Two-dimensional Valence Bond Solid (AKLT) states from $t_{2g}$ electrons

2.7 MB·English

by  Maciej Koch-Janusz

Two-dimensional Valence Bond Solid (AKLT) states from t electrons
2g
Maciej Koch-Janusz
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot IL-76100, Israel
D. I. Khomskii
II.Physikalisches Institut, Universitaet zu Koeln, Germany
Eran Sela
Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel Aviv, 69978, Israel
Two-dimensionalAKLTmodelonahoneycomblatticehasbeenshowntobeauniversalresourcefor
5 quantumcomputation. Inthisvalencebondsolid,however,thespininteractionsinvolvehigherpowers
1 of the Heisenberg coupling (S~i S~j)n, making these states seemingly unrealistic on bipartite lattices,
·
0 whereoneexpectsasimpleantiferromagneticorder. Weshowthatthoseinteractionscanbegenerated
2 by orbital physics in multiorbital Mott insulators. We focus on t electrons on the honeycomb lattice
2g
and propose a physical realization of the spin-3/2 AKLT state. We find a phase transition from the
n
AKLT to the Neel state on increasing Hund’s rule coupling, which is confirmed by density matrix
a
J renormalization group (DMRG) simulations. An experimental signature of the AKLT state consists of
protected, free spins-1/2 on lattice vacancies, which may be detected in the spin susceptibility.
4
1
PACSnumbers: 75.10.Nr,75.10.Jm,75.10.Hk

]
l
e
Introduction - Spin liquids are ground states of mag-
- a) b) c)
r netic systems which do not exhibit magnetic symmetry
t
s breaking down to zero temperature [1]; in contrast to
.
t conventional ordered states like ferromagnets and antifer- z
a
m romagnets (AF), they host fractional excitations [2]. A x y
classical example is the spin-1 Heisenberg chain which, as
-
d predicted by Haldane [3, 4] and observed in experiment
n [5], does not order and exhibits fractionalized spin-1/2 d) e)
o excitations at its boundaries.
c
[
In 1987, Affleck, Kennedy, Lieb, and Tasaki (AKLT)

1 formulated a family of exactly solvable valence-bond-solid
v (VBS) models which (i) account for the Haldane phase
2
in one dimension [6], but also (ii) generalize to higher Figure 1. (Color online) Nearest-neighbout t hoppings, cor-
1 2g
dimensions [7]. In these VBS states, the spin-S at each responding to (a) xy, (b) yz, and (c): zx orbitals, and (d) the
5
3 site is partitioned into 2S spins-1/2, and the latter form a resulting directional-hopping model, (e) free spins (denoted by
0 solid of singlet valence bonds, see Fig 1d for the S =3/2 stars) around lattice vacancy in the AKLT phase.
1. case. As a result the spin-singlet ground state exhibits
0 spin-1/2 excitations at edges or boundaries, whenever
5 valence bonds are broken. The model Hamiltonians which the absence of magnetic ordering at T=0 is guaranteed by
1 strongquantumfluctuations,thegroundstatesonbipartite
realize VBSs as exact ground states consist of a sum of
: lattices break the spin SU(2) symmetry and order antifer-
v operators projecting the total spin on neighbouring sites
i to the maximal possible value of 2S. These projectors romagnetically [7]. As a result, a physical realization of
X
may be expanded in powers of (S~ S~ ); for the AKLT the 2D AKLT phase in bipartite systems remained elusive.
r i · j The importance of just such states has been underlined
a model of spins-3/2 on the honeycomb lattice which will be
recentlybythediscoverythattheyareauniversalresource
considered here it reads:
for measurement-based quantum computation [8].
