Exact ground states with deconfined gapless excitations for the 3 leg spin-1/2 tube

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by  M. Lajko

Exact ground states with deconfined gapless excitations for the 3 leg spin–1/2 tube
Miklo´s Lajk´o,1,2 Philippe Sindzingre,3 and Karlo Penc1
1Research Institute for Solid State Physics and Optics, H-1525 Budapest, P.O.B. 49, Hungary
2Department of Physics, Budapest University of Technology and Economics, Budafoki u´t 8, H-1111 Budapest, Hungary
3Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, Universit´e P. et M. Curie, Paris, France
(Dated: January 9, 2012)
Weconsideraspin-1/2tube(athree-legladderwithperiodicboundaryconditions)withaHamil-
tonian given by two projection operators — one on the triangles, and the other on the square
plaquetteson thesideof thetube— that can bewritten in termsof Heisenberg and four-spin ring
exchange interactions. We identify 3 phases: (i) for strongly antiferromagnetic exchange on the
2
triangles, anexactgroundstatewithagappedspectrumcanbegiven asan alternation ofspinand
1
chiralitysingletbondsbetweennearesttriangles;(ii)forferromagneticexchangeonthetriangles,we
0
recoverthephaseofthespin-3/2Heisenbergchain;(iii) betweenthesetwophases,agaplessincom-
2
mensuratephaseexists. Weconstructanexactgroundstatewithtwodeconfineddomainwalls and
n a gapless excitation spectrum at the quantum phase transition point between the incommensurate
a
and dimerized phase.
J

6 PACSnumbers: 75.10.Jm,75.10.Kt,75.30.Kz

]
l
e The projection operator approachto spin models pro- subspaceofstateswherethetotalspinoftheplaquetteα
-
vided significant results on the ground state properties is2,andgives0ifthespinis0or1(i.e. ifapairofspins
r
t of quantum magnets. Examples include the Majumdar– on the α square form a valence bond). The expanded
s
. Ghosh Hamiltonian [1], a spin–1/2 antiferromagnetic Hamiltonian using the spin operators reads
t
a Heisenbergchainwhere the twoexactgroundstate wave
m L 3
functions are given by a product of purely nearest- = J S S +J S S
- neighbor valence bonds (pairs of S=1/2 spins forming H { ⊥ i,j · i,j+1 1 i,j · i+1,j
d Xi=1Xj=1
a singlet) with a gapped excitation spectrum, in accor-
n +J (S S +S S )
o dancewiththe Lieb–Schultz–Mattistheorem[2]. The ex- 2 i,j · i+1,j+1 i,j · i+1,j−1
c act “valence bond solid” ground state in the Affleck– +JRE[(Si,j Si+1,j)(Si,j+1 Si+1,j+1)
· ·
[ Kennedy–Lieb–Tasaki(AKLT)model[3]withgappedex- +(Si,j Si,j+1)(Si+1,j Si+1,j+1)
· ·
3 citationsisanexplicitrealizationofHaldane’sconjecture +(S S )(S S )] (2)
i,j i+1,j+1 i,j+1 i,j+1
v forS =1Heisenbergchains[4]. Furtherexamplesinclude · · }
1 the two–dimensionalShastry–Sutherlandmodel[5]which wheretheintra-triangleJ⊥ =5K(cid:3)/6+2K△/3,theinter-
0
5 has been realized in SrCu2(BO3)2 [6]. In the pyrochlore triangle J1 = 5K(cid:3)/6 and J2 = 5K(cid:3)/12, and the ring
5 lattice, Yamashita and Ueda have introduced a model exchange interaction JRE = K(cid:3)/3. We set K(cid:3) = 1 in
. with a macroscopically degenerate ground state [7]. In the following.
7
allthesecasestheHamiltonianisasumofprojectionop- Spin tubes [9–16] are interesting not only since there
0
1 erators [8] and positive semidefinite by construction, so exist experimental realizations [17] but also as they are
1 thatanystatethathas0energyisanexactgroundstate. the next step after spin ladders towards two dimension
: Hereweextendthis approachtoamodelofspin–1/2’s (2D). Batista and Trugman, have recently studied the
v
i arrangedin a 3 leg tube geometry, given by the Hamiltonian that is a sum of the Rα operators over all
X thesquaresofthesquarelatticeandshownthataclassof
r L L 3 states consisting of nearestneighbor valence bond cover-
a
H=K△ Pi+K(cid:3) R(i,j)(i+1,j)(i+1,j+1)(i,j+1) ings,where eachsquareplaquette sharesa valence bond,
Xi=1 Xi=1Xj=1 have zero energy, i.e. are exact ground states[18, 19].
