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Two dimensional staggered current phase
Congjun Wu
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).
Kavli Institute for Theoretical Physics, UCSB
UCSB, 01/13/2006
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Collaborators
• S. C. Zhang, Stanford.
• S. Capponi, Université Paul Sabatier, Toulouse, France.
Many thanks to D. Ceperley, D. Scalapino for helpful discussions.
• J. Zaanen, Instituut-Lorentz for Theoretical Physics, Leiden University, the Netherlands.
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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).
• Estimated orbital AF magnetic moment per plaquette .
B21
• Neutron scattering results are controversial.
H. A. Mook et al., PRB 69, 134509 (2004). C. Stock et al., PRB 66, 024505 (2002).
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Background: T=17K transition in URu2Si2
P. Coleman, et al., Nature 417, 831 (2002).
O. O. Bernal, PRL 87, 196402 (2001).
• AF moments are too small to explain the specific heat anomaly.• Hidden order. Incommensurate orbit current state?• NMR line-width broadening below Tc.
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Background: two-leg ladder systems
• Analytical results: Bosonization + RG.
• 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).
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)
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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.
• Ferromagnetic moments ( ) along [110] direction.
B21
C. Wu, J. Zaanen, and S. C. Zhang, Phys. Rev. Lett. 95, 247007 (2005).
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Reliable 2D QMC results without the sign problem!
top view d-density wave
S. Capponi, C. Wu and S. C. Zhang, PRB 70, 220505 (R) (2004).
• 2D staggered currents in a bi-layer model.
• Alternating sources and drains; curl free v.s. source free
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Outline
• The 2D staggered ground state current phase in a bi-layer model.
• Spin-orbit coupling induced ferromagnetism in the DDW phase in La2-xBaxCu2O4.
• T-invariant decomposition and the sign problem in quantum Monte Carlo simulations.
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The tilt distortion in La2-xBaxCuO4
• Low temperature orthorhombic (LTO) phase at doping<0.12.
,0 .}.)({
ijjij chcitcH
• Spin processes as electron hops in the lattice. Time reversal invariance requires the appearance of “i”.
• The Dzyaloshinskii-Moriya type SO coupling appears in the band structure.
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Pattern of the DM vector N. E. Bonesteel et al., PRL 68, 2684 (1992).
• 2-fold rotations around c axis on O sites.
• Inversion respect to Cu sites.
• Reflection respect to the [110] direction.
)0,,()(,)0,,()( 12ˆ,21ˆ, i
eiii
eii yx
0, zji
zijzzji
jiij
• Hermitian.
yxyx eiiieiyxyx eiei,, ˆ,,ˆ,, ,
xyeii
yxeiixyyxyx yx
eiei ,ˆ,
,ˆ,,,,
• DM vectors:
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SO coupling induced ferromagnetism in the DDW phase
• DM coupling as staggered spin flux. Assume 21
.).
()(Im
,
,,
int,
chcc
cciH
y
x
eii
eiij
ii
MF
• DDW as staggered charge flux.
.).
()(
,,1
,,,1
chcc
cciH
y
x
eii
ijeii
iDM
2/)(2,1 yx
• Ferromagnetic moments appear with doping.
//k
mev)20,mev2,mev100( t
1.0// S
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• Spin polarization is fixed along the [110] direction regardless of the ration of .
General pattern of the DM vector
)( yxddw SSgOF
Sx+Sy Sx-Sy Sz DDW
TR odd odd odd odd
Two-fold rotation
odd odd even odd
reflection odd even even odd
21 /
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• In realistic systems, .
S is only suppressed 15% compared to the value at .
General pattern of DM vectors 21
1221 , • Define
chckkkicH Qkk
kk .)(2 ,//21//,
• The magnitude of ferromagnetic moment is also robust due to the large anisotropy of the Dirac cones.
k
//k
)2
,2
(
mev221
21
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• Ferromagnetic moments should be easy to detect by neutron scattering, muon spin relaxation, hysteresis behavior etc. So far, no such moments are reported.
Experiment proposal
• SO coupling by itself does not induce spin moments in superconducting phase due to the TR invariance.
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• If the DDW phase does not exists, a spin polarization along the [110] direction can induce a DDW orbital moment.
Staggered spin galvanic effect
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• 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.
YBCO system (under investigation)
• 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.
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Outline
• The 2D staggered ground state current phase in a bi-layer model.
• Spin-orbit coupling induced ferromagnetism in the DDW phase in La2-xBaxCu2O4.
• T-invariant decomposition and the sign problem in quantum Monte Carlo simulations.
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The bi-layer Scalapino-Zhang-Hanke Model
D. Scalapino, S. C. Zhang, and W. Hanke, PRB 58, 443 (1998).
)1()1()())((
)(}.{}.{
,,2
1,,2
1,,
//
dii
cicii
ciij
idic
iji
ijij
iji
nnVdcnnUSSJ
inchdctchddcctH
//t
t V J
U c
d
• U, V, J are interactions within the rung.
• No inter-rung interaction.
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Reliable 2D QMC results without the sign problem!
top view d-density waveS. Capponi, C. Wu and S. C. Zhang, PRB 70, 220505 (R) (2004).
• T=Time reversal operation
*flipping two layers
• Alternating sources and drains; curl free v.s. source free
• T-invariant decomposition in quantum Monte Carlo (QMC) simulations.
