The following is a certainly incomple list of papers and theses which may be helpful in understanding bits and pieces of the toolkit better. This list is not intended as a historical summary of the field but only intends to collect pedagogically useful papers which are relevant to specific implementations in the SyTen toolkit. Nevertheless, any addition to this list is very welcome, some topics are still missing completely!

In addition to the papers linked here, see also the folder documentation-input/relevant-slides for some slides of talks given at various points in time with relevance to the toolkit. documentation-input/relevant-papers contains freely available PDFs of some of the papers linked below.

Furthermore, for a collection of articles describing tensor network methods, you may also wish to have a look at tensornetwork.org.

Symmetries in Tensor Networks

• For a generic introduction to symmetries in MPS: I. P. McCulloch, J. Stat. Mech 2007: From density-matrix renormalization group to matrix product states [arxiv]. The last part on non-abelian symmetries is helpful to further the understanding, but does not apply 1-to-1 to the implementation in SyTen.
• Specifically for the choice of implementation for non-abelian symmetries in SyTen: A. Weichselbaum, Ann. Phys. 2012: Non-abelian symmetries in tensor networks: A quantum symmetry space approach [arxiv] and the second chapter of Claudius' thesis.

MPS, error estimates etc.

• U. Schollwöck, Ann. Phys. 2011: The density-matrix renormalization group in the age of matrix product states [arxiv].
• For the MPO construction method used: C. Hubig, I. P. McCulloch, U. Schollwöck, Phys. Rev. B 2017: Generic construction of efficient matrix product operators [arXiv].
• For error estimates and extrapolations \( m \to \infty \): C. Hubig, J. Haegeman and U. Schollwöck, Phys. Rev. B 2018: Error estimates for extrapolations with matrix-product states [arXiv].

DMRG, also on cylinders and in 2D

• For single-site DMRG: C. Hubig, I. P. McCulloch, U. Schollwöck and F. A. Wolf, Phys. Rev. B 2015: Strictly single-site DMRG algorithm with subspace expansion [arxiv] and also the third chapter of Claudius' thesis with some additional updates.
• For an introduction of the snaking on cylinders: E. M. Stoudenmire and S. R. White, Ann. Rev. Cond. Mat. Phys. 2012: Studying Two-Dimensional Systems with the Density Matrix Renormalization Group [arxiv].
• For hybrid space DMRG on cylinders: G. Ehlers, S. R. White and R. M. Noack: Phys. Rev. B. 2017, Hybrid-space density matrix renormalization group study of the doped two-dimensional Hubbard model [arxiv]; Leo Stenzel's master thesis has many interesting bits, Claudius' thesis has some in chapter 5.

Time evolution with MPS

• The recent review: S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck and C. Hubig: Time-evolution methods for matrix-product states [arXiv].
• The original paper on TDVP with MPS as implemented now: J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken and F. Verstraete, Phys. Rev. B 2016: Unifying time evolution and optimization with matrix product states [arXiv].
• The original paper on the MPO \( W^\mathrm{II} \) method: M. P. Zaletel, R. S. K. Mong, C. Karrasch, J. E. Moore and F. Pollmann, Phys. Rev. B 2015: Time-evolving a matrix product state with long-ranged interactions [arXiv]
• For the matrix orthogonalisation mode in syten-krylov (and also DMRG): P. E. Dargel, A. Wöllert, A. Honecker, I. P. McCulloch, U. Schollwöck and T. Pruschke, Phys. Rev. B 2012: Lanczos algorithm with matrix product states for dynamical correlation functions [arXiv].
• For purifications, in addition to the review above, see the references therein.

Infinite projected entangled pair states (iPEPS)

• For symmetries specifically when used in iPEPS (describing the iPEPSv1 implementation): C. Hubig, SciPost Physics, 2018: Abelian and non-abelian symmetries in infinite projected entangled pair states [arXiv]
• For time-dependent iPEPS simulations: C. Hubig and J. I. Cirac, SciPost Physics, 2019: Time-dependent study of disordered models with infinite projected entangled pair states [arXiv] and C. Hubig, A. Bohrdt, M. Knap, F. Grusdt, J. I. Cirac, arXiv 2019: Evaluation of time-dependent correlators after a local quench in iPEPS: hole motion in the t-J model
• For the WithQR CTM contraction method in the iPEPSv2 implementation: P. Corboz, T. M. Rice, and M. Troyer, Phys. Rev. L 2014: Competing States in the t-J Model: Uniform d-Wave State versus Stripe State [arXiv] (in the supplemental material)
• For the SVDOnly CTM contraction method in the iPEPSv2 implementation: P. Corboz, J. Jordan, and G. Vidalm Phys. Rev. B 2010: Simulation of fermionic lattice models in two dimensions with projected entangled-pair states: Next-nearest neighbor Hamiltonians [arXiv] (Fig. 11b)
• For the SVDLiao CTM contraction method in the iPEPSv2 implementation: H.-J. Liao, J.-G. Liu, L. Wang, T. Xiang, arXiv 2019: Differentiable Programming Tensor Networks

Works using the toolkit

If your paper/thesis/preprint is missing, please add it. The list is sorted chronologically, with the newest paper last, by the earlier of arXiv or publication date.

