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arXiv:2403.11588

Numerical observation of $\mathrm{SU}(N)$ Nagaoka ferromagnetism

Thomas Botzung, Pierre Nataf

Published Mar 18, 2024• 0 citations
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Abstract

We provide numerical evidence of the Nagaoka's theorem in the $\mathrm{SU}(N)$ Fermi-Hubbard model on various cluster geometries, such as the square, the honeycomb and the triangular lattices. In particular, by diagonalizing several finite-size clusters, we show that for one hole away from filling $1/N$, the itinerant ferromagnetism arises for $U$ (the positive on-site interaction) larger than $U_c$ (the value at the transition), which strongly depends on the coordination number $z$ and on $N$, the number of degenerate orbitals, that we vary from $N=2$ to $N=6$ in our simulations. We prove that $U_c$ is a non decreasing function of $N$. In addition, we find that the lattice dependency is rooted in the kinetic energy of the hole. We find that large coordination numbers $z$ lower the value of $U_c$. Complementary, we explore the effect of long-range hopping on the appearance of itinerant ferromagnetism and demonstrate that it acts as an increased coordination number, protecting the ferromagnetic phase at small $U$. Finally, both the effects of the presence of some additional holes and of the finite size of the clusters are briefly discussed.

DOI
10.48550/arxiv.2403.11588

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Numerical observation of SU(N)\mathrm{SU}(N)SU(N) Nagaoka ferromagnetism
We provide numerical evidence of the Nagaoka's theorem in the SU(N)\mathrm{SU}(N)SU(N) Fermi-Hubbard model on various cluster geometries, such as the square, the honeycomb and the triangular lattices. In particular, by diagonalizing several finite-size clusters, we show that for one hole away from filling 1/N1/N1/N, the itinerant ferromagnetism arises for UUU (the positive on-site interaction) larger than UcU_cUc​ (the value at the transition), which strongly depends on the coordination number zzz and on NNN, the number of degenerate orbitals, that we vary from N=2N=2N=2 to N=6N=6N=6 in our simulations. We prove that UcU_cUc​ is a non decreasing function of NNN. In addition, we find that the lattice dependency is rooted in the kinetic energy of the hole. We find that large coordination numbers zzz lower the value of UcU_cUc​. Complementary, we explore the effect of long-range hopping on the appearance of itinerant ferromagnetism and demonstrate that it acts as an increased coordination number, protecting the ferromagnetic phase at small UUU. Finally, both the effects of the presence of some additional holes and of the finite size of the clusters are briefly discussed.
We provide numerical evidence of the Nagaoka's theorem in the SU(N)\mathrm{SU}(N)SU(N) Fermi-Hubbard model on various cluster geometries, such as the square, the honeycomb, and the triangular lattices. In particular, by diagonalizing several finite-size clusters, we show that for one hole away from filling 1/N1/N1/N, the itinerant ferromagnetism arises for UUU (the positive on-site interaction) larger than Uc{U}_{c}Uc​ (the value at the transition), which depends strongly on the coordination number zzz and on NNN, the number of degenerate orbitals, that we vary from N=2N=2N=2 to 6 in our simulations. We prove that Uc{U}_{c}Uc​ is a nondecreasing function of NNN. In addition, we find that the lattice dependency is rooted in the kinetic energy of the hole. We find that large coordination numbers zzz lower the value of Uc{U}_{c}Uc​. Complementary, we explore the effect of long-range hopping on the appearance of itinerant ferromagnetism, and we demonstrate that it acts as an increased coordination number, protecting the ferromagnetic phase at small UUU. Finally, the effects of both the presence of some additional holes and the finite size of the clusters are briefly discussed.
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.
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.
We investigate an SU(3) Fermi-Hubbard model on a hypercubic lattice at finite temperatures, combining dynamical mean-field theory with continuous-time quantum Monte Carlo simulations. Taking strong correlations into account carefully, we find a ferromagnetically ordered state, in which one of the three components becomes dominant, when holes are doped away from one-third filling. Furthermore, we demonstrate that this ferromagnetically ordered phase undergoes a first-order transition to a paramagnetic state. We clarify the stability of the ferromagnetically ordered state against interaction strength, hole doping, and temperatures. The relevance of generalized Nagaoka ferromagnetism is also addressed, by comparing the results on the Bethe lattice.
