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

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

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.

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

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:

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

Acknowledgments to:

Tokyo: N.Nagaosa ， H.Fukuyama ， M.Saito K. Uchinokura

Tsukuba: M. Hase, N. Taniguchi, T. Hikihara

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

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?

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?

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

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 ）

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 ]

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

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

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

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)

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?

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)

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

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

FET technique may provide realization of superconductivity In quasi-1d. Attempts are now being made for CuGeO3.

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

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 .

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.