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Time Dependent Variational Principle for Tree Tensor Networks
by Daniel Bauernfeind, Markus Aichhorn
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Submission summary
| Authors (as registered SciPost users): | Daniel Bauernfeind |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/1908.03090v3 (pdf) |
| Date accepted: | 2020-01-07 |
| Date submitted: | 2019-12-16 01:00 |
| Submitted by: | Bauernfeind, Daniel |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
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 spin-orbit coupling effects are important, or when distortions of the crystal lattice are present.
List of changes
We added "with short range couplings" in the introduction according to the first referee's remark.
We added the citations mentioned by all three referees. In particular, we mentioned the MPO-based time evolution by Zaletel et al. in the text and added a sentence about (SIAM J. Matrix Anal. Appl., 34(2), 470–494).
We added two sentences in the introduction mentioning Multi-layer Multi-Configurational Time-Dependent Hartree methods used in quantum chemistry and point out similarities to TDVP.
Changed SVD to QR decomposition in Sec. 2.2.
Changed the sentence regarding orthogonality centers allowing for easy calculation of local observables.
Rephrased the sentence about start- and end-points as requested by the third referee in Sec. 3.1.
Clarified the appearance of first- and second order Trotter decomposition. Clarified the sentence regarding the "reversed order of updates" in Sec. 3.2.
Removed "(QR decomposition suffices)" in Sec. 3.3 and mentioned that this is the point where the tensor network is truncated.
Added a figure showing the update sequence of two-site TDVP.
Added the example TTN to Fig. 4 and Fig. 5 to remind the reader of the structure.
Defined $H_{int}$ in Sec. 4.
Added a remark that a backwards time-step and a forwards step cancel in Sec. 4.
We added more information on the bond-dimensions and truncated weights to the captions of the figures in Sec. 4.
Improved the error discussion in Sec. 4, including a new figure, showing convergence with respect to the control parameters as well as the expected $\sim \Delta t^2$ scaling of the error in Sec. 4.
We clarified the sentence in Sec. 4 stating that "non-interacting systems are already entangled" as requested by the second referee.
Corrected typos.
Published as SciPost Phys. 8, 024 (2020)