arXiv:0808.0008 · Physical Review B

Wigner crystal versus Friedel oscillations in the one-dimensional Hubbard model

Stefan A. Söffing, Michael Bortz, Imke Schneider, Alexander Struck, Michael Fleischhauer, Sebastian Eggert

Published May 18, 2009• 59 citations

Abstract

We analyze the fermion density of the one-dimensional Hubbard model using bosonization and numerical density matrix renormalization group calculations. For finite systems we find a relatively sharp crossover even for moderate short-range interactions into a region with $4{k}_{F}$ density waves as a function of density. The results show that the unstable fixed point of a spin-incoherent state can dominate the physical behavior in a large region of parameter space in finite systems. The crossover may be observable in ultracold fermionic gases in optical lattices and in finite quantum wires.

DOI: 10.1103/physrevb.79.195114

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arXiv:0808.0008 · Physical Review B

Wigner crystal versus Friedel oscillations in the one-dimensional Hubbard model

Stefan A. Söffing, Michael Bortz, Imke Schneider, Alexander Struck, Michael Fleischhauer, Sebastian Eggert

Published May 18, 2009• 59 citations

arXiv HTML PDF

Abstract

4{k}_{F}

density waves as a function of density. The results show that the unstable fixed point of a spin-incoherent state can dominate the physical behavior in a large region of parameter space in finite systems. The crossover may be observable in ultracold fermionic gases in optical lattices and in finite quantum wires.

DOI: 10.1103/physrevb.79.195114

Related papers

Wigner crystal vs. Friedel oscillations in a finite quantum wire

Stefan Soeffing, Michael Bortz, Sebastian Eggert

We analyze the fermion density of the one-dimensional Hubbard model using bosonization and numerical DMRG calculations. For finite systems we find a relatively sharp crossover even for moderate short range interactions into a region with $4k_F$ density waves as a function of density. The results show that the unstable fixed point of a spin-incoherent state can dominate the physical behavior in a large region of parameter space in finite systems. The crossover may be observable in ultra cold fermionic gases in optical lattices and in finite quantum wires.We analyze the fermion density of the one-dimensional Hubbard model using bosonization and numerical DMRG calculations. For finite systems we find a relatively sharp crossover even for moderate short range interactions into a region with $4k_F$ density waves as a function of density. The results show that the unstable fixed point of a spin-incoherent state can dominate the physical behavior in a large region of parameter space in finite systems. The crossover may be observable in ultra cold fermionic gases in optical lattices and in finite quantum wires.

Aug 1, 20080 citations

2kF -Friedel to 4kF -Wigner oscillations in one-dimensional Fermi gases under confinement

Gao Xianlong

Density oscillations of confined one-dimensional Fermi gases of contact repulsive interactions in a continuous space are discussed within Bethe-ansatz-based spin-density-functional theory. The results are compared against the exact analytical and the exact diagonalization method. For an unpolarized system, the number of peaks in the density profiles is doubled, signaling the crossover of the $2{k}_{F}$ -Friedel to $4{k}_{F}$ -Wigner oscillations (with ${k}_{\mathrm{F}}$ being the Fermi wave vector). For both unpolarized and polarized systems, a threshold of the short-range interaction strength can be found where ${N}_{f}$ -peak Wigner oscillations of a $4{k}_{F}$ wave vector appear in the density profile ( ${N}_{f}$ is the total particle number).Density oscillations of confined one-dimensional Fermi gases of contact repulsive interactions in a continuous space are discussed within Bethe-ansatz-based spin-density-functional theory. The results are compared against the exact analytical and the exact diagonalization method. For an unpolarized system, the number of peaks in the density profiles is doubled, signaling the crossover of the $2{k}_{F}$-Friedel to $4{k}_{F}$-Wigner oscillations (with ${k}_{\mathrm{F}}$ being the Fermi wave vector). For both unpolarized and polarized systems, a threshold of the short-range interaction strength can be found where ${N}_{f}$-peak Wigner oscillations of a $4{k}_{F}$ wave vector appear in the density profile (${N}_{f}$ is the total particle number).

