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D. A. Garanin and V. S. Lutovinov
In the framework of the spin operator diagram technique the dynamical properties of the easy axis type antiferromagnets at high temperatures are considered. The diagonalization procedure generalizing the u-v transformation of the spin operators is presented. The spin wave (SW) spectrum allowing for the temperature and the magnetic field dependence of the sublattice magnetization at T ~ T_N is obtained. The SW relaxation frequencies due to the processes of the SW scattering on the thermal spin fluctuations are calculated. The latter are in agreement with the data obtained on MnF_2 near the spin-flop transition. Comment: The procedure of the spin
Hamiltonian diagonalization is proposed. The relaxation frequencies of
spin waves in the easy-axis antiferromagnets are calculated.
D. A. Garanin and V. S. Lutovinov
Relaxation processes in the nuclear subsystem of antiferromagnets of the easy-plane type are investigated when the dynamic frequency shift (DFS) in the nuclear spin wave (NSW) spectrum is arbitrary. The diagram technique for spin operators was used to calculate the NSW relaxation frequencies both in the region T >>\omega_n and T << \omega_n, where \omega_n is the unshifted NMR frequency. It is shown that when the DFS is large, one must allow for many-spin interactions in addition to the pair (Suhl-Nakamura) interaction of nuclear spins with each other. When many-spin interactions are properly accounted for, in the presence of the symmetry in the basal plane there appears in the NSW spectrum a hydrodynamic region (\omega_k \propto k), where the damping of the NSWs is determined by the four-magnon scattering processes: \gamma_k \propto k^2. The main contribution to the NSW damping in the principal part of the phase space of the nuclear subsystem is provided by the scattering of NSWs on thermal fluctuations of the longitudinal components of nuclear spins: \gamma_k \propto kT \omega_k^3. The characteristic temperature T* >>\omega_n is determined, at which the perturbation theory in the number of loops in the diagrams ceases to be valid and the spin-wave picture in the nuclear subsystem vanishes. The effect of the dipole-dipole interactions on the fluctuational damping of NSWs is taken into account. Comment: One of my best works. Maybe
it was worth to show it to Harry Suhl at the proper time.
D. A. Garanin and V. S. Lutovinov
In the framework of the standard basis operators method for magnetic
systems with single-sight anisotropy, the dynamic properties of the easy-axis
type ferromagnet are investigated. The basis used in the paper makes it
possible to consider systems with arbitrary values of the spin. It is shown
that for the values of the anisotropy large enough (of the order of the
exchange interaction), the energy spectrum of the system includes in addition
to the acoustic branch also the optical branches - magnetic excitons associated
with multiple excitations of the isolated ion and stable with respect to
the decay into the acoustic magnons. The excitons corresponding to the
transitions from the ground to the second excited state of an isolated
ion are stable also for small values of the anisotropy, if their wave vector
is large enough. The quasiparticle lifetimes due to the processes of their
scattering on each other are calculated. The scattering amplitudes involving
the acoustic magnons satisfy the Adler's principle.
D. A. Garanin and V. S. Lutovinov
The spin Hamiltonian diagonalization procedure developed earlier [Garanin and Lutovinov, 1982] is extended for the systems with nonequidistant spectrum. The consideration is carried out in the framework of the universal basis operators diagram technique [see, e.g., Garanin and Lutovinov, 1983] and on the example of the nuclear subsystem of antiferromagnets with quadrupole interaction playing the role of single-sight easy-axis anisotropy. The spectrum of quadrupole splitted nuclear spin waves (NSW) and their relaxation frequencies due to the processes of the NSW scattering on the thermal fluctuations of the longitudinal spin components are calculated. Comment: This is the title of my PhD Thesis, as well.
D. A. Garanin and V. S. Lutovinov
The modified version of the spin operator diagram technique is presented with regard to the dynamical properties of the easy-plane ferromagbets. The non-interacting spin wave (SW) spectrum and scattering amplitudes are obtained with the help of the spin Hamiltonian diagonalization procedure. The SW relaxation frequencies due to processes of the SW scattering on each other and on the thermal fluctuations of the longitudinal spin components are calculated.
