Theory of ground states for classical Heisenberg spin systems V

Schmidt, Heinz-Jürgen; Florek, Wojciech

doi:10.48550/arxiv.2002.12705

Cited by 3 publications

(6 citation statements)

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“…We will analytically calculate the saturation susceptibility and check it by numerical calculations. This example has also be considered in [22] with a general ferromagnetic bond strength. Its homogeneously gauged J-matrix has the form…”

Section: Almost Regular Cubementioning

confidence: 99%

“…Here we have adopted the notation Root [p(x), n] for the nth root of the polynomial p(x) analogous to the similar MATHEMATICA command. In passing we note the special form of ξ with alternating components due to the reflectional symmetry of the almost regular cube, see [22] for details. We conclude…”

Section: B Isosceles Trianglementioning

confidence: 99%

“…The isosceles triangle, Section VI A, has coplanar ground states for all values of B that can be analytically calculated and hence directly compared with the corresponding perturbation series results. The almost regular cube, Section VI C, also considered in [22] for other reasons, has coplanar ground states close to the saturation field. Its saturation susceptibility can be determined as the root of a third order equation and checked…”

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confidence: 99%

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Theory of ground states for classical Heisenberg spin systems VI

Schmidt¹,

Florek²

2020

Preprint

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We formulate part VI of a rigorous theory of ground states for classical, finite, Heisenberg spin systems. After recapitulating the central results of the parts I -V previously published we consider a magnetic field and analytically calculate the susceptibility at the saturation point. To this end we have to distinguish between parabolic and non-parabolic systems, and for the latter ones between two-and three-dimensional ground states. These results are checked for a couple of examples. I. INTRODUCTIONThe ground state of a spin system and its energy represent valuable information, e. g., about its low temperature behaviour. Most research approaches deal with quantum systems, but also the classical limit has found some interest and applications, see, e. g., [1] -[16]. For classical Heisenberg systems, including Hamiltonians with a Zeeman term due to an external magnetic field, a rigorous theory has been recently established [17] -[22] that yields, in principle, all ground states. However, two restrictions must be made: (1) the dimension m of the ground states found by the theory is per se not confined to the physical case of m ≤ 3, and (2) analytical solutions will only be possible for special couplings or small numbers N of spins. A first application of this theory to frustrated systems with wheel geometry has been given in [23] and [24].The purpose of the present paper is to give a concise review of the central results of [17] -[22] and to apply the methods outlined there to describe the magnetic behaviour of a spin system subject to a magnetic field close to the saturation point. For each field larger than the saturation field B sat all spins will point into the direction of the field (or opposite the direction, depending on the sign of the Zeeman term), but for values of B slightly below B sat the spins will form an "umbrella" with infinitesimal spread, see, e. g., Figure 9. It is an obvious goal to calculate that umbrella in lowest order w. r. t. some sensible expansion parameter t. Another physically interesting property in this connection would be the saturation susceptibility χ 0 , that is the limit of the susceptibility for B ↑ B sat . Note that numerical calculations close to the saturation point are difficult and do not yield precise estimates for the spin system's behaviour in lowest order.In order to investigate the reaction of the spin system to magnetic fields near the saturation point, some case distinctions prove to be necessary. According to the general theory outlined in [17] -[22] the various ground states can be obtained by means of linear combinations of eigenvectors of a so-called dressed J-matrix corresponding to its minimal eigenvalue. The first case distinction refers to whether the ground state at the saturation point is essentially unique (non-parabolic case) or not (parabolic case). In the parabolic case the minimal energy E will be a quadratic function of the magnetization M (hence the name) and consequently the susceptibility will be constant for a certain domain. In the non-parabo...

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Section: Almost Regular Cubementioning

confidence: 99%

Section: B Isosceles Trianglementioning

confidence: 99%

mentioning

confidence: 99%

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Theory of ground states for classical Heisenberg spin systems VI

Schmidt¹,

Florek²

2020

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“…Our approach will be based on the 'Lagrange-variety theory' of classical ground states given in [6,11,[19][20][21][22][23] which is intended to apply to general Heisenberg systems and is not specifically tailored for spin lattices. The basic result of this theory is that the ground states can be constructed by superpositions of the eigenvectors corresponding to the minimal eigenvalue not of the original J-matrix but of the 'dressed J-matrix'.…”

Section: Introductionmentioning

confidence: 99%

“…The paper is organized as follows. In section 2.1 we recapitulate the general definitions and results that have already appeared in [6,11,[19][20][21][22][23], while the specializations to spin lattices are presented in section 2.2. The proofs of two propositions appearing in this section are moved to the appendix.…”

Section: Introductionmentioning

confidence: 99%

Classical ground states of spin lattices

Schmidt¹,

Richter²

2022

J. Phys. A: Math. Theor.

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We present a generalization of the Luttinger-Tisza-Lyons-Kaplan (LTLK) theory of classical ground states of Bravais lattices with Heisenberg coupling to non-Bravais lattices. It consists of adding certain Lagrange parameters to the diagonal of the Fourier transformed coupling matrix analogous to the theory of the general ground state problem already published. This approach is illustrated by an application to a modified honeycomb lattice, which has exclusive three-dimensional ground states as well as a classical spin-liquid ground state for different values of the two coupling constants. Another example, the modified square lattice, shows that we can also obtain so-called incommensurable ground states by our method.

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Discontinuous quantum and classical magnetic response of the pentakis dodecahedron*

Konstantinidis¹

2021

J. Phys.: Condens. Matter

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The pentakis dodecahedron, the dual of the truncated icosahedron, consists of 60 edge-sharing triangles. It has 20 six-fold and 12 five-fold coordinated vertices, with the former forming a dodecahedron, and each of the latter connected to the vertices of one of the 12 pentagons of the dodecahedron. When spins mounted on the vertices of the pentakis dodecahedron interact according to the nearest-neighbor antiferromagnetic Heisenberg model, the two different vertex types necessitate the introduction of two exchange constants. As the relative strength of the two constants is varied the molecule interpolates between the dodecahedron and a molecule consisting only of quadrangles. The competition between the two exchange constants, frustration, and an external magnetic field results in a multitude of ground-state magnetization and susceptibility discontinuities. At the classical level the maximum is ten magnetization and one susceptibility discontinuities when the 12 five-fold vertices interact with the dodecahedron spins with approximately one-half the strength of their interaction. When the two interactions are approximately equal in strength the number of discontinuities is also maximized, with three of the magnetization and eight of the susceptibility. At the full quantum limit, where the magnitude of the spins equals 1 2 , there can be up to three ground-state magnetization jumps that have the z-component of the total spin changing by ΔS z = 2, even though quantum fluctuations rarely allow discontinuities of the magnetization. The full quantum case also supports a ΔS z = 3 discontinuity. Frustration also results in nonmagnetic states inside the singlet-triplet gap. These results make the pentakis dodecahedron the molecule with the largest number of magnetization and susceptibility discontinuities from the quantum to the classical level, taking its size also into account.

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Theory of ground states for classical Heisenberg spin systems V

Cited by 3 publications

References 13 publications

Theory of ground states for classical Heisenberg spin systems VI

Theory of ground states for classical Heisenberg spin systems VI

Classical ground states of spin lattices

Discontinuous quantum and classical magnetic response of the pentakis dodecahedron*

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