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A Generalized Quantum Dimer Model Applied to the Frustrated Heisenberg Model on the Square Lattice:
Emergence of a Mixed Columnar-Plaquette Phase
Arnaud Ralko, Matthieu Mambrini, Didier Poilblanc
To cite this version:
Arnaud Ralko, Matthieu Mambrini, Didier Poilblanc. A Generalized Quantum Dimer Model Applied
to the Frustrated Heisenberg Model on the Square Lattice: Emergence of a Mixed Columnar-Plaquette
Phase. Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical
Society, 2009, 80, pp.184427. �10.1103/PhysRevB.80.184427�. �hal-00387382�
Generalized quantum dimer model applied to the frustrated Heisenberg model on the square lattice: Emergence of a mixed columnar-plaquette phase
A. Ralko,
1M. Mambrini,
2and D. Poilblanc
21
Institut Néel, CNRS and Université de Grenoble, F-38000 Grenoble, France
2
Laboratoire de Physique Théorique, CNRS and Université de Toulouse, F-31062 Toulouse, France
共Received 13 May 2009; revised manuscript received 31 August 2009; published 30 November 2009
兲Aiming to describe frustrated quantum magnets with nonmagnetic singlet ground states, we have extended the Rokhsar-Kivelson
共RK
兲loop expansion to derive a generalized quantum dimer model containing only connected terms up to arbitrary order. For the square-lattice frustrated Heisenberg antiferromagnet
共J
1-J
2-J
3model
兲, an expansion up to eighth order shows that the leading correction to the original RK model comes from dimer moves on length-6 loops. This model free of the original sign problem is treated by advanced numerical techniques. The results suggest that a rotationally anisotropic plaquette phase
关A. Ralko, D. Poilblanc, and R.
Moessner, Phys. Rev. Lett. 100, 037201
共2008
兲兴is the ground state of the Heisenberg model in the parameter region of maximum frustration.
DOI: 10.1103/PhysRevB.80.184427 PACS number
共s
兲: 75.10.Jm, 05.30. ⫺ d, 05.50. ⫹ q I. INTRODUCTION
Over the last decades theoretical efforts have been de- voted to study quantum phases of two-dimensional frustrated quantum magnets, motivated by the discovery of experimen- tal antiferromagnets showing the absence of long-range mag- netic ordering down to very low temperatures.
1–5In such systems a gap to magnetic excitation traditionally opens up while the spin-SU
共2
兲symmetry remains unbroken. Two classes of “singlet” phases have been distinguished. Valence- bond crystals
共VBC
兲where some spatial symmetries are spontaneously broken and spin liquid for which all symme- tries remain unbroken
共e.g., the resonating valence-bond liquid.
6兲.
However, it is usually difficult to characterize the singlet phases in simple microscopic S = 1/ 2 models. For example, in the well-known J
1-J
2Heisenberg S = 1/ 2 antiferromagnet on the square lattice, where frustration is controlled by the next-nearest interaction J
2, no definitive answer has been given on the nature of the nonmagnetic Ground State
共GS兲for maximal frustration at J
2/J
1⬃0.5. Mambrini et al.
7re- cently addressed a work to this task, studying the J
1-J
2-J
3model, containing an extra next-next-nearest neighbor J
3frustrating antiferromagnetic,
H = J
1具兺
i,j典
S
i. S
j+ J
2具具兺
i,j典典
S
i. S
j+ J
3具具具兺
i,j典典典
S
i. S
j.
共1
兲Interestingly, in this model the description in terms of nearest-neighbor valence-bond
共NN VB
兲coverings
8is excel- lent in some extended region of parameter space, in particu- lar, around the point J
2= J
3= 1 / 4, with some significant ex- tension along the line
共J2+ J
3兲= J
1/2. This model is therefore one simple canonical case where a truncation within the nearest-neighbor singlet configuration basis is legitimate and can be used as a simpler and convenient framework.
In this paper, we have extended the Rokhsar-Kivelson
共RK兲loop expansion to derive a generalized quantum dimer model
共QDM兲acting in the space of hardcore NN dimer coverings of the lattice. We show that this expansion based on the hierarchy of the overlap matrix elements between the dimer coverings leads to an effective Hamiltonian that con- tains a sum of dimer moves, each involving only a single closed loop or loops at finite distances
共connected term兲. Inother words, all disconnected terms cancel out order by or- der. We apply this procedure to the J
1-J
2-J
3model and show that the leading contributions are of the form of a simple generalized QDM on the square lattice which, in addition to original QDM,
9contains an additional loop-6 term which brings kinetic competition in the system. The effective Hamiltonian then reads
H
eff= v
p p
− t
4p p
− t
6p p
.
共2兲where the sums run, respectively, over all square or rectan- gular plaquettes of the square lattice. Here we use the fol- lowing convention:
共i兲white plaquettes denote kinetic
共off-diagonal兲 operators that flip dimers around the thick contour and
共ii兲yellow plaquettes stand for potential
共diagonal兲op- erators that leave configurations unchanged with a factor 1 if it is flippable around the thick contour and 0 in the opposite
case. In the following, the omission of p indices in the dia- grammatic notation is a shortcut notation for an implicit summation over all inequivalent plaquettes with a given shape. For example,
=
p
p
.
