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Spin waves in rings of classical magnetic dipoles
Authors:
H. -J. Schmidt,
C. Schröder,
M. Luban
Abstract:
We theoretically and numerically investigate spin waves that occur in systems of classical magnetic dipoles that are arranged at the vertices of a regular polygon and interact solely via their magnetic fields. There are certain limiting cases that can be analyzed in detail. One case is that of spin waves as infinitesimal excitations from the system's ground state, where the dispersion relation can…
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We theoretically and numerically investigate spin waves that occur in systems of classical magnetic dipoles that are arranged at the vertices of a regular polygon and interact solely via their magnetic fields. There are certain limiting cases that can be analyzed in detail. One case is that of spin waves as infinitesimal excitations from the system's ground state, where the dispersion relation can be determined analytically. The frequencies of these infinitesimal spin waves are compared with the peaks of the Fourier transform of the thermal expectation value of the autocorrelation function calculated by Monte Carlo simulations. In the special case of vanishing wave number an exact solution of the equations of motion is possible describing synchronized oscillations with finite amplitudes. Finally, the limiting case of a dipole chain with $N\longrightarrow \infty$ is investigated and completely solved.
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Submitted 31 October, 2016; v1 submitted 23 September, 2016;
originally announced September 2016.
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Dynamics and thermodynamics of a pair of interacting magnetic dipoles
Authors:
Heinz-Jürgen Schmidt,
Christian Schröder,
Eva Hägele,
Marshall Luban
Abstract:
We consider the dynamics and thermodynamics of a pair of magnetic dipoles interacting via their magnetic fields. We consider only the "spin" degrees of freedom; the dipoles are fixed in space. With this restriction it is possible to provide the general solution of the equations of motion in analytical form. Thermodynamic quantities, such as the specific heat and the zero field susceptibility are c…
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We consider the dynamics and thermodynamics of a pair of magnetic dipoles interacting via their magnetic fields. We consider only the "spin" degrees of freedom; the dipoles are fixed in space. With this restriction it is possible to provide the general solution of the equations of motion in analytical form. Thermodynamic quantities, such as the specific heat and the zero field susceptibility are calculated by combining low temperature asymptotic series and a complete high temperature expansion. The thermal expectation value of the autocorrelation function is determined for the low temperature regime including terms linear in $T$. Furthermore, we compare our analytical results with numerical calculations based on Monte Carlo simulations.
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Submitted 14 October, 2014;
originally announced October 2014.
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Exchange randomness and spin dynamics in the frustrated magnetic Keplerate {W72V30}
Authors:
J. Schnack,
A. M. Todea,
A. Müller,
H. Nojiri,
S. Yeninas,
Y. Furukawa,
R. Prozorov,
M. Luban
Abstract:
The magnetic properties and spin dynamics of the spin frustrated polyoxometalate {W72V30}, where 30 V^{4+} ions (s = 1/2) occupy the sites of an icosidodecahedron, have been investigated by low temperature magnetization, magnetic susceptibility, proton and vanadium nuclear magnetic resonance, and theoretical studies. The field-dependent magnetization at 0.5 K increases monotonically up to 50 T wit…
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The magnetic properties and spin dynamics of the spin frustrated polyoxometalate {W72V30}, where 30 V^{4+} ions (s = 1/2) occupy the sites of an icosidodecahedron, have been investigated by low temperature magnetization, magnetic susceptibility, proton and vanadium nuclear magnetic resonance, and theoretical studies. The field-dependent magnetization at 0.5 K increases monotonically up to 50 T without any sign of staircase behavior. This low-temperature behavior cannot be explained by a Heisenberg model based on a single value of the nearest-neighbor exchange coupling. We analyze this behavior upon assuming a rather broad distribution of nearest-neighbor exchange interactions. Slow spin dynamics of {W72V30} at low temperatures is observed from the magnetic field and temperature dependence of nuclear spin-lattice relaxation rate 1/T_1 measurements.
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Submitted 9 April, 2013;
originally announced April 2013.
