Hierarchical mean-field approach to the $J_1$-$J_2$ Heisenberg model on a square lattice

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by  L. Isaev

Hierarchical mean-field approach to the J -J Heisenberg model on a square lattice
1 2
L. Isaev1, G. Ortiz1, and J. Dukelsky2
1Department of Physics, Indiana University, Bloomington IN 47405, USA
2Instituto de Estructura de la Materia - CSIC, Serrano 123, 28006 Madrid, Spain
We study the quantum phase diagram and excitation spectrum of the frustrated J1-J2 spin-1/2
Heisenberg Hamiltonian. A hierarchical mean-field approach, at the heart of which lies the idea
of identifying relevant degrees of freedom, is developed. Thus, by performing educated, manifestly
symmetry preserving mean-field approximations, we unveil fundamental properties of the system.
Wethencomparevariouscoveringsofthesquarelatticewithplaquettes,dimersandotherdegreesof
9 freedom,andshowthatonlythesymmetricplaquettecovering,whichreproducestheoriginalBravais
0 lattice,leadstotheknownphasediagram. Theintermediatequantumparamagneticphaseisshown
0 to be a (singlet) plaquette crystal, connected with the neighbouring N´eel phase by a continuous
2 phase transition. We also introduce fluctuations around the hierarchical mean-field solutions, and
demonstrate that in theparamagnetic phase theground and first excited states are separated by a
n
finitegap,whichclosesintheN´eelandcolumnarphases. Ourresultssuggestthatthequantumphase
a
J transition between N´eel and paramagnetic phases can be properly described within the Ginzburg-
Landau-Wilson paradigm.
3
1
PACSnumbers: 05.30.-d,75.10.Jm,64.70.Tg

]
l
e
I. INTRODUCTION As a result, several possible candidate ground states
-
r were proposed, namely: spin liquid7, preserving trans-
t
s lational and rotational symmetries of the lattice, as well
One of the primary goals of modern condensed mat-
at. ter physics is the characterization of strongly correlated athsevadriimouers1l2a,1t4ti,caensdymthmeeptlrayqubertetaekrinesgopnhaatisnegs,voauletnocfewbhoinchd
m quantumsystems. Alargeclassofsuchmaterialsisrepre-
phases6 are worth mentioning.
sentedbyfrustratedantiferromagnets,whicharebelieved
-
d toexhibitavarietyofnovelstatesofmatteratsufficiently Not surprisingly, the nature of the quantum phase
n strong coupling. Growing experimental evidence indi- transition separating the N´eel-ordered and quantum
o cates that layered materials such as Li VO(Si,Ge)O 1, paramagnetic phases is also under scrutiny. The most
2 4
c VOMoO 2 and BaCdVO(PO ) 3 can be adequately de- dramatic,andatthesametimeoriginal,scenario15 isbe-
[ 4 4 2
scribed by an antiferromagnetic Heisenberg model with lieved to violate the Ginzburg-Landau-Wilson paradigm

2 frustratingnext-andnext-next-nearestneighborinterac- ofphasetransitions16 whichrevolvesaroundthe concept
v tions. As a result, the study of low-dimensionalmagnets of an order parameter. Such point of view is based on
5
and their frustration-driven quantum phase transitions the observation that there are different spontaneously
3
have attracted a lot of theoretical attention in the last broken symmetries in the N´eel and quantum paramag-
0
4 decade4,5. neticphases,whichthuscannotbeconnectedbyagroup-
. A paradigmatic system, illustrating the effects of frus- subgroup relation. The former, of course, breaks the
9
SU(2) invariance of the Hamiltonian and lattice trans-
0 trating couplings, is the spin-1/2 Heisenberg model on
lational symmetry T17,18, but preserves the four-fold ro-
8 a square lattice with competing nearest (J1), and next-
0 nearest (J ) neighbor antiferromagnetic (AF) interac- tationalsymmetry of the square C4. On the other hand,
2
: tions (J -J model). Despite numerous analytical and the paramagnetic phase is known to restore the spin-
v 1 2
rotational symmetry and is believed to break T and C ,
i numericalefforts,itsphasediagram,whichexhibitsatwo 4
X due to spontaneous formation of dimers along the links
sublattice N´eel AF, quantum paramagnetic, and a four
r sublattice columnar AF states, continues to stir certain of the lattice12,14. It follows then, that these two phases
a cannotbejoinedbytheusualLandausecond-ordercriti-
controversy(forareviewofrecentachievements,seeRef.
calpoint. Thisphasetransitioncaneitherbeofthefirst-
5). While existence of the N´eel-ordered phase at small
order19(thelatestcoupledclustercalculations9,however,
frustrationratio J /J , and of the columnar AF state at
2 1
seem to rule out this possibility), or represent an exam-
largeJ /J is widely established, properties of the inter-
2 1
ple of a second order critical point, which cannot be de-
mediate non-magnetic phase, which occurs around the
scribedintermsofabulk orderparameter,butratherin
maximum frustration value J /J = 1/2, are still under
2 1
termsofemergentfractionalexcitations(spinons),which
debate. Particularly, the correlated nature of the inter-
become deconfined right at the critical point15.
mediate state and the kind of quantum phase transition
separatingitfromthe N´eelstate,attractmostattention. However,evidence regarding the structure of the non-
Various methods have been recently applied to charac- magnetic phase is quite controversial. Indeed, the re-
terize the quantum paramagneticphase, such as Green’s sults of spin-wave calculations12, large-N expansions14,
function Monte Carlo6,7,8, coupled cluster methods9, se- and calculations using the density matrix renormaliza-
ries expansions10 and field-theoretical methods11,12,13. tion group combined with Monte-Carlo simulations8 are
2
believed to indicate the emergence of a dimer order.
