Quantum Spin Pump in S=1/2 antiferromagnetic chains -Holonomy of phase operators in sine-Gordon theory-

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by  Ryuichi Shindou

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Quantum Spin Pump in S = 1/2 antiferromagnetic chains
-Holonomy of phase operators in sine-Gordon theory-
Ryuichi Shindou∗
5
0
Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
0
2 RIKEN (The Institute of the Physical and Chemical Research), 2-1 Hirosawa, Wako, Saitma 351-0198, Japan
n (ReceivedFebruary6,2008)
a Inthispaper,weproposethequantumspinpumpinginquantumspinsystemswhereanap-
J pliedelectricfield(E)andmagneticfield(H)causeafinitespingaptoitscriticalgroundstate.

8 Whenthesesystemsaresubjecttoalternatingelectromangeticfields;(E,H)=(sin2πt,cos2πt)
T T
2 and travel along the loop Γ which encloses their critical ground state in this E-H phase
loop

diagram,thelockingpotentialinthesine-Gordonmodelslidesandchangesitsminimum.Asa
]
l result,thephaseoperatoracquires2πholonomyduringonecyclealongΓloop,whichmeansthat
l
a thequantizedspincurrenthasbeentransportedthroughthebulksystemsduringthisadiabatic
h process. The relevance to real systems such as Cu-benzoate and Yb4As3 is also discussed.
-
s
KEYWORDS: field induced spin gap systems, quantized spin transport, Dirac monopole, sine Gordon
e
m theory, phaseoperator, topological stability, LandauZener tunneling
.
t
a
1. Introduction valueobservedintheseexperimentsshouldbeattributed
m
Transport phenomena in magnets such as the colos- totheCoulombblockade23,25,26 andisdifferentformthe
-
d sal magnetoresistance1 and anomalous Hall effect2–8 aforementioned Thouless quantization, which is assured
n are long-standing subjects in solid state physics. Re- only in the thermodynamic limit.23
o cently, exotic characters of the anomalous Hall effect Inthispaper,basedontheoriginalideaoftheQAPT,
c
[ are closed up in ferromagnets,9–12 antiferromagnets13 we propose the quantized spin transport (quantum spin
and spin glass14,15 with non-coplanar spin configura- pump)inthemacro-scalemagnets.Therewediscusssys-
3 tions and collinear ferromagnets with strong spin-orbit tematically the topological stability of its quantization.
v
couplings.16–20 Their common origin is gradually rec- Specifically,weclarifyinthis paperthe relationbetween
8
6 ognized as the topological character of magnetic Bloch its quantization and holonomy of the phase operator of
6 wavefunctions in its ordered phase, sharing the same sine-Gordon theory in 1D quantum field theory. By us-
2 physics with the two dimensional (2D) quantum Hall ing this relation, its topological stability can be quanti-
1 tatively judged by seeing whether the expectation value
effect. Thereby the spontaneous Hall conductivity is re-
3
latedtothefictitiousmagneticfielddefinedinthecrystal of this phase operator acquires 2π holonomy during the
0
cyclic process or not.
/ momentum space and its quantized value is directly re-
t Asfortheexperimentalrelevantsystemstoourtheory
a latedtothefirstChernnumberassociatedwiththefiber
m bundlewhosebasespaceisspannedbycrystalmomenta. of quantized spin transports, we have the S = 12 quan-
- Meanwhile, in early 80’s, Thouless and Niu proposed tumspinchainsuchasCu-benzoateandYb4As3 (charge
d the quantized adiabatic particle transport (QAPT) in orderedphase),whereitsunitcellcontainstwocrystallo-
n
gappedfermisystems,whereanadiabaticslidingmotion graphicallyinequivalentsites.Accuratelyspeaking,both
co ofthe periodic electrostaticpotentialin one dimensional the translationalsymmetry by one site (T :Sj →Sj+1)
: (1D) system pumps up an integer number of electrons and the bond-centered inversion symmetry which ex-
iv per cycle.21,22 In fact, this quantized particle transport changes the nearest neighboring sites (Ibond : Si−j ↔
X isanotherphysicalmanifestationofthetopologicalchar- Si+j+1 are crystallographically broken. As a result of
r acter associated with the Bloch wavefunctions. Specifi- this peculiar crystal symmetry, the g-tensors of its two
a
cally this particle (polarization)currentis relatedto the sublattices are in general different in these systems;
fictitious magnetic field defined in the generalized crys- N
tal momentum space which is in turn spanned by the H =J S ·S +
i i+1
crystalmomentumanddeformed parameters.Becauseof i=1
X
this analogy to the 2D quantum Hall currents, this 1D
H· [g ]+(−1)i[g ] ·S , (J >0)(1)
quantized particle current is known to be topologically u a i
i
protected against other perturbations such as disorders X (cid:2) (cid:3)
and electron-electron correlations.22 Due to this pecu- where [gu] and [ga] represent the uniform and staggered
liar topological protection, the QAPT became recently component of the g-tensors respectively.27 Late 90’s,
highlighted in the realm of the spintronics, where elec- these quantum spin systems, whose ground states are
tronpumpingshavebeenexperimentallyrealizedinmeso nearly critical because of their strong one dimensionali-
and/or nano-scale systems.23–26 However the quantized ties,areexperimentallyrevealedtoshowthespingapbe-
haviors under the external magnetic field. Furthermore,
followingtheoreticalworksshowedthatthefield-induced
∗E-mailaddress:[email protected].
2 J.Phys.Soc.Jpn. FullPaper Online-JournalSubcommittee
gapbehaviors(∆∼H2/3) are originatedfrom this stag- against it, where it turns out that the sweeping velocity
geredcomponentoftheg-tensor,wheretheeffectivestag- should be appropriately chosen so as to avoid the relax-
gered magnetic field in the eq. (1) endows its critical ation process via this uniform Zeeman field.
ground state with a finite spin gap.29,30
2. Bosonization
On the other hand, the exchange interaction J in
these S = 1/2 quantum spin chains, J S · S , For clarity, let us first illustrate the physics of our
i i i+1
does not have an alternating component, since these quantum spin pumping in a simple limiting case;
P
systems are invariant under the site-centered inversion
N
J
symmetry which exchanges the nearest neighbor bonds H = − (Sˆ+Sˆ− +c.c.), (4)
(I :S ↔S ).However,whenwebreakthissym- XY 2 i i+1
site i−j i+j i=1
metry by applying anelectric field E alongan appropri- X
N
ate direction,31 the exchange interaction in general does H = ∆ (−1)i Sˆ+Sˆ− +Sˆ−Sˆ+ , (5)
acquire a staggered component; dim 2 i i+1 i i+1
Xi=1 (cid:16) (cid:17)
N
H = J −(−1)i∆ Si·Si+1. (2) Hst = hst (−1)iSˆiz, (6)
i
i=1 X
X(cid:2) (cid:3)
wherewetakethedirectionofthestaggeredZeemanfield
Futhermore a site-centered inversion operation with the
sign of E reversed requiresthat∆ mustbe anodd func- as the z-direction.32 In this simplification, we neglected
tion of E. This dimerizing field ∆ is also expected to theexchangecouplingbetweenthez-componentofspins.