D. A. Garanin and V. S. Lutovinov
In the framework of the spin operator diagram technique the method of taking into account correlation effects in magnetic systems based on the idea of the 1/z expansion (z is the number of nearesr neighbours) is proposed. An equation of state of 3-dimensional Ising systems with the arbitrary spin values S in the first order in 1/z is presented, which considerably improves the mean field theory (being the zeroth-order approximation). For various lattices the calculated values of T_c coincide within 1% with the results of the high temperature expansions. Comment: The system of equations derived here was in fact obtained earlier by Horwitz and Callen (1961), as well as by others. Here the equations have been solved numerically, and the physical community has seen for the first time, how good the self-consistent Gaussian approximation (SCGA) is.
D. A. Garanin and V. S. Lutovinov
The equation of state obtained earlier for the 3-dimensional Ising ferromagnets in the first order in 1/z (z is the number of the nearest neighbours) is generalized for the classical D-component vector model. The results obtained are valid in a wide temperature range |\tau| >= [z^2(D+2)]^{-1} [\tau = (T_c-T)/T_c] and become exact in the limit D \to \infty. Comment: Here the anisotropic spherical model (ASM), which, in particular, describes ordering in low dimensions, appears for the first time.
D. A. Garanin, V. V. Ishchenko, and L. V. Panina
The dynamics of an ensemble of noninteracting single-domain magnetic particles is investigated both on the basis of analytic solution of the Fokker-Planck equation and in the framework of the reduced-description method. It is shown that in the general case the shape of the resonance and relaxation curves is not Lorentzian. In the isotropic case, the deviations from Lorentzian form reach 7%. In the presence of anisotropy, the main source of broadening of the resonance is thermal spread of the precession frequencies of the magnetic moments. An exact expression is obtained for the integral time of the longitudinal relaxation of magnetic particles with uniaxial anisotropy; it is valid for any value of the potential barrier. It is shown that for isotropic particles the distribution function is in good agreement with the obtained exact results. In the first approximation of the moment method a generalized equation of Landau-Lifshitz-Bloch type is obtained; it gives a reduced description of the dynamics of the ensemble of magnetic particles in the general nonlinear case. Comment: Here the Landau-Lifshitz-Bloch equation appears for the first time. Equations of motion for ferromagnets of such a type, which contain longitudinal relaxation terms, have been written phenomenologically by Bar'yakhtar in his pioneering works starting from 1984.
D. A. Garanin
It is shown that the tunneling splitting of the energy levels of the easy-axis spin system in a small transverse magnetic field can be calculated perturbatively for arbitrary S. The results are exact, in contrast to those obtained previously in the same limit by indirect and rather difficult methods. Comment: The same for the ground state has been already done by Korenblit and Shender (1978).
D. A. Garanin
We consider a diffusion of a charged particle over a three-dimensional barrier in a magnetic field. By the solution of the Fokker-Planck equation it is shown that the magnetic field causes the diminishing of thre prefactor |\Gamma_0 in the expression for the particle's thermoactivation rate: \Gamma =\Gamma_0 \exp(-U_0/T), T << U_0. Unlike the two-dimensional case considered earlier, in the general situation this diminishing is non-uniform with respect to the orientation of the vector B, with a saturation at some non-zero level for some range of orientations. The latter causes the splitting of the relaxation peak in the bulk materials containing randomly oriented two-well relaxators at high magnetic field.