PHYSICAL REVIEW B 80, 184427
共2009
兲1098-0121/2009/80
共18
兲/184427
共10
兲184427-1 ©2009 The American Physical Society
Guided by a variational approach and combining numeri- cal techniques such as exact diagonalizations
共ED兲and zero- temperature Green’s-function Monte Carlo, we compute the phase diagram of this model. More specifically, we give evi- dence in favor of a mixed columnar-plaquette phase first pro- posed in Ref. 10 and, since, evidenced in number of other contexts.
11Remarkably, this phase is found to be stable even in the presence of the loop-6 fluctuations. Hereafter, using the relation between the effective and microscopic models, we argue in favor of the SU共2兲-invariant version
共i.e., appli-cable to a spin-1/2 model instead of a dimer model兲 of the above mixed columnar-plaquette phase in the J
1-J
2-J
3micro- scopic model along the maximally frustrated line J
2+ J
3= J
1/ 2 where an approach restricting to the short-range VB basis has been justified previously.
7II. DERIVATION OF THE MODEL
A systematic way to derive the generalized QDM Hamil- tonian
共1兲consists of
共i兲projecting the Heisenberg model in the manifold formed by NN VB coverings of the square lat- tice and
共ii
兲perform a transformation that turns the nonor- thogonal VB basis into the orthogonal quantum dimer basis.
The key ingredient of the calculation is the overlap matrix O
,=
具
兩
典, where兩
典and
兩
典are two NN VB states. Step
共i兲is equivalent to solving H兩
典=E兩
典, where 兩
典is re-
stricted to the NN VB subspace. Writing
兩
典=兺i␣
i兩
i典,where the sum runs over the NN VB states
兩
i典, the originaleigenvalue problem turns into a generalized one
兺j具
i兩H兩
j典␣
j= E兺
j具
i兩
j典␣
j. In turn, this problem is equiva- lent to the conventional eigenvalue problem H
eff兩
典= E兩
典with H
eff= O
−1/2HO
−1/2which provides the transformation required by step
共ii兲.Using a convention where all bond singlets are oriented from sites A to sites B according to the canonical bipartition of the square lattice, the overlap matrix can be written as O
,= ␣
N−2nl共,兲, where N is the size of the system, n
l共 ,
兲are the number of loops in the overlap diagram obtained by superimposing the two configurations, and ␣ = 1/
冑2. On the other hand,
具
兩Si. S
j兩
典= 具
兩
典with = −3 / 4
共respectively, = +3 / 4兲 if i and j belongs to the same loop of the overlap diagram but belong to distinct sublattices
共respectively, be- long to the same sublattice兲 and = 0 if i and j belongs to two distinct loops. Using a convenient scaling and shifting H
→
共4/ 3兲H + J
1N /2 of Hamiltonian
共1兲, the matrix elements 具
兩H兩
典can be expressed as H
,= h
,O
,, where h
,only depends on the loops configuration. In particular, this convention enforces h
,= 0 for all .
It is then possible to expand O and H in powers of ␣ and compute H
eff= O
−1/2HO
−1/2accordingly as shown in Table I up to ␣
6. The expansion up to ␣
8as well further technical details of the calculation are given in the Appendix B. It is TABLE I.
共Color online
兲Expansion of O , H , and H
effup to order ␣
6. Note that the effective Hamiltonian
contains only local
共connected
兲processes. Some processes
共marked as 0 ”兲 does not appear in O or H but are produced in H
effby contractions of the generically noncommuting terms of the expansion
共see Appendix B for details
兲.
Processes O H H
eff= O
−1/2HO
−1/2Id 1 0 0
∅ ∅ 2(J
1− J
2)α
4α
22( − J
1+ J
2)α
2− 2 (J
1− J
2) α
21 + α
4α
42( − 2J
1+ 2J
2+ J
3)α
42 ( − J
1+ J
2+ J
3) α
4α
44( − J
1+ J
2)α
40
α
62( − 3J
1+ 3J
2+ J
3)α
60
α
6( − 6J
1+ 7J
2+ 4J
3)α
6( − 2J
1+ 3J
2+ 2J
3)α
6α
62( − 3J
1+ 3J
2+ 2J
3)α
6( − J
1+ J
2+ 2J
3) α
6α
66( − J
1+ J
2)α
60
∅ ∅ (J
1− J
2− J
3) α
6worth mentioning two peculiarities of this expansion:
共i
兲contrary to several previous approaches
9,12,13our expansion is not controlled by the length of the loops but by the actual amplitudes of the overlap matrix elements that only depend on the overall number of loops in the overlap diagrams and
共ii
兲all nonlocal and disconnected processes appearing in both H and O cancel in the expression of H
eff.
