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Journal of the Physical Society of Japan

Logarithmic Correction Terms of the Magnetic Susceptibility in Highly Correlated Electron Systems

Kenro Kawano, Minoru Takahashi

Published Nov 15, 1995• 5 citations
Publisher

Abstract

The logarithmic correction terms in the repulsive Hubbard chain is investigated by using the Bethe Ansatz . The magnetic susceptibility χ of this model has a field dependent logarithmic correction term in small magnetic field at zero-temperature. This term causes ∂χ/∂ h ∣ h =0 =∞ . In arbitrary n and U the existence of the logarithmic correction term is shown for the susceptibility. We consider how this term depends on n and U . We also discuss the logarithmic correction term of the super-symmetric t-J model.

DOI
10.1143/jpsj.64.4331

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The logarithmic correction terms in the repulsive Hubbard chain is investigated by using the Bethe Ansatz . The magnetic susceptibility χ of this model has a field dependent logarithmic correction term in small magnetic field at zero-temperature. This term causes ∂χ/∂ h ∣ h =0 =∞ . In arbitrary n and U the existence of the logarithmic correction term is shown for the susceptibility. We consider how this term depends on n and U . We also discuss the logarithmic correction term of the super-symmetric t-J model.
The magnetic susceptibility of the one-dimensional Hubbard model with open boundary conditions at arbitrary filling is obtained from field theory at low temperatures and small magnetic fields, including leading and next-leading orders. Logarithmic contributions to the bulk part are identified as well as algebraic-logarithmic divergences in the boundary contribution. As a manifestation of spin-charge separation, the result for the boundary part at low energies turns out to be independent of filling and interaction strength and identical to the result for the Heisenberg model. For the bulk part at zero temperature, the scale in the logarithms is determined exactly from the Bethe ansatz. At finite temperature, the susceptibility profile as well as the Friedel oscillations in the magnetisation are obtained numerically from the density-matrix renormalisation group applied to transfer matrices. Agreement is found with an exact asymptotic expansion of the relevant correlation function.
We study the leading low-temperature dependence of magnetic susceptibility of one-dimensional electrons with a fixed total number of particles at the magnetic fields equal to zero-temperature critical values where the magnetic field induces commensurate-incommensurate quantum phase transitions. For free and repulsively interacting electrons there is only one such critical field corresponding to the transition to the fully polarized state. For attractively interacting electrons besides saturation field there is another critical field equal to the spin gap where zero-temperature magnetization sets in. For all cases, except for the lattice models at half filling, the magnetic susceptibility at critical values of magnetic field has a universal logarithmic temperature dependence, $\ensuremath{\chi}(T)=\ensuremath{\chi}(0)(1+2/ln\phantom{\rule{0.16em}{0ex}}T+\ensuremath{\cdots})$ for $T\ensuremath{\rightarrow}0$.
We present a general method for the calculation of correlation functions in the repulsive one-dimensional Hubbard model at less than half-filling in a magnetic field h. We describe the dependence of the critical exponents that drive their long-distance asymptotics on the Coulomb coupling, the density, and h. This dependence can be described in terms of a set of coupled Bethe-Ansatz integral equations. It simplifies significantly in the strong-coupling limit, where we give explicit formulas for the dependence of the critical exponents on the magnetic field. In particular, we find that at small field the functional dependence of the critical exponents on h can be algebraic or logarithmic---depending on the operators involved. In addition, we evaluate the singularities of the Fourier images of the correlation functions. It turns out that switching on a magnetic field gives rise to singularities in the dynamic field-field correlation functions that are absent at h=0.
We present a general method for the calculation of correlation functions in the repulsive one-dimensional Hubbard model at less than half-filling in a magnetic field h. We describe the dependence of the critical exponents that drive their long-distance asymptotics on the Coulomb coupling, the density, and h. This dependence can be described in terms of a set of coupled Bethe-Ansatz integral equations. It simplifies significantly in the strong-coupling limit, where we give explicit formulas for the dependence of the critical exponents on the magnetic field. In particular, we find that at small field the functional dependence of the critical exponents on h can be algebraic or logarithmic---depending on the operators involved. In addition, we evaluate the singularities of the Fourier images of the correlation functions. It turns out that switching on a magnetic field gives rise to singularities in the dynamic field-field correlation functions that are absent at h=0.
We consider the integrable SU(N)-invariant Heisenberg model in one dimension with degrees of freedom in the symmetric rank-m tensor representation. It is shown that for antiferromagnetic coupling the zero-temperature susceptibility to an arbitrary field breaking the SU(N) invariance of the internal degrees of freedom has logarithmic singularities as the field tends to zero. The logarithms arise from the interference of the two ``Fermi points'' of the spin-wave spectrum and do not depend on the level splitting scheme. These properties hold for a variety of models, e.g., the SU(N)-invariant Heisenberg chain of spin S=(N-1)/2, the Babujian-Takhtajan model, the N-component fermion gas interacting in one dimension by a repulsive \ensuremath{\delta}-function potential, the N-component supersymmetric t-J model in one dimension, the Gross-Neveu model, and the SU(N) generalization of the Hubbard chain.