D. A. Garanin
We derive microscopically an equation of motion for a ferromagnetic substance at nonzero temperature allowing for both transverse and longitudinal relaxation and generalizing the Landau-Lifshitz equation. The consideration starts from the density matrix equation for a quantum spin interacting with the environment, which is within 7% accuracy reduced to the closed equation for the first moment of the distribution function -- the magnetization. The latter interpolates between the Landau-Lifshitz equation (S >> 1 and low temperatures) and the Bloch equation (S=1/2 or high temperatures). For condensed magnetic media (i.e., a ferromagnet) one can replace in the spirit of the mean field theory the magnetic field by the molecular one containing the exchange field acting on a given magnetic ion from its neighbours, which results in a Landau-Lifshitz type equation of motion with a longitudinal relaxation term providing the Curie-Weiss static solution. Further we consider the mobility of a domain wall (DW) in a uniaxial ferromagnet at nonzero temperatures where the magnetization in the middle of the domain boundary is smaller than in the domains (the elliptic DW transforms into the linear one near T_c). It is shown that longitudinal relaxation plays a crucial role in DW dynamics in a wide range of elevated temperatures. Fazit: the LLB equation for quantum ferromagnets; the linear mobility of domain walls across the Bloch-wall phase transition.
D. A. Garanin
With the use of the generalized Landau-Lifshitz-Bloch equation of motion for a ferromagnet at finite temperatures we investigate the dynamic properties of domain walls (DWs) taking into account their ellipticity, i.e., the deficit of the magnetization value M in the DW in comparison with that in domains. The translational motion of elliptic DWs is accompanied by longitudinal relaxation which governs the DW dynamics even in the case of small ellipticity, if the relaxation constants are small. The linear DW mobility \mu calculated in the whole temperature range 0 =< T =< T_c shows a singular behavior in the point T=T* where the elliptic DW restructures into a linear one. At low temperatures the calculated dependence of \mu on the transverse field H_x is consistent with the experimental observations. The longitudinal relaxation mechanism of the DW damping switches off for the DW velocities v >= v_0 ~ \Gamma_1 \delta (\delta is the DW width, \Gamma_1 is the longitudinal relaxation rate), and the dependence v(H) may show a hysteresis. We also derive Slonczewsky-like reduced equations of motion for the DW parameters, which contain an additional equation for the magnetization deficite. With the use of these equations the stability of the stationary DW motion is investigated. It is shown that in some cases there are three separate stable sections of the curve v(H). Finally, the dynamic susceptibility of a ferromagnetic sample due to the diaplacement of elliptic domain walls is calculated. Comment: The effect predicted in this and the preceding papers has been experimentally observed [J. Kötzler, D. A. Garanin, M. Hartl, and L. Jahn, 1993], which gave me a postdoc position for two years in Hamburg. (Anything less would not do, I have tried to apply before.)
D. A. Garanin
The nonlinear mobility of a domain wall in an idealized model of a RE ferrrite-garnet is considered, the thermally agitated RE sublattice being deascibed by the Landau-Lifshitz-Bloch equation with longitudinal relaxation terms. It is shown that in some field intervals the DW velocity is small and governed by the longitudinal relaxation of the RE ions. This effect may be observed in sufficiently low magnetic fields near the compensation point of ferrites. If the dynamics of the RE spins is dissipationless, there is a non-analytic contrubution to the DW damping due to the "wake " in the RE subsystem. Comment: The only application of the LLB equation in its full form. The experimental predictions has not yet been tested experimentally.
D. A. Garanin, V. S. Lutovinov, and L. V. Panina
In the framework of the spin operator diagram technique, the processes of relaxation of the lower branch magnons in the two-sublattice easy-plane antiferromagnet are investigated. The procedure of the spin Hamiltonian diagonalization allowing to obtain compact expressions for magnon interaction amplitudes is proposed. The relaxation frequencies due to three-and four-magnon processes, processes of elastic scattering and interbranch transformation by thermal and concentrational magnetization fluctuations as well as the three-magnon confluence processes with rescattering by fluctuations are calculated. The results obtained are consistent with the experimental data on parametric excitation of spin waves in the low-temperature antiferromagnets MnCO_3 and CsMnF_3. Comment: The most advanced version of the spin Hamiltonian diagonalization procedure.