N´eel Plaquette Crystal Columnar
On the other hand, Monte-Carlo6 and coupled cluster
calculations9, and analytical results11 seem to support
the presence of C symmetry (plaquette-type ordering)
4
in the paramagnetic phase. In the absence of a reliable
numerical or analytical proof of existence of any partic-
ular order in the non-magnetic region, there is no ap-
parent reason to believe in the exotic deconfined quan-
tum criticality scenario. Although there apparently ex-
ists numerical evidence20, at the moment of writing the Txy ⊗C4⊗SU(2) Txy ⊗C4⊗SU(2) Ty ⊗C2⊗ SU(2)
authors are unaware of a local Hamiltonian in space di-
J = 0 J
mensions larger than one, rigorously proven to exhibit 2 Landau 2nd order Landau 1st order 2
the type of quantum critical point discussed in Ref. 15.
Interestingly,it wasdemonstratedin Ref. 21 that a two-
dimensional (2D) lattice model can possess a first order FIG.1: AschematicphasediagramoftheJ1-J2 model,sum-
marizing our results. In each phase we show spontaneously
quantumcriticalpoint,whichexhibitsdeconfinedexcita-
broken (framed symbols) and unbroken symmetries (usual
tions.
symbols). The translational invariance is broken along both
Allinall,thecomplexityofmethodsusedtoinferprop- directions in the N´eel and paramagnetic phases, and only
erties of the paramagnetic phase and the variety of dif- along the y–direction in the columnar phase. This fact is
ferent conclusions have created a certain degree of con- indicated by the subscripts xy and y after T. Conclusions
fusion. Our goal in the present paper is to try to clarify regarding the order of the phase transition, separating N´eel
andplaquettecrystalphases,aswellassymmetriesofvarious
some of this controversy by proposing a controlled and
phases, are based upon extrapolation of our results towards
manifestly symmetry preservingmethod, gearedto com-
thethermodynamic limit.
puting ground state properties of the J -J model. Our
1 2
approach is based on the recently proposed systematic
methodologyto investigatethe behaviorofstronglycou-
pled systems22, whose main idea consists of identifying
neticstatewiththecolumnarantiferromagneticphaseat
relevant degrees of freedom and performing an educated
(π,0) and (0,π) for J /J &0.66.
approximation,calledthehierarchicalmean-field(HMF), 2 1
touncoverthephasediagramandotherpropertiesofthe We emphasize thatour investigationprimarily focuses
system of interest. In a future work these ideas will be on the symmetry analysis of the various phases. Out of
coupled to a new, variationalwith respect to the energy, many possible coarse graining scenarios, such as cover-
renormalization group approach, which thus adapts to ingofthe2Dlatticewithplaquettes,dimersandcrosses,
the concept of relevant degrees of freedom. onlythe C -symmetry preserving plaquette(orsuperpla-
4
InthepresentworkweconstructHMFapproximations quette) covering (which reproduces the original Bravais
forthe J -J model. Thecruxofourmethodistheiden- lattice) displays the correct phase diagram. In particu-
1 2
tification of a plaquette (spin cluster 2 2 or even larger lar, the intermediate paramagnetic phase is shown to be
4 4 (superplaquette) symmetry-prese×rving cluster) as a plaquette crystal, which preserves spin and lattice ro-
th×e relevant elementary degree of freedom, which cap- tational symmetries. For all other scenarios, including
tures necessary quantum correlationsto representessen- dimerized (bond-ordered) phases, we were unable to re-
tial features of the phase diagram. The importance of produce all known quantum phase transition points of
this degree of freedom was realized only recently in the the model.
present context11, and somewhat earlier in connection
We notice that the HMF coarse graining procedure
with SU(4) spin-orbital23, and Hubbard24 models. Be-
leadsto anexplicit breakingofa particulartranslational
sides being variational, our formalism has the attractive
symmetry. Asaresult,onecannotdrawrigorousconclu-
feature of preserving fundamental lattice point symme-
sions on the order of the phase transitions, based solely
tries and the SU(2) symmetry of the Hamiltonian, by
on a fixed coarse graining. Nevertheless, it is still possi-
utilizing the Schwinger boson-type representation and
ble to make some predictions, using a finite-size scaling
Racah algebra technology. Remarkably, such simple
of the relevant degree of freedom towards the thermo-
mean-field calculation already yields all known results,
dynamic limit, where the effects associated with coarse
concerningthephasediagramoftheJ -J model,witha
1 2 graining should disappear.
goodaccuracy,namely: existenceofaN´eel-orderedphase
with antiferromagnetic wavevector (π,π) and spin-wave Next two sections are devoted to the formulation of
type excitations for J /J . 0.42, a non-magnetic in- the HMF approach. Then, we present results of our cal-
2 1
termediate gapped phase, separated by a second order culations and close the paper with a discussion. Our
quantum phase transition, and a first order transition main conclusions are summarized in Fig. 1, which em-
point, whichis characterizedby the discontinuousdisap- phasizes symmetry relations between different phases of
pearance of the energy gap and connects the paramag- the model.
3
3 4
II. THE PLAQUETTE DEGREE OF FREEDOM
J
1
Weconsiderthespin-1/2antiferromagneticHeisenberg
modelwithfrustratednext-nearestneighborinteractions
J , defined on a 2D bipartite lattice with N sites:
2
J J
1 1
H =J S S +J S S . (1)
1 i j 2 i j J J
· ·
Xi,j Xi,j 2 2
h i hh ii
As mentionedalreadyinthe Introduction,we choosethe
J
plaquette, Fig. 2, as our elementary degree of freedom.
1
Then, assuming that N is chosen appropriately, the en- 1 2
tire lattice canbe coveredwith such plaquettes in a sub-
exponentially23 ( 2√N) large number of ways.
Aiming at illus∼trating the main idea of the method, FIG. 2: A single 2×2 plaquette has each vertex occupied
by a S = 1/2 spin. The diagonal spins interact through a
in this section we consider in detail only the symmetric
covering of the lattice with 2 2 plaquettes, which pre- Heisenberg term of strength J2, while nearest neighbor spins
servestheC latticesymmetry×,seeFig. 3,althoughlater interact with strength J1.
4
the displaced covering (Fig. 4), which breaks C down
4
to C (two-fold symmetry axis), and the case of larger
2
plaquettes(superplaquettes, Fig. 13)willbe analyzedas
well.
