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Journal of Physics A Mathematical and Theoretical

The absence of logarithmic divergences in the spin and charge density correlations of the 1d Hubbard model

Vieri Mastropietro

Published Mar 14, 2007• 3 citations
Publisher

Abstract

We perform a rigorous computation of the spin and charge density correlations of the 1d repulsive Hubbard model at weak coupling, focusing on the properties of the Fourier transform at momentum 0 and ±2 p F , if p F is the Fermi momentum. We prove that the interaction changes the singularity at ±2 p F (the discontinuity in the derivative becomes a power law singularity) while the singularity at 0 is essentially unchanged. Our results show that the logarithmic divergences at zero momentum recently found in [7], which would be in contrast with Luttinger liquid behaviour, are indeed spurious.

DOI
10.1088/1751-8113/40/13/004

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The electronic system of the 1D Hubbard model is not stable due to Peierls instability; the correlations are strong even for the weak Coulomb interaction. The resulting strongly correlated state without Landau quisi--particle excitations is known as the Luttinger liquid. Critical exponents of a power--law dependence of correlation functions at low energies differ substantially for the Luttinger and Fermi liquids. In this paper we evaluate two critical exponents that define non--trivial behavior of the density of electronic states at low frequences and the momentum distribution of the occupation number nearby kFk_FkF​.
We present analytical results of fundamental properties of one-dimensional (1D) Hubbard model with a repulsive interaction, ranging from fractional excitations to universal thermodynamics, interaction-driven criticality, correlation functions, Contact susceptibilities and quantum cooling. Using the exact solutions of the Bethe Ansatz equations of the Hubbard model, we first rigorously calculate the gapless spin and charge excitations, exhibiting exotic features of fractionalized spinons and holons. Based on the analysis on the fractional charge and spin excitations, the spin-incoherent Luttinger liquid with only the charge propagation mode is elucidated by the asymptotic of the two-point correlation functions with the help of the conformal field theory. Near quadruple critical point, we then further analytically obtain the thermodynamical properties, dimensionless ratios and scaling functions near quantum phase transitions in terms of chemical potential, magnetic field and interaction. In particular, we determine additivity rules of spin and charge susceptibilities, and derive explicit forms of thermodynamics of spin-incoherent Luttinger liquid. Finally, in order to capture deeper insight into the Mott insulator and interaction driven criticality, we further study the double occupancy and its associated Contact and Contact susceptibilities through which an adiabatic cooling scheme upon the quantum criticality is introduced.