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]. [cirac2020] is a recent, broad review of MPS and MPS with a focus on analytical theorems. [stoudenmire2012] reviews the use of DMRG for 2D systems. [cirac2009] discusses the different groups of tensor network states. [vanderstraeten2019] is a great review on tangent space methods for infinite, uniform MPS.

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]. [vanderstraeten2019] gives a very good introductin to infinite, uniform MPS.

References

[affleck1987]

Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki. Rigorous results on valence-bond ground states in antiferromagnets. Physical Review Letters, 59(7):799–802, August 1987. doi:10.1103/PhysRevLett.59.799.

[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.

[cirac2020]

Ignacio Cirac, David Perez-Garcia, Norbert Schuch, and Frank Verstraete. Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems. arXiv:2011.12127 [cond-mat, physics:hep-th, physics:quant-ph], November 2020. arXiv:2011.12127.

[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.

[haegeman2012]

Jutho Haegeman, Bogdan Pirvu, David J. Weir, J. Ignacio Cirac, Tobias J. Osborne, Henri Verschelde, and Frank Verstraete. Variational matrix product ansatz for dispersion relations. Physical Review B, 85(10):100408, March 2012. doi:10.1103/PhysRevB.85.100408.

[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.

[mcculloch2002]

I. P. McCulloch and M. Gulácsi. The non-Abelian density matrix renormalization group algorithm. EPL (Europhysics Letters), 57(6):852, March 2002. doi:10.1209/epl/i2002-00393-0.

[mcculloch2008]

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

[milsted2013]

Ashley Milsted, Jutho Haegeman, Tobias J. Osborne, and Frank Verstraete. Variational Matrix Product Ansatz for Nonuniform Dynamics in the Thermodynamic Limit. Physical Review B, 88(15):155116, October 2013. arXiv:1207.0691, doi:10.1103/PhysRevB.88.155116.

[motruk2016]

Johannes Motruk, Michael P. Zaletel, Roger S. K. Mong, and Frank Pollmann. Density matrix renormalization group on a cylinder in mixed real and momentum space. Physical Review B, 93(15):155139, April 2016. arXiv:1512.03318, doi:10.1103/PhysRevB.93.155139.

[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.

[phien2012]

Ho N. Phien, Guifre Vidal, and Ian P. McCulloch. Infinite boundary conditions for matrix product state calculations. Physical Review B, 86(24):245107, December 2012. arXiv:1207.0652, doi:10.1103/PhysRevB.86.245107.

[phien2013]

Ho N. Phien, Guifré Vidal, and Ian P. McCulloch. Dynamical windows for real-time evolution with matrix product states. Physical Review B, 88(3):035103, July 2013. arXiv:1207.0678, doi:10.1103/PhysRevB.88.035103.

[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.

[suzuki1991]

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.

[vanderstraeten2019] (1,2)

Laurens Vanderstraeten, Jutho Haegeman, and Frank Verstraete. Tangent-space methods for uniform matrix product states. SciPost Physics Lecture Notes, pages 7, January 2019. arXiv:1810.07006, doi:10.21468/SciPostPhysLectNotes.7.

[verstraete2008]

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.

[vidal2004]

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.

[vidal2007]

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

[white1992] (1,2)

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.

[white1993]

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.

[white2005]

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.

[white2009]

Steven R. White. Minimally Entangled Typical Quantum States at Finite Temperature. Physical Review Letters, 102(19):190601, May 2009. arXiv:0902.4475, doi:10.1103/PhysRevLett.102.190601.

[yang2012]

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.

[zauner-stauber2018]

V. Zauner-Stauber, L. Vanderstraeten, M. T. Fishman, F. Verstraete, and J. Haegeman. Variational optimization algorithms for uniform matrix product states. Physical Review B, 97(4):045145, January 2018. arXiv:1701.07035, doi:10.1103/PhysRevB.97.045145.

[zauner2015]

V. Zauner, M. Ganahl, H. G. Evertz, and T. Nishino. Time Evolution within a Comoving Window: Scaling of signal fronts and magnetization plateaus after a local quench in quantum spin chains. Journal of Physics: Condensed Matter, 27(42):425602, October 2015. arXiv:1207.0862, doi:10.1088/0953-8984/27/42/425602.

PhD theses

For inspiration, we also (somewhat randomly) list a few PhD theses related to tensor networks (accessible online) - they often provide a very good introduction to some specific topic.

[kollath2005]

Corinna Kollath. The Adaptive Time-Dependent Density Matrix Renormalization Group Method : Development and Applications. PhD thesis, RWTH Aachen University, 2005. URL: https://publications.rwth-aachen.de/record/62126.

[Eisert2006]

J. Eisert. Entanglement in Quantum Information Theory. PhD thesis, University of Potsdam, October 2006. URL: https://arxiv.org/abs/quant-ph/0610253.

[schuch2007]

Norbert Schuch. Quantum Entanglement: Theory and Applications. PhD thesis, Technische Universität München, 2007. URL: https://mediatum.ub.tum.de/?id=620846.

[Bauer2011]

Roman Bela Bauer. Tensor Network States. PhD thesis, ETH Zürich, 2011. URL: https://doi.org/10.3929/ethz-a-006698826.

[singh2012a]

Sukhwinder Singh. Tensor Network States and Algorithms in the Presence of Abelian and Non-Abelian Symmetries. PhD thesis, University of Queensland, March 2012. URL: https://arxiv.org/abs/1203.2222.

[Hubig2017]

Claudius Hubig. Symmetry-Protected Tensor Networks. PhD thesis, Ludwig-Maximilians-Universität München, 2017. URL: https://edoc.ub.uni-muenchen.de/21348/.

[RubenPHD]

Ruben Verresen. Topology and Excitations in Low-Dimensional Quantum Matter. PhD thesis, Technische Universität Dresden, Dresden, 2019. URL: https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-356075.

[eberharter2023a]

Alexander A Eberharter. Investigation of Quantum Dynamics and Phase Transitions Using Tensor Network Algorithms. PhD thesis, Universität Innsbruck, September 2023. URL: https://ulb-dok.uibk.ac.at/ulbtirolhs/content/titleinfo/9000350.

[secular2024a]

Paul Secular. Parallel Tensor Network Methods for Quantum Lattice Systems. PhD thesis, Bath, 2024. URL: https://researchportal.bath.ac.uk/en/studentTheses/parallel-tensor-network-methods-for-quantum-lattice-systems.