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Chin. Phys. B, 2021, Vol. 30(3): 030205    DOI: 10.1088/1674-1056/abd74a
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Delta-Davidson method for interior eigenproblem in many-spin systems

Haoyu Guan(关浩宇) and Wenxian Zhang(张文献)†
1 School of Physics and Technology, Wuhan University, Wuhan 430072, China
Abstract  Many numerical methods, such as tensor network approaches including density matrix renormalization group calculations, have been developed to calculate the extreme/ground states of quantum many-body systems. However, little attention has been paid to the central states, which are exponentially close to each other in terms of system size. We propose a delta-Davidson (DELDAV) method to efficiently find such interior (including the central) states in many-spin systems. The DELDAV method utilizes a delta filter in Chebyshev polynomial expansion combined with subspace diagonalization to overcome the nearly degenerate problem. Numerical experiments on Ising spin chain and spin glass shards show the correctness, efficiency, and robustness of the proposed method in finding the interior states as well as the ground states. The sought interior states may be employed to identify many-body localization phase, quantum chaos, and extremely long-time dynamical structure.
Keywords:  numerical exact method      interior eigenvalue      delta function filter      subspace diagonalization  
Received:  25 August 2020      Revised:  11 December 2020      Accepted manuscript online:  30 December 2020
PACS:  02.70.-c (Computational techniques; simulations)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 91836101, U1930201, and 11574239).
Corresponding Authors:  Corresponding author. E-mail:   

Cite this article: 

Haoyu Guan(关浩宇) and Wenxian Zhang(张文献) Delta-Davidson method for interior eigenproblem in many-spin systems 2021 Chin. Phys. B 30 030205