It is convenient to take as a basis the states
a = l l LM , (2)
1 2
| i | i
where l = S +S and l = S +S are total spins of
1 1 4 2 2 3
the plaquette diagonals, while L = l +l is the total
1 2
spin of the entire plaquette and M is its z-component.
In this basis the Hamiltonian of a single plaquette,
H(cid:3) =J1 S1+S4 S2+S3 +J2 S1 S4+S2 S3 (3)
· ·
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)
is diagonal with eigenvalues
J
1
ǫ = L L+1 l l +1 l l +1 +
l1l2L 2 − 1 1 − 2 2
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
J
2
+ l l +1 +l l +1 3 . (4)
1 1 2 2
2 −
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1) (cid:3)
7 8 3 4 7 8
We note that the basis of Eq. (2) is a natural one and
allows us to explicitly label states with corresponding
representations of SU(2).
The next step is to establish how a plaquette couples
to the rest of the system. In Fig. 3 we show the sym- 5 6 1 2 5 6
metric plaquette coveringof the 2Dlattice. In the figure
the vertices of every non-central plaquette are similarly
labeledbythenumbers5,6,7,8,andtotalspinsofdiago- FIG. 3: Symmetriccovering of the 2D lattice with 2×2 pla-
nalsarel =S +S andl =S +S . Intheuncoupled quettes. Eachplaquetteisconnectedto4nearestand4next-
3 5 8 4 6 7
nearest neighbors.
basis matrix elements of the inter-plaquette interaction
are:
(cid:0)Hiσnt(cid:1)aa′11aa′22 =XLM(cid:10)λ′1λ′2,LM|Hiσnt|λ1λ2,LM(cid:11)× (5) L = L1+L2 is their total angular momentum. In this
× L′1M1′L′2M2′|L′1L′2LM L1M1L2M2|L1L2LM , equationwehaveintroducedthenotationsλ1 ={l1l2L1},
(cid:10) (cid:11)(cid:10) (cid:11) λ2 = l3l4L2 and ai = λiMi , and similarly for the
{ } { }
where σ = 1 (σ = 2) corresponds to the nearest (next- primed indices. Because each plaquette has 4 nearest
nearest) neighbor interaction, L ,L (L ,L ) represent neighbors and 4 next-nearest neighbors (see Fig. 3), the
1 2 ′1 ′2
initial(final)angularmomentaofthetwoplaquettesand symmetrized next-nearest neighbor interaction may be
4
written as: present section we discuss only the simplest one – a
λ λ ,LM H2 λ λ ,LM =J ρL′1L′2(L) (6) Hatree-Fock like (HF) approximation. A possible way
′1 ′2 | int| 1 2 2 L1L2 × to include fluctuation corrections is presented in Ap-
(cid:10) Sλ′1λ1Sλ′2λ2+Sλ′1λ1Sλ′2λ2(cid:11)+Sλ′1λ1Sλ′2λ2+Sλ′1λ1Sλ′2λ2 . pendix B. The HF approximation introduces the mix-
×(cid:18) 3 6 1 8 2 7 4 5 (cid:19) ing of single-plaquette states which minimizes the total
energy of the system and is based on a canonical trans-
while the symmetrized nearest neighbor plaquette inter- formationamongthebosons,whichwewillrestricttobe
action has the form: uniform (plaquette independent):
λ′1λ′2,LM|Hi1nt|λ1λ2,LM =J1ρLL1′1LL2′2(L)× (7) γia =RanΓin. (9)
(cid:10) × S1λ′1λ1 +S4λ′1λ1 S6λ(cid:11)′2λ2 +S7λ′2λ2 + The real matrix R satisfies canonical orthogonality and
(cid:2)(cid:0)+ Sλ′1λ1 +Sλ(cid:1)′1(cid:0)λ1 Sλ′2λ2 +Sλ(cid:1)′2λ2 + completeness relations:
2 3 5 8
+(cid:0)2 λ′1λ′2,LM|Hi2n(cid:1)t(cid:0)|λ1λ2,LM , (cid:1)(cid:3) RanRan′ =δnn′; RanRan′ =δaa′
(cid:10) (cid:11)
InEqs. (6)and(7)thesymbolsSλ′λ = λ S λ denote A translationally invariant variational ansatz for the
n h ′k nk i
reduced matrix elements of the n-th spin operator, and: groundstate (vacuum) is a bosoncondensate inthe low-
est HF single-particle energy state (n=0):
ρL′1L′2(L)= 1( 1)L+L′2+L1 L′1 L′2 L ,
L1L2 4 − (cid:26)L2 L1 1 (cid:27) |HFi= Γ†i0|0i, (10)
Yi
where are Wigner 6j symbols (or Racah
coefficien{t·s·)·2}5. and since it has one boson per plaquette, there is no
Let us now identify the plaquette degree of freedom need to impose the Schwinger boson constraint in the
with a Schwinger boson which creates a specific state of calculation.
the plaquette. Then, the Hamiltonian of Eq. (1) in the Minimizing the total energy with respect to R, we ar-
plaquette basis can be expressed as: rive at the self-consistent equation:
H =Xi,a ǫaγi†aγia+Xij (cid:0)Hi1nt(cid:1)aa′11aa′22γi†a′1γj†a′2γia1γja2+ (cid:26)ǫaδaa′ +Xσ zσ(cid:0)Hiσnt(cid:1)aa′aa12Ra01Ra02(cid:27)Ran′ =εnRan, (11)
h i
+ Hi2nt aa′11aa′22γi†a′1γj†a′2γia1γja2, (8) where z1 = z2 = 4 are the nearest- and next-nearest
Xij (cid:0) (cid:1) coordination numbers. The ground state energy (GSE)
hh ii
per spin is then given by the expression:
where the operator γi†a creates a boson on site i of the
plaquette lattice (which contains N(cid:3) =N/4 sites) in the E0 = hHF|H|HFi = 1 ε + ǫ R0 2 , (12)
state, denotedby an index a, running throughthe entire N N 8(cid:18) 0 a a (cid:19)
Xa (cid:0) (cid:1)
single-plaquette Hilbert space (of dimension 24 = 16)
and the summation is performed over doubly repeated with ε0 being the lowest eigenvalue of Eq. (11).
dummy indices. The unphysicalstates are eliminated by Another fundamental quantity to compute is the po-
enforcing the local constraint aγi†aγia = 1. In what larization of spins within a plaquette:
fpollalqowuest,tweelaitmtipceo.se periodic bouPndary conditions on the hHF|Sizn|HFi=(Snz)a′aRa0′Ra0,
The bosonic operators γ define the hierarchical
ia where n = 1,...,4 is the spin index, and the matrix
language22 for our problem. It will be used in the next
elements (determined from the Wigner-Eckart theorem)
section, where we develop an approximation scheme for
are:
diagonalizing the Hamiltonian of Eq. (8).
