Literature and References

This is a (by far non-exhaustive) list of some references for the various ideas behind the code. They can be cited like this:

  • [TeNPyNotes] for TeNPy/software related sources

  • [white1992] (lowercase first-author + year) for entries from literature.bib.

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]. [paeckel2019] 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 [verstraete2008], [eisert2013] and [orus2014]. [stoudenmire2012] reviews the use of DMRG for 2D systems. [cirac2009] discusses the different groups of tensor network states.

Algorithm developments

[white1992][white1993] 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. [singh2010][singh2011] explain how to incorporate Symmetries. [haegeman2011] introduced TDVP, again explained more accessible in [haegeman2016]. [zaletel2015] is another standard method for time-evolution with long-range Hamiltonians. [karrasch2013] gives some tricks to do finite-temperature simulations (DMRG), which is a bit extended in [hauschild2018a]. [vidal2007] introduced MERA. The scaling \(S=c/6 log(\chi)\) at a 1D critical point is explained in [pollmann2009].



Thomas Barthel. Matrix product purifications for canonical ensembles and quantum number distributions. Physical Review B, 94(11):115157, September 2016. arXiv:1607.01696, doi:10.1103/PhysRevB.94.115157.


Thomas Barthel and Yikang Zhang. Optimized Lie-Trotter-Suzuki decompositions for two and three non-commuting terms. Annals of Physics, 418:168165, July 2020. arXiv:1901.04974, doi:10.1016/j.aop.2020.168165.


Pasquale Calabrese and John Cardy. Entanglement Entropy and Quantum Field Theory. Journal of Statistical Mechanics: Theory and Experiment, 2004(06):P06002, June 2004. arXiv:hep-th/0405152, doi:10.1088/1742-5468/2004/06/P06002.


Lukasz Cincio and Guifre Vidal. Characterizing topological order by studying the ground states of an infinite cylinder. Physical Review Letters, 110(6):067208, February 2013. arXiv:1208.2623, doi:10.1103/PhysRevLett.110.067208.


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


J. Eisert. Entanglement and tensor network states. arXiv:1308.3318 [cond-mat, physics:quant-ph], September 2013. arXiv:1308.3318.


Adolfo G. Grushin, Johannes Motruk, Michael P. Zaletel, and Frank Pollmann. Characterization and stability of a fermionic \nu=1/3 fractional Chern insulator. Physical Review B, 91(3):035136, January 2015. arXiv:1407.6985, doi:10.1103/PhysRevB.91.035136.


Jutho Haegeman, J. Ignacio Cirac, Tobias J. Osborne, Iztok Pizorn, Henri Verschelde, and Frank Verstraete. Time-dependent variational principle for quantum lattices. Physical Review Letters, 107(7):070601, August 2011. arXiv:1103.0936, doi:10.1103/PhysRevLett.107.070601.


Jutho Haegeman, Christian Lubich, Ivan Oseledets, Bart Vandereycken, and Frank Verstraete. Unifying time evolution and optimization with matrix product states. Physical Review B, 94(16):165116, October 2016. arXiv:1408.5056, doi:10.1103/PhysRevB.94.165116.


Johannes Hauschild, Eyal Leviatan, Jens H. Bardarson, Ehud Altman, Michael P. Zaletel, and Frank Pollmann. Finding purifications with minimal entanglement. Physical Review B, 98(23):235163, December 2018. arXiv:1711.01288, doi:10.1103/PhysRevB.98.235163.


Johannes Hauschild and Frank Pollmann. Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy). SciPost Physics Lecture Notes, pages 5, October 2018. arXiv:1805.00055, doi:10.21468/SciPostPhysLectNotes.5.


Claudius Hubig, Ian P. McCulloch, Ulrich Schollwöck, and F. Alexander Wolf. A Strictly Single-Site DMRG Algorithm with Subspace Expansion. Physical Review B, 91(15):155115, April 2015. arXiv:1501.05504, doi:10.1103/PhysRevB.91.155115.


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


I. P. McCulloch. Infinite size density matrix renormalization group, revisited. arXiv:0804.2509 [cond-mat], April 2008. arXiv:0804.2509.


V. Murg, J. I. Cirac, B. Pirvu, and F. Verstraete. Matrix product operator representations. New Journal of Physics, 12(2):025012, February 2010. arXiv:0804.3976, doi:10.1088/1367-2630/12/2/025012.


Titus Neupert, Luiz Santos, Claudio Chamon, and Christopher Mudry. Fractional quantum Hall states at zero magnetic field. Physical Review Letters, 106(23):236804, June 2011. arXiv:1012.4723, doi:10.1103/PhysRevLett.106.236804.


