A possible spin liquid phase and topological entanglement entropy for J1‐J2 Heisenberg model on the honeycomb lace Donna Sheng CSUN Shoushu Gong (CSUN) Olexei I. Motrunich (Caltech) M. P. A. Fisher (UCSB) Outline: Introducon and movaon; DMRG and VMC results; Phase diagram; Magnec order; Spin gap, decay of VBS order (for torus and cylinder) Entanglement entropy, spectra (not complete) DMRG and VMC comparisons; Open issues‐‐‐Spin liquid or VBS Introducon and movaon: Resonang valence bond (RVB) state and quantum spin liquid. (P. W. Anderson, 1972). Spin liquid: no symmetry breaking even at zero temperature and has topological order, at least for gapped spin liquid. Z2 spin liquid found in contrived (dimer) models, kagome extended spin models with large Jz (three neighboring couplings) Heisenberg model on spin‐1/2 kagome and J1‐J2 square lace, Hubbard model on the honeycomb lace. H. C. Jiang, et. al., Phys. Rev. Le. 101, 117203 (2008). S. Yan, et. al., Science 332, 1173 (2011). H. C. Jiang, et. al., Phys. Rev. B 86, 024424 (2012). L. Wang, et. al., arxiv: 1112.3331, (2011). Z. Y. Meng, et. al. Nature (London) 464, 847 (2010). S. Sorella, et. al. arxiv: 1207.1783, (2012). …… Entanglement entropy (to posively idenfy quantum spin liquid): For a gapped phase universal constant E(LB)=aLB‐γ A B A. Kitaev and J. Preskill, PRL 96, 110404 (2006), M. Levin and X.‐G. Wen, PRL 96, 110405 (2006). For a topological ordered phase, γ=ln(D). D is the total quantum dimension. DMRG in cylinder systems. H. C. Jiang, Z. Wang, L. Balents, arxiv:1205. 4289. Depenbrock, McCulloch, Schollwoeck, PRL 109, 067201 (2012). Topological Entanglement Entropy and Minimum Entangled states‐‐‐‐Y. Zhang, T. Grover, A. Turner, M. Oshikawa, A. Vishwanath, arXiv:1111.2342 (2011). Entanglement Spectra: Hui Li and F. D. M. Haldane Phys. Rev. Le. 101, 010504 No topological degenerang states have been found yet Introducon and movaon: J1‐J2 Heisenberg model on the honeycomb lace ED: J. B. Fouet, et. al., EPJB 20, 241 (2001); A. F. Albuquerque, et. al., PRB 84, 024406 (2011); H. Mosadeq, et. Al., JPCM 23, 226006 (2011). VMC: B. K. Clark, et. al., PRL 107, 087204 (2011). Analyc: F. Wang, PRB 82, 024419 (2010); Y. M. Lu, et. al., PRB 84, 024420 (2011). …… P l a q u e ^ e VBS or spin Neel phase liquid A. F. Albuquerque, et. al. PRB 84, 024406 ? (2011). 0 Jc1~0.2 Jc2~0.35 ? Paramagnec Stripe or staggered dimer phase Introducon and movaon: J1‐J2 Heisenberg model on the honeycomb lace A. F. Albuquerque, et. al. PRB 84, 024406 (2011). J3~0, J2 from 0.2 to 0.35, both the plaquee/columnar and staggered VBS orders are weak, might be a phase without symmetry breaking !!! Introducon and movaon: J1‐J2 Heisenberg model on the honeycomb lace F. Wang, PRB 82, 024419 (2010); Y. M. Lu, et. al., PRB 84, 024420 (2011); B. K. Clark, et. al., PRL 107, 087204 (2011). Gapped spin liquid has a low variaonal energy close to the ED or DMRG results Both the slave fermion and Schwinger boson lead to a gapped spin liquid for J2≥0.2; Around J2=0.2, the energy of VMC for spin liquid states are close to the exact DMRG results. Introducon and movaon: J1‐J2 Heisenberg model on the honeycomb lace I. Kimchi, et. al., arxiv: 1207. 