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Frame et al. (Frame D, He R Z, Ipsen I, Lee D, Lee D, Rrapaj E
2018 Phys. Rev. Lett. 121 032501 ) proposed to use eigenvector continuation to solve high-dimensional many-body wavefunctions of relevant quantum models. When a model’s Hamiltonian matrix includes smoothly varying parameters, the corresponding eigenvector trajectory spans only a low-dimensional subspace. Therefore, it is possible to simplify the calculations by projecting the Hamiltonian onto a set of basis vectors of this subspace. However, the dimension of the trajectory subspace and its relationship with the size of the model are still unclear. In this paper, we study the antiferromagnetic Heisenberg chain models of different sizes systematically; their exchange interactions change with parameters smoothly. We first use principal component analysis to determine the subspaces of ground state many-body wavefunction vector trajectories of a 4-spin model and a 6-spin model, and plot the trajectories in the subspaces, respectively; we then analyze the principal components of ground state vector trajectories of models including$8,\cdots ,14$ spins, and reveal that when using eigenvector continuation to solve the ground state of an antiferromagnetic Heisenberg chain model, the number of basis vectors required increases with the number of spins in the model increasing. Our study can guide the application of eigenvector continuation in solving the Hamiltonian of a Heisenberg chain model containing more spins.-
Keywords:
- antiferromagnetic Heisenberg chain models /
- principal component analysis /
- eigenvector continuation
[1] Mezzacapo F, Schuch N, Boninsegni M, Cirac J I 2009 New J. Phys. 11 083026
Google Scholar
[2] Beach M J S, Melko R G, Grover T, Hsieh T H 2019 Phys. Rev. B 100 094434
Google Scholar
[3] Anders S, Plenio M B, Dür W, Verstraete F, Briegel H J 2006 Phys. Rev. Lett. 97 107206
Google Scholar
[4] Girardeau M D 1990 Phys. Rev. A 42 3303
Google Scholar
[5] Liu J W, Qi Y, Meng Z Y, Fu L 2017 Phys. Rev. B 95 041101
Google Scholar
[6] Elhatisari S, Epelbaum E, Krebs H, Lahde T A, Lee D, Li N, Lu B N, Meibner U G, Rupak G 2017 Phys. Rev. Lett. 119 222505
Google Scholar
[7] Baumgärtner A, Burkitt A N, Burkitt A N, Ceperley D M, De Raedt H, Ferrenberg A M, Heermann D W, Herrmann H J, Landau D P, Levesque D, Linden W, Reger F D, Schmidt K E, Selke W, Stauffer D, Swendsen R H, Wang J S, Weis J J, Young A P 2012 The Monte Carlo Method in Condensed Matter Physics (Springer Science & Business Media) pp23−51
[8] Dawson C M, Eisert J, Osborne T J 2008 Phys. Rev. Lett. 100 130501
Google Scholar
[9] Needs R J, Towler M D, Drummond N D, López Ríos P 2009 J. Phys.: Condens. Matter 22 023201
Google Scholar
[10] Ceperley D M, Alder B J 1980 Phys. Rev. Lett. 45 566
Google Scholar
[11] Sandvik A W, Kurkijärvi J 1991 Phys. Rev. B 43 5950
Google Scholar
[12] Gelfand M P, Singh R R P 2000 Adv. Phys. 49 93
Google Scholar
[13] Oitmaa J, Hamer C, Zheng W 2006 Series Expansion Methods for Strongly Interacting Lattice Models (Cambridge: Cambridge University Press) pp99−123
[14] Lanczos C 1950 J. Res. Nat. Bur. Stand. 45 255
Google Scholar
[15] Saad Y 2011 Numerical Methods for Large Eigenvalue Problems, Classics in Applied Mathematics (Philadelphia: Society for Industrial and Applied Mathematic) pp125−162
[16] Frame D, He R Z, Ipsen I, Lee D, Lee D, Rrapaj E 2018 Phys. Rev. Lett. 121 032501
Google Scholar
[17] Wang L 2016 Phys. Rev. B 94 195105
Google Scholar
[18] Hu W J, Singh R R P, Scalettar R T 2017 Phys. Rev. E 95 062122
Google Scholar
[19] Wang C, Zhai H 2017 Phys. Rev. B 96 144432
Google Scholar
[20] Wang C, Zhai H 2018 Front. Phys. 13 130507
Google Scholar
-
图 1 一维Heisenberg链模型示意图
Figure 1. Schematic diagram of a one-dimensional Heisenberg chain model.
