Effective field theories for two-component repulsive bosons on lattice and their phase diagrams

0.59 MB·English

by  Y. Kuno

Effective field theories for two-component repulsive bosons on lattice
and their phase diagrams
Yoshihito Kuno, Keisuke Kataoka, and Ikuo Ichinose
Department of Applied Physics, Nagoya Institute of Technology, Nagoya, 466-8555 Japan
(Dated: November1, 2012)
Inthispaper,weconsiderthebosonict-Jmodel,whichdescribestwo-componenthard-corebosons
with a nearest-neighbor (NN)pseudo-spin interaction and a NN hopping. To study phase diagram
2
of this model, we derive effective field theories for low-energy excitations. In order to represent
1
the hard-core nature of bosons, we employ a slave-particle representation. In the path-integral
0
quantization, we first integrate our the radial degrees of freedom of each boson field and obtain
2
the low-energy effective field theory of phase degrees of freedom of each boson field and an easy-

t plane pseudo-spin. Coherent condensates of the phases describe, e.g., a “magnetic order” of the
c
O pseudo-spin,superfluidityof hard-core bosons, etc. This effective field theory is a kindof extended
quantum XY model, and its phase diagram can be investigated precisely by means of the Monte-

1 Carlo simulations. We then apply a kind of Hubbard-Stratonovichtransformation to the quantum
3 XY model and obtain the second-version of the effective field theory, which is composed of fields
describingthepseudo-spindegreesoffreedom andboson fieldsoftheoriginaltwo-componenthard-

] corebosons. Asapplicationoftheeffective-fieldtheoryapproach,weconsiderthebosonict-Jmodel
s
on the square lattice and also on the triangular lattice, and compare the obtained phase diagrams
a
g with the results of the numerical studies. We also study low-energy excitations rather in detail in
- the effective field theory. Finally we consider the bosonic t-J model on a stacked triangular lattice
t and obtain its phase diagram. We compare the obtained phase diagram with that of the effective
n
a field theory to find close resemblance.
u
q PACSnumbers: 67.85.Hj,75.10.-b,03.75.Nt
.
t
a
m I. INTRODUCTION the slave-particlerepresentation. By integratingoutam-
plitude modes of the slave particles, we obtained an ex-
-
d tendedquantumXYmodel,whichdescribeBoseconden-
In recent years, systems of cold atoms have attracted
n sationof two atoms and pseudo-spindegrees of freedom.
o interest of many condensed-matter physicists. In par- Numericalstudy by the Monte-Carlo(MC) methods can
c ticular, cold atoms put on an optical lattice sometimes
be performed without any difficulties and the phase dia-
[ regarded as a “simulator” to study canonical models
gramis obtained for the B-t-J model on both the square
of strongly correlated electron systems1. These sys-
2 and triangular lattices in Sec.III. In Sec.IV, we shall de-
v tems are quite controllable and impurity effects are sup- rive second-version of the effective field theory from the
0 pressed. From this point of view, we have started study
extended quantum XY model by introducing collective
1 of the bosonic t-J model2,3, which is a bosonic counter-
fields for the two bosons and the pseudo-spin degrees of
9 part of the t-J model for the high-T superconducting
7 phenomena4. The bosonic t-J modelc(B-t-J model) de- freedom. By using this effective field theory, we obtain
. phase diagram of the B-t-J model and verify the consis-
0 scribes two-component bosonic cold atoms in an optical
tency of the results. Furthermore we study low-energy
1 latticewithstronginterandintra-speciesrepulsionsthat
excitations, i.e., the Nambu-Goldstone bosons in various
2 can be controlled in the experiments.
1 phases in the phase diagram. In Sec.V, we numerically
: Our previous studies on the B-t-J model mostly em- shall the B-t-J model in the stackedtriangular lattice at
v
ployednumericalMonte-Carlo(MC) simulations to clar- finiteT. Obtainedphasediagramhasasimilarstructure
i
X ify its phase diagramat finite temperature (T)2. In con- of the model in the triangular lattice at T = 0. Section
r trasttothefermionict-Jmodel,thenumericalstudycan VI is devoted for conclusion and discussion.
a bedonewithoutanydifficultiesforsomecasesoftheB-t-
Jmodel. Inthispaper,weshallderiveeffectivefieldsthe-
oriesoftheB-t-Jmodelandstudythembybothanalyti- II. BOSONIC t-J MODEL AND DERIVATION
calandnumericalmethods. WemostlyfocusontheB-t-J OF EFFECTIVE MODEL
modelonsquareandtriangularlatticesatT =0andthe
phase diagram for quantum phase transitions. The ob- Hamiltonian ofthe B-t-J model, which will be studied
tained results are compared with the previous findings in this paper, is given as2–5,
and numerical study of finite-T systems.
