arXiv subjects

Heinz-Jürgen Schmidt

Publications and source records attributed to Heinz-Jürgen Schmidt.

At least 55 records · Page 3Linked to original sources

arXiv2018

The Floquet theory of the two level system revisited

We reconsider the periodically driven two level system and especially the Rabi problem with linear polarization. The Floquet theory of this problem can be reduced to its classical limit, i.e., to the investigation of periodic solutions of the classical Hamiltonian equations of motion in the Bloch sphere. The quasienergy is essentially the action integral over one period and the resonance condition due to J.H. Shirley is shown to be equivalent to the vanishing of the time average of a certain component of the classical solution. This geometrical approach is applied to obtain analytical approximations to physical quantities of the Rabi problem with linear polarization as well as asymptotic formulas for various limit cases.

quant-ph↗

arXiv2017

Theory of ground states for classical Heisenberg spin systems IV

We extend the theory of ground states of classical Heisenberg spin systems previously published to the case where the interaction with an external magnetic field is described by a Zeeman term. The ground state problem for the Heisenberg-Zeeman Hamiltonian can be reduced first to the relative ground state problem, and, in a second step, to the absolute ground state problem for pure Heisenberg Hamiltonians depending on an additional Lagrange parameter. We distinguish between continuous and discontinuous reduction. Moreover, there are various general statements about Heisenberg-Zeeman systems that will be proven under most general assumptions. One topic is the connection between the minimal energy functions $E_{min}$ for the Heisenberg energy and $H_{min}$ for the Heisenberg-Zeeman energy which turn out to be essentially mutual Legendre-Fenchel transforms. This generalization of the traditional Legendre transform is especially suited to cope with situations where the function $E_{min}$ is not convex and consequently there is a magnetization jump at a critical field. Another topic is magnetization and the occurrence of threshold fields $B_{thr}$ and saturation fields $B_{sat}$, where we provide a general formula for the latter. We suggest a distinction between ferromagnetic and anti-ferromagnetic systems based on the vanishing of $B_{sat}$ for the former ones. Parabolic systems are defined in such a way that $E_{min}$ and $H_{min}$ have a particularly simple form and studied in detail. For a large class of parabolic systems the relative ground states can be constructed from the absolute ground state by means of a so-called umbrella family. Finally we provide a counter-example of a parabolic system where this construction is not possible.

cond-mat.other↗

arXiv2017

Theory of ground states for classical Heisenberg spin systems II

We apply the theory of ground states for classical, finite, Heisenberg spin systems previously published to a couple of spin systems that can be considered as finite models $K_{12},\,K_{15}$ and $K_{18}$ of the AF Kagome lattice. The model $K_{12}$ is isomorphic to the cuboctahedron. In particular, we find three-dimensional ground states that cannot be viewed as resulting from the well-known independent rotation of subsets of spin vectors. For a couple of ground states with translational symmetry we calculate the corresponding wave numbers. Finally we study the model $K_{12w}$ without boundary conditions which exhibits new phenomena as, e.~g., two-dimensional families of three-dimensional ground states.

cond-mat.other↗

arXiv2017

Theory of ground states for classical Heisenberg spin systems III

We extend the theory of ground states of classical Heisenberg spin systems published previously by a closer investigation of the convex Gram set of $N$ spins. This is relevant for the present purpose since the set of ground states of a given spin system corresponds to a face of Gram set. The investigation of the Gram set is facilitated by the determination of its symmetry group. The case of the general spin triangle is completely solved and illustrated by a couple of examples.

cond-mat.other↗

arXiv2017

Theory of ground states for classical Heisenberg spin systems I

We formulate part I of a rigorous theory of ground states for classical, finite, Heisenberg spin systems. The main result is that all ground states can be constructed from the eigenvectors of a real, symmetric matrix with entries comprising the coupling constants of the spin system as well as certain Lagrange parameters. The eigenvectors correspond to the unique maximum of the minimal eigenvalue considered as a function of the Lagrange parameters. However, there are rare cases where all ground states obtained in this way have unphysical dimensions $M>3$ and the theory would have to be extended. Further results concern the degree of additional degeneracy, additional to the trivial degeneracy of ground states due to rotations or reflections. The theory is illustrated by a couple of elementary examples.

cond-mat.other↗

arXiv2017

Interpolation between low and high temperatures of the specific heat for spin systems