H = [S~ S~ +c (S~ S~ )2+c (S~ S~ )3], (1) In this paper we study theoretically multiorbital insula-
AKLT i j 2 i j 3 i j
J · · ·
tors described by Hubbard models with orbital-dependent
hXi,ji
directional nearest-neighbour hopping, and propose that
with c = 116 and c = 16 . The multi-quadratic in- theyarea platform forrealizationoftwo dimensionalVBS
2 243 3 243
teractions are typically not found in real materials e.g. states. The choice of honeycomb geometry is motivated by
insulators; usually the dominant spin-spin interaction is recent activity on the layered hexagonal Iridates such as
generated by superexchange, leading to AF Heisenberg Na IrO and Li IrO . In these spin-orbit entangled Mott
2 3 2 3
interactions. Consequently,exceptforthe1Dcaseinwhich insulatorsthet electronsleadto[9]anisotropicspin-spin
2g
2
interactions of the Kitaev-Heisenberg model. We consider U t
a similar Hubbard model, however in a d3 configuration,
i.e. havingthreeelectronspersite(halffilling);anexample
H
eff
is Li MnO . While Li MnO shows magnetic order at low
2 3 2 3 (cid:144)
Gapped Spin-3 2
temperatures [10], variants of this material may be in the 1
VBS phase. AKLT Neel
We show that the VBS state is realized in our model J t
H
~1
and is stable against interactions; it is adiabatically con- (cid:144)
nected to the AKLT state. Using analytic arguments and
Figure 2. Phase diagram of the system Eq.(2) for U0 <U.
DMRG simulations [11] we find that while strong Hund’s
(cid:144)
rule coupling leads to bulk Neel order, an extended region
in the phase diagram at small J exists, with properties
H sites connected by a single orbital γ, i.e. it factorizes into
identical to the AKLT state (see Fig. 2): an energy gap,
a sum of decoupled dimer Hamiltonians H . One can
exponentially decaying correlations, a singlet groundstate, hi,jiγ
show that the ground state remains a product of singlets
and, notably, symmetry protected free spin-1/2 excita-
for any value of U/t.
tions at lattice vacancies, which provide an experimental
InthelargeU limitthreespin-1/2statesareoccupiedat
signature.
each site, in distinct orbitals, since U <U. The electron
Model - We consider a tight-binding model of t elec- 0
2g hopping generates an interaction between the spin degrees
trons on a honeycomb lattice, arising for example in the
of freedom connected by the same orbital γ, as shown in
basal plane of the Iridate compounds, as seen in Fig. 1:
Fig.1(d): H = J ~s ~s with J = t2. Thus
at J = 0, efoffr any U,hit,hjieγgri,oγu·ndj,γstate is a prUoduct of
H =−t c†i,γ,σcj,γ,σ+H.c. singlHets with an enePrgy gap E = 2t for t U and
gap
σX=↑,↓hXi,jiγ E = t2 = J for U t. While one|c|annot(cid:29)solve the
+ U n n J ~s 2, (2) gap U (cid:29)
γγ0 i,γ,σ i,γ0,σ0 − H i model analytically for intermediate U, we expect a gapped
Xi γσX6=γ0σ0 Xi phase all along the line JH =0, as illustrated in Fig. 2.
Furthermore, at large U/t and J U we can write
where γ = xy,yz,zx 1,2,3 is the orbital index, H
(cid:28)
≡ down an effective Hamiltonian for three spins-1/2 per site,
ni,γ = σc†i,γ,σci,γ,σ and ~si = γ~si,γ with ~si,γ =
as a function of J :
σ,σ0c†i,γP,σ~σσ2σ0ci,γ,σ0. The notatioPn hi,jiγ implies that H
the hopping between nearest-neighbours i and j involves H =J ~s ~s J ( ~s )2. (3)
P eff iγ jγ H iγ
γ orbital only. · −
The directional hopping originates from the anisotropy hXi,jiγ Xj Xγ
ofthet orbitals,whicharepredominantlyconfinedtothe In the limit of large J /t a spin-3/2 degree of freedom
2g H
xy,yz,andzxplanes. Consequentlythereisanappreciable is formed on each site i, denoted by T~ = ~s . They
i γ i,γ
spatialoverlapofwavefunctionsonneighbouringsitesalong are coupled via a Heisenberg interaction H =J T~ T~
preferred directions only, as shown in Figs. 1(abc), and P eff i· j
with J = J = t2 , since in the spin-3/2 subspace of the
the hopping matrix elements in nonequivalent directions eff 9 9U
Hilbert space of three spins-1/2 we have~s = 1T~. The
on the honeycomb lattice are dominated by distinct t2g iγ 3 i
spins-3/2 live on a bipartite lattice and will thus order
orbitals, see Fig. 1(d). For simplicity we neglect hopping
antiferromagneticallyinthislimitandbreakthespinSU(2)
elements other than those in Eq. (2), which we don’t
symmetry. Since at small J /t the system is a gapped
expect to change the picture. H
singlet, there is necessarily a phase transition as function
TheinteractiontermscontainaHund’srulecouplingJ
H of J . The transition should occur at J J = t2.
