arXiv:1108.2327 · Physical Review A

SU (N) magnetism in chains of ultracold alkaline-earth-metal atoms: Mott transitions and quantum correlations

Salvatore R. Manmana, Kaden R. A. Hazzard, Gang Chen, Adrian E. Feiguin, Ana María Rey

Published Oct 3, 2011• 84 citations

Abstract

We investigate one-dimensional SU$(N)$ Hubbard chains at zero temperature, which can be emulated with ultracold alkaline-earth-metal atoms, by using the density matrix renormalization group (DMRG), Bethe ansatz (BA), and bosonization. We compute experimental observables and use the DMRG to benchmark the accuracy of the Bethe ansatz for $N>2$ where the BA is only approximate. In the worst case, we find a relative error $\ensuremath{\varepsilon}\ensuremath{\lesssim}4%$ in the BA ground-state energy for $N\ensuremath{\leqslant}4$ at filling $1/N$, which is due to the fact that BA improperly treats the triply and higher occupied states. Using the DMRG for $N\ensuremath{\leqslant}4$ and the BA for large $N$, we determine the regimes of validity of strong- and weak-coupling perturbation theory for all values of $N$ and, in particular, the parameter range in which the system is well described by a SU$(N)$ Heisenberg model at filling $1/N$. We find this depends only weakly on $N$. We investigate the Berezinskii-Kosterlitz-Thouless phase transition from a Luttinger liquid to a Mott insulator by computing the fidelity susceptibility and the Luttinger parameter ${K}_{\ensuremath{\rho}}$ at $1/N$ filling. The numerical findings give strong evidence that the fidelity susceptibility develops a minimum at a critical interaction strength which is found to occur at a finite positive value for $N>2$.

DOI: 10.1103/physreva.84.043601

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Quantum magnetism in ultracold alkali and alkaline-earth fermion systems with symplectic symmetry

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Density matrix renormalization group simulations of the SU( N ) Fermi-Hubbard chain implementing the full SU( N ) symmetry via semistandard Young tableaux and unitary group subduction coefficients

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Metal-insulator transition and magnetism of SU(3) fermions in the square lattice

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arXiv:1108.2327 · Physical Review A

SU (N) magnetism in chains of ultracold alkaline-earth-metal atoms: Mott transitions and quantum correlations

Salvatore R. Manmana, Kaden R. A. Hazzard, Gang Chen, Adrian E. Feiguin, Ana María Rey

Published Oct 3, 2011• 84 citations

arXiv HTML PDF

Abstract

We investigate one-dimensional SU

(N)

Hubbard chains at zero temperature, which can be emulated with ultracold alkaline-earth-metal atoms, by using the density matrix renormalization group (DMRG), Bethe ansatz (BA), and bosonization. We compute experimental observables and use the DMRG to benchmark the accuracy of the Bethe ansatz for

N>2

where the BA is only approximate. In the worst case, we find a relative error

\ensuremath{\varepsilon}\ensuremath{\lesssim}4%

in the BA ground-state energy for

N\ensuremath{\leqslant}4

at filling

1/N

, which is due to the fact that BA improperly treats the triply and higher occupied states. Using the DMRG for

N\ensuremath{\leqslant}4

and the BA for large

N

, we determine the regimes of validity of strong- and weak-coupling perturbation theory for all values of

N

and, in particular, the parameter range in which the system is well described by a SU

(N)

Heisenberg model at filling

1/N

. We find this depends only weakly on

N

. We investigate the Berezinskii-Kosterlitz-Thouless phase transition from a Luttinger liquid to a Mott insulator by computing the fidelity susceptibility and the Luttinger parameter

{K}_{\ensuremath{\rho}}

1/N

filling. The numerical findings give strong evidence that the fidelity susceptibility develops a minimum at a critical interaction strength which is found to occur at a finite positive value for

N>2

DOI: 10.1103/physreva.84.043601

Related papers

Exact numerical and analytical results for correlated lattice electrons in one dimension