Nagaoka ferromagnetism (NF) is a long-predicted example of itinerant ferromagnetism (IF) in the Hubbard model that has been studied theoretically for many years. The condition for NF, an infinite on-site Coulomb repulsion and a single hole in a half-filled band, does not arise naturally in materials. NF was only realized recently for the first time in experiments on a $2\ifmmode\times\else\texttimes\fi{}2$ array of gated quantum dots. Dopant arrays and gated quantum dots in Si allow for engineering controllable systems with complex geometries. This makes dopant and quantum dot arrays good candidates to study NF in different array geometries through analog quantum simulation. Here we present theoretical simulations done for $3\ifmmode\times\else\texttimes\fi{}3$ arrays and larger $N\ifmmode\times\else\texttimes\fi{}N$ arrays and predict the emergence of different forms of ferromagnetism in different geometries. We find NF in perfect $3\ifmmode\times\else\texttimes\fi{}3$ arrays, as well as in $N\ifmmode\times\else\texttimes\fi{}N$ arrays for one hole doping of a half-filled band. The ratio of the hopping ttt to Hubbard on-site repulsion UUU that defines the onset of NF scales as 1/N41/{N}^{4}1/N4 as NNN increases, approaching the bulk limit of infinite UUU for large NNN. Additional simulations are done for geometries made by removing sites from $N\ifmmode\times\else\texttimes\fi{}N$ arrays. Different forms of ferromagnetism are found for different geometries. Loops show ferromagnetism, but only for three electrons. For loops, the critical t/Ut/Ut/U for the onset of ferromagnetism scales as NNN as the loop length increases. We show that the different dependencies on size for loops and $N\ifmmode\times\else\texttimes\fi{}N$ arrays can be understood by scaling arguments that highlight the different energy contributions to each different form of ferromagnetism. Our results show how analog quantum simulation with small arrays can elucidate the role of effects including wave-function connectivity; system geometry, size, and symmetry; bulk and edge sites; and kinetic energy in determining quantum magnetism of small systems.
Nagaoka Ferromagnetism in 3×33 \times 33×3 Arrays and Beyond
Nagaoka ferromagnetism (NF) is a long-predicted example of itinerant ferromagnetism (IF) in the Hubbard model that has been studied theoretically for many years. The condition for NF, an infinite on-site Coulomb repulsion and a single hole in a half-filled band, does not arise naturally in materials. NF was only realized recently for the first time in experiments on a 2×22\times 22×2 array of gated quantum dots. Dopant arrays and gated quantum dots in Si allow for engineering controllable systems with complex geometries. This makes quantum dot arrays good candidates to study NF in different geometries through analog quantum simulation. Here we present theoretical simulations done for 3×33\times 33×3 arrays and larger N×NN\times NN×N arrays, and predict the emergence of different forms of ferromagnetism in different geometries. We find NF in perfect 3×33\times 33×3 arrays, and in N×NN\times NN×N arrays for one hole doping of a half-filled band. The ratio of the hopping ttt to Hubbard on-site repulsion UUU that defines the onset of NF scales as 1/N41/N^{4}1/N4 as NNN increases, approaching the bulk limit of infinite UUU for large NNN. Additional simulations are done for geometries made by removing sites from N×NN\times NN×N arrays. Different forms of ferromagnetism are found for different geometries. Loops show ferromagnetism, but only for three electrons. For loops, the critical t/Ut/Ut/U for the onset of ferromagnetism scales as NNN as the loop length increases. We show that the different dependences on size for loops and N×NN\times NN×N arrays can be understood by scaling arguments that highlight the different energy contributions to each form of ferromagnetism. Our results show how analog quantum simulation with small arrays can elucidate the role of effects including wavefunction connectivity; system geometry, size and symmetry; bulk and edge sites; and kinetic energy in determining quantum magnetism of small systems.