arXiv:cond-mat/0607374 · Physical Review B

Mott transition and dimerization in the one-dimensional SU(n) Hubbard model

K. Buchta, Örs Legeza, E. Szirmai, Jenő Sólyom

Published Apr 19, 2007• 48 citations

Abstract

The one-dimensional $\mathrm{SU}(n)$ Hubbard model is investigated numerically for $n=2$, 3, 4, and 5 at half-filling and $1∕n$ filling using the density-matrix renormalization-group method. The energy gaps and various quantum information entropies are calculated. In the half-filled case, finite spin and charge gaps are found for arbitrary positive $U$ if $n>2$. Furthermore, it is shown that the transition to the gapped phase at ${U}_{c}=0$ is of Kosterlitz-Thouless type and is accompanied by a bond dimerization both for even and odd $n$. In the $1∕n$-filled case, the transition has similar features as the metal-insulator transition in the half-filled SU(2) Hubbard model. The charge gap opens exponentially slowly for $U>{U}_{c}=0$, the spin sector remains gapless, and the ground state is nondimerized.

DOI: 10.1103/physrevb.75.155108

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arXiv:cond-mat/0607374 · Physical Review B

Mott transition and dimerization in the one-dimensional SU(n) Hubbard model

K. Buchta, Örs Legeza, E. Szirmai, Jenő Sólyom

Published Apr 19, 2007• 48 citations

arXiv HTML PDF

Abstract

The one-dimensional

\mathrm{SU}(n)

Hubbard model is investigated numerically for

n=2

, 3, 4, and 5 at half-filling and

1∕n

filling using the density-matrix renormalization-group method. The energy gaps and various quantum information entropies are calculated. In the half-filled case, finite spin and charge gaps are found for arbitrary positive

U

n>2

. Furthermore, it is shown that the transition to the gapped phase at

{U}_{c}=0

is of Kosterlitz-Thouless type and is accompanied by a bond dimerization both for even and odd

n

. In the

1∕n

-filled case, the transition has similar features as the metal-insulator transition in the half-filled SU(2) Hubbard model. The charge gap opens exponentially slowly for

U>{U}_{c}=0

, the spin sector remains gapless, and the ground state is nondimerized.

DOI: 10.1103/physrevb.75.155108

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We use the density matrix renormalization group to study the quantum transitions that occur in the half-filled one-dimensional fermionic Hubbard model with onsite potential disorder. We find a transition from the gapped Mott phase with algebraic spin correlations to a gapless spin-disordered phase beyond a critical strength of the disorder $Δ_c \approx U/2$ . Both the transitions in the charge and spin sectors are shown to be coincident. We also establish the finite-size corrections to the charge gap and the spin-spin correlation length in the presence of disorder and using a finite-size-scaling analysis we obtain the zero temperature phase diagram of the various quantum phase transitions that occur in the disorder-interaction plane.We use the density matrix renormalization group to study the quantum transitions that occur in the half-filled one-dimensional fermionic Hubbard model with onsite potential disorder. We find a transition from the gapped Mott phase with algebraic spin correlations to a gapless spin-disordered phase beyond a critical strength of the disorder $Δ_c \approx U/2$. Both the transitions in the charge and spin sectors are shown to be coincident. We also establish the finite-size corrections to the charge gap and the spin-spin correlation length in the presence of disorder and using a finite-size-scaling analysis we obtain the zero temperature phase diagram of the various quantum phase transitions that occur in the disorder-interaction plane.

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The one-dimensional SU $(n)$ Hubbard model is investigated using an analytical, perturbative renormalization-group treatment in the fermionic formulation of the model for $n>2$ . It is found that in the half-filled case, the umklapp processes couple the spin and charge degrees of freedom and generate a finite gap for arbitrary finite positive $U$ . This behavior is thus different from that at $1∕n$ filling, where there is one particle per site, in which case the dynamics of the two degrees of freedom can be separated and a critical ${U}_{\mathrm{c}}$ is needed for the Mott transition.The one-dimensional SU$(n)$ Hubbard model is investigated using an analytical, perturbative renormalization-group treatment in the fermionic formulation of the model for $n>2$. It is found that in the half-filled case, the umklapp processes couple the spin and charge degrees of freedom and generate a finite gap for arbitrary finite positive $U$. This behavior is thus different from that at $1∕n$ filling, where there is one particle per site, in which case the dynamics of the two degrees of freedom can be separated and a critical ${U}_{\mathrm{c}}$ is needed for the Mott transition.

