Exact results for SU(3) spin chains: trimer states, valence bond solids, and their parent Hamiltonians

0.12 MB·English

by  Martin Greiter

Exact results for SU(3) spin chains:
trimer states, valence bond solids, and their parent Hamiltonians
Martin Greiter, Stephan Rachel, and Dirk Schuricht
Institut fu¨r Theorie der Kondensierten Materie, Universit¨at Karlsruhe, Postfach 6980, 76128 Karlsruhe, Germany
We introduce several exact models for SU(3) spin chains: (1) a translationally invariant parent
Hamiltonianinvolvingfour-siteinteractionsforthetrimerchain,withathree-folddegenerateground
7
state. Weprovidenumericalevidencethattheelementaryexcitationsofthismodeltransformunder
0 representation ¯3 of SU(3) if the original spins of the model transform underrep. 3. (2) a family of
0
2 parent Hamiltonians for valence bond solids of SU(3) chains with spin reps. 6, 10, and 8 on each
lattice site. We argue that of these three models, only the latter two exhibit spinon confinement
n
and a Haldane gap in the excitation spectrum.
a
J
PACSnumbers: 75.10.Jm,75.10.Pq,75.10.Dg,32.80.Pj
5
1
Beginning with the invention of the Bethe Ansatz in however,cannotbe directly generalizedto SU(n) chains,
] 1931[1]asamethodtosolvetheS = 1 Heisenbergchain asthereisnodirectequivalentoftheCP1 representation
el with nearest neighbor interactions, a s2ignificant share of used in Haldane’s analysis.
-
r theentireeffortincondensedmatterphysicshasbeende- In recent years, ultracold atoms in optical lattices
t
s votedto the study ofquantumspinchains. Faddeev and haveprovideda frameworkfor model realizationsofvar-
t. Takhtajan[2] discoveredin1981thatthe elementaryex- ious problems of condensed matter physics, including
a citations(nowcalledspinons)ofthe spin-1/2Heisenberg the phase transition from a superfluid to a Mott insu-
m
chain carry spin 1/2 while the Hilbert space is spanned lator[26, 27], the fermionic Hubbard model[28, 29], and
d- by spin flips, which carry spin 1. The fractional quanti- SU(2) spin chains [30, 31]. In particular, the Hamiltoni-
n zation of spin in spin chains is comparable to the frac- ansfor spinlattice models maybe engineeredwith polar
o tionalquantizationofchargeinquantizedHallliquids[3]. moleculesstoredinopticallattices,wherethespinisrep-
c In 1982, Haldane [4] identified the O(3) nonlinear sigma resented by a single-valence electron of a heteronuclear
[ model as the effective low-energy field theory of SU(2) molecule [32]. Systems of ultracold atoms in optical lat-
1 spinchains,andarguedthatchainswithintegerspinpos- tices may further provide an experimantal realization of
v sessagapintheexcitationspectrum,whileatopological SU(3) spin systems, and in particular antiferromagnetic
4 term renders half-integer spin chains gapless. SU(3) spin chains, in due course. A simple and intrigu-
5
Thegeneralmethods—theBetheAnsatzandtheuseof ing possibility is to manipulate an atomic system with
3
effective field theories including bosonization—are com- total angular momentum F = 3 such that it simulates
1 2
plemented by a number ofexactly solvable models, most anSU(3)spin. Forsuchatoms,oneeffectivelysuppresses
0
7 prominently among them the Majumdar–Ghosh (MG) the occupation of one of the “middle” states, say the
0 Hamiltonian [5] for the S = 1 dimer chain, the AKLT Fz = −1 state, by shifting it to a higher energy while
2 2
/ model [6] as a paradigm of the gapped S =1 chain, and keeping the other states approximately degenerate. At
t
a the Haldane–Shastry model (HSM) [7–9]. In the HSM sufficientlylowtemperatures,oneisleftwiththreeinter-
m thewavefunctionsforthegroundstateandsingle-spinon nal states Fz = −3,+1,+3, which one identifies with
2 2 2
- excitations are of a simple Jastrow form, elevating the the colors “blue”, “red”, and “green” of an SU(3) spin.
d
conceptual similarity to quantized Hall states to a for- In leading order, the number of particles of each color
n
mal equivalence. One of the unique features of the HSM is now conserved, as required by the SU(3) symmetry.
o
c is that the spinons are free in the sense that they only If one places one of these atoms at each site of an opti-
: interact through their half-Fermi statistics [10–12]. The callattice andallowsfor aweakhopping,one obtainsan
v
HSM has been generalizedfrom SU(2) to SU(n) [13–16]. SU(3)antiferromagnetforsufficientlylargeon-siterepul-
i
X For the MG and the AKLT model, only the ground sions U.