Inthislimit,wecangetaquadraticformofHamiltonian
induce a finite mass to its critical ground state, causing
intermsoftheJordan-Wigner(JW)fermionSˆz =f†f −
the spin-Peierls state. i i i
Based on these observations, we propose a method of 12,Sˆi+ = fi†exp(iπ ij−=11fj†fj).33,34 Then HXY forms a
generatingthequantizedspincurrentinthistypeofspin cosine band in its momentum space, ǫ(k) = −Jcoskα
P
systems (T,I : broken, I : unbroken) by using (k,α are crystalmomentum andlattice constant).In the
bond site
electromagnetic fields. Specifically, we study a following absenceof∆andhst,thefermipointslocateatk =±2πα
S = 1/2 Heisenberg model with time-dependent stag- since fermions inthe groundstate fill up allthe k points
gered field and bond alternation: but those with positive energy ǫ(k).
Nonzero ∆ and/or h introduce a finite gap at these
st
N
Hˆ(t) = J −(−1)i∆(t) S ·S two Fermi points. The dimer state induced by a finite ∆
i i+1 can be understood as a Peierls insulator, where the JW
i=1
X(cid:2) (cid:3) fermions occupy the bonding orbitals between the two
N neighboringsitesandformavalencebandinitsk-space.
+ h (t) (−1)iSz. (3)
st i On the contrary, the antiferromagnetic state induced by
Xi=1 the effective staggered magnetic field hst along the z-
where staggered Zeeman field hst(t) and bond alterna- direction canbe interpreted as an ionic insulator, where
tions ∆(t) are supposed to be controlled by the applied the JW fermions stay on every other site. The conduc-
electromagnetic fields. In § 2, we will argue by using tion band and the valence band touch at k =± π , only
2α
bosonization technique that the quantized number of z- when (∆,h ) is taken at the origin in this ∆-h plane.
st st
component of spins are transported from one end of the Becauseoftheperiodicityof π alongthek-axis,thedou-
α
systemtotheotherduringonecyclealongtheloopwhich bledegeneracypointat(k,∆,h )=( π ,0,0)isidentical
st 2α
encloses the critical ground state at (∆,hst) = (0,0) in to that of (k,∆,hst)=(−2πα,0,0).
the ∆-hst plane. Elementary analyses show that this double degener-
Inordertoupholdthisbosonizationargument,wealso acy point becomes the source (sink) of the vector field
demonstrate the numerical calculations in § 3, where we B (B )definedinthegeneralizedcrystalmomentum
+1 −1
evolve the ground state wavefunction according as the space (k-∆-h space);
st
abovetime-dependentHamiltonian.Therebyweattibute
the quantized spin transport to the Landau-Zener tun- Bn(K) = ∇K ×An(K),
nelings40 which indeed happen between several energy i
levels during this cycle process. An(K) = 2πhn(K)|∇K|n(K)i,
nenInt oafddZieteiomnantofieeqld.(s3i)n, wreealalssyostheamvse,a uniform compo- w|nh(Kere)iKrep=res(ekn,t∆s,thhest)pearnioddi∇cpKart=of(t∂hke,∂B∆lo,c∂hhsftu).ncHtieorne
N for the valence band (n=+1) and the conduction band
H′(t)= H(t)·Sj, (n=−1).The vectorfieldAn,µ changesby 21π∇Kµφun-
j=1 der the U(1) gauge transformation of these Bloch wave-
X
whose magnitude is usually largerthan that of the stag- functions; |n(K)i → exp[iφ(K)]|n(K)i, while its rota-
geredcomponenthst(t).Thisuniformfield,especiallyits tion, i.e., Bn remains invariant. Because of this gauge
x or y component, seems to easily flip the spins accu- invariance, we often call the latter vector field Bn as
mulatedatbothboundariesandmightspoilthephysical a fictitious magnetic field or flux. Correspondingly, the
consequence of the spin transport. Then we also discuss double degeneracy point located at (±2πα,0,0) will be
in § 3 the effect of this uniform field and the remedy referred to as the fictitious magnetic charge (magnetic
J.Phys.Soc.Jpn. FullPaper Online-JournalSubcommittee 3
monopole in the generalized momentum space), whose fermion fields R(x ) and L(x ):
j j
magnetic unit can be shown to be 1:
1
Sˆz =f†f −
dS·B =±1. (7) j j j 2
±1
ZS1 ≃:R†(xj)R(xj):+:L†(xj)L(xj):
Here S represents the arbitrary closed surface which
1
+(−1)j R†(x )L(x )+L†(x )R(x ) , (12)
encloses the double degeneracy point at (k,∆,h ) = j j j j
st
(+ π ,0,0)≡(− π ,0,0). h i
2α 2α Sˆ+Sˆ− +Sˆ−Sˆ+ =f†f +f† f
Based on these observations, let us consider the j j+1 j j+1 j j+1 j+1 j
adiabatic process where the two parameters (∆,hst) ≃iα· :R†(x )∂ R(x ):−:L†(x )∂ L(x ):−H.c.
