The Hubbard model

2023

1. PRB

   Ferromagnetic domains in the large-U Hubbard model with a few holes: A full configuration interaction quantum Monte Carlo study

   Sujun Yun, Werner Dobrautz, Hongjun Luo, Vamshi Katukuri, Niklas Liebermann, and 1 more author

   Phys. Rev. B, Feb 2023

   Abs arXiv Bib HTML

   Two-dimensional Hubbard lattices with two or three holes are investigated as a function of U in the large-U limit. In the so-called Nagaoka limit (one-hole system at infinite U), it is known that the Hubbard model exhibits a ferromagnetic ground state. Here, by means of exact full configuration interaction quantum Monte Carlo simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing U. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centered on individual holes. The overall total spin of the wave functions remains the lowest possible (0 or 12, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than 1) number of holes.

   @article{PhysRevB.107.064405,
     title = {Ferromagnetic domains in the large-$U$ Hubbard model with a few holes: A full configuration interaction quantum Monte Carlo study},
     author = {Yun, Sujun and Dobrautz, Werner and Luo, Hongjun and Katukuri, Vamshi and Liebermann, Niklas and Alavi, Ali},
     journal = {Phys. Rev. B},
     volume = {107},
     issue = {6},
     pages = {064405},
     numpages = {9},
     year = {2023},
     month = feb,
     publisher = {American Physical Society},
     doi = {10.1103/PhysRevB.107.064405},
     url = {https://link.aps.org/doi/10.1103/PhysRevB.107.064405},
     dimensions = {true},
     project_guga = {true},
     project_hubbard = {true}
   }

2. PRResearch

   Orders of magnitude increased accuracy for quantum many-body problems on quantum computers via an exact transcorrelated method

   Igor O. Sokolov, Werner Dobrautz, Hongjun Luo, Ali Alavi, and Ivano Tavernelli

   Phys. Rev. Res., Jun 2023

   Abs arXiv Bib HTML

   Transcorrelated methods provide an efficient way of partially transferring the description of electronic correlations from the ground-state wave function directly into the underlying Hamiltonian. In particular, Dobrautz et al. [Phys. Rev. B 99, 075119 (2019)] have demonstrated that the use of momentum-space representation, combined with a nonunitary similarity transformation, results in a Hubbard Hamiltonian that possesses a significantly more “compact” ground-state wave function, dominated by a single Slater determinant. This compactness/single-reference character greatly facilitates electronic structure calculations. As a consequence, however, the Hamiltonian becomes non-Hermitian, posing problems for quantum algorithms based on the variational principle. We overcome these limitations with the Ansatz-based quantum imaginary-time evolution algorithm and apply the transcorrelated method in the context of digital quantum computing. We demonstrate that this approach enables up to four orders of magnitude more accurate and compact solutions in various instances of the Hubbard model at intermediate interaction strength (U/t=4), enabling the use of shallower quantum circuits for wave-function Ansätzes. In addition, we propose a more efficient implementation of the quantum imaginary-time evolution algorithm in quantum circuits that is tailored to non-Hermitian problems. To validate our approach, we perform hardware experiments on the ibmq_lima quantum computer. Our work paves the way for the use of exact transcorrelated methods for the simulations of ab initio systems on quantum computers.

   @article{PhysRevResearch.5.023174,
     title = {Orders of magnitude increased accuracy for quantum many-body problems on quantum computers via an exact transcorrelated method},
     author = {Sokolov, Igor O. and Dobrautz, Werner and Luo, Hongjun and Alavi, Ali and Tavernelli, Ivano},
     journal = {Phys. Rev. Res.},
     volume = {5},
     issue = {2},
     pages = {023174},
     numpages = {19},
     year = {2023},
     month = jun,
     publisher = {American Physical Society},
     doi = {10.1103/PhysRevResearch.5.023174},
     url = {https://link.aps.org/doi/10.1103/PhysRevResearch.5.023174},
     dimensions = {true},
     project_quantum = {true},
     project_tc = {true},
     project_hubbard = {true}
   }

