**Frustrated Magnetism and Orbital Physics**

**Coupled Ladders and Chains**

My involvement in frustrated magnetism grew first out of a materials problem in the search for

*S = 1/2*spin-ladder materials. A number of these, including SrCu2O3, have the "trellis lattice" structure, with a frustrated zig-zag coupling between the ladders. We mapped out [1] a triangular phase diagram which interpolates between the limits of ladders, chains and zig-zag chains, and used it [2] to place the CaV2O5 and MgV2O5 systems. We also invoked frustrated interchain coupling as a possible reason for the mysterious properties of (VO)2P2O7 [3].**Probing Frustrated Systems with Nonmagnetic Impurities**

This topic [4-7] has its own separate page.

**Extreme Frustration and High-Dimensional Spinons**

Our studies with nonmagnetic impurities [5] gave some hints that sufficiently highly frustrated systems could be associated with the presence of elementary

*S = 1/2*excitations (spinons) beyond one dimension. Some work on the square lattice with a frustration which makes the singlets and triplets of each elementary plaquette degenerate led to the concept of*dimensional reduction*. We put this Klein-type spin model on the checkerboard (2D) and pyrochlore (3D) lattices [8], where its ground state is spanned by the set of singlet dimer coverings and thus possesses an extensive ground-state degeneracy. This corresponds to complete dimensional reduction,*i.e.*to point-like excitations in the form of deconfined fractional spinons which propagate through the entire system.## Although we showed that a Hubbard-type electronic model on

the pyrochlore lattice can lie close to the Klein point, we also showed that, in two dimensions, this point is inherently unstable in that any perturbation restores spinon confinement at zero temperature. At finite temperatures, the deconfined phase survives as a dilute Coulomb gas of thermally excited spinons. At

*T = 0*and away from the Klein point, we used a variational approach based on the singlet dimer coverings of the pyrochlore lattice (and on dealing with their nonorthogonality) to show that the ground states are valence-bond crystal (VBC) phases. Our spinons can be considered as the quantum analogue of the classical "magnetic monopoles in pyrochlore antiferromagnets" which have received a lot of press since 2008.**Orbital Physics and Extreme Frustration**

The physics of orbital degeneracy forms the basis for the science and technology of colossal magnetoresistance in manganites. In fact it is a generic feature of all transition metals except for those at the edges of the sequence, and it adds a fundamental new parameter to the Hamiltonian of charge, spin and lattice degrees of freedom. My studies in orbital physics have been focussed on using orbital degeneracy to enhance frustration in magnetic systems, with the aim of finding liquid rather than ordered states. By this is meant states where spin and orbital order are sufficiently frustrated and unfavourable that the kinetic terms dominate instead, producing resonating ground states.

## We decided that absolutely the most frustrated model

realisable in a material must combine the geometrical frustration of the triangular lattice with the threefold orbital degeneracy of the

*t2g*manifold and the quantum nature of*S = 1/2*spins. This might be found on the <111> planes of NaTiO2 if structural distortion could be avoided. In a very long analysis of this model [9], we showed that ordered states are all very frustrated and spin-orbital dimer states are more favourable. We computed correlation functions for (very) small clusters and obtained energy gains due to resonance about valence-bond configurations (which are in fact exact in the direct-exchange limit of the model). However, we could not prove that the large resonance energies in the superexchange limit of the model would drive a spin-orbital liquid state. In a later analysis [10], I extracted quantum dimer models (QDMs) for the three cases of most interest, to find out if these might fall close to the known resonating valence-bond (RVB, liquid) states of triangular-lattice QDMs. I had a lot of fun playing with high-order hopping processes, multi-coloured QDM s, Majorana-fermion representations and triangular plaquette models, and I discovered the "colour vison" excitation, but in the end all the models lay far from the spin-liquid regime: they all favored local fluctuation processes about relatively small numbers of fixed VBC configurations.**Frustration: the Review Article**

I was asked by Contemporary Physics to write an article making my work on frustrated systems accessible to an advanced undergraduate student. Unfortunately this was a little bit of an oxymoron, so instead I wrote a review article [11] focussing on the current (as of early 2009) hot topics in frustration, in the three different areas of

- materials: the preparation of frustrated systems;
- theory: the search for and characterisation of the spin liquid;
- numerics: the solution of frustrated models.

**Further Forms of Frustration**

Frustration can involve a lot more than merely competing Heisenberg spin interactions. Without taking the definition too broadly, three further pieces of work can be classified in this section.