j x j j x j
are changed along a loop Γ enclosing the origin
loop
((∆,h )=(0,0)); (cid:2) (cid:3)
st −2i(−1)j R†(x )L(x )−L†(x )R(x ) , (13)
j j j j
(h ,∆)≡R(cosϕ,sinϕ) (8) (cid:20) (cid:21)
st
where : ... : stands for the normal order.36 Here we re-
R6=0 , ϕ:0→2π. (9)
tain the lowest order terms with respect to the lattice
Then,accordingtotheoriginalideaoftheQAPT,21,22,35 constant α both for the uniform and staggered compo-
the total number of JW fermions (I) which are trans- nents. According to the bosonization recipe,37 we can
ported from one side of this system to the other (in the rewritethesefermionoperatorsbyusingthephaseopera-
positive direction) during this adiabatic process is equal torθˆ (x)anditscanonicalconjugatefieldΠˆ(x).Namely,
+
tothetotalfluxforthevalenceband(n=1),B ,which the spin operators in eqs.(12) and (13) read;
+1
penetratesthe2DclosedspherespannedbyΓ andthe
loop ∂ θˆ (x ) 1
Brillouin zone : Sˆz = x + j −(−1)j sinθˆ (x ), (14)
j 2π πα + j
I = dS·B+1 =+1. (10) Sˆ+Sˆ− +Sˆ−Sˆ+
ZΓloop×[−2πα,2πα] j j+1 j j+1
The physicalmeaning ofthis quantizedfermiontrans- =−α 1 ∂ θˆ (x ) 2+4πΠˆ(x )2
x + j j
port is nothing but the quantized spin transport, since 4π
h i
the JW fermion density is related to the z-component (cid:0)2 (cid:1)
−(−1)j cosθˆ (x ). (15)
of the spin density. In other words, the total Sz around πα + j
one end of this system decreases by 1 while that of the
Then we substitute these equations into eqs.(4)−(6)
otherendincreasesby1duringthisadiabaticcycle.This
and (11) and obtain the following expressions for
quantization is topologically protected against the other
H ,H ,H and H in the continuum limit, where
XY Z dim st
perturbations as long as the gap along the loop remains
we neglect the rapid varying components such as
finite,22,35inotherwords,asfarasthedoubledegeneracy (−1)jcosθˆ (x ) and etc.:
j + j
pointsdonotgetoutof(enterinto)the2Dclosedsurface
Γloop ×[−2πα,2πα]. Then, we naturally expect that this P H = dx 2πJΠˆ(x)2+ J (∂ θˆ (x))2 , (16)
quantized spin transport is stable against the weak ex- XY x +
8π
" #
change interactions between the z-components of spins; Z
HZ = |γ| J +(−1)j∆ SˆjzSˆjz+1. (11) HZ = |αγ| dx 4Jπ2(∂xθˆ+(x))2+ 2(πJα)2 cos2θˆ+(x)
Xj (cid:2) (cid:3) Z "
In the following, we will prove that this is indeed the
∆
case, by introducing the exchange interactions between −(πα)2 cosθˆ+(x) , (17)
Sz. As a first step, we will treat H term as a non- #
XY
perturbedtermandreviewwhichperturbationsarerele- ∆
H =− dxcosθˆ (x) (18)
vantamongHZ,Hdim andHst byusingthebosonization dim πα2 +
technique. Namely, we first introduce the slowly varying Z
h
fields, R(x),L(x): H =− st dxsinθˆ (x), (19)
st πα2 +
f ≃ R(x )eikFxj +L(x )e−ikFxj, Z
j j j Here the origin of the third term of eq.(17) were dis-
R(x ) = f ei(k−kF)xj, cussed elsewhere,38 which we will omit by redefining ∆
j k
|k−kXF|≪α−1 in eq. (18) in the followings. Then our final expression
for Hamiltonian reads,
L(x ) = f ei(k+kF)xj.
j k 1
|k+kXF|≪α−1 H = dx v πηΠˆ2+ 4πη(∂xθˆ+)2
Then we rewrite the spin operator in terms of these Z (cid:26) h i
R |γ|J
− sin(θˆ +ϕ)+ cos2θˆ , (20)
πα2 + 2π2α3 +
(cid:27)
4 J.Phys.Soc.Jpn. FullPaper Online-JournalSubcommittee
where the velocity v is given by v = J2 (1+ 2π|γα|). We (I) Spin Gap H ~ <hsθ t+ >=π/2
also introduced the quantum parameterqη as phase E ~ ∆
1 critical ground state
η =2 (21)
s 1+ 2π|γα| (II) <θ+ >=0
which measures the stre(cid:0)ngth of(cid:1)the quantum fluctua-
tion. Namely, when |γ| in eq.(11) grows from 0, this pa-
rameter decreases monotonically from 2. Even though
(III)
eqs. (16)−(21) were derived in the weak coupling limit
(|γ|≪1), the final form of eq. (20) is known to be valid <θ+ >=−π/2
until |γ| reaches 1 (isotropic Heisenberg model), where
the SU(2) rotational symmetry of correlation functions
(hSxSxi ∼ hSzSzi) requires that η = 1 at |γ| = 1.36 (IV)
j i j i
By using this quantum parameter η, the renormaliza- <θ >=+−π
tion group(RG) eigenvalueof sin(θˆ +ϕ) is represented
by 2− η while that of cos2θˆ is g+iven by 2−2η. This <θ+ >=-3π/2
2 + (I)
indicates that, as long as |γ| ≤ 1 (1 ≤ η ≤ 2) , the
exchange coupling between the z-component of spins -3π/2−π/2 π/2
(2|πγ2|αJ3 cos2θˆ+)is alwaysirrelevantin the senseof renor- −π 0
malization group analyses, while the dimerizing field ∆
and staggeredfield h ( R sin(θˆ +ϕ)) are equally rel-
st πα2 + Fig. 1. Asinecurverepresentsthelockingpotentialwhichslides
evant and lock the phase operator θˆ+ on π2 −ϕ+2πn. adiabatically. As long as the sliding speed is low enough, the
As the system is always locked by sin(θˆ + ϕ) for system(shadedcircle)stays thesamevalleyanddoes notjump
+
intoitsneighboringvalleys.
|γ| ≤ 1, we naturally expect that the quantized spin
transport in the case of |γ| = 0 could be generalized
into the case of finite |γ|, at least up to |γ| = 1. This
the spin state within the “edge” region is strongly af-
expectation is easily verified when we notice the physi-
fected by the boundaries. Then, by definition, the spins
cal meaning of the phase operator θˆ+. Since the spatial withintheinteriorwouldturnbacktothesamespincon-
2π
derivative of the phase operator corresponds to the z- figuration as that of the initial state after an adiabatic
componentofspindensity,thisphaseoperatorisnothing evolution along Γ ,
loop
but minus of the spatial polarization of the z-component
δPin =0.