Let us discuss the results of this expansion. When trun- cated up to order ␣
2, we recover the usual Hamiltonian ob- tained in Ref. 9 with v /
兩t
4兩= ␣
2= 1 / 2. Note that such a drastic truncation appears a bit pathological in the sense that it does not capture any aspect of the frustration of the original model, v/
兩t4兩is indeed independent of J
2/ J
1. In the perspec- tive of a justification of QDM model from the Heisenberg model, nontrivial effects emerge from order ␣
4. Furthermore, considering the last column of Table I it is quite easy to see that, in the maximally frustrated region of the phase diagram
共J2+ J
3⬃J
1/ 2兲, where the validity of the NN VB approach have been established,
7the three processes retained in Eq.
共
2
兲are larger by at least a factor of 2 compared to the higher- order kinetic processes
共see TableIV兲. Importantly, we find that t
4⬎ 0 and t
6⬎ 0 which enable the use of efficient sto- chastic methods not applicable to the original frustrated spin model which suffers from the so-called “minus sign” prob- lem. Note that the next-leading corrections are diagonal terms
共see TableIV兲 and, hence, could also be added in future developments.
III. VARIATIONAL ANALYSIS
We now turn to the investigation of the effective Hamil- tonian
共2兲. We start with some discussion of the expectedVBC phases shown in Fig. 1. Regular columnar and plaquette phases have been introduced in the context of the frustrated J
1-J
2model and of the QDM.
14More recently, an anisotropic mixed columnar-plaquette phase has been introduced.
10We consider here the possibility of such phases which interpolate between the simple higher-symmetry VBC
共such as columnar or plaquette兲. Because of the presence ofloop-6 dimer moves, we also consider the possibility of a trimerization of the columns of dimers. We summarize the quantum numbers of the degenerate GS of the various VBC in Table II. This will be used further in this paper to analyze the low-energy spectrum of Hamiltonian
共2兲.Before showing the results of an extensive numerical analysis, we first start with a simple variational analysis. In- deed, variational ansatze for the VBC phases of Fig. 1 can be easily constructed as tensor products of resonating plaquette states
共see Appendix A兲and the knowledge of their relative stability provides a useful guide for the numerical search of VBC
共but is also subject to some artifact of the variationalmethod
兲. For convenience, let us map the two-dimensional parameter space on a sphere by expressing the Hamiltonian parameters in terms of two Euler angles and , as v
= cos共
兲sin共
兲,t
4= cos共
兲cos共
兲, andt
6= sin共
兲. The varia-tional phase diagram
共Appendix A兲in the
共 ,
兲plane con- TABLE II. Quantum numbers of the eigenstates collapsing to- ward the same degenerate GS for each of the ordered phases con- sidered in this paper. When applicable, we used the standard nota- tions for the irreducible representations of the C
4vand C
2vpoint groups, whose elements are defined w.r.t. a plaquette center. Defi- nitions of the ⌫ , M, and K points in the Brillouin zone are given in Fig. 2.
共⫻2
兲denotes an additional first-excited level
共denoted by
ⴱin the text
兲in the
共M , A
1兲sector. The states with momenta Q
B=
共⫾2 / 3 , 0
兲,
共0 , ⫾ 2 / 3
兲,
共, ⫾ 2 / 3
兲,
共, ⫾ 2 / 3
兲, Q
2=
共⫾/2 , 兲 ,
共, ⫾ / 2
兲, and Q
3=
共⫾/2 , ⫾ / 2
兲are even under reflection w.r.t. the momentum directions. The degeneracy of the pure columnar or plaquette
共mixed
兲phases is 4
共8
兲and it is 12 for the trimerized phase.
⌫ , A
1⌫ , B
1M , A
1K , A
1K , B
1Q
BQ
2Q
3Col.
冑 冑 冑Pla.
冑 冑 冑CP
1 冑 冑 冑 共⫻2
兲 冑 冑CP
2 冑 冑 冑 冑CP
3 冑 冑 冑 冑T
冑 冑 冑 冑CP3 CP1 CP2 T
Col.
Plaq.
FIG. 1.
共Color online
兲. VBC states considered in this work.
Generalized anisotropic VBC states labeled by CP
a 共mixed columnar-plaquette phases
兲and T
共trimerized phase
兲interpolate be- tween the most symmetric limits shown in the figure.
crossing
aΘ0.3Π
18 14 38 12
0.00 0.01 0.02 0.03 0.04
E
crossing
bΘatan0.5
18 14 38 12
0.00 0.01 0.02 0.03 0.04
E
ΦΠ ΦΠ
c
M K
FIG. 2.
共Color online
兲. Typical ED low-energy spectra on a 8
⫻ 8 cluster
共the GS energy is set to zero
兲for
共a
兲 = −0.3 and
共b
兲 = tan
−1共0.5
兲as a function of . Levels of special symmetries
共see text
兲are highlighted as colored symbols, from bottom to top:
共⌫
, A
1兲 共corresponding to the GS
兲,
共M , A
1兲,
共⌫, B
1兲,
共K, A
1兲, and
共M , A
1兲ⴱ. The arrow indicates the level crossing that marks the limit of the region where the latter levels correspond to the lowest exci- tations. Inset: Brillouin zone and its high-symmetry points ⌫
=
共0 , 0
兲, M =
共, 0
兲, and K=
共, 兲 .