favoring large spin formation, and a Hubbard interaction H H ∼ U
The spin-3/2 formation holds at large J for any U. In
Uinγtγe0ra=ctiUonδsγγb0e+twUee0(n1e−lecδtγrγo0n).s rHeesirdeinUg ainndthUe0saarmeethoer the limit of U (cid:28)JH their effective couplinHg is Jeff = 72JtH2
distinct orbitals, respectively; we assume U0 < U. We (from 2nd order perturbation theory). At U =0 the full
focus on a configuration with 3 electrons per site, as in model (2) contains only two parameters (J ,t), hence we
H
Li MnO [10]. expect that the phase transition will occur at J t. A
2 3 H
∼
Phase diagram - Consider first the case of vanishing schematic phase diagram is drawn in Fig. 2. We support
Hund’s rule coupling J =0. In the noninteracting limit it using DMRG simulations.
H
U = U 0 the system trivially forms a dimer band- Numerical simulations - To study the phase diagram
0
≡
insulator (six electrons per unit cell); the ground state is we performed density matrix renormalization group (finite
a product of spin-singlets on links with an energy gap of DMRG)calculationsusingtheITensor(http://itensor.org)
2t. Allowing U =0 while keeping U =0 does not alter package. We simulated model Eq. (3) on a decorated
0
6
the picture, since the Hamiltonian only couples pairs of honeycomb lattice using cylindrical boundary conditions
3
1 8 1 0.8
Egap 126, impu12r621it066y ξ−1 122166 7 0 log<Sz> (a) Se (b)
0.8 216, impurity -1 0.7
0.2 6
-2
0.15 5 0.6
0.6 -3
0.1 E 4 -4 0.5
0.4 0.05 3 -5
0.2 0 0 0.21/L 0.4 0.6 2 --76 000J...123H/J 0.4
0.5 0.3
1 0.7
∆ E -8 0.9
(a) (b) 1.0 126
0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 0 -9 04.0 1 2 3 4 5 6 7 8 021 60.5 1 1.5 2 2.5 3 3.5 4 0.2
JH/J JH/J r JH/J
Figure 3. (Color online) (a) Energy gap of model Eq. (3) in Figure4. (a)Expectationvalue sz ononeofthesublat-
units of J with and without an impurity. Inset: Finite size γh j,γi
tices,alongthecylinder,8hexagonsinlength(b)Entanglement
scaling of E for J /J =10, (b) Inverse correlation length P
gap H entropy S of the ground state of model Eq. (3) normalized by
ξ−1 extractedfromtheexponentialdecayof sz (seeFig. 4a). the numbeer of cut bonds.
h i
and system sizes of up to 216 spins-1/2, keeping a fixed transition: a singularity develops at J = J, as seen in
H
aspect ratio of the cylinder. Fig. 4b.