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Determination of exchange energies in the sawtooth spin ring {Mo75V20} by ESR
Authors:
Yugo Oshima,
Hiroyuki Nojiri,
Jürgen Schnack,
Paul Kögerler,
Marshall Luban
Abstract:
The magnetism of the polyoxometalate cluster {Mo75V20}, containing a sawtooth ring of 10 corner-sharing triangles located on the equator of the barrel-shaped molecule, has remained debatable since it is masked by contributions from impurities as well as temperature-independent paramagnetism. In this article we demonstrate the usefulness of ESR measurements since the temperature dependence of the E…
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The magnetism of the polyoxometalate cluster {Mo75V20}, containing a sawtooth ring of 10 corner-sharing triangles located on the equator of the barrel-shaped molecule, has remained debatable since it is masked by contributions from impurities as well as temperature-independent paramagnetism. In this article we demonstrate the usefulness of ESR measurements since the temperature dependence of the ESR intensity can discriminate between impurity and molecular contributions. We determine the exchange parameters and therefore also the low-lying spectrum of {Mo75V20}, especially the low-lying singlet states which so far have been probed solely by specific heat measurements.
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Submitted 19 July, 2011;
originally announced July 2011.
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Level crossings and zero-field splitting in the \{Cr$_8$\}-cubane spin-cluster by inelastic neutron scattering and magnetization studies
Authors:
D. Vaknin,
V. O. Garlea,
F. Demmel,
E. Mamontov,
H. Nojiri,
C. Martin,
I. Chiorescu,
Y. Qiu,
P. Koegerler,
J. Fielden,
L. Engelhardt,
C. Rainey,
M. Luban
Abstract:
Inelastic neutron scattering in variable magnetic field and high-field magnetization measurements, at the milikelvin temperature range, were performed to gain insight into the low-energy magnetic excitation spectrum and the field-induced level crossings in the molecular spin cluster \{Cr$_8$\}-cubane. These complementary techniques provide consistent estimates of the lowest level-crossing field. T…
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Inelastic neutron scattering in variable magnetic field and high-field magnetization measurements, at the milikelvin temperature range, were performed to gain insight into the low-energy magnetic excitation spectrum and the field-induced level crossings in the molecular spin cluster \{Cr$_8$\}-cubane. These complementary techniques provide consistent estimates of the lowest level-crossing field. The overall features of the experimental data are explained using an isotropic Heisenberg model, based on three distinct exchange interactions linking the eight Cr$^{\text{III}}$ paramagnetic centers (spins $s = 3/2$), that is supplemented with a relatively large molecular magnetic anisotropy term for the lowest S=1 multiplet. It is noted that the existence of the anisotropy is clearly evident from the magnetic field dependence of the excitations in the INS measurements, while the magnetization measurements are not sensitive to its effects.
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Submitted 1 November, 2010;
originally announced November 2010.
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Theory of severe slowdown in the relaxation of rings and clusters with antiferromagnetic interactions
Authors:
Ioannis Rousochatzakis,
Andreas Laeuchli,
Ferdinando Borsa,
Marshall Luban
Abstract:
We show that in the severe slowing down temperature regime the relaxation of antiferromagnetic rings and similar magnetic nanoclusters is governed by the quasi-continuum portion of their quadrupolar fluctuation spectrum and not by the lowest excitation lines. This is at the heart of the intriguing near-universal power-law temperature dependence of the electronic correlation frequency $ω_c$ with…
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We show that in the severe slowing down temperature regime the relaxation of antiferromagnetic rings and similar magnetic nanoclusters is governed by the quasi-continuum portion of their quadrupolar fluctuation spectrum and not by the lowest excitation lines. This is at the heart of the intriguing near-universal power-law temperature dependence of the electronic correlation frequency $ω_c$ with an exponent close to 4. The onset of this behavior is defined by an energy scale which is fixed by the lowest spin gap $Δ_0$. This explains why experimental curves of $ω_c$ for different cluster sizes and spins nearly coincide when $T$ is rescaled by $Δ_0$.
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Submitted 31 January, 2009; v1 submitted 3 November, 2008;
originally announced November 2008.