D. A. Garanin and V. S. Lutovinov
The attenuation of the low frequency longitudinal and transverse sound waves in the isotrope elastic body is calculated in the framework of the diagrammatic Green's function approach taking into account three-phonon and defect scattering processes. It is shown that the system of integral equations for the renormalized three-phonon confluence vertices is a generalization of the system of the linearized Boltzmann equations for the phonon gas considered by Akhiezer. An accurate solution of these equations gives the microscopic expressions for the kinetic coefficients (first- and second viscosities, heat conductivity) of the isotrope elastic media entering macroscopic formulae for the sound attenuation. Comment: Interesting methodical work, but no essentially new results.In the Appendix a useful Compendium of the Keldysh Diagram Technique can be found.
J. Kötzler, D. A. Garanin, M. Hartl, and L. Jahn
The temperature variation of the relaxation rate \Gamma_w of the domain walls about their transition at T=T* from the Bloch to the linear Bulaevskii-Ginzburg structure has been investigated in the uniaxial ferrimagnet SrFe_12 O_19. The significant suppression of T* below the mean-field approximation (MFA) value along with the speeding up of \Gamma_w (T =< T*), which is much stronger than that predicted by the MFA, reveals the first evidence for critical fluctuations in a domain wall. The speeding up can be almost quantitatively explained by taking two dimensional fluctuations of the transverse magnetization into account.
D. A. Garanin
The physical characteristics of two-dimensional ferro- and antiferromagnets have been calculated in the whole temperature range by an analytical approach based on the expansion in powers of 1/D, where D is the number of spin components. This approach works rather well since it yields exact results for antiferromagnetic susceptibility and specific heat at T=0 already in the first order in 1/D and it is consistent with HTSE at high temperatures. For the quantities singular at T=0, such as ferromagnetic susceptibility and correlation length, the 1/D expansion supports their general-D functional form obtained by Fukugita and Oyanagi. The critical index \eta calculated in the first order in 1/D proves to be temperature dependent: \eta = 2\theta/(\pi D) [\theta =T/T_c^{MFA}, T_c^{MFA} =J_0/D, J_0 is the Fourier component of the exchange interaction].
M. Hartl-Malang, J. Kötzler, and D. A. Garanin
By measuring the linear ac susceptibility of single crystals of SrFe_12 O_19 between 10 Hz and 20 MHz, the domain-wall dynamics has been investigated near the continuous phase transition from Bloch to linear walls at T* \approx 0.99 T_c. There the kinetic coefficient of the wall relaxation L_w has a deep minimum, the essential features of which indicate the presence of strong, two-dimensional Ising-like fluctuations of the order parameter of the Bloch walls. Magnetic fields, applied transverse to the easy c axis, increase L_w, in almost quantitative agreement with predictions of the Landau-Lifshitz-Bloch approach for the mean-field region away from T*. Effects of fluctuations in the critical region of the disordered phase remain unexplained. Comment: The extended version of the previous PRL [J. Kötzler, D. A. Garanin, M. Hartl, and L. Jahn, 1993].
D. A. Garanin
The field-dependent magnetization m(H,T) of 1- and 2-dimensional classical magnets described by the D-component vector model is calculated analytically in the whole range of temperature and magnetic fields with the help of the 1/D expansion. In the 1-st order in 1/D the theory reproduces with a good accuracy the temperature dependence of the zero-field susceptibility of antiferromagnets \chi with the maximum at T =< |J_0|/D (J_0 is the Fourier component of the exchange interaction) and describes for the first time the singular behavior of \chi(H,T) at small temperatures and magnetic fields: \lim_{T\to 0}\lim_{H\to 0} \chi(H,T)=1/(2|J_0|)(1-1/D) and \lim_{H\to 0}\lim_{T\to 0} \chi(H,T)=1/(2|J_0|). Comment: Even for the zero-field case, I was unable to find MC simulation results for the susceptibility of classical Heisenberg antiferromagnets on the square lattice to compare with my analytical results. Shenker and Tobochnik (1980) simulated only the energy. For the susceptibility only quantum MC simulations for S=1/2 are available. Is it not strange?
