DOI: 10.1088/1742-5468/2004/04/p04005. Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces

Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces

10.1088/1742-5468/2004/04/p04005

Crossref journal-article

IOP Publishing

Journal of Statistical Mechanics: Theory and Experiment (266)

Abstract

An algorithm for the simulation of the evolution of slightly entangled quantum states has been recently proposed as a tool to study time-dependent phenomena in one-dimensional quantum systems. Its key feature is a time-evolving block-decimation (TEBD) procedure to identify and dynamically update the relevant, conveniently small, subregion of the otherwise exponentially large Hilbert space. Potential applications of the TEBD algorithm are the simulation of time-dependent Hamiltonians, transport in quantum systems far from equilibrium and dissipative quantum mechanics. In this paper we translate the TEBD algorithm into the language of matrix product states in order to both highlight and exploit its resemblances to the widely used density-matrix renormalization-group (DMRG) algorithms. The TEBD algorithm, being based on updating a matrix product state in time, is very accessible to the DMRG community and it can be enhanced by using well-known DMRG techniques, for instance in the event of good quantum numbers. More importantly, we show how it can be simply incorporated into existing DMRG implementations to produce a remarkably effective and versatile ‘adaptive time-dependent DMRG’ variant, that we also test and compare to previous proposals.

Bibliography

A J Daley, C Kollath, U SchollwÃ¶ck, & G Vidal. (2004). Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces. Journal of Statistical Mechanics: Theory and Experiment, 2004(04), P04005.

Authors 4

A J Daley (first)
C Kollath (additional)
U Schollwöck (additional)
G Vidal (additional)

References 45 Referenced 918

10.1103/PhysRevLett.69.2863 / Phys. Rev. Lett. (1992)
10.1103/PhysRevB.48.10345 / Phys. Rev. (1993)
{'year': '2000', 'key': '3-1'} (2000)
10.1023/A:1019601218492 / Quant. Inf. Proc. (2002)
{'year': '2004', 'key': '5-1'} (2004)
{'year': '2003', 'key': '6-1'} (2003)
10.1103/PhysRevLett.90.227902 / Phys. Rev. Lett. (2003)
{'year': '2003', 'key': '8-1'} (2003)
{'year': '2003', 'key': '9-1'} (2003)
10.1038/35005001 / Nature (2000)
10.1103/PhysRevA.66.032110 / Phys. Rev. (2002)
10.1038/416608a / Nature (2002)
10.1103/PhysRevB.68.195116 / Phys. Rev. (2003)
{'year': '2004', 'key': '14-1'} (2004)
10.1103/PhysRevLett.92.027901 / Phys. Rev. Lett. (2004)
10.1103/PhysRevLett.92.087201 / Phys. Rev. Lett. (2004)
{'year': '2003', 'key': '17-1'} (2003)
10.1007/BFb0106062 (1999)
{'journal-title': 'Rev. Mod. Phys.', 'issn-type': 'print', 'year': '2004', 'key': '19-1', 'ISSN': 'https://id.crossref.org/issn/0034-6861'} / Rev. Mod. Phys. (2004)
10.1103/PhysRevLett.88.256403 / Phys. Rev. Lett. (2002)
10.1038/415039a / Nature (2002)
{'year': '2003', 'key': '22-1'} (2003)
10.1103/PhysRevLett.91.147902 / Phys. Rev. Lett. (2003)
{'year': '2004', 'key': '24-1'} (2004)
{'year': '2004', 'key': '25-1'} (2004)
10.1209/0295-5075/24/4/010 / Europhys. Lett. (1993)
10.1007/BF02099178 / Commun. Math. Phys. (1992)
10.1103/PhysRevLett.77.3633 / Phys. Rev. Lett. (1996)
{'year': '1992', 'key': '29-1'} (1992)
10.1103/PhysRevB.52.R9827 / Phys. Rev. (1995)
10.1103/PhysRevB.60.335 / Phys. Rev. (1999)
10.1103/PhysRevB.66.045114 / Phys. Rev. (2002)
10.1103/PhysRevLett.91.049701 / Phys. Rev. Lett. (2003)
10.1103/PhysRevLett.91.049702 / Phys. Rev. Lett. (2003)
10.1103/PhysRevLett.75.3537 / Phys. Rev. Lett. (1995)
10.1209/epl/i1998-00381-x / Europhys. Lett. (1998)
10.1103/PhysRevLett.59.799 / Phys. Rev. Lett. (1987)
10.1007/BF01218021 / Commun. Math. Phys. (1988)
10.1209/0295-5075/10/7/005 / Europhys. Lett (1989)
10.1143/PTP.56.1454 / Prog. Theor. Phys. (1976)
10.1143/JPSJ.68.1537 / J. Phys. Soc. Japan (1999)
10.1016/0375-9601(90)90092-3 / Phys. Lett. (1990)
10.1016/S0010-4655(98)00009-5 / Comput. Phys. Commun. (1998)
10.1103/PhysRevB.44.2681 / Phys. Rev. (1991)
{'year': '2004', 'key': '45-1'} (2004)

Dates

Type	When
Created	21 years, 4 months ago (April 21, 2004, 11:14 p.m.)
Deposited	8 months, 2 weeks ago (Dec. 12, 2024, 6:13 a.m.)
Indexed	4 hours, 33 minutes ago (Aug. 26, 2025, 3:03 a.m.)
Issued	21 years, 4 months ago (April 1, 2004)
Published	21 years, 4 months ago (April 1, 2004)
Published Online	21 years, 4 months ago (April 21, 2004)
Published Print	21 years, 4 months ago (April 1, 2004)

Funders 0

None

BibTeX

@article{A_J_Daley_2004, title={Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces}, volume={2004}, ISSN={1742-5468}, url={http://dx.doi.org/10.1088/1742-5468/2004/04/p04005}, DOI={10.1088/1742-5468/2004/04/p04005}, number={04}, journal={Journal of Statistical Mechanics: Theory and Experiment}, publisher={IOP Publishing}, author={A J Daley and C Kollath and U Schollwöck and G Vidal}, year={2004}, month=apr, pages={P04005} }

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