This paper is organized as follows. In Sec.II, we shall HtJ =− t(a†iaj +b†ibj +h.c.)+Jz SizSjz
introduce the B-t-J model and derive first-version of an hXi,ji hXi,ji
effective field theory by using the path-integral meth- +J (SxSx+SySy) (2.1)
i j i j
ods. The local constraint is faithfully treated by using
hXi,ji
2
where a†i and b†i are boson creation operators6 at site i, nary time, i.e.,
pseudo-spin operator S~i = 21Bi†~σBi with Bi = (ai,bi)t,
~σ are the Pauli spin matrices, and i,j denotes nearest-
h i Z = [DφDϕ1Dϕ2]exp dτ ϕ¯1i(τ)∂τϕ1i(τ)
neighbor(NN)sitesofthelattice. Weshallconsiderboth −
the square and triangular lattices in the following study. Z +ϕ¯ (τ)∂ ϕ (hτ)+Zφ¯(τ(cid:16))∂ φ (τ)
2i τ 2i i τ i
Physical Hilbert space of the system consists of states
with total particle number at each site less than unity +HtJ+HV , (2.5)
(the local constraint). In order to incorporate the local (cid:17)i
constraint faithfully, we use the following slave-particle where H is expressed by the slave particles and the
tJ
representation2,3, above path integral is calculated under the constraint
(2.3). (In this paper we set ¯h = 1.) For the existence
ai =φ†iϕi1, bi =φ†iϕi2, (2.2) HV, we separatethe path-integralvariables ϕ’s and φ as
φ†iφi+ϕ†i1ϕi1+ϕ†i2ϕi2−1 |physi=0, (2.3) ϕ = ρ +ℓ exp(iω ),
1i 1i 1i 1i
(cid:16) (cid:17)
whereφi is abosonoperatorthatannihilates hole atsite ϕ2i =pρ2i+ℓ2iexp(iω2i), (2.6)
i, whereas ϕ and ϕ are bosons that represent the
pseudo-spin d1eigrees of2ifreedom. phys is the physical φi = pρ3i+ℓ3iexp(iω3i),
| i
state of the slave-particle Hilbert space. p
andweintegrateoutthe(fluctuationof)radialdegreesof
The previous numerical study of the B-t-J model2,3,7
freedom. Thereexists a constraintlike ℓ +ℓ +ℓ =0
show that there appear various phases including super- 1i 2i 3i
on performing the path-integral over the radial degrees
fluid with Bosecondensation,state withthe pseudo-spin
of freedom, i.e., ℓ (σ = 1,2,3). This constraint can
long-rangeorder,etc. Forthemostofthem,theMCsim- σi
be readily incorporated by using a Lagrange multiplier
ulationsshowthatdensityfluctuationateachlatticesite
λ (τ),
is not large even in the spatially inhomogeneous states i
like a phase-separated state. From this observation, we
expect that there appears the following term effectively, δ(ℓ +ℓ +ℓ )= dλ eiRdτ(ℓ1i+ℓ2i+ℓ3i)λi.
