*Frustrated Spin-Systems*

Magnetic interactions are frustrated, if a spin cannot arrange its orientation such that it profits from the interaction with its neighbors. The logo shows the example of Ising spins ( which can only point up or down ) with antiferromagnetic interactions on a triangle. The third spin cannot gain energy, i.e., here spins with antiferro interactions are frustrated.

Frustration can prevent magnetic order; an outstanding example is the Kitaev model on the honeycomb lattice which forms a spin liquid ground state. Here our research focuses on questions as: What is the effect of Heisenberg-type perturbations in this model, or what are the consequences of magnetic fields and defects in the Kitaev-Heisenberg model?

Further research topics include: the Compass-Heisenberg model on the square lattice; moreover we are interested in spin chains formed by copper-oxygen units with frustrated interactions and helical magnetic order.

Another research topic are one-dimensional Wigner lattices and their magnetic interactions. Finally it should be stressed, that there is a close connection to the research on generalized spin-orbital models, which are generically frustrated.

*Compass-Heisenberg System / Nanoscale Physics*

Frustrated quantum magnetism belongs to the very active research areas in condensed matter theory. In this context we study the compass-Heisenberg model of spins-1/2 on a square lattice, generalizing the quantum compass model via the addition of perturbing Heisenberg interactions between nearest neighbors, and investigate its phase diagram( see Figure 1) and magnetic excitations.

This model has motivations both from the field of strongly correlated systems with orbital degeneracy and from that of solid-state based devices proposed for quantum computing. We find that the high degeneracy of ground states of the compass model is fragile and changes into twofold degenerate ground states for any finite amplitude of Heisenberg coupling. By computing the spin structure factors of finite clusters with Lanczos diagonalization, we evidence a rich variety of phases characterized by Z_{2} symmetry, that are either ferromagnetic, C-type antiferromagnetic, or of Neel type, and analyze the effects of quantum fluctuations on phase boundaries.

**Figure 1:** Phase diagram of the compass-Heisenberg model in the plane spanned by the compass interaction J_{x}/J_{z} and Heisenberg interaction I/J_{z} for antiferromagnetic J_{z}=1. While the quantum compass model (I=0) on a LxL lattice has a 2^{L} degeneracy, this degeneracy is lifted for any finite Heisenberg coupling and a phase with Z_{2} symmetry is selected.

In the ordered phases the anisotropy of compass interactions leads to a finite excitation gap to spin waves. Furthermore we have shown that for small nanoscale clusters with large anisotropy gap the lowest excitations are column-flip excitations that emerge due to Heisenberg perturbing interactions from the manifold of degenerate ground states of the compass model.

The low energy column-flip or compass-type excitations are of particular interest, as they are robust against decoherence processes and are therefore well designed for storing information in quantum computing. We have also pointed out that the dipolar interactions between nitrogen-vacancy centers forming a rectangular lattice in a diamond matrix may permit a solid-state realization of the anisotropic compass-Heisenberg model.

*Kitaev-Heisenberg model on honeycomb lattice*

Besides the mathematical beauty of the Kitaev model, recent studies were motivated by its possible relevance for orbitally degenerate systems with strong spin-orbit coupling such as layered iridates Na_{2}IrO_{3} and Li_{2}IrO_{3}. These applications require to consider the extension, namely the Kitaev-Heisenberg model on honeycomb lattice, which reveals in its phase diagram apart of a liquid phase also several ordered (e.g. stripe or zig-zag) phases.

In particular we have explored the ground state properties of the Kitaev-Heisenberg model in a magnetic field, as well as the evolution of spin correlations in the presence of non-magnetic vacancies. By means of exact diagonalizations, the phase diagram without vacancies was determined as a function of the magnetic field and the ratio between Kitaev and Heisenberg interactions.

**Figure 2:** Vacancy induced spin density * δσ _{j}^{z}* in the Kitaev-Heisenberg model. The pattern of the z-component of spin around the vacany site (for

*α*=0.9 and an infintesimal magnetic field applied along z) reflects the peculiar spin correlation functions of the model in the Kitaev limit, where only the nearest-neighbor correlations are finite.

