Physical review. B./Physical review. B

Numerical observation of SU( N ) Nagaoka ferromagnetism

Thomas Botzung, Pierre Nataf

Published Jun 17, 2024• 7 citations

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 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.

DOI: 10.1103/physrevb.109.235131

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Physical review. B./Physical review. B

Numerical observation of SU( N ) Nagaoka ferromagnetism

Thomas Botzung, Pierre Nataf

Published Jun 17, 2024• 7 citations

Publisher

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 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.

DOI: 10.1103/physrevb.109.235131

Related papers

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

Ferromagnetic domains in the large- $U$ Hubbard model with a few holes: an FCIQMC studyFerromagnetic domains in the large-$U$ Hubbard model with a few holes: an FCIQMC study

Yun, Sujun, Werner Dobrautz, Luo Hongjun +3 more

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 FCIQMC 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, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or $\frac{1}{2}$ , 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 $U$ 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 FCIQMC 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, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or $\frac{1}{2}$, 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.

arXiv:2210.12049Oct 21, 20221 citation

Nagaoka ferromagnetism in 3 × 3 arrays and beyond

Yan Li, Keyi Liu, Garnett W. Bryant

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.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.

Physical review. B./Physical review. BDec 26, 20242 citations

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

Su-Jun Yun, Werner Dobrautz, Hongjun Luo +3 more

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 $\frac{1}{2}$ , 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.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 $\frac{1}{2}$, 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.

Physical review. B./Physical review. BFeb 7, 20238 citations

Itinerant ferromagnetism in an SU(3) Fermi-Hubbard model at finite temperatures: A dynamical mean-field theory study

Juntaro Fujii, Kazuki Yamamoto, Akihisa Koga

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.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.

arXiv:2505.12704Physical review. B./Physical review. BSep 3, 20252 citations

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

Sujun Yun, Werner Dobrautz, Hongjun Luo +1 more

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.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.

arXiv:2105.06802Physical review. B./Physical review. BDec 2, 202113 citations

We provide numerical evidence of the Nagaoka's theorem in the

\mathrm{SU}(N)

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}

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.

Ferromagnetic domains in the large-

U

Hubbard model with a few holes: an FCIQMC study

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 FCIQMC 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, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or

\frac{1}{2}

, 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.

t

to Hubbard on-site repulsion

U

that defines the onset of NF scales as

1/{N}^{4}

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.

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

\frac{1}{2}

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.