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Two dimensional staggered current phase

Two dimensional staggered current phase. Congjun Wu. Kavli Institute for Theoretical Physics, UCSB. Reference: C. Wu, J. Zaanen, and S. C. Zhang, Phys. Rev. Lett. 95, 247007 (2005). C. Wu and S. C. Zhang, Phys. Rev. B 71, 155115(2005);

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Two dimensional staggered current phase

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  1. Two dimensional staggered current phase Congjun Wu Kavli Institute for Theoretical Physics, UCSB Reference: C. Wu, J. Zaanen, and S. C. Zhang, Phys. Rev. Lett. 95, 247007 (2005). C. Wu and S. C. Zhang,Phys. Rev. B 71, 155115(2005); S. Capponi, C. Wu and S. C. Zhang, Phys. Rev. B 70, 220505(R) (2004). UCSB, 01/13/2006

  2. Collaborators • S. Capponi, Université Paul Sabatier, Toulouse, France. • J. Zaanen, Instituut-Lorentz for Theoretical Physics, Leiden University, the Netherlands. • S. C. Zhang, Stanford. Many thanks to D. Ceperley, D. Scalapino for helpful discussions.

  3. Estimated orbital AF magnetic moment per plaquette . Background: pseudogap in high Tc superconductivity • D-density wave state? It is related but different from the staggered flux phase. Chakravarty, et. al., PRB 63, 94503 (2000); M. Hermele, T. Senthil, M. P. A. Fisher, PRB 72, 104404 (2005). Affleck and Marston, PRB 37, 3774 (1988); Lee and Wen, PRL 76, 503 (1996). • Neutron scattering results are controversial. H. A. Mook et al., PRB 69, 134509 (2004). C. Stock et al., PRB 66, 024505 (2002).

  4. Background: T=17K transition in URu2Si2 • AF moments are too small to explain the specific heat anomaly. • Hidden order. Incommensurate orbit current state? • NMR line-width broadening below Tc. P. Coleman, et al., Nature 417, 831 (2002). O. O. Bernal, PRL 87, 196402 (2001).

  5. Background: two-leg ladder systems • Analytical results: Bosonization + RG. H. H. Lin, L. Balents, and M. P. A. Fisher, Phys. Rev. B 58, (1998) J. Fjarestad, and J. B. Marston, Phys. Rev. B 65, 125106 (2002). C. Wu , W. V. Liu, and E. Fradkin, Phys. Rev. B 68, 115104(2003) • Numerical results: DMRG. • Marston et. al., PRL 89, 56404, (2002); U. Schollwöck et al., PRL 90, 186401, (2003).D. Scalapino, S. White, and I. Affleck, Phys. Rev. B 64, 100506 (2001).

  6. Ferromagnetic moments ( ) along [110] direction. Use spin-orbit coupling to probe the DDW phase • SO coupling induced ferromagnetism in the DDW phase in La2-xBaxCu2O4. • The DDW state: staggered orbital moments. • The ferromagnetic state: uniform spin moments. • Staggered Dzyaloshinskii-Moriya SO coupling. C. Wu, J. Zaanen, and S. C. Zhang, Phys. Rev. Lett. 95, 247007 (2005).

  7. Reliable 2D QMC results without the sign problem! • 2D staggered currents in a bi-layer model. • Alternating sources and drains; curl free v.s. source free S. Capponi, C. Wu and S. C. Zhang, PRB 70, 220505 (R) (2004). top view d-density wave

  8. Outline • Spin-orbit coupling induced ferromagnetism in the DDW phase in La2-xBaxCu2O4. • The 2D staggered ground state current phase in a bi-layer model. • T-invariant decomposition and the sign problem in quantum Monte Carlo simulations.

  9. The tilt distortion in La2-xBaxCuO4 • Low temperature orthorhombic (LTO) phase at doping<0.12. • The Dzyaloshinskii-Moriya type SO coupling appears in the band structure. • Spin processes as electron hops in the lattice. Time reversal invariance requires the appearance of “i”.

  10. Pattern of the DM vector N. E. Bonesteel et al., PRL 68, 2684 (1992). • Hermitian. • 2-fold rotations around c axis on O sites. • Inversion respect to Cu sites. • Reflection respect to the [110] direction. • DM vectors:

  11. » l » c » ( t mev , 2 mev , 20 mev) 100 SO coupling induced ferromagnetism in the DDW phase • DDW as staggered charge flux. • DM coupling as staggered spin flux. Assume • Ferromagnetic moments appear with doping.

  12. General pattern of the DM vector • Spin polarization is fixed along the [110] direction regardless of the ration of .

  13. Define General pattern of DM vectors • The magnitude of ferromagnetic moment is also robust due to the large anisotropy of the Dirac cones. • In realistic systems, . • S is only suppressed 15% compared to the value at .

  14. Experiment proposal • Ferromagnetic moments should be easy to detect by neutron scattering, muon spin relaxation, hysteresis behavior etc. So far, no such moments are reported. • SO coupling by itself does not induce spin moments in superconducting phase due to the TR invariance.

  15. Staggered spin galvanic effect • If the DDW phase does not exists, a spin polarization along the [110] direction can induce a DDW orbital moment.

  16. YBCO system (under investigation) • Due to the CuO pyramid, the inversion symmetry is broken in each layer. SO coupling is the uniform Rashba type but with opposite sign for two adjacent layers. • Pairing structure: mixed singlet and triplet pairing. • Rashba coupling effect in the DDW phase. • No spin moments on Cu sites, but AF moments can appear on O sites. C. Wu, J. Zaanen, in preparation.