of spins, i.e., −PˆSz ≡ −N1 Nj=1jSˆjz. The equivalence Sz
betweenthese twoquantities is discussedindetailin the Therefore the difference of the spatial polarization of
appendix. Then, through thPe adiabatic process eqs.(8) hSˆzi should be attributed to the change of spin configu-
j
and(9), hθˆ i decreases monotonically and acquires −2π rations in the edge region,
+
holonomy after one cycle (see Fig. 1). In other words, 1
δPedge = j·δhSˆzi=1. (23)
PSz increases by 1 per one cycle, i.e., Sz N j
j∈XΩedge
1
δPSz ≡ dPSz =− dλ·∂λhθˆ+i=1. (22) In eq.(23),we nextapproximatej for sites aroundthe
2π
IΓloop IΓloop right end by N/2 and those of left by −N/2;
This relationalwaysholdsas farasthe systemis locked 1 1
by the sliding potential sin(θˆ++ϕ(t)), which is true for −2 δhSjzi+ 2 δhSjzi≈1, (24)
|γ|≤1. j≃X−N2 jX≃N2
Then we will argue the physical consequence of eq. where we take the origin of site index j at the center of
(22). Generally speaking, when the bulk system has a thesystem.Thisapproximationisallowed,sincetheedge
finite spin gap, the effects of boundaries range over its region ranges over the magnetic correlation length from
magneticcorrelationlengthfromthebothends.Thenwe both ends, which is at most several sites in gapped spin
naturally divide the contribution to PSz into following systems.Therebythe semi-equalsignineq.(24)couldbe
two parts; safely replaced by the equal sign in the thermodynamic
1 limit. When we bear in mind that the system has been
PSz = PSinz +PSedzge, PSinz(edge) = N j·hSˆjzi stayingontheeigenspaceofSˆtzot =0duringthisprocess;
j∈ΩXin(edge) δhSzi+ δhSzi=0,
j j
where the j-summation in Pin are restricted within the
interiorofthe systems(Ω ),Swzhile spinswithin the edge j≃X−N2 jX≃N2
in
region(Ω ) contribute to Pedge. The “interior” of the eq.(24) indicates that the total Sz around the right end
edge Sz (j ≃N/2) increases by 1 while that of the left end (j ≃
system is defined as the region where the spin state is
−N/2) decreases by 1 along this cyclic evolution:
sameasthatoftheperiodicboundarycondition(p.b.c.).
Namely, the presence of the boundaries does not influ- δhSzi=−1, δhSzi=1.
j j
enceonthespinstateoftheinterior.Ontheotherhand,
j∈left end j∈right end
X X
J.Phys.Soc.Jpn. FullPaper Online-JournalSubcommittee 5
Namely, the spin current quantized to 1 flows from the 0.4 (a) Open boundary: Periodic boundary:
left end of the system to the right end during this one 0.3
cycle.
0.2
In summary, the quantized spin transport is always
0.1
awsistuhrethdeafisnloitnegsalisdi|nγg|≤sp1e.edThci=sst2Toπr,yadsofeasrnaostitalitsesrloewveenr z > S<T=tj 0
than the height of this locking potential, in other words, -0.1
the spin gap along the loop.
-0.2
3. Numerical Analyses -0.3
Inordertoconfirmtheaboveargument,weperformed -0.4 2 4 6 8 10 12 14
j : site number
the numericalcalculation in the case of |γ|=1. Namely,
0.5 (b) Open boundary: Periodic boundary:
we generated numerically the temporal evolution of the
0.4
ground state wavefunction under the following time-
0.3
dependent Hamiltonian:
0.2
t 0.1
|φ(t)i=T exp[i dt′Hˆ(t′)] ·|giI, >T=t 0
n Z0 o z S<j-0.1
N
Hˆ(t)= J −(−1)i∆(t) S ·S -0.2
i i+1 -0.3
i=1
X(cid:2) (cid:3) -0.4
N -0.5
+ hst(t) (−1)iSiz, 2 4 6j : site n8umber 10 12 14
i=1
X
(hst(t),∆(t))=R(cos2Tπt,sin2Tπt), (25) FigδP.S2z. ≡(a)h:PˆShSzˆjziti=t=TT−fohPrˆSJzi=t=01.5=,R0.6=230,.3(ba):ndhSˆTjzit==T40f,orwJher=e
where T represents the time order operator, T is the 0.6,R=0.3andT =80,whereδPSz =0.862.
cycleperiodduringwhichthe systemhasswepttheloop
(Γ )onetimeand|gi isthegroundstatewavefunction
loop I and the four states39 including the ground state are
at (h ,∆)=(R,0).
InsFtig. 2, we show the expectation value of Sˆz taken nearly degenerate at ϕ= 32π for N =14.
j Let us discuss the physical meanings of this quasi-
over the final state, |φ(t=T)i, both with the p.b.c. and
degeneracy.At(h ,∆)=(0,−R),theexchangecoupling
with the open boundary condition. The parameter R is st
between S and S (n = 1,2,...) are strengthened
fixed to be 0.3 and J is taken as 1.5 (0.6) in Fig. 2a(b). 2n 2n+1
(see eq.(25)) and dimers are formed between them, i.e.,
ThecycleperiodT istakenas40fortheformerand80for
the latter, both of which are sufficiently long compared S −S =S −···−S =S −S
1 2 3 N−2 N−1 N
withthe inverseofthe spingapobservedalongthe loop.
where S = S represents the singlet bond. Then, in
i i+1
thecaseoftheo.b.c.,thespinatj =1andthatofj =N
For the p.b.c., as far as the sweeping velocity is low
cannot form a spin singlet due to the absence of S ·S
enough, the final state gives completely same configu- 1 N
rations of hSˆzi as that of the initial antiferromagnetic term. Namely, in the limit of J = R, the following four
j states would be degenerated,
groundstate.Howeverinthe caseofthe o.b.c.,the spins
around both boundaries are clearly modified as seen in |↓i ⊗(dimer chain)⊗|↑i , (26)
1 N
Fig. 2. When we read the total Sz around the left end
( 3 hSˆzi ) and that of the right end ( 14 hSˆzi ) |↑i1⊗(dimer chain)⊗|↓iN, (27)
j=1 j j=12 j
from Fig. 2a(b), the former increases by 0.67 (0.9) while |↓i ⊗(dimer chain)⊗|↓i , (28)
P P 1 N
thelatterdecreasesby0.67(0.9)afterthiscyclicprocess.