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Modulated spin waves and robust quasi-solitons in classical Heisenberg rings
Authors:
Heinz-Jürgen Schmidt,
Christian Schröder,
Marshall Luban
Abstract:
We investigate the dynamical behavior of finite rings of classical spin vectors interacting via nearest-neighbor isotropic exchange in an external magnetic field. Our approach is to utilize the solutions of a continuum version of the discrete spin equations of motion (EOM) which we derive by assuming continuous modulations of spin wave solutions of the EOM for discrete spins. This continuous EOM…
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We investigate the dynamical behavior of finite rings of classical spin vectors interacting via nearest-neighbor isotropic exchange in an external magnetic field. Our approach is to utilize the solutions of a continuum version of the discrete spin equations of motion (EOM) which we derive by assuming continuous modulations of spin wave solutions of the EOM for discrete spins. This continuous EOM reduces to the Landau-Lifshitz equation in a particular limiting regime. The usefulness of the continuum EOM is demonstrated by the fact that the time-evolved numerical solutions of the discrete spin EOM closely track the corresponding time-evolved solutions of the continuum equation. Of special interest, our continuum EOM possesses soliton solutions, and we find that these characteristics are also exhibited by the corresponding solutions of the discrete EOM. The robustness of solitons is demonstrated by considering cases where initial states are truncated versions of soliton states and by numerical simulations of the discrete EOM equations when the spins are coupled to a heat bath at finite temperatures.
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Submitted 28 January, 2008;
originally announced January 2008.
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Low temperature magnetization and the excitation spectrum of antiferromagnetic Heisenberg spin rings
Authors:
Larry Engelhardt,
Marshall Luban
Abstract:
Accurate results are obtained for the low temperature magnetization versus magnetic field of Heisenberg spin rings consisting of an even number N of intrinsic spins s = 1/2, 1, 3/2, 2, 5/2, 3, 7/2 with nearest-neighbor antiferromagnetic (AF) exchange by employing a numerically exact quantum Monte Carlo method. A straightforward analysis of this data, in particular the values of the level-crossin…
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Accurate results are obtained for the low temperature magnetization versus magnetic field of Heisenberg spin rings consisting of an even number N of intrinsic spins s = 1/2, 1, 3/2, 2, 5/2, 3, 7/2 with nearest-neighbor antiferromagnetic (AF) exchange by employing a numerically exact quantum Monte Carlo method. A straightforward analysis of this data, in particular the values of the level-crossing fields, provides accurate results for the lowest energy eigenvalue E(N,S,s) for each value of the total spin quantum number S. In particular, the results are substantially more accurate than those provided by the rotational band approximation. For s <= 5/2, data are presented for all even N <= 20, which are particularly relevant for experiments on finite magnetic rings. Furthermore, we find that for s > 1 the dependence of E(N,S,s) on s can be described by a scaling relation, and this relation is shown to hold well for ring sizes up to N = 80 for all intrinsic spins in the range 3/2 <= s <= 7/2. Considering ring sizes in the interval 8 <= N <= 50, we find that the energy gap between the ground state and the first excited state approaches zero proportional to 1/N^a, where a = 0.76 for s = 3/2 and a = 0.84 for s = 5/2. Finally, we demonstrate the usefulness of our present results for E(N,S,s) by examining the Fe12 ring-type magnetic molecule, leading to a new, more accurate estimate of the exchange constant for this system than has been obtained heretofore.
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Submitted 1 December, 2005;
originally announced December 2005.
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Observation of field-dependent magnetic parameters in the magnetic molecule {Ni4Mo12}
Authors:
J. Schnack,
M. Brueger,
M. Luban,
P. Koegerler,
E. Morosan,
R. Fuchs,
R. Modler,
Hiroyuki Nojiri,
Ram C. Rai,
Jinbo Cao,
J. L. Musfeldt,
Xing Wei
Abstract:
We investigate the bulk magnetic, electron paramagnetic resonance, and magneto-optical properties of {Ni4Mo12}, a magnetic molecule with antiferromagnetically coupled tetrahedral {Ni4Mo12} in a diamagnetic molybdenum matrix. The low-temperature magnetization exhibits steps at irregular field intervals, a result that cannot be explained using a Heisenberg model even if it is augmented by magnetic…
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We investigate the bulk magnetic, electron paramagnetic resonance, and magneto-optical properties of {Ni4Mo12}, a magnetic molecule with antiferromagnetically coupled tetrahedral {Ni4Mo12} in a diamagnetic molybdenum matrix. The low-temperature magnetization exhibits steps at irregular field intervals, a result that cannot be explained using a Heisenberg model even if it is augmented by magnetic anisotropy and biquadratic terms. Allowing the exchange and anisotropy parameters to depend on the magnetic field provides the best fit to our data, suggesting that the molecular structure (and thus the interactions between spins) may be changing with applied magnetic field.
