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arXiv:2208.07620 · Physical review. B./Physical review. B

Accurate localization of Kosterlitz-Thouless-type quantum phase transitions for one-dimensional spinless fermions

Florian Gebhard, Kevin Bauerbach, Örs Legeza

Published Nov 18, 2022• 8 citations
arXivHTMLPDF

Abstract

Spinless fermions in one dimension with nearest-neighbor transfer amplitude $\ensuremath{-}t$ and interaction $V$ display a Kosterlitz-Thouless-type quantum phase transition from a Luttinger liquid metal to a charge density wave insulator at finite interaction strength ${V}_{\mathrm{c}}=2t$. The authors use the density matrix renormalization group (DMRG) method on rings with $L\ensuremath{\le}512$ sites for an accurate localization of the transition. The maximum of the quasiparticle density in the upper Hartree-Fock band, ${n}_{\ensuremath{\beta}}^{\mathrm{max}}(L,V)={\mathrm{max}}_{k}\phantom{\rule{0.167em}{0ex}}{n}_{k,\ensuremath{\beta}}(L,V)$, develops a peak close to the critical interaction whose position extrapolates to the exact result in the thermodynamic limit, ${lim}_{L\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}{V}_{\ensuremath{\beta}}^{\mathrm{max}}(L,V)={V}_{\mathrm{c}}$, with an accuracy of better than one percent. Monitoring the Luttinger parameter permits to locate the transition with an accuracy of three percent.

DOI
10.1103/physrevb.106.205133

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The one-dimensional Holstein model of spinless fermions interacting with dispersionless phonons is studied using a new variant of the density matrix renormalization group. By examining various low-energy excitations of finite chains, the metal-insulator phase boundary is determined precisely and agrees with the predictions of strong coupling theory in the antiadiabatic regime and is consistent with renormalization group arguments in the adiabatic regime. The Luttinger liquid parameters, determined by finite-size scaling, are consistent with a Kosterlitz-Thouless transition.
We examine the ground-state properties of the one-dimensional Edwards spinless fermion transport model by means of large-scale density-matrix renormalization-group calculations. Determining the single-particle gap and the Tomonaga-Luttinger liquid parameter (KρK_ρKρ​) at zero temperature, we prove the existence of a metal-to-insulator quantum phase transition at one-third band filling. The insulator---established by strong correlation in the background medium---typifies a charge density wave (CDW) that is commensurate with the band filling. Kρ=2/9K_ρ=2/9Kρ​=2/9 is very small at the quantum critical point, and becomes KρCDW=1/9K_ρ^{\rm CDW}=1/9KρCDW​=1/9 in the infinitesimally doped three-period CDW, as predicted by the bosonization approach.
The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground-state energy, first fermionic excited state, density wave correlations, and mean lattice displacements are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of t=0.1\ensuremath{\omega}, \ensuremath{\omega}, and 10\ensuremath{\omega}, where \ensuremath{\omega} is the phonon frequency. At a finite fermion-phonon coupling g there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite-size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$ and ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, the velocity of charge excitations, and the correlation exponent, respectively. For free fermions (g=0) and for strong coupling (g2{\mathit{g}}^{2}g2\ensuremath{\gg}t\ensuremath{\omega}) our results agree well with known analytic results. \textcopyright{} 1996 The American Physical Society.
We investigate the local density of states of the one-dimensional half-filled spinless fermion model with nearest-neighbour hopping t > 0 and interaction V in its Luttinger liquid phase -2t < V ≤ 2t. The bulk density of states and the local density of states in open chains are calculated over the full band width ∼4t with an energy resolution ≤0.08t using the dynamical density-matrix renormalization group (DDMRG) method. We also perform DDMRG simulations with a resolution of 0.01t around the Fermi energy to reveal the power-law behaviour D(ϵ) ∼ |ϵ - ϵ(F)|(α) predicted by the Luttinger liquid theory for bulk and boundary density of states. The exponents α are determined using a finite-size scaling analysis of DDMRG data for lattices with up to 3200 sites. The results agree with the exact exponents given by the Luttinger liquid theory combined with the Bethe Ansatz solution. The crossover from boundary to bulk density of states is analysed. We have found that boundary effects can be seen in the local density of states at all energies, even far away from the chain edges.
An interacting spinless fermion wire coupled to a three-dimensional (3D) semiconducting substrate is approximated by a narrow ladder model (NLM) with varying number of legs. We compute density distributions, gaps, charge density wave (CDW) order parameters, correlation functions, and the central charge using the density-matrix renormalization group method. Three ground-state phases are observed: a one-component Luttinger liquid, a quasi-one-dimensional (1D) CDW insulator, and a band insulator. We investigated the convergence of the NLM properties with increasing number of legs systematically and confirm that the NLM is a good approximation for the quasi-1D phases (Luttinger liquid and CDW) of the 3D wire-substrate model. The quantum phase transitions between these phases are investigated as function of the coupling between wire and substrate. The critical nearest-neighbor interaction increases with increasing coupling between wire and substrate and thus the substrate stabilizes the Luttinger liquid in the wire. Our study confirms that a Luttinger liquid or CDW insulator phase could occur in the low-energy properties of atomic wires deposited on semiconducting substrates.
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.