The high temperature expansion (HTE) of the specific heat of a spin system fails at low temperatures, even if it is combined with a Padé approximation. On the other hand we often have information about the low temperature asymptotics (LTA) of the system. Interpolation methods combine both kind of information, HTE and LTA, in order to obtain an approximation of the specific heat that holds for the whole temperature range. Here we revisit the entropy method that has been previously published and propose two variants that better cope with problems of the entropy method for gapped systems. We compare all three methods applied to the antiferromagnetic Haldane spin-one chain and especially apply the second variant, called Log Z method, to the cuboctahedron for different spin quantum numbers. In particular, we demonstrate that the interpolation method is able to detect an extra low-temperature maximum in the specific heat that may appear if a separation of two energy scales is present in the considered system. Finally we illustrate how interpolation also works for classical spin systems.

cond-mat.str-el↗

arXiv2016

Ground states of classical magnetic dipole rings

We investigate the two well-known ground states of rings of $N$ classical magnetic dipoles that are given by clockwise or anti-clockwise spin orientations tangent to the circle encompassing the dipole ring. In particular, we formulate a rigorous proof of the ground state property of the states in question. The problem can be reduced to the determination of the lowest eigenvalue of a $3N\times 3N$ matrix ${\mathbf J}$. We show that all eigenvalues of ${\mathbf J}$ can be analytically calculated and, at least for $N=3,\ldots,8$, the lowest one can be directly determined. The main part of the paper is devoted to the completion of the proof for $N\ge 9$ based on various estimates and case distinctions. We also discuss the question to what extent computer-algebraic results should be allowed to contribute to a mathematical proof.

cond-mat.other↗

arXiv2016

High-temperature expansion for frustrated magnets: Application to the J1-J2 model on the BCC lattice

We present the high-temperature expansion up to 11th order for the specific heat $C$ and the uniform susceptibility $χ_0$ and up to 9th order for the structure factor $S_{\bf Q}$ of the frustrated spin-half $J_1$-$J_2$ Heisenberg model on the BCC lattice. We consider ferromagnetic as well as antiferromagnetic nearest-neighbor exchange $J_1$ and frustrating antiferromagnetic next-nearest-neighbor exchange $J_2$. We discuss the influence of frustration on the temperature dependence of these quantities. Furthermore, we use the HTE series to determine the critical temperature $T_c$ as a function of the frustration parameter $J_2$.

cond-mat.str-el↗

arXiv2015

Fourier-Taylor series for the figure eight solution of the three body problem

We provide an analytical approximation of a periodic solution of the three body problem in celestial mechanics, the so-called figure eight solution, discovered 1993 by C. Moore. This approximation has the form of a Fourier series whose components are in turn Taylor series w. r. t. some parameter. The method is first illustrated by application to two other problems, (1) the problem of oscillations of a particle in a cubic potential that has a well-known analytic solution in terms of elliptic functions and (2) periodic solutions and corresponding eigenvalues of a generalized Mathieu equation that cannot be solved analytically. When applied to the three body problem it turns out that the Fourier-Taylor series, evaluated up to 30th order, represents un-physical solutions except for a particular value of the series parameter. For this value the series approximates the numerical solution known from the literature up to a relative error of $1.6\times 10^{-3}$.

physics.class-ph↗

arXiv2014

Dynamics and thermodynamics of a pair of interacting magnetic dipoles

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.

cond-mat.stat-mech↗

arXiv2014

Tenth-order high-temperature expansion for the susceptibility and the specific heat of spin-s Heisenberg models with arbitrary exchange patterns: Application to pyrochlore and kagome magnets

We present the high-temperature expansion (HTE) up to 10th order of the specific heat C and the uniform susceptibility χfor Heisenberg models with arbitrary exchange patterns and arbitrary spin quantum number s. We encode the algorithm in a C++ program which allows to get explicitly the HTE series for concrete Heisenberg models. We apply our algorithm to pyrochlore ferromagnets and kagome antiferromagnets using several Padé approximants for the HTE series. For the pyrochlore ferromagnet we use the HTE data for χto estimate the Curie temperature T_c as a function of the spin quantum number s. We find that T_c is smaller than that for the simple cubic lattice, although both lattices have the same coordination number. For the kagome antiferromagnet the influence of the spin quantum number s on the susceptibility as a function of renormalized temperature T/s(s+1) is rather weak for temperatures down to T/s(s+1) \sim 0.3. On the other hand, the specific heat as a function of T/s(s+1) noticeably depends on s. The characteristic maximum in C(T) is monotonously shifted to lower values of T/s(s+1) when increasing s.