In the liquid phase, the magnetization pattern around a single vacancy in a small field is determined (see Figure 2), and its spatial anisotropy is related to that of non-zero further neighbor correlations induced by the field and/or Heisenberg interactions. In the stripe phase, the combination of a vacancy and a small field breaks the six-fold symmetry of the model and stabilizes a particular stripe pattern. Similar symmetry-breaking effects occur even at zero field due to interaction effects between vacancies. This selection mechanism and intrinsic randomness of vacancy positions may lead to spin-glass behavior.

*Magnetic frustration in cuprate chain componds*

Research of edge-sharing Cu-O chain compounds like LiCuVO_{4} and NaCu_{2}O_{2} was stimulated by the discovery of their helical spin structure. The origin of the non-collinear arrangement of spins lies in the frustrated nature of magnetic superexchange interactions J_{1}, J_{2} and J_{3} in these compounds. The source of the frustration lies in the relative importance of further neighbor hopping (see Figure 3).

The remarkable magnetoelectric properties are another interesting feature of these compounds. Simultaneously with the helical spin arrangement in LiCuVO_{4} appears a homogeneous electric polarization field that can be controlled by a magnetic field via the helix! Such a control of electrial polarization by magnetic field or magnetization by an electric field is characteristic for multiferroic compounds.

**Figure 3:** Effective hopping processes that contribute to superexchange in edgesharing Cu-O chains: In the ground state of the (undoped) chain all Cu ions are in the d^{9} configuration. By an effective hopping process (which involves intermediate O states) a d^{9}d^{9} pair is virtually excited into pair formed by a d^{10} configuration and a d^{9} ligand-hole pair. The intermediate excitations are singlet configurations and therefore favor antiferromagnetism. Importantly in the edgesharing compounds the 2nd neighbor process, via the Cu-O-O-Cu path, leads to a large AF superexchange integral J_{2}. Due to further processes that involve oxygen J_{1} is negative. Typically these interactions lead to frustration and a helical spin arrangement.

Excitons in the Mott-Hubbard gap: Edgesharing chains have very small hopping matrix elements as compared to the effective Hubbard U and the longer range Coulomb interactions (V_{1},V_{2},V_{3}....). We have shown that in this extreme narrow band case excitons in the Hubbard gap are expected at energies U-V_{1} and U-V_{2}. This exciton doublet has been seen by ellipsometry experiments in LiCuVO_{4} and NaCu_{2}O_{2}.

Importantly the intensity of the absorptions is determined by the nearest and next-nearest neighbor spin correlation functions. Thus the optical absorption can be used to study the strength and the temperature dependence of the spin-correlations of the frustrated spin chains.

*One-dimensional Wigner lattice*

Motivated by the finding that the intrinsically **doped** edge-sharing Cu-O chain compounds Na_{3}Cu_{2}O_{4} and Na_{8}Cu_{5}O_{10}, that have been synthesized in Jansen's department, are realizations of one-dimensional (1D) Wigner crystals, we have investigated the electronic properties of these systems. Due to the 90 degree Cu-O-Cu bonds in this structure the kinetic energy is small and the electrons are strongly correlated and their positions controlled by the long-range Coulomb interaction. In particular we studied the optical and the density fluctuation spectra, the single particle spectral functions, and the phase diagram of such systems. Charge excitations in 1D Wigner crystals are described in terms of domain-wall excitations with fractional charge, rather than by particle-hole excitations as in conventional insulators.

A very interesting aspect of the edge-sharing compounds is the relevance of both nearest- and next-nearest neighbor hopping, t_{1} and t_{2}, which allows to distinguish charge modulations due to Fermi surface instabilities and due to Coulomb interactions, i.e., Wigner lattice formation. In particular we have shown that a soft domain-wall exciton at sufficiently large t_{2} drives an instability of the Wigner lattice towards a charge-density wave with a modulation period distinct from that of the Wigner lattice.

