¶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 J1 and NNN J2 couplings denoted by solid and dashed lines, respectively. Below the ab plane containing V4+ S=1/2 ions is shown.

Phase Diagram of the Spin-1/2 Phase Diagram of the Spin-1/2 Frustrated Square LatticeFrustrated Square Lattice

Bond NematicBond Nematic Spin LiquidSpin Liquid

PO4

O

VO5

J1

J2

Fig.1

IntroductionIntroduction

The experimental study of a The experimental study of a series of vanadates which series of vanadates which represent prototypes of the spin represent prototypes of the spin 1/2 frustrated square lattice 1/2 frustrated square lattice (FSL) model (Fig.1) is presented. (FSL) model (Fig.1) is presented. From NMR, From NMR, μSR, magnetization μSR, magnetization and specific heat measurements and specific heat measurements [2-5] we derive information on [2-5] we derive information on how the static and dynamic how the static and dynamic properties evolve as a function properties evolve as a function of the ratio r=Jof the ratio r=J22/J/J11..

In particular we shall address:In particular we shall address:

The r and T dependence of the order The r and T dependence of the order parameter.parameter.

The r and T dependence of the low-The r and T dependence of the low-energy dynamics.energy dynamics.

The effect of spin dilution on the FSLThe effect of spin dilution on the FSL

The effect of high pressure on r The effect of high pressure on r

Fig.2: Temperature dependence of the Fig.2: Temperature dependence of the local field at the muon normalized to local field at the muon normalized to its value for Tits value for T0 in the collinear phase 0 in the collinear phase of three different vanadates. The of three different vanadates. The temperature is normalized to Ttemperature is normalized to TNN

Several vanadates as Several vanadates as LiLi22VO(Si,Ge)OVO(Si,Ge)O44 and Pb and Pb22VO(POVO(PO44))22 are characterized by a collinear are characterized by a collinear ground-state, as initially ground-state, as initially confirmed from the analysis of confirmed from the analysis of 77Li Li NMR spectra [2]. Zero-field NMR spectra [2]. Zero-field μSRμSR (Fig. 2)measurements show that (Fig. 2)measurements show that the order parameter has a critical the order parameter has a critical exponent exponent ββ0.24 [2,3], 0.24 [2,3], characteristic of acharacteristic of a 2DXY system. 2DXY system.

For a 2D S=1/2 AF For a 2D S=1/2 AF [6][6]

In LiIn Li22VOSiOVOSiO44 for T>T for T>TNN 1/T1 1/T1 increases exponentially on increases exponentially on cooling, as expected for a 2D cooling, as expected for a 2D S=1/2 AF, but with a reduced spin-S=1/2 AF, but with a reduced spin-stiffness (Fig.4) [4].stiffness (Fig.4) [4].

Low energy excitations can be Low energy excitations can be probed by means of nuclear or probed by means of nuclear or muon spin-lattice relaxation rate.muon spin-lattice relaxation rate.

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.00.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

1/

T1 (

ms-1

)

T/(1-x)2 (K)

Li2V

1-xTi

xSiO

5

x=0 x=0.5 x=0.11 x=0.2

0 10 20 30 40 500.01

0.1

1

0 2 4 6 8 10 12

0.1

Sr2CuO

2Cl

2

Pb2VO(PO

4)

2

(s

-1)

[JC/(T-T

N)]1/2

Pb2VO(PO

4)

2

(s

-1)

T(K)

0.1 1

0.1

0.2

0.3

0.4

0.1 1 100.0

0.2

0.4

0.6

0.8

1.0

(s

-1)

1/T (K-1)

(s

-1)

T(K)

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

1.2

=0.24

B(T

)/B

(0)

T/TN

Li2VOGeO

4

Li2VOSiO

4

Pb2VO(PO

4)

2

withwith

for for Heisenberg Heisenberg

2DXY 2DXY In PbIn Pb22VO(POVO(PO44))22 the T the T

dependence of dependence of λ is consistent λ is consistent with a 2DXY behaviour for Twith a 2DXY behaviour for TTTN N

(Fig. 3) [3].(Fig. 3) [3].

0.2 0.4 0.6 0.80.0

0.5

1.0

1.5

2.0

Li2VOSiO

4 , = J

1+J

2

La2CuO

4 , = J

T1(T

>>)

/T1(T

)

T/

In BaCdVO(POIn BaCdVO(PO44))22, a compound , a compound which is very close to the which is very close to the boundary to the non-magnetic boundary to the non-magnetic bond-nematic phase (Fig. 1), one bond-nematic phase (Fig. 1), one notices that notices that λλ increases increases logarithmically with decreasing logarithmically with decreasing temperature (Fig. 5) [3]. This temperature (Fig. 5) [3]. This behaviour suggests the onset of behaviour suggests the onset of 1D stripy-like correlations, as 1D stripy-like correlations, as expected for the nematic phase expected for the nematic phase [3]. [3].

