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arXiv:cond-mat/9312046

Anomalous low-temperature specific heat of the antiferromagnetic SU(N) Heisenberg model in a field

K. Lee

Published Dec 11, 1993• 0 citations
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Abstract

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-$m$ 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.

DOI
10.48550/arxiv.cond-mat/9312046

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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\gam\gam is a function of a field which breaks the SU(N) invariance of the internal degrees of freedom. We calculate the \gam\gam\gam in a zero field and in a small field. The in-field \gam\gam\gam is less than the zero-field \gam\gam\gam as expected since the entropy is reduced in the ordered system. The zero-field \gam\gam\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.
It is expected that the specific heat of the SU(2)-invariant 1D Heisenberg antiferromagnet of spin S is linear to the temperature at low T with the linear coefficient gamma s being a function of field H. The author's main result is that lim H to O gamma s = 1 / 3 (1+ square root S Gamma (S)(e/S) s / pi ) not= lim T to O lim H to O gamma s =2S/(1+S). This extends the previous result for S= 1 / 2 to other spin values. The author also provides an approximate interpolation formula between these two limits as a function of H/T for very small H and T.
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 consider the one-dimensional SU(N)-invariant t-J model, which consists of electrons with N spin components on a lattice with nearest-neighbor hopping t constrained by the excluded multiple occupancy of the sites and spin-exchange J between neighboring lattice sites. The model is integrable and has been diagonalized in terms of nested Bethe ansatze at the supersymmetric point t=J. The low-T specific heat is proportional to T. The γ-coefficient is extracted from the thermodynamic Bethe-ansatz equations and is expressed in terms of the spin wave velocities and the group velocity of the charges for arbitrary N, band filling, and splitting of the levels (magnetic and crystalline fields). Our results contain the following special cases: (i) For N=2 the traditional spin-1/2 supersymmetric t-J model, (ii) for exactly one electron per site the SU(N)-Heisenberg chain, and (iii) for N=4 the two-band supersymmetric t-J model with crystalline field splitting.
Thermodynamic properties of the SU(nnn) Heisenberg model in one dimension is studied by means of high-temperature expansion for arbitrary nnn. The specific heat up to O[(βJ)23]O[(βJ)^{23}]O[(βJ)23] and the correlation function up to O[(βJ)18]O[(βJ)^{18}]O[(βJ)18] are derived with βJβJβJ being the antiferromagnetic exchange in units of temperature. It is found for n>2n>2n>2 that the specific heat shows a shoulder in the high-temperature side of a peak. The origin of this structure is clarified by deriving the temperature dependence of the correlation function. With decreasing temperature, the short-range correlation with two-site periodicity develops first, and then another correlation with nnn-site periodicity at lower temperature. This behavior is in contrast to that of the inverse square interaction model, where the specific heat shows a single peak according to the exact solution. Our algorithm has an advantage that neither computational time nor memory depends on the multiplicity nnn per site; the series coefficients are obtained as explicit functions of nnn.
We have extracted the leading low-temperature contributions to the specific heat and the magnetic susceptibility from the thermodynamic Bethe-ansatz equations of the SU(2) invariant ferromagnetic Heisenberg chain of arbitrary spin S. This extends previous analytic results for S=1/2 to other spin values. The critical exponent for the specific heat is \ensuremath{\alpha}=-1/2, that for the susceptibility is \ensuremath{\gamma}=2, and that for the correlation length is \ensuremath{\nu}=1. The susceptibility and the correlation length exhibit logarithmic corrections, reflecting the analogy to the Kondo problem. These logarithmic corrections are quenched by relatively small magnetic fields. In large fields the energy required to flip a spin gives rise to an exponential activation.
We present a brief review on the recent theoretical developement about boundary effects in one-dimensional quantum spin systems focusing on the application of boundary conformal field theory. We discuss about low-temperature behaviors of the boundary contributions of the spin susceptibility and the specific heat coefficient. The anomalous behaviors at boundaries are deeply related with finite-temperature corrections of the boundary entropy or the ground state degeneracy, which yields a highly sensitive response of spin excitations in the vicinity of open boundaries.