GENERALIZED QUANTUM DIMER MODEL APPLIED TO … PHYSICAL REVIEW B 80, 184427
共2009
兲184427-3
tains three phases,
共i兲a RK region,
共ii兲the well-known com- pletely isotropic four-site plaquette phase, and
共iii兲a large domain covered by the trimerized VBC with a six-site unit- cell interpolating between the columnar phase and the pure six-site plaquette phase. For zero loop-6 kinetic term
共
= 0兲, the phase diagram is the same than the one obtained in Ref. 11
共for ⬎ −0.25
兲. Once is turned on, kinetic fluc- tuations due to 6-loop kinetic terms suppress, at a given point, the standard four-site plaquette and RK phases. Nev- ertheless, the latter phases are rather robust under the addi- tional t
6kinetic term in the vicinity of the RK point
共seethick lines in Fig. 5
兲. However, the variational approach
overestimate the stability of the RK phase
共in fact limited toa single point with algebraic dimer correlations
兲and of the trimerized phase
共the corresponding wave function has morefour-site flippable plaquettes than plaquette counterpart兲. In contrast, no mixed anisotropic VBC phase is found. A careful numerical approach is therefore necessary.
IV. NUMERICAL RESULTS
We now move to ED of clusters up to 8 ⫻ 8 sites. More especially, we compute the lowest-energy spectrum in each symmetry sectors, using both translations and point-group symmetries. Our analysis is based on the symmetry classifi- cation of the tour of states in the
共 ,
兲plane. In other words, each symmetry-breaking VBC phase is characterized by a finite degeneracy of the GS with a set of well-defined quan- tum numbers
共see table兲separated by a gap from the con- tinuum. On a finite-size cluster, the degenerate GS is split but a close inspection of the low-energy spectrum can provide informations on the VBC phase
共if the cluster is largeenough兲. For t
6= 0
共i.e., = 0兲, previous results
共extrapolatedto the thermodynamic limit兲 show that a phase transition between the columnar phase and a mixed columnar-plaquette phase
共in fact the CP
1phase of this paper
兲occurs around
⬃
−0.03 . Such a phase can be obtained via an in-phase spontaneous dimerization in the direction of the columns of dimers of the columnar phase or, equivalently, via a sponta- neous rotation symmetry breaking of the pure plaquette phase. To simplify the discussion, we describe here two rep- resentative set of parameters that contain these two phases,
= a tan共0.5兲 共v/ t
4= 0.5兲 and = −0.3 , for which we have computed, as a function of , the spectrum of the effective model by full ED. The spectra
共defined w.r.t. the respective GS energies兲 are displayed in Fig. 2. Special symbols have been used to label five of the six low-energy levels
共sevenstates over eight兲 associated to the mixed CP
1columnar- plaquette phase. We do not consider the last one, the
共K,B
1兲
0.00 0.05 0.10 0.15 0.00
0.02 0.04 0.06 0.08 0.10M
0.00 0.05 0.10 0.15 0.00
0.01 0.02 0.03 0.04 0.05
MK
Θatan0.5
1L 1L
0.00 0.05 0.10 0.15 0.20 0.25
ΦΠ
0.000 0.001 0.002 0.003 0.004 0.005 0.006
M
Αq
FIG. 3.
共Color online
兲. Extrapolations in the thermodynamic limit of the order parameters defined in the text characterizing the mixed CP
1columnar-plaquette phase. Insets: finite-size scaling of both M
+共, 兲 and M
−共0 , 0
兲as a function of the inverse linear cluster size, using 36 sites
共6 ⫻ 6
兲, 64 sites
共8 ⫻ 8
兲, 100 sites
共10
⫻ 10
兲, 144 sites
共12 ⫻ 12
兲, and 196 sites
共14 ⫻ 14
兲square clusters.
The chosen value of corresponds to the case of the J
1-J
2-J
3model studied in this work.
ΦΠ
14
18 38
12 14
ΘΠ
0.0 14?
J1J2J3
FIG. 4.
共Color online
兲. Phase diagram in the
共, 兲 plane ob- tained from numerical simulations. The previous knowledge of the
= 0 case
共Ref. 10
兲has been used, in particular, to estimate the limit of the columnar phase as an approximate
共black
兲straight line.
The boundary of the pure plaquette
共CP
1mixed
兲VBC phase ob- tained from the finite-size scaling of the associated order parameter is indicated by large circles
共triangles
兲. The
共red
兲thick segment corresponds to the parameter region of the frustrated quantum anti- ferromagnet with J
2+J
3=J
1/ 2 according to the mapping described in the text. Crude ED estimates of the boundaries of the columnar phase and of the CP
1phase are indicated by a thin dashed line and by small
共blue
兲circles, respectively. The
共light blue
兲region marked by a question mark has not been identified.
Plaq. RK
Col.
T
b
Trimerized
J1J2J3
0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.0
0.1 0.2 0.3 0.4 0.5
ΘΠ
ΦΠ a
CP3 CP1 CP2 T
FIG. 5.
共Color online
兲 共a
兲Variational wave functions
共associated
to their respective VBC states
兲considered in this appendix. Gener-
alized anisotropic VWF labeled by
兩CP
a典and
兩T
典interpolate be-
tween the most symmetric limits shown in the figure.