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Submitted 14 December, 2005; v1 submitted 19 September, 2005;
originally announced September 2005.
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Magnetic excitations in {Mo72Fe30}
Authors:
V. O. Garlea,
S. E. Nagler,
J. L. Zarestky,
C. Stassis,
D. Vaknin,
P. Kogerler,
D. F. McMorrow,
C. Niedermayer,
D. A. Tennant,
B. Lake,
Y. Qiu,
M. Exler,
J. Schnack,
M. Luban
Abstract:
We report cold-neutron inelastic neutron scattering measurements on deuterated samples of the giant polyoxomolybdate magnetic molecule {Mo72Fe30}. The 30 s = 5/2 Fe+3 ions occupy the vertices of an icosidodecahedron, and interact via antiferromagnetic nearest neighbor coupling. The measurements reveal a band of magnetic excitations near E ~ 0.6 meV. The spectrum broadens and shifts to lower ener…
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We report cold-neutron inelastic neutron scattering measurements on deuterated samples of the giant polyoxomolybdate magnetic molecule {Mo72Fe30}. The 30 s = 5/2 Fe+3 ions occupy the vertices of an icosidodecahedron, and interact via antiferromagnetic nearest neighbor coupling. The measurements reveal a band of magnetic excitations near E ~ 0.6 meV. The spectrum broadens and shifts to lower energy as the temperature is increased, and also is strongly affected by magnetic fields. The results can be interpreted within the context of an effective three-sublattice spin Hamiltonian.
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Submitted 3 May, 2005;
originally announced May 2005.
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Master Equations for pulsed magnetic fields: Application to magnetic molecules
Authors:
Ioannis Rousochatzakis,
Marshall Luban
Abstract:
We extend spin-lattice relaxation theory to incorporate the use of pulsed magnetic fields for probing the hysteresis effects and magnetization steps and plateaus exhibited, at low temperatures, by the dynamical magnetization of magnetic molecules. The main assumption made is that the lattice degrees of freedom equilibrate in times much shorter than both the experimental time scale (determined by…
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We extend spin-lattice relaxation theory to incorporate the use of pulsed magnetic fields for probing the hysteresis effects and magnetization steps and plateaus exhibited, at low temperatures, by the dynamical magnetization of magnetic molecules. The main assumption made is that the lattice degrees of freedom equilibrate in times much shorter than both the experimental time scale (determined by the sweep rate) and the typical spin-lattice relaxation time. We first consider the isotropic case (a magnetic molecule with a ground state of spin $S$ well separated from the excited levels and also the general isotropic Heisenberg Hamiltonian where all energy levels are relevant) and then we include small off-diagonal terms in the spin Hamiltonian to take into account the Landau-Zener-Stückelberg (LZS) effect. In the first case, and for an $S=1/2$ magnetic molecule we arrive at the generalized Bloch equation recently used for the magnetic molecule \{V$_6$\} in Phys. Rev. Lett. 94, 147204 (2005). An analogous equation is derived for the magnetization, at low temperatures, of antiferromagnetic ring systems. The LZS effect is discussed for magnetic molecules with a low spin ground state, for which we arrive at a very convenient set of equations that take into account the combined effects of LZS and thermal transitions. In particular, these equations explain the deviation from exact magnetization reversal at $B\approx 0$ observed in \{V$_6$\}. They also account for the small magnetization plateaus (``magnetic Foehn effect''), following the LZS steps, that have been observed in several magnetic molecules. Finally, we discuss the role of the Phonon Bottleneck effect at low temperatures and specifically we indicate how this can give rise to a pronounced Foehn effect.
