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 depends strongly on the coordination number $z$ and on $N$ , the number of degenerate orbitals, that we vary from $N=2$ to 6 in our simulations. We prove that ${U}_{c}$ is a nondecreasing 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 we demonstrate that it acts as an increased coordination number, protecting the ferromagnetic phase at small $U$ . Finally, the effects of both the presence of some additional holes and the finite size of the clusters are briefly discussed.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 depends strongly on the coordination number $z$ and on $N$, the number of degenerate orbitals, that we vary from $N=2$ to 6 in our simulations. We prove that ${U}_{c}$ is a nondecreasing 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 we demonstrate that it acts as an increased coordination number, protecting the ferromagnetic phase at small $U$. Finally, the effects of both the presence of some additional holes and the finite size of the clusters are briefly discussed.

Physical review. B./Physical review. BJun 17, 20247 citations

Ferromagnetism in the Hubbard Model

Wolfgang Nolting, Michael Potthoff, T. Herrmann +1 more

arXiv:cond-mat/0107255Lecture notes in physicsJan 1, 20015 citations

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

Thomas Botzung, Pierre Nataf

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.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.

arXiv:2403.11588Mar 18, 20240 citations

Describing ferromagnetism in the Hubbard model

Ben Hur Bernhard

Journal of Magnetism and Magnetic MaterialsMar 7, 20084 citations

Variational cluster approach to ferromagnetism in infinite dimensions and in one-dimensional chains

Matthias Balzer, Michael Potthoff

The variational cluster approach (VCA) is applied to study spontaneous ferromagnetism in the Hubbard model at zero temperature. We discuss several technical improvements of the numerical implementation of the VCA which become necessary for studies of a ferromagnetically ordered phase, e.g., more accurate techniques to evaluate the variational ground-state energy, improved local as well as global algorithms to find stationary points, and different methods to locate the magnetic phase transition. Using the single-site VCA, i.e., the dynamical impurity approximation (DIA), the ferromagnetic phase diagram of the model in infinite dimensions is worked out. The results are compared with previous dynamical mean-field studies for benchmarking purposes. The DIA results provide a unified picture of ferromagnetism in the infinite-dimensional model by interlinking different parameter regimes that are governed by different mechanisms for ferromagnetic order. Using the DIA and the VCA, we then study ferromagnetism in one-dimensional Hubbard chains with nearest- and next-nearest-neighbor hopping ${t}_{2}$ . In comparison with previous results from the density-matrix renormalization group, the phase diagram is mapped out as a function of the Hubbard- $U$ , the electron filling, and ${t}_{2}$ . The stability of the ferromagnetic ground state against local and short-range nonlocal quantum fluctuations is discussed.The variational cluster approach (VCA) is applied to study spontaneous ferromagnetism in the Hubbard model at zero temperature. We discuss several technical improvements of the numerical implementation of the VCA which become necessary for studies of a ferromagnetically ordered phase, e.g., more accurate techniques to evaluate the variational ground-state energy, improved local as well as global algorithms to find stationary points, and different methods to locate the magnetic phase transition. Using the single-site VCA, i.e., the dynamical impurity approximation (DIA), the ferromagnetic phase diagram of the model in infinite dimensions is worked out. The results are compared with previous dynamical mean-field studies for benchmarking purposes. The DIA results provide a unified picture of ferromagnetism in the infinite-dimensional model by interlinking different parameter regimes that are governed by different mechanisms for ferromagnetic order. Using the DIA and the VCA, we then study ferromagnetism in one-dimensional Hubbard chains with nearest- and next-nearest-neighbor hopping ${t}_{2}$. In comparison with previous results from the density-matrix renormalization group, the phase diagram is mapped out as a function of the Hubbard-$U$, the electron filling, and ${t}_{2}$. The stability of the ferromagnetic ground state against local and short-range nonlocal quantum fluctuations is discussed.

arXiv:1007.2517Physical Review BNov 30, 201021 citations

Ferromagnetism in the Hubbard model

Andreas Mielke, Hal Tasaki

Communications in Mathematical PhysicsNov 1, 1993369 citations

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 depends strongly on the coordination number

z

and on

N

, the number of degenerate orbitals, that we vary from

N=2

to 6 in our simulations. We prove that

{U}_{c}

is a nondecreasing 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 we demonstrate that it acts as an increased coordination number, protecting the ferromagnetic phase at small

U

. Finally, the effects of both the presence of some additional holes and the finite size of the clusters are briefly discussed.

Numerical observation of

\mathrm{SU}(N)

Nagaoka ferromagnetism

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

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.

{t}_{2}

. In comparison with previous results from the density-matrix renormalization group, the phase diagram is mapped out as a function of the Hubbard-

U

, the electron filling, and

{t}_{2}

. The stability of the ferromagnetic ground state against local and short-range nonlocal quantum fluctuations is discussed.