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arXiv:2105.06802 · Physical review. B./Physical review. B

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

Sujun Yun, Werner Dobrautz, Hongjun Luo, Ali Alavi

Published Dec 2, 2021• 13 citations
arXivHTMLPDF

Abstract

We investigate Nagaoka ferromagnetism in the two-dimensional Hubbard model with one hole using the spin-adapted [$\text{SU}(2)$ 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, ${U}_{c}$, 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 ${U}_{c}$, the overall ground states are found to favour the minimal total spin ($S=\frac{1}{2}$), and no intermediate spin state is found to be the overall ground state on lattices larger than 16 sites. However, at ${U}_{c}$, 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.

DOI
10.1103/physrevb.104.235102

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We investigate Nagaoka ferromagnetism in the two-dimensional Hubbard model with one hole using the spin-adapted [SU(2)\text{SU}(2)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 SSS of finite Hubbard lattices, here obtained for lattices up to 26 sites for various interaction strengths (UUU). The critical interaction strength, Uc{U}_{c}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{U}_{c}Uc​, the overall ground states are found to favour the minimal total spin (S=12S=\frac{1}{2}S=21​), and no intermediate spin state is found to be the overall ground state on lattices larger than 16 sites. However, at Uc{U}_{c}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.
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of UUU in the large-UUU limit. In the so-called Nagaoka limit (one-hole system at infinite UUU), 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 UUU. 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\frac{1}{2}21​, 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.
Ferromagnetic domains in the large-UUU Hubbard model with a few holes: an FCIQMC study
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of UUU in the large-UUU limit. In the so-called Nagaoka limit (one-hole system at infinite UUU), it is known that the Hubbard model exhibits a ferromagnetic ground-state. Here, by means of exact FCIQMC simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing UUU. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or 12\frac{1}{2}21​, 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 one) number of holes.
We present different numerical calculations based on variational quantum Monte Carlo simulations supporting a ferromagnetic ground state for finite and small hole densities in the two-dimensional infinite-U Hubbard model. Moreover, by studying the energies of different total spin sectors, these calculations strongly suggest that the paramagnetic phase is unstable against a phase with a partial polarization for large hole densities delta approximately 0.40 with evidence for a second-order transition to the paramagnetic large doping phase.
The infinite-U Hubbard model on finite two-dimensional square lattices is investigated by the use of exact numerical diagonalization. Configurations with one spin flip away from the ferromagnetic state and up to three holes are studied for lattice sizes ranging from 4*4 to 12*12. For all these situations, the state with one spin flip is found to be more stable than the Nagaoka ferromagnetic state. The crystal momentum of the ground state in this spin sector depends on the system size. Systematic comparison with the Gutzwiller wavefunction shows that it reproduces qualitatively the crystal momentum dependence of the energy, but not the exact correlation between the spin flip and the holes.
To validate the ability of full configuration interaction quantum Monte Carlo (FCIQMC) for studying the 2D Hubbard model near half-filling regime, the ground state energies of a 4 × 4 square lattice system with various interaction strengths are calculated. It is found that the calculated results are in good agreement with those obtained by exact diagonalization (i.e., the exact values for a given basis set) when the population of psi particles (psips) is higher than the critical population required to correctly sample the ground state wave function. In addition, the variations of the average computational time per 20 Monte Carlo cycles with the coupling strength and the number of processors are also analyzed. The calculated results show that the computational efficiency of an FCIQMC calculation is mainly affected by the total population of psips and the communication between processors. These results can provide useful references for understanding the FCIQMC algorithm, studying the ground state properties of the 2D Hubbard model for the larger system size by the FCIQMC method and using a computational budget as effectively as possible.
We study the magnetic properties of itinerant quantum magnetic particles, described by a generalized Hubbard model with large spin (S>1/2)(S>1/2)(S>1/2), which may be realized in optical lattices of laser-cooled atom systems. In fermion systems (half-integer spins), an extended form of Nagaoka ferromagnetism may be realized. However, as novel aspects of the large-spin cases, we found that the condition on the lattice connectivity is more stringent than in the case of S=1/2S=1/2S=1/2 particles and that the system shows a peculiar degenerate structure of the ground state in which the ferromagnetic state is included. In contrast, it turns out that the ground state of itinerant bosonic systems (integer spins) has a degenerate structure similar to that of fermion system with S>1/2S>1/2S>1/2 regardless of the shape, connectivity, or filling of the lattice, and that the state with the maximum total spin is always one of the ground states. Because the system consists of 2S+12S+12S+1 types of particles and we study a SU(2S+1)\text{SU}(2S+1)SU(2S+1) invariant model, the degeneracy of the ground state is given by the multiplets of the fully symmetric Young tableau of SU(2S+1)\text{SU}(2S+1)SU(2S+1) if the state with maximum total spin belongs to the ground state.