(Sz) = l l LM Sz l l LM = (13)
n a′a h1′ 2′ ′ ′| n|1 2 i
III. HIEARPAPRRCOHXICIMAALTMIOENAN-FIELD =(−1)L+L′+1δMM′h10L√M2L|1′L+L1′Mihl1′l2′L′||Sn||l1l2Li.
This enables us to define the staggered and collinear
As it follows from Eq. (4), the lowest single-plaquette (along x and y axes) magnetizations:
state has the energy ǫ /4 = J /2+J /8 per spin,
1100 1 2
which, when J = 0, gives only t−he energy of a classical M =(1/4) HFSz+Sz Sz Sz HF ; (14)
2 stag h | 1 4 − 2 − 3| i
2D antiferromagnet. Thus, it is necessary to take into M (x,y)=(1/4) HFSz Sz +Sz Sz HF .
account the interaction term in Eq. (8). col h | 1 − 4 2,3− 3,2| i
The HMF approximation is a mean-field approach, Notice the extreme simplicity of the HMF approxima-
performed on the relevant degrees of freedom. In the tion. The reason why it is able to realize meaningful
5
FIG.4: Thedisplacedplaquettecovering. NoticethattheC4
symmetry is broken down to C2.
FIG. 6: Covering of the lattice with crosses – arrays of five
spins. Sinceonecrosscannotformasinglet,itisnecessaryto
double the unit cell, as indicated by the gray shading. This
choiceofadegreeoffreedomclearlypreservestheC4 symme-
try,but theresulting lattice breaks it.
IV. GROUND STATE PROPERTIES AND
EXCITATION SPECTRUM OF THE MODEL
Our choice of the plaquette as an elementary degree
of freedom remains unjustified at this point. In order to
show its relevance we applied the analysis of two previ-
oussectionstoseveralothercoarsegrainings(besidesthe
symmetricplaquettecovering,case(a),showninFig. 3):
(b) superplaquette (spin cluster 4 4) degree of freedom,
×
covering the lattice in such a way that C is preserved
4
(seeAppendixAfordetails);(c)displacedplaquettecov-
ering of the lattice, Fig. 4; (d) symmetric and displaced
dimercoverings,shownrespectivelyinrightandleftpan-
FIG.5: Connectivityofthedimerlatticeforsymmetric(right els of Fig. 5; (e) cross degree of freedom, Fig. 6. One
panel) and displaced dimer coverings (left panel). The rota- should observe that symmetries of the original Bravais
tional C4 symmetry is lowered to C2 in both cases. lattice are preserved only in cases (a) and (b). In cases
(c)and(d)thelatticerotationalsymmetryC islowered
4
to C . Case (e) is special in the sense that an isolated
2
degreeoffreedomdoesnotpossessasingletgroundstate.
The information about a particular configuration is en-
coded in matrix elements of Hσ , whose calculation is
int
results is that the plaquette degree of freedom seems to elementary. Other equations, presented in Secs. II and
containthemaincorrelationsdefiningthephysicsbehind III, retain their form.
the Hamiltonian of Eqs. (1), and (8). To avoid confu- ForeachoftheabovecasesweiterativelysolveEq. (11)
sion, we emphasize that the HF approximation and the and compute the GSE (12), and staggered and collinear
fluctuationtheoryofAppendixBarephysically(andob- magnetizations(14). Themainmessage,whichwewould
viously mathematically) different from the spin-wave or like to convey in this section is that only the plaquette
canonical Schwinger-Wigner boson mean-field approach degreeoffreedom(ofanysize)isrelevantforconstructing
tospinsystems26. Inparticular,wemakenoassumption the phase diagram of the Hamiltonian of Eq. (1).
about the underlying ground state, thus allowing for an
interplay of various quantum phases. Moreover, it will
be demonstrated, that the collective excitation spectra A. Symmetry preserving plaquette configurations
in each phase consistently reflect spontaneously broken
symmetries, unlike the usual Schwinger boson case26, in Let us focus first on cases (a) and (b), i.e. symmetry-
which one obtains gapped excitations. preserving coverings of the lattice with plaquette, Fig.
1 6
-0.4 0.5
2x2
4x4
0
-0.45
-0.5
2
2
-0.5 J
J1 -0.58 Nd
E/N0 -0.55 E(J=0)/NJ021 ---000-0...665.6219 2JdE/10 -1-.15 g ----1111-....18642
-0.6 -0.63 -2 -2
-0.64 -2.2 2x2
0 0.05 0.11/N 0.15 0.2 0.25 -2.5 0 0.05 0.11/N 0.15 0.2 0.25 4x4
-0.65
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
J /J
J /J 2 1
2 1
FIG. 7: Ground state energy per spin computed at the HF FIG.8: Second-orderderivatived2E0/dJ22forthe2×2and4×
level for the 2×2 and 4×4 plaquette elementary degrees of 4plaquettedegreesoffreedom. ThediscontinuityatJ2/J1 ≈
0.42isindicativeofasecond-orderquantumphasetransition.
freedom. The inset shows finite-size scaling in the AF phase
In the inset we present finite-size scaling for the jump g ≡
at J2 =0. `J1d2E0/NdJ22´JJ22cc11−+00.