r statesareknownexactly. Nonetheless,thesemodelshave Motivated by both this prospect as well as the math-
a
amplycontributedtoourunderstandingofmanyaspects ematical challenges inherent to the problem, we propose
ofspinchains,eachofthemthroughthespecificconcepts severalexactmodelsforSU(3)spinchainsinthisLetter.
captured in its ground state [17–24]. The models aresimilar in spiritto the MG orthe AKLT
In the past, the motivation to study SU(n) spin sys- modelforSU(2),andconsistofparentHamiltoniansand
tems with n > 2 has been mainly formal. The Bethe their exact ground states.
Ansatz has been generalized to multiple component sys- Consider a chain with N lattice sites, where N has to
tems by Sutherland [25], and has been applied to the be divisible by three, and periodic boundary conditions
SU(n) HSM [13, 14]. The effective field theory descrip- (PBCs). On each lattice site we place an SU(3) spin
tion of Haldane yielding the distinction between gapless which transforms under the fundamental representation
half-integer spin chains and gapped integer spin chains, 3 = (1,0), i.e., the spin can take the values (or colors)
blue (b), red (r), or green (g). (We label the represen- 20 20
exact energy state for N=13 exact energy state for N=14
tations of SU(3) by their dimensions (the bold numbers) trial state trial state
or their Dynkin coordinates(a pair of non-negative inte-
15 15
gtsahicerneergssgl)cpeitv[i3ne13csn].=o.b)nTy(Te0ha,hec0eh)t,httrhwriermheeieeclrihnnesewtiagaerhtlcebyasolirlaniarndegetprosibeimtnteaedsrientnaoetndfdotbrrsmiykmeraetenrcqhSustUiitari(tbn3eygs) energy [arbitrary units] 10 10
5 5
c c c c c c c c c (1)
0 0
andtwomorestatesobtainedby shifting this oneby one 0 2 4 6 8 10 12 0 2 4 6 8 10 12 14
momenta [2 pi/N] momenta [2 pi/N]
or two lattice sites, respectively. Introducing operators
c† which create a fermion of color σ (σ = b,r,g) at FIG. 1: Dispersions of the rep. 3 (left) and the rep. ¯3 trial
iσ
states (right) in comparison to the exact excitation energies
lattice site i, the trimer states can be written as
of (5). The lines are a guide to theeye.
ψ(µ) = sign(π)c† c† c† |0i,
trimer iα i+1β i+2γ
(cid:12)(cid:12)(cid:12) E(i−3µYiniteger)(cid:16)(πα(,Xbβ,,rγ,g)=) (cid:17) (where ¯3 = (0,1)), i.e., the total SU(3) spin on four
neighboring sites can only transform under representa-
(2)
tions 3 or ¯6. The eigenvalues of the quadratic Casimir
where µ = 1,2,3 labels the three degenerate ground
operator for these representations are 4 and 10, respec-
states, i runs over the lattice sites subject to the con- 3 3
straint that i−µ is integer, and the sum extends over all tively. The auxiliary operators
3
six permutations π of the three colors b, r, and g.
2 4 2 10
asThe SU(3) generators at each lattice site i are defined Hi =(cid:18)(cid:16)J(i4)(cid:17) − 3(cid:19)(cid:18)(cid:16)J(i4)(cid:17) − 3 (cid:19) (7)
1 henceannihilatethetrimerstatesforallvaluesofi,while
Jia = 2 c†iσλaσσ′ciσ′, a=1,...,8, (3) theyyieldpositiveeigenvaluesfor15or15′,i.e.,allother
σ,σX′=b,r,g states. Summing H over all lattice sites i yields (5).
i
where the λa are the Gell-Mann matrices [33]. The op- There are two different kinds of domain walls between
erators (3) satisfy the commutation relations Ja,Jb = the degenerate ground states. The first kind consists of
i j
an individual SU(3) spin, which transforms under rep-
δ fabcJc, a,b,c=1,...,8, with fabc the stru(cid:2)cture c(cid:3)on-
ij i resentation 3; the second kind consists of two antisym-
stants of SU(3). We further introduce the total SU(3)
metrically coupled spins on two neighboring sites, and
spin of ν neighboring sites i,...,i+ν−1, hence transformsunder representation¯3. Since they can
i+ν−1 decay into each other, only one of these domain walls
Ji(ν) = Jj, (4) can constitute an approximate eigenstate of the trimer
Xj=i model. We have performed numerical studies on chains
withN =13andN =14,whichclearlyindicatethatthe
where J is the eight-dimensional vector formed by its
i elementary excitations of the trimer chain (5) transform
components (3). The parent Hamiltonian for the trimer under ¯3 (see Fig. 1). This result appears to be a general
states (2) is given by
feature of rep. 3 SU(3) spin chains, as it was recently
N shown explicitly to hold for the HSM as well [16]. The
4 14 2 40
H = J(4) − J(4) + . (5) elementaryexcitationsofthetrimerchainaredeconfined,
trimer Xi=1(cid:18)(cid:16) i (cid:17) 3 (cid:16) i (cid:17) 9 (cid:19) meaningthattheenergyoftwolocalizedrepresentation¯3
domainwallsorcoloronsdoesnotdependonthedistance
To verify this Hamiltonian, note that since the spins on
between them.