共b
兲Variational
phase diagram as a function of
共x axis
兲and
共y axis
兲. The
topographic map of ␣ is depicted by the continuous gray lines. The
line joining the Columnar phase and the trimerized one means that
these regions are continuously connected in terms of ␣ . The thick
red line shows the region of parameters of the effective model that
maps to the J
1-J
2-J
3model along the J
2+J
3=J
1/ 2 line for which
the ground state is well described by singlet coverings.
level, which is believed to be more affected by finite-size effects. The plots show wide intervals of , where the above five levels are the true lowest eigenenergies, hence pointing toward the mixed phase as a possible GS. The level crossing at which the low-energy spectrum becomes not anymore compatible with such a phase is indicated by an arrow. This level crossing can be used as a first crude estimator of the
range of stability of the mixed phase. Surprisingly enough, the spectrum at ⬎ 0 is not drastically affected by a finite value of t
6 共 ⬎ 0兲, showing that the mixed phase is rather stable w.r.t an extra loop-6 term, up to
⯝0.3 . This range of stability will be corroborated by our thermodynamic limit extrapolations of the order parameters
共see below兲. In con-trast, at = −0.3 and small , one can see that the very lowest levels of the spectrum
共i.e., those really separated bya sizable gap from the rest兲 are compatible with the columnar phase whose symmetries are given in Table II. A narrow region of mixed columnar-plaquette phase might however exist at intermediate values before the level crossing in- volving the
共M,A
1ⴱ兲state occurs. To finish this ED analysis, let us mention that, apart from the pure columnar and the mixed CP
1columnar-plaquette phases, no region in param- eter space could be found where the low-energy spectrum is compatible with the other VBC phases described above. In particular, for large
共relative兲t
6the spectrum becomes quite dense at low energies preventing any VBC phase identification.
15In order to give a more quantitative determination of the region of stability of the mixed CP
1columnar-plaquette or- der, we have computed the related plaquette structure factors,
I
共q兲=
具⌿0兩P共−q兲P
共q兲兩
0典具⌿0兩⌿0典
,
共3兲where P
共q兲is a diagonal operator with the same symmetry as the degenerate GS listed in Table II that we aim to target, defined as Fourier transform of plaquette operators intro- duced in Ref. 10. For positive t
4and t
6values, such quanti- ties can be computed efficiently using Green’s-function quantum Monte Carlo
共GFQMC兲. A Bragg peak ofI
+共q兲at momentum
共 ,
兲 共K point
兲and a divergence of I
−共q = 0
兲 共⌫point
兲reflect spontaneous translation and rotation symmetry breaking of the mixed phase, respectively. Related order pa- rameters M
共q
兲=
冑I
共q
兲/L can be conveniently defined and results are displayed in Fig. 3 showing size scalings of both M
+共 ,
兲and M
−共0 , 0兲up to cluster size of 196 sites. These data correspond to the line v/ t
4= 0.5, i.e., originating from the expansion of the microscopic model J
1-J
2-J
3under con- sideration here. Our results reveal that the Bragg peak at the K point survives up to ⬍ 0.3 , in good agreement with the ED criterion above. Interestingly, for increasing , the co- lumnar order parameter I
−vanishes before the plaquette one;
hence rotation symmetry is first recovered and a narrow re- gion of pure plaquette order is stabilized between the mixed phase region and the more complicated
共unknown兲phase at larger t
6. The extension of the GFQMC calculation to the
⬎ 0 region which do not have ergodicity problems limita- tions, has enable to draw the phase diagram depicted in Fig. 4.
V. CONCLUDING REMARKS
To finish, let us summarize our findings and their impli- cations. First we have extended the QDM to the case with a finite t
6amplitude for the loop-6 kinetic processes. Although we have also extended the search for VBC phases, we found TABLE III. Expansions of O and H up to order ␣
8.
Processes O H
Id 1 0
α
22 (J
2− J
1) α
2α
44 (J
2− J
1) α
4α
42 ( − 2J
1+ 2J
2+ J
3) α
4α
66 (J
2− J
1) α
6α
62 ( − 3J
1+ 3J
2+ J
3) α
6α
6( − 6J
1+ 7J
2+ 4J
3)α
6α
62 ( − 3J
1+ 3J
2+ 2J
3) α
6α
88 (J
2− J
1) α
8α
82 ( − 4J
1+ 4J
2+ J
3) α
8α
84 ( − 2J
1+ 2J
2+ J
3) α
8α
84 ( − 2J
1+ 2J
2+ J
3) α
8α
8( − 8J
1+ 9J
2+ 4J
3)α
8α
82 ( − 4J
1+ 4J
2+ 3J
3) α
8α
8( − 8J
1+ 9J
2+ 6J
3)α
8α
8( − 8J
1+ 10J
2+ 6J
3)α
8α
8( − 8J
1+ 10J
2+ 6J
3)α
8GENERALIZED QUANTUM DIMER MODEL APPLIED TO … PHYSICAL REVIEW B 80, 184427
共2009
兲184427-5
that the previously known columnar, plaquette, and mixed
共here called CP1兲columnar-plaquette phases are stable when a moderate t
6is added. This conclusion is first obtained by a close inspection of the low-energy spectrum of the model on finite clusters
共ED
兲. The quantitative determination of the extensions of the three previous phases is made possible by GFQMC simulations which do not suffer from the sign prob- lem when t
6⬎ 0. Still, it has not been possible to characterize the GS in the whole parameter space, in particular, when t
6dominates
共accumulation of low-energy states兲or when v
⬍ 0
共i.e., ⬍ 0兲 where the GFQMC encounters numerical limitations
共both cases being physically irrelevant anyway兲.Our present work on the generalized QDM turns out to be very useful to make progress in the understanding of the frustrated S = 1/ 2 quantum antiferromagnet on the square lat- tice. Indeed, it was previously argued that, in the vicinity of J
2⯝J
3⯝J
1/ 4 and along the J
2+ J
3= J
1/ 2 line, a truncation in the NN SU共2兲-singlet basis was legitimate. Using this argu- TABLE IV.
