role of quantal phases in low-dimensional correlated electrons -nonperturbative approach
DESCRIPTION
Talk at National Center of Theoretical Sciences, Hsinchu, Taiwan Feb.1, 2002. Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach. Computational Materials Research Center National Institute for Materials Research ( 物質・材料研究機構 ) - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/1.jpg)
Role of quantal phases in low-dimensional correlated electrons
-nonperturbative approach
Computational Materials Research CenterNational Institute for Materials Research ( 物質・材料研究機構 ) Akihiro TANAKA (田中秋広) , Xiao Hu( 胡暁 ) http://www.nims.go.jp/cmsc/scm/index.html
Talk at National Center of Theoretical Sciences, Hsinchu, Taiwan Feb.1, 2002
![Page 2: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/2.jpg)
Outline of Talk
1.Examples of quantal (Berry) phases for spins:
-quantum tunneling in nanomagnets and JJ s
2.Impurity effects in spin gapped systems and superconductivity:
-role of quantal phases
3. Related physics in stripes in superconductors
![Page 3: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/3.jpg)
What is the Berry phase of a spin?
…it records the history of its directional fluctuation.
ω
)()( :
)()( : thatus llusually te textbooks
)1|(|n orientatiospin :)( ))( ,(
)( teettbut
tett
ttothersHH
tHiti
tHi
(t))for angle solid: ( ,)(
(t)T)(twhen
ST
Quantization of spin follows from ambiguity of ω mod 4π:
integer-halfor integer )4( See iSiS
Geometric nature of Berry phases can lead to far-reaching consequences.
![Page 4: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/4.jpg)
Illustration: tunneling in nano-size molecular magnets
eg. Fe 8 (Wernsdorfer et al, Science 284, 133 (1999)) Mn12 acetate (S=10)(Wernsdorfer et al cond-mat/0109066)
0)( 22 yzyyzz KKKKH
consider an easy-plane easy-axis magnet
Tunneling amp. between the 2 low energy states
z
x
y
|1>
|2>Path A Path B
■Amp=0 for half-integer S : destructive interference between paths A, B ⇒ absence of splitting of the classical levels■For integer S: constructive interference
2)()(
)(1||2 0)()A(
SSAB
eeeeAmp SBiiH
![Page 5: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/5.jpg)
Application to Josephson junctions (lattice superconductor)
continuum notation
ˆ (r ) (
r ),(
r )
The spin tunneling analysis can be carried over to the Cooper pair tunneling problem.
“Geometrically controlled quibits” Makhlin et al Rev. Mod. Phys. 73, 357(2001).
Φ
θ ˆ
present Sz , absent Sz [ Sz 1
2(n -1) ]
S Sx iSy absent present
pseudo-spin (S: controlled by gate-voltage) description of JJ arrays
Charging energy Ec →Kz
Josephson energy EJ→Ky
For large Ec:
![Page 6: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/6.jpg)
Competing/Coexisting Orders in Correlated Electrons
物質・材料研究機構 (NIMS) 田中秋広、胡暁
http://www.nims.go.jp/cmsc/scm/index.html
□ MC simulation of SO(5) theory of high Tc superconductivity
□ Nonmagnetic impurity effects and quantal phase interference
![Page 7: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/7.jpg)
Acknowledgments to:
Tokyo: N.Nagaosa , H.Fukuyama , M.Saito K. Uchinokura
Tsukuba: M. Hase, N. Taniguchi, T. Hikihara
![Page 8: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/8.jpg)
Experiments and Backgrounds
Masuda et al (CuGeO3)
1. Nonmagnetic impurity in singlet spin-gapped systems
■Spin-Peierls compound CuGeO3
AF with less than 1%Zn (Si, Mn)doping
■Spin-ladder compound SrCu2O3
AF order with ~ 1% Zn doping(Azuma et al ’ 97)
■Pseudo-gap phase of underdoped cupratesCu→Zn subst. in YBCO; weight transfer to low energy(Kitaoka et al ’93)
Basic picture
before
after
![Page 9: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/9.jpg)
Questions
These analogies have received attentions:
■Sensitivity of 40meV magnetic resonance mode to Cu→Zn subst. (Keimer, Fukuyama)
■Zn doping into staggered flux state (Pepin and P.A.Lee)
But viewing impurities (site depletion)as the static limit of mobile holes, what information can this provide for the hole-doped system and its SC?