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Submitted 21 August, 2005; v1 submitted 30 March, 2005;
originally announced March 2005.
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Hysteresis loops and adiabatic Landau-Zener-Stückelberg transitions in the magnetic molecule V$_6$
Authors:
I. Rousochatzakis,
Y. Ajiro,
H. Mitamura,
P. Kögerler,
M. Luban
Abstract:
We have observed hysteresis loops and abrupt magnetization steps in the magnetic molecule V$_6$, where each molecule comprises a pair of identical spin triangles, in the temperature range 1-5 K for external magnetic fields $B$ with sweep rates of several Tesla/ms executing a variety of closed cycles. The hysteresis loops are accurately reproduced using a generalization of the Bloch equation base…
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We have observed hysteresis loops and abrupt magnetization steps in the magnetic molecule V$_6$, where each molecule comprises a pair of identical spin triangles, in the temperature range 1-5 K for external magnetic fields $B$ with sweep rates of several Tesla/ms executing a variety of closed cycles. The hysteresis loops are accurately reproduced using a generalization of the Bloch equation based on direct one-phonon transitions between the instantaneous Zeeman-split levels of the ground state (an $S=1/2$ doublet) of each spin triangle. The magnetization steps occur for $B\approx 0$ and they are explained in terms of adiabatic Landau-Zener-Stückelberg transitions between the lowest magnetic energy levels as modified by inter-triangle anisotropic exchange of order 0.4 K.
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Submitted 3 February, 2005;
originally announced February 2005.
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Metamagnetic phase transition of the antiferromagnetic Heisenberg icosahedron
Authors:
Christian Schröder,
Heinz-Jürgen Schmidt,
Jürgen Schnack,
Marshall Luban
Abstract:
The observation of hysteresis effects in single molecule magnets like Mn$_{12}$-acetate has initiated ideas of future applications in storage technology. The appearance of a hysteresis loop in such compounds is an outcome of their magnetic anisotropy. In this Letter we report that magnetic hysteresis occurs in a spin system without any anisotropy, specifically, where spins mounted on the vertice…
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The observation of hysteresis effects in single molecule magnets like Mn$_{12}$-acetate has initiated ideas of future applications in storage technology. The appearance of a hysteresis loop in such compounds is an outcome of their magnetic anisotropy. In this Letter we report that magnetic hysteresis occurs in a spin system without any anisotropy, specifically, where spins mounted on the vertices of an icosahedron are coupled by antiferromagnetic isotropic nearest-neighbor Heisenberg interaction giving rise to geometric frustration. At T=0 this system undergoes a first order metamagnetic phase transition at a critical field $\Bcrit$ between two distinct families of ground state configurations. The metastable phase of the system is characterized by a temperature and field dependent survival probability distribution.
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Submitted 24 January, 2005;
originally announced January 2005.
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Competing Spin Phases in Geometrically Frustrated Magnetic Molecules
Authors:
C. Schroeder,
H. Nojiri,
J. Schnack,
P. Hage,
M. Luban,
P. Koegerler
Abstract:
We have found a class of zero-dimensional geometrically frustrated Heisenberg spin systems exhibiting anomalous behavior in an external magnetic field B similar to that occuring in geometrically frustrated planar antiferromagnetic lattices. Our calculations for both the classical and quantum isotropic Heisenberg models show the emergence of a pronounced minimum in the differential susceptibility…
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We have found a class of zero-dimensional geometrically frustrated Heisenberg spin systems exhibiting anomalous behavior in an external magnetic field B similar to that occuring in geometrically frustrated planar antiferromagnetic lattices. Our calculations for both the classical and quantum isotropic Heisenberg models show the emergence of a pronounced minimum in the differential susceptibility dM/dB at B_sat/3 as the temperature T is raised from 0K for structures based on corner-sharing triangles, specifically the octahedron, cuboctahedron, and icosidodecahedron. Low temperature measurements of magnetization M versus B for the giant Keplerate magnetic molecule Mo72Fe30 (Fe(3+) ions with spin s=5/2 on the 30 vertices of an icosidodecahedron) are consistent with our calculational results. The minimum in dM/dB is due to the fact that for low temperatures when B is approx. B_sat/3 there exist two competing families of spin configurations of which one behaves magnetically 'stiff' leading to a reduction of the susceptibility.
