1. ¶ CNISM-Dipartimento di Fisica “A. Volta,” Università di Pavia, 27100 Pavia, (Italy) ║ Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, (Germany) ‡ Dipartimento di Scienze della Terra, Università di Pavia,27100 Pavia, (Italy) $ Dipartimento di Chimica Fisica, Università di Pavia, 27100 Pavia, (Italy) * Institut für Festkörperforschung, Forschungzentrum Jülich, 52425 Jülich, (Germany) # CEMES, CNRS, 31055 Toulouse Cedex (France) Fig.1: Fig.1: Phase diagram of the frustrated square-lattice (FSL) model and the corresponding model compounds. CAF represents the collinear order, FM the ferromagnetic one and NAF the Nèel antiferromagnetic order. The inset shows the regular FSL with the NN J 1 and NNN J 2 couplings denoted by solid and dashed lines, respectively. Below the ab plane containing V 4+ S=1/2 ions is shown. Phase Diagram of the Spin-1/2 Frustrated Square Lattice Bond Nematic Spin Liquid PO 4 O VO 5 J1J1 J2J2 Fig.1 Introduction The experimental study of a series of vanadates which represent prototypes of the spin 1/2 frustrated square lattice (FSL) model (Fig.1) is presented. From NMR, μSR, magnetization and specific heat measurements [2-5] we derive information on how the static and dynamic properties evolve as a function of the ratio r=J 2 /J 1. In particular we shall address:  The r and T dependence of the order parameter.  The r and T dependence of the low-energy dynamics.  The effect of spin dilution on the FSL  The effect of high pressure on r

2. Fig.2: Temperature dependence of the local field at the muon normalized to its value for T  0 in the collinear phase of three different vanadates. The temperature is normalized to T N  Several vanadates as Li 2 VO(Si,Ge)O 4 and Pb 2 VO(PO 4 ) 2 are characterized by a collinear ground-state, as initially confirmed from the analysis of 7 Li NMR spectra [2]. Zero-field μSR (Fig. 2)measurements show that the order parameter has a critical exponent β  0.24 [2,3], characteristic of a 2DXY system. For a 2D S=1/2 AF [6] In Li 2 VOSiO 4 for T>T N 1/T1 increases exponentially on cooling, as expected for a 2D S=1/2 AF, but with a reduced spin-stiffness (Fig.4) [4].  In Li 2 VOSiO 4 for T>T N 1/T1 increases exponentially on cooling, as expected for a 2D S=1/2 AF, but with a reduced spin-stiffness (Fig.4) [4].  Low energy excitations can be probed by means of nuclear or muon spin-lattice relaxation rate. with for Heisenberg for Heisenberg 2DXY 2DXY  In Pb 2 VO(PO 4 ) 2 the T dependence of λ is consistent with a 2DXY behaviour for T  T N (Fig. 3) [3]. In BaCdVO(PO 4 ) 2, a compound which is very close to the boundary to the non-magnetic bond-nematic phase (Fig. 1), one notices that λ increases logarithmically with decreasing temperature (Fig. 5) [3]. This behaviour suggests the onset of 1D stripy-like correlations, as expected for the nematic phase [3].  In BaCdVO(PO 4 ) 2, a compound which is very close to the boundary to the non-magnetic bond-nematic phase (Fig. 1), one notices that λ increases logarithmically with decreasing temperature (Fig. 5) [3]. This behaviour suggests the onset of 1D stripy-like correlations, as expected for the nematic phase [3]. Fig.3: Temperature dependence of the muon longitudinal relaxation rate λ in Pb 2 VO(PO 4 ) 2. The solid line shows the behaviour expected for a 2D XY model. In the inset the same data are reported for T → T N as a function of J C / ( T − T N ), together with the data derived for Sr2CuO 2 Cl 2. Fig.4: Temperature dependence of 1/T 1, normalized to its high temperature value, in Li 2 VOSiO 4 compared to La 2 CuO 4, a prototype of 2D Heisenberg AF. The temperature is normalized to the Curie-Weiss temperature. Fig.5: Temperature dependence of λ in BaCdVO(PO 4 ) 2. In the inset the same data are reported vs. 1 /T in a linear-log scale in order to evidence the logarithmic increase of λ above T N. Fig.2 Fig.3 Fig.4 Fig.5 At variance with non-frustrated 2D S=1/2 AF, a simple dilution model  At variance with non-frustrated 2D S=1/2 AF, a simple dilution model does not apply to Li 2 V 1-x Ti x SiO 4 (Fig. 6) [4]. Fig.6 Order Parameter Low-energy excitations

3. References: [1] [1] P. Chandra and B. Doucot., Phys. Rev. B 38, 9335 (1988). [2] R. Melzi P. Carretta [2] R. Melzi et al., Phys.Rev.Lett. 85, 1318 (2000). P. Carretta et al., Phys.Rev.Lett. 88, 047601 (2002). [3] [3] P. Carretta et al., Phys. Rev. B 79, 224432 (2009). [4] N. Papinutto [4] N. Papinutto et al., Phys. Rev. B 71, 174425 (2005). [5] [5] P. Carretta et al., J.Phys. Condens. Matter 16, S849 (2004). [6] [6] P. Carretta et al., Phys. Rev. Lett. 84, 366 (2000). [7] [7] E. Pavarini et al., Phys. Rev. B 77, 014425 (2008). Tuning frustration with pressure Fig.7: Electron density distribution in Li 2 VOSiO 4 in the ac plane, at y / b =0.25 level, derived from the structure factors measured by XRD after the high pressure experiments. Relative distances in Å are reported on the axes. Li x, z coordinates are also indicated Fig.7: Electron density distribution in Li 2 VOSiO 4 in the ac plane, at y / b =0.25 level, derived from the structure factors measured by XRD after the high pressure experiments. Relative distances in Å are reported on the axes. Li x, z coordinates are also indicated. Hopping integrals t lmn in meV from site i to site j distant l a+ m b+ n c and for different pressures in GPa. P =0 corresponds to ambient pressure. The hopping integrals are given up to the fourth nearest neighbors. Further hopping integrals are small and may be neglected. The magnetic couplings J i in K are obtained by standard superexchange theory as J i  t i 2 / U, using U =5eV for the screened Coulomb interaction. Notice that the ratio J 2 /J 1 is independent of U up to order t i / U. The application of pressures of the order of a few GPa modifies the structure and accordingly the ratio r=J 2 /J 1. By means of XRD we have investigated the modification in the crystal structure up to P= 7.6 GPa in a Li 2 VOSiO 4 single crystal [7]. Then, by performing band structure calculations it was possible to derive how the nearest neighbour (t 1 ) and next-nearest neighbour (t 2 ) hopping integrals change with pressure. Since those systems are Mott insulator with U  t one has that J 2 /J 1 = (t 2 /t 1 ) 2.  The application of pressures of the order of a few GPa modifies the structure and accordingly the ratio r=J 2 /J 1. By means of XRD we have investigated the modification in the crystal structure up to P= 7.6 GPa in a Li 2 VOSiO 4 single crystal [7]. Then, by performing band structure calculations it was possible to derive how the nearest neighbour (t 1 ) and next-nearest neighbour (t 2 ) hopping integrals change with pressure. Since those systems are Mott insulator with U  t one has that J 2 /J 1 = (t 2 /t 1 ) 2.