![Page 10: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/10.jpg)
2. Spin-Peierls like (bond-centered density) order in underdoped cuprates?
Softening of LO phonon at q=(π/2,0,0)
McQueeney et al PRL 82 (1999)628( La 1.85 Sr 0.15 CuO4) & unpublished 2001(YBa2Cu3O 6.95 ) questions: ■ role of AF fluctuation on stripes? ■relation to superconductivity?
![Page 11: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/11.jpg)
System quasi-1d spin-Peierls state
Static vacancy→induced AF Mobile vacancy(hole)→superconductivity
Same origin: spin-charge phase interferenceFindings
Step1: local weight transfer: singlet→AF
A by-product SU(2) invariant phase-Hamiltonian approach
Step2: t’-type hopping + intersublattice attraction
![Page 12: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/12.jpg)
The Main IdeaIn a non-singlet state (SDW+directional fluctuations),
In a singlet state, the spin moments are quenched (no “arrows”)-> Berry phase effects should be absent
Conventional bosonization: does not give complete description of
Semiclassical methods: can only immitate as
Wanted: a method which incorporates both including space and/or time nonuniformity of singlet pair formations
( e.g. RVB )
![Page 13: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/13.jpg)
Sugawara form : Hamiltonian for free fermionU(1)
SU(2)
directional fluctuations
g e i U(1), g e i U(1)
JR ig g , JL ig g
H free JRJR JLJL 1
2[()2 (x)2 ]
g e i Q cos iQsin+ SU(2)
g e i Q cos iQsin+ SU(2)
(Q =n
, |
n |= 1)
J R ig g,
J L ig g
H free J R
J R
J L
J L
=1
2[( )2 (x)2 ]
sin2
2[(
n )2 (x
n )2 ]
![Page 14: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/14.jpg)
Starting Hamiltonian (Peierls-Hubbard model) H (t ( 1)i
i t)(ci
†ci1 h.c.) U ni n
i i
ni n
i 1
2(n
i ni )
1
6(ci
† ci )
2
exp[U
6(ci
† ci)
2 ] di exp[ 2U
3(i
2 i ci
†
2
ci)]
H [R†L†] ivF x 4Um
3Q i2 t
4Um3
Q i2 t ivF x
R
L
[R† L† ]
ivF x 0Qe-iQ
2
0QeiQ
2 ivF x
R
L
,
with i = (-1)i m
n i , Q =
n
, 0 (
4Um
3)2 (4t)2 ,
2
tan 1(3t
2Um)
Semiclassical decoupling (SDW)
Linearize: 4x4 Dirac fermion
![Page 15: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/15.jpg)
2i-1 2i+1 2i+3 2i 2i+2
2
a2
a
a
V 20 sin(2kFx + (
2
+2
)) (for = )
2 0 sin(2kFx - (2
-2
)) (for = )
0 = (
4Um
3)2 (2 t)2 , / 2 = tan -1(
3 t
2Um)
Minima of spin-dependent effective potential when Q≡σZ
Hoff -diagonal V (ei 2k F xL
†R e i 2k F xR† L ),
◆Physical picture (valid for ξspin>>a)
A bond-centered density wave
![Page 16: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/16.jpg)
Generalization of phase Hamiltonians
R e ikF x + L e-ik Fx
R ~ exp[i2
( ) +i2
( )]
L ~ exp[i2
(- ) +i2
(- )]
Phase fieldscharge
spin
Replacement
Parametrization of level 1 SU(2)WZW field as g = exp(-i+Q)
Q =n
(e.g. L†
R ~ exp[i+ + iQ+])
a = -iU5 U5 , U 5 exp(
i
2Q 5)
...
More rigorous identification:view as chiral transform:
Zspin Dn D+ DD e
- ddx[ +ia ]
Dn D+ exp( Swzw [g]) |g= exp(-i +Q) Zwzw
See eg. A. Tsvelik’s textbook
![Page 17: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/17.jpg)
Abelian bosonization vs Rotating frame bosonization comparision of dictionaries:
R L 1
x same
■charge
■spin
Sz(k = 0) 1
2x
Sz(k = 2kF ) sin(2kFx + +)sin
S (k = 0)
J R
J L 1
2x
n
1
2cos sinx
n
1
2sin2
n x
n
S (k = 2kF) sin(2kFx ++)sin
n
■free fermion action
L 1
4()2
1
4()2 L
1
4()2 Lwzw(g) |gexp(-i+Q)
![Page 18: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/18.jpg)
Lwzw(g) |gexp( i+ Q) 1
4()2 1
8(sin2 )(
n )2 i(2+ - sin2+)qx
qx 1
4n
n x
n ・・・ densityof instantons
If interaction pins φ+ → O(3)NLσ-like model +θ-termBulk case 2 ways to treat interaction:
Lint( , ,
n )
2
sin cos(
2
)< >
< >
semiclassical
backscatteringincorporated
eff
eff sin
spin-singlet 0 : + = eff = 0
= 0 (no dimer) : eff unfixed:RVB (Inagaki-Fukuyama)
Vacancy: depleted charge local spin moment
Lint( , ,
n )
2
sin sin
2
cos Dcos2
How will this effect the Berry phase term?
