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Physical review. B, Condensed matter

Logarithmic singularities in the susceptibility of the antiferromagnetic SU(N) Heisenberg model

P. Schlottmann

Published Mar 1, 1992• 20 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.45.5293

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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.
We discuss the low-temperature specific heat of the integrable SU(N)- invariant Heisenberg model in one dimension with degrees of freedom in the symmetric rank-mmm tensor representation, especially for the antiferromagnetic coupling. It is known that the linear specific heat coefficient $\gam$ is a function of a field which breaks the SU(N) invariance of the internal degrees of freedom. We calculate the $\gam$ in a zero field and in a small field. The in-field $\gam$ is less than the zero-field $\gam$ as expected since the entropy is reduced in the ordered system. The zero-field $\gam$ is the same as the one obtained by the prediction of the critical behavior of 1D quantum spin systems via conformal field theory. This extends the previous results for N=2 to an arbitrary N.
Unusual properties of the Hubbard chain of arbitrary degeneracy N and excluded-site occupations of more than two electrons are presented: (i) In zero field and in the limit of large N the ground state resembles an interacting Bose gas. The 1/N contributions vanish identically, so that corrections to leading order are 1/N2{\mathrm{N}}^{2}N2. (ii) The susceptibility at T=0 shows logarithmic singularities as H\ensuremath{\rightarrow}0. (iii) The Hubbard model and the Heisenberg chain have the same low-energy spin excitations for all U. (iv) The specific-heat coefficient \ensuremath{\gamma} is singular. These properties hold for arbitrary band filling and (ii)--(iv) hold for all N\ensuremath{\ge}2.
The degenerate Hubbard chain with N internal degrees of freedom is integrable if site occupations of more than two electrons are excluded. The ground state of the model has several unusual properties; e.g., the magnetic susceptibility has logarithmic singularities as the field H\ensuremath{\rightarrow}0 and also the specific-heat coefficient is singular. An extensive discussion of ground-state properties as a function of U and the band filling is presented. The elemental charge and spin excitations are derived for arbitrary U and band filling. If, on average, there is exactly one electron per site as a function of U, a qualitative change in the low-energy charge excitations is found at a critical value Uc{\mathit{U}}_{\mathit{c}}Uc​. This change is probably associated with a Mott metal-insulator transition (Uc{\mathit{U}}_{\mathit{c}}Uc​=0 for N=2). The Fermi velocity is finite for UUc{\mathit{U}}_{\mathit{c}}Uc​ but vanishes for U>Uc{\mathit{U}}_{\mathit{c}}Uc​ (for one electron per site). The spin-wave velocity is inversely proportional to the magnetic susceptibility for all U and band fillings. There are N-1 elemental spin-wave branches. For arbitrary band filling there are then N-1 soft modes at different finite momenta, all related to the Fermi surface. For exactly one electron per site, the spin-excitation spectrum can be mapped onto the one of the SU(N) Heisenberg chain. The implications of the hole states on superconductivity are also discussed.
The authors considers the one-dimensional SU(N)-invariant t-J model which consists of electrons with N spin components on a lattice with nearest-neighbour hopping t constrained by the excluded multiple occupancy of the lattice sites and spin-exchange J between neighbouring lattice sites. The model is integrable at the supersymmetric point t=J. The ground state Bethe ansatz equations are analysed and solved numerically for arbitrary band filling and several values of N. The ground state energy, the chemical potential and the spin susceptibility are obtained as a function of band filling. The elemental charge and spin excitations are derived for arbitrary N and band filling. The energy of the charge excitations vanishes at the Fermi surface. The Fermi velocity has a maximum as a function of band filling, vanishing for the empty and full bands. The spinwave velocity is inversely proportional to the susceptibility. For exactly one electron per site the charge fluctuations are suppressed and the Bethe ansatz equations map onto those of the SU(N)-invariant Heisenberg chain.
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.