1i 2i 3i i
τ Z
V Y
HV = 40 (ϕ†1iϕ1i−ρ1i)2+(ϕ†2iϕ2i−ρ2i)2 The variablesℓ (σ =1,2,3)alsoappearinH , but we
Xi (cid:16) σi tJ
+(φ†iφi−ρ3i)2 , (2.4) aignndortehetnhewme bhyavseimply replacing ϕσi → √ρσiexp(iωσi),
(cid:17)
where ρ etc are the parameter that controls the den-
sities of1ai-atom and b-atom at site i, and V (> 0) con- dλidℓieRdτP3σ=1(−V0(ℓσ,i)2+iℓσ,i(∂τωσ,i+λi))
0
trolstheirfluctuationsaroundthemeanvalues. Itshould Z
be remarked here that the expectation value of the par- = dλie−4V10 RdτPσ(∂τωσ,i+λi)2, (2.7)
ticle numbers in the physical state phys are given as Z
| i
N1 iha†iaii≡ N1 iTrphys(a†iai)= N1 iTrphys(ϕ†1iϕ1i) where we have ignored the terms like dτ∂τωσ,i by the
aNndPisstihmeilnaurlmybN1er Pofihsbit†iebsiio=f thN1e latitTicrephaPyns(dϕT†2irϕph2yi)s,dwenhoertee preesruioltdainctbqouuanndtaitryyocnonthdeitiRoHnSfoorft(h2e.7im)iaRsgpinoasirtyivteimdeefi.nTithee,
P P
the trace over the states satisfying the local constraint, andthereforethe numericalstudy by the MC simulation
i.e.,thephysical-statecondition(2.3). Thereforethecon- canbedonewithoutanydifficulty. Itshouldberemarked
straint (2.3) requires 3 ρ = 1 at each site i. The that the Lagrange multiplier λ in Eq.(2.7) behaves as a
σ=1 σi i
values of V and ρ (σ =1,2,3) are to be determined in gauge field, i.e., the RHS of (2.7) is invariant under the
0 σi
P
principle by t, J , J and filling factor, but here we add following“gaugetransformation”,ω ω +α , λ
z σ,i σ,i i i
→ →
H to H by hand and regard parameters in H as a λ ∂ α . In the practical calculation, we shall show
V tJ V i τ i
−
free parameter. In this sense, we are considering an ex- thatallphysicalquantitiesareinvariantunderthe above
tendedbosonict-Jmodel. Itshouldbestressedherethat gauge transformation.
the strong on-site repulsions between atoms enhance the Various phases can form in the system (2.1) at T =0,
effects of spatial lattice and as a result quasi-excitations e.g., states with long-range spin orders (FM, AF, spi-
with Lorentz-invariant dispersion can appear in a non- ral, etc), superfluid with Bose condensation of a and/or
relativisticoriginalmodelliketheBose-Hubbardmodel8. b atoms, and superposition of them (the supersolid
The existence of HV in Eq.(2.4) is very useful for (SS))7,9. In the following sections, we shall consider sev-
study of the quantum many-particle systems. In eralspecific casesofthe extendedB-t-Jmodel H +H
tJ V
the path-integral representation of the partition func- and derive effective field-theory model. Then we clarify
tion Z, the action contains the imaginary terms like the phase diagram of the B-t-J model by studying the
dτφ¯i(τ)∂τφi(τ), where φ¯i is for φ†i and τ is the imagi- effective model.
R
3
III. PHASE DIAGRAM OF THE EXTENDED 1st order
2nd order
QUANTUM XY MODEL: NUMERICAL STUDY
2
1.8
1.6 FM FM+ a SF+ b SF
A. FM coupling on square lattice 1.4
1.2
1 1
C 0.8
Inthissection,weshallstudythe effective fieldtheory 0.6 PM
obtained in the previous section on the two-dimensional 0.4
0.2
(2D) square lattice as the first example. One of the sim-
0 0.5 1 1.5 2
plest case is xy-FM system corresponding to J = 0
z C3
and J < 0 in Eq.(2.1), though an extension to the
case J < J is rather straightforward. In this case
z
the p|seu|do-s−pin symmetry is O(2), and then we put 2
1.8
ρ1i = ρ2i = ρ and ρ3i = 1 − 2ρ. The partition func- 11..46 FM FM+ a SF+ b SF
tfrioonmZEqqXsY.(2o.f5)thaendeff(e2c.7ti)v,e theory is obtained as follows C111.2
0.8
0.6
0.4 PM
0.2
ZqXY = [ Dωσ]exp Aτ A(eiΩσ,e−iΩσ) ,
− − 0 0.5 1 1.5 2
Z σ=Y1,2,3 h i C3
(3.1)
where
2
1.8
1 1.6
Aτ = dτ (∂τωσi+λi)2, 1.4 FM FM+ a SF+ b SF
4V0 Z Xi,σ C1 011..82
A(eiΩσ,e−iΩσ)= dτ 00..46 PM
0.2
Z h
c cos(Ω Ω )+cos(Ω Ω ) 0 0.5 1 1.5 2
h 2,i 2,j 3,i 3,j
− − − C3
hXi,ji (cid:16) (cid:17)
FIG.1. (Coloronline)Phasediagramoftheeffectivefieldthe-
c cos(Ω Ω ) (3.2)
− s 1,i− 1,j oryALxyon3Dcubiclatticeforvariousvaluesofcτ. ρ=0.35.
hXi,ji i Dots denote the observed phase transition point. Thick line
denotes first-order phase transition line whereas the others
where
are second-order transition lines. System size L=24.