Satoshi Ejima

Since the advent of high- $T_c$ cuprate superconductors in 1986, strongly correlated electron systems have attracted much attention. Since the cuprates are essentially two-dimensional, low-dimensional systems have moved into the focus of condensed-matter theory. From a theoretical point of view, one-dimensional systems are of particular interest because there are exact numerical and analytical methods which permit detailed studies and deep insights into the many-body problem. In the first part of this thesis, using the numerically exact methods Exact Diagonalization and the Density-Matrix Renormalization Group (DMRG), we investigate the properties of the Tomonaga–Luttinger liquid which is the generic metallic state of matter in one dimension. In particular, we concentrate on the investigation of the so-called Tomonaga–Luttinger liquid parameter which determines the critical exponent $\alpha$ for the density of states near the Fermi energy. Experimental results for some quasi one-dimensional materials report $\alpha \gtrsim 1$ , which would imply $K_\rho \lesssim 0.17$ , a value which cannot be reconciled with the bare Hubbard model where $K_\rho^H \geq 0.5$ , i.e., $\alpha^H \lesssim 1/8$ . We develop new accurate numerical methods to obtain $K_\rho$ and investigate how to obtain such small values for $K_\rho$ for slightly doped charge-density-wave insulators. In the second part of this thesis, using the Thermodynamic Bethe Ansatz (TBA) as exact analytical method, we investigate the one-dimensional Hubbard model in the spindisordered regime, which is characterized by the temperature being much larger than the magnetic energy scale but small compared to the Mott–Hubbard gap. Our study is motivated by the controversy about the Mott–Hubbard insulator in infinite dimensions whose ground state is also spin-disordered. In this system the determination of the precise value of the critical interaction strength Uc where the Mott–Hubbard gap closes is still unsolved. Therefore, we provide an example of a Hubbard-type model with a disordered spin background which can be solved exactly. The thermodynamics of our model can be understood in terms of gapped charged excitations with an effective dispersion which are decoupled form the spin degrees of freedom; the latter contribute only entropically. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal-insulator transition at a finite interaction strength above which the gap opens linearly. Our exact results indicate that the strong-coupling expansion of the ground-state energy cannot be used to locate $U_c$ . However, the strong-coupling expansion of the gap permits a reliable extrapolation of the critical interaction strength.Since the advent of high-$T_c$ cuprate superconductors in 1986, strongly correlated electron systems have attracted much attention. Since the cuprates are essentially two-dimensional, low-dimensional systems have moved into the focus of condensed-matter theory. From a theoretical point of view, one-dimensional systems are of particular interest because there are exact numerical and analytical methods which permit detailed studies and deep insights into the many-body problem. In the first part of this thesis, using the numerically exact methods Exact Diagonalization and the Density-Matrix Renormalization Group (DMRG), we investigate the properties of the Tomonaga–Luttinger liquid which is the generic metallic state of matter in one dimension. In particular, we concentrate on the investigation of the so-called Tomonaga–Luttinger liquid parameter which determines the critical exponent $\alpha$ for the density of states near the Fermi energy. Experimental results for some quasi one-dimensional materials report $\alpha \gtrsim 1$, which would imply $K_\rho \lesssim 0.17$, a value which cannot be reconciled with the bare Hubbard model where $K_\rho^H \geq 0.5$, i.e., $\alpha^H \lesssim 1/8$. We develop new accurate numerical methods to obtain $K_\rho$ and investigate how to obtain such small values for $K_\rho$ for slightly doped charge-density-wave insulators. In the second part of this thesis, using the Thermodynamic Bethe Ansatz (TBA) as exact analytical method, we investigate the one-dimensional Hubbard model in the spindisordered regime, which is characterized by the temperature being much larger than the magnetic energy scale but small compared to the Mott–Hubbard gap. Our study is motivated by the controversy about the Mott–Hubbard insulator in infinite dimensions whose ground state is also spin-disordered. In this system the determination of the precise value of the critical interaction strength Uc where the Mott–Hubbard gap closes is still unsolved. Therefore, we provide an example of a Hubbard-type model with a disordered spin background which can be solved exactly. The thermodynamics of our model can be understood in terms of gapped charged excitations with an effective dispersion which are decoupled form the spin degrees of freedom; the latter contribute only entropically. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal-insulator transition at a finite interaction strength above which the gap opens linearly. Our exact results indicate that the strong-coupling expansion of the ground-state energy cannot be used to locate $U_c$. However, the strong-coupling expansion of the gap permits a reliable extrapolation of the critical interaction strength.