The calculated energy gap is shown in Fig. 3a: the Boundary excitations- One of defining properties of
systemisgappednotonlyinthetrivialcaseJ =0,where the AKLT state are the emergent degrees of freedom on
H
thegapisequalto t2 =J, butalsooverafiniteparameter system boundaries. Consider a lattice vacancy obtained
U
rangeofJ <J. UponincreasingJ thesystemreachesa by removing a single site from the bulk of the honeycomb
H H
transitionatJ J. Forfinitesizesystemswesimulated, lattice; see Fig. 1(e). In the AKLT state, this breaks three
H
theenergygapsa∼turatesatafinitevalueintheNeelphase, valence bonds hence creating three unpaired spins-1/2
corresponding to the finite size spectrum of spin wave surrounding the vacancy. Those are generically coupled,
excitations. It vanishes, however, in the thermodynamic however, as long as the system respects time reversal sym-
limit, consistent with the infinite size extrapolation shown metry a Kramers-degenerate spin-1/2 degree of freedom
in the inset. At small J the gap behaves as E = must remain. This effect is reproduced in our numerical
H gap
J J + (J2). This linear dependence can be derived simulations of the model Eq. (3) with a vacancy: we
an−alytHicallyO: coHnsider an elementary excitation of the VBS remove a site and compute the ground state energy in
state, which is a triplet on some link with an energy cost sectors whose total Sz differs by one. The light blue and
E = J. In 1st order perturbation theory J induces purple curves in Fig. 3a represent the energy difference
gap H
hopping of such excitations on the Kagome lattice formed ∆E of those two states: they are degenerate in the AKLT
by the links, hence reducing the energy gap as shown. The phase. In contrast, the two states split in the magnetically
phase transition may be thus thought of as condensation ordered Neel phase, where ∆E becomes energy to create a
of such spin-triplet excitations. magnon excitation.
The free spin-1/2 degrees of freedom emerging at lattice
While in principle it is not obvious that there is a direct
vacancies in the AKLT phase can be detected experimen-
AKLT-Neel transition, the absence of an intermediate
tallyduetotheircontributiontothefreespinsusceptibility,
phase can be tested by comparing the gap closing with
which will be proportional to the impurity density.
spin-spin correlations. Those can be numerically extracted
by applying a strong magnetic field on both boundaries of Two site toy model - The mechanism by which inter-
the cylinder and measuring the AF order parameter sz actions of the form of Eq. (1) arise from the directional
h i
as function of distance from the boundaries. This is shown nature of the orbitals may be intuitively understood by
in Fig. 4a for various values of J . For small J there is considering a two-site toy version of the model Eq. (3),
H H
exponentialdecayasexpectedfortheAKLTstate[6]while H =−JH( γ~s1,γ)2−JH( γ~s2,γ)2+J~s1,1~s2,1,seeinset
for large J there is no decay as expected for the long of Fig. 5. There is only one J coupling, which we take to
H P P
range ordered Neel state. The extracted inverse AF spin- be in the γ =xy 1 orbital.
≡
correlation length ξ 1 is plotted in Fig. 3b; its vanishing The Hilbert space of three spins-1/2 at each site decom-
−
coincides with the vanishing of the gap, ruling out the poses into two doublets and a quartet. As we discussed
possibility of an intermediate phase. The behaviour of the before for large J /J the spin-3/2 quartets are energeti-
H
entanglement entropy S when partitioning the system cally favored by the Hund’s rule coupling, and they are
e
intotwohalvesprovidesanothersignatureofasinglephase coupledbyaneffectiveHeisenbergAFinteractionJ = J.
eff 9
4
The Hilbert space of those two spins-3/2 decomposes in Hamiltonian, without closing the gap, as we now demon-
turn as follows: 3 3 =0 1 2 3, and consequently strate. No other phase transitions intervening, the gapped
2 ⊗ 2 ⊕ ⊕ ⊕
the Heisenberg term may be written in terms of projectors phaseofourdiagramisthereforeintheAKLTuniversality
to subspaces of total spin: class. The result is similar in spirit to Refs. [12, 13] con-
necting S =1 Heisenberg chain and the antiferromagnetic
15 11 3 9
T~ T~ = + . (4) S = 1/2 Heisenberg ladder, though in our intermediate
1 2 0 1 2 3
· − 4 P − 4 P − 4P 4P transformations we break translation invariance.