Ω =ω ω , Ω =ω ω , Ω =ω ω ,
1i 1i 2i 2i 1i 3i 3i 2i 3i
− − −
where AL(eiΩσ,e−iΩσ) corresponds to A(eiΩσ,e−iΩσ) in
and Eq.(3.2),
t
ch = ρ(1 2ρ), cs =4Jρ4. (3.3) AL(eiΩσ,e−iΩσ)=
2 −
C cos(Ω Ω )+cos(Ω Ω )
3 2,r 2,r′ 3,r 3,r′
It is obvious that the integrand in Eq.(3.1) is positive − − −
definite, andthereforethe numericalcalculationofZ hXr,r′i (cid:16) (cid:17)
qXY
by means of the MC simulation is possible without any C1 cos(Ω1,r Ω1,r′), (3.6)
− −
difficulty. Themodel(3.1)isakindoflatticerotormodel. hXr,r′i
Forpracticalcalculation,weintroducealatticeforthe
whereC =c ∆τ andC =c ∆τ,and r,r denotesthe
imaginary-timedirectionandusethefollowinglatticeac- 1 s 3 h ′
h i
NN sites in the 2D spatial lattice. We fixed the value of
tion A corresponding to A
Lτ τ
c and calculated the partition function Z, the “internal
τ
3 energy”E and “specific heat” C as a function of C and
1
ALτ =cτ cos(ωσ,r+τˆ−ωσ,r+λr), (3.4) C3,
r σ=1
XX
wherer denotessite ofthe space-timecubic lattice, cτ = Z = [dω]e−ALxy,
1 and∆τ isthelatticespacingoftheimaginary-time Z
V0∆τ E = A /L3,
direction. Lxy
h i
We numerically studied the lattice model defined by C = (A E)2 /L3, (3.7)
Lxy
h − i
the lattice action
where L is the linear size of the 3D cubic lattice. In
ALxy =ALτ +AL(eiΩσ,e−iΩσ), (3.5) order to identify various phases, we also calculated the
4
following pseudo-spin and boson correlationfunctions, 1st order
2nd order
1 2
GS(r)= L3 heiΩ1,ie−iΩ1,i+ri, 1.8 A
Xi 1.6 120 o B
1 1.4
Ga(r)= L3 i heiΩ2,ie−iΩ2,i+ri, C1 11.2
1 X 0.8 C
Gb(r)= L3 heiΩ3,ie−iΩ3,i+ri, (3.8) 00..46 PM D
i
X 0.2
where sites i and i+r are located in the same spatial
0 0.5 1 1.5 2
2D lattice, i.e., the equal-time correlations. For exam-
C3
ple, G (r) finite as r indicates Bose-Einstein
a
→ → ∞
condensation (BEC) of the a-atom. A; 120+(aSF or bSF) C; PM+2SF
For numerical simulations, we employ the standard
B; 120+2SF D; FM(XY)+2SF
Monte-Carlo Metropolis algorithm with local update10.
Thetypicalsweepsformeasurementis(30000 50000) FIG. 2. (Color online) Phase diagram of the quantum XY
∼ × model on triangular lattice. Density of each boson ρ = 0.3.
(10 samples), and the acceptance ratio is 40% 50%.
∼ System size L=18. There are six phases. Phase of 120o de-
Errors are estimated from 10 samples with the jackknife
notes thestate with 120o pseudo-spin long-range order with-
methods.
out atomic BEC. SFstands for superfluid.
We show the obtained phase diagram for ρ = 0.35 in
Fig.1. There are three phases and they are separated
with each other by first or second-order phase transi- in the C3 C1 plane. For small C3, the pseudo-spin
tion lines. It is obvious that for small C and C , there degrees of−freedom exhibits the 120o long-range order.
1 3
exists a “paramagnetic state” (PM state) without any As we assume a homogeneoushole distribution, a super-
long-range orders (LRO’s) and its domain is decreased posedstateofatomsandholeisrealizedateachsite,but
for increasing c . In this PM state, a superposition of coherentcondensationofatomsdoesnottakesplaceyet.