May 3, 20060 citations

One-dimensional Bose-Hubbard model with local three-body interactions

Satoshi Ejima, Florian Lange, Holger Fehske +2 more

We employ the (dynamical) density-matrix renormalization-group technique to investigate the ground-state properties of the Bose-Hubbard model with nearest-neighbor transfer amplitudes $t$ and local two-body and three-body repulsion of strength $U$ and $W$ , respectively. We determine the phase boundaries between the Mott-insulating and superfluid phases for the lowest two Mott lobes from the chemical potentials. We calculate the tips of the Mott lobes from the Tomonaga-Luttinger liquid parameter and confirm the positions of the Kosterlitz-Thouless points from the von Neumann entanglement entropy. We find that physical quantities in the second Mott lobe such as the gap and the dynamical structure factor scale almost perfectly in $t/(U+W)$ , even close to the Mott transition. Strong-coupling perturbation theory shows that there is no true scaling but deviations from it are quantitatively small in the strong-coupling limit. This observation should remain true in higher dimensions and for not too large attractive three-body interactions.We employ the (dynamical) density-matrix renormalization-group technique to investigate the ground-state properties of the Bose-Hubbard model with nearest-neighbor transfer amplitudes $t$ and local two-body and three-body repulsion of strength $U$ and $W$, respectively. We determine the phase boundaries between the Mott-insulating and superfluid phases for the lowest two Mott lobes from the chemical potentials. We calculate the tips of the Mott lobes from the Tomonaga-Luttinger liquid parameter and confirm the positions of the Kosterlitz-Thouless points from the von Neumann entanglement entropy. We find that physical quantities in the second Mott lobe such as the gap and the dynamical structure factor scale almost perfectly in $t/(U+W)$, even close to the Mott transition. Strong-coupling perturbation theory shows that there is no true scaling but deviations from it are quantitatively small in the strong-coupling limit. This observation should remain true in higher dimensions and for not too large attractive three-body interactions.

arXiv:1310.4719Physical Review ADec 12, 201314 citations

Quantum magnetism in ultracold alkali and alkaline-earth fermion systems with symplectic symmetry

Hsiang-Hsuan Hung, Yupeng Wang, Congjun Wu

We numerically study the quantum magnetism of ultracold alkali and alkaline-earth fermion systems with large hyperfine spin $F=\frac{3}{2}$ , which are characterized by a generic $Sp(N)$ symmetry with $N=4$ . The methods of exact diagonalization (ED) and density matrix renormalization group are employed for the large size one-dimensional (1D) systems, and the ED method is applied to a two-dimensional (2D) square lattice on small sizes. We focus on the magnetic exchange models in the Mott-insulating state at quarter-filling. Both 1D and 2D systems exhibit rich phase diagrams depending on the ratio between the spin exchanges ${J}_{0}$ and ${J}_{2}$ in the bond spin singlet and quintet channels, respectively. In one dimension, the ground states exhibit a long-range-ordered dimerization with a finite spin gap at ${J}_{0}/{J}_{2}>1$ and a gapless spin-liquid state at ${J}_{0}/{J}_{2}\ensuremath{\leqslant}1$, respectively. In the former and latter cases, the correlation functions exhibit the two-site and four-site periodicities, respectively. In two-dimensions, various spin-correlation functions are calculated up to the size of $4\ifmmode\times\else\texttimes\fi{}4$. The N\'eel-type spin correlation dominates at large values of ${J}_{0}/{J}_{2}$ , while a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette correlation is prominent at small values of this ratio. Between them, a columnar spin-Peierls dimerization correlation peaks. We infer the competition among the plaquette ordering, the dimer ordering, and the N\'eel ordering in the 2D system.We numerically study the quantum magnetism of ultracold alkali and alkaline-earth fermion systems with large hyperfine spin $F=\frac{3}{2}$, which are characterized by a generic $Sp(N)$ symmetry with $N=4$. The methods of exact diagonalization (ED) and density matrix renormalization group are employed for the large size one-dimensional (1D) systems, and the ED method is applied to a two-dimensional (2D) square lattice on small sizes. We focus on the magnetic exchange models in the Mott-insulating state at quarter-filling. Both 1D and 2D systems exhibit rich phase diagrams depending on the ratio between the spin exchanges ${J}_{0}$ and ${J}_{2}$ in the bond spin singlet and quintet channels, respectively. In one dimension, the ground states exhibit a long-range-ordered dimerization with a finite spin gap at ${J}_{0}/{J}_{2}>1$ and a gapless spin-liquid state at ${J}_{0}/{J}_{2}\ensuremath{\leqslant}1$, respectively. In the former and latter cases, the correlation functions exhibit the two-site and four-site periodicities, respectively. In two-dimensions, various spin-correlation functions are calculated up to the size of $4\ifmmode\times\else\texttimes\fi{}4$. The N\'eel-type spin correlation dominates at large values of ${J}_{0}/{J}_{2}$, while a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette correlation is prominent at small values of this ratio. Between them, a columnar spin-Peierls dimerization correlation peaks. We infer the competition among the plaquette ordering, the dimer ordering, and the N\'eel ordering in the 2D system.

arXiv:1103.1926Physical Review BAug 2, 201137 citations

Density matrix renormalization group simulations of the SU( N ) Fermi-Hubbard chain implementing the full SU( N ) symmetry via semistandard Young tableaux and unitary group subduction coefficients

Pierre Nataf

We have developed an efficient method for performing density matrix renormalization group (DMRG) simulations of the SU( $N$ ) Fermi-Hubbard chain, fully leveraging the SU( $N$ ) symmetry of the problem. This method extends a previously developed approach for the SU( $N$ ) Heisenberg model and relies on the systematic use of the semistandard Young tableaux (SSYT) basis in a DMRG algorithm ``\`a la White.'' Specifically, the method aligns the site-by-site growth process of the infinite-size part of the DMRG, in its original formulation, with the site-by-site construction of the SSYT (or Gelfand-like) basis, based on the chain of unitary subgroups $U(1)\ensuremath{\subset}U(2)\ensuremath{\subset}U(3)\ensuremath{\subset}U(4)\ensuremath{\cdots}$. We give special emphasis to the calculation of the symmetry-resolved reduced matrix elements of the hopping terms between the left and the right block, which makes direct use of the basis of SSYT and of the Gelfand-Tsetlin coefficients, offering a computational advantage in scaling with $N$ compared to alternative methods that rely on summing over Clebsch-Gordan coefficients. Focusing on the model with homogeneous hopping between nearest neighbors, we have calculated the ground-state energy as a function of $U$ , i.e., the atom-atom interaction amplitude, up to $N=6$ for filling $1/N$ (one particle per site in average), and for one atom (respectively, hole) away from filling $1/N$ . It allows us to compute the charge gaps and to estimate in the thermodynamical limit, the critical value ${U}_{c}$ (separating the Mott insulator from the metallic phase), which is proved to be finite (i.e., ${U}_{c}>0$ ) and to increase with $N$ . Central charges $c$ are also extracted from the entanglement entropy using the Calabrese-Cardy formula and are consistent with the theoretical predictions: $c=N\ensuremath{-}1$, expected from the $\mathrm{SU}{(N)}_{1}$ Wess-Zumino-Witten conformal field theory in the spin sector for the Mott phase, and $c=N$ in the metallic phase, reflecting the presence of one additional (charge) gapless critical mode.We have developed an efficient method for performing density matrix renormalization group (DMRG) simulations of the SU($N$) Fermi-Hubbard chain, fully leveraging the SU($N$) symmetry of the problem. This method extends a previously developed approach for the SU($N$) Heisenberg model and relies on the systematic use of the semistandard Young tableaux (SSYT) basis in a DMRG algorithm ``\`a la White.'' Specifically, the method aligns the site-by-site growth process of the infinite-size part of the DMRG, in its original formulation, with the site-by-site construction of the SSYT (or Gelfand-like) basis, based on the chain of unitary subgroups $U(1)\ensuremath{\subset}U(2)\ensuremath{\subset}U(3)\ensuremath{\subset}U(4)\ensuremath{\cdots}$. We give special emphasis to the calculation of the symmetry-resolved reduced matrix elements of the hopping terms between the left and the right block, which makes direct use of the basis of SSYT and of the Gelfand-Tsetlin coefficients, offering a computational advantage in scaling with $N$ compared to alternative methods that rely on summing over Clebsch-Gordan coefficients. Focusing on the model with homogeneous hopping between nearest neighbors, we have calculated the ground-state energy as a function of $U$, i.e., the atom-atom interaction amplitude, up to $N=6$ for filling $1/N$ (one particle per site in average), and for one atom (respectively, hole) away from filling $1/N$. It allows us to compute the charge gaps and to estimate in the thermodynamical limit, the critical value ${U}_{c}$ (separating the Mott insulator from the metallic phase), which is proved to be finite (i.e., ${U}_{c}>0$) and to increase with $N$. Central charges $c$ are also extracted from the entanglement entropy using the Calabrese-Cardy formula and are consistent with the theoretical predictions: $c=N\ensuremath{-}1$, expected from the $\mathrm{SU}{(N)}_{1}$ Wess-Zumino-Witten conformal field theory in the spin sector for the Mott phase, and $c=N$ in the metallic phase, reflecting the presence of one additional (charge) gapless critical mode.

arXiv:2412.13320Physical review. B./Physical review. BMay 27, 20251 citation

Mott insulators of ultracold fermionic alkaline earth atoms in three dimensions

Hao Song, Michael Hermele

We study a class of $\text{SU}(N)$ Heisenberg models, describing Mott insulators of fermionic ultracold alkaline earth atoms on the three-dimensional simple cubic lattice. Based on an earlier semiclassical analysis, magnetic order is unlikely, and we focus instead on a solvable large- $N$ limit designed to address the competition among nonmagnetic ground states. We find a rich phase diagram as a function of the filling parameter $k$ , composed of a variety of ground states spontaneously breaking lattice symmetries, and in some cases also time-reversal symmetry. One particularly striking example is a state spontaneously breaking lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state.We study a class of $\text{SU}(N)$ Heisenberg models, describing Mott insulators of fermionic ultracold alkaline earth atoms on the three-dimensional simple cubic lattice. Based on an earlier semiclassical analysis, magnetic order is unlikely, and we focus instead on a solvable large-$N$ limit designed to address the competition among nonmagnetic ground states. We find a rich phase diagram as a function of the filling parameter $k$, composed of a variety of ground states spontaneously breaking lattice symmetries, and in some cases also time-reversal symmetry. One particularly striking example is a state spontaneously breaking lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state.

arXiv:1301.3470Physical Review BApr 29, 201312 citations

Metal-insulator transition and magnetism of SU(3) fermions in the square lattice

Eduardo Ibarra-García-Padilla, Chunhan Feng, Giulio Pasqualetti +4 more

We study the SU(3) symmetric Fermi-Hubbard model (FHM) in the square lattice at $1/3$ -filling using numerically exact determinant quantum Monte Carlo and numerical linked-cluster expansion techniques. We present the different regimes of the model in the $T\text{\ensuremath{-}}U$ plane, which are characterized by local and short-range correlations, and capture signatures of the metal-insulator transition and magnetic crossovers. These signatures are detected as the temperature scales characterizing the rise of the compressibility, and an interaction-dependent change in the sign of the diagonal spin-spin correlation function. The analysis of the compressibility estimates the location of the metal-insulator quantum critical point at ${U}_{c}/t\ensuremath{\sim}6$, and provides a temperature scale for observing Mott physics at finite $T$ . Furthermore, from the analysis of the spin-spin correlation function we observe that for $U/t\ensuremath{\gtrsim}6$ and $T\ensuremath{\sim}J=4{t}^{2}/U$ there is a development of a short-range two-sublattice (2SL) antiferromagnetic structure, as well as an emerging three-sublattice (3SL) antiferromagnetic structure as the temperature is lowered below $T/J\ensuremath{\lesssim}0.57$. This crossover from 2SL to 3SL magnetic ordering agrees with Heisenberg limit predictions, and has observable effects on the density of on-site pairs. Finally, we describe how the features of the regimes in the $T\ensuremath{-}U$ plane can be explored with alkaline-earth-like atoms in optical lattices with currently achieved experimental techniques and temperatures. The results discussed in this paper provide a starting point for the exploration of the SU(3) FHM upon doping.We study the SU(3) symmetric Fermi-Hubbard model (FHM) in the square lattice at $1/3$-filling using numerically exact determinant quantum Monte Carlo and numerical linked-cluster expansion techniques. We present the different regimes of the model in the $T\text{\ensuremath{-}}U$ plane, which are characterized by local and short-range correlations, and capture signatures of the metal-insulator transition and magnetic crossovers. These signatures are detected as the temperature scales characterizing the rise of the compressibility, and an interaction-dependent change in the sign of the diagonal spin-spin correlation function. The analysis of the compressibility estimates the location of the metal-insulator quantum critical point at ${U}_{c}/t\ensuremath{\sim}6$, and provides a temperature scale for observing Mott physics at finite $T$. Furthermore, from the analysis of the spin-spin correlation function we observe that for $U/t\ensuremath{\gtrsim}6$ and $T\ensuremath{\sim}J=4{t}^{2}/U$ there is a development of a short-range two-sublattice (2SL) antiferromagnetic structure, as well as an emerging three-sublattice (3SL) antiferromagnetic structure as the temperature is lowered below $T/J\ensuremath{\lesssim}0.57$. This crossover from 2SL to 3SL magnetic ordering agrees with Heisenberg limit predictions, and has observable effects on the density of on-site pairs. Finally, we describe how the features of the regimes in the $T\ensuremath{-}U$ plane can be explored with alkaline-earth-like atoms in optical lattices with currently achieved experimental techniques and temperatures. The results discussed in this paper provide a starting point for the exploration of the SU(3) FHM upon doping.

arXiv:2306.10644Physical Review ANov 13, 202325 citations

We investigate one-dimensional SU

(N)

N>2

where the BA is only approximate. In the worst case, we find a relative error

\ensuremath{\varepsilon}\ensuremath{\lesssim}4%

in the BA ground-state energy for

N\ensuremath{\leqslant}4

at filling

1/N

, which is due to the fact that BA improperly treats the triply and higher occupied states. Using the DMRG for

N\ensuremath{\leqslant}4

and the BA for large

N

, we determine the regimes of validity of strong- and weak-coupling perturbation theory for all values of

N

and, in particular, the parameter range in which the system is well described by a SU

(N)

Heisenberg model at filling

1/N

. We find this depends only weakly on

N

. We investigate the Berezinskii-Kosterlitz-Thouless phase transition from a Luttinger liquid to a Mott insulator by computing the fidelity susceptibility and the Luttinger parameter

{K}_{\ensuremath{\rho}}

1/N

filling. The numerical findings give strong evidence that the fidelity susceptibility develops a minimum at a critical interaction strength which is found to occur at a finite positive value for

N>2

Since the advent of high-

T_c

cuprate superconductors in 1986, strongly correlated electron systems have attracted much attention. Since the cuprates are essentially two-dimensional, low-dimensional systems have moved into the focus of condensed-matter theory. From a theoretical point of view, one-dimensional systems are of particular interest because there are exact numerical and analytical methods which permit detailed studies and deep insights into the many-body problem. In the first part of this thesis, using the numerically exact methods Exact Diagonalization and the Density-Matrix Renormalization Group (DMRG), we investigate the properties of the Tomonaga–Luttinger liquid which is the generic metallic state of matter in one dimension. In particular, we concentrate on the investigation of the so-called Tomonaga–Luttinger liquid parameter which determines the critical exponent