3, and superplaquette degrees of freedom. The resulting
GSE as a function of J2/J1 is shown in Fig. 7. One im- 0.5
mediatelyobservesalevel-crossingatJc2 0.67J ,indi-
2 ≈ 1 0.45
catingthefirst-ordertransitionandasecond-orderquan-
tum critical point at Jc1 0.42J , which is supported 0.4
by a jump of the sec2ond≈order 1derivative d2E0/dJ22, 0.35 0 .004..344534
Fig. 8. Both N´eel and columnar phases are charac- 0.3 0.425
terized by spontaneously broken SU(2) symmetry. The =0) 0.42
former exhibits a nonvanishing staggeredmagnetization, M 0.25 M(Jz2 0 .04.1451
0.2 0.405
M , while the latter has nonzero collinear magnetiza- 0.4
stag
tion along the x-direction, Mcol(x). Both order parame- 0.15 0 .03.9359
0 0.05 0.1 0.15 0.2 0.25
ters become zero in the paramagnetic phase, suggesting 0.1
1/N
that SU(2) is restored. These results are summarized in 0.05 2x2
Fig. 9, from which it also follows that the phase tran- 4x4
0
sition at J2c1 is continuous, while J2c2 corresponds to a 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
first-order transition point. We remind, in this connec- J /J
2 1
tion,thatourapproachdoesnotexplicitlybreakthespin
rotational symmetry, thus allowing for the treatment of
competing ground states. FIG. 9: Staggered magnetization, Mstag, for J2 ≤ J2c1 and
As expected,consideringalargerelementarydegreeof collinear magnetization along the x-direction, Mcol(x), for
freedom–superplaquette–leadstoasignificantimprove- J2 ≥ J2c2 (for the 2×2 and 4×4 plaquette degrees of free-
dom),computedattheHFlevel. Noticethecontinuousphase
mentoftheGSEandreductionofthemagnetization,due
to larger quantum fluctuations. The finite-size scaling transition at J2/J1 ≈ 0.42 and a first order transition at
J2/J1 ≈ 0.68 (2×2) and J2/J1 ≈ 0.66 (4×4). The inset
(insets in Figs. 7 and 9), using these two sizes (2 2
× shows finite-size scaling of Mz at J2 =0.
and 4 4), indicates that E (J =0)/N 0.64J and
0 2 1
× →−
M (J = 0) 0.39 in the thermodynamic limit, a sat-
z 2
→
isfying result for a HF approximation, which completely
ignores fluctuations (these numbers should be compared servethecolumnarordering,whichspontaneouslybreaks
with well-known results of Monte-Carlo simulations27: C down to C , and SU(2), but partially (i.e., along
4 2
E0/N 0.67J1 and Mz 0.31). one direction) restores the lattice translational symme-
≈− ≈
Next,wediscussinmoredetailsymmetrypropertiesof try. We present a more detailed discussionof the spatial
thevariousphasesinFigs. 7,9. AtallvaluesofJ Jc2, symmetries later in this section. In the intermediate re-
thelatticetranslationalsymmetryT isbroken18,2b≤utt2he gionJ (Jc1,Jc2) the spin SU(2)rotationalsymmetry
rotational C symmetry is preserved. For J Jc1 this is resto2r∈ed. I2n th2is paramagnetic phase the groundstate
4 2 ≤ 2
correspondstoaN´eel-typelong-rangeorderwithsponta- wavefunction is a tensor product of individual plaque-
neously broken SU(2). At large values J Jc2, we ob- tte ground states (with quantum numbers l = l = 1,
2 ≥ 2 1 2
7
L=M =0): 0.6
0
-0.05
-0.1
1 0.4 -0.15
1100 = 2 + -0.2
| i 2√3(cid:20) |↑1↑4↓2↓3i |↓1↓4↑2↑3i − F4 -0-.02.53
(cid:0) (cid:1) -0.35
1 4 2 3 + 1 4 2 3 + (15) 0.2 -0-.04.54
− |↑ ↓ ↑ ↓ i |↑ ↓ ↓ ↑ i -0.5
(cid:0) 0 0.05 0.1 0.15 0.2 0.25
+ 1 4 2 3 + 1 4 2 3 . Fi 0 1/N
|↓ ↑ ↑ ↓ i |↓ ↑ ↓ ↑ i (cid:21)
(cid:1)
-0.2
This ground state necessarily breaks the lattice transla-
tional symmetry, but preserves C4. In fact, the param- F1(2x2)
agnetic region on the phase diagram of Figs. 7 and 9 is -0.4 F2(2x2)
F (2x2)
a trivial plaquette crystal: a set of non-interacting pla- F4(4x4)
4
quettes, because the expectation value of the plaquette -0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
interaction(see Eq. (8)) in the singletstate iγi†,1100|0i J2/J1
vanishes. An analogous situation is realizeQd when the
superplaquetteis chosenasanelementarydegreeoffree-
dom: theparamagneticphaseisacrystalofsuperplaque- FIG. 10: HF ground state expectation values of the
ttes. It is interesting to note that in Ref. 6 a “plaquette symmetry-breakingperturbations,givenbyEq. (16),plotted
resonating-valence-bondstate”,exactlyequalto(15),has as functions of J2/J1 for the plaquette and superplaquette
been proposed. However, later7 the intermediate phase degreesoffreedom. DuetotheunbrokenC4 symmetryinthe
wasarguedtobeaspinliquid,i.e.,astatethatpreserves N´eel and paramagnetic phases, values of F4 are twice larger
the lattice translational symmetry. than the corresponding values of F1, except in the columnar
In order to learn about spatial symmetries in various phase, where C4 is spontaneously broken down to C2. The
phases, we compare magnitudes of the several lattice insetshowsfinite-sizescalingforF4 forthreevaluesofJ2/J1:
0 (circles), 0.504 (triangles) and 0.997 (rhombs).
symmetry-breaking observables proposed in the litera-
ture. We considerthe followingthree,introducedinRef.