theindividualsitestransformunderthefundamentalrep-
resentation 3, the SU(3) content of four sites is We now introduce a family of exactly soluble valence
bondmodelsfor SU(3)chainsofvariousspinrepresenta-
3⊗3⊗3⊗3 = 3·3 ⊕ 2·¯6 ⊕ 3·15 ⊕ 15′, (6) tionsoftheSU(3)spinsateachlatticesite. Toformulate
i.e., we obtain representations 3, ¯6=(0,2), and the two t(hbelusee),mro,dre†ls(,rewde),wainlldugse,gS†U((g3r)eeSnc)hw[3i5n]g,ewrhbiochsoanrsebd,be†-
non-equivalent representations 15 = (2,1) and 15′ =
fined by |bi = c†|0i = b†|0i, |ri = c†|0i = r†|0i,
(4,0). All these representations can be distinguished by b r
and |gi = c†|0i = g†|0i, and satisfy b,b† = r,r† =
their eigenvalues of the quadratic Casimir operator [33]. g
Forthe trimerstates(1), the situationsimplifies aswe g,g† = 1 while all other commutat(cid:2)ors v(cid:3)anis(cid:2)h. T(cid:3)he
only have the two possibilities (cid:2)Schwi(cid:3)nger bosons can be used to combine spins trans-
forming under the fundamental representation3=(1,0)
c c c c =ˆ 1 ⊗ 3 = 3, symmetrically, and hence to construct representations
c c c c =ˆ ¯3 ⊗ ¯3 = 3 ⊕ ¯6, like 6=(2,0) and 10=(3,0).
2
The trimer states (2) can be rewritten using SU(3) and27=(2,2). With the eigenvaluesofthe Casimirop-
Schwinger bosons as ψ(µ) =Ψµ b†,r†,g† |0i with erator, which are 6 and 8, respectively, we obtain the
(cid:12) trimerE parent Hamiltonian for (12)
(cid:12) (cid:2) (cid:3)
(cid:12)
Ψµ b†,r†,g† = sign(π)α†β† γ† . N
(cid:2) (cid:3) Yi (cid:16)(α,Xβ,γ)= i i+1 i+2(cid:17) H10VBS = JiJi+1 2+5JiJi+1+6 . (14)
(i−3µinteger) π(b,r,g) Xi=1 (cid:16)(cid:0) (cid:1) (cid:17)
(8) The Hamiltonian (14) provides the equivalent of the
We obtain a representations 6 VBS from two trimer AKLT model [6], whose unique ground state is con-
states by projecting the tensor product of two funda- structed from dimer states by projectiononto spin 1, for
mental representations 3 onto the symmetric subspace, SU(3) spin chains.
i.e., onto the 6 in the tensor product 3 ⊗ 3 = ¯3 ⊕ 6. Since the 10 VBS state (12) is unique, domain walls
Graphically, we illustrate this as follows: connectingdifferentgroundstatesdonotexist. Wehence
one site expectthecoloronandanti-coloronexcitationstobecon-
c c c c c c c c c finedinpairs,asillustratedbelow. Thestatebetweenthe
c c c c c c c c c excitations is no longer annihilated by (14), as there are
pairs of neighboring sites containing representations dif-
projection onto rep. 6=(2,0) (9) ferent from 10 and 27, as indicated by the dotted box
below. As the number of such pairs increases linearly
This construction yields three linearly independent 6
with the distance between the excitation, the confine-
VBS states, which are readily written out using (8),
ment potential depends linearly on this distance.