\alpha

for the density of states near the Fermi energy. Experimental results for some quasi one-dimensional materials report

\alpha \gtrsim 1

, which would imply

K_\rho \lesssim 0.17

, a value which cannot be reconciled with the bare Hubbard model where

K_\rho^H \geq 0.5

, i.e.,

\alpha^H \lesssim 1/8

. We develop new accurate numerical methods to obtain

K_\rho

and investigate how to obtain such small values for

K_\rho

for slightly doped charge-density-wave insulators. In the second part of this thesis, using the Thermodynamic Bethe Ansatz (TBA) as exact analytical method, we investigate the one-dimensional Hubbard model in the spindisordered regime, which is characterized by the temperature being much larger than the magnetic energy scale but small compared to the Mott–Hubbard gap. Our study is motivated by the controversy about the Mott–Hubbard insulator in infinite dimensions whose ground state is also spin-disordered. In this system the determination of the precise value of the critical interaction strength Uc where the Mott–Hubbard gap closes is still unsolved. Therefore, we provide an example of a Hubbard-type model with a disordered spin background which can be solved exactly. The thermodynamics of our model can be understood in terms of gapped charged excitations with an effective dispersion which are decoupled form the spin degrees of freedom; the latter contribute only entropically. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal-insulator transition at a finite interaction strength above which the gap opens linearly. Our exact results indicate that the strong-coupling expansion of the ground-state energy cannot be used to locate

U_c

. However, the strong-coupling expansion of the gap permits a reliable extrapolation of the critical interaction strength.

We employ the (dynamical) density-matrix renormalization-group technique to investigate the ground-state properties of the Bose-Hubbard model with nearest-neighbor transfer amplitudes

t

and local two-body and three-body repulsion of strength

U

and

W

, respectively. We determine the phase boundaries between the Mott-insulating and superfluid phases for the lowest two Mott lobes from the chemical potentials. We calculate the tips of the Mott lobes from the Tomonaga-Luttinger liquid parameter and confirm the positions of the Kosterlitz-Thouless points from the von Neumann entanglement entropy. We find that physical quantities in the second Mott lobe such as the gap and the dynamical structure factor scale almost perfectly in

t/(U+W)

, even close to the Mott transition. Strong-coupling perturbation theory shows that there is no true scaling but deviations from it are quantitatively small in the strong-coupling limit. This observation should remain true in higher dimensions and for not too large attractive three-body interactions.

We numerically study the quantum magnetism of ultracold alkali and alkaline-earth fermion systems with large hyperfine spin

F=\frac{3}{2}

, which are characterized by a generic

Sp(N)

symmetry with

N=4

. The methods of exact diagonalization (ED) and density matrix renormalization group are employed for the large size one-dimensional (1D) systems, and the ED method is applied to a two-dimensional (2D) square lattice on small sizes. We focus on the magnetic exchange models in the Mott-insulating state at quarter-filling. Both 1D and 2D systems exhibit rich phase diagrams depending on the ratio between the spin exchanges

{J}_{0}

and

{J}_{2}

in the bond spin singlet and quintet channels, respectively. In one dimension, the ground states exhibit a long-range-ordered dimerization with a finite spin gap at

{J}_{0}/{J}_{2}>1

and a gapless spin-liquid state at ${J}_{0}/{J}_{2}\ensuremath{\leqslant}1$, respectively. In the former and latter cases, the correlation functions exhibit the two-site and four-site periodicities, respectively. In two-dimensions, various spin-correlation functions are calculated up to the size of $4\ifmmode\times\else\texttimes\fi{}4$. The N\'eel-type spin correlation dominates at large values of

{J}_{0}/{J}_{2}

, while a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette correlation is prominent at small values of this ratio. Between them, a columnar spin-Peierls dimerization correlation peaks. We infer the competition among the plaquette ordering, the dimer ordering, and the N\'eel ordering in the 2D system.