  17. Outline • Spin-orbit coupling induced ferromagnetism in the DDW phase in La2-xBaxCu2O4. • The 2D staggered ground state current phase in a bi-layer model. • T-invariant decomposition and the sign problem in quantum Monte Carlo simulations.

  18. The bi-layer Scalapino-Zhang-Hanke Model D. Scalapino, S. C. Zhang, and W. Hanke, PRB 58, 443 (1998). • U, V, J are interactions within the rung. • No inter-rung interaction.

  19. Reliable 2D QMC results without the sign problem! • Alternating sources and drains; curl free v.s. source free • T-invariant decomposition in quantum Monte Carlo (QMC) simulations. top view d-density wave • T=Time reversal operation • *flipping two layers S. Capponi, C. Wu and S. C. Zhang, PRB 70, 220505 (R) (2004).

  20. Fermionic auxiliary field QMC results at T=0K • The equal time staggered current-current correlations • Finite scaling of J(Q)/L2 v.s. 1/L. • True long range Ising order. S. Capponi, C. Wu and S. C. Zhang, PRB 70, 220505 (R) (2004).

  21. Disappearance of the staggered current phase i) increase ii) increase iii) increase doping

  22. + Strong coupling analysis at half-filling • The largest energy scale J>>U,V. • Project out the three rung triplet states. • Low energy singlet Hilbert space: doubly occupied states, rung singlet state. - =

  23. rung current bond strength cdw Pseudospin SU(2) algebra • The pseudospin SU(2) algebra v.s. the spin SU(2) algebra. • Pseudospin-1 representation. • Rung current states

  24. Pseudospin-1 AF Heisenberg Hamiltonian • t// induces pseudospin exchange. • Anisotropic terms break SU(2) down to Z2 .

  25. rung singlet staggered current staggered bond order Competing phases • Neel order phases and rung singlet phases. CDW

  26. favors the easy plane of staggered current and CDW. favors the easy plane of staggered current and bond order. the easy axis of the staggered current SU(2)Z2 Competing phases • 2D spin-1 AF Heisenberg model has long range Neel order. • Subtle conditions for the staggered current phase. • is too large  polarized pseudospin along rung bond strength • is too large  rung singlet state

  27. Outline • Spin-orbit coupling induced ferromagnetism in the DDW phase in La2-xBaxCu2O4. • The 2D staggered ground state current phase in a bilayer model. • T-invariant decomposition and the sign problem in quantum Monte Carlo simulations.

  28. Auxiliary field QMC Probability: positive number Fermions: Grassmann number Auxiliary Field QMC Blankenbecer, Scalapino, and Sugar. PRD 24, 2278 (1981) • Using path integral formalism, fermions are represented as Grassmann variables. • Transform Grassmann variables into probability. • Decouple interaction terms using Hubbard-Stratonovich (H-S) bosonic fields. • Integrate out fermions and the resulting fermion functional determinants work as statistical weights.

  29. Absence of the sign problem in the negative U Hubbard model • HS decoupling in the density channel. • B is the imaginary time evolution operator. • Factorize the fermion determinant into two identical real parts.

  30. The sign (phase) problem!!! • Generally, the fermion functional determinants are not positive definite. Sampling with the absolute value of fermion functional determinants. • Huge cancellation in the average of signs. • Statistical errors scale exponentially with the inverse of temperatures and the size of samples. • Finite size scaling and low temperature physics inaccessible.

  31. A general criterion: symmetry principle • Need a general criterion independent of factorizibility of fermion determinants. The T (time-reversal) invariant decomposition. • Applicable in a wide class of multi-band and high models at any doping level and lattice geometry. Reference: C. Wu and S. C. Zhang,Phys. Rev. B 71, 155115(2005); C. Capponi, C. Wu, and S. C. Zhang, Phys. Rev. B 70, 220505(R) (2004). C. Wu and S. C. Zhang, Phys. Rev. Lett. 91, 186402 (2003).

  32. T-invariant decomposition CW and S. C. Zhang, PRB 71, 155115 (2005); E. Koonin et. al., Phys. Rep. 278 1, (1997) • Theorem: If there exists an anti-unitary transformation T for any H-S field configuration, then Generalized Kramer’s degeneracy • I+B may not be Hermitian, and even not be diagonalizable. • Eigenvalues of I+B appear in complex conjugate pairs (l, l*). • If l is real, then it is doubly degenerate. • T may not be the physical time reversal operator.

  33. The sign problem in spin 1/2 Hubbard model • U<0: H-S decoupling in the density channel. • T-invariant decomposition  absence of the sign problem • U>0: H-S decoupling in the spin channel. • Generally speaking, the sign problem appears. • The factorizibility of fermion determinants is not required. • Validity at any doping level and lattice geometry. • Application in multi-band, high spin models.

  34. Distribution of eigenvalues

  35. T=Time-reversal*flip two layers • T-invariant operators: total density, total density; • bond AF, bond current. • Absence of the sign problem at g, g’, gc>0,. .

  36. Summary • Spin-orbit coupling induced ferromagnetism in the DDW phase in La2-xBaxCu2O4. • The 2D staggered ground state current phase in a bi-layer model. • T-invariant decomposition and the sign problem in quantum Monte Carlo simulations.

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