(1) The spin tube is identical to the square latice with pe-
riod three wrapped in one direction.
Hamiltonian (see Fig. 1). The tube has L triangles and Exactdiagonalization(ED) of the Hamiltonian (1) up
periodic boundary conditions are assumed. The projec- to L=12 triangles (36 sites) showed for K =0 an un-
tor Pi = (4S˜i ·S˜i −3)/12, where S˜i = 3j=1Si,j is the usual result: for an odd number of triangle△s it revealed
spin operator on the ith triangle, gives P1 if the triangle a 0 energy spin–1/2 doublet at k = 0 momentum, while
has a total spin of 3/2, and 0 if the spin is 1/2. The for evennumber we find three singletgroundstates: one
projectionR actsonthesquaresthatareonthesurface at k = 0 and two at k = π, as shown in Fig. 2(a).
α
of the tube. We denote S = S as the sum of The appearance of 0 energy eigenstates means that they
α (i,j)∈α i,j
the spin operators belonging tPo the α square plaquette, have zero projections with all of the Rα operatorsin the
then R = (S S )(S S 2)/24 projects onto the Hamiltonian. As there is no static covering of valence
α α · α α · α −
2
(5,1)
2
(4,1) A1
(3,1) 1.5 A2
(2,1) E
(1,1)
1
E
0.5
(2,2)
(1,2) (2,3) 0 ((aa))
(1,3)
3
L IR
FIG.1. Thespintubewithasnapshotofvalencebondcover- 2.5 6 A1
ingfromtheexactspin–chiralitydimerizedgroundstate. The 2 8 A2
shaded plaquettesare not satisfied in this particular configu- 10 A1
ration,andonlyquantumresonancewithotherconfigurations E 1.5 12 A2
will make all the plaquettessatisfied.
1
0.5
0 ((bb))
bondsonthespin–tubewhereeachplaquetteissatisfied, 0 π/5 2π/5 3π/5 4π/5 π
this points to a feature that was not encounteredin pro-
k
jection Hamiltonians so far. Even more striking is the
appearance of the 0 energy ground state for the tubes FIG.2. (a) Energy spectrum from exact diagonalization of a
with odd number of triangles, as in this case necessarily tube with 10 triangles and K△ =0 as a function of momen-
aspinisleftunpaired. Inthefollowing,wewillshowthat tumalongthetube. Theempty(filled)symbolsdenotespin–
singlet (triplet) states classified according to the irreducible
this apparent contradiction is resolved due to the quan-
tum mechanical nature of the valence bonds, and that representations (IR) of D3. (b) Low energy two domain wall
excitationsinthethermodynamiclimitcomparedtoEDspec-
theexistenceofthe0energygroundstateintheoddsize
traofsmallclusterswithsymmetrycompatiblewiththevari-
systemisrelatedwiththefactthatK△ =0isaquantum ationalsolution. Thethicklinebelowthe(shaded)continuum
critical point with gapless deconfined excitations. is thebound state.
Let us begin by considering weakly coupled triangles.
The P splits the 8 states of a triangle into a S˜ = 3/2
i
quadruplet and two degenerate S˜ = 1/2 dublets. The
lattersetoffourstatesconstitutethekerneloftheP that that the spectrum is gapped. Since the energy between
i
can be classified according to the spin σ and chirality τ two neighboring triangles is 0 if either the spins or the
degrees of freedom, with an orthonormal basis spanned chiralitiesformasinglet,thepositivesemidefinite ′ has
H
by the σzτz , where a twofold degenerate exact ground state of alternating
| i ii
spin and chirality singlet bonds [22], shown in Fig. 3(a).
|σi,±1/2ii=|νiσ,1i+e±2π3i|νiσ,2i+e±4π3i|νiσ,3i. (3) Actuallythesetwostatesbreakingtranslationalinvari-
ance are ground states not only of the effective model
The νσ,j are states with a valence bond between sites
| i i (5), but are exact eigenstates of Eq. (1) for any value of
j+1,j+2andanunpairedspinσ onsite j ofatriangle:
K , and are ground states for K 0, when expressed
△ △
≥
1 as a linear superpositions of valence bonds [23] . In
νσ,j = (σ σ ), (4)
| i i √2 | i,j ↑i,j+1↓i,j+2i−| i,j ↓i,j+1↑i,j+2i this wave function each triangle contains a valence bond
– this makes the P projections on the triangles happy.
i
observingtheperiodicboundaryconditiononj. Forcon-
The unpaired spins of the triangles form valence bonds
venience, we use r and l for the τz = 1/2 chirality.
that connect pairs of neighboring triangles (they map to
| i | i ±
In the K + limit the low energy space is
△ the spin–singlet bonds of the effective model), as shown
→ ∞
spanned by the spin–1/2 states given by Eq. (3) and the
in Fig. 1 between triangles 2 and 3, and 4 and 5. The
effective spin–chirality Hamiltonian reads
plaquettes between these connected triangles all have a
valencebond,sothecorrespondingR sgive0. However,
L α
5 3
′ = +σˆ σˆ 1+τˆ+τˆ− +τˆ−τˆ+ , outofthethreeplaquettesbelongingtoachiralitysinglet
H 9Xi=1(cid:18)4 i· i+1(cid:19)(cid:0) i i+1 i i+1(cid:1) (e.g. triangle 1,2 or 3,4 in Fig. 1) two have valence bond
(5) andthethirdoneisseeminglynotsatisfied. Theproblem
where σˆ are the spin–1/2, and τˆ± are the chirality can be resolved if we realize that the three νσ,j states
i i | i i
(pseudospin–1/2) raising/lowering operators of the ith usedinthedefinitionof σ,τ inEq.(3)arenotindepen-
triangle. This is a particular case of the effective model dent, namely νσ,1 + νσ|,2 +i νσ,3 =0. So the chirality
| i i | i i | i i
studied in Refs. [10–14, 20, 21], where it has been shown singlet between the ith and i+1th triangles can be ex-
3
(a)
(a)
i−2 i−1 i i+1 i+2 i+3
i−2 i−1 i i+1 i+2 i+3
(b) (b)
(c)
(c)
FIG.3. (a)Schematicrepresentationofoneofthetwoground
states of H with alternating spin and chirality valence bonds (d)
(a translation by a lattice vector gives the other one). The
small dots inside the circle represent the individual spins of
a triangle, thick solid lines denote valence bonds. The arcs
(e)
connectingtwocirclesstandforachiralitysingletbond|rli−
|lri. In(b)and(c)weshowtherelevantdomainwallsinodd
lengthtubes: (b)|ξ↑i,atrianglewithS˜=3/2(hexagon);(c)
i (f)
theellipsecontainingthreespin–1/2triangleswithchiralities
|rrri+|llli and centered at i+1 denotes |η↑ i. Both |ξ↑i
i+1
and |η↑i havespin–1/2 degrees of freedom.
FIG. 4. Relevant two domain wall configurations in the spin
singlet sector for L even. In (a) the small ellipse denotes a
chirality triplet (|rli+|lri), that breaks up into two domain
walls. (c), (e) and (f) are generally given as |ξ↑η↓ −ξ↓η↑i,
i j i j
|η↑η↓−η↓η↑i,and|ξ↑ξ↓−ξ↓ξ↑i,wherethedomainwallscanbe
pressedas νσi,1νσi+1,2 νσi,2νσi+1,1 . Inthisexpression aribitjrariilyjseparatedi.j(b)ianjd (d) shows overlapping domain
the plaque|tties bei+tw1eeni−t|hei legsi+j1=i1 and 3, and j = 2 walls, (b) is actually |ξ↑ξ↓ −ξ↓ξ↑ i, in (d) the chirality
i i+1 i i+1
and 3 both have valence bonds explicitly in each term, configurationis|llrri−|rrlli. Arrowsconnectstatesbetween
thustheyaresatisfied. Usingthelineardependence,this whichtheHamiltonianHK∆=0hasanonzeromatrixelement,
is exactly the same as νσi,2νσi+1,3 νσi,3νσi+1,2 , that the position of the arrows corresponds to the position of R
| i i+1 i−| i i+1 i relevant in theoverlap.