We have developed an efficient method for performing density matrix renormalization group (DMRG) simulations of the SU(

N

) Fermi-Hubbard chain, fully leveraging the SU(

N

) symmetry of the problem. This method extends a previously developed approach for the SU(

N

) Heisenberg model and relies on the systematic use of the semistandard Young tableaux (SSYT) basis in a DMRG algorithm ``\`a la White.'' Specifically, the method aligns the site-by-site growth process of the infinite-size part of the DMRG, in its original formulation, with the site-by-site construction of the SSYT (or Gelfand-like) basis, based on the chain of unitary subgroups $U(1)\ensuremath{\subset}U(2)\ensuremath{\subset}U(3)\ensuremath{\subset}U(4)\ensuremath{\cdots}$. We give special emphasis to the calculation of the symmetry-resolved reduced matrix elements of the hopping terms between the left and the right block, which makes direct use of the basis of SSYT and of the Gelfand-Tsetlin coefficients, offering a computational advantage in scaling with

N

compared to alternative methods that rely on summing over Clebsch-Gordan coefficients. Focusing on the model with homogeneous hopping between nearest neighbors, we have calculated the ground-state energy as a function of

U

, i.e., the atom-atom interaction amplitude, up to

N=6

for filling

1/N

(one particle per site in average), and for one atom (respectively, hole) away from filling

1/N

. It allows us to compute the charge gaps and to estimate in the thermodynamical limit, the critical value

{U}_{c}

(separating the Mott insulator from the metallic phase), which is proved to be finite (i.e.,

{U}_{c}>0

) and to increase with

N

. Central charges

c

are also extracted from the entanglement entropy using the Calabrese-Cardy formula and are consistent with the theoretical predictions: $c=N\ensuremath{-}1$, expected from the

\mathrm{SU}{(N)}_{1}

Wess-Zumino-Witten conformal field theory in the spin sector for the Mott phase, and

c=N

in the metallic phase, reflecting the presence of one additional (charge) gapless critical mode.

We study a class of

\text{SU}(N)

Heisenberg models, describing Mott insulators of fermionic ultracold alkaline earth atoms on the three-dimensional simple cubic lattice. Based on an earlier semiclassical analysis, magnetic order is unlikely, and we focus instead on a solvable large-

N

limit designed to address the competition among nonmagnetic ground states. We find a rich phase diagram as a function of the filling parameter

k

, composed of a variety of ground states spontaneously breaking lattice symmetries, and in some cases also time-reversal symmetry. One particularly striking example is a state spontaneously breaking lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state.

We study the SU(3) symmetric Fermi-Hubbard model (FHM) in the square lattice at

1/3

-filling using numerically exact determinant quantum Monte Carlo and numerical linked-cluster expansion techniques. We present the different regimes of the model in the $T\text{\ensuremath{-}}U$ plane, which are characterized by local and short-range correlations, and capture signatures of the metal-insulator transition and magnetic crossovers. These signatures are detected as the temperature scales characterizing the rise of the compressibility, and an interaction-dependent change in the sign of the diagonal spin-spin correlation function. The analysis of the compressibility estimates the location of the metal-insulator quantum critical point at ${U}_{c}/t\ensuremath{\sim}6$, and provides a temperature scale for observing Mott physics at finite

T

. Furthermore, from the analysis of the spin-spin correlation function we observe that for $U/t\ensuremath{\gtrsim}6$ and $T\ensuremath{\sim}J=4{t}^{2}/U$ there is a development of a short-range two-sublattice (2SL) antiferromagnetic structure, as well as an emerging three-sublattice (3SL) antiferromagnetic structure as the temperature is lowered below $T/J\ensuremath{\lesssim}0.57$. This crossover from 2SL to 3SL magnetic ordering agrees with Heisenberg limit predictions, and has observable effects on the density of on-site pairs. Finally, we describe how the features of the regimes in the $T\ensuremath{-}U$ plane can be explored with alkaline-earth-like atoms in optical lattices with currently achieved experimental techniques and temperatures. The results discussed in this paper provide a starting point for the exploration of the SU(3) FHM upon doping.