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Submitted 13 December, 2004; v1 submitted 18 May, 2004;
originally announced May 2004.
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Classical ground states of symmetrical Heisenberg spin systems
Authors:
Heinz-Juergen Schmidt,
Marshall Luban
Abstract:
We investigate the ground states of classical Heisenberg spin systems which have point group symmetry. Examples are the regular polygons (spin rings) and the seven quasi-regular polyhedra including the five Platonic solids. For these examples, ground states with special properties, e.g. coplanarity or symmetry, can be completely enumerated using group-theoretical methods. For systems having copl…
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We investigate the ground states of classical Heisenberg spin systems which have point group symmetry. Examples are the regular polygons (spin rings) and the seven quasi-regular polyhedra including the five Platonic solids. For these examples, ground states with special properties, e.g. coplanarity or symmetry, can be completely enumerated using group-theoretical methods. For systems having coplanar (anti-) ground states with vanishing total spin we also calculate the smallest and largest energies of all states having a given total spin S. We find that these extremal energies depend quadratically on S and prove that, under certain assumptions, this happens only for systems with coplanar S=0 ground states. For general systems the corresponding parabolas represent lower and upper bounds for the energy values. This provides strong support and clarifies the conditions for the so-called rotational band structure hypothesis which has been numerically established for many quantum spin systems.
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Submitted 11 September, 2002; v1 submitted 6 September, 2002;
originally announced September 2002.
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Approximating parabolas as natural bounds of Heisenberg spectra: Reply on the comment of O. Waldmann
Authors:
H. -J. Schmidt,
J. Schnack,
Marshall Luban
Abstract:
O. Waldmann has shown that some spin systems, which fulfill the condition of a weakly homogeneous coupling matrix, have a spectrum whose minimal or maximal energies are rather poorly approximated by a quadratic dependence on the total spin quantum number. We comment on this observation and provide the new argument that, under certain conditions, the approximating parabolas appear as natural boun…
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O. Waldmann has shown that some spin systems, which fulfill the condition of a weakly homogeneous coupling matrix, have a spectrum whose minimal or maximal energies are rather poorly approximated by a quadratic dependence on the total spin quantum number. We comment on this observation and provide the new argument that, under certain conditions, the approximating parabolas appear as natural bounds of the spectrum generated by spin coherent states.
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Submitted 30 November, 2001;
originally announced November 2001.
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Quantum rotational band model for the Heisenberg molecular magnet Mo72Fe30
Authors:
J. Schnack,
M. Luban,
R. Modler
Abstract:
We derive the low temperature properties of the molecular magnet Mo72Fe30, where 30 Fe(3+) paramagnetic ions occupy the sites of an icosidodecahedron and interact via isotropic nearest-neighbour antiferromagnetic Heisenberg exchange. The key idea of our model (J.S. & M.L.) is that the low-lying excitations form a sequence of rotational bands, i.e., for each such band the excitation energies depe…
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We derive the low temperature properties of the molecular magnet Mo72Fe30, where 30 Fe(3+) paramagnetic ions occupy the sites of an icosidodecahedron and interact via isotropic nearest-neighbour antiferromagnetic Heisenberg exchange. The key idea of our model (J.S. & M.L.) is that the low-lying excitations form a sequence of rotational bands, i.e., for each such band the excitation energies depend quadratically on the total spin quantum number. For temperatures below 50 mK we predict that the magnetisation is described by a staircase with 75 equidistant steps as the magnetic field is increased up to a critical value and saturated for higher fields. For higher temperatures thermal broadening effects wash out the staircase and yield a linear ramp below the critical field, and this has been confirmed by our measurements (R.M.). We demonstrate that the lowest two rotational bands are separated by an energy gap of 0.7 meV, and this could be tested by EPR and inelastic neutron scattering measurements. We also predict the occurrence of resonances at temperatures below 0.1 K in the proton NMR spin-lattice relaxation rate associated with level crossings. As rotational bands characterize the spectra of many magnetic molecules our method opens a new road towards a description of their low-temperature behaviour which is not otherwise accessible.