1. Generic construction of efficient matrix product operators: Phys. Rev. B 95, 2017 [arXiv]
   Claudius Hubig, Ian P. McCulloch and Ulrich Schollwöck
2. A DMRG study of the Fermi-Hubbard model in hybrid space: LMU Munich, 2017
   Leo Stenzel, Master Thesis
3. Spinon confinement in a quasi one dimensional anisotropic Heisenberg magnet: Phys. Rev. B 96, 2017 [arXiv]
   Anup K. Bera, Bella Lake, Fabian H. L. Essler, Laurens Vanderstraeten, Claudius Hubig, Ulrich Schollwöck, A. T. M. Nazmul Islam, Astrid Schneidewind and Diana L. Quintero-Castro
4. Doped Kondo chain, a heavy Luttinger liquid: PNAS 115, 2018 [arXiv]
   Ilia Khait, Patrick Azaria, Claudius Hubig, Ulrich Schollwöck and Assa Auerbach
5. Symmetry-Protected Tensor Networks: LMU Munich, 2017
   Claudius Hubig, PhD Thesis
6. Error estimates for extrapolations with matrix-product states: Phys. Rev. B 97, 2018 [arXiv]
   Claudius Hubig, Jutho Haegeman and Ulrich Schollwöck
7. Kinetic-Energy Density-Functional Theory on a Lattice: J. Chem. Theory Comput. 14, 2018 [arXiv]
   Iris Theophilou, Florian Buchholz, F. G. Eich, Michael Ruggenthaler, Angel Rubio
8. Abelian and non-abelian symmetries in infinite projected entangled pair states: SciPost Physics 5, 2018 [arXiv]
   Claudius Hubig
9. Interaction quench and thermalization in a one-dimensional topological Kondo insulator: Phys. Rev. B 99, 2019 [arXiv]
   Imre Hagymási, Claudius Hubig and Ulrich Schollwöck
10. Density-matrix embedding theory study of the one-dimensional Hubbard-Holstein model: J. Chem. Theory Comput. 15 (4), 2019 [arXiv]
    Teresa E. Reinhard, Uliana Mordovina, Claudius Hubig, Joshua S. Kretchmer, Ulrich Schollwöck, Heiko Appel, Michael A. Sentef and Angel Rubio
11. Thermal control of spin excitations in the coupled Ising-chain material RbCoCl3: arXiv, 2018
    Mattia Mena, Nora Hänni, Simon Ward, Eva Hirtenlechner, Robert Bewley, Claudius Hubig, Ulrich Schollwöck, Bruce Normand, Karl W. Krämer, Des F. McMorrow and Christian Rüegg
12. Thermalization of a Cavity Mode in the Presence of a Dye Molecule: LMU Munich, 2018
    David Wierichs, Master Thesis
13. Time-dependent study of disordered models with infinite projected entangled pair states: SciPost Physics 6, 2019 [arXiv]
    Claudius Hubig and J. Ignacio Cirac
14. Density Matrix Embedding Theory: Foundations, Applications and Connection to Functional Theories: MPSD Hamburg, 2019
    Teresa E. Reinhard, PhD Thesis
15. Dynamical Mean-Field Theory Studies on Real Materials: LMU Munich, 2019
    Nils-Oliver Linden, PhD Thesis
16. Time-evolution methods for matrix-product states: Annals of Physics 411, 2019 [arXiv]
    Sebastian Paeckel, Thomas Köhler, Andreas Swoboda, Salvatore R. Manmana, Ulrich Schollwöck and Claudius Hubig
17. Quantum phases and topological properties of interacting fermions in one-dimensional superlattices: Phys. Rev A 99, 2019 [arXiv]
    Leo Stenzel, Andrew L. C. Hayward, Claudius Hubig, Ulrich Schollwöck, Fabian Heidrich-Meisner
18. Dynamical topological quantum phase transitions in nonintegrable models: Phys. Rev. Lett 122, 2019 [arXiv]
    Imre Hagymási, Claudius Hubig, Örs Legeza, Ulrich Schollwöck
19. Gaussian time-dependent variational principle for the Bose-Hubbard model: Phys. Rev. B 100, 094529 [arXiv]
    Tommaso Guaita, Lucas Hackl, Tao Shi, Claudius Hubig, Eugene Demler, J. Ignacio Cirac
20. Use and implementation of autodifferentiation in tensor network methods with complex scalars: arXiv, 2019
    Claudius Hubig
21. Imaginary-time matrix product state impurity solver in a real material calculation: Spin-orbit coupling in Sr2RuO4: Phys. Rev. B 101(R), 2020 [arXiv]
    Nils-Oliver Linden, Manuel Zingl, Claudius Hubig, Olivier Parcollet, Ulrich Schollwöck
22. Absence of superconductivity in the pure two-dimensional Hubbard model: arXiv, 2019
    Mingpu Qin, Chia-Min Chung, Hao Shi, Ettore Vitali, Claudius Hubig, Ulrich Schollwöck, Steven R. White, Shiwei Zhang
23. Open quantum systems in thermal non-ergodic environments: arXiv, 2019
    Carlos A. Parra-Murillo, Max Bramberger, Claudius Hubig, Inés de Vega
24. Evaluation of time-dependent correlators after a local quench in iPEPS: hole motion in the t-J model: arXiv, 2019
    Claudius Hubig, Annabelle Bohrdt, Michael Knap, Fabian Grusdt, J. Ignacio Cirac
25. Sr2MoO4 and Sr2RuO4: Disentangling the Roles of Hund's and van Hove Physics: arXiv, 2020
    Jonathan Karp, Max Bramberger, Martin Grundner, Ulrich Schollwöck, Andrew J. Millis, Manuel Zingl
26. Concept of orbital entanglement and correlation in quantum chemistry
    Lexin Ding, Sam Mardazad, Sreetama Das, Szilárd Szalay, Ulrich Schollwöck, Zoltán Zimborás, Christian Schilling
27. Topological phases in the Fermi-Hofstadter-Hubbard model on hybrid-space ladders
    Leo Stenzel, Andrew L. C. Hayward, Ulrich Schollwöck, and Fabian Heidrich-Meisner