Time dependent variational principle for tree Tensor Networks

Abstract

We present a generalization of the Time Dependent Variational Principle (TDVP) to any finite sized loop-free tensor network. The major advantage of TDVP is that it can be employed as long as a representation of the Hamiltonian in the same tensor network structure that encodes the state is available. Often, such a representation can be found also for long-range terms in the Hamiltonian. As an application we use TDVP for the Fork Tensor Product States tensor network for multi-orbital Anderson impurity models. We demonstrate that TDVP allows to account for off-diagonal hybridizations in the bath which are relevant when spinorbit coupling effects are important, or when distortions of the crystal lattice are present.arXiv:1908.03090v2 [cond-mat.str-el] 6 Sep 2019 SciPost Physics Submission method to calculate dynamical properties [21][22][23][24]. Approaches to perform the real-time evolution include, among others, the Time-dependent Density Matrix Renormalization Group (tDMRG) [25,26], the closely related Time Evolving Block Decimation (TEBD) [27,28] as well as the Time Dependent Variational Principle (TDVP) [29,30]. An in depth comparison of several time evolution algorithms performed in Ref. [31] came to the conclusion that while all approaches have strengths and weaknesses, TDVP is among the most reliable methods to perform the time evolution. While time evolution approaches for MPS are well established, much less has been done for general tensor networks. So far, mostly TEBD (and minor variations) have been used, for example for the MERA network [32], for PEPS [33] and for TTNs [9,17,18,34,35]. The advantage of TEBD is its relative simplicity, since it effectively boils down to a repeated application of short range operators obtained from a Suzuki-Trotter decomposition [36] of the full time-evolution operator. However, one of the major disadvantages of TEBD is that it can become difficult to implement for more complicated Hamiltonians, especially when long-range couplings are present. TDVP to some degree circumvents this problem by only demanding a Hamiltonian represented in the same tensor network structure as the state which is often easy to find. Additionally, TDVP in its single-site variant exactly respects conserved quantities of the Hamiltonian like energy or magnetization [30]. Although some works applied TDVP to more general tensor networks [37,38], it is not obvious how these algorithms work in detail and how it can be generalized. A more practical motivation for the formulation of TDVP for TTNs are Dynamical Mean-Field Theory (DMFT) calculations using the FTPS tensor network. So far, this approach has been used for so-called diagonal hybridizations only. On the other hand, real materials often exhibit off-diagonal hybridizations, which can for example come from spin-orbit coupling, or from distortions of the crystal lattice. For off-diagonal hybridizations, the TEBD approach used so far [18,19] is difficult to generalize and we hence choose to use TDVP in these situations. Although part of the motivatio...

Studying dynamics in two-dimensional quantum lattices using tree tensor network states

“…We introduced an algorithm based on the timedependent variational principle for arbitrary TTNS and benchmarked it on systems of non- interacting fermions and interacting hard-core bosons in two dimensions, comparing the performance to previously published results using matrix product states. During the preparation of the manuscript we became aware of a recent complementary work introducing a similar versions of the algorithm, which were applied in rather different settings (as an impurity solver [62], and in a more formal derivation of the algorithm [63]).…”

“…The main differences between the algorithm of Ref. [62] and the one presented here are in the definition of the walk on the tree and in the absence of a top-node, including it's separate propagation routine.…”

“…Recently, similar versions of this algorithm were reported in detail in Refs. [62,63], which we became aware of during the preparation of this manuscript. While in our work we focus on two-dimensional systems, Ref.…”

“…While in our work we focus on two-dimensional systems, Ref. [62] showcases a promising application of a TTNS as an impurity-solver, which is an effectively zero-dimensional problem. On the other hand, Ref.…”

Computing vibrational eigenstates with tree tensor network states (TTNS)

“…Furthermore, DMRG-based algorithms for time evolution 35,36,38,109,110 can straightforwardly be applied to TTNS. 39,111 It will be very interesting to see how they compare to the MCTDH-based algorithms. Also, we believe that the diagrammatic notation used in the DMRG community and in this work will highlight new facets of established MCTDH methodology.…”