Matrix product state applications for the ALPS project. (English) Zbl 1360.81016
Summary: The density-matrix renormalization group method has become a standard computational approach to the low-energy physics as well as dynamics of low-dimensional quantum systems. In this paper, we present a new set of applications, available as part of the ALPS package, that provide an efficient and flexible implementation of these methods based on a matrix product state (MPS) representation. Our applications implement, within the same framework, algorithms to variationally find the ground state and low-lying excited states as well as simulate the time evolution of arbitrary one-dimensional and two-dimensional models. Implementing the conservation of quantum numbers for generic Abelian symmetries, we achieve performance competitive with the best codes in the community. Example results are provided for (i) a model of itinerant fermions in one dimension and (ii) a model of quantum magnetism.
MSC:
| 81-04 | Software, source code, etc. for problems pertaining to quantum theory |
| 81V70 | Many-body theory; quantum Hall effect |
| 81-08 | Computational methods for problems pertaining to quantum theory |
| 82D40 | Statistical mechanics of magnetic materials |
| 81T17 | Renormalization group methods applied to problems in quantum field theory |
References:
| [1] | White, S. R., Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett., 69, 19, 2863-2866 (1992) |
| [2] | Östlund, S.; Rommer, S., Thermodynamic limit of density matrix renormalization, Phys. Rev. Lett., 75, 3537 (1995) |
| [3] | Vidal, G., Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett., 93, 4, 040502 (2004) |
| [4] | White, S. R.; Feiguin, A. E., Real-time evolution using the density matrix renormalization group, Phys. Rev. Lett., 93, 7, 076401 (2004) |
| [5] | Daley, A. J.; Kollath, C.; Schollwöck, U.; Vidal, G., Time-dependent density-matrix renormalization-group using adaptive effective hilbert spaces, J. Stat. Mech.-Theory E., 2004, 04, P04005 (2004) · Zbl 1075.82015 |
| [6] | Verstraete, F.; García-Ripoll, J. J.; Cirac, J. I., Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett., 93, 207204 (2004) |
| [7] | Feiguin, A. E.; White, S. R., Finite-temperature density matrix renormalization using an enlarged hilbert space, Phys. Rev. B, 72, 220401 (2005) |
| [8] | Zwolak, M.; Vidal, G., Mixed-state dynamics in one-dimensional quantum lattice systems: A time-dependent superoperator renormalization algorithm, Phys. Rev. Lett., 93, 207205 (2004) |
| [9] | Barthel, T.; Schollwöck, U.; White, S. R., Spectral functions in one-dimensional quantum systems at finite temperature using the density matrix renormalization group, Phys. Rev. B, 79, 245101 (2009) |
| [10] | I. Pizorn, V. Eisler, S. Andergassen, M. Troyer, Real time evolution at finite temperatures with operator space matrix product states, ArXiv e-prints arXiv:1305.0504; I. Pizorn, V. Eisler, S. Andergassen, M. Troyer, Real time evolution at finite temperatures with operator space matrix product states, ArXiv e-prints arXiv:1305.0504 |
| [11] | Stoudenmire, E.; White, S. R., Studying two-dimensional systems with the density matrix renormalization group, Ann. Rev. Cond. Mat. Phys., 3, 1, 111-128 (2012) |
| [12] | Liang, S.; Pang, H., Approximate diagonalization using the density-matrix renormalization-group method: A two-dimensional perspective, Phys. Rev. B, 49, 13, 9214 (1994) |
| [13] | White, S. R.; Chernyshev, A. L., Neél order in square and triangular lattice heisenberg models, Phys. Rev. Lett., 99, 12, 127004 (2007) |
| [14] | Yan, S.; Huse, D. A.; White, S. R., Spin-liquid ground state of the s=1/2 kagome Heisenberg antiferromagnet, Science, 332, 6034, 1173-1176 (2011) |
| [15] | Bauer, B.; Corboz, P.; Läuchli, A. M.; Messio, L.; Penc, K.; Troyer, M.; Mila, F., Three-sublattice order in the su(3) Heisenberg model on the square and triangular lattice, Phys. Rev. B, 85, 125116 (2012) |
| [16] | Depenbrock, S.; McCulloch, I. P.; Schollwöck, U., Nature of the spin-liquid ground state of the \(s = 1 / 2\) Heisenberg model on the kagome lattice, Phys. Rev. Lett., 109, 067201 (2012) |
| [17] | Jiang, H.-C.; Wang, Z.; Balents, L., Identifying topological order by entanglement entropy, Nat. Phys., 8, 12, 902-905 (2012) |
| [18] | B. Bauer, L. Cincio, B.P. Keller, M. Dolfi, G. Vidal, S. Trebst, A.W.W. Ludwig, Chiral spin liquid and emergent anyons in a Kagome lattice Mott insulator, ArXiv e-prints arXiv:1401.3017; B. Bauer, L. Cincio, B.P. Keller, M. Dolfi, G. Vidal, S. Trebst, A.W.W. Ludwig, Chiral spin liquid and emergent anyons in a Kagome lattice Mott insulator, ArXiv e-prints arXiv:1401.3017 |
| [19] | Albuquerque, a.; Alet, F.; Corboz, P.; Dayal, P.; Feiguin, a.; Fuchs, S.; Gamper, L.; Gull, E.; Gurtler, S.; Honecker, a., The ALPS project release 1.3: Open-source software for strongly correlated systems, J. Magn. Magn. Mater., 310, 2, 1187-1193 (2007) |