2021

1. PRB

   Benchmark study of Nagaoka ferromagnetism by spin-adapted full configuration interaction quantum Monte Carlo

   Sujun Yun, Werner Dobrautz, Hongjun Luo, and Ali Alavi

   Phys. Rev. B, Dec 2021

   Abs arXiv Bib HTML

   We investigate Nagaoka ferromagnetism in the two-dimensional Hubbard model with one hole using the spin-adapted [SU(2) conserving] full configuration interaction quantum Monte Carlo method. This methodology gives us access to the ground-state energies of all possible spin states S of finite Hubbard lattices, here obtained for lattices up to 26 sites for various interaction strengths (U). The critical interaction strength, Uc, at which the Nagaoka transition occurs is determined for each lattice and is found to be proportional to the lattice size for the larger lattices. Below Uc, the overall ground states are found to favour the minimal total spin (S=12), and no intermediate spin state is found to be the overall ground state on lattices larger than 16 sites. However, at Uc, the energies of all the spin states are found to be nearly degenerate, implying that large fluctuations in total spin can be expected in the vicinity of the Nagaoka transition.

   @article{PhysRevB.104.235102,
     title = {Benchmark study of Nagaoka ferromagnetism by spin-adapted full configuration interaction quantum Monte Carlo},
     author = {Yun, Sujun and Dobrautz, Werner and Luo, Hongjun and Alavi, Ali},
     journal = {Phys. Rev. B},
     volume = {104},
     issue = {23},
     pages = {235102},
     numpages = {6},
     year = {2021},
     month = dec,
     publisher = {American Physical Society},
     doi = {10.1103/PhysRevB.104.235102},
     url = {https://link.aps.org/doi/10.1103/PhysRevB.104.235102},
     dimensions = {true},
     project_guga = {true},
     project_hubbard = {true}
   }

2019

1. PRB

   Compact numerical solutions to the two-dimensional repulsive Hubbard model obtained via nonunitary similarity transformations

   Werner Dobrautz, Hongjun Luo, and Ali Alavi

   Phys. Rev. B, Feb 2019

   Abs arXiv Bib HTML

   Similarity transformation of the Hubbard Hamiltonian using a Gutzwiller correlator leads to a non-Hermitian effective Hamiltonian, which can be expressed exactly in momentum-space representation and contains three-body interactions. We apply this methodology to study the two-dimensional Hubbard model with repulsive interactions near half filling in the intermediate interaction strength regime (U/t=4). We show that at optimal or near optimal strength of the Gutzwiller correlator, the similarity-transformed Hamiltonian has extremely compact right eigenvectors, which can be sampled to high accuracy using the full configuration interaction quantum Monte Carlo (FCIQMC) method and its initiator approximation. Near-optimal correlators can be obtained using a simple projective equation, thus obviating the need for a numerical optimization of the correlator. The FCIQMC method, as a projective technique, is well suited for such non-Hermitian problems, and its stochastic nature can handle the three-body interactions exactly without undue increase in computational cost. The highly compact nature of the right eigenvectors means that the initiator approximation in FCIQMC is not severe and that large lattices can be simulated, well beyond the reach of the method applied to the original Hubbard Hamiltonian. Results are provided in lattice sizes up to 50 sites and compared to auxiliary-field QMC. New benchmark results are provided in the off-half-filling regime, with no severe sign problem being encountered. In addition, we show that methodology can be used to calculate excited states of the Hubbard model and lay the groundwork for the calculation of observables other than the energy.

   @article{PhysRevB.99.075119,
     title = {Compact numerical solutions to the two-dimensional repulsive Hubbard model obtained via nonunitary similarity transformations},
     author = {Dobrautz, Werner and Luo, Hongjun and Alavi, Ali},
     journal = {Phys. Rev. B},
     volume = {99},
     issue = {7},
     pages = {075119},
     numpages = {20},
     year = {2019},
     month = feb,
     publisher = {American Physical Society},
     doi = {10.1103/PhysRevB.99.075119},
     url = {https://link.aps.org/doi/10.1103/PhysRevB.99.075119},
     dimensions = {true},
     project_tc = {true},
     project_hubbard = {true}
   }