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Journal of Physics Condensed Matter

The low-temperature specific heat anomaly of the SU(2) invariant 1D Heisenberg antiferromagnet of spin S in a small magnetic field

Kong-Ju-Bock Lee

Published Mar 18, 1991• 4 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/3/11/025

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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 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.
Low-temperature specific heat of Heisenberg-Ising ring at | Δ| ≪ 1 is calculated by the method of non-linear integral equations. Specific heat in constant magnetic field (CH) is proportional to temperature (T) at |2µ0H| ≪ (1+Δ)J. In particular lim H →0 lim T →0CH/T is 2θ/(3Jsin θ) where θ= cos -1Δ, ≤θ≪ π. It is conjectured that lim T→0 lim H→0CH/T = lim H→0 lim T→0CH/T from the result of numerical calculation. Low-temperature specific heat of the one-dimensional X-Y-Z model is proportional to T-3/2 exp (-α/T) in the case of Jz > Jy > 0, Jy ≧Jx > -Jy.
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 construct an integrable SU(2)-invariant model consisting of the Heisenberg chain of spin 1 interacting with an impurity of spin S. This generalizes previous results by Andrei and Johannesson for the spin-(1/2 chain. The model Hamiltonian is diagonalized and the thermodynamics is obtained. For ferromagnetic coupling at low temperatures the impurity susceptibility diverges as ${T}^{\mathrm{\ensuremath{-}}2}$ and the impurity specific heat is proportional to T1/2{T}^{1/2}T1/2 for all values of the impurity spin S. For antiferromagnetic coupling and T=0 the impurity susceptibility diverges proportionally to \ensuremath{\Vert}lnH\ensuremath{\Vert} as H\ensuremath{\rightarrow}0 if S=(1/2, while if S>1 the impurity spin is only partially compensated for by the S=1 chain.
Results are presented for the low-temperature specific heat of a disordered Heisenberg chain with random $\ifmmode\pm\else\textpm\fi{}J$ interactions in the large-spin limit. The specific heat varies at T2/3{T}^{2/3}T2/3 with a coefficient that depends on the relative fractions of positive and negative bonds. The coefficient is calculated exactly for the equal-probability distribution. There is also a brief analysis of the destabilizing effects of an applied magnetic field.
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.