| [20] | Bauer, B., The ALPS project release 2.0: open source software for strongly correlated systems, J. Stat. Mech., 2011, 05, P05001 (2011) |
| [21] | http://alps.comp-phys.org; http://alps.comp-phys.org |
| [22] | http://www.hp2c.ch; http://www.hp2c.ch |
| [23] | Bauer, B.; Huijse, L.; Berg, E.; Troyer, M.; Schoutens, K., Supersymmetric multicritical point in a model of lattice fermions, Phys. Rev. B, 87, 165145 (2013) |
| [24] | M. Cheng, M. Becker, B. Bauer, R.M. Lutchyn, Interplay between Kondo and Majorana interactions in quantum dots, ArXiv e-prints arXiv:1308.4156; M. Cheng, M. Becker, B. Bauer, R.M. Lutchyn, Interplay between Kondo and Majorana interactions in quantum dots, ArXiv e-prints arXiv:1308.4156 |
| [25] | Shinaoka, H.; Dolfi, M.; Troyer, M.; Werner, P., Hybridization expansion monte carlo simulation of multi-orbital quantum impurity problems: matrix product formalism and improved sampling, J. Stat. Mech.-Theory E., 2014, 6, P06012 (2014) · Zbl 1456.82979 |
| [26] | Dolfi, M.; Bauer, B.; Troyer, M.; Ristivojevic, Z., Multigrid algorithms for tensor network states, Phys. Rev. Lett., 109, 2, 020604 (2012) |
| [27] | ITensor, http://itensor.org; ITensor, http://itensor.org |
| [28] | Open Source MPS, http://sourceforge.net/projects/openmps; Open Source MPS, http://sourceforge.net/projects/openmps |
| [29] | Alvarez, G., The density matrix renormalization group for strongly correlated electron systems: A generic implementation, Comput. Phys. Commun., 180, 9, 1572-1578 (2009) |
| [30] | BLOCK, http://www.princeton.edu/chemistry/chan/software/dmrg; BLOCK, http://www.princeton.edu/chemistry/chan/software/dmrg |
| [31] | Wouters, S.; Poelmans, W.; Ayers, P. W.; Van Neck, D., CheMPS2: A free open-source spin-adapted implementation of the density matrix renormalization group for ab initio quantum chemistry, Comput. Phys. Commun., 185, 6, 1501-1514 (2014) · Zbl 1348.81029 |
| [32] | Schollwöck, U., The density-matrix renormalization group in the age of matrix product states, Ann. Phys., NY, 326, 1, 96-192 (2011) · Zbl 1213.81178 |
| [33] | Hastings, M. B., An area law for one-dimensional quantum systems, J. Stat. Mech.-Theory E., 2007, 08, P08024 (2007) · Zbl 1456.82046 |
| [34] | Schuch, N.; Wolf, M. M.; Verstraete, F.; Cirac, J. I., Entropy scaling and simulability by matrix product states, Phys. Rev. Lett., 100, 3, 030504 (2008) · Zbl 1228.82014 |
| [35] | Verstraete, F.; Cirac, J. I., Matrix product states represent ground states faithfully, Phys. Rev. B, 73, 094423 (2006) |
| [36] | Eisert, J.; Cramer, M.; Plenio, M. B., Colloquium: Area laws for the entanglement entropy, Rev. Modern Phys., 82, 277-306 (2010) · Zbl 1205.81035 |
| [37] | Crosswhite, G. M.; Bacon, D., Finite automata for caching in matrix product algorithms, Phys. Rev. A, 78, 012356 (2008) |
| [38] | Jordan, P.; Wigner, E., Über das Paulische Äquivalenzverbot, Z. Phys., 47, 9-10, 631-651 (1928) · JFM 54.0983.03 |
| [39] | White, S. R., Density matrix renormalization group algorithms with a single center site, Phys. Rev. B, 72, 180403 (2005) |
| [40] | McCulloch, I. P., From density-matrix renormalization group to matrix product states, J. Stat. Mech.-Theory E., 2007, 10, P10014 (2007) |
| [41] | Vidal, G., Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett., 91, 14, 147902 (2003) |
| [42] | Trotter, H. F., On the product of semi-groups of operators, Proc. Amer. Math. Soc., 10, 545 (1959) · Zbl 0099.10401 |
| [43] | Suzuki, M., Generalized trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems, Comm. Math. Phys., 51, 2, 183-190 (1976) · Zbl 0341.47028 |
| [44] | Bauer, B.; Corboz, P.; Orús, R.; Troyer, M., Implementing global abelian symmetries in projected entangled-pair state algorithms, Phys. Rev. B, 83, 125106 (2011) |
| [45] | Silva, C. T.; Freire, J.; Callahan, S. P., Provenance for visualizations: Reproducibility and beyond, Comput. Sci. Eng., 9, 5, 82-89 (2007) |
| [46] | http://vistrails.org; http://vistrails.org |
| [47] | Noack, R.; White, S.; Scalapino, D., The ground state of the two-leg hubbard ladder a density-matrix renormalization group study, Physica C, 270, 3-4, 281-296 (1996) |
| [48] | Luther, A.; Emery, V., Backward scattering in the one-dimensional electron gas, Phys. Rev. Lett., 33, 10, 589-592 (1974), Cited By (since 1996)247, http://dx.doi.org/10.1103/PhysRevLett.33.589 |
| [49] | Antal, T.; Rácz, Z.; Rákos, A.; Schütz, G. M., Transport in the xx chain at zero temperature: Emergence of flat magnetization profiles, Phys. Rev. E, 59, 4912-4918 (1999) |
| [50] | Kitaev, A. Y., Unpaired majorana fermions in quantum wires, Phys.-Usp., 44, 10S, 131 (2001) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