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Submitted 8 October, 2001;
originally announced October 2001.
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Heisenberg exchange parameters of molecular magnets from the high-temperature susceptibility expansion
Authors:
H. -J. Schmidt,
J. Schnack,
Marshall Luban
Abstract:
We provide exact analytical expressions for the magnetic susceptibility function in the high temperature expansion for finite Heisenberg spin systems with an arbitrary coupling matrix, arbitrary single-spin quantum number, and arbitrary number of spins. The results can be used to determine unknown exchange parameters from zero-field magnetic susceptibility measurements without diagonalizing the…
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We provide exact analytical expressions for the magnetic susceptibility function in the high temperature expansion for finite Heisenberg spin systems with an arbitrary coupling matrix, arbitrary single-spin quantum number, and arbitrary number of spins. The results can be used to determine unknown exchange parameters from zero-field magnetic susceptibility measurements without diagonalizing the system Hamiltonian. We demonstrate the possibility of reconstructing the exchange parameters from simulated data for two specific model systems. We examine the accuracy and stability of the proposed method.
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Submitted 5 July, 2001;
originally announced July 2001.
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Zero-field Time Correlation Functions of Four Classical Heisenberg Spins on a Ring
Authors:
Richard A. Klemm,
Marshall Luban
Abstract:
A model relevant for the study of certain molecular magnets is the ring of N=4 classical spins with equal near-neighbor isotropic Heisenberg exchange interactions. Assuming classical Heisenberg spin dynamics, we solve explicitly for the time evolution of each of the spins. Exact triple integral representations are derived for the auto, near-neighbor, and next-nearest-neighbor time correlation fu…
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A model relevant for the study of certain molecular magnets is the ring of N=4 classical spins with equal near-neighbor isotropic Heisenberg exchange interactions. Assuming classical Heisenberg spin dynamics, we solve explicitly for the time evolution of each of the spins. Exact triple integral representations are derived for the auto, near-neighbor, and next-nearest-neighbor time correlation functions for any temperature. At infinite temperature, the correlation functions are reduced to quadrature. We then evaluate the Fourier transforms of these functions in closed form, which are double integrals. At low temperatures, the Fourier transform functions explicitly demonstrate the presence of magnons. Our exact results for the infinite temperature correlation functions in the long-time asymptotic limit differ qualitatively from those obtained assuming diffusive spin dynamics. Whether such explicitly non-hydrodynamic behavior would be maintained for large-N rings is discussed.
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Submitted 3 May, 2001;
originally announced May 2001.
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Bounding and approximating parabolas for the spectrum of Heisenberg spin systems
Authors:
H. -J. Schmidt,
J. Schnack,
Marshall Luban
Abstract:
We prove that for a wide class of quantum spin systems with isotropic Heisenberg coupling the energy eigenvalues which belong to a total spin quantum number S have upper and lower bounds depending at most quadratically on S. The only assumption adopted is that the mean coupling strength of any spin w.r.t. its neighbours is constant for all N spins. The coefficients of the bounding parabolas are…
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We prove that for a wide class of quantum spin systems with isotropic Heisenberg coupling the energy eigenvalues which belong to a total spin quantum number S have upper and lower bounds depending at most quadratically on S. The only assumption adopted is that the mean coupling strength of any spin w.r.t. its neighbours is constant for all N spins. The coefficients of the bounding parabolas are given in terms of special eigenvalues of the N times N coupling matrix which are usually easily evaluated. In addition we show that the bounding parabolas, if properly shifted, provide very good approximations of the true boundaries of the spectrum. We present numerical examples of frustrated rings, a cube, and an icosahedron.
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Submitted 16 January, 2001;
originally announced January 2001.