The energy splitting of those four multiplets is linear in At the noninteracitng point the ground state of our
J, as can be seen from dashed lines in bottom-left part of model Eq. (2) is a product of decoupled singlets. We
Fig. 5, wherethespectrumofthis6-spinmodel, computed show that the AKLT state can be connected to such a
via exact diagonalisation, is shown. As expected, the product of singlets. The procedure works for AKLT spin
ground state is a singlet. Note, however, that this result is models in arbitrary dimension. We work in the enlarged
valid for small J/J only, when the doublets do not mix Hilbert space, where each AKLT spin-3/2 is composed of
H
considerably with the spin-3/2 quartets. spin-1/2 degrees of freedom. The decoupling into singlets
is performed repeating the following steps: (i) Choose a
Upon increasing J/J the states belonging to doublets
H
site i, and let ij for j = 1,2,3 denote the three links
andquartetswillmix;thiscanbeseeninFig. 5intheform
h i
incident on i. (ii) Adiabatically weaken the coupling
of level repulsion between the states of like total S =
tot
0,1,2(seecolorcode)originatingin 3 3,1 3 and 1 1 constants ij,untilsiteiisalmostdecoupled. Theeffective
2⊗2 2⊗2 2⊗2 HamiltoniJanforsiteiisgivenbyspin-3/2coupledtothree
subspaces. Furthermore, the lowest S = 0,1,2 levels
tot
effective spins-1/2 emerging at the boundary of the AKLT
are approximately degenerate – they form a groundstate
state around site i. (iii) This Hamiltonian is adiabatically
manifold, while the unique state of S = 3 becomes
tot
connected to three decoupled singlets, each made up of
an excited state. This mimics the spectral properties
one of the effective boundary spins-1/2 and one of the
of the AKLT model Eq. (1) which on two sites reads
three spins-1/2 which build the spin-3/2 on site i (see
H = , in which all states but those forming a
AKLT 3
JP supplemental material for a generic example of a path in
maximal total spin-3 between the two sites, are exactly
Hamiltonian space). (iv) Turn back the couplings.
degenerate. ij
J
Thethreespins-1/2atsiteiarenowcompletelydecoupled
from each other (and only coupled to the spins-3/2 at
neighbouring sites j). (v) Move to the next site.
Uponvisitingallsites,theHamiltonianwillhavebecome
a sum of decoupled singlets on individual links, without
ever closing the gap. We have thus managed to find
a path in hamiltonian space between H and the
AKLT
noninteracting limit of our model Eq. (2).
Conclusions and outlook - We considered states which
canberealizedinelectronicsystemswithorbitaldegreesof
freedom, in which electron hopping, due to the directional
character of the orbitals, is highly anisotropic. For a half-
filled 2D honeycomb lattice of t electrons we have shown
2g
that a 2D AKLT state may be realized as the ground
state of the system. This state is known to be a universal
Figure5. Eigenvaluesofthetwo-siteHamiltonianasafunction resource for quantum computation [8].
of J/J . Inset above: two-site toy model.