![Page 19: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/19.jpg)
Neel I SP I
Neel II SP II
SP II[θcl=Φ-sinΦ+2π→θ=2π]
Φ+
Neel I [θcl=π→unfixedθ( RVB like)]
Neel II [θcl=-π→unfixedθ( RVB like)]
SP I[θcl=Φ-sinΦ→θ=0]
Physical View of the Spin Phase Field direct relation to θ-angle (Haldane gap physics)
![Page 20: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/20.jpg)
Effect of dilute vacancies
Put
Implement sublattice structure via 2 charge fields A,
B .
If
+ (A
B )X imp
:
1)Static vacancies
i(
A B )qx
i
2( 1)
Ximp / a
Ximp
[n (X imp, )]
Random exchange model(c.f. spin ladder case: Nagaosa et al)
Berry phase
Seff ( 1)
X j / a[n j ( )]
j dJeff e
X j X j1 / s n j
n j 1 , Jeff sin2
2
Spin correlation ξ ~ T-2α(α ~ 0.22),χunif ~ 1/T,χstagg ~ 1/T1+2α
Spectral weight transfer
Im (k,)gapped ~cos 2
(k - )2 m 2, Im (k, )spin wave ~ sin /(k - )
c.f. Saito, Saito-Fukuyama
+ SP1SP2
c s , Ltop = i(2 sin 2)qx 2i(A
B)q x
![Page 21: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/21.jpg)
2)mobile vacancies (holes)Terms related to :
A, B
L(A,
B) 1
8[(
A B)]2
1
8[(
A B)]2 2i(
A B)qx
cf. Shankar Refermionize (spinless fermions):
Lfermion A( i A ) A B( i A ) B
A n
a ,
n a
n spin gauge field
Enhanced intrasublattice hopping (t’-term)Effective attraction between A-holes and B-holes
A
B : can be shown to be massive
Singlet pairing susceptibility~1/r
RA (x) L
B (x) ~ ei
2(
A B )
ei
2(
A B )
e i
![Page 22: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/22.jpg)
FET technique may provide realization of superconductivity In quasi-1d. Attempts are now being made for CuGeO3.
![Page 23: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/23.jpg)
What can be said for 2d systems?
Stripe order, AF fluctuations and Superconductivity
Zhou et al, Noda et al, Science 286, 265 (1999) ARPES: low energy=stripes, high energy=dSC-like
Zaanen et al,D.H.Lee: SC-stripe duality question: how can nodal fermions arise from stripes?
Momoi: melting transition of stripes via dislocations
AF Merons with winding number Qxy=1/2
![Page 24: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/24.jpg)
X’
X’
X’
τ
How to calculate momentum carried by AF topological defects
eg. 2d Heisenberg modelView [11] direction as time.→ maps into 1+1d AF chain.
T[11]=exp(iPa) =exp(i2πSQxy)
Preliminary results on similar methods applied to stripes suggestthat condensation on AF merons are related to nodal fermions .
![Page 25: Role of quantal phases in low-dimensional correlated electrons -nonperturbative approach](https://reader031.vdocuments.mx/reader031/viewer/2022013011/56813b49550346895da43407/html5/thumbnails/25.jpg)
Conclusion:
Disorder-induced AF in spin-gapped systems: direct relation to pairing via spin-gauge field when doped with dilute amount of holes.
Relevance to
Underdoped Cuprate c.f. McQueeny et al : coexistence of SP order and d-wave SC (LSCO and YBCO ) Spin-Peierls compound CuGeO3 (FET?) Spin Ladder SrCu2O4
c.f.Y2BaNiO5 (Ito et al 2001) disorder induced AFLRO X enhanced conductivity X (charge gap~spin gap)
Quantum melting of stripes via AF fluctuations can be related to nodal structures of the dSC.