0.5
9 (in the notation of that paper):
0.5
0.45 0.45
1 0.4
F1 =N Xx,y(−1)xSx,ySx+1,y; 0 0.3.45 ∆/J1 000 ...00123..55523
F = 1 S S S ; 0.3 0 .00 .051 ∆∆12
2 x,y x+1,y x,y+1 0 0.2 0.4 0.6 0.8 1
N Xx,y (cid:0) − (cid:1) ∆/J1 0.25 J2/J1
F = 1 S ( 1)xS +( 1)yS , 0.2
4 x,y x+1,y x,y+1
N Xx,y (cid:2) − − (cid:3) 0.15
0.1
where indices x,y specify a spin in the 2D lattice. The
operator F4 probes the plaquette ordering, which pre- 0.05 ∆∆1
2
serves the lattice rotational symmetry, while F and F 0
1 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
correspondto the columnar ordering. We note,however,
J /J
that F is already non-zero for an isolated plaquette (or 2 1
1
superplaquette). These functions can be combined in
the complex “order parameter”, introduced in Ref. 15.
FIG.11: Thetwolowestexcitationenergiestakenatthecen-
HereweshowdetailsofthecalculationoffunctionsF
1,2,4 ter of the plaquette Brillouin zone. The main panel shows
for the plaquette degree of freedom, case (a), and only
the self-consistent solution to Bogoliubov’s equations, while
present the result for F4 for the superplaquette case (b). theinsetcorrespondstothetime-dependentGross-Pitaevskii
In the plaquette representation the above operators are equation (weak coupling). Since wavefunctions of collective
written as: excitations in the N´eel and columnar phases have different
symmetries, there are level crossings in the non-magnetic
1
F = S S +S S phase (cuspsin theplot).
1 1;i,j 2;i,j 3;i,j 4;i,j
N −
Xi,j (cid:2)(cid:0) (cid:1)
S S +S S ;
2;i,j 5;i+1,j 4;i,j 7;i+1,j
− In this equation the indices i,j are coordinates of a pla-
1 (cid:0) (cid:1)(cid:3)
F = S S S S (16) quette in the lattice.
2,4 1;i,j 4;i,j 2;i,j 3;i,j
N ∓ ∓ ±
Xi,j (cid:2)(cid:0) (cid:1)(cid:0) (cid:1) ExpectationvaluesofthefunctionsEq. (16)intheHF
S S +S S ground state are shown in Fig. 10. Both phase transi-
± 2;i,j 5;i+1,j 4;i,j 7;i+1,j∓ tion points Jc1 and Jc2 are clearly seen from this plot.
(cid:0) S S S S . 2 2
3;i,j 5;i,j+1 4;i,j 6,i,j+1 Allfunctionschangecontinuouslyacrossthesecond-order
∓ ∓
(cid:1)(cid:3)
8
critical point Jc1 and jump at the first-order transition -0.25
2
point Jc2. Except in the columnar phase the values of
2 -0.3
F are everywhere exactly twice larger than those of F ,
4 1
whichisanindicationoftheunbrokenfour-foldrotational -0.35
symmetryofthelatticeintheseregions. Inthecolumnar
1 -0.4
phase, on the other hand, this symmetry is broken and NJ
the above relation does not hold. While in the N´eel and E/0 -0.45
columnar phases nonlocal terms in Eq. (16) are impor-
-0.5
tant, in the paramagnetic state the only contribution to
either expectation value comes from isolated plaquettes -0.55
disp. plaq.
(localtermsinEq. (16)),orsuperplaquettes. Thisobser- cross
-0.6
vationisconsistentwithpropertiesofthegroundstatein
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
the non-magneticphase,discussedearlierin this section. J /J
2 1
AsmentionedintheIntroduction,anychoiceofdegree
-0.36
offreedombreaksexplicitlythelatticetranslationalsym-
-0.38
metry T, with the result that links in the lattice become
-0.4
inequivalent. Indeed, the functions of Eq. (16), defined
-0.42
on the links, have non-zero values even in the AF phase
-0.44
at J2 = 0. However, this effect vanishes in the ther- J1
N -0.46
modynamic limit (i.e., as the size of the degree of free- /0
E
dom is increased). The finite-size scaling for F (J /J ) -0.48
4 2 1
is presented in the inset to Fig. 10. The three extrap- -0.5
olated values: F (0) = 0.075, F (0.504) = 0.245 and -0.52
4 4
− −
F4(0.997)= 0.004,suggestthatthe“link-wise”transla- -0.54 disp. dim.
− sym. dim.
tional invariance is restored in the thermodynamic limit -0.56
in the N´eel and columnar phases, but not in the plaque- 0 0.2 0.4 0.6 0.8 1
J /J
tte crystal phase. Moreover, the value of the jump g 2 1
J1d2E0 J2c1+0, extrapolated to the thermodynamic lim≡it
(sNedeJ2i2ns(cid:12)Je2tc1−to0Fig. 8) remains finite: g(Jc1)= 1.006. In FIG. 12: HF ground state energy per spin for the displaced
(cid:12) 2 − plaquette and cross (upper panel), and dimer (lower panel)
other words, these results imply that the critical point
Jc1 correspondsto the usual Landausecond-orderphase coverings.
2
transition.
should be determined self-consistently, and is a measure
oftheapplicabilityoftheentireapproximation: itshould
B. Excitations in the plaquette crystal phase satisfy the inequality n 1 1. Once the condensa-
0
| − | ≪
tion part is separated from γ , what remains describes
ia
Until now we have consideredonly ground-state prop- fluctuation corrections to the HF ground state. These
erties of the model Eq. (1). However, low-lying excited fluctuations have rather strong effects near the phase
states are also of considerable interest. In particular, transition points, leading to the modification of Jc1 to
2
the paramagnetic phase is known to have gapped exci- a λ-point and its considerable shift. The value of Jc2
2
tations, while N´eel and columnar phases exhibit Gold- alsochanges,butmuchlesssignificantly. Thesefactsim-
stone modes. Thus, the phase transition points Jc1 and ply that our approximation breaks down near the phase
2
Jc1 must be accompanied by the opening of a gap in transition points. Indeed, in Appendix B it is shown
2
the excitation spectrum: the former in a continuous and that close to the transition, the condensate is strongly
the latter in a discontinuous fashion. In Appendix B suppressed. However, deep in each phase n0 0.9, thus
∼
we present a particular method to obtain the collective allowingus to drawconclusionsaboutgeneralproperties
spectrum of the system. The main idea of this approx- of the collective spectrum.