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Fermionic auxiliary field QMC results at T=0K
0.2,5.0,0,1.0,1// JVUtt
),()()(
)()(1
)(2
QrJeQJ
rrnrnL
rJ
r
rQi
iicurricurr
• Finite scaling of J(Q)/L2 v.s. 1/L.
• True long range Ising order.
• The equal time staggered current-current correlations
S. Capponi, C. Wu and S. C. Zhang, PRB 70, 220505 (R) (2004).
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Disappearance of the staggered current phase
ti) increase
ii) increase
)(4
3 JVUE
iii) increase doping
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Strong coupling analysis at half-filling
• Low energy singlet Hilbert space: doubly occupied states, rung singlet state.
• The largest energy scale J>>U,V.
UJVU4
3:E
• Project out the three rung triplet states.
-=
+
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Pseudospin SU(2) algebra
)()(2
1 iiii ddcciQ
)()(2 iiiii
curt cddcin
)()(2
1 iiiibd dccdin
rung current
bond strength
cdw
• The pseudospin SU(2) algebra v.s. the spin SU(2) algebra.
c
d
• Pseudospin-1 representation.
down,up|
• Rung current states
UJVU4
3:E
1,0;1: Q
cba ;2
1
2
i
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• Anisotropic terms break SU(2) down to Z2 .
Pseudospin-1 AF Heisenberg Hamiltonian
• t// induces pseudospin exchange.
)}()()()(
)()({
jQiQjnin
jninJH
bdbd
ijcurtcurtpseudoex
field external uniform:t
)2
1)(()(2 2
ibd iQEintH
c
d
t
//t //t
E
anisotropy site-on :E)(
4
3 JVUE
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Competing phases
rung singlet
staggered current
• Neel order phases and rung singlet phases.
CDW
staggered bond order
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Competing phases
ppz zJEzJt 0),4(2
• Subtle conditions for the staggered current phase.
is too large polarized pseudospin along rung bond strength
is too large rung singlet state
the easy axis of the staggered current
SU(2)Z2
favors the easy plane of staggered current and CDW.
t
0E favors the easy plane of staggered current and bond order.
t
E
• 2D spin-1 AF Heisenberg model has long range Neel order.
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Outline
• The 2D staggered ground state current phase in a bilayer model.
• Spin-orbit coupling induced ferromagnetism in the DDW phase in La2-xBaxCu2O4.
• T-invariant decomposition and the sign problem in quantum Monte Carlo simulations.
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Auxiliary Field QMC Blankenbecer, Scalapino, and Sugar. PRD 24, 2278 (1981)
Probability: positive number
Fermions:Grassmann
number
Auxiliary field QMC
• Decouple interaction terms using Hubbard-Stratonovich (H-S) bosonic fields.
• Integrate out fermions and the resulting fermion functional determinants work as statistical weights.
• Using path integral formalism, fermions are represented as Grassmann variables.
• Transform Grassmann variables into probability.
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Absence of the sign problem in the negative U Hubbard model
)2
1)()(
2
1)((||)(}.{
ininUinchcctHii
jij
i
)}det()1),((exp{ 2
02
|| BIinddnZi
U
0)det()det()det( BIBIBI
• Factorize the fermion determinant into two identical real parts.
• HS decoupling in the density channel.
),()()(||)(
})(exp{
,,
0
inccUH
HHdB
ii
iI
IK
• B is the imaginary time evolution operator.
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The sign (phase) problem!!!
• Huge cancellation in the average of signs.
• Generally, the fermion functional determinants are not positive definite. Sampling with the absolute value of fermion functional determinants.
signsign /OO
• Statistical errors scale exponentially with the inverse of temperatures and the size of samples.
• Finite size scaling and low temperature physics inaccessible.
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The T (time-reversal) invariant decomposition.
• Applicable in a wide class of multi-band and high models at any doping level and lattice geometry.
• Need a general criterion independent of factorizibility of fermion determinants.
A general criterion: symmetry principle
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).
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• Eigenvalues of I+B appear in complex conjugate pairs (l, l*).
• If l is real, then it is doubly degenerate.
T-invariant decompositionCW and S. C. Zhang, PRB 71, 155115 (2005); E. Koonin et. al., Phys. Rep. 278 1, (1997)
0)())(()det( **22
*11 nnBI
• Theorem: If there exists an anti-unitary transformation T
IIKK HTTHHTTHT 112 ,,1
for any H-S field configuration, then
0)det( BI
• T may not be the physical time reversal operator.
Generalized Kramer’s degeneracy
• I+B may not be Hermitian, and even not be diagonalizable.
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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
STSTnTnT
11 ,
• 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.
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Distribution of eigenvalues
35.
T=Time-reversal*flip two layers
JVUgJ
VUgJVUg
ingingining
inintchddcctH
c
i icAF
icurtbond
iibondjij
ijiSZH
4
334,
44,
4
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)2)(()()}()({
)()(}.{
2222
//
• Absence of the sign problem at g, g’, gc>0, .
)()(2
1 iiiiAF ddccin
)()(2 iiiii
curt cddcin
)()(2
1 iiiibond dccdin )()( iiii ddccin
c
d
• T-invariant operators: total density, total density;
bond AF, bond current.
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Summary
• The 2D staggered ground state current phase in a bi-layer model.
• Spin-orbit coupling induced ferromagnetism in the DDW phase in La2-xBaxCu2O4.
• T-invariant decomposition and the sign problem in quantum Monte Carlo simulations.