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Continuous families of isospectral Heisenberg spin systems and the limits of inference from measurements
Authors:
Heinz-Juergen Schmidt,
Marshall Luban
Abstract:
We investigate classes of quantum Heisenberg spin systems which have different coupling constants but the same energy spectrum and hence the same thermodynamical properties. To this end we define various types of isospectrality and establish conditions for their occurence. The triangle and the tetrahedron whose vertices are occupied by spins 1/2 are investigated in some detail. The problem is al…
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We investigate classes of quantum Heisenberg spin systems which have different coupling constants but the same energy spectrum and hence the same thermodynamical properties. To this end we define various types of isospectrality and establish conditions for their occurence. The triangle and the tetrahedron whose vertices are occupied by spins 1/2 are investigated in some detail. The problem is also of practical interest since isospectrality presents an obstacle to the experimental determination of the coupling constants of small interacting spin systems such as magnetic molecules.
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Submitted 13 December, 2000;
originally announced December 2000.
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Rotational modes in molecular magnets with antiferromagnetic Heisenberg exchange
Authors:
J. Schnack,
Marshall Luban
Abstract:
In an effort to understand the low temperature behavior of recently synthesized molecular magnets we present numerical evidence for the existence of a rotational band in systems of quantum spins interacting with nearest-neighbor antiferromagnetic Heisenberg exchange. While this result has previously been noted for ring arrays with an even number of spin sites, we find that it also applies for ri…
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In an effort to understand the low temperature behavior of recently synthesized molecular magnets we present numerical evidence for the existence of a rotational band in systems of quantum spins interacting with nearest-neighbor antiferromagnetic Heisenberg exchange. While this result has previously been noted for ring arrays with an even number of spin sites, we find that it also applies for rings with an odd number of sites as well as for all of the polytope configurations we have investigated (tetrahedron, cube, octahedron, icosahedron, triangular prism, and axially truncated icosahedron). It is demonstrated how the rotational band levels can in many cases be accurately predicted using the underlying sublattice structure of the spin array. We illustrate how the characteristics of the rotational band can provide valuable estimates for the low temperature magnetic susceptibility.
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Submitted 21 September, 2000;
originally announced September 2000.
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Transition from quantum to classical Heisenberg trimers: Thermodynamics and time correlation functions
Authors:
D. Mentrup,
H. -J. Schmidt,
J. Schnack,
Marshall Luban
Abstract:
We focus on the transition from quantum to classical behavior in thermodynamic functions and time correlation functions of a system consisting of three identical quantum spins s that interact via isotropic Heisenberg exchange. The partition function and the zero-field magnetic susceptibility are readily shown to adopt their classical forms with increasing s. The behavior of the spin autocorrelat…
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We focus on the transition from quantum to classical behavior in thermodynamic functions and time correlation functions of a system consisting of three identical quantum spins s that interact via isotropic Heisenberg exchange. The partition function and the zero-field magnetic susceptibility are readily shown to adopt their classical forms with increasing s. The behavior of the spin autocorrelation function (ACF) is more subtle. Unlike the classical Heisenberg trimer where the ACF approaches a unique non-zero limit for long times, for the quantum trimer the ACF is periodic in time. We present exact values of the time average over one period of the quantum trimer for s less or equal 7 and for infinite temperature. These averages differ from the long-time limit, (9/40)\ln3+(7/30), of the corresponding classical trimer by terms of order 1/(s*s). However, upon applying the Levin u-sequence acceleration method to our quantum results we can reproduce the classical value to six significant figures.
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Submitted 7 October, 1999;
originally announced October 1999.
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Spin dynamics of quantum and classical Heisenberg dimers
Authors:
D. Mentrup,
J. Schnack,
Marshall Luban
Abstract:
Analytical solutions for the time-dependent autocorrelation function of the classical and quantum mechanical spin dimer with arbitrary spin are presented and compared. For large spin quantum numbers or high temperature the classical and the quantum dimer become more and more similar, yet with the major difference that the quantum autocorrelation function is periodic in time whereas the classical…
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Analytical solutions for the time-dependent autocorrelation function of the classical and quantum mechanical spin dimer with arbitrary spin are presented and compared. For large spin quantum numbers or high temperature the classical and the quantum dimer become more and more similar, yet with the major difference that the quantum autocorrelation function is periodic in time whereas the classical is not.
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Submitted 13 April, 1999;
originally announced April 1999.