1. Integrable Models and Applications Florence, 15-20 September 2003 G. Morandi F. Ortolani E. Ercolessi C. Degli Esposti Boschi F. Anfuso S. Pasini P. Pieri (Univ. Camerino) L. Campos Venuti (Univ. Stuttgart) Spin-1 Chains: Critical Properties and CFT Condensed Matter Theory Group in Bologna M. Roncaglia

2. Integrable Models and Applications Florence, 15-20 September 2003 1D arrangement of spins with interactions Exact solution for S=½ Heisenberg model (Bethe, 1931) From spin ½ to fermions (Jordan-Wigner, 1928) Quantum fluctuations prevent from LRO even at T=0 Algebraic decay of correlation functions Gapless excitations carrying spin ½

3. Integrable Models and Applications Florence, 15-20 September 2003 S=½ XXZ model FMAFM 01 BKTFDSD Gaussian model Continuum limit: = compactified boson

4. Integrable Models and Applications Florence, 15-20 September 2003 Spin-1 models b b=1 BA integrable (Lai-Sutherland, 1974) b=  1 BA integrable (Babudjian-Takhtajan, 1982) SU(3) k=1 SU(2) k=2 WZNW model Effective 3 free Majorana fermions, c=3/2 b=1/3 Only GS is known, short ranged (AKLT) Typical configurations String order parameter

5. Integrable Models and Applications Florence, 15-20 September 2003 Spin-1 Heisenberg model Integer-spin chains are gapped, with finite correlation length  (Haldane’s conjecture,1983) Verified numerically and experimentally Mapping onto an O(3) nonlinear sigma model Is the Haldane state destroyed by anisotropies? Non vanishing string order No Neel order

6. Integrable Models and Applications Florence, 15-20 September 2003 Inclusion of anisotropies ( D ) D Lower symmetry: from SU(2) to U(1) x Z 2 Quantum number conserved =Ising-like D = single ion =spin1/2 Bosonization on a two-leg ladder (Schulz, 1986) S=1

7. Integrable Models and Applications Florence, 15-20 September 2003 Phase diagram Large-D: unique GS, gapped Neel: double GS, gapped Haldane: unique GS, gapped XY1: gapless with spin 1 excitations XY2: gapless with spin 2 excitations From W. Chen et al., PRB 67, 104401 (2003)

8. Integrable Models and Applications Florence, 15-20 September 2003 Critical phases First order c=1/2 Haldane-Neel (c=1/2) c=1 Haldane-Large D + XY1 + XY2 (c=1) c=3/2 ? Tricritical point SU(2) k=2 WZNW (c=3/2)

9. Integrable Models and Applications Florence, 15-20 September 2003 Haldane-Large D Classically, we have a planar phase for D > Ansatz: inclusion of small fluctuations along z-axis Path integral approach with coherent states Integration over the field l Continuum limit treating  (x,  as a slow variable

10. Integrable Models and Applications Florence, 15-20 September 2003 Gaussian model (c=1) Compactification radius where The vertex operators V mn have conformal dimensions The total magnetization coincides with the winding number m K=  g The mass generation term is =dual field

11. Integrable Models and Applications Florence, 15-20 September 2003 Finite size effects CFT on a finite size chain of length L The excited states are related to the dimensions label the secondary states The levels E mn are calculated numerically Two parameters ( v,K ) to be fixed Only K determines the universality class BKTFDSD 1/2 1 2 K

12. Integrable Models and Applications Florence, 15-20 September 2003 Numerical calculations Exact diagonalization small systems Density matrix renormalization group “DMRG” (White,1992) Usually works best for gapped systems, but very accurate if combined with CFT High accuracy approx. method (<0.0001% on GS energy) Multi-target algorithm that converges on the first low-lying states Every calculation is done within a given sector of the total magnetization along z-axis Reference: “Density Matrix Renormalization”, Peschel et al. Eds. (Springer, 1998)

13. Integrable Models and Applications Florence, 15-20 September 2003 Along the Haldane-Large D line ( D=0.99 ) m=1,n=0 m=0,n=0 secondary m=2,n=0 m=0, n=1 v(th) = 2.45, K(th) = 1.285v = 2.58, K = 1.328

14. Integrable Models and Applications Florence, 15-20 September 2003  D  v(num)K(num)v(th)K(th) (0.5, 0.65)2.1971.5802.071.52 (1, 0.99)2.5881.3282.451.285 (2.59, 2.3)3.700.853.430.91 (3.2, 2.9)4.450.5263.770.83 Tricritical point ? Conjecture: K=1/2 (SD) instead of K=1 (FD) Towards the tricritical point

15. Integrable Models and Applications Florence, 15-20 September 2003 XY2 phase Perturbative approach for large negative D No 0’s Spin 1Spin 1/2 where Effective S=1/2 XXZ 2nd order

16. Integrable Models and Applications Florence, 15-20 September 2003  eff = 0 K(num) = 0.998 Free Dirac

17. Integrable Models and Applications Florence, 15-20 September 2003

18. Integrable Models and Applications Florence, 15-20 September 2003 Open problems Mass generation away from the H-D critical line is described by a sine-Gordon theory. What is the string-order parameter in the continuum? What is the nature of the transition between the two critical phases XY1 and XY2? Tricritical point?

19. Integrable Models and Applications Florence, 15-20 September 2003 Conclusions Reference: C. Degli Esposti Boschi, E. Ercolessi, F. Ortolani and M. Roncaglia, “On c=1 Critical Phases in Anisotropic Spin-1 Chains”, to appear in EPJ B, cond-mat/0307396. A combined use of both analytical and numerical calculations gives interesting quantitative results for the study of critical phases. We have found that the c=1 phases in the S=1 -D model are described by a pure Gaussian theory (without any orbifold) There are strong indications that the tricritical point at large D and does not belong to the universality class described by a SU(2) k=2 WZNW theory.