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Physical Review B

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

E. Szirmai, Jenő Sólyom

Published May 16, 2005• 21 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.71.205108

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Physical review. B, Condensed matter•Dec 1, 1980•112 citations
The one-dimensional SU(n)(n)(n) Hubbard model is investigated using an analytical, perturbative renormalization-group treatment in the fermionic formulation of the model for n>2n>2n>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 UUU. This behavior is thus different from that at 1∕n1∕n1∕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 Uc{U}_{\mathrm{c}}Uc​ is needed for the Mott transition.
The one-dimensional SU(n)\mathrm{SU}(n)SU(n) Hubbard model is investigated numerically for n=2n=2n=2, 3, 4, and 5 at half-filling and 1∕n1∕n1∕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 UUU if n>2n>2n>2. Furthermore, it is shown that the transition to the gapped phase at Uc=0{U}_{c}=0Uc​=0 is of Kosterlitz-Thouless type and is accompanied by a bond dimerization both for even and odd nnn. In the 1∕n1∕n1∕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>Uc=0U>{U}_{c}=0U>Uc​=0, the spin sector remains gapless, and the ground state is nondimerized.
The one-dimensional Mott metal-insulator transition is a typical strong correlation effect triggered by the Umklapp scattering. However, in a physical system, the Umklapp scattering coexists with the normal scattering, including both forward and backward scattering, which conserves the total momentum of scattered electrons. Therefore, it is not easy to quantify the contribution of the Umklapp scattering in a Mott metal-insulator transition. To resolve this difficulty, we propose to explore these scattering processes separately. We study the contribution of each scattering process in the one-dimensional Hubbard model using the momentum-space density-matrix renormalization group (kDMRG) and bosonization methods. Our kDMRG calculation confirms that the Mott charge gap results from the Umklapp scattering, but the normal scattering processes strongly renormalize its value. Furthermore, we present a scaling analysis of the Mott charge gap in the bosonization theory and show that the interplay between the Umklapp and forward scattering dictates the charge dynamics in the half-filled Hubbard model.
The one-dimensional Hubbard model with different on-site interactions is investigated by renormalization group technique. In the case of a 1/4-filled band the dynamical nonequivalence of sites leads to the appearance of Umklapp processes in the system and to the dynamical generation of a gap in the charge excitation spectrum for Uanot=UbU_{a}\\not=U_{b}Ua​not=Ub​, Ua>0U_{a}>0Ua​>0 or Ub>0U_{b}>0Ub​>0. The ground-state phase diagram is obtained in the limit of second order renormalization. Depending on the sign and relative values of the bare coupling constants, there is a gap in the spin or charge excitation spectrum and the model system tends to superconducting or antiferromagnetic order at T=0, with doubled period. The role of interaction between particles on nearest and next-nearest neighbor sites is also considered.
The one-dimensional Hubbard model with different on-site interactions is investigated by renormalization group technique. In the case of a 1/4-filled band the dynamical nonequivalence of sites leads to the appearance of Umklapp processes in the system, and to the dynamical generation of a gap in the charge excitation spectrum for Ua≠Ub, Ua>0 or Ub>0. The ground-state phase diagram is obtained in the limit of second order renormalization. Depending on the sign and relative values of the bare coupling constants, there is a gap in the spin or charge excitation spectrum, and the model system tends to superconducting or antiferromagnetic order at T=0, with doubled period. The role of interaction between particles on nearest and next-nearest neighbor sites is also considered.
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 Δc≈U/2Δ_c \approx U/2Δc​≈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.
The ground state of the half-filled Hubbard model is studied using a real-space renormalization-group technique. We first study in detail the one-dimensional case. We find the ground state to be insulating for all finite coupling, in agreement with the exact solution. We compute the ground-state energy, localization length, energy gap, magnitude of the local moment, and spin-density autocorrelation function. For those quantities that are exactly known we find good agreement with the exact results. Using a simple extension of our one-dimensional calculation, we are able to study approximately two- and three-dimensional lattices. We find a Mott transition at finite interaction for these cases. The critical exponents for these transitions are found to satisfy an approximate interdimensional scaling law.