D. A. Garanin
The self-consistent Gaussian approximation (SCGA) for classical spin systems described by a completely anisotropic D-component vector model is proposed, which takes into account fluctuations of the molecular field and thus is a next step beyond the molecular field approximation. The SCGA is sensitive to the lattice dimension and structure and to the form of spin interactions and yields rather accurate values of the field-dependent magnetization m(H,T) and other thermodynamic functions in the whole plane (H,T) excluding the vicinity of the critical point (0,T_c), where the SCGA breaks down, showing a first-order phase transition. The values of T_c themselves can be determined in the SCGA with an accuracy better than 1% for actual 3-dimensional structures. At low and high temperatures the SCGA recovers the leading terms of the spin-wave theory, the low- and high-temperature series expansions, respectively. The accuracy of the SCGA increases with the increase of the spin dimension D, and in the limit D \to \infty the exact solution for the spherical model is recovered. Comment: At last the comprehensive paper on the SCGA, with numerical calculations and many figures.
D. A. Garanin
The temperature-induced second-order phase transition from Bloch to linear (Ising-like) domain walls in uniaxial ferromagnets is investigated for the model of D-component classical spin vectors in the limit D \to \infty. This exactly soluble model is equivalent to the standard spherical model in the homogeneous case, but deviates from it and is free from unphysical behavior in a general inhomogeneous situation. It is shown that the thermal fluctuations of the transverse magnetization in the wall (the Bloch-wall order parameter) result in the diminishing of the wall transition temperature T_B in comparison to its mean-field value, thus favouring the existence of linear walls. For finite values of T_B an additional anisotropy in the basis plane x,y is required; in purely uniaxial ferromagnets a domain wall behaves like a 2-dimensional system with a continuous spin symmetry and does not order into the Bloch one. Comment: In the domain-wall geometry the "unsolvable" ASM system of equations has the exact solution which can be guessed. In other geometries, as films and surfaces, the numerical solution is needed, in general.
D. A. Garanin
The corrections to the Curie temperature T_c of a ferromagnetic film consisting of N layers are calculated for N >> 1 for the model of D-component classical spin vectors in the limit D \to \infty, which is exactly soluble and close to the spherical model. The present approach accounts, however, for the magnetic anisotropy playing the crucial role in the crossover from 3 to 2 dimensions in magnetic films. In the spatially inhomogeneous case with free boundary conditions the D=\infty model is nonequivalent to the standard spherical one and always leads to the diminishing of T_c(N) relative to the bulk.
D. A. Garanin
The integral relaxation time \tau_{int} of thermoactivating noninteracting single-domain ferromagnetic particles is calculated analytically in the geometry with a magnetic field H applied parallel to the easy axis. It is shown that the drastic deviation of \tau_{int}^{-1} from the lowest eigenvalue of the Fokker-Planck equation \Lambda_1 at low temperatures, starting from some critical value of H, is the consequence of the depletion of the upper potential well. In these conditions the integral relaxation time consists of two competing contributions corresponding to the overbarrier and intrawell relaxation processes. Comment: A remake of a part of my hardly accessible paper D. A. Garanin, V. V. Ishchenko, and L. V. Panina, 1990.A new idea about the depletion of the upper potential well, which explains the numerical findings of Coffey et. al. (1995), is present, however.
D. A. Garanin
The static transverse and longitudinal correlation functions (CF) of a 3-dimensional ferromagnet are calculated for the exactly solvable anisotropic spherical model (ASM) determined as the limit D \to \infty of the classical D-component vector model. The results are nonequivalent to those for the standard spherical model of Berlin and Kac even in the isotropic case. Whereas the transverse CF has the usual Ornstein-Zernike form for small wave vectors, the longitudinal CF shows a nontrivial behavior in the ordered region caused by spin-wave fluctuations. In particular, in the isotropic case below T_c one has S_{zz}(k) \propto 1/k (the result of the spin-wave theory) for k =< \kappa_m \propto T_c-T.