ψ6(µV)BS =Ψµ b†,r†,g† ·Ψµ+1 b†,r†,g† |0i (10) colo¯3ron anti-c3oloron
(cid:12)(cid:12) E (cid:2) (cid:3) (cid:2) (cid:3) c c c sc sc c c c c c c sc c c c
for(cid:12)µ = 1, 2, or 3. These states are zero-energy ground c c c c c c c c c c c c c c c
statesoftheparentHamiltonianH6VBS = Ni=1Hi with c c c c c c c c c c c cc cc cc c
P (cid:27) -
H = J(4) 2− 4 J(4) 2− 10 J(4) 2− 16 energy cost ∝ distance (15)
i (cid:18)(cid:16) i (cid:17) 3(cid:19)(cid:18)(cid:16) i (cid:17) 3 (cid:19)(cid:18)(cid:16) i (cid:17) 3 (cid:19) The confinement force between the pair induces a linear
(11)
oscillator potential for the relative motion of the con-
(see [34]). Note that the operators Ja, a = 1,...,8, are
i stituents. The zero-point energy of this oscillator gives
now given by 6×6 matrices, as the Gell-Mann matrices
rise to a Haldane-type energy gap (see [36] for a similar
only provide the generators (3) of the fundamental rep-
discussion in the two-leg Heisenberg ladder). We expect
resentation3. As inthe trimermodel,twodistincttypes this gap to be a generic feature of rep. 10 spin chains
ofdomainwallsexist,whichtransformaccordingtoreps.
3 and¯3. Since both excitations are merely domainwalls witFhinsahlolyr,t-wraengceonasnttriufecrtroamraegpnreetsiecntinatteioranct8ioVnsB.S state,
between different ground states, there is no confinement where 8 = (1,1) is the adjoint representation of SU(3).
between them. Consider first a chain with alternating reps. 3 and ¯3 on
Let us now turn to the 10 VBS chain. By combining
neighboring sites, which we combine into singlets. This
the three different trimer states (8) symmetrically,
can be done in two ways, yielding the two states
|ψ10VBSi=Ψ1 b†,r†,g† ·Ψ2 b†,r†,g† ·Ψ3 b†,r†,g† |0i, c e c e c e and c e c e c e .
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (12) 3 ¯3
we automatically project out the rep. 10 in the tensor We then combine one 3-¯3 state with the one shifted by
product 3⊗3⊗3 = 1⊕2·8⊕10 generated on each one lattice spacing. This yields representations 3⊗¯3 =
lattice site by the three trimer chains. This construction 1⊕8ateachsite. The8VBSstateisobtainedbyproject-
yields a unique state: ingontothe adjointrepresentations8. Correspondingto
the two 3-¯3 states illustrated above, we obtain two lin-
one site
earlyindependent 8 VBS states,ΨL andΨR, whichmay
c c c c c c c c c c c c
c c c c c c c c c c c c be visualized as
c c c c c c c c c c c c one site
c e c e c e c e c e c e
projection onto 10=(3,0) (13) e c e c e c and e c e c e c .
The parent Hamiltonian acts on pairs of neighboring
projection onto 8=(1,1) (16)
sites. It is constructed by noting that the only rep-
resentations which are included in both 10 ⊗ 10 and These states transform into each other under parity or
¯3⊗¯3⊗3⊗3 (which is the representationcontent of the color conjugation (interchange of 3 and ¯3). The corre-
totalspinoftwoneighboringsitesoftheVBSstateasin- sponding states may be formulated as a matrix prod-
dicatedbythedashedboxabove)arethereps.10=(0,3) uct [34].
3
TheparentHamiltonianforthesestatesisconstructed one may easily infer from a cartoon similar to the one
along the same lines as above, yielding above. We hence expect a Haldane gap for each individ-
ual domain wall as well.
N
H8VBS = JiJi+1 2+ 9JiJi+1+ 9 . (17) The results regarding confinement and deconfinement
Xi=1(cid:18)(cid:0) (cid:1) 2 2(cid:19) oftheexcitationsofthe6,10,and8VBSpresentedhere
are consistent with a rigorous theorem by Affleck and
ΨL and ΨR are the only (zero-energy) ground states of Lieb[37]ontheexistenceofenergygapsinthespectrum
(17) for N ≥3. of SU(n) nearest-neighbor Heisenberg spin chains.
The low-energy excitations of the 8 VBS model are
Inconclusion,wehaveformulatedseveralexactmodels
given by coloron–anti-coloronbound states:
ofSU(3)spinchains. We firstintroducedatrimermodel
anti-coloron coloron
3 ¯3 and presented evidence that the elementary excitations
of the model transform under the SU(3) representations
c e c e c e c e c e c e
c e sc e c e c ue c e conjugatetotherepresentationoftheoriginalspinonthe
chain. We further introduced three SU(3) valence bond
(cid:27) -
ΨL energy cost ∝ distance ΨL (18) solid chains with representation 6, 10, and 8, respec-
tively,oneachlattice site. Theelementaryexcitationsof
We find a linear confinement potential between the exci- the 10 and the 8 valence bond solid chain were found to
tations, and hence a Haldane-type gap in the spectrum.
be confined.
