressing it in terms of local tensors, which are contracted according to the network structure This reduces the complexity from an exponential to a polynomial number of variables. Tensor network states are successful in one dimension, where they are known as “matrix-product states” (MPS) [4], which have become state-of-the-art machinery for the classical simulation of one-dimensional (1D) manybody systems. This popularity rests primarily on the existence of powerful algorithms to variationally optimize the energy of the state [e.g., the density matrix renormalization group (DMRG) [5]] and on the ability to compute ma-
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