1. THE ISING PHASE IN THE J1-J2 MODEL Valeria Lante and Alberto Parola

2. OUTLINE: introduction to the model our aim analytical approach numerical approach Conclusions the future the model motivation phase diagram { non linear sigma model Lanczos exact diagonalizations

3. What is the J 1 - J 2 model ? J1J1 J2J2 INTRODUCTION: Why the J 1 - J 2 model ? ? Can quantum fluctuations stabilize a disordered phase in spin systems at T=0 ? relevance of low dimensionality relevance of small spin spin systems: symmetry breaking (magnetization) (no 1D for Mermin-Wagner theorem) T = 0

4. Simple quantum spin model: Heisenberg model (J 2 =0) ≠ 0 at T=0 in 2D Frustration may enhance quantum fluctuations J 1 - J 2 model ~0.4~0.6 J 2 /J 1 N é el phaseCollinear phase Paramagnetic phase T=0

5. Connection with high temperature superconductivity  AF 0.040.05 M ? SC T = 0 Holes moving in a spin disordered background (?) It is worth studying models with spin liquid phases   J 1 - J 2 model Vanadate compounds VOMoO 4 Li 2 VOSiO 4 Li 2 VOGeO 4

6. 2D Heisenberg model at T=0 (J 2 =0) Some definitions    S  S  i ii {  i = 1 i  + -1 i  - Néel state GS { =   classical (S→  ) ≠    quantum  { m /S s ~ 0.6 quantum (S=1/2) = 1 classical S tot S z

7. PHASE DIAGRAM OF THE J 1 -J 2 MODEL

8. Classical (S  ) ground state (GS) at T=0 classical energy minimized by if J(q) is minimum S = e cos(q·r) + e sin(q·r) r 21 0.5 J 2 /J 1 J 2 /J 1< 0.5 : J(q) minimum at q=(  ) J 2/ J 1> 0.5 : two independent AF sublattices * J 2/ J 1= 0.5 : J(q) minimum at q=(q x  ) and q=(  q y ) * thermal or quantum fluctuations select a collinear phase (CP) with q=(0  ) or q=(  )

9. Quantum ground state at T=0 ~0.4~0.6 J 2 /J 1 O(3) X Z 2 O(3) broken symmetries  < ~ 0.4 long range N é el ordered phase   > ~ 0.6 collinear order (Ref. 1). ~    < ~ 0.6 Magnetically disordered region (Ref. 1). m ≠ 0 s s + s - n  + L / S ++ -- -n  + L / S ++ -n  + L / S --  = n · n ≠ 0 + -

10. = VBC: valence bond crystal RVB SL: resonating valence bond spin liquid | RVB > =  A(C )|C > C i ii C = dimer covering i no long-ranged order no SU(2) symmetry breaking no long-ranged spin-spin correlations dimer = 1 / √2 ( |  > - |    VBC = regular pattern of singlets at nearest neighbours: dimers or plaquettes long-ranged dimer-dimer or plaquette-plaquette order no SU(2) symmetry breaking no long-ranged spin-spin correlations

11. OUR AIM: ~0.6 J 2 /J 1 ≠ 0 collinear = 0 “disorder” ~0.6 J 2 /J 1 ≠ 0 collinear ≠ 0 = 0 “Ising” = 0 “disorder” ?

12. ANALITYCAL APPROACH : 2D Quantum model at T=0 2+1 D Classical model at T eff ≠0 Haldane mapping Non Linear Sigma Model method for  =  J 2 /J 1 > 1/2 I. The partition function Z is written in a path integral representation on a coherent states basis. II. For each sublattice every spin state is written as the sum of a “Néel” field and the respective fluctuation.

13. III. In the continuum limit, to second order in space and time derivatives and to lowest order in 1/S, Z results :  = Ising order parameter  = n · n + -

14. checks: classical limit (S → ∞ ) saddle point approximation for large  : n = n +  n + - + -- + 0 static and homogeneous same results of spin wave theory Collinear long range ordered phase

15. NUMERICAL APPROACH: Lanczos diagonalizations: On the basis of the symmetries of the effective model, an intermediate phase with = 0 and finite Ising order parameter ≠ 0 may exist. + - It can be either a: VB nematic phase, where bonds display orientational ordering VBC ( translational symmetry breaking) Analysis of the phase diagram for values of  around 0.6 for a 4X4 and a 6X6 cluster

16. Lowest energy states referenced to the GS ordered phases and respective degenerate states collinear (0,0) s S=0 (0,0) d S=0 (0,  ) S=1 ( ,0) S=1 { columnar VBC (0,0) s S=0 (0,0) d S=0 (0,  ) S=0 ( ,0) S=0 { plaquette VBC (0,0) s S=0 (  0) S=0 (0,  ) S=0 ( ,  ) S=0 { conclusions:  0.60 : (0,0) s and (0,0) d singlets quasi degenerate → Z 2 breaking  0.62 :(0,  ) S=0 higher than (0,  ) S=1 →  no columnar VBC    (    ) S=0 higher than the others →  no plaquette VBC  0.62 triplet states are gapped 4X4 6X6

17. Order Parameter 0.6 and |d> quasi-degenerate  s> + |d>)/√2 breaks Z 2     Ô r = Ŝr · Ŝ r+y - Ŝ r · Ŝ r+x lim ≠ 0 and |s> and |d> degenerate (N → ∞)   Z 2 symmetry breaking PxPx PyPy conclusions:  0.60 : P y compatible with a disordered configuration  0.60 : P x  P x for Heisenberg chains As   grows   P y   0 : vertical triplets  collinear phase ≠ 0

18. Structure factor S(k) = Fourier transform of the spin-spin correlation function Blue (cyan) triangles: S(k) on the lowest s-wave (d-wave) singlet for a 4x4 cluster. Red (green) dots: The same for a 6x6 cluster.  = (0,0) M= (0,  ) X= ( ,  ) conclusions: S(k) on |s >  S(k) on |d >    same physics  0.70 : S( ,0) grows with size   collinear order 0.60  0.62 : S(k) flat + no size dependence 0.62<   towards transition to collinear phase numerical data fitted by a SW function except at single points

19. From the symmetries of the non linear sigma model: ● At T=0 possibility of : CONCLUSIONS: Isingdisordercollinear ISING PHASE = VB nematic phase Isingdisordercollinear ? 0.62 ≠ 0 = 0 ≠ 0 = 0 The Lanczos diagonalizations at T = 0 ● Ising phase for ? <  < 0.62 ●  ~ 0.60: collection of spin chains weakly coupled in the transverse direction.

20. THE FUTURE: About the J 1 -J 2 model on square lattice Monte Carlo simulation of the NLSM action Numerical analysis (LD) of the phase: looking for a chiral phase: Ŝr · (Ŝ r+y  Ŝ r+x) About the J 1 -J 2 model on a two chain ladder Numerical analysis (LD) of the phase diagram “novel” phase diagram proposed by Starykh and Balents PRL (2004)