τ
a-atomandb-atomisrealizedateachsite,butcoherence As C3 is increased, phase transition to the states with
oftherelativephaseinthesuperpositiondoesnotexists. the 120o long-range order and Bose condensation take
It should be notice that the atomic BEC always accom- place. For example in the phase B in Fig.2, the both
panies the pseudo-spinLRO.The obtainedresults arein atomsBosecondense,as the correlationfunctions shown
good agreementwith the result of the previous study on in Fig.3 exhibit. In the mean-field approximation, the
some related model on 3D cubic lattice at finite T3. wave function of that state is given by
|Ψi= [a†i +b†i +cA]
B. AF coupling on triangular lattice i A
Y∈
× [a†i +ei23πb†i +cB]
In this subsection, we shall study the B-t-J model on i B
Y∈
tinhaer2yDtitmriea,ntghuelamroladtetlicAe. AsinweEaql.s(o3.d5)iscisredteizfientehdeoinmtahge- × [a†i +e−i23πb†i +cC]|0i,
Lxy
i C
3Dstackedtriangularlattice. In this sectionweconsider Y∈
the case of the xy-AF case with O(2) symmetry. There- wherec’saresomecomplexnumber and 0 isthe empty
| i
fore we setJ >0 andJ =0 in Eq.(2.1). As there exists state of a and b-atoms. This phase will be discussed by
z
the frustration, we expect that various phases appear in using the effective field theory in Sec.IV.C and also by
contrasttothexy-FMcase. Inlatersection,wealsoshow the MC simulations in Sec.V. As the parameter C is
3
the result of the numerical study of the B-t-J model on increased further, order of the pseudo-spin is destroyed
thestackedtriangularlatticeat finite but low T,whichis first (phase C) and then changes to the FM one in the
closelyrelatedtothe2DmodelatT =0. Thiscloserela- Sx Sy plane (phase D).
−
tion between3D model atlow T and2D model atT =0
has been previously observed in various systems11. Ori-
gin of this similarity is somewhat obvious from A in IV. EFFECTIVE FIELD THEORY FOR BOSE
Lτ
Eq.(3.4) as it can be regardedas an inter-layercoupling. CONDENSATION AND PSEUDO-SPIN ORDER
To perform the numerical calculation, we first assign
values of the parameters ρ ,ρ (ρ = 1 ρ ρ ). In the previous sections, we numerically studied lat-
1i 2i 3i 1i 2i
− −
Hereweassumeahomogeneousstateandputρ =ρ = ticequantumXYmodelthatdescribethetwo-component
1i 2i
ρ (ρ =1 2ρ) as in the previous case. Results of more bosons with strong repulsions. The obtained phase dia-
3i
−
general cases will be reported in a future publication. grams show that there exist Bose-condensed phases as
In Fig.2, we show the obtained phase diagram for well as the state of simple pseudo-spin LRO. In this sec-
ρ = 0.3 and the hole density= 0.4. There are six phases tion, we shall derive an effective field theory from the
5
and the fact that other correlators like eiωi(τ)eiωi(τ′)
1 1 are vanishing. RHS of Eq.(4.3) can be exhpressed by ini-
0.5 0.5 troducing a complex boson field Φi as
0 0 eRdτRdτ′η¯i(τ)e−V0|τ−τ′|ηi(τ′)
-0-.15 0 1 2 3 4GGGabs(((rrr )))5 6 -0-.15 0 1 2 3 4GGGabs(((rrr )))5 6 =Z [dΦ]exph− V10 Z dτΦ∗i(−∂τ2+V02)Φi
r r
+ dτ(η Φ +η¯Φ ) . (4.5)
i i i ∗i
FIG. 3. (Color online) Correlation functions in the phase Z i
diagram. C3 = 0.65 and C1 =1.6 (left), C3 =1.0 and C1 = By inserting Eq.(4.5) into Eq.(4.1), we obtain the effec-
1.6 (right). tive field theory of the rotor model with the following
action A ,
rotor
1
qcounadnetunmsatXioYn amndodsepl,inwohridcehr.deTshcrisibfieesldditrhecetolryytnhoetBonoslye Arotor = dτ V0 Φ∗i(−∂τ2+V02)Φi
explainsthenