Dynamic generation of spin-orbit coupling

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
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 T17K 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
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
  dopinglt0.12.

• 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
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 .

SxSy Sx-Sy Sz DDW
TR odd odd odd odd
Two-fold rotation odd odd even odd
reflection odd even even odd
13
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

• TTime 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 T0K

• 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 JgtgtU,V.

• Project out the three rung triplet states.

• Low energy singlet Hilbert space
  doubly occupied states, rung singlet state.

-

23
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
Competing phases

• Neel order phases and rung singlet phases.

CDW
26
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 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 Kramers degeneracy

• IB may not be Hermitian, and even not be
  diagonalizable.

• Eigenvalues of IB 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

• Ult0 H-S decoupling in the density channel.
• T-invariant decomposition ? absence of the sign
  problem

• Ugt0 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
TTime-reversalflip two layers

• T-invariant operators total density, total
  density
• bond AF, bond
  current.

• Absence of the sign problem at g, g, gcgt0, .

.
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.