H The physics described in this paper, in particular the
competition between the gapped singlet state of a valence
ThebunchingofthelowestStot =0,1,2levels,atotalof bondsolid(AKLT)typeandthatwithaconventionallong-
9 states, can be intuitively understood in the limit of large range magnetic order, may be relevant for real correlated
J/JH, when a singlet forms between the spins coupled by electron honeycomb systems. For example Na2IrO3 [14]
J. The remaining two pairs of spins-1/2 form two spins-1 and Na RuO [15] exhibit long-range magnetic order, but
2 3
due to the presence of JH, whose Hilbert space is: 1⊗1= some samples of Li2RuO3 have singlet valence bonds [16,
0 1 2. The approximate degeneracy between these 17]. Though this state is at a different filling to the one
⊕ ⊕
Stot =0,1,2 states is broken once we include an isotropic, we consider, it raises the hope to realize the VBS state in
all orbital-to-all orbital coupling J γ,γ0~s1γ ·~s2γ0. similarcompounds. Wenotethatweneglectedtheeffective
Adiabatic connection of band insulator and the AKLT dd hopping via ligand (e.g. oxygen p-orbitals), which may
P
state - We refer to the gapped phase in the diagram as be relevant in some materials; also the electron-lattice
the “AKLT” phase since at U =J =0 the model can be coupling may change the situation, leading to structural
H
smoothly connected in Hamiltonian space to the AKLT transitionswithalossofhexagonalsymmetry,whichseems
5
to be the case in MoCl3 [18]. [7] I. Affleck, T. Kennedy, E. Lieb, and H. Tasaki, Commu-
The general picture described above (possibility of for- nications in Mathematical Physics 115, 477 (1988).
mationofAKLT-likestateinamulti-orbitalsystem)canbe [8] T.-C.Wei,I.Affleck,andR.Raussendorf,Phys.Rev.Lett.
106, 070501 (2011).
realizedalsoinothergeometries,beyondthe2Dhoneycomb
[9] J. c. v. Chaloupka, G. Jackeli, and G. Khaliullin, Phys.
lattice: it may apply to recently synthesized hyperhoney-
Rev. Lett. 105, 027204 (2010).
comb lattices or to a new 3D structure of γ-Li IrO [19],
2 3 [10] S. Lee, S. Choi, J. Kim, H. Sim, C. Won, S. Lee, S. A.
and to novel hyperoctagon lattice structures proposed by Kim, N. Hur and J.-G. Park, J. Phys.: Condens. Matter
M. Hermanns and S. Trebst [20]. In all these cases, for 24, 456004 (2012).
ions in d3 configuration, we anticipate the possibility of [11] S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
existence of two competing states, the gapless state with a [12] S. R. White, Phys. Rev. B 53, 52 (1996).
[13] F. Anfuso and A. Rosch, Phys. Rev. B 75, 144420 (2007).
long-range magnetic order, and the gapful state of a VBS
[14] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster,
type.
I. I. Mazin, S. J. Blundell, P. G. Radaelli, Y. Singh,
We thank Erez Berg, Eli Eisenberg and Ady Stern for
P. Gegenwart, K. R. Choi, S.-W. Cheong, P. J. Baker,
useful discussions. MKJ thanks E.M. Stoudenmire and C. Stock, and J. Taylor, Phys. Rev. Lett. 108, 127204
Anna Kesselman for the helpful discussions on numerics. (2012).
This work was supported by ISF and Marie Curie CIG [15] J. C. Wang, J. Terzic, T. F. Qi, F. Ye, S. J. Yuan,
grants (ES). S. Aswartham, S. V. Streltsov, D. I. Khomskii, R. K.
Kaul, and G. Cao, Phys. Rev. B 90, 161110 (2014).
[16] Y. Miura, Y. Yasui, M. Sato, N. Igawa, and K. Kakurai,
Journal of the Physical Society of Japan 76 (2007).
[17] G. Jackeli and D. Khomskii, Physical review letters 100,
147203 (2008).
[1] L. Balents, Nature 464, 199 (2010).
[18] H. Schafer, H.-G. V. Schnering, J. Tillack, F. Kuhnen,
[2] X. G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B 39,
H. Wohrle, and H. Baumann, Z.Anorg.Allg.Chem. 353,
11413 (1989).
281 (1967).
[3] F. Haldane, Physics Letters A 93, 464 (1983).
[19] A.Biffin,R.D.Johnson,I.Kimchi,R.Morris,A.Bombardi,
[4] F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).
