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arXiv:cond-mat/9910471 · Physical review. B, Condensed matter

Quantum phase transitions and collapse of the Mott gap in the d=1+ε dimensional Hubbard model with 2kF umklapp scattering

Jun‐ichiro Kishine

Published Jul 15, 2000• 4 citations
arXivHTMLPDF

Abstract

We study the low-energy asymptotics of the $d=1+\ensuremath{\epsilon}$ dimensional Hubbard model with a circular Fermi surface where there is ${2k}_{F}$ umklapp scattering present a priori. Peculiarity of the $d=1+\ensuremath{\epsilon}$ dimensions is incorporated through the imbalance between the elementary particle-particle and particle-hole (PH) loops: infrared logarithmic singularity of the PH loop is smeared for $\ensuremath{\epsilon}>0.$ The one-loop renormalization-group flows indicate that a quantum phase transition from a metallic phase to the Mott insulator phase occurs at a finite on-site Coulomb repulsion U for $\ensuremath{\epsilon}>0.$ We also discuss effects of randomness.

DOI
10.1103/physrevb.62.2377

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We study the low-energy asymptotics of the d=1+\ensuremathϵd=1+\ensuremath{\epsilon}d=1+\ensuremathϵ dimensional Hubbard model with a circular Fermi surface where there is 2kF{2k}_{F}2kF​ umklapp scattering present a priori. Peculiarity of the d=1+\ensuremathϵd=1+\ensuremath{\epsilon}d=1+\ensuremathϵ dimensions is incorporated through the imbalance between the elementary particle-particle and particle-hole (PH) loops: infrared logarithmic singularity of the PH loop is smeared for \ensuremathϵ>0.\ensuremath{\epsilon}>0.\ensuremathϵ>0. The one-loop renormalization-group flows indicate that a quantum phase transition from a metallic phase to the Mott insulator phase occurs at a finite on-site Coulomb repulsion U for \ensuremathϵ>0.\ensuremath{\epsilon}>0.\ensuremathϵ>0. We also discuss effects of randomness.
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 UUU equals the bandwidth Uc=W{U}_{\mathrm{c}}=WUc​=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 nk{n}_{k}nk​ at the Fermi wave number kF=0{k}_{\mathrm{F}}=0kF​=0 cannot be used to locate accurately Uc{U}_{\mathrm{c}}Uc​ from finite-size systems. However, the slope of nk{n}_{k}nk​ 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 Uc{U}_{\mathrm{c}}Uc​ reliably from nk{n}_{k}nk​ using accurate finite-size data.
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.
Abstract We present a T = 0 K renormalization group (RG) phase diagram for the electronic Hubbard model in two dimensions on the square lattice at half filling. The RG procedure treats quantum fluctuations in the single particle occupation number nonperturbatively via the unitarily decoupling of one electronic state at every RG step. The resulting phase diagram thus possesses the quantum fluctuation energy scale (ω) as one of its axes. A relation is derived between ω and the effective temperature scale upto which gapless, as well as emergent gapped phases can be obtained. We find that the normal and insulating phases of the half-filled Hubbard model correspond, for any on-site repulsion, to a marginal Fermi liquid normal phase and a topologically-ordered gapped Mott insulating liquid respectively. The marginal Fermi liquid is found to arise from singular forward scattering in directions normal to the nested Fermi surface, while singular backscattering events lead to Mott liquid state. The transition between these two phases involves passage through a pseudogapped phase bookended by Fermi surface topology-changing Lifshitz transitions. The pseudogap phase is observed to arise from the electronic differentiation encoded within the nested Fermi surface, and involves the gradual gapping of the Fermi surface from antinodes to nodes via charge and spin excitations that are mutually entangled. We obtain effective Hamiltonians for various phases, as well as wavefunctions for the low-energy many-body eigenstates of the Mott liquid. Benchmarking of the ground-state energy per particle and the double-occupancy fraction for the Mott liquid against existing numerical results yields excellent agreement. Presence of a Néel ordering symmetry-breaking perturbation in the RG leads to an antiferromagnetic spin-ordered charge insulating Mott state. Our results thus offer novel insights on a variety of aspects of the Mott–Hubbard problem, and can be extended to the doped system.
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.
Effects of randomness on interacting fermionic systems in one dimension are investigated by quantum Monte Carlo techniques. At first, interacting spinless fermions are studied whose ground state shows charge ordering. Quantum phase transition due to randomness is observed associated with the collapse of the charge ordering. We also treat random Hubbard model focusing on the Mott gap. Although the randomness closes the Mott gap and low-lying states are created, which is observed in the charge compressibility, no (quasi-)Fermi-surface singularity is formed. It implies localized nature of the low-lying states.
The spectrum of charge excitations, Im N ( q , ω), has been evaluated near the Mott transition in the one-dimensional Hubbard model with a particular value of the Coulomb interaction. The present theory based on a continuum limit describes the smooth change between the Tomonaga-Luttinger liquid state with a q -linear mode and the Mott insulator with a charge gap, by considering Umklapp scattering. The charge susceptibility is seen to diverge proportional to the inverse of the doping rate as the Mott insulator state is approached, in agreement with Bethe-ansatz result. It is shown that the q region where the renormalization group theory and conformal field theory are valid becomes vanishingly small as the transition is approached.