D. A. Garanin
A macroscopic equation of motion for the magnetization of a ferromagnet at elevated temperatures should contain both transverse and longitudinal relaxation terms and interpolate between Landau-Lifshitz equation at low temperatures and the Bloch equation at high temperatures. It is shown that for the classical model where spin-bath interactions are described by stochastic Langevin fields and spin-spin interactions are treated within the mean-field approximation (MFA), such a ``Landau-Lifshitz-Bloch'' (LLB) equation can be derived exactly from the Fokker-Planck equation, if the external conditions change slowly enough. For weakly-anisotropic ferromagnets within the MFA the LLB equation can be written in a macroscopic form based on the free-energy functional interpolating between the Landau free energy near T_c and the ``micromagnetic'' free energy, which neglects changes of the magnetization magnitude |M|, at low temperatures.
D. A. Garanin
The integral relaxation time describing the thermoactivated escape of a uniaxial quantum spin system interacting with a boson bath is calculated analytically in the whole temperature range. For temperatures T much less than the barrier height \Delta U, the level quantization near the top of the barrier and the strong frequency dependence of the one-boson transition probability can lead to the regularly spaced deep minima of the thermoactivation rate as a function of the magnetic field applied along the z axis. Comment: This paper has been rejected from PRL: "It is mathematics, not physics at all". I do not agree, it is not mathematics. What is it then?
D. A. Garanin and E. M. Chudnovsky
The dynamical theory of thermally activated resonant magnetization tunneling in uniaxially anisotropic magnetic molecules such as Mn$_{12}$Ac ($S=10$) is developed. The observed slow dynamics of the system is described by master equations for the populations of spin levels. The latter are obtained by the adiabatic elimination of fast degrees of freedom from the density matrix equation with the help of the perturbation theory developed earlier for the tunneling level splitting [D. A. Garanin, J. Phys. A 24, L61 (1991)]. There exists a temperature range (thermally activated tunneling) where the escape rate follows the Arrhenius law, but has a nonmonotonic dependence on the bias field due to tunneling at the top of the barrier. At lower temperatures this regime crosses over to the non-Arrhenius law (thermally assisted tunneling). The transition between the two regimes can be first or second order, depending on the transverse field, which can be tested in experiments. In both regimes the resonant maxima of the rate occur when spin levels in the two potential wells match at certain field values. In the thermally activated regime at low dissipation each resonance has a multitower self-similar structure with progressively narrowing peaks mounting on top of each other. Comment: The non-continuation of my position in Hamburg was celebrated in New York. The dissipative tunneling is considered here without recourse to the Caldeira-Leggett formalism. The relation to the latter should be cleared up in the future.
E. M. Chudnovsky and D. A. Garanin
We have found a novel feature of the bistable large-spin model described by the Hamiltonian {\cal H} = -D S_z^2 - H_xS_x. The crossover from thermal to quantum regime for the escape rate can be either first (H_x<SD/2) or second (SD/2<H_x<2SD) order, that is, sharp or smooth, depending on the strength of the transverse field. This prediction can be tested experimentally in molecular magnets like Mn_12 Ac. Comment: The main observation has been actually done in our preceding paper. Here the more accurate consideration in the large-spin limit is given. The paper has been written between Rioja and paella, Montserrat and Gaudi in Barcelona, the city which attracts everybody who has seen it at least one time in the life.
D. A. Garanin, X. Martinez Hidalgo, and E. M. Chudnovsky
The escape rate \Gamma of the large-spin model described by the Hamiltonian
\cal H = -D Sz2 - Hz
Sz - Hx Sx
Comment: The extended version of our previous PRL.