imation is borrowed from the Bogoliubov-Fetter theory The complete summary of the results is given in Ap-
of superfluidity28. Namely, assume that on each plaque- pendix B, here we only presentthe mostinteresting one:
tte the majority of Schwinger bosons form a condensate The gap in the excitation spectrum as a function of
in an appropriately chosen lowest energy state and ne- J /J . Althoughwefocusonlyoncase(a)(2 2plaque-
2 1
×
glect fluctuations in the number of condensed particles. tte), the superplaquette degree of freedom can be con-
We note, however,that due to the Schwinger bosoncon- sidered in a similar manner. The gap always occurs in
straint, this quantity has the meaning of a probability momentum space at k = 0, which reflects translational
to find a given plaquette in the lowest energy HF state, invariance of the plaquette lattice. Below we focus only
rather than the number of particles. Nevertheless, we on this point in the plaquette Brillouin zone. In fact,
willcallitthecondensatefractionn ,which,inprinciple, there are 16 1 = 15 collective branches and only some
0
−
9
ofthembecomegaplessinthephaseswithspontaneously invariant under reflection in the plane J = Jc1. These
2 2
broken SU(2). However, in the paramagnetic phase all observationsimplythatthecoarsegrainingprescriptions
branches develop a gap. In Fig. 11 we show the en- (c)-(e) areprobablya badstartingpoint for anyapprox-
ergy gap ∆(J ) = ω(k = 0,J ) for the two lowest ex- imation scheme.
2 2
citation branches in a system of 100 100 plaquettes,
×
which approximates well the thermodynamic limit. The
mainpanelshowsresultsoftheself-consistentsolutionof V. DISCUSSION
Bogoliubov’s equations. The inset compares it with the
solution of the time-dependent Gross-Pitaeveskii equa- Inthissectionwewouldliketoputourmainresultsin
tion (25), which corresponds to the weak-coupling ap- perspective by making several summarizing remarks. It
proximation. In the N´eel and columnar phases there are shouldbeemphasized,thatthediscussionbelowisbased
two spin wave-type Goldstone modes, both of which ac- on our finite-size scaling results.
quire a gap in the paramagnetic phase, at Jc1 and Jc2. First of all, from our analysis it follows that dimer
2 2
However, as it follows from Fig. 11, positions of these (bond) order is always unfavorable in the non-magnetic
Jp2oc2int≈sc0h.a6n5Jge1.froTmhethceriirtiHcaFl vpaoliunetsJt2oc1: Jw2ca1s≈ob0t.3a3inJe1danbdy pwhearesec.onNsoidteicreedthsautchevaennowrdheerndtihdenpoltaqoucectutre,caolvtheoriungghs
extrapolationof the staggeredmagnetizationcurve(Fig. spontaneous dimerization was not explicitly prohibited.
16)tozero,whilethe first-orderpointJc2 byextrapolat- Instead, the quantum paramagnetic phase prefers to
2
ingthetwoGSEcurvesinFig. 14untilintersection. The preserve the lattice rotational symmetry, which makes
single-plaquette physical picture, discussed previouslyin the phase transition separating it from the N´eel phase
connection with the paramagnetic phase, remains valid, fit perfectly well within the Ginzburg-Landau-Wilson
e.g. the condensationoccursagaininthe plaquette state paradigm. Thedatapresentedforthestaggeredmagneti-
1100 . Using this observation and symmetries of the zation, Fig. 9, and symmetry-breaking observables, Fig.
|matriix elements Hσ a′b′, one can rigorously show the 10,indicatethatthesymmetrygroupoftheN´eelstateis
existenceofagap(cid:0)initnht(cid:1)eanbon-magneticregion. Infact,we a subgroup of the symmetry group in the paramagnetic
cansay,thatitisapropertyofourHMFapproximation, phase, as both phases break T and preserve C4, but the
rather than a numerical evidence. latter also preservesSU(2). On the contrary,there is no
such group-subgroup relation between the paramagnetic
and columnar antiferromagnetic phases. Consequently,
C. Other degrees of freedom the transition between these two states is first order.
These observations are summarized in Fig. 1. Indeed,
startingfromthe knownsymmetry inthe N´eelstate and
Finally, we comment on the results for cases (c)-(e),
assuming validity of the Landau theory, one can unam-
which, contrary to the configurations considered before,
biguously rule out dimerized structures in the paramag-
explicitly break lattice rotational symmetry, see Figs. 4-
netic phase, since they break lattice rotational symme-
6. ThecorrespondingGSEsareshowninFig. 12. Incon-
try. Therefore, our results do not favor the scenario of
trast to the previously considered scenarios, these cases
deconfined quantum criticality, advocated in Refs. 9,15.
give qualitatively wrong phase diagrams. Indeed, if we
Asalreadydiscussedintheprevioussection,ourmethod
cover the lattice with displaced plaquettes or crosses,
explicitly breaks a particular lattice translational sym-
there exists no classical spin configuration, which gives
metry: The ground state in Eq. (10) at J = 0 (AF
the long-rangeN´eel order. On the other hand, such con- 2
phase), is not invariant under a [11] lattice translation.
figurationexistsforthecolumnarstate. Forthedisplaced
One way to cure this problem is to consider variational
plaquettecoveringthelow-J phaseJ Jc2isanSU(2)
2 2 ≤ 2 wavefunctions of the “resonating plaquette” type:
singlet,andspatiallyisasetofnon-interactingplaquettes
(notice the coincidence of2 2–plaquetteenergiesinthe
Ψ = 1+T HF ,
× 11
paramagneticphasesofFigs. 7and12). Thus,thephase | i | i
(cid:0) (cid:1)
transitionto the columnar state is of the first-order. For whichrestorethatsymmetry,withT thetranslationop-
11
the cross covering, on the other hand, SU(2) is explic- erator along the [11] direction in the lattice. This state
itly broken for all values of J , but since the columnar describestworesonatingplaquetteconfigurations,shifted
2
phase partially restores the lattice translational invari- with respect to eachother along [11]. Results of calcula-
ance, it is again separated from the non-magnetic state tionsusingthiswavefunctionforsystemsupto6 6spins
by a first-order phase transition point. The two dimer indicate that for J < Jc1 the ground state h×as long-
2 2
configurations,case(d),arecomplementarytoeachother rangeN´eelorderandisparamagneticforJ (Jc1,Jc2).