共Color online
兲Expansion of H
effup to order ␣
8.
Processes
H
eff= O
−1/2HO
−1/2Analytic J
3= J
1/2 J
2= J
1/4 J
2= J
1/2 expression J
2= 0 J
3= J
1/4 J
3= 0 2 (J
1− J
2) α
41 + α
40.625 0.46875 0.3125
− 2 (J
1− J
2) α
21 + α
4-1.25 -0.9375 -0.625
+ -0.226562 -0.242187 -0.257812
( − 2J
1+ 3J
2+ 2J
3)α
6-0.125 -0.09375 -0.0625
( − J
1+ J
2+ 2J
3) α
60. -0.03125 -0.0625 (J
1− J
2− J
3) α
60.0625 0.0625 0.0625 (3J
1− 3J
2− J
3) α
80.15625 0.125 0.09375
1
2
( − 5J
1+ 5J
2+ J
3) α
8-0.140625 -0.109375 -0.078125
4 (J
1− J
2) α
80.25 0.1875 0.125
1
4
(3J
1− 5J
2− 3J
3) α
80.0234375 0.015625 0.0078125
1
4
(2J
1− 2J
2− 3J
3) α
80.0078125 0.0117187 0.015625
1
4
(6J
1− 6J
2− 5J
3) α
80.0546875 0.0507812 0.046875
1
4
(7J
1− 7J
2− 3J
3) α
80.0859375 0.0703125 0.0546875 (3J
1− 3J
2+ J
3) α
80.21875 0.15625 0.09375
−
14(3J
1− 5J
2− 6J
3) α
80. -0.00390625 -0.0078125
1
4
( − 5J
1+ 9J
2+ 6J
3) α
8-0.03125 -0.0195312 -0.0078125
1
2
(3J
1− J
2+ 4J
3) α
80.15625 0.117187 0.078125 α
42 (J
3− J
1+ J
2)
α
4(
145 J
3− J
1+ J
2) )
ment and generalizing the RK expansion in terms of the magnitudes of the overlaps of the elements of the truncated
共nonorthogonal
兲basis we have made a link between the mi- croscopic model and some small region of parameter space of the generalized QDM where, fortunately, a precise char- acterization of the phase can be made. We can therefore de- duce that the frustrated S = 1 /2 quantum antiferromagnet ex- hibits in the vicinity of J
2⯝J
3⯝J
1/ 4 the same type of lattice symmetry breaking as the mixed columnar-plaquette phase
共CP1兲of the QDM. In the language of the quantum antifer- romagnet, it is a eightfold degenerate SU共2兲-symmetric phase with a 2 ⫻ 2 unit cell
共such as the plaquette phase兲and rotation symmetry breaking
共such as the columnar phase兲.While a previous investigation of the microscopic model on a 6 ⫻ 6 cluster has indeed provided evidence of plaquette correlations,
7only the mapping to the effective model
共whichcan be studied on much larger clusters兲 provides enough ac- curacy to show the spatially anisotropic nature of this spin- singlet plaquette VBC phase.
ACKNOWLEDGMENTS
D.P. and A.R. are indebted to F. Becca for useful discus- sions at an early stage of this work. M.M. would like to thank D. Schwandt for his careful reading of the fusion rules.
APPENDIX A: VARIATIONAL ANALYSIS
The effective generalized quantum dimer model on the square lattice originated from the microscopic frustrated Heisenberg Hamiltonian studied here can be first investi- gated by a variational method. Indeed, it is possible to con- struct variational ansatze for valence-bond crystal phases
共see inset of Fig.5兲 as simple tensor products of resonating plaquette states, extending the number of possible phases arising in the standard QDM. Variational energies can then be computed analytically. Despite its simplicity, this ap- proach reveals itself to be a useful guide for the numerical search of VBC
共although artifacts due to its variational na-ture are expected兲 and can easily incorporate the symmetry analysis provided in the paper. In other words, there is a one-to-one correspondence between these variational wave functions
共VWF兲and the VBC states defined in the paper by the set of spontaneously broken symmetries.
