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arXiv:2212.10972 · Physical review. B./Physical review. B

Umklapp scattering in the one-dimensional Hubbard model

Tong Liu, Kang Wang, Runze Chi, Yang Liu, Haijun Liao, Tao Xiang

Published Sep 21, 2023• 1 citation

Abstract

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.

DOI: 10.1103/physrevb.108.125134

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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.

We apply the density-matrix renormalization group (DMRG) method to a one-dimensional Hubbard model that lacks Umklapp scattering and thus provides an ideal case to study the Mott-Hubbard transition analytically and numerically. The model has a linear dispersion and displays a metal-to-insulator transition when the Hubbard interaction

U

equals the bandwidth

{U}_{\mathrm{c}}=W

, where the single-particle gap opens linearly, $\mathrm{\ensuremath{\Delta}}(U\ensuremath{\ge}W)=U\ensuremath{-}W$. The simple nature of the elementary excitations permits us to determine numerically with high accuracy the critical interaction strength and the gap function in the thermodynamic limit. The jump discontinuity of the momentum distribution

{n}_{k}

at the Fermi wave number

{k}_{\mathrm{F}}=0

cannot be used to locate accurately

{U}_{\mathrm{c}}

from finite-size systems. However, the slope of

{n}_{k}

at the band edges ${k}_{\mathrm{B}}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$, reveals the formation of a single-particle bound state which can be used to determine

{U}_{\mathrm{c}}

reliably from

{n}_{k}

using accurate finite-size data.

The competition between the Mott transition and the Anderson localization in one-dimensional electron systems is studied with the bosonization and renormalization-group methods. The $\ensuremath{\beta}$ function is calculated up to second order in the strength of diagonal disorder by using a replica trick. It is found that sufficiently strong forward scattering by random impurities destroys the Mott-Hubbard gap, and backward scattering gives rise to Anderson localization for the resulting gapless state. On the other hand, if the umklapp interaction is strong enough, the Mott insulating state still overwhelms the Anderson localization.

We determine the ground-state phase diagram for the

1/r

-Hubbard model with repulsive nearest-neighbor interaction at half band filling using the density-matrix renormalization group method. Due to the absence of Umklapp scattering, the phase diagram displays finite regions for the three generic phases, namely, a Luttinger liquid metal for weak interactions, a Mott-Hubbard insulator for dominant Hubbard interactions, and a charge-density-wave insulator for dominant nearest-neighbor interactions. Up to moderate interaction strengths, the quantum phase transitions between the metallic and insulating phases are continuous, i.e., the gap opens continuously as a function of the interaction strength. We conclude that generic short-range interactions do not change the nature of the Mott transition qualitatively.

The effect of an alternating potential on a one-dimensional half-filled Hubbard model with repulsive interaction has been examined by applying the renormalization group method to the bosonized Hamiltonian. The electronic state, which is determined by the competition between alternating potential and umklapp scattering, is calculated where the relevance and the irrelevance of the alternating potential leads to the band insulator and the Mott insulator respectively. The excitation gaps for charge and spin fluctuations are calculated for both states.

The low--energy excited states of a system of interacting one--dimensional fermions in a conducting state are collective charge and spin density oscillations. The unusual physical properties of such a system (called ``Luttinger liquid'') are characterized by the velocities

u_ρ

and

u_σ

of the charge and spin excitations, as well as by a parameter

K_ρ

that determines the power law behavior of correlation functions. Umklapp scattering occuring at half--filling or other commensurate bandfilling can lead to a transition into an insulating state, characterized in particular by a gap in the charge excitations (the Mott--Hubbard gap). The properties in the vicinity of the transition are shown to depend on both the way the transition is approached (constant bandfilling and varying interaction, or constant interaction and varying bandfilling) and on the ``order'' of the commensurability. In particular, even and odd fractional fillings show quite different behavior. This behavior is illustrated in detail using lattice models like the Hubbard model and its extensions.

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.