makes the third plaquette explicitly satisfied. Thus we
have shown that even though there is no valence bond
covering that satisfies all the plaquettes, a linear combi-
nation of ‘imperfect’ coverings still constitutes a ground the domain walls in an infinite system is given by the
state(i.e. foranyplaquettewecanchooseabasiswherea
valence bond is explicitly present on that plaquette[24]). 5
E±(k)= 4 √10+6cos2k (6)
1 36(cid:16) ± (cid:17)
For tubes of odd length with periodic boundary con-
ditions the system cannot be covered with alternating gaplessdispersion. Thisisuncommonforawavefunction
spin and chirality singlets, yet ED shows a ground state consistingofshortrangevalencebonds,andisalsopossi-
with zero energy. A study of finite size wave functions bly observed in Ref. [25]. From the Hellmann–Feynman
disclosedthatthefollowingtwotypesofdomain–wallex- theorem we get that ∂EGS/∂K△ = 3/4, i.e. the energy
citationofthespin–chiralitysingletwavefunctionarerel- of the domain wall becomes negative for K△ < 0, lower
evant: one is a S˜ = 3/2 triangle connected with valence than the energy of the dimerized state. This indicates a
bonds to the two sides, the other is a S˜ = 1/2 triangle phase transition at K△ = 0 between the dimerized and
connectedwithchiralitytripletstothesides,asshownin a gapless phase.
Fig. 3. We use the notation ξσ and ησ respectively, Letus nowreturnto tubes with evennumberofspins.
| ii | ii
whereidenotesthepositionofthedomainwall. ξσ and In this case the relevant excitations are pairs of domain
| i i
ησ have0eigenvaluewithalltheR intheHamiltonian, walls that originate from promoting a chirality singlet
| ii
exceptthosethatincludespinsontheith triangle. Using bond into a chirality triplet, as shown in Fig. 4. In large
these states as a variational basis, we get the following systems the domain walls can propagate independently
non-vanishing overlaps in Fourier-space: ησ ησ = if they are far from each other. When the domain walls
h k|H| ki
5(1 a cosk)/6, ξσ ξσ = 5(1 a cosk)/18, and get close they can ‘overlap’ spatially, these components
− L h k|H| ki − L
ξσ ησ = 5√3(cosk a )/18, while the overlaps can again be retrieved from results of ED for small sys-
h k|H| ki − − L
are ησ ησ = (1 a cosk) and ξσ ξσ = 1, where tems. Oncealltherelevantstatesareidentified,theover-
h k | ki − L h k | ki
a = 8/2L vanishes for L . It turns out that for lap matrix and the matrix of the Hamiltonian between
L
→ ∞
K = 0 the √3ξσ + ησ is an eigenstate with 0 these states can be calculated analytically for any finite
△ | k=0i | k=0i
energy, thus an exact ground state. The propagation of length systems, allowing us to take the L limit
→ ∞
4
∆=0 ∆=0 ∆>0
isfy all the projection operators in the Hamiltonian, a
S=3/2 chain incommensurate dimerized
linear superposition of such coverings is a good ground
−0.23 0 K /K state. Furthermore,wehaveconstructedanexactground
state at a quantum phase transition point that involves
FIG. 5. Schematic phase diagram of themodel.
short range valence bonds and deconfined domain walls
with a gapless excitation spectrum. Identifying the rel-
evant excitations and combining with exact diagonaliza-
tion studies, we have conjectured the phase diagram of
[23] . At k = π, a particular linear combination, featur-
themodel(Fig.5),withthreephasesasafunctionofthe
ingdeconfineddomainwalls,has0energyindependentof
interactionstrengthinthetriangles: agappeddimerized,
the system size, and is therefore the third exact ground
agaplessincommensurate,andthegaplessLuttingerliq-
state. The low energy part of the two domain wall spec-
uid phase of the spin–3/2 Heisenberg chain.
trum is shownin Fig. 2(b): the two independently prop-
agating walls form a continuum, while the interaction Wearepleasedtoacknowledgestimulatingdiscussions
between the walls leads to a bound state. The bound with A. Kolezhuk, Z. Nussinov, and S. Miyashita. This
state is present for any value of k, whereas the energy work was supported by Hungarian OTKA under Grant
gap between the bound state and the continuum is k4 No. K73455. Large scale numerical computations were
near k=0 and π, indicating a fine tuned interaction∝be- performed at IDRIS (Orsay).
tween the domain walls. We note that a finite size gap
is closing as 1/L2 at k =0 in accordance with the Lieb-
Schultz-Mattistheorem,andthatthedomainwallsinthe
triplet spin sector arealso gapless. The two domainwall
[1] C. K. Majumdar and D. Ghosh,
spectrum differs from similar spectra in the Majumdar-
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Ghosh and AKLT model: in the first one a bound state [2] E. Lieb, T. Schultz, and D. Mattis,
emerges from a gapped two spinon continuum at finite Annals of Physics 16, 407 (1961).
