Literature

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.

- 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.

- 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].

- 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.

- 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.

- 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*

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.

*Generic construction of efficient matrix product operators: Phys. Rev. B 95, 2017*[arXiv]

Claudius Hubig, Ian P. McCulloch and Ulrich Schollwöck*A DMRG study of the Fermi-Hubbard model in hybrid space: LMU Munich, 2017*

Leo Stenzel, Master Thesis*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*Doped Kondo chain, a heavy Luttinger liquid: PNAS 115, 2018*[arXiv]

Ilia Khait, Patrick Azaria, Claudius Hubig, Ulrich Schollwöck and Assa Auerbach*Symmetry-Protected Tensor Networks: LMU Munich, 2017*

Claudius Hubig, PhD Thesis*Error estimates for extrapolations with matrix-product states: Phys. Rev. B 97, 2018*[arXiv]

Claudius Hubig, Jutho Haegeman and Ulrich Schollwöck*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*Abelian and non-abelian symmetries in infinite projected entangled pair states: SciPost Physics 5, 2018*[arXiv]

Claudius Hubig*Interaction quench and thermalization in a one-dimensional topological Kondo insulator: Phys. Rev. B 99, 2019*[arXiv]

Imre Hagymási, Claudius Hubig and Ulrich Schollwöck*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*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*Thermalization of a Cavity Mode in the Presence of a Dye Molecule: LMU Munich, 2018*

David Wierichs, Master Thesis*Time-dependent study of disordered models with infinite projected entangled pair states: SciPost Physics 6, 2019*[arXiv]

Claudius Hubig and J. Ignacio Cirac*Density Matrix Embedding Theory: Foundations, Applications and Connection to Functional Theories: MPSD Hamburg, 2019*

Teresa E. Reinhard, PhD Thesis*Dynamical Mean-Field Theory Studies on Real Materials: LMU Munich, 2019*

Nils-Oliver Linden, PhD Thesis*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*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*Dynamical topological quantum phase transitions in nonintegrable models: Phys. Rev. Lett 122, 2019*[arXiv]

Imre Hagymási, Claudius Hubig, Örs Legeza, Ulrich Schollwöck*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*Use and implementation of autodifferentiation in tensor network methods with complex scalars: arXiv, 2019*

Claudius Hubig*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*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*Open quantum systems in thermal non-ergodic environments: arXiv, 2019*

Carlos A. Parra-Murillo, Max Bramberger, Claudius Hubig, Inés de Vega*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