Numerical studies on a chain with N = 8 sites provide
evidenceinsupportofthisconclusion[34]. Inadditionto We would like to thank Peter Zoller, Ronny Thomale,
the bound state (18), the model allows for domain wall and Peter W¨olfle for valuable discussions of various as-
between the two ground states ΨL and ΨR. They con- pects of this work. SR was supported by a Ph.D. schol-
sist of bound states of either two colorons or two anti- arship of the Cusanuswerk, and DS by the German Re-
colorons, which are confined through the same mecha- searchFoundation(DFG) throughGK284andthe Cen-
nisms as the coloron–anti-coloron bound state (18), as ter for Functional Nanostructures Karlsruhe.
[1] H.Bethe, Z. Phys. 71, 205 (1931). [20] S.R.WhiteandI.Affleck,Phys.Rev.B54,9862(1996).
[2] L. D. Faddeev and L. A. Takhtajan, Phys. Lett. A 85, [21] U. Schollw¨ock et al.,Phys.Rev.B 53, 3304 (1996).
375 (1981). [22] A. Kolezhuk et al.,Phys. Rev.Lett. 77, 5142 (1996).
[3] R.B. Laughlin, Phys.Rev.Lett. 50, 1395 (1983). [23] A.KolezhukandU.Schollw¨ock,Phys.Rev.B65,100401
[4] F. D. M. Haldane, Phys. Lett. A 93, 464 (1983); Phys. (2002).
Rev.Lett. 50, 1153 (1983). [24] B. Normand and F. Mila, Phys. Rev. B 65, 104411
[5] C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (2002).
1399 (1969). [25] B. Sutherland,Phys.Rev.B 12, 3795 (1975).
[6] I. Affleck et al., Phys. Rev. Lett. 59, 799 (1987); Com- [26] D. Jaksch et al.,Phys. Rev.Lett. 81, 3108 (1998).
mun.Math. Phys. 115, 477 (1988). [27] M. Greiner et al.,Nature 415, 39 (2002).
[7] F. D.M. Haldane, Phys.Rev.Lett. 60, 635 (1988). [28] C. Honerkamp and W. Hofstetter, Phys. Rev. Lett. 92,
[8] B. S. Shastry,Phys.Rev.Lett. 60, 639 (1988). 170403 (2004); Phys.Rev.B 70, 094521 (2004).
[9] F. D.M. Haldane, Phys.Rev.Lett. 66, 1529 (1991). [29] A. Rapp et al.,cond-mat/0607138.
[10] F. D.M. Haldane, Phys.Rev.Lett. 67, 937 (1991). [30] L.-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev.
[11] F. H.L. Eßler, Phys. Rev.B 51, 13357 (1995). Lett. 91, 090402 (2003).
[12] M. Greiter and D. Schuricht, Phys. Rev. B 71, 224424 [31] J. J. Garc´ıa-Ripoll, M. A. Martin-Delgado, and J. I.
(2005). Cirac, Phys.Rev.Lett. 93, 250405 (2004).
[13] N. Kawakami, Phys. Rev. B 46, 1005 (1992); ibid. 46, [32] A. Micheli, G. K. Brennen, and P. Zoller, Nature Phys.
R3191 (1992). 2, 341 (2006).
[14] Z. N. C. Ha and F. D. M. Haldane, Phys. Rev. B 46, [33] J.F.Cornwell,Grouptheoryinphysics(AcademicPress,
9359 (1992); ibid. 47, 12459 (1993). London, 1984), Vol.II.
[15] P.BouwknegtandK.Schoutens,Nucl.Phys.B482,345 [34] M. Greiter and S.Rachel, in preparation.
(1996). [35] see e.g. A. Auerbach, Interacting electrons and quantum
[16] D. Schuricht and M. Greiter, Europhys. Lett. 71, 987 magnetism (Springer, NewYork,1994).
(2005); Phys.Rev. B 73, 235105 (2006). [36] M. Greiter, Phys. Rev. B 65, 134443 (2002); ibid. 66,
[17] I.Affleck, J. Phys.: Condens. Matter 1, 3047 (1989). 054505 (2002).
[18] K. Okamoto and K. Nomura, Phys. Lett. A 169, 433 [37] I.AffleckandE.H.Lieb,Lett.Math.Phys.12,57(1986).
(1992).
[19] S.Eggert, Phys. Rev.B 54, 9612 (1996).
4