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is well separated from the continuum[27], even if the in- Communications in Mathematical Physics 115, 477 (1988),
10.1007/BF01218021.
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△ [5] B. S. Shastry and B. Sutherland,
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successionoflevelcrossingswherethegroundstatealter- [6] H.Kageyama,K.Yoshimura,R.Stern,N.V.Mushnikov,
nates between the A and A symmetry as we decrease K.Onizuka,M.Kato,K.Kosuge,C.P.Slichter,T.Goto,
1 2
K from 0. If we recall that the symmetry of a chi- and Y.Ueda, Phys. Rev.Lett. 82, 3168 (1999).
△
[7] Y. Yamashita and K. Ueda,
rality valence bond is A , the alternation is related to
2 Phys. Rev.Lett. 85, 4960 (2000).
introducinga pairofdomainwallsateachlevelcrossing.
[8] P isanorthogonalprojectionifitisself-adjointandP =
This phase is gapless and incommensurate, however its P2. This implies that P has only two eigenvalues, 0 for
precise characterization we leave for a future work. At the states in the kernel of P, and 1 for the orthogonal
K 0.23 the alternation terminates before all the L subspace it projects onto.
△
≈ −
possible domain walls are added, and the incommensu- [9] D. C. Cabra, A. Honecker, and P. Pujol,
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[10] K. Kawano and M. Takahashi,
For K . 0.23 the model is in the universality class
△ − J. Phys.Soc. Jpn.66, 4001 (1997).
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→ −∞
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[13] S. Nishimoto and M. Arikawa,
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Phys. Rev.B 78, 054421 (2008).
ergy spectrum is adiabatically connected to the one of
[14] T.Sakai,M.Sato,K.Okunishi,Y.Otsuka,K.Okamoto,
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5
J. Schnack, R. Schnalle, J. Richter, P. K¨ogerler, G. N. states in thecase K△ =0 .
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explicit form of the ground states and the matrix ele-
ments of the Hamiltonian between the two domain wall
SUPPLEMENTARY MATERIAL
dimerized ground state
The explicit form of the dimerized wave function of alternating spin and chirality singlets reads as
L/2
( 1)(1/2−σ2zi) ν−σ2i−2,1νσ2i,2 ν−σ2i−2,2νσ2i,1 . (7)
{σ2,σX4,...,σL}Oi=1 − (cid:16)| 2i−1 2i i−| 2i−1 2i i(cid:17)
The ν−σ2i−2,1νσ2i,2 ν−σ2i−2,2νσ2i,1 term is a chirality singletbetweentriangles 2i 1 and2i. The spin degreesof
| 2i−1 2i i−| 2i−1 2i i −
freedomof triangles2i and2i+1 forma singletbond , thereforethe spin oftriangle 2i+1 is the opposite
|↑↓i−|↓↑i
to that of triangle 2i, and the 1 factor takes care of the antisymmetrization.
−
the ground state with 2–domain walls and η↑ η↓ η↓ η↑ m n is odd.
m n− m n −
W(cid:12)hen the two d(cid:11)omain walls get close, i.e they over-
(cid:12)
Most of the matrix elements of the Hamiltonian be- lap spatially, both the diagonal and off-diagonal matrix
tween2-domainwallstates,showninFig. 4inthepaper, element may differ from the general case, and they read
can be derived from the one domain wall overlaps. As a
reminder,theonedomainwallonsiteterms ξσ ξσ =
5/18 hηmσ|H|ηmσi = 5/6 and the propagahtiomn|Hof|amido- Dηm↑ ηm↓+1−ηm↓ ηm↑+1(cid:12)H(cid:12)ηm↑ ηm↓+1−ηm↓ ηm↑+1E=5/6,
main wall is described by ξσ ησ = 5√3/36. (cid:12) (cid:12)
h m|H m±1 − ξ↑ ξ↓ ξ↓ ξ↑ (cid:12) (cid:12)ξ↑ ξ↓ ξ↓ ξ↑ =5/12,
These overlaps are valid for the in(cid:12)(cid:12)finitel(cid:11)y long systems D m m+1− m m+1(cid:12)H(cid:12) m m+1− m m+1E
(aLs 2→−L∞. ), for finite systems we have other terms scaling Dξm↑ ηm↓+2−ξm↓ ηm↑+2(cid:12)(cid:12)(cid:12)H(cid:12)(cid:12)(cid:12)ξm↑ ηm↓+2−ξm↓ ηm↑+2E=5/4,
(cid:12) (cid:12)
For tubes of even length, the nonzero overlaps in the ξ↑ ξ↓ ξ↓ ξ↑ (cid:12) (cid:12)ξ↑ η↓ ξ↓ η↑ = 5/18,
thermodynamic limit for m and n are sufficiently sepa- D m m+1− m m+1(cid:12)H(cid:12) m m+2− m m+2E −
(cid:12) (cid:12)
rated are: (cid:12) (cid:12) ζ ζ =5/6,
D m+12(cid:12)H(cid:12) m+12E
(cid:10)ηξm↑m↑ηξn↓n↓−−ηξm↓m↓ηξn↑n↑(cid:12)(cid:12)HH(cid:12)(cid:12)ηξm↑m↑ηξn↓n↓−−ηξm↓m↓ηξnn↑↑(cid:11)==55//93,, Dξm↑ ξm↓+1−ξm↓ ξm↑+1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)H(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ζm+21E=−5√6/36.
(cid:10)ξm↑ ηn↓ −ξm↓ ηn↑(cid:12)(cid:12)H(cid:12)(cid:12)ξm↑ ηn↓ −ξm↓ ηn↑(cid:11)=10/9, These states are orthonormal (i.e. the overlap matrix is
(cid:10)ηξ↑m↑ηξ↓(cid:10)n↓ −ηξ↓m↓ηξn↑↑(cid:12)(cid:12)H(cid:12)(cid:12)(cid:12)ξη(cid:12)(cid:12)m↑↑ηξn↓↓(cid:12)(cid:12)±1−ξηm↓↓ηξn↑↑±1E(cid:11)==−55√√33//3366,. ia2db(ebon)vt.eitwy)erineuthseedLto→pr∞odluimceitt.hTehvearmiaattiorinxaellreemsuelnttssignivFeing
For(cid:10) ξm↑ηn↓− ξm↓ ηn↑(cid:12)(cid:12)Hm(cid:12)(cid:12)(cid:12) mnins±e1v−en,mwhnil±e1fEor ξ−↑ ξ↓ ξ↓ ξ↑ asThe third exact ground state of HK∆=0 is then given
m n− m n − m n− m n
(cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12)
6
L/2 L−2 L/2
Ψ2dw =√2ζ(π)+2√3 ξξ(π,1) +3 ( 1)l−21 ξξ(π,l) √3i ( 1)2l ξη(π,l) ( 1)l−21 ηη(π,l)
| i − | i− − | i− − | i
l=X4,odd l=X2,even l=X1,odd
(8)
This wave function is actually valid for a system of arbitrary length L. Above we use the Fourier transform with the
following phase convention
l
ξξ(k,l)= ξ↑ ξ↓ ξ↓ ξ↑ exp ik m+
Xm (cid:12)(cid:12) m m,m+l− m m,m+lE (cid:18) (cid:18) 2(cid:19)(cid:19)
(cid:12) l
ηη(k,l)= η↑ η↓ η↓ η↑ exp ik m+
Xm (cid:12)(cid:12) m m,m+l− m m,m+lE (cid:18) (cid:18) 2(cid:19)(cid:19) (9)
(cid:12) l
ξη(k,l)= ξ↑ ξ↓ ξ↓ ξ↑ exp ik m+
Xm (cid:12)(cid:12) m m,m+l− m m,m+lE (cid:18) (cid:18) 2(cid:19)(cid:19)
(cid:12) 1
ζ(k)= ζ exp ik(m+ )
Xm (cid:12)(cid:12) m+12E (cid:18) 2 (cid:19)
(cid:12)
for ξξ(k,l) and ηη(k,l) l is odd and 1 l L/2 since ξξ(k,l) = ξξ(k, l) , while for ξη(k,l) l is even and runs
≤ ≤ − −
from 2 to L 2. m+l/2 corresponds to the center of mass of the two domain walls, and k is the total wave number.
−