2 ∈ 2 2
in the sense that one of them has only the classical N´eel The intermediate phase has a plaquette crystal order,
state andanother–only columnarphase. Itfollowsthat but with partially restoredtranslationalinvariance. The
these configurationscanhaveonlyonesecond-ordercrit- phase transitionat Jc1 is stillof the second-order,which
2
ical point at which SU(2) is restored and other symme- isnotsurprising,asitcanbe describedsolelyintermsof
tries remain broken. As a result, one obtains the phase theSU(2)orderparameter. However,basedonthesesys-
diagram, shown in the lower panel of Fig. 12, which is tem sizes, we can not definitively conclude whether this
10
phase transition remains of the second-order or becomes 33 34 43 44
weakly first-order in the thermodynamic limit.
3 4
Next, we observethat despite profounddifferencesbe-
tween the 2D and 1D equivalent J -J models, their
1 2 31 32 41 42
non-magnetic phases present some similarities. The
one-dimensional model is known to be quasi-exactly
solvable29 atthe point J =0.5J and exhibits a param-
2 1
agnetic groundstate with short-rangecorrelationsfor J
2
above the critical value30 Jc 0.24J (however, due to
the peculiar physics in one2di≈mension1, the critical point 13 14 23 24
Jc is an essential singularity and, therefore, not obvi- 1 2
2
ously accessible for the HMF approximation of the type
presented here). In this non-magnetic region the ground
11 12 21 22
state is doubly degenerate, corresponding to two possi-
ble coverings of a 1D lattice with dimers, in accordance
with the Lieb-Schultz-Mattis theorem31. Unfortunately,
FIG. 13: The 4×4 superplaquette degree of freedom. Each
a finite-size scaling calculation for the gap between the
spin carries two indices: a 2× 2 plaquette number and a
lowest and first excited energy levels, based on exact di-
coordinate within this plaquette.
agonalization of the 2 2 and 4 4 clusters with peri-
× ×
odic boundary conditions, does not provide a definitive
answer to the question on whether the ground state of
such an algorithm.
the 2D J -J model becomes degenerate in the region
1 2
Jc1 J Jc2. This is indeed what one would ex-
2 ≤ 2 ≤ 2
pect on the basis of a generalizationof the Lieb-Schultz-
Mattistheoremtohigherspacedimensions(see,e.g. Ref. VI. CONCLUSION
32). At the HF level, it is true that different plaque-
tte coverings of the lattice have the same energy (simply Insummary,weanalyzedthe phasediagramofthe 2D
because each plaquette is in its singlet ground state). J -J model on a square lattice, focusing on symmetries
1 2
However, the total number of such configurations grows of the various phases.
sub-exponentially 2√N, which should be contrasted We showed that in this model the hierarchical
∼
with the dimer covering problem, where this number is language22 is defined by identifying the plaquette as a
known33 tobeexponentiallylarge. Basedonthisdistinc- relevant degree of freedom. Using anunbiasedand mani-
tion, one may speculate that if our plaquette picture is festlysymmetry-conservingmean-fieldapproachwecom-
valid,there arenotenoughdifferentplaquetteconfigura- pared it with several other possible candidates: dimer
tionsforthe paramagneticphasetobecomeaspinliquid andcrossdegreesoffreedom,as wellasdifferent waysto
(i.e., a resonating plaquette state). This statement, cer- cover a 2D lattice with plaquettes. Our results indicate
tainly, requires a separate investigation. that the plaquette (and superplaquette) covering, which
Finally,weemphasizethatthemaingoalofthisworkis preservesthelatticerotationalsymmetryhasthebesten-
to investigate the fundamental symmetries of the phases ergyamongconsideredcoarsegrainingscenariosanditis
exhibited by the J -J model. Although the energies theonlyonetoreproduceknownfacts,suchasthe inter-
1 2
presented for the 2 2 and 4 4 plaquette cases are mediate phase with gapped excitations, concerning the
× ×
different from those, obtained by more sophisticated nu- phase diagram of the model, while other configurations
merical methods, they can be systematically improved fail to exhibit all quantum phase transition points.
by consideringcorrelatedtrialwavefunctionsor by using Consistent with the previous work on the subject, we
more complex methods, which build upon the results re- found the quantum paramagnetic phase in the interval
portedhere. However,weexpectthatsymmetry-wiseour 0.42 J /J 0.66. Main controversies revolve around
2 1
≤ ≤
conclusionswillremainunchanged. Oneofsuchmethods the nature of this intermediate non-magnetic phase and
suitedforcomputingthephasediagramofthemodel(1), ofthe quantumcriticalpointseparatingit fromthe N´eel
which received significantly less attention, amounts to – ordered state. We found that the paramagnetic phase
applying the Wilson renormalization group procedure34 is a plaquette crystal, preserving both lattice and spin
and the density matrix renormalization group (DMRG) rotational symmetries. Extrapolation of our numerical
method8. Application of the latter faces serious diffi- results to the thermodynamic limit suggests that the
culties in 2D (see Ref. 35 for a related discussion) and Ginzburg-Landau-Wilson paradigm of phase transitions
requiresamappingofthe 2Dsystemontoachain,which is perfectly applicable in this case. Indeed, within the
introduces a certain bias to the final results8. While for- HMF, there is a group-subgroup relation between sym-
mulation of a practical and efficient DMRG approach to metries of the non-magnetic and N´eel phases, which are
2D systems is yet to be developed, we note that our re- thus separated by a second-order phase transition. On
sults may provide a useful and rational initial input for thecontrary,suchrelationdoesnotexistbetweenthepla-