|↑i ⊗(dimer chain)⊗|↑i . (29)
This resultis qualitativelyconsistentwith ourpreceding 1 N
arguments. The last two states ((28),(29)) do not belong to the
In order to understand the difference between the re- eigenspace of Sz =0 and we ignore them henceforth.39
tot
sultoftheo.b.c.andthatofthep.b.c.,wealsocalculated When we introduce finite J − R > 0, the N-th order
2
the instantaneous eigenenergy as a function of ϕ ≡ 2Tπt perturbation in terms of Sˆ1+Sˆ2−, Sˆ3+Sˆ4−,..., and SˆN+−1SˆN−
in Fig. 3, where we take the offset as the ground state
(andtheirHermiteconjugate)liftsthisdegeneracy.Then
energy. For the p.b.c. (inset of Fig. 3a) , there is always
theresultinggapshouldbescaledasexp(−Nα/l),where
a finite energy gap from the ground state for all ϕ. On
l isexpectedtobeamagneticcorrelationlength.Infact,
the contrary, the gap for the o.b.c. reduces strongly at
we can fit the size dependence of ∆ by this exponential
1
ϕ= 3π (Figs. 3a and3b). Furthermore,this reduceden-
2 function as in the inset of Fig. 3b, fromwhich the corre-
ergy gap ∆1 becomes smaller and smaller when we take lation length at ϕ= 3π is estimated around 3 sites. Be-
the system size N larger (inset of Fig. 3b,N=6,8,10,12), 2
cause of this size dependence, we expect that ∆ would
1
6 J.Phys.Soc.Jpn. FullPaper Online-JournalSubcommittee
2 1
(a) n=4 (a)
0.8
1.5 n=3
2
E >|^ 0.6 : |<n=0|φ(t )>|^2
2.0 n=2 1 ng -E φ()|<n|t 0.4 :: +||<< nn|<==n21=||φφ4((|ttφ ())t>> )||>^^22|^2
0.5
1.0
0.2
n=1
3.0 6.0
0
0 1 2 ϕ = 23πt/T 4 5 6 04.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2
2πt/T (=ϕ)
1
(b) ∆ 2 (b)
2
n=4 n=3 1.6 E 0>|^2ϕ=2π 00..68 cc==22..78 ::
1.2 =
n n : N=8
∆ n=2 00..68 ∆∆2 0.8 g -E φ( )t=T|0.4 ::: NNN===111024
1 n=1 00..24 1 0.4 1 - |<0.2
0.0 c=2.6 :
6 8 10 12 14
system size N c=2.4 :
0 0
4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 0 1000 2000 3000 4000 5000 6000
ϕ = 2πt/T cycle period T
Fig. 4. (a): Projected weight of φ(t) onto each instantaneous
Fig. 3. (a): Energy levels of all the eigenstates as a function of eigenstate nφ(t) 2 for n = 0,|1,2,i4 where the w.f. φ(t) is
ϕ ≡ 2Tπt for the o.b.c. (inset: p.b.c.). J = 1.5, R = 0.3 and always alm|ohst|perip|endicular to the excited state with|n =i 3:
N =14.(b):EnergylevelsofthoseeigenstateswithSˆz =0for n = 3φ(t) 2 < 10−7. J = 1.5,R = 0.3 and N = 14. (b):
tot |h | i|
theo.b.c.J =1.5,R=0.3andN =14.(inset:Size-dependence Transition probability Pn=0→n=1 as a function of cycle period
of the minimum energy gaps ∆1 minϕ En=1(ϕ) Eg(ϕ) T. Every data points are fitted by the Landau-Zener function
aarennsddpe∆∆ct22ivae≡rley.m)0.i3n2ϕe(cid:2)xEpn(=−2((Nϕ)−−6)E/n3)=1a≡(nϕd)(cid:3)0..9F2i5(cid:2)tteixnpg(−cu(Nrve−−s 6fo)r/4∆.31(cid:3)) eRxp=(cid:0)−0.3c×.π2∆π21/T(cid:1), where c is estimated around 2.5. J = 1.5 and
However, as in the inset of Fig. 3b, its size dependence
vanish when we took the system size sufficiently large
is also scaled by
compared with this correlation length.
Because of this crossing character of the energy spec- ∆2 ∼exp(−Nα/l), (30)
trum at ϕ = 3π, the system which has resided on the
2 where the magnetic correlation length l is estimated
ground state (n = 0) transits, at ϕ = 3π, into the first
2 around 4.6 sites. Therefore, when we take the system
excitedstatewithn=1whichisrepresentedbythebold
size large enough, we can naturally expect that the gap
solidlineinFig.3b.WecanseethistransitioninFig.4a,
∆ wouldreduceexponentially,whichenhancesthetran-
2
where the projected weights of |φ(t)i onto each instan-
sition probability P drastically.
taneous eigenstate are given as a function of ϕ = 2πt. n=1→n=2
T Inordertoverifythisexpectation,weperformanother
Futhermore,itstransitionprobability,P ,iswell
n=0→n=1 numerical calculations. As it is very difficult to calcu-
fitted by the Landau-Zener formula40,41
late numerically with larger system size (N > 14), we,
π∆2 instead, decrease the ratio J/R in order to reduce the
P =exp(− 1 ),
n=0→n=1 c×2π/T magnetic correlation length l, which is also expected to
reduce these gaps according to eq.(30). The result for
as in Fig. 4b, where we plotted P for various
n=0→n=1 J = 0.6,R = 0.3 and N = 14 is summarized in Fig.
cycleperiodT andvarioussystemsize.Fromthisfitting,
5. Figure5a indicates the energy spectrum for the lower
cintheLandau-Zenerfunctionisestimatedaround2.6± four excited states with Sˆz = 0 and Fig. 5b shows the
0.2.42 tot
projected weight, |hn|φ(t)i|2 for n = 0,1,2,4. Figure5c
As shownin the Fig. 4a,after the firsttransition took represents the expectation value of Sˆz taken over the
place at ϕ = 3π, the second transition from the first j
2 |n = 4i as a function of site index j. Then, as is
ϕ=2π≡0
excited state to the second excited state with n = 2
expected, the gaps we observed in Fig. 3 (∆ ,∆ ) de-
1 2
(representedbytheboldbrokenline)happensatϕ≃5.9.
crease drastically. In addition to them, there appears a
However its transition probability P is around
n=1→n=2 new gap ∆ between the excited state with n = 2 and
3
70% and not so perfect compared with P . This
n=0→n=1 that with n = 4 at ϕ = 6.15. This gap ∆ is also very
3
is mainly because the gap ∆ around ϕ ≃ 5.9 between
2 tiny and scales as exp(−N/2.25).
the first excited state and the second excited state is
Asaresultofthesecrossingcharactersofenergyspec-
a substantial amount compared with ∆ (see Fig. 3b).