Fig.3: Temperature dependence of the Fig.3: Temperature dependence of the muon longitudinal relaxation rate muon longitudinal relaxation rate λ in λ in PbPb22VO(POVO(PO44))22. . The solid line shows the The solid line shows the behaviour expected for a 2D XY model. behaviour expected for a 2D XY model. In the inset the same data are reported In the inset the same data are reported for for T → TT → TNN as a function of Jas a function of JCC//((T − TT − TNN), ), together with the data derived for together with the data derived for Sr2CuOSr2CuO22ClCl22..

Fig.4: Temperature dependence of Fig.4: Temperature dependence of 1/T1/T11, normalized to its high , normalized to its high temperature value, in Litemperature value, in Li22VOSiOVOSiO44 compared to Lacompared to La22CuOCuO44 , a prototype of , a prototype of 2D Heisenberg AF. The temperature is 2D Heisenberg AF. The temperature is normalized to the Curie-Weiss normalized to the Curie-Weiss temperature.temperature.

Fig.5: Temperature dependence of Fig.5: Temperature dependence of λ in λ in BaCdVO(POBaCdVO(PO44))22. . In the inset the same data In the inset the same data are reported vs. 1are reported vs. 1/T /T in a linear-log scale in in a linear-log scale in order to evidence the logarithmic increase order to evidence the logarithmic increase of λof λ above Tabove TNN..

Fig.2

Fig.3

Fig.4

Fig.5

At variance with non-frustrated 2D At variance with non-frustrated 2D S=1/2 AF, a simple dilution model S=1/2 AF, a simple dilution model

does not apply to Lidoes not apply to Li22VV1-x1-xTiTixxSiOSiO4 4 (Fig. (Fig. 6) [4].6) [4].

Fig.6

Order Order ParameterParameter

Low-energy excitationsLow-energy excitations

References:References:[1] [1] P. Chandra and B. Doucot., Phys. Rev. B 38, 9335 (1988).

[[2] 2] R. Melzi R. Melzi et al., Phys.Rev.Lett. 85, 1318 (2000). P. Carretta 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 pressureTuning frustration with pressure

Fig.7: Fig.7: Electron density distribution in Electron density distribution in LiLi22VOSiOVOSiO44 in the in the ac ac plane, at plane, at y y //bb=0.25 =0.25 level, derived from the structure factorslevel, derived from the structure factors measured by XRD after the high pressuremeasured by XRD after the high pressure experiments. Relative distances in Å are experiments. Relative distances in Å are reported on the axes. Li reported on the axes. Li xx, , z z coordinates coordinates are also indicatedare also indicated.

Hopping integrals Hopping integrals ttlmnlmn in meV from site in meV from site i i to site to site j j distant distant lla+a+mmb+b+nnc and for different pressures in GPa. c and for different pressures in GPa. PP=0 corresponds to ambient pressure. The hopping =0 corresponds to ambient pressure. The hopping integrals are given up to the fourth nearest integrals are given up to the fourth nearest neighbors. Further hopping integrals are small and neighbors. Further hopping integrals are small and may be neglected. The magnetic couplings Jmay be neglected. The magnetic couplings J i i in K are in K are obtained by standard superexchange theory as obtained by standard superexchange theory as JJiittii

22 / /UU, using , using UU=5eV for the screened Coulomb =5eV for the screened Coulomb interaction. Notice that the ratio Jinteraction. Notice that the ratio J22 /J /J11 is independent is independent of of U U up to order up to order tti i //UU..

The application of pressures of The application of pressures of the order of a few GPa modifies the order of a few GPa modifies the structure and accordingly the the structure and accordingly the ratio ratio r=Jr=J22/J/J11 . By means of XRD we . By means of XRD we have investigated the have investigated the modification in the crystal modification in the crystal structure up to P= 7.6 GPa in a structure up to P= 7.6 GPa in a LiLi22VOSiOVOSiO44 single crystal [7]. Then, single crystal [7]. Then, by performing band structure by performing band structure calculations it was possible to calculations it was possible to derive how the nearest neighbour derive how the nearest neighbour (t(t11) and next-nearest neighbour ) and next-nearest neighbour (t(t22) hopping integrals change with ) hopping integrals change with pressure. Since those systems are pressure. Since those systems are Mott insulator with UMott insulator with Ut one has t one has that that JJ22/J/J11= (t= (t22/t/t11))22 . .