# Literature¶

This is a (by far non-exhaustive) list of some references for the various ideas behind the code.
They can be cited from the python doc-strings using the format `[Author####]_`

.
Within each category, we sort the references by year and author.

## General reading¶

[Schollwoeck2011] is an extensive introduction to MPS, DMRG and TEBD with lots of details on the implementations, and a classic read, although a bit lengthy. Our [TeNPyNotes] are a shorter summary of the important concepts, similar as [Orus2014]. [Hubig2019] is a very good, recent review focusing on time evolution with MPS. The lecture notes of [Eisert2013] explain the area law as motivation for tensor networks very well. PEPS are for example reviewed in [Verstraete2009], [Eisert2013] and [Orus2014]. [Stoudenmire2011] reviews the use of DMRG for 2D systems. [Cirac2009] discusses the different groups of tensor network states.

- Cirac2009
“Renormalization and tensor product states in spin chains and lattices” J. I. Cirac and F. Verstraete, Journal of Physics A: Mathematical and Theoretical, 42, 50 (2009) arXiv:0910.1130 doi:10.1088/1751-8113/42/50/504004

- Verstraete2009
“Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems” F. Verstraete and V. Murg and J.I. Cirac, Advances in Physics 57 2, 143-224 (2009) arXiv:0907.2796 doi:10.1080/14789940801912366

- Schollwoeck2011
“The density-matrix renormalization group in the age of matrix product states” U. Schollwoeck, Annals of Physics 326, 96 (2011), arXiv:1008.3477 doi:10.1016/j.aop.2010.09.012

- Stoudenmire2011
“Studying Two Dimensional Systems With the Density Matrix Renormalization Group” E.M. Stoudenmire, Steven R. White, Ann. Rev. of Cond. Mat. Physics, 3: 111-128 (2012), arXiv:1105.1374 doi:10.1146/annurev-conmatphys-020911-125018

- Eisert2013(1,2)
“Entanglement and tensor network states” J. Eisert, Modeling and Simulation 3, 520 (2013) arXiv:1308.3318

- Orus2014(1,2)
“A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States” R. Orus, Annals of Physics 349, 117-158 (2014) arXiv:1306.2164 doi:10.1016/j.aop.2014.06.013

- Hubig2019
“Time-evolution methods for matrix-product states” S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck, C. Hubig, arXiv:1901.05824

## Algorithm developments¶

[White1992] is the invention of DMRG, which started everything. [Vidal2004] introduced TEBD. [White2005] and [Hubig2015] solved problems for single-site DMRG. [McCulloch2008] was a huge step forward to solve convergence problems for infinite DMRG. [Singh2009], [Singh2010] explain how to incorporate Symmetries. [Haegeman2011] introduced TDVP, again explained more accessible in [Haegeman2016]. [Karrasch2013] gives some tricks to do finite-temperature simulations (DMRG), which is a bit extended in [Hauschild2018]. [Vidal2007] introduced MERA.

- White1992
“Density matrix formulation for quantum renormalization groups” S. White, Phys. Rev. Lett. 69, 2863 (1992) doi:10.1103/PhysRevLett.69.2863, S. White, Phys. Rev. B 84, 10345 (1992) doi:10.1103/PhysRevB.48.10345

- Vidal2004
“Efficient Simulation of One-Dimensional Quantum Many-Body Systems” G. Vidal, Phys. Rev. Lett. 93, 040502 (2004), arXiv:quant-ph/0310089 doi:10.1103/PhysRevLett.93.040502

- White2005
“Density matrix renormalization group algorithms with a single center site” S. White, Phys. Rev. B 72, 180403(R) (2005), arXiv:cond-mat/0508709 doi:10.1103/PhysRevB.72.180403

- Vidal2007
“Entanglement Renormalization” G. Vidal, Phys. Rev. Lett. 99, 220405 (2007), arXiv:cond-mat/0512165, doi:10.1103/PhysRevLett.99.220405

- McCulloch2008
“Infinite size density matrix renormalization group, revisited” I. P. McCulloch, arXiv:0804.2509

- Singh2009
“Tensor network decompositions in the presence of a global symmetry” S. Singh, R. Pfeifer, G. Vidal, Phys. Rev. A 82, 050301(R), arXiv:0907.2994 doi:10.1103/PhysRevA.82.050301

- Singh2010
“Tensor network states and algorithms in the presence of a global U(1) symmetry” S. Singh, R. Pfeifer, G. Vidal, Phys. Rev. B 83, 115125, arXiv:1008.4774 doi:10.1103/PhysRevB.83.115125

- Haegeman2011
“Time-Dependent Variational Principle for Quantum Lattices” J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pizorn, H. Verschelde, F. Verstraete, Phys. Rev. Lett. 107, 070601 (2011), arXiv:1103.0936 doi:10.1103/PhysRevLett.107.070601

- Karrasch2013
“Reducing the numerical effort of finite-temperature density matrix renormalization group calculations” C. Karrasch, J. H. Bardarson, J. E. Moore, New J. Phys. 15, 083031 (2013), arXiv:1303.3942 doi:10.1088/1367-2630/15/8/083031

- Hubig2015
“Strictly single-site DMRG algorithm with subspace expansion” C. Hubig, I. P. McCulloch, U. Schollwoeck, F. A. Wolf, Phys. Rev. B 91, 155115 (2015), arXiv:1501.05504 doi:10.1103/PhysRevB.91.155115

- Haegeman2016
“Unifying time evolution and optimization with matrix product states” J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, F. Verstraete, Phys. Rev. B 94, 165116 (2016), arXiv:1408.5056 doi:10.1103/PhysRevB.94.165116

- Hauschild2018
“Finding purifications with minimal entanglement” J. Hauschild, E. Leviatan, J. H. Bardarson, E. Altman, M. P. Zaletel, F. Pollmann, Phys. Rev. B 98, 235163 (2018), arXiv:1711.01288 doi:10.1103/PhysRevB.98.235163