1 Nielsen M A and Chuang I L2011 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
2 Sachdev S2011 Quantum Phase Transitions (Cambridge: Cambridge University Press)
3 Bogolubov N N and Bogolubov J N N2009 Introduction to Quantum Statistical Mechanics (Singapore: World Scientific Publishing Co Pte Ltd)
4 Dutta A, Aeppli G, Chakrabarti B K, Divakaran U, Rosenbaum T F and Sen D2015 Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information (Cambridge: Cambridge University Press)
5 Bortz M, Eggert S, Schneider C, St\"ubner R and Stolze J 2010 Phys. Rev. B 82 161308
6 Dobrovitski V V and De Raedt H A 2003 Phys. Rev. E 67 056702
7 Sandvik A W 2010 Phys. Rev. Lett. 104 137204
8 Blankenbecler R and Sugar R L 1983 Phys. Rev. D 27 1304
9 Sandvik A W and Kurkij\"arvi J 1991 Phys. Rev. B 43 5950
10 White S R 1992 Phys. Rev. Lett. 69 2863
11 Vidal G 2004 Phys. Rev. Lett. 93 040502
12 Vidal G 2007 Phys. Rev. Lett. 99 220405
13 Verstraete F, Murg V and Cirac J I 2008 Adv. Phys. 57 143
14 Verstraete F, Murg V and Cirac J J2008 Adv. Phys. 57 143
15 Or\'us R 2014 Ann. Phys. 349 117
16 Eisert J, Cramer M and Plenio M B 2010 Rev. Mod. Phys. 82 277
17 Loh E Y, Gubernatis J E, Scalettar R T, White S R, Scalapino D J and Sugar R L 1990 Phys. Rev. B 41 9301
18 Troyer M and Wiese U 2010 Phys. Rev. Lett. 94 170201
19 Lanczos C 1950 J. Res. Natl. Bur. Stand. 45 255
20 Vidmar L, Hackl L, Bianchi E and Rigol M 2010 Phys. Rev. Lett. 121 220602
21 Kj\"all J A, Bardarson J H and Pollmann F 2014 Phys. Rev. Lett. 113 107204
22 Ng N and Kolodrubetz M 2019 Phys. Rev. Lett. 122 240402
23 Bohigas O, Giannoni M J and Schmit C 1984 Phys. Rev. Lett. 52 1
24 Georgeot B and Shepelyansky D L 1998 Phys. Rev. Lett. 81 5129
25 Ericsson T and Ruhe A 1980 Math. Comp. 35 1251
26 Ericsson T and Ruhe A1980 Math. Comput. 35 1251
27 Wyatt R E 1995 Phys. Rev. E 51 3643
28 Minehardt T J, Adcock J D and Wyatt R E 1997 Phys. Rev. E 56 4837
29 Bai Z, Demmel J, Dongarra J, Ruhe A and Vorst H2000 Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide(Philadelphia: SIAM)
30 Pietracaprina F, Mac\'e N, Luitz D J and Alet F 2018 SciPost Phys. 5 45
31 Davidson E R 1989 Comput. Phys. Commun. 53 49
32 Saad Y2011 Numerical Methods for Large Eigenvalue Problems(Philadelphia: SIAM)
33 Dorando J J, Hachmann J and Chan G K 2007 J. Chem. Phys. 127 084109
34 Dorando J J, Hachmann J and Chan G K2007 J. Chem. Phys. 127 401
35 Jordan G, Marsman M, Kim Y S and Kresse G 2012 J. Comput. Phys. 231 4836
36 Neuhauser D 1990 J. Chem. Phys. 93 2611
37 Neuhauser D 1991 J. Chem. Phys. 95 4927
38 Santra R, Breidbach J, Zobeley J and Cederbaum L S 2000 J. Chem. Phys. 112 9243
39 Santra R, Breidbach J, Zobeley J and Cederbaum L S2000 J. Chem. Phys. 112 9243
40 Vijay A and Wyatt R E 2000 Phys. Rev. E 62 4351
41 Pieper A, Kreutzer M, Alvermann A, Galgon M, Fehske H, Hager G, Lang B and Wellein G 2016 J. Comput. Phys. 325 226
42 Fang H R and Saad Y 2012 SIAM J. Sci. Comput. 34 A2220
43 Li R P, Xi Y Z, Vecharynski E, Yang C and Saad Y 2016 SIAM J. Sci. Comput. 38 A2512
44 Zhou Y K 2010 J. Comput. Phys. 229 9188
45 Zhou Y K and Saad Y 2007 SIAM J. Matrix Anal. Appl. 29 954
46 Mason J C and Handscomb D C2003 Chebyshev polynomials (Boca Raton: CRC Press LLC)
47 Boyd J P2000 Chebyshev and Fourier Spectral Methods (New York: Dover Publications)
48 Georgeot B and Shepelyansky D L 2007 Phys. Rev. E 62 3504
49 Daniel J W, Gragg W B, Kaufman L and Stewart G W 1976 Math. Comp. 30 772
50 Daniel J W, Gragg W B, Kaufman L and Stewart G W1976 Math. Comput. 30 772
51 Zhou Y K and Li R C 2011 Linear Algebra Appl. 435 480
52 Lehoucq R B, Sorensen D C and Yang C1998 ARPACK User's Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods(Philadelphia: SIAM)
53 Hams A and De Raedt H A 2000 Phys. Rev. E 62 4365
54 Fisher D S 1995 Phys. Rev. B 51 6411
55 Schollw\"ock U 2005 Rev. Mod. Phys. 77 259
56 Anderson E, Bai Z, Bischof C, Blackford L S, Demmel J, Dongarra J J, Croz J D, Hammarling S, Greenbaum A and Mckenney A1999 LAPACK Users' guide, 3rd Ed. (Philadelphia: SIAM)
57 Sorensen D C 1992 SIAM J. Matrix Anal. Appl. 13 357
58 Stewart G W 2001 SIAM J. Matrix Anal. Appl. 23 601
59 Luitz D J, Laflorencie N and Alet F 2015 Phys. Rev. B 91 081103
60 Sierant P and Zakrzewski J 2020 Phys. Rev. B 101 104201
61 Hopjan M and Fabian H M 2020 Phys. Rev. A 101 063617
62 Montangero S2018 Introduction to Tensor Network Methods: Numerical simulations of low-dimensional many-body quantum systems (Switzerland: Springer International Publishing)
63 Sierant P, Lewenstein M and Zakrzewski J 2020 Phys. Rev. Lett. 125 156601
64 Abanin D A, Altman E, Bloch I and Serbyn M 2019 Rev. Mod. Phys. 91 021001
65 Mondal D, Sinha S and Sinha S 2020 Phys. Rev. E 102 020101(R)
66 Debabrata M, Sudip S and Sinha S Phys. Rev. E 102 020101(R)
67 Guo A Y, Tran M C, Childs A M, Gorshkov A V and Gong Z X 2020 Phys. Rev. A 102 010401
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