D. A. Garanin
The ordinary surface magnetic phase transition is studied for the exactly solvable anisotropic spherical model (ASM), which is the limit D \to \infty of the D-component uniaxially anisotropic classical vector model. The bulk limit of the ASM is similar to that of the spherical model, apart from the role of the anisotropy stabilizing ordering for low lattice dimensions, d =< 2, at finite temperatures. The correlation functions and the energy density profile in the semi-infinite ASM are calculated analytically and numerically for T >= Tc and 1 =< d =< \infty. As the lattice dimensions d=1, 2, 3, and 4 are marginal, the lattice continuous with respect to the d'=d-1 dimensions parallel to the surface is introduced. Preserving the dimension perpendicular to the surface discrete helps to avoid non-physical singularities and facilitates the numerical solution. The results obtained generalize the isotropic-criticality results for 2<d<4 of Bray and Moore [Phys. Rev. Lett. 38, 735 (1977); J. Phys. A 10, 1927 (1977)]. Comment: The work on this problem continued about two years with breaks. This is my the most labour-consuming project. I hope, I will be able to "harvest" some further results now, before the others come to the idea.
D. A. Garanin and Benjamin Canals
Thermodynamic quantities and correlation functions (CFs) of the classical antiferromagnet on the kagome' lattice are studied for the exactly solvable infinite-component spin-vector model, D \to \infty. In this limit, the critical coupling of fluctuations dies out and the critical behavior simplifies, but the effect of would be Goldstone modes preventing ordering at any nonzero temperature is properly accounted for. In contrast to conventional two-dimensional magnets with continuous symmetry showing extended short-range order at distances smaller than the correlation length, r < xic ~ exp(T*/T), correlations in the kagome'-lattice model decay already at the scale of the lattice spacing due to the strong degeneracy of the ground state characterized by a macroscopic number of strongly fluctuating local degrees of freedom. At low temperatures, spin CFs decay as <S0Sr> ~ 1/r2 in the range a0 << r << xic ~ T-1/2, where a0 is the lattice spacing. Analytical results for the principal thermodynamic quantities in our model are in fairly good quantitative agreement with the MC simulations for the classical Heisenberg model, D=3. The neutron scattering cross section has its maxima beyond the first Brillouin zone; at T \to 0 it becomes nonanalytic but does not diverge at any q. D. A. Garanin
Dimensional crossover of ordering in ferromagnetic films with both periodic and free boundary conditions is studied for the exactly solvable uniaxial model of classical D-component spin vectors in the limit D \to \infty. Analytical and numerical solution of the exact equations describing this model shows that for lattice dimensionalities d>4, finite-size corrections to the bulk values of Tc are characterized by the mean-field exponents and anisotropy-dependent amplitudes. For d =< 4, the mean-field behavior is only realized in the region kappacN >> 1, where kappac is the dimensionlesss inverse transverse (with respect to the easy axis) bulk correlation length at Tc and N is the number of layers in the film. In the region kappacN << 1 and the dimensionality range 3 < d =< 4, finite-size corrections are described by the universality class of the isotropic D=\infty model. For d =< 3, magnetic ordering vanishes in the isotropic limit, kappac \to 0, since the film behaves as an object of dimensionality d'=d-1 =< 2 and long-wavelength fluctuations destroy the order. Here the suppression of Tc of a film can be substantial, depending on the competition between the weakening anisotropy and the increasing film thickness. For thick films, Tc becomes small only for very small anisotropy. D. A. Garanin and E. M. Chudnovsky
Phys. Rev. B (submitted August 1998)The quantum-classical transition of the escape rate of the spin model \cal H= -D Sz2 - Hz Sz + B Sx2 is investigated by a perturbative approach with respect to B [D. A. Garanin, J. Phys A 24, L61 (1991)]. The transition is first order for B<Bc(Hz), the boundary line going to zero as Bc/D ~ 1 - Hz/(2SD) in the strongly biased limit. The range of the first-order transition is thus larger than for the model \cal H = -D Sz2 - Hz Sz - Hx Sx studied earlier, where in the strongly biased case Hxc/(2SD) ~ [1-Hz/(2SD)]3/2. The temperature of the quantum-classical transition, T0, behaves linearly in the strongly biased case for both models: T0 ~ 2SD -Hz. D. A. Garanin