1 trum at ϕ= 3π,5.97and 6.15,three Landau-Zenertype
2
J.Phys.Soc.Jpn. FullPaper Online-JournalSubcommittee 7
0.062*exp(-(N-8)/1.8) 0.05*exp(-(N-8)/2.25) wavefunction.
∆ 0.05 ∆
0.05 2 3 To summarize, the numerical observations found in
0.03
1.4 00..0031 0.01 nn==12(a) Fig. 2 can be ascribed to the difference between the en-
8 10 12 14 8 10 12 14 n=4 ergyspectrumwiththep.b.c.andthatwiththeo.b.c.In
1.2 system size system size thelattercase,acertainexcitedstatedecreasesitsenergy
1.0 n=4 n=3 level until ϕ = 3π and then picks up the system which
2
Eg0.8 ∆ ∆3 has been staying on the ground state. Then, through
- E n0.6 n=2 2 1.4 tthheeLsyasntdemau-hZaennderintuhnannedlinpgesr,fescetvleyraulpeoxnciatepdasrttaicteuslacrarerxy-
0.4 01..60 cited state at ϕ = 0, whose PˆSz increases by almost 1
0.2 ∆ n=1 0.2 in comparison with that of the initial state. These ob-
1
0.0 4.6 5.0 5.4 5.8 6.2 servations remain unchanged if the sweeping velocity is
φ()|<t|n=0>|^2 n=0 (b) 100000......086420 always higher thT1an>πi∆c=m2i1a;,2x,3 π∆c2i . (31)
φ()|<t|n=1>|^2 n=1 Tvahniisshlowinerthliemtihtefromrothdeynswame(cid:16)eipcinligm(cid:17)vite,loscinitcyeitsheex∆pe1c,2t,e3drteo-
ducesexponentiallytozeroasthesystemsizeN becomes
φ()|<t|n=2>|^2 n=2 lwairtFghienrta.hlTelyhnleuertmefueosrriemcatelhnreteicsouonnltacsbluoosfuiotthntihsiensreeecqlte.ivo(an2n.2c)eisofcoounrsissttoernyt
φ()|<t|n=4>|^2 n=4 tcthooemrepeaafflnesicyetssitvteehmesstuasnugicgfhoerramesdCefiffuee-lcbdteivnaezlooZnaetgeemtahanendfizYe-bdld4irsAeasc3ltoi.onTnghteahrcee-
4.4 4.6 4.8 5 ϕ5 =.2 2πt/5T.4 5.6 5.8 6 6.2 x(or y)-direction, Hux(y). Furthermore, since these sys-
tems breakthe bond-centeredinversionsymmetry,there
>0=π=ϕ2 0.4 |n=4(ϕ=0=2π)> isaDzyaloshinskii-Moriya(DM)interaction.ItsDMvec-
4=n 0.2 tor must alternate bond by bond in the absence of the
z| S|4=nj-00..02 (c) enleencttriocf fitehled.DIMn avdedcittoiornwtooultdhebme,atlhseouinndifuocrmedcboympthoe-
< -0.4
applied electric field in general. Even if these terms are
2 4 6 8 10 12 14
j gradually introduced to our model calculations, δθ+ re-
mains invariantas far as the gap along the loop remains
open, in other words,the system along the loop remains
Fig. 5. J = 0.6,R = 0.3 and N = 14. (a): Energy spec-
trum for the eigenstates with Sˆtzot = 0 as a function of ϕ,(b): locked by the sliding potential sin(θˆ+ +ϕ(t)). This is a
Projected weight of φ(t) onto the lower five eigenstates, i.e., manifestation of the fact that the quantized δθ (in the
|hφ(t)|ni|2(n=0,1,2,|4),(ic):ExpectationvalueofSˆjz takenover thermodynamic limit) is a topologically protect+ed quan-
thefourthexcitedstatewithn=4|n=4iϕ=2π tity. However, detail studies on the effect of these terms
might be remaining interesting problems.
When the system travels around this critical ground
tunnelings take place as shown in Fig. 5b. Namely, the
stateM times,totalz-componentofspinaroundoneend
wavefunction |φ(t)i changes its weight from the ground
increases by M, while that on the other end decreases
state with n = 0 to the first excited state with n = 1
by a same amount. Such an inhomogeneous magnetic
(represented by the bold solid line in Fig. 5a) at ϕ= 3π
2 structure can be detectable around sample boundaries
, from the first excited state to the excited state with
and/or magnetic domain boundaries by using some op-
n=2 (bold brokenline) at ϕ=5.97 and lastly from the
ticalprobes.However,wemustmentionthatthe excited
excited state with n=2 to the fourth excited state with
state with PSz = m (In the case of m = 1, this corre-
n = 4 (bold dotted line) at ϕ = 6.15. All these tran-
spondsto|n=4i inFig.5)mightfallintotheground
ϕ=0
sition probabilities are almost 100% in accordance with
state |n= 0i by way of the x-component of the uni-
our previous expectations.43 ϕ=0
form magnetic field Hx at around t = mT, when we
After all, through these transitions, the wavefunction u
sweep the system slower than 2µ Hx. In order to avoid
is raised from the ground state onto the fourth excited B u
this relaxation process, the sweeping velocity should be
state (n=4)during this lastquarterof the cycle.When
taken faster than 2µ Hx.
