¶ CNISM-Dipartimento di Fisica “A. Volta,” Università di Pavia, 27100 Pavia, (Italy)

1. PO4 O VO5 J1 J2 Phase Diagram of the Spin-1/2 Frustrated Square Lattice ¶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: Phase diagram of the frustrated square-lattice (FSL) modeland 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 J1and NNN J2couplings denoted by solid and dashed lines, respectively. Below the ab plane containing V4+ S=1/2 ions is shown. Introduction 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 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=J2/J1. Bond Nematic Spin Liquid Fig.1

2. Low-energy excitations Order Parameter • Low energy excitations can be probed by means of nuclear or muon spin-lattice relaxation rate. • In Li2VOSiO4 for T>TN 1/T1 increases exponentially on cooling, as expected for a 2D S=1/2 AF, but with a reduced spin-stiffness (Fig.4) [4]. Fig.5 • Several vanadates as Li2VO(Si,Ge)O4 and Pb2VO(PO4)2 are characterized by a collinear ground-state, as initially confirmed from the analysis of 7Li 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] with for Heisenberg 2DXY Fig.2 • In Pb2VO(PO4)2 the T dependence of λ is consistent with a 2DXY behaviour for TTN (Fig. 3) [3]. Fig.4 Fig.5: Temperature dependence of λ in BaCdVO(PO4)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 λabove TN. • At variance with non-frustrated 2D S=1/2 AF, a simple dilution model Fig.4: Temperature dependence of 1/T1, normalized to its high temperature value, in Li2VOSiO4 compared to La2CuO4 , a prototype of 2D Heisenberg AF. The temperature is normalized to the Curie-Weiss temperature. does not apply to Li2V1-xTixSiO4 (Fig. 6) [4]. • In BaCdVO(PO4)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.6 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 TN Fig.3: Temperature dependence of the muon longitudinal relaxation rate λ in Pb2VO(PO4)2. The solid line shows the behaviour expected for a 2D XY model. In the inset the same data are reported for T → TNas a function of JC/(T − TN), together with the data derived for Sr2CuO2Cl2. Fig.3

3. Tuning frustration with pressure References: [1] P. Chandra and B. Doucot.,Phys. Rev. B 38, 9335 (1988). [2] R. Melzi et al.,Phys.Rev.Lett. 85, 1318 (2000). P. Carretta et al.,Phys.Rev.Lett. 88, 047601 (2002). [3] P. Carretta et al.,Phys. Rev. B 79, 224432 (2009). [4] N. Papinuttoet al.,Phys. Rev. B 71, 174425 (2005). [5] P. Carretta et al.,J.Phys. Condens. Matter 16, S849 (2004). [6] P. Carretta et al.,Phys. Rev. Lett. 84, 366 (2000). [7] E. Pavarini et al.,Phys. Rev. B 77, 014425 (2008). • The application of pressures of the order of a few GPa modifies the structure and accordingly the ratio r=J2/J1 . By means of XRD we have investigated the modification in the crystal structure up to P= 7.6 GPa in a Li2VOSiO4 single crystal [7]. Then, by performing band structure calculations it was possible to derive how the nearest neighbour (t1) and next-nearest neighbour (t2) hopping integrals change with pressure. Since those systems are Mott insulator with Ut one has that J2/J1= (t2/t1)2 . Hopping integrals tlmnin meV from site i to site j distant la+mb+nc 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 Ji in K are obtained by standard superexchange theory as Jiti2 /U, using U=5eV for the screened Coulomb interaction. Notice that the ratio J2 /J1 is independent of U up to order ti /U. Fig.7: Electron density distribution in Li2VOSiO4 in the ac plane, at y /b=0.25 level, derived from the structure factorsmeasured by XRD after the high pressureexperiments. Relative distances in Å are reported on the axes. Li x, z coordinates are also indicated.