**Further Projects**

**Magnetoelastic Coupling in (VO)2P2O7**

Vanadyl pyrophosphate [(VO)2P2O7, or VOPO] is a black powder of considerable industrial importance as a catalyst for methanol production, and a reliable source informs the author that DuPont has a VOPO factory somewhere in Spain. For the physicist, theorist or experimentalist, the more one looks at VOPO, the more one finds. It is a system of dimerised chains (not ladders) of

*S*=*1/2*V(4+) ions. Our early work [1] showed that, assuming a purely magnetic Hamiltonian, the thermodynamic and dynamic measurements could only be explained by a significant, and frustrating, interchain coupling.## Detailed NMR and neutron scattering experiments

then found that a consistent explanation would need a rather complex magnetic structure. Raman light scattering measurements indicated in addition a strong magnetoelastic coupling. Thus we then presented [2] qualitative arguments for why the extremely small structural distortions should have such significant magnetic effects, and combined these with a study of the coupled spin and lattice systems. Using the theoretical method of continuous unitary transformations, in which the respective magnon and phonon excitations are systematically separated and transformed, we found good qualitative and quantitative agreement with all the thermodynamic quantities already measured for the material. We also predicted the finite-temperature dynamical properties, namely thermal renormalisation of the magnon dispersion by phonons, which as far as we know still await measurement.

**Charge Order in Manganites**

Colossal magnetoresistive (CMR) manganites formed the basis for a large amount of both fundamental science and techniological application. The observation of charge-inhomogeneous stripe ordering in two classes of manganites made us investigate this phenomenon [3] within the general but minimal "Ferromagnetic Kondo Lattice model:" manganites are characterised by the competition between a double-exchange hopping interaction

*t*, driven by the kinetics of the conduction electrons and favouring ferromagnetic core-spin alignment, and a superexchange*J*between these core spins which favours antiferromagnetism. We analysed the ground-state spin and charge configurations using a self-consistent combination of analytical expressions for the conductions electron contribution and classical Monte Carlo simulations for the core spins, to map out the phase diagram in*J/t*and the filling*x*. We found a wide variety of novel charge "island" phases and spin-flux phases, with wide regions of phase separation among these. Most of the island phases indeed have striped charge order, indicating that the basic features of the materials may already be traced to the fundamental competition of*t*and*J*.**Solution Physics**

The solvation of non-polar molecules in a polar solvent like water is both a complex problem in the physics of hydrogen bonding and a process fundamental to a bunch of industries, to cooking, to the function of soap, and indeed to a number of biological processes which sustain life itself. Some of my colleagues had a very elegant model based only on the energy and entropy of the water molecules in a shell next to the polar surface, and how these compared to their energy in the bulk solvent (the energy goes up but the entropy comes down). We used this to show how further results in the solution physics of hydrophobic molecules energe naturally. One was the formation in solution of ordered structures -- micelles -- of soap-like molecules, those with a polar head but long, non-polar tails [4]. The other was the action of "third-party" cosolvent molecules which can alter the solution process of the non-polar species: they do this by interfering with (enhancing or scrambling) the structure of the bulk solvent, changing those energy and entropy terms and thus altering the balance in temperature and concentration where the large molecule can be dissolved [5].

**Ionic Hubbard Model**

The ionic Hubbard model is the minimal model for the transition from a band insulator (dictated by one-particle energy terms) to a Mott insulator (dictated by electron interaction terms). It has a generalisation away from half-filling which should remove the Umklapp terms and might create metallic rather than insulating phases. However, we investigated the phase diagram [6] for three- and four-site unit cells and found that it has the same generic properties as the two-site version, albeit more weakly and as a consequence of generated interaction terms. These properties are a band insulator (BI) for large ionicity separated from a (Mott-like) correlated insulator (CI) for large on-site repulsion by an intermediate phase (FI) notable for spontaneous symmetry breaking in its bond-order, which makes it ferroelectric and gives it fractional quasiparticles with an incommensurate (interaction-tunable) fraction.