## i) The cyclic ring-exchange interaction on a four-site plaquette is non-negligible in cuprate materials and

competes with Heisenberg interactions to form a number of different types of ordered phase, including spontanously dimerised, scalar chiral and vector chiral. We investigated these states, and the quantum phase transitions between them, in a spin ladder [12]. We used the existence of a number of exactly soluble points in a slightly generalised Hamiltonian to draw the complete phase diagram and to conduct a detailed analysis of the effects of ring-exchange terms in this one-dimensional chain of plaquettes.

## ii) Effective staggered magnetic field terms can arise in a spin

system due to Dyzaloshinskii-Moriya interactions or

*g*-tensor anisotropies. We used quantum Monte Carlo simulations of coupled spin chains to analyse the competition of the field term with the Heisenberg interactions, studying the transverse moment of the ordered phases and the gap of the disordered ones. We found some novel enhancement of magnetic order by a competing field at low interchain coupling and quasi-one-dimensionality at high coupling in a field of sufficient strength [13].## iii) The single-ion anisotropy *D* of spins with *S > 1/2* also competes with Heisenberg interactions,

preferring to form a nonmagnetic or low-spin state. For

*S = 1*, the Gaussian transition from the Haldane to the large-*D*phase has been a long-running saga on which many authors have tested different techniques. We used [14] an improved DMRG algorithm which controls the absolute (rather than relative) error very strictly, and applied it to the bulk entropy of the system to fix the transition at*D/J = 0.96845(8)*. With this we calculated the ground-state energy, gap, velocity, transverse string-order parameter, Luttinger parameter and critical exponent to extremely high accuracy. We used our results to deduce quantitatively the logarithmic divergence of the entropy at the critical point and confirm the topological nature of the transition.*Phase Diagram of the S = 1/2 Frustrated Coupled Ladder System*

B. Normand, K. Penc, M. Albrecht and F. Mila,

Phys. Rev. B**56**, R5736 (1997).*Magnetic Properties of the Coupled Ladder System MgV2O5*

P.Millet, C. Satto, J. Bonvoisin, B. Normand, K. Penc, M. Albrecht and F. Mila,

Phys. Rev. B**57**, R5005 (1998).*Magnetic Properties of (VO)2P2O7 from Frustrated Interchain Coupling*

G. S. Uhrig and B. Normand,

Phys. Rev. B**58**, R14705 (1998).*Absence of Effective Spins 1/2 Induced by Nonmagnetic Impurities in a Class of Low-Dimensional Magnets*

B. Normand and F. Mila,

Phys. Rev. B**65**, 104411 (2002).*Static Impurities in the Kagome Lattice: Dimer Freezing and Mutual Repulsion*

S. Dommange, M. Mambrini, B. Normand and F. Mila,

Phys. Rev. B**68**, 224416 (2003).*Nonmagnetic Impurities in the S = 3/2 Kagome Antiferromagnet*

A. Läuchli, S. Dommange, B. Normand and F. Mila,

Phys. Rev. B**76**, 144413 (2007).*Dzyaloshinskii-Moriya Anisotropy and Nonmagnetic Impurities in the S = 1/2 Kagome System ZnCu3(OH)6Cl2*

I. Rousochatzakis, S. R. Manmana, A. M. Läuchli, B. Normand and F. Mila,

Phys. Rev. B**79**, 214415 (2009).*High-dimensional Fractionalization and Spinon Deconfinement in Pyrochlore Antiferromagnets*

Z. Nussinov, B. Normand, C. D. Batista and S. A. Trugman,

Phys. Rev. B**75**, 094411 (2007).*Frustration and Entanglement in the t2g Spin-Orbital Model on the Triangular Lattice: Valence-Bond and Generalized Liquid States*

B. Normand and A. M. Oleś,

Phys. Rev. B**78**, 094427 (2008).*Multicolored Quantum Dimer Models, Resonating Valence-Bond States, Color Visons, and the Triangular-Lattice t2g Spin-Orbital System*

B. Normand,

Phys. Rev. B**83**, 064413 (2011).*Frontiers in Frustrated Magnetism*

B. Normand,

Contemporary Physics**50**, 4, 533 (2009).*Phase Diagram of the Heisenberg Spin Ladder with Ring Exchange*

V. Gritsev, B. Normand and D. Baeriswyl,

Phys. Rev. B**69**, 094431 (2004).*Low-Energy Properties of Anisotropic Two-Dimensional Spin-1/2 Heisenberg Models in Staggered Magnetic Fields*

B. Xi, S. Hu, J. Z. Zhao, G. Su, B. Normand and X. Q. Wang,

Phys. Rev. B**84**, 134407 (2011).*Accurate Determination of the Gaussian Transition in Spin-1 Chains with Single-Ion Anisotropy*

S. Hu, B. Normand, X. Q. Wang and L. Yu,

Phys. Rev. B**84**, 224402 (2011).