weseetheexpectationvalue ofSˆz takenoverthis fourth B u
j
1
excited state at ϕ = 0 ≡ 2π (see Fig. 5c), it is almost >2µ Hx, (32)
identical to the solid line of the lower panel in Fig. 2, T B u
where the total Sˆz onthe left end( 3 hSˆzi) decreases However,inordertokeepthesystemonaparticularmin-
j=1 j
by 0.9 while that of the rightend (P1j=412hSˆjzi) increases imumofthelockingpotentialcos(θ++2Tπt),the velocity
by0.9incomparisonwiththatoftheinitialgroundstate mustbealsoslowerthanthespingapobservedalongthe
P
8 J.Phys.Soc.Jpn. FullPaper Online-JournalSubcommittee
loop, locked on a particular valley of the relevant potential
h2 +∆2sin(θˆ +ϕ) : hθˆ i= π −ϕ+2πn. When the
1 <J1/3R2/3. (33) systsetm is deform+ed slowly a+long2the loop which encloses
T p
the critical ground state in the E-H plane, the lock-
Therefore we need to have a finite window between ing potential h2 +∆2sin(θˆ +ϕ) in the sine-Gordon
the lower limit of the sweeping velocity and the up- st +
modelslidesgradually(seeFig.1).After onecyclealong
per limit. In the case of Cu-benzoate, the spin gap of p
thisloop,theexpectationvalueofthephaseoperatorθˆ
0.1meV is induced by the applied magnetic field of 1T +
for the ground state acquires 2π holonomy. This means
(2µ ×1T ∼ 0.1meV). Thereby, the lower limit of this
B that a spatial polarization of z-component of spins P
sweeping velocity is unhappily comparable to the higher Sz
increases by 1 after this cycle.56 In other words, the z-
limit in this system. However,spin gaps around 0.1meV
component of the spin current quantized to be 1 flows
couldbe alsoinducedbythe magneticfieldsmaller than
through the bulk system during this cycle.
1T in those quantum spin chains with relatively larger
Theseargumentsaresupportedbythenumericalanal-
J, where substantialamount of the window between the
yses,whereweperformedtheexactdiagonalizationofthe
upper limit and the lower limit would exist. Therefore,
finite size system and developed the ground state wave-
it is not so hard to realize our story experimentally on
function along this loop in a thermally insulating way.
the quasi-1Dquantumspin systemswith relativelylarge
intrachain interaction and with two crystallographically Acknowledgments
inequivalent sites in its unit cell.
The author acknowledges A. V. Balatskii, Masaaki
We also want to mention about the magnitude of the
Nakamura, M. Tsuchiizu, K. Uchinokura, T. Hikihara
electricfieldrequiredinordertoinducethedimerization
and A. Furusaki, S. Miyashita, S. Murakamiand N. Na-
gapenoughto be observableandalsoenoughto be com-
gaosa for their fruitful discussions and for their critical
parable to the spin gap induced by the magnetic field,
readings of this manuscript. The author is very grateful
which is around 0.1meV. Since it goes beyond the scope
to K. Uchinokura for his critical discussions which are
of this paper to quantify microscopically the change of
essential for this work.
the exchange interaction induced by external electric
fields, we will pick up some reference data from other Appendix: Equivalencebetweenthephaseoper-
materials.Let’sseemagneto-electricmaterials,wherethe ator and the spatial polarization op-
applied electric field often changes its effective exchange erator
interactionsJ duetoitspeculiarcrystalstructureandits
In this appendix, we will show that the contin-
change ∆ could be quantitatively estimated via the re-
uum limit of Pˆ corresponds to the phase operator
sulting magnetization.45 In the case of famous magneto- Sz
− 1 θˆ+(y)dy, where related work was done by Naka-
electricmaterialCr2O3,theintra-sublatticeexchangein- Nα 2π
mura and Voit.47 Historically the polarization operator
teraction for the + sublattice and that for the − sublat-
R
is known to be an ill-defined operator on the Hilbert
ticeshouldbeequaltoeachotherbecauseoftheinversion
space with the periodic boundary condition (p.b.c.) and
symmetry (J =J =J). Then an electric field applied
+ −
isconsequentlyformidabletotreatinsolidstatephysics.
alongits principle axisbreaksthis symmetryandinduce
However, recently King-Smith and Vanderbilt51,52 and
a finite difference ∆ = J −J , where ∆/J was esti-
+ −
mated around 10−5 under E ∼1kV/cm.45 Namely, Resta48–50,53 showed that the derivative of the polariza-
tion with respect to some external parameter λ can be
∆
=cE. (34) expressed by the current operator which is in turn well-
J defined in the Hilbert space with the p.b.c.. Based on
with its linear coefficient 10−5[cm/kV]. When we apply thisobservation,theyquantitativelyestimatedelectronic
this coefficienttoourHeisenbergmodel,eq.(2)withJ ∼ contributions to macroscopic polarizations in dielectrics
0.1eV, we need an electric field of order of 1kV/cm in andsemiconductorssuchasGaAs,KNbO andIII-Vni-
3
order to induce the dimerizing gap J1/3∆2/3 of order of tride. Following their strategies, we will show the equiv-
0.1meV, which is still one order of magnitude smaller alence between the derivative of − 1 dyhθˆ+(y)i with
Nα 2π
than the typical value of the Zener’s breakdownfield (≥ respect to λ and that of hP i.
Sz
104V/cm at T ≤100K) of 1D Mott insulators.46 According to the standard perturbatRion theory,49 the
derivative of the polarization with respect to some me-
4. Conclusion and discussion
chanical parameter λ is given by
In this paper, we propose the quantum spin pump-
ing in S = 1/2 quantum spin chains with two crys- ∂hPˆSzi
=−
tallographically inequivalent sublattices in its unit cell ∂λ
N6=g
X
(T,I : broken and I : unbroken) and with rela-
bond site
tively large intrachain exchange interactions. Due to its hg(λ)|[PˆSz,Hˆ]|N(λ)ihN(λ)|∂Hˆ∂λ(λ)|g(λ)i +c(.cA.·1,)
peculiar crystal structure, an applied electric field (E) (E (λ)−E (λ))2
g N
and magnetic field (H) endow its critical ground state h i
where we assume the system has a finite spin gap from
with the finite spin gapvia the dimerizing field(∆∼E)
the ground state. The time derivative of P in the nu-
andthestaggeredmagneticfield(h ∼H),respectively, Sz
st
merator reduces to the summation of the current opera-
where the phase operator θˆ in sine-Gordon theory is
+
J.Phys.Soc.Jpn. FullPaper Online-JournalSubcommittee 9
tor over all bonds; that this is indeed the case, we will carefully reexamine
the bosonization of Sˆ+Sˆ− +Sˆ+ Sˆ−55 :
−i PˆSz,H =−i PˆSz,HXY+Hdim j j+1 j+1 j
Sˆ+Sˆ− +H.c.
h i h i j j+1
N N−1
1 1
=iN jSˆjz,−2 J +(−1)k∆ (Sˆk+Sˆk−+1+c.c.) = R†(xj)e−ikFxj +L†(xj)eikFxj
Xj=1 Xk=1h (cid:2) (cid:3) i h i
i N−1 × R(xj+1)eikFxj+1 +L(xj+1)e−ikFxj+1 +H.c.
= J +(−1)j∆ (Sˆ−Sˆ+ −Sˆ+Sˆ− ). (A·2)
2N j j+1 j j+1 h α2 i
Xj=1(cid:2) (cid:3) ≈ (r.h.s of eq.(13))−i(−1)j · R†(xj)∂x2L(xj)
2
Thereby the contribution of every single bond is always
h
O(1/N), which is not the case with PˆSz = N1 Nj=1jSˆjz. −L†(xj)∂x2R(xj)−H.c.
As a result, we are allowed to estimate with the periodic
P i
boundary condition the r.h.s. of eq. (A·2) in the thermo- = (r.h.s. of eq. (15))
dynamic limit, which we should have figured out with
1
the open boundary. This is because their difference is + (−1)jα − (∂xθˆ+(xj))2cosθˆ+(xj)
4π
attributed to the spin currents on the several bonds54 (cid:26)
around the boundaries, which is at most O(1/N) in our + π Πˆ2(x )·cosθˆ (x )
j + j
spingappedsystem.Thenwewillconsidereq.(A·2)with
h
thep.b.c.andrewriteitintermsofthebosonizationlan-
+ Πˆ(x )·cosθˆ (x )·Πˆ(x )+H.c. . (A·7)
guage.Thatis to say,we expressthe currentoperatorin j + j j
eq. (A·2) in terms of the phase operator θˆ (x) and its i(cid:27)
+ Then,bytakingitsstaggeredcomponents,weobtainthe
conjugate field Πˆ(x),
following expression for H , instead of eq.(18),
dim
Sˆ+ Sˆ−−Sˆ+Sˆ− =f† f −f†f
j+1 j j j+1 j+1 j j j+1 H =
dim
≈ R†(xj+1)e−ikFxj+1 +L†(xj+1)eikFxj+1 ∆ dx − 1 cosθˆ − 1 ∂ θˆ 2cosθˆ
h i ( πα2 + 8π x + +
× R(x )eikFxj +L(x )e−ikFxj −H.c. Z (cid:16) (cid:17)
j j
≈ −2hi· R†(xj)R(xj)−L†(xj)L(xij) + π2(cid:20)Πˆ2·cosθˆ++Πˆ ·cosθˆ+·Πˆ +H.c.(cid:21)).(A·8)
h i
+ iα·(−1)j R†(x )∂ L(x )−L†(x )∂ R(x )+H.c. Here we wantto mentionthat, irrespectiveof this modi-
j x j j x j fication, cosθˆ in H always locks the phase operator
+ dim
= 4iΠˆ(xj) h iin combination with sinθˆ+ in hst as far as |γ| ≤ 1 (see
thetext).ThisisbecausetheRGeigenvaluesoftheaddi-
+ i(−1)j 2Πˆ(xj)cosθˆ+(xj)+H.c. . (A·3)tional terms such as (∂xθˆ+)2cosθˆ+, (Πˆ)2cosθˆ+ and etc.
areallnegativeandtherebyirrelevantinthesenseofthe
h i
Accordingly,thetimederivativeofP inthecontinuum
Sz renormalization group study. However these additional
limit can be expressed in the following form,
terms in H cannot be discarded, since the commuta-
dim
−i Pˆ ,H torbetweenthesetermsandthephaseoperatorproduces
Sz
the last two term of eq.(A·4):
h i
≈−N1α dx 2JΠˆ +∆ Πˆcosθˆ++H.c. (A.·4) i θˆ+(x),Hdim
Z (cid:26) h i(cid:27) 2π
Fromnowon,wewillshowthatthisexpressionisiden- (cid:2) (cid:3)
tical to the time derivative of the phase operator: = −∆ Πˆ(x)cosθˆ+(x)+cosθˆ+(x)Πˆ(x) .(A·9)
(cid:20) (cid:21)
1 θˆ (x)
(r.h.s. of eq.(A·4))=i[ dx + ,Hˆ]. (A·5) After all, by comparing eqs.(A·6) and (A·9)
Nα 2π with eq.(A·4), we can safely replace −i[P ,H] by
Z Sz
The first term in the r.h.s. of eq.(A·4) comes from the i dx[θˆ+,H] in the continuum limit and rewrite
Nα 2π
commutator between θˆ and H given in eq.(16), eq. (A·1) into a following simple form by using the
+ XY
R
bosonization language:
θˆ (x)
i +2π ,HXY =−2JΠˆ(x). (A·6) ∂hPˆ i 1 hg|[θˆ+(x),Hˆ]|NihN|∂Hˆ|gi
h i Sz ≈ dx 2π ∂λ +c.c.
On the other hand, the commutator between the phase ∂λ Nα (E −E )2
N6=gZ " g N #
operator and H given in eq.(18) vanishes and does X
dim
not produce the last two terms of eq. (A·4). This is be- 1 θˆ (x) ∂
+
= − dx hg| |NihN| |gi+c.c.
cause eq.(18) is not an accurate expression for Hdim in Nα 2π ∂λ
the continuum limit and needs some additional terms, XN Z h i
whose commutators with the phase operator correctly 1 ∂ θˆ (x)
+
= − dx h i. (A·10)
yield the last two terms of eq. (A·4). In order to prove Nα ∂λ 2π
Z
10 J.Phys.Soc.Jpn. FullPaper Online-JournalSubcommittee
Here we used hN|∂∂Hλˆ|gi=(Eg −EN)hN|∂∂λ|gi. 33) P.JordanandE.Wigner:Z.Phys.47(1928) 631
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Phys.Rev.Lett.90(2003) 206601 tot
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Wetakethissignconvention onlyin 2.
§ 54) Inspin-gappedsystems,theeffectsoftheboundaryrangeonly