**Cold Bosons in a Trap**

The preparation of condensates of ultra-cold atoms in tunable potentials has truly opened up a whole new chapter in many-body physics. We wanted to investigate exactly how the superfluid phase at low interaction strength

*U*goes over to the Mott insulator phase at high*U*in an experiment. Because these are done by first trapping a fixed number of atoms, the ensemble is canonical rather than grand-canonical, and the presence of 3 energy scales (*U*, the kinetic term*t*, the trap term*V*) implies 2 energy ratios, 2 transitions and 3 phases. We studied this process [7] for interacting bosons on an optical lattice in a one-dimensional, harmonic trap by high-precision DMRG calculations. We indeed found an intermediate regime between the SF and the MI, characterised by a cascade of microscopic steps. These steps are a consequence of individual boson ``squeezing'' events, where atoms are pushed from the centre to the edge of the trap with increasing*U*, and they even show an even--odd alternation which depends on how the trap is centred relative to the optical lattice. This sort of behaviour is quite generic in an external trap, and we calcualted the parameters for observing it.**Potts Models on Irregular Lattices**

Potts models are an excellent example of classical statistical mechanics at high entropies. A curiously large number of Potts models with a value of the site "direction"

*q*somehow corresponding to the connectivity*z*of the lattice only order at*T = 0*. However, the*q = 3*Potts model on the diced lattice (b) is an exception here, ordering at a finite temperature. We were working on applications of tensor-based numerical methods in classical systems and decided to investigate this type of transition by finding another such "exception." The Union-Jack lattice (a), like the diced lattice, is "irregular" in the sense that there are different types of site with different coordination numbers. We evaluated [8] the thermodynamic properties (entropy, specific heat, "magnetisation", susceptibility) of the 4-state antiferromagnetic Potts model on the Union-Jack lattice and found a previously unknown, ``entropy-driven,'' finite-temperature phase transition to a partially ordered state, similar to that in the diced lattice. With this insight we also computed the thermodynamics of Potts models on the diced and centred diced (c) lattices, the latter a third example in this class, and suggested that finite-temperature transitions and partially ordered states are ubiquitous on irregular lattices.*Magnetic Properties of (VO)2P2O7 from Frustrated Interchain Coupling*

G. S. Uhrig and B. Normand,

Phys. Rev. B**58**, R14705 (1998).*Magnetic Properties of (VO)2P2O7: Two-Plane Structure and Spin-Phonon Coupling*

G. S. Uhrig and B. Normand,

Phys. Rev. B**63**, 134418 (2001).*Island Phases and Charge Order in Two-Dimensional Manganites*

H. Aliaga, B. Normand, K. Hallberg, M. Avignon and B. Alascio,

Phys. Rev. B**64**, 024422 (2001).*Solvent-Induced Micelle Formation in a Hydrophobic Interaction Model*

S. Moelbert, B. Normand and P. de los Rios,

Phys. Rev. E**69**, 061924 (2004).*Cosmotropes and Chaotropes: Modelling Preferential Exclusion, Binding and Aggregate Stability*

S. Moelbert, B. Normand and P. de los Rios,

Biophys. Chem.**112**, 45 (2004)*Quantum Phase Diagram of the Generalized Ionic Hubbard Model for ABn Chains*

M. E. Torio, A. A. Aligia, G. I. Japaridze and B. Normand,

Phys. Rev. B**73**, 115109 (2006).*Quantized Squeezing and Even-Odd Asymmetry of Trapped Bosons*

S. Hu, Y. C. Wen, Y. Yu, B. Normand and X. Q. Wang,

Phys. Rev. A**80**, 063624 (2009).*Finite-Temperature Phase Transitions in Potts Models on Irregular Lattices*

Q. N. Chen, M. P. Qin, J. Chen, Z. C. Wei, H. H. Zhao, B. Normand and T. Xiang,

Phys. Rev. Lett.**107**, 165701 (2011).