Effects of the surface exchange anisotropy on ordering of ferromagnetic films are studied for the exactly solvable classical spin-vector model with D \to \infty components. For small surface anisotropy \eta's << 1 (defined relative to the exchange interaction), the shift of Tc in a film consisting of N >> 1 layers behaves as Tcbulk - Tc(N) ~ (1/N) ln(1/\eta's) in three dimensions. The finite-size-scaling limit Tcbulk - Tc(N) ~ 1/( \eta'1/2 N^2), which is realized for the model with a bulk anisotropy \eta' << 1 in the range N \eta'1/2 >~ 1, never appears for the model with the pure surface anisotropy. Here for N exp(-1/\eta's) >~ 1 in three dimensions, film orders at a temperature above Tcbulk (the surface phase transition). In the semi-infinite geometry, the surface phase transition occurs for whatever small values of \eta's (i.e., the special phase transition corresponds to Tcbulk) in dimensions three and lower. K. Kladko, P. Fulde, and D. A. Garanin
A method for obtaining a systematic expansion of thermodynamic functions of spin systems with large spin S in powers of 1/S is proposed. This method uses the cumulant technique and a coherent-state representation of the partition function Z. The expansion of Z in terms of cumulants yields an effective classical Hamiltonian with temperature-dependent quantum corrections. For the Heisenberg quantum Hamiltonian, they have a non-Heisenberg form. The effective Hamiltonian can be solved by methods familiar for classical systems. D. A. Garanin, E. C. Kennedy, D. S. F. Crothers, and W. T. Coffey
Classical escape rates of uniaxial spin systems are characterized by
a prefactor differing from and much smaller than that of the particle problem,
since the maximum of the spin energy is attained everywhere on the line
of constant latitude: theta=const, 0 =< phi =< 2*pi. If a transverse
field is applied, a saddle point of the energy is formed, and high, moderate,
and low damping regimes (similar to those for particles) appear. Here we
present the first analytical and numerical study of crossovers between
the uniaxial and other regimes for spin systems. It is shown that there
is one HD-Uniaxial crossover, whereas at low damping the uniaxial and LD
regimes are separated by two crossovers.
D. A. Garanin
The integral relaxation time for the difference of the number of particles
in the two potential wells (IRT-N) for double-well classical spin
systems is introduced. For the uniaxial symmetry, it is given by a quadrature.
Unlike the previously introduced integral relaxation time for the magnetization,
the
D. A. Garanin, K. Kladko, and P. Fulde
We propose a method for obtaining effective classical Hamiltonians \cal H for many-body quantum spin systems with large spins. This method uses the coherent-state representation of the partition function Z and the cumulant expansion in powers of 1/S. For the quantum Hamiltonian \hat H of a Heisenberg form, the 1/S corrections in \cal H have a non-Heisenberg many-spin form. The effective Hamiltonian \cal H can be treated by methods familiar for classical systems. The non-Heisenberg terms in \cal H may be responsible for such effects as spin-Peierls transition and uplifting of the classical degeneracy by quantum fluctuations. D. Hinzke, U. Nowak, and D. A. Garanin
We simulated the field-dependent magnetization m(H,T) and the uniform susceptibility chi(H,T) of classical Heisenberg antiferromagnets in the chain and square-lattice geometry using Monte Carlo methods. The results confirm the singular behavior of chi(H,T) at small T,H: limT \to 0 limH \to 0 chi(H,T)=1/(2J0)(1-1/D) and limH \to 0 limT \to 0 chi(H,T)=1/(2J0), where D=3 is the number of spin components, J0=zJ, and z is the number of nearest neighbors. A good agreement is achieved in a wide range of temperatures T and magnetic fields H with the first-order 1/D expansion results [D. A. Garanin, J. Stat. Phys. 83, 907 (1996)].
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