Excitation Spectrum and Thermodynamic Properties of the Ising-Heisenberg Linear Ferromagnet

New analytic results are presented for the low-T thermodynamics of the Ising-Heisenberg linear ferromagnetic in a magnetic field H/sub 0/. For small H/sub 0/ the thermodyanmic functions show unexpected and interesting structure as a function of H/sub 0/ and the anisotropy ..delta... The thermal and magnetic energy gaps have singularities, not necessarily at the same ..delta..-H/sub 0/ location, as changes occur in the type of excitation dominating the low-T behavior. The results may relate to quantum solitons in the linear ferromagnet.

Fermi-and Bose-gas models, ' the linear Hub- bard model of a metal-insulator transition, ' and the linear, spin-2, Ising-Heisenberg XY contin- uum model. 4Exact solutions for a continuum electron gas' and an electron gas on a lattice ' are  known.These are relevant to the important field of 1D organic conductors.' Models for organic charge-transfer salts can be mapped into a quan-!tum magnetic chain' which in the antiferromag-netic limit corresponds to the Hubbard dimer gas.' The exact solutions of the 1D quantum- mechanical sine-Gordon and related equations (solitons) have been extensively applied to chargedensity waves in 1D conductors.' " Very recent- ly the Bethe's Ansata techniques have been used to solve the massive Thirring model."' Fa- deev's review presents a unified approach to all the models discussed above." Sutherland gives an overview of the quantum soliton concept and its connections to Bethe's Ansat~."   In this Letter we present new, unanticipated, and interesting exact results for the 1D, spin-2, ferromagnetic, Ising-Heisenberg model.The Hamiltonian" is N N H =g fS, "S, +, "+S,'S, +, '+~(S,'S, +, '-4)]-H, g S, .'.
The S's are 2 the respective Pauli matrices, and there are periodic boundary conditions on the system.
6-l except, if H, is in a small, order T, neighborhood of zero, we restrict 6&l.(T is tempera- ture. ) We set Boltzmann's constant to l throughout the body of this paper.
This system was first studied" in the 1930's; a formalism for the thermodynamics was derived by Gaudin much later." In Gaudin's work assumptions were made which are difficult to verify directly.
In this and previous work" we have made comparisons of the predictions of Gaudin's formalism to nu- merical results on finite systems." All comparisons are favorable thus enhancing our faith in the as- sumptions contained in the thermodynamic formalism.
We have performed low-temperature expansions of Gaudin's formalism to derive all of our results.We will not present this approach in this paper, however, since it is a fairly long and detailed deriva- tion.We will give a "physical" argument for the results which shows the connection between the low- temperature thermodynamics and the excitations of the system.
It is known that the zero-temperature dispersion curves for this system" are given by E"(P)=nH, + sinhC'(coshn4 -cosP)/sinhnc ',   where 6, =cosh@, 0-P-2&, andn =1, 2, . . . .The n =1 excitations and linear combination of the n =1 excitations are spin waves, and the higher-n excitations are bound states of spin waves.The P s are distributed uniformly between 0 and 2& and, for a given n, obey a Fermi-like exclusion principle.
The energies of the first excited states are E(q) =H, +b.-cosq.There are N such states with q =2mm/N, 0 & q & 2m.The states we first sum to derive the partition function are these N states, the 2N(N -1) states with energies E(q,) +E(q,), ... , the N!/[I!(N -l)] states with energies E(q,) +E(q,) +... +E(q, ), etc.These are all the spin-wave excitations, and they provide a contribution" to E(T,cr) where a is the magnetization per spin.
For Ho far enough away from zero, this is all we need to obtain the low-temperature thermoydnam- ics to exponential accuracy in T.However, for small H, other excitations, the high-lying bound states, can dominate.For large n E"(P) -nHO+sinhC'; note that the E"(P) are independent of P and, according- ly, are effectively just the energies of a 1D Ising model with exchange constant J = sinh@.Therefore, we add to Eq. ( 3) the Ising free energy for this J. " We obtain, after some simplification for low T of the Ising-model result, (4) This is our basic result and is the same result as obtained by the low-T expansion of the Gaudin formalism.It is valid for low T, O(T) &H, -0 and b, & 1.The correction, E, is exponentially higher or- der in T than the larger of the two terms on the right-hand side of Eq. ( 4) even after taking an arbi- trary number of T derivatives or up to and including two H, derivatives.(Note that this means, in particular, that if one expands the square root for H, exponentially larger or smaller than exp[-(4' -1)'/'/2T], one should retain two terms in the expansion.Both terms are significant, and E is expo- nentially higher order than the integral or the second term of the square-root expansion, whichever is larger.}" We now discuss the detailed behavior of Eq. ( 4) in terms of the susceptibility y =--[ 8'(F -aH, )/BH, '] r and specific heat Cz-= -T[8'(F -crHo)/BT']z .We find from Eq. ( 4 (5) (6a) Ex is exponentially higher order in T than the larger of the first two terms.E" is exponentially higher order than the first term of Eq. (6a) or O(T) higher order than the second term, whichever is larger.Similarly, for C~, we obtain E, is O(T) higher order than the larger of the first two terms.The first terms of both y and C~are bound-state contributions and the second terms are spin-wave contributions.
For X we redefine variables to H, =e '; Fig.  ' '] the spin-wave term domi- nates the bound-state term.For n & a, the bound- state term dominates.Obvious simplif ications can be made to either Eq. ( 5) or (6a) by dropping appropriate terms in these cases.The.bound- state region subdivides into n & a, = -(b.' -1)'"/2 and n & n, .For o. & n"Eq.( 5) simplifies to X =(4T) 'exp[(&' -1) ' '/(2T)]. (7a) For a, &a&o." Corrections to both these equations are exponen- tially higher order in T. We thus have three separate regions for y with different exponential be- havior in each.for y, the susceptibility.The bottom portion illustrates the character of Cz, the specific heat.In both cases S-W labels the spin-wave regions, 8-S labels the bound- state regions, and ~=-1.
Again one has three separate regions with different exponential behavior in each (as for y), but the details are different from X.
2. We plot the effective energy gap for G~.Shown are the gaps for the ferromagnet and antiferro- magnet.The ferromagnet curve iQustrates the singu- larity (kink) described in the text.The antiferromag net gap is given by a single expression over the whole anisotropy r~~ge.
for -, ' &a&1.Corrections to Etl. ( 9) are 0(T) higher order.Thus X has a single effective gap for all L at Hp 0 while CH has two effective gaps with a crossover between the two at 4 =-, '. " The bound states dominate for large 4 while the spin waves dominate for small &.This is shown in Fig. 2, where the notation is that of Ref. 14.
It would be of considerable interest to investi- gate these crossover effects experimentally.
Farinfrared studies like those of Torrance and Tink- ham" on CoCl, ~2H, O might be performed on the Ising-like linear ferromagnet cobalt chloride dipyridine (CoCl, ~2NC, H, ).The very recent dis- covery of a family of good Heisenberg-like ferromagnets" offers the possibility of studies by neu- trons or other means of the more isotropic region.Finally, we note that an understanding of the excitations of the linear ferromagnet may be important for the quantum soliton problem.'4   This work was supported in part by the U. S.
FIG. 1.The top portion presents the various regions