J.G. Analytis, A. Vishwanath, R. Coldea, arXiv:1407.3954
[5] M. Hagiwara, K. Katsumata, I. Affleck, B. I. Halperin,
(unpublished).
and J. P. Renard, Phys. Rev. Lett. 65, 3181 (1990).
[20] M. Hermanns, S. Trebst, Phys. Rev. B 89, 235102 (2014).
[6] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys.
Rev. Lett. 59, 799 (1987).
Supplemental material: Two-dimensional Valence Bond Solid (AKLT) states from t
2g
electrons
fact that the lowest excitation carries spin-1 for both the
8 AKLT and the Neel phase.
Adiabatic connection Whileitisnotessential,forread-
6 ers’ interest, we show an explicit example of a path in
E Hamiltonian space mentioned in the main text, by means
4 of exact diagonalization. The path itself is generic.
We write the central spin-3/2 as S~ = 3 ~σ and
5 2 the three effective boundary spins we denoteγb=y1~τ γ. The
γ
1 P
projector into the total spin of 3/2+1/2 between S and
0 0
2 0.0 0.5 1.0 1.5 2.0 one of the τ spins is:
J Λ(cid:61)1
H

n
a Figure 1. Eigenvalues of H[JH,λ] in Eq. (2) as function of JH 1 1
4 J at λ=1, showing a singlet grou(cid:72)nd(cid:76)state for any JH. P32+12(S~,~τγ)= 4(S~ +~τγ)2− 2. (1)
Defining the interpolating Hamiltonian as:
1
DMRG simulations - We include here some tech-
]
el nuiscinalgdtehtaeilsdeavbeoloupterthebrDanMchRGofsimthuelaItTioennss,orperpfaocrkmaegde H[J ,λ]= J ( 3 ~σ )2 3(3 +1)
r- (http://itensor.org)ontheWeizmannInstitutecluster. We H − H γ=1 γ − 2 2 !
t X
s have extended the functionality of the package by writing 3 3
t. a new matrix-product Hamiltonian file for a decorated +(1 λ) (S~,~τ )+λ (~σ ~τ + 3), (2)
a − P32+21 γ γ · γ 4
m honeycomb lattice with and without a vacancy (see Figs. γX=1 γX=1
1d,e in the main text). The computations were performed
- the initial Hamiltonian is H =H[ ,0], and the final
d for cylindrical geometry, keeping the aspect ratio of cylin- init ∞
n der length to circumference (in units of hexagons) equal Hamiltonian is Hfinal = H[0,1]. A path in the (JH,λ)
o to 2. The system sizes simulated were 60, 126 and 216 parameter space should now connect these two points
c without closing the gap.
spin-1/2 degrees of freedom, which corresponds to 4x2,
[
6x3 and 8x4 cylinders. The initial state for the DMRG A convenient path is via (JH,λ)=(∞,0)→(∞,1)→
1 algorithm was chosen to be a classical Neel state with (0,1). It is obvious that along the first transformation
v
2 Stzot = 0. The free spin-1/2 degrees of freedom on the (JH,λ)=(∞,0)→(∞,1) the ground state remains un-
edges of the cylinder were frozen using a strong magnetic changedandthegapremainsfinite,sinceS~ islockedintoa
1
5 field applied there. These dangling free spins on the two spin-3/2 state by Jh and the Hamiltonian only undergoes
3 edges of the cylinder live on two distinct sublattices of a change by a constant and by a positive proportional-
0 the Neel AF state, hence opposite magnetic fields were ity coefficient. The results of the second transformation
.
1 applied on different edges. We have kept up to m=1600 ( ,1) (0,1) is shown in Fig. 1, where the spectrum
∞ →
0 states in the 8x4 computation and up to m=800 in the of the exactly diagonalized Hamiltonian is plotted as a
5
smaller ones. The gap was obtained by comparing the function of J : the gap remains finite. We stress that this
1 H
ground state energies in sectors with Sz =0,1 using the particular path was chosen for illustrative purposes only.
: tot
v
i
X
r
a