0498. Featureless bose insulator without topological order on the honeycomb lace is also possible. One more challenge to detect the topological order. Paramagnec phase Plaquee VBS Trivial featureless state Topological ordered SL Large scale DMRG (with SU(2) symmetry) calculaons and VMC studies on this model! Outline: Introducon and movaon; DMRG and VMC results; Phase diagram; Magnec order; Spin gap, decay of VBS order (for some geometries) Entanglement entropy; spectra (not complete) DMRG and VMC comparisons; Conclusion and open issues‐‐‐Spin liquid or VBS Phase diagram: Staggered dimer Jc1=0.17J1 Jc2=0.35J1 Staggered magnec moment, spin structure factor: Jc1=0.17J1 ED data is from A. F. Albuquerque, et. al. PRB 84, 024406 (2011). DMRG in PBC, kept 20000~40000 U(1) equivalent states, truncaon error<1.0‐6 . Dimer structure factor: Spin gap: for J2=0.25, 0.3, N starts from N=2*4*4 N=2*3*3, 2*4*3, 2*4*4, 2*5*4, 2*5*5, 2*6*5, 2*6*6 Dimer correlaon in zigzag cylinder: N=2*18*6 for blue and red bonds, respecvely. J2=0.25: J2=0.3: Response to VBS pertubaon (zigzag): Plaquee pinning Jpin<J1 Columnar pinning Jpin>J1 Response to VBS pertubaon (zigzag): Jpin=0.5 J2=0.25: J2=0.3: (L2=6) ξdec ~2.4 (L2=9) Jpin=2.0 J2=0.25: J2=0.3: (L2=6) ξdec ~2.5 (L2=9) Decay faster in wider system Outline: Introducon and movaon; DMRG and VMC results; Phase diagram; Magnec order; Spin gap, decay of VBS order (for some geometries) Entanglement entropy DMRG and VMC comparisons; Conclusion and open issues‐‐‐Spin liquid or VBS Entanglement entropy: For a gapped phase E(LB)=aLB‐γ universal constant A B A. Kitaev and J. Preskill, PRL 96, 110404 (2006), M. Levin and X.‐G. Wen, PRL 96, 110405 (2006). For a topological ordered phase, γ=ln(D). D is the total quantum dimension. DMRG in cylinder systems. H. C. Jiang, Z. Wang, L. Balents, arxiv:1205. 4289. In large Lx limit, DMRG gets the minimal entangled state with vercal cut in cylinder Entanglement Entropy (in armchair cut): L1 dependence of entropy Entanglement Entropy (in zigzag cut): 1/N dependence of entropy for give L2 L2=even L2=5 The fings of entropy in large L1 limit by L2=4, 6 lead to the TEE consistent with the armchair results. Entanglement Entropy (in armchair cut): J2=0.25, γ=0.51 J2=0.3, γ=0.675 a. Topological ordered phase b. γ is close to the value ln2 of Z2 spin liquid Coupling dependence of entanglement spectrum (some large eigenvalues) in zigzag cylinder N=2*12*6. Coupling dependence of entanglement spectrum in armchair cylinder N=2*12*6 DMRG and VMC comparison: Y. M. Lu, et. al. PRB 84, 024420 (2011); B. K. Clark, et. al. PRL 107, 087204 (2011). Z2 spin liquid for this VMC wave funcon The agreement between the DMRG and VMC is striking! Conclusions: The intermediate phase of this honeycomb J1‐J2 model is possibly a gapped spin liquid with vanishing magnec and dimer orders, but very strong pinning strengthens VBS order, there could be a real phase transion between SL & VBS ‐‐ Need Further Study. The topological entanglement entropy and the DMRG‐VMC comparisons support the Z2 spin liquid for this phase. It could sele the issue if we can obtain the near degenerang states, and compare their entanglement spectra. Thank you for your aenon!.