We shall consider a set of five different VWF,
共i兲the well-known
兩RK典one provided by Rokhsar and Kivelson
9as an equal weight superposition of all dimer configurations,
共ii
兲three VWF based on four-site plaquette tensor products
共兩CPa典,a = 1 , 2 , 3兲 and
共iii兲one with a six-site unit-cell inter- polating between the columnar phase and the pure six-site plaquette phase
共e.g., 兩T典in Fig. 5兲. Excepted for
兩RK典, allthese VWF depend on a parameter ␥ which allows a continu- ous interpolation between pure singlet crystals and highly resonating VBC. To illustrate this construction, here is the expression of one of the anisotropic four-site plaquette phases and the above-mentioned six-site plaquette one,
|CP
1=
⊗p
cos( )|
p+ sin( )|
p|T =
⊗p
sin( )|
p+ cos( ) √
2 (|
p+ |
p) ,
␥
␥
␥
␥
共
A1
兲where the product is performed over the set of separate plaquettes p or p ⬘ as suggested in Fig. 5.
In order to describe the stability of these phases w.r.t. the parameters, the expectation value of the effective generalized QDM Hamiltonian is computed. The only off-diagonal terms of H contributing to
具⌫兩H兩⌫典, with 兩⌫典being one of those VWF, are plaquette flips on occupied plaquettes and the di- agonal potential term. This leads to contributions propor- tional to cos共 ␥
兲sin共␥
兲for all the VWF, plus one in cos
2共␥
兲for the six-site plaquette one. For the diagonal terms, both occupied and nonoccupied plaquettes yield nonzero contri- butions to the expectation value of H . It is worth to empha- size that the
兩RK典WF requires a special analysis. Using well- defined Pfaffian techniques, one can compute analytically the probability of flipping a four-site and a six-site plaquette, which are, respectively, P
4= 1 /4 and P
6= 0.0330112共1兲. Fi- nally, these expectation values, as a function of the Hamil- tonian parameters v共 ,
兲,t
4共 ,
兲, andt
6共 ,
兲, are given byE
RK= v − t
4− 4t
6P
6, E
CP1
= v关1 + cos
4共␥
兲+ sin
4共␥
兲兴− 2t
4cos共 ␥
兲sin共␥
兲,E
CP2
= v关1 + cos
4共␥
兲兴− 2t
4cos共 ␥
兲sin共␥
兲, E
CP3
= v关1 + 2 cos
2共␥
兲sin2共␥
兲兴− 2t
4cos共 ␥
兲sin共␥
兲, E
T= v关2 cos
2共␥
兲+ 4 sin
2共␥
兲+ 3 cos
4共␥
兲/2 + 2 sin4共␥
兲兴/3− t
48 cos共 ␥
兲sin共␥
兲3
冑2 − t
62 cos
2共␥
兲3 .
共A2兲These energies are then minimized w.r.t. ␥ and the varia- tional phase diagram displayed in Fig. 5 as obtained in the
共 ,
兲plane. This phase diagram is discussed in the paper.
The colors correspond to different values of the ␥ parameter,
␥ = / 2
共blue兲for the pure columnar phase, ␥ = 0
共purple兲for the pure six-site resonating plaquette phase and ␥ = / 4
共or- ange
兲for the isotropic four-site plaquette phase. The RK re- gion has an arbitrary color.
APPENDIX B: DERIVATION OF THE HAMILTONIAN This appendix is devoted to a technical description of the generalized QDM derivation scheme.
1. Choice of a small parameter and disconnected processes As briefly described in the paper, the generalized QDM is obtained by developing H
eff= O
−1/2HO
−1/2according to the matrix element hierarchy of both O and H in the VB basis.
Indeed, the amplitude of O
, 共with and two VB con- figurations兲 only depends on the number of loops n
lof the overlap diagram g =
共 ,
兲obtained by superimposing the
GENERALIZED QUANTUM DIMER MODEL APPLIED TO … PHYSICAL REVIEW B 80, 184427
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兲184427-7
two configurations: O
g= ␣
N−2nl共g兲with ␣ = 1/
冑2 and N the total number of sites
共see Fig.6兲.
The maximal number of loops is obtained for = . In this case g is just a collection of N / 2 trivial loops with length L
l= 2 and O
g= 1. The next term is obtained by considering one nontrivial loop with length L
l= 4共
兲while all remaining
共N− 4兲/ 2 loops are chosen trivial which leads to O
g= ␣
2. In the same spirit, one L
l= 6 loop共
兲and
共N− 6兲/2 trivial loops is an ␣
4-order process
共see thefourth line of Tables I and III兲. Such a construction suggests that a quite natural driving parameter for the expansion is the length L
lof a unique nontrivial loop surrounded by
共N−L
l兲/ 2 trivial loops: such a process indeed appears at order ␣
Ll−2. However the total length of loops is a constraint quantity
共兺l苸gL
l= N兲 and other
共nonconnected
兲terms indeed appear in O at the same order.
For example, formed by two disconnected squares also occurs in O with the amplitude ␣
4despite the fact the nontrivial contour length is different from, e.g., .