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Jianhui Dai, Xiaoyong Feng, Tao Xiang +1 more

We study the phase diagram of a one-dimensional extended Hubbard model with antiferromagnetic exchange interaction analytically and numerically. The bosonization and transfer-matrix renormalization group methods are used in the corresponding coupling regimes. At half-filling, the system is a Mott insulator with a finite spin excitation gap if the on-site Coulomb repulsion is fairly smaller than the antiferromagnetic exchange $J$ . This Mott insulator is characterized by the bond-charge-density-wave order or spontaneously dimerization. In the weak-coupling regime where the spin-charge separation holds approximately, the critical point separating the gapless and gapped spin liquid phases is ${U}_{c}\ensuremath{\sim}J∕2$. However, as $J$ increases, the spin-charge couplings become important and the critical point ${U}_{c}$ is significantly suppressed and eventually tends to zero as $J\ensuremath{\rightarrow}\ensuremath{\infty}$. Away from half-filling, the charge gap completely collapses but the spin gap persists.We study the phase diagram of a one-dimensional extended Hubbard model with antiferromagnetic exchange interaction analytically and numerically. The bosonization and transfer-matrix renormalization group methods are used in the corresponding coupling regimes. At half-filling, the system is a Mott insulator with a finite spin excitation gap if the on-site Coulomb repulsion is fairly smaller than the antiferromagnetic exchange $J$. This Mott insulator is characterized by the bond-charge-density-wave order or spontaneously dimerization. In the weak-coupling regime where the spin-charge separation holds approximately, the critical point separating the gapless and gapped spin liquid phases is ${U}_{c}\ensuremath{\sim}J∕2$. However, as $J$ increases, the spin-charge couplings become important and the critical point ${U}_{c}$ is significantly suppressed and eventually tends to zero as $J\ensuremath{\rightarrow}\ensuremath{\infty}$. Away from half-filling, the charge gap completely collapses but the spin gap persists.

arXiv:cond-mat/0407114Physical Review BAug 31, 200432 citations

Spatially nonuniform phases in the one-dimensional SU(n) Hubbard model for commensurate fillings

E. Szirmai, Örs Legeza, Jenő Sólyom

The one-dimensional repulsive $\mathrm{SU}(n)$ Hubbard model is investigated analytically by bosonization approach and numerically using the density-matrix renormalization-group method for $n=3$ , 4, and 5 for commensurate fillings $f=p∕q$ , where $p$ and $q$ are relatively primes. It is shown that the behavior of the system is drastically different depending on whether $q>n$ , $q=n$ , or $q<n$ . When $q>n$ , the umklapp processes are irrelevant and the model is equivalent to an $n$ -component Luttinger liquid with central charge $c=n$ . When $q=n$ , the charge and spin modes are decoupled, the umklapp processes open a charge gap for finite $U>0$ , whereas the spin modes remain gapless and the central charge $c=n\ensuremath{-}1$. The translational symmetry is not broken in the ground state for any $n$ . On the other hand, when $q<n$ , the charge and spin modes are coupled, the umklapp processes open gaps in all excitation branches, and a spatially nonuniform ground state develops. Bond-ordered dimerized, trimerized, or tetramerized phases are found depending on the filling.The one-dimensional repulsive $\mathrm{SU}(n)$ Hubbard model is investigated analytically by bosonization approach and numerically using the density-matrix renormalization-group method for $n=3$, 4, and 5 for commensurate fillings $f=p∕q$, where $p$ and $q$ are relatively primes. It is shown that the behavior of the system is drastically different depending on whether $q>n$, $q=n$, or $q<n$. When $q>n$, the umklapp processes are irrelevant and the model is equivalent to an $n$-component Luttinger liquid with central charge $c=n$. When $q=n$, the charge and spin modes are decoupled, the umklapp processes open a charge gap for finite $U>0$, whereas the spin modes remain gapless and the central charge $c=n\ensuremath{-}1$. The translational symmetry is not broken in the ground state for any $n$. On the other hand, when $q<n$, the charge and spin modes are coupled, the umklapp processes open gaps in all excitation branches, and a spatially nonuniform ground state develops. Bond-ordered dimerized, trimerized, or tetramerized phases are found depending on the filling.

arXiv:0708.3799Physical Review BJan 8, 200830 citations

The one-dimensional

\mathrm{SU}(n)

Hubbard model is investigated numerically for

n=2

, 3, 4, and 5 at half-filling and

1∕n

U

n>2

. Furthermore, it is shown that the transition to the gapped phase at

{U}_{c}=0

is of Kosterlitz-Thouless type and is accompanied by a bond dimerization both for even and odd

n

. In the

1∕n

-filled case, the transition has similar features as the metal-insulator transition in the half-filled SU(2) Hubbard model. The charge gap opens exponentially slowly for

U>{U}_{c}=0

, the spin sector remains gapless, and the ground state is nondimerized.

Δ_c \approx U/2

. Both the transitions in the charge and spin sectors are shown to be coincident. We also establish the finite-size corrections to the charge gap and the spin-spin correlation length in the presence of disorder and using a finite-size-scaling analysis we obtain the zero temperature phase diagram of the various quantum phase transitions that occur in the disorder-interaction plane.

We investigate the metal-insulator transition of the one-dimensional

\mathrm{SU}(N)

Hubbard model for repulsive interaction. Using the bosonization approach a Mott transition in the charge sector at half filling ${(k}_{F}=\ensuremath{\pi}{/Na}_{0})$ is conjectured for

N>2.

Expressions for the charge and spin velocities as well as for the Luttinger-liquid parameters and some correlation functions are given. The theoretical predictions are compared with numerical results obtained with an improved zero-temperature quantum Monte Carlo approach. The method used is a generalized Green's function Monte Carlo scheme in which the stochastic time evolution is partially integrated out. Very accurate results for the gaps, velocities, and Luttinger-liquid parameters as a function of the Coulomb interaction

U

are given for the cases

N=3

and

N=4.

Our results strongly support the existence of a Mott-Hubbard transition at a nonzero value of the Coulomb interaction. We find ${U}_{c}\ensuremath{\sim}2.2$ for

N=3

and ${U}_{c}\ensuremath{\sim}2.8$ for

N=4.

We study a one-dimensional SU(N) Hubbard model with an attractive on-site interaction and

N>2

at half-filling on the bipartite lattice using density-matrix renormalization-group method and a perturbation theory. We find that the ground state of the SU(N) Hubbard model is a charge density wave state with two-fold degeneracy. All the excitations are found to be gapful, resulting in an insulating ground state, on contrary to that in the SU(2) case. Moreover, the charge gap is equal to the Cooperon gap, which behaves as

-2Nt^2/(N-1)U

in the strong coupling regime. However, the spin gap

Δ_{s}

and the quasiparticle gap

Δ_{1}

as well open exponentially in the weak coupling region, while in the strong coupling region, they linearly depend on

U

such that

Δ_{s}\sim -U(N-1)

and

Δ_{1}\sim -U(N-1)/2

The one-dimensional SU

(n)

Hubbard model is investigated using an analytical, perturbative renormalization-group treatment in the fermionic formulation of the model for

n>2

. It is found that in the half-filled case, the umklapp processes couple the spin and charge degrees of freedom and generate a finite gap for arbitrary finite positive

U

. This behavior is thus different from that at

1∕n

filling, where there is one particle per site, in which case the dynamics of the two degrees of freedom can be separated and a critical

{U}_{\mathrm{c}}

is needed for the Mott transition.

J

. This Mott insulator is characterized by the bond-charge-density-wave order or spontaneously dimerization. In the weak-coupling regime where the spin-charge separation holds approximately, the critical point separating the gapless and gapped spin liquid phases is ${U}_{c}\ensuremath{\sim}J∕2$. However, as

J

increases, the spin-charge couplings become important and the critical point

{U}_{c}

is significantly suppressed and eventually tends to zero as $J\ensuremath{\rightarrow}\ensuremath{\infty}$. Away from half-filling, the charge gap completely collapses but the spin gap persists.

The one-dimensional repulsive

\mathrm{SU}(n)

Hubbard model is investigated analytically by bosonization approach and numerically using the density-matrix renormalization-group method for

n=3

, 4, and 5 for commensurate fillings

f=p∕q

, where

p

and

q

are relatively primes. It is shown that the behavior of the system is drastically different depending on whether

q>n

q=n

, or

q<n

. When

q>n

, the umklapp processes are irrelevant and the model is equivalent to an

n

-component Luttinger liquid with central charge

c=n

. When

q=n

, the charge and spin modes are decoupled, the umklapp processes open a charge gap for finite

U>0

, whereas the spin modes remain gapless and the central charge $c=n\ensuremath{-}1$. The translational symmetry is not broken in the ground state for any

n

. On the other hand, when

q<n

, the charge and spin modes are coupled, the umklapp processes open gaps in all excitation branches, and a spatially nonuniform ground state develops. Bond-ordered dimerized, trimerized, or tetramerized phases are found depending on the filling.