cond-mat.str-el↗

arXiv2012

The general spin triangle

We consider the Heisenberg spin triangle with general coupling coefficients and general spin quantum number $s$. The corresponding classical system is completely integrable. In the quantum case the eigenvalue problem can be reduced to that of tridiagonal matrices in at most $2s+1$ dimensions. The corresponding energy spectrum exhibits what we will call spectral symmetries due to the underlying permutational symmetry of the considered class of Hamiltonians. As an application we explicitly calculate six classes of universal polynomials that occur in the high temperature expansion of spin triangles and more general spin systems. Aspects of quantum integrability are also discussed in this context.

math-ph↗

arXiv2012

Exact ground state of a frustrated integer-spin modified Shastry-Sutherland model

We consider a two-dimensional geometrically frustrated integer-spin Heisenberg system that admits an exact ground state. The system corresponds to a decorated square lattice with two coupling constants J1 and J2, and it can be understood as a generalized Shastry-Sutherland model. Main elements of the spin model are suitably coupled antiferromagnetic spin trimers with integer spin quantum numbers s and their ground state Phi will be the product state of the local singlet ground states of the trimers. We provide exact numerical data for finite lattices as well as analytical considerations to estimate the range of the existence in dependence on the ratio of the two couplings constants J2 and J1 and on the spin quantum number s. Moreover, we find that the magnetization curves as a function of the applied magnetic field shows plateaus and jumps. In the classical limit s \to \infty the model exhibits phases of three- and two-dimensional ground states separated by a one-dimensional (collinear) plateau state at 1/3 of the saturation magnetization.

cond-mat.str-el↗

arXiv2011

Eighth-order high-temperature expansion for general Heisenberg Hamiltonians

We explicitly calculate the moments $t_n$ of general Heisenberg Hamiltonians up to eighth order. They have the form of finite sums of products of two factors. The first factor is represented by a (multi-)graph which has to be evaluated for each particular system under consideration. The second factors are well-known universal polynomials in the variable $s(s+1)$, where $s$ denotes the individual spin quantum number. From these moments we determine the corresponding coefficients of the high-temperature expansion of the free energy and the zero field susceptibility by a new method. These coefficients can be written in a form which makes explicit their extensive character. Our results represent a general tool to calculate eighth-order high-temperature series for arbitrary Heisenberg models. The results are applied to concrete systems, namely to magnetic molecules with the geometry of the icosidodecahedron, to frustrated square lattices, and to the pyrochlore magnets. By comparison with other methods that have been recently applied to these systems, we find that the typical susceptibility maximum of the spin-$s$ Heisenberg antiferromagnet is well described by the eighth-order high-temperature series.

cond-mat.stat-mech↗

arXiv2010

Exact ground states for coupled spin trimers

We consider a class of geometrically frustrated Heisenberg spin systems which admit exact ground states. The systems consist of suitably coupled antiferromagnetic spin trimers with integer spin quantum numbers $s$ and their ground state $Φ$ will be the product state of the local singlet ground states of the trimers. We provide linear equations for the inter-trimer coupling constants which are equivalent to $Φ$ being an eigenstate of the corresponding Heisenberg Hamiltonian and sufficient conditions for $Φ$ being a ground state. The classical case $s\to\infty$ can be completely analyzed. For the quantum case we consider a couple of examples, where the critical values of the inter-trimer couplings are numerically determined. These examples include chains of corner sharing tetrahedra as well as certain spin tubes. $Φ$ is proven to be gapped in the case of trimer chains. This follows from a more general theorem on quantum chains with product ground states.

cond-mat.str-el↗

arXiv2008

Coexistence of qubit effects

Two quantum events, represented by positive operators (effects), are coexistent if they can occur as possible outcomes in a single measurement scheme. Equivalently, the corresponding effects are coexistent if and only if they are contained in the ranges of a single (joint) observable. Here we give several equivalent characterizations of coexistent pairs of qubit effects. We also establish the equivalence between our results and those obtained independently by other authors. Our approach makes explicit use of the Minkowski space geometry inherent in the four-dimensional real vector space of selfadjoint operators in a two-dimensional complex Hilbert space.

quant-ph↗

arXiv2008

Modulated spin waves and robust quasi-solitons in classical Heisenberg rings

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.

cond-mat.other↗

arXiv2007

Universal properties of highly frustrated quantum magnets in strong magnetic fields

The purpose of the present paper is two-fold. On the one hand, we review some recent studies on the low-temperature strong-field thermodynamic properties of frustrated quantum spin antiferromagnets which admit the so-called localized-magnon eigenstates. One the other hand, we provide some complementary new results. We focus on the linear independence of the localized-magnon states, the estimation of their degeneracy with the help of auxiliary classical lattice-gas models and the analysis of the contribution of these states to thermodynamics.

cond-mat.str-el↗