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Physical review. B./Physical review. B

Nagaoka ferromagnetism in 3 × 3 arrays and beyond

Yan Li, Keyi Liu, Garnett W. Bryant

Published Dec 26, 2024• 2 citations
Publisher

Abstract

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 $t$ to Hubbard on-site repulsion $U$ that defines the onset of NF scales as $1/{N}^{4}$ as $N$ increases, approaching the bulk limit of infinite $U$ for large $N$. 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 $t/U$ for the onset of ferromagnetism scales as $N$ 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.

DOI
10.1103/physrevb.110.245141

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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×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 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×\else\texttimes\fi33\ifmmode\times\else\texttimes\fi{}33\ifmmode×\else\texttimes\fi3 arrays and larger N\ifmmode×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN arrays and predict the emergence of different forms of ferromagnetism in different geometries. We find NF in perfect 3\ifmmode×\else\texttimes\fi33\ifmmode\times\else\texttimes\fi{}33\ifmmode×\else\texttimes\fi3 arrays, as well as in N\ifmmode×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN 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×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN 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×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN 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.
It is believed that strong ferromagnetic interactions in some solids are generated by subtle interplay between quantum many-body effects and spin-independent Coulomb interactions between electrons. It is a challenging problem to verify this scenario in the Hubbard model, which is an idealized model for strongly interacting electrons in a solid. Nagaoka's ferromagnetism is a well-known rigorous example of ferromagnetism in the Hubbard model. It deals with the limiting situation in which there is one fewer electron than in the half-filling and the on-site Coulomb interaction is infinitely large. There are relatively new rigorous examples of ferromagnetism in Hubbard models called flat-band ferromagnetism. Flat-band ferromagnetism takes place in carefully prepared models in which the lowest bands (in the single-electron spectra) are “flat.” Usually, these two approaches are regarded as two complimentary routes to ferromagnetism in the Hubbard model. In the present paper we describe Nagaoka's ferromagnetism and flat-band ferromagnetism in detail, giving all the necessary background as well as complete (but elementary) mathematical proofs. By studying an intermediate model called the long-range hopping model, we also demonstrate that there is indeed a deep relation between these two seemingly different approaches to ferromagnetism. We further discuss some attempts to go beyond these approaches. We briefly discuss recent rigorous example of ferromagnetism in the Hubbard model which has neither infinitely large parameters nor completely flat bands. We give preliminary discussion regarding possible experimental realizations of the (nearly-)flat-band ferromagnetism. Finally, we focus on some theoretical attempts to understand metallic ferromagnetism. We discuss three artificial one-dimensional models in which the existence of metallic ferromagnetism can be easily proved. We have tried to make the present paper as self-contained as possible, keeping in mind readers who are new to the field. Although the present paper is written as a review, it contains some material which appears for the first time.
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.
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.
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.
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.