图 2
$ L=4 $ 的Heisenberg链 (a) 基态矢量随$ \theta $ 变化的轨迹(从红到蓝是0°到90°); (b) 在6维和2维空间中分别求得的基态能量Figure 2. The Heisenberg chain with
$ L=4 $ : (a) Ground state vector trajectory (from red to blue: θ = 0° to θ = 90°); (b) ground state energies calculated in a 6-dimensional space and in a 2-dimensioanl space, respectively.
图 3
$ L=6 $ 的Heisenberg链 (a) 基态矢量随$ \theta $ 变化的轨迹(从红到蓝是0°到90°); (b) 在20维和3维空间中分别求得的基态能量Figure 3. The Heisenberg chain with
$ L=6 $ : (a) Ground state vector trajectory (from red to blue: θ = 0° to θ = 90°); (b) ground state energies calculated in a 20-dimensional space and in a 3-dimensioanl space, respectively.
图 4 (a) 采用特征矢量延拓方法求解随
$ \theta $ 变化的Heisenberg链基态能量所需基矢数目与其所含自旋个数之间的关系; (b)$ L=16 $ 的Heisenberg链在10维和6维空间中分别求得的基态能量Figure 4. (a) Relationship between the number of basis vectors needed to calculate
$ \theta $ -dependent ground state energies of the Heisenberg chain by eigenvector continuation and the number of spins in the chain; (b) ground state energies of the Heisenberg chain with$ L=16 $ calculated in a 10-dimensional space and in a 6-dimensioanl space, respectively. -
[1] Mezzacapo F, Schuch N, Boninsegni M, Cirac J I 2009 New J. Phys. 11 083026
Google Scholar
[2] Beach M J S, Melko R G, Grover T, Hsieh T H 2019 Phys. Rev. B 100 094434
Google Scholar
[3] Anders S, Plenio M B, Dür W, Verstraete F, Briegel H J 2006 Phys. Rev. Lett. 97 107206
Google Scholar
[4] Girardeau M D 1990 Phys. Rev. A 42 3303
Google Scholar
[5] Liu J W, Qi Y, Meng Z Y, Fu L 2017 Phys. Rev. B 95 041101
Google Scholar
[6] Elhatisari S, Epelbaum E, Krebs H, Lahde T A, Lee D, Li N, Lu B N, Meibner U G, Rupak G 2017 Phys. Rev. Lett. 119 222505
Google Scholar
[7] Baumgärtner A, Burkitt A N, Burkitt A N, Ceperley D M, De Raedt H, Ferrenberg A M, Heermann D W, Herrmann H J, Landau D P, Levesque D, Linden W, Reger F D, Schmidt K E, Selke W, Stauffer D, Swendsen R H, Wang J S, Weis J J, Young A P 2012 The Monte Carlo Method in Condensed Matter Physics (Springer Science & Business Media) pp23−51
[8] Dawson C M, Eisert J, Osborne T J 2008 Phys. Rev. Lett. 100 130501
Google Scholar
[9] Needs R J, Towler M D, Drummond N D, López Ríos P 2009 J. Phys.: Condens. Matter 22 023201
Google Scholar
[10] Ceperley D M, Alder B J 1980 Phys. Rev. Lett. 45 566
Google Scholar
[11] Sandvik A W, Kurkijärvi J 1991 Phys. Rev. B 43 5950
Google Scholar
[12] Gelfand M P, Singh R R P 2000 Adv. Phys. 49 93
Google Scholar
[13] Oitmaa J, Hamer C, Zheng W 2006 Series Expansion Methods for Strongly Interacting Lattice Models (Cambridge: Cambridge University Press) pp99−123
[14] Lanczos C 1950 J. Res. Nat. Bur. Stand. 45 255
Google Scholar
[15] Saad Y 2011 Numerical Methods for Large Eigenvalue Problems, Classics in Applied Mathematics (Philadelphia: Society for Industrial and Applied Mathematic) pp125−162
[16] Frame D, He R Z, Ipsen I, Lee D, Lee D, Rrapaj E 2018 Phys. Rev. Lett. 121 032501
Google Scholar
[17] Wang L 2016 Phys. Rev. B 94 195105
Google Scholar
[18] Hu W J, Singh R R P, Scalettar R T 2017 Phys. Rev. E 95 062122
Google Scholar
[19] Wang C, Zhai H 2017 Phys. Rev. B 96 144432
Google Scholar
[20] Wang C, Zhai H 2018 Front. Phys. 13 130507
Google Scholar
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