# 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

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

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]. is a very good, recent review focusing on time evolution with MPS. The lecture notes of explain the area law as motivation for tensor networks very well. PEPS are for example reviewed in , and [orus2014]. reviews the use of DMRG for 2D systems. discusses the different groups of tensor network states.

## Algorithm developments¶

is the invention of DMRG, which started everything. introduced TEBD. and solved problems for single-site DMRG. was a huge step forward to solve convergence problems for infinite DMRG. explain how to incorporate Symmetries. introduced TDVP, again explained more accessible in . is another standard method for time-evolution with long-range Hamiltonians. gives some tricks to do finite-temperature simulations (DMRG), which is a bit extended in . introduced MERA. The scaling $$S=c/6 log(\chi)$$ at a 1D critical point is explained in .

## References¶

barthel2016

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.

barthel2020

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.

calabrese2004

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.

cincio2013

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.

cirac2009

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.

eisert2013(1,2)

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

grushin2015

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.

haegeman2011

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.

haegeman2016

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.

hauschild2018

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.

hauschild2018a(1,2)

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.

hubig2015

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.

karrasch2013

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.

mcculloch2008

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

murg2010

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.

neupert2011

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.

orus2014(1,2)

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.

paeckel2019

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.

pollmann2009

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.

pollmann2012

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.

resta1998

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

schollwoeck2011

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.

schuch2013

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

shapourian2017

Hassan Shapourian, Ken Shiozaki, and Shinsei Ryu. Many-body topological invariants for fermionic symmetry-protected topological phases. Physical Review Letters, 118(21):216402, May 2017. arXiv:1607.03896, doi:10.1103/PhysRevLett.118.216402.

singh2010

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.

singh2011

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.

stoudenmire2010

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.

stoudenmire2012

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.

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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.

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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.

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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.

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

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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.

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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.

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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.

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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.

zaletel2015

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.