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Stefan A. Söffing, Michael Bortz, Sebastian Eggert

The density distribution of the one-dimensional Hubbard model in a harmonic trapping potential is investigated in order to study the effect of the confining trap. Strong superimposed oscillations are always present on top of a uniform density cloud, which show universal scaling behavior as a function of increasing interactions. An analytical formula is proposed on the basis of bosonization, which describes the density oscillations for all interaction strengths. The wavelength of the dominant oscillation changes with interaction, which indicates the crossover to a spin-incoherent regime. Using the Bethe ansatz the shape of the uniform fermion cloud is analyzed in detail, which can be described by a universal scaling form.The density distribution of the one-dimensional Hubbard model in a harmonic trapping potential is investigated in order to study the effect of the confining trap. Strong superimposed oscillations are always present on top of a uniform density cloud, which show universal scaling behavior as a function of increasing interactions. An analytical formula is proposed on the basis of bosonization, which describes the density oscillations for all interaction strengths. The wavelength of the dominant oscillation changes with interaction, which indicates the crossover to a spin-incoherent regime. Using the Bethe ansatz the shape of the uniform fermion cloud is analyzed in detail, which can be described by a universal scaling form.

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Gerald Bedürftig, B. Brendel, Holger Frahm +1 more

Using the density-matrix renormalization-group (DMRG) technique, we calculate critical exponents for the one-dimensional Hubbard model with open boundary conditions with and without additional boundary potentials at both ends. A direct comparison with open boundary condition Bethe ansatz calculations provides a good check for the DMRG calculations on large system sizes. On the other hand, the DMRG calculations provide an independent check of the predictions of conformal field theory, which are needed to obtain the critical exponents from the Bethe ansatz. From the Bethe ansatz we predict the behavior of the $1/L$ -corrected mean value of the Friedel oscillations (for the density and the magnetization) and the characteristic wave vectors, and show numerically that these conjectures are fulfilled with and without boundary potentials. The quality of the numerical results allows us to determine the behavior of the coefficients of the Friedel oscillations as a function of the Hubbard interaction.Using the density-matrix renormalization-group (DMRG) technique, we calculate critical exponents for the one-dimensional Hubbard model with open boundary conditions with and without additional boundary potentials at both ends. A direct comparison with open boundary condition Bethe ansatz calculations provides a good check for the DMRG calculations on large system sizes. On the other hand, the DMRG calculations provide an independent check of the predictions of conformal field theory, which are needed to obtain the critical exponents from the Bethe ansatz. From the Bethe ansatz we predict the behavior of the $1/L$-corrected mean value of the Friedel oscillations (for the density and the magnetization) and the characteristic wave vectors, and show numerically that these conjectures are fulfilled with and without boundary potentials. The quality of the numerical results allows us to determine the behavior of the coefficients of the Friedel oscillations as a function of the Hubbard interaction.

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4{k}_{F}

4k_F

density waves as a function of density. The results show that the unstable fixed point of a spin-incoherent state can dominate the physical behavior in a large region of parameter space in finite systems. The crossover may be observable in ultra cold fermionic gases in optical lattices and in finite quantum wires.

2{k}_{F}

-Friedel to

4{k}_{F}

-Wigner oscillations (with

{k}_{\mathrm{F}}

being the Fermi wave vector). For both unpolarized and polarized systems, a threshold of the short-range interaction strength can be found where

{N}_{f}

-peak Wigner oscillations of a

4{k}_{F}

wave vector appear in the density profile (

{N}_{f}

is the total particle number).

{k}_{F}

of the minimum of the dispersion.

We study the dynamical response of the half-filled one-dimensional Hubbard model for a range of interaction strengths

U

and temperatures

T

by a combination of numerical and analytical techniques. Using time-dependent density matrix renormalization group computations we find that the single-particle spectral function undergoes a crossover to a spin-incoherent Luttinger liquid regime at temperatures $T\ensuremath{\sim}J=4{t}^{2}/U$ for sufficiently large

U>4t

. At smaller values of

U

and elevated temperatures the spectral function is found to exhibit two thermally broadened bands of excitations, reminiscent of what is found in the Hubbard-I approximation. The dynamical density-density response function is shown to exhibit a finite-temperature resonance at low frequencies inside the Mott gap, with a physical origin similar to the Villain mode in gapped quantum spin chains. We complement our numerical computations by developing an analytic strong-coupling approach to the low-temperature dynamics in the spin-incoherent regime.

1/L

-corrected mean value of the Friedel oscillations (for the density and the magnetization) and the characteristic wave vectors, and show numerically that these conjectures are fulfilled with and without boundary potentials. The quality of the numerical results allows us to determine the behavior of the coefficients of the Friedel oscillations as a function of the Hubbard interaction.