Prof. Dr. Jürgen Schnack Universität Bielefeld
Fakultät für Physik
D-33501 Bielefeld


Magnon crystallization in the kagome lattice antiferromagnet

kagome, specific heat Numerical evidence for the crystallization of magnons below the saturation field at nonzero temperatures for the highly frustrated spin-half kagome Heisenberg antiferromagnet is presented. This phenomenon can be traced back to the existence of independent localized magnons or, equivalently, flatband multimagnon states. We present a loop-gas description of these localized magnons and a phase diagram of this transition, thus providing information for which magnetic fields and temperatures magnon crystallization can be observed experimentally. The emergence of a finite-temperature continuous transition to a magnon crystal is expected to be generic for spin models in dimension D>1 where flatband multimagnon ground states break translational symmetry.
Jürgen Schnack, Jörg Schulenburg, Andreas Honecker, and Johannes Richter, Magnon crystallization in the kagome lattice antiferromagnet, Phys. Rev. Lett. 125 (2020) 117207

Magnetism of the N=42 kagome lattice antiferromagnet

kagome, specific heat For the paradigmatic frustrated spin-half Heisenberg antiferromagnet on the kagome lattice large-scale numerical investigations of thermodynamic functions by means of the finite-temperature Lanczos method for system sizes of up to N=42 have been performed. We present the dependence of magnetization as well as specific heat on temperature and external field and show in particular that a finite-size scaling of specific heat supports the appearance of a low-temperature shoulder below the major maximum. This seems to be the result of a counterintuitive motion of the density of singlet states towards higher energies. Other interesting features that we discuss are the asymmetric melting of the 1/3 magnetization plateau as well the field dependence of the specific heat that exhibits characteristic features caused by the existence of a flat one-magnon band. By comparison with the unfrustrated square-lattice antiferromagnet the tremendous role of frustration in a wide temperature range is illustrated.
Jürgen Schnack, Jörg Schulenburg, Johannes Richter, Magnetism of the N=42 kagome lattice antiferromagnet, Phys. Rev. B 98 (2018) 094423

High spin cycles: topping the spin record for a single molecule verging on quantum criticality

Mn6Mn The cyclisation of a short chain into a ring provides fascinating scenarios in terms of transforming a finite array of spins into a quasi-infinite structure. If frustration is present, theory predicts interesting quantum critical points, where the ground state and thus low-temperature properties of a material change drastically upon even a small variation of appropriate external parameters. We report a new mixed 3d/4f cyclic coordination cluster that turns out to be very near or even at such a quantum critical point. It has a ground state spin of S=60, the largest ever observed for a molecule (120 times that of a single electron). The molecule forms a nano-torus with alternating gadolinium and iron ions with a nearest neighbour Fe-Gd coupling and a frustrating next-nearest neighbour Fe-Fe coupling. Such a spin arrangement corresponds to a cyclic delta or saw-tooth chain, which can exhibit unusual frustration effects. In the present case, the quantum critical point bears a 'flatland' of tens of thousands of energetically degenerate states between which transitions are possible at no energy costs with profound caloric consequences. Entropy-wise the energy flatland translates into the pointed summit overlooking the entropy landscape. Going downhill several target states can be reached depending on the applied physical procedure which offers new prospects for addressability.
Amer Baniodeh, Nicola Magnani, Yanhua Lan, Gernot Buth, Christopher E. Anson, Johannes Richter, Marco Affronte, Jürgen Schnack, A.K. Powell, High spin cycles: topping the spin record for a single molecule verging on quantum criticality, Nature Partner Journal Quantum Materials 3 (2018) 10

Quantum signatures of a molecular nanomagnet in direct magnetocaloric measurements

Mn6Mn Geometric spin frustration in low-dimensional materials, such as the two-dimensional kagome or triangular antiferromagnetic nets, can significantly enhance the change of the magnetic entropy and adiabatic temperature following a change in the applied magnetic field, that is, the magnetocaloric effect. In principle, an equivalent outcome should also be observable in certain high-symmetry zero-dimensional, that is, molecular, structures with frustrated topologies. Here we report experimental realization of this in a heptametallic gadolinium molecule. Adiabatic demagnetization experiments reach approximately 200 mK, the first sub-Kelvin cooling with any molecular nanomagnet, and reveal isentropes (the constant entropy paths followed in the temperature-field plane) with a rich structure. The latter is shown to be a direct manifestation of the trigonal antiferromagnetic net structure, allowing study of frustration-enhanced magnetocaloric effects in a finite system.
Joseph W. Sharples, David Collison, Eric J. L. McInnes, Jürgen Schnack, Elias Palacios, Marco Evangelisti, Quantum signatures of a molecular nanomagnet in direct magnetocaloric measurements, Nature Communications 5 (2014) 5321

Systematic investigations of the single-molecule magnet family [M1III6M2III]3+

Mn6Mn By means of rational design strategies single-molecule magnets are synthesized in the group of Prof. Thorsten Glaser (Bielefeld). The family of [M1III6M2III]3+ contains two kinds of metal centers M1 and M2 which may exhibit large local zero-field splittings. By means of complete exact diagonalization of the appropriate spin Hamiltonians magnetic properties are modelled and magneto-chemical correlations are investigated.
Veronika Hoeke, Maik Heidemeier, Erich Krickemeyer, Anja Stammler, Hartmut Bögge, Jürgen Schnack, Thorsten Glaser, Structural influences on the exchange coupling and zero-field splitting in the single-molecule magnet [MnIII6MnIII]3+ triplesalen single-molecule magnet, Dalton. Trans. 41 (2012) 12942-12959

Effects of frustration on magnetic molecules

icosahedron In magnetism, of which molecular magnetism is a part, the term frustration is used rather sloppily. Sometimes one gains the impression that if the reason for some phenomenon is not quite clear then it is attributed to frustration. In this paper a discussion of the effects of frustration that are relevant for the field of molecular magnetism is presented. As will become clear later these effects indeed lead to a variety of unusual magnetic properties.
J. Schnack, Effects of frustration on magnetic molecules: a survey from Olivier Kahn until today, Dalton Trans. 39 (2010) 4677 - 4686

Complete diagonalization studies of an anisotropic molecule

hysteresis Complete diagonalization studies have been carried out for the anisotropic molecule MnIII6FeIII. Besides a Heisenberg superexchange interaction local anisotropy tensors were used. These tensors are parameterized by their dominant term, i.e. a local axis (not collinear) and strength. Thus, a determination of ionic parameters has become possible via the study of the molecular magnetization.
Numerically, the S6 symmetry of the magnetic core was employed in order to reduce the size of the Hamiltonian matrices.
T. Glaser, M. Heidemeier, E. Krickemeyer, H. Bögge, A. Stammler, R. Fröhlich, E. Bill, J. Schnack, Exchange Interactions and Zero-Field Splittings in C3-Symmetric MnIII6FeIII: Using Molecular Recognition for the Construction of a Series of High Spin Complexes Based on the Triplesalen Ligand, Inorg. Chem. 48 (2009) 607-620

Hysteresis without anisotropy

hysteresis Normally hysteretic behaviour of magnetic systems is the outcome of anisotropic terms in the Hamiltonian. In a recent work we could show that the classical Heisenberg icosahedral antiferromagnet exhibits a hysteresis loop when the external field is swept (see figure). In this hypothetical magnetic molecule the spins are mounted at the vertices of an icosahedron and interact solely via antiferromagnetic nearest neighbor coupling. In the corresponding quantum system the (T=0) magnetization curve exhibits anusual jumps. The whole behavior can be charcterized as a first order metamagnetic phase transition.
C. Schröder, H.-J. Schmidt, J. Schnack, and M. Luban, Metamagnetic phase transition of the antiferromagnetic Heisenberg icosahedron, Phys. Rev. Lett. 94 (2005) 207203

Competing Spin Phases in Fe30

dip Frustrated spin systems such as the kagome lattice antiferromagnet show a pronounced (T=0) magnetization plateau at one third saturation magnetization. Since the magnetic molecule Fe30 is built of spins mounted at the vertices of a perfect icosidodecahedron it is structurally a "little brother" of the kagome antiferromagnet and shares several properties. In a recent experiment we could show that the differential suscptibility dM/dB features a pronounced minimum around one third of the saturation field which gives clear evidence that in this zero-dimensional system the related plateau is produced by competing spin phases just in the same way as on the two-dimensional kagome lattice.
C. Schröder, H. Nojiri, J. Schnack, P. Hage, M. Luban, P. Kögerler, Competing Spin Phases in Geometrically Frustrated Magnetic Molecules, Phys. Rev. Lett. 94 (2005) 017205

Giant magnetization jumps

dip For a class of frustrated spin lattices including the Kagome lattice we construct exact eigenstates consisting of several independent, localized one-magnon states and argue that they are ground states for high magnetic fields. If the maximal number of local magnons scales with the number of spins in the system, which is the case for the Kagome lattice, the effect persists in the thermodynamic limit and gives rise to a macroscopic jump in the zero-temperature magnetization curve just below the saturation field. The effect decreases with increasing spin quantum number and vanishes in the classical limit. Thus it is a true macroscopic quantum effect.
J. Schulenburg, A. Honecker, J. Schnack, J. Richter, H.-J. Schmidt, Macroscopic magnetization jumps due to independent magnons in frustrated quantum spin lattices, Phys. Rev. Lett. 88 (2002) 167207

Fermionic Molecular Dynamics

dip The time-dependent variational principle for many-body trial states is used to discuss the relation between the approaches of different molecular-dynamics models that describe indistinguishable fermions. Early attempts to include effects of the Pauli principle by means of nonlocal potentials, as well as more recent models that work with antisymmetrized many-body states, are reviewed under these premises.
We discuss how heavy-ion collisions as well as quasi equlibrium situations can be described on the same footing.
H. Feldmeier, J. Schnack, Molecular Dynamics for Fermions, Rev. Mod. Phys. 72 (2000) 655-688

The synthesis of molecular magnets has undergone rapid progress in recent years. Each of the identical molecular units can contain as few as two and up to several dozens of paramagnetic ions (spins). Although these materials appear as macroscopic samples, i.e. crystals or powders, the intermolecular magnetic interactions are utterly negligible as compared to the intramolecular interactions. Therefore, measurements of their magnetic properties reflect mainly ensemble properties of single molecules.

We are interested in static and dynamical properties of magnetic molecules, our research focuses on:

  • application of group theoretical methods to spin systems;
  • properties and modelling of anisotropic molecules;
  • investigations of deposited magnetic molecules;
  • properties of spin rings like the popular ferric wheels;
  • rigorous and numerical results on properties of the spectrum for Heisenberg spin arrays, like bounding parabolas, rotational bands, quantum numbers of low-lying states;
  • high N limits for spin chains;
  • frustration effects in antiferromagnetic spin systems;
  • comparison of classical and quantum Heisenberg model;
  • spin-spin correlation functions and related quantities like spin-lattice relaxation times or neutron scattering cross sections;
  • approximate methods for spin systems like DMRG, high temperature expansion etc.
Klaus Bärwinkel - Professor, Senior Fellow @ UOS
Euan Brechin - Professor, Senior Fellow @ The University of Edinburgh, UK
Lee Cronin - Professor, Senior Fellow @ The University of Glasgow, UK
Thorsten Glaser - Professor, Senior Fellow @ Bielefeld University
Nedko Ivanov - Professor, Senior Fellow @ Bulgarian Academy of Sciences, Bulgaria
Paul Kögerler - Professor, Senior Fellow @ RWTH Aachen
Ulrich Kortz - Professor, Senior Fellow @ Jacobs University Bremen
Marshall Luban - Professor, Senior Fellow @ Ames Lab, Iowa
Eric McInnes - Professor, Senior Fellow @ The University of Manchester, UK
Hiroyuki Nojiri - Professor @ Tohoku University, Japan
Andrei Postnikov - Professor @ Metz University, France
Johannes Richter - Professor @ Magdeburg University
Heinz-Jürgen Schmidt - Professor, Senior Fellow @ UOS
Christian Schröder - Professor, Senior Fellow @ FH Bielefeld
Richard Winpenny - Professor, Senior Fellow @ The University of Manchester, UK

Free-standing carbon nanomembranes (CNM) with molecular thickness and macroscopic size are fascinating objects both for fundamental reasons and for applications in nanotechnology. Although being made from simple and identical precursors their internal structure is not fully known and hard to simulate due to the large system size that is necessary to draw definite conclusions. We perform large-scale classical molecular dynamics investigations of e.g. biphenyl-based carbon nanomembranes. We show that one-dimensional graphene-like stripes constitute a highly symmetric quasi one-dimensional energetically favorable ground state. This state does not cross-link. Instead cross-linked structures are formed from highly excited precursors with a sufficient amount of broken phenyls.

Armin Gölzhäuser - Professor, Senior Fellow @ Bielefeld University

Statistical properties of finite interacting systems are of great interest. The aim is to describe the behaviour of systems like atomic clusters, atomic vapours or atomic nuclei at finite temperatures and to investigate properties like the specific heat or phase transitions. For realistic systems like atomic clusters or nuclei where the Hamilton function or operator contains a (two-body) interaction it is hard or impossible to evaluate the partition function especially for the quantum description.

Equations of motion for the investigated system are often much easier; either they are exactly known and can be integrated at least numerically as it is the case with the classical Hamilton's equation or they can be approximated with standard methods like Time-dependent Hartree-Fock (TDHF) or quantum molecular dynamics methods as it is the case on the quantum side. The idea then is to extract the desired thermodynamic quantities from the time evolution of the system. If the system is ergodic, ensemble averages can be replaced by time averages.

Our current research focuses on:

  • thermodynamic equlibrium properties of small quantum systems by time averaging together with Fermionic Molecular Dynamics (FMD);
  • determination of the caloric curve of finite nuclei and investigation of the nuclear liquid-gas phase transition;
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  • thermalization of a quantum system with the help of additional degrees of freedom and complex time steps;
  • Nose-Hoover-like thermostat for ideal quantum gases in harmonic oscillator potentials;
  • Nose-like thermostat for general quantum systems.
Detlef Mentrup - Philips Research, Hamburg
Hans Feldmeier - Professor, Senior Fellow @ GSI/TUD

Ideal quantum gases are usually treated in the thermodynamic limit, i.e. occupying an infinite volume but maintained at a given density, since all applications which were important in the past, like the electron gas, phonons or photons, deal with huge particle numbers. Only the experimental attempts of the last years to investigate finite Fermi and Bose systems and to describe them in terms of thermodynamics called for new theoretical effort. Interesting finite Fermi systems are for instance nuclei, which behave like a liquid drop and therefore can undergo a first order liquid-gas-like phase transition. On the low excitation site of the caloric curve the nuclear systems can be very often well described as an ideal Fermi gas in a common harmonic oscillator potential (shell model). Small Bose systems became available through the development of traps. Here the focus is on the Bose-Einstein condensation which for instance could be found investigating dilute atomic vapours (alkali atoms) in magnetic traps. Again the system can be well described as an ideal quantum gas contained in an external harmonic oscillator potential.

With the help of recursion formulae analytical and approximative results are obtained for small non-interacting Fermi and Bose systems.

A closer inspection of the canonical partition function uncovers a surprising symmetry property which connects fermions and bosons contained in harmonic oscillator potentials of odd space dimensions. Simply speaking, it turns out that the properties of N fermions at temperature T are related to the properties of N bosons at the respective negative temperature -T.

Heinz-Jürgen Schmidt - Professor, Senior Fellow @ UOS

A quasi-particle theory for monatomic gases in equilibrium is formulated and evaluated to yield the exact virial contributions to the thermodynamic state functions in lowest order of the density. Van der Waals blocking has necessarily to be accounted for in occupation number statistics. The quasi-particle distribution function differs from the Wigner function by a bilinear functional thereof. The progress made so far is promising with respect to a corresponding version of kinetic theory.
Klaus Bärwinkel - Professor, Senior Fellow @ UOS

FMD (1990-1997)
A new type of molecular dynamics is proposed to solve approximately the many-body problem of interacting identical fermions with spin 1/2 using variational principles. The interacting system is represented by an antisymmetrized many-body wave function consisting of single-particle states which are localized in phase space. The equations of motion for the parameters characterizing the many-body state (e.g. position, momentum, width and spin of the particles) are derived from a quantum variational principle. The model is designed to describe ground state properties of nuclii as well as heavy ion reactions. Therfore the ansatz is extended towards correlated many-body states, in order to include short-range correlations. Due to its non-linear equations of motion the model shows large fluctuations in the final stage as it is seen in fragmentation reactions. Not only heavy-ion reactions may be addressed, but also properties of excited nuclii like the nuclear liquid-gas phase transition.
detailed description (PDF with hyperlinks)
Hans Feldmeier - Professor, Senior Fellow @ GSI/TUD
Thomas Neff - PostDoc @ GSI/TUD
Robert Roth - Prof. @ TUD

UCOM (1993-1997)
The short range repulsion between nucleons is treated by a unitary correlation operator which shifts the nucleons away from each other whenever their uncorrelated positions are within the replusive core. By formulating the correlation as a transformation of the relative distance between particle pairs, general analytic expressions for the correlated wave functions and correlated operators are given. The decomposition of correlated operators into irreducible n-body operators is discussed. The one- and two-body-irreducible parts are worked out explicitly and the contribution of three-body correlations is estimated to check convergence. Ground state energies of nuclei up to mass number A=48 are calculated with a spin-isospin-dependent potential and single Slater determinants as uncorrelated states. They show that the deduced energy- and mass-number-independent correlated two-body Hamiltonian reproduces all "exact" many-body calculations surprisingly well.
Hans Feldmeier - Professor, Senior Fellow @ GSI/TUD
Thomas Neff - PostDoc @ GSI/TUD
Robert Roth - Prof. @ TUD

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