**Figure 4:** (a) Schematic representation of Wigner charge order in the edge-sharing CuO chain compound Na_{5}Cu_{3}O_{6}. In this case the unit cell involves 3 Cu ions, where filled circles represent Cu d^{9} configurations, i.e., sites with spin 1/2, while the open circle corresponds to a non-magnetic Cu site with a hole (Zhang-Rice singlet). The relevant magnetic superexchange interactions J_{1}, J_{2} and J_{3} are indicated; (b) and (c) are effective spin models where the sites with holes are eliminated. Such effective spin models, also augmented by relevant interchain couplings, are being used to calculate the magnetic properties.

More recently we discovered that virtual excitations across the Wigner charge gap Δ_{W} contribute to the magnetic superexchange. For example J_{2} is proportional to t_{1}^{2}t_{2}/Δ_{W}^{2} due to constructive interference in the singlet channel. As these terms are odd in t_{2} they favor effectively the ferromagnetic state for one sign of t_{2}. These exchange interactions are controlled by the Wigner charge gap Δ_{W} and not by the much larger Mott-Hubbard gap Δ_{MH} as the usual AF interactions in the system, thus the new terms become more relevant in systems with decreasing Δ_{W}. Recently, in Jansen's department another doped chain compound Na_{5}Cu_{3}O_{6} was synthesized whose susceptibility could only be explained by invoking the above mechanism that amplifies the tendency towards ferromagnetism (see Figure 4).

*Local experts:*

*External collaborators:*

- Maria Daghofer (Dresden)
- R. M. Noack (Marburg)
- A. M. Oles (Krakau)
- Jesko Sirker (Kaiserslautern)

*Key publications:*

- Fabien Trousselet, Mona Berciu, Andrzej M. Oles, and Peter Horsch
**Hidden Quasiparticles and Incoherent Photoemission Spectra in Na**_{2}IrO_{3}

Phys. Rev. Lett. 111, 037305 (2013)cond-mat/1302.0187 - F. Trousselet, A.M. Oles, and P. Horsch

**Magnetic properties of nanoscale compass-Heisenberg planar clusters**

Phys. Rev. B**86**, 134412 (2012) cond-mat/1207.0102 - Alexander Herzog, Peter Horsch, Andrzej M. Oles, and Jesko Sirker

**Dimerized ferromagnetic Heisenberg chain**

Phys. Rev.**B**84, 134428 (2011) cond-mat/1107.2772 - Fabien Trousselet, Giniyat Khaliullin, and Peter Horsch

**Effects of spin vacancies on magnetic properties of the Kitaev-Heisenberg model**

Phys. Rev.**B**84, 054409 (2011) cond-mat/1104.4707 - Naveed Zafar Ali, Jesko Sirker, Juergen Nuss, Peter Horsch, and Martin Jansen

**Spin exchange dominated by charge fluctuations of the Wigner lattice in the chain cuprate Na5Cu3O6**

Phys. Rev.**B**84, 035113 (2011) cond-mat/1103.1588 - J. Sirker

**Thermodynamics of multiferroic spin chains**

Phys. Rev.**B**81, 014419 (2010) cond-mat/0909.4579 - F. Trousselet, A.M. Oles, and P. Horsch

**Compass-Heisenberg model on the square lattice: Spin order and elementary excitations**

Europhysics Letters EPL**91**, 40005 (2010) cond-mat/1005.1508 - Y. Matiks, P. Horsch, R. K. Kremer, B. Keimer, and A. V. Boris

**Exciton Doublet in the Mott-Hubbard Insulator LiCuVO4 Identified by Spectral Ellipsometry**

Phys. Rev. Lett. 103, 187401 (2009) cond-mat/1103.1588 - M. Daghofer, R. M. Noack, and P. Horsch

**Magnetism of one-dimensional Wigner lattices and its impact on charge order**

Phys. Rev.**B**78 , 205115 (2008) cond-mat/0711.1990

Last modified: 13. August 2013