For this reason, in the derivation scheme presented here, we chose the overall number of loops as the driving parameter for the expansion of O rather than the commonly used length of the loops.
9,12,13The key difference lies in the presence of disconnected processes such as : as we will see, they cancel at every order of the final effective Hamiltonian but are crucial in the calculation because they are responsible for the emergence of nontrivial diagonal and off-diagonal connected processes.
In the expansion of O, we use the following notation:
O =
p兺
ⱖ0␣
2p
p,
共B1兲where
pare combinations of
pg
process on graphs g,
p= 兺g
pg.
共B2兲
For a full list of
pg
up to order ␣
8, see Table III.
2. Heisenberg Hamiltonian expansion
The action of each term of the Heisenberg Hamiltonian
共1兲of a VB state consist in a reconfiguration of dimers. Thus,
具
兩H兩
典can be deduced form inspecting the topology of the overlap diagram
具
兩
典as recalled in Fig. 7. This allow H to be expanded in power of ␣
共see Ref.16兲 similarly to O,
H =
p兺
ⱖ0
␣
2p兺
gh
pg
pg.
共B3兲Note that it is convenient here to rescale the Hamiltonian by a factor 4/3. Furthermore, evaluating
具
兩共4/ 3兲S
i.S
j兩
典ge- nerically involves an extensive number of length-2 loops which only effect is to produce a trivial extensive contribu- tion to the matrix element. This can be removed by scaling and shifting H to
共4 / 3
兲H+ J
1N / 2. Using this convention, the expansion of H contains only kinetic terms. For a full list of h
pgup to order ␣
8, see the last column of Table III.
3. Fusion rules and effective Hamiltonian
The first step to compute the effective Hamiltonian H
eff= O
−1/2HO
−1/2is to derive the expression of O
−1/2. To achieve this, we use the formal expansion,
O
=
k兺
ⱖ0
⌫共1 +
兲⌫共1 + − k兲⌫共1 + k兲
共O− 1兲
k.
共B4兲Powers of O and products with H generically involve symmetric products of diagrams
共see TableIII兲 that do not commute, e.g,
,
= ⊗ + ⊗ .
Evaluating these products requires establishing fusion rules that
共i兲governs algebraic properties of diagrams and
共ii兲gen- erate, order by order, extra diagonal and off-diagonal pro- cesses. We give below, the minimal set of rule that are needed up to order ␣
8.
Order 4 fusion rule,
1 2
,
= + + 2
0 +1
-1
FIG. 7.
共Color online
兲. The matrix element
具兩共4 / 3
兲S
i. S
j兩典ex- pressed as f
ij具兩典 . Bond
共i, j
兲is represented as a solid black line.
Red and blue bonds represent
兩典and
兩典. f
ij= +1
共respectively, f
ij= −1
兲if i and j belong to the same loop at even
共respectively odd
兲distance along the loop. f
ij= 0 if
共i , j
兲connects two sites on distinct loops
共see Ref. 16
兲.
ϕ ϕ|ψ ψ
FIG. 6.
共Color online
兲. Overlap
具兩典between two VB configu-
rations ad . Closed loops that appear in the overlap diagram g
are represented as yellow shades.
Order 6 fusion rules,
,
= 2 + 6 + 2 +
,
= 2 + 2 +
,
= + 2 + 2 + 2
Order 8 fusion rules,
1 2
,
= + 2
+ 2
,
= 4 + 2 + +
1 2
,
= + 2 + + 6
+ 2 + + +
1 2
,
= + + 2 +
+
+
,
= + + 2 + 2
+ 2 + + + +
,
= 2 + 8 + 2 +
,
= + 2 + 4 + 4
+ 2 + 2 + 2 +
,
= + + 2
+ 2 + +
,
= + 2 + + 2 + 2
,
= 4 + 4 + 4 +
+ 2 +
,
= + 2 + 2 + + +
= +
,
= 2
+ +
+ 2
GENERALIZED QUANTUM DIMER MODEL APPLIED TO … PHYSICAL REVIEW B 80, 184427
共2009
兲184427-9
Note that contractions of diagrams not only produce poten- tial
共diagonal兲terms but also nontrivial assisted off-diagonal operators such as, e.g., that flip a plaquette only if a dimer is present next to it. Special process also appear such
as
共respectively 兲which simultaneously
flip two neighboring plaquettes with parallel
共respectively,perpendicular兲 dimers. In Table IV, we summarize the result
of the expansion up to order ␣
8. Interestingly enough, all disconnected terms vanish after simplifying the product O
−1/2HO
−1/2. The demonstration of this generic property is beyond the scope of the present paper and will be presented elsewhere.
17At this level, let us remark that this absence of nonlocal terms in the generalized QDM Hamiltonian is physically satisfactory and is a strong indication of the internal consistency of the derivation scheme presented here.
1
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兲; V. Murg, F. Verstraete, and J. Cirac, ibid. 79, 195119
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8
A valence bond is a
共spatially antisymmetric
兲singlet pair of two S = 1 / 2 spins.
9
D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376
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1988
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Note that the trimerized phase is frustrated on the 8 ⫻ 8 cluster.
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16
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17