Roman Orus. A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States. Annals of Physics, 349:117–158, October 2014. arXiv:1306.2164, doi:10.1016/j.aop.2014.06.013.


Sebastian Paeckel, Thomas Köhler, Andreas Swoboda, Salvatore R. Manmana, Ulrich Schollwöck, and Claudius Hubig. Time-evolution methods for matrix-product states. Annals of Physics, 411:167998, December 2019. arXiv:1901.05824, doi:10.1016/j.aop.2019.167998.


Frank Pollmann, Subroto Mukerjee, Ari Turner, and Joel E. Moore. Theory of finite-entanglement scaling at one-dimensional quantum critical points. Physical Review Letters, 102(25):255701, June 2009. arXiv:0812.2903, doi:10.1103/PhysRevLett.102.255701.


Frank Pollmann and Ari M. Turner. Detection of Symmetry Protected Topological Phases in 1D. Physical Review B, 86(12):125441, September 2012. arXiv:1204.0704, doi:10.1103/PhysRevB.86.125441.


Raffaele Resta. Quantum-Mechanical Position Operator in Extended Systems. Physical Review Letters, 80(9):1800–1803, March 1998. doi:10.1103/PhysRevLett.80.1800.


Ulrich Schollwoeck. The density-matrix renormalization group in the age of matrix product states. Annals of Physics, 326(1):96–192, January 2011. arXiv:1008.3477, doi:10.1016/j.aop.2010.09.012.


Norbert Schuch. Condensed Matter Applications of Entanglement Theory. arXiv:1306.5551 [cond-mat, physics:quant-ph], June 2013. arXiv:1306.5551.


Sukhwinder Singh, Robert N. C. Pfeifer, and Guifre Vidal. Tensor network decompositions in the presence of a global symmetry. Physical Review A, 82(5):050301, November 2010. arXiv:0907.2994, doi:10.1103/PhysRevA.82.050301.


Sukhwinder Singh, Robert N. C. Pfeifer, and Guifre Vidal. Tensor network states and algorithms in the presence of a global U(1) symmetry. Physical Review B, 83(11):115125, March 2011. arXiv:1008.4774, doi:10.1103/PhysRevB.83.115125.


E. M. Stoudenmire and Steven R. White. Minimally Entangled Typical Thermal State Algorithms. New Journal of Physics, 12(5):055026, May 2010. arXiv:1002.1305, doi:10.1088/1367-2630/12/5/055026.


E. M. Stoudenmire and Steven R. White. Studying Two Dimensional Systems With the Density Matrix Renormalization Group. Annual Review of Condensed Matter Physics, 3(1):111–128, March 2012. arXiv:1105.1374, doi:10.1146/annurev-conmatphys-020911-125018.


Masuo Suzuki. General theory of fractal path integrals with applications to many-body theories and statistical physics. Journal of Mathematical Physics, 32(2):400–407, February 1991. doi:10.1063/1.529425.


F. Verstraete, J. I. Cirac, and V. Murg. Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems. Advances in Physics, 57(2):143–224, March 2008. arXiv:0907.2796, doi:10.1080/14789940801912366.


G. Vidal. Efficient simulation of one-dimensional quantum many-body systems. Physical Review Letters, 93(4):040502, July 2004. arXiv:quant-ph/0310089, doi:10.1103/PhysRevLett.93.040502.


G. Vidal. Entanglement Renormalization. Physical Review Letters, 99(22):220405, November 2007. doi:10.1103/PhysRevLett.99.220405.


Steven R. White. Density matrix formulation for quantum renormalization groups. Physical Review Letters, 69(19):2863–2866, November 1992. doi:10.1103/PhysRevLett.69.2863.


Steven R. White. Density-matrix algorithms for quantum renormalization groups. Physical Review B, 48(14):10345–10356, October 1993. doi:10.1103/PhysRevB.48.10345.


Steven R. White. Density matrix renormalization group algorithms with a single center site. Physical Review B, 72(18):180403, November 2005. arXiv:cond-mat/0508709, doi:10.1103/PhysRevB.72.180403.


Shuo Yang, Zheng-Cheng Gu, Kai Sun, and S. Das Sarma. Topological flat band models with arbitrary Chern numbers. Physical Review B, 86(24):241112, December 2012. arXiv:1205.5792, doi:10.1103/PhysRevB.86.241112.


Michael P. Zaletel, Roger S. K. Mong, Christoph Karrasch, Joel E. Moore, and Frank Pollmann. Time-evolving a matrix product state with long-ranged interactions. Physical Review B, 91(16):165112, April 2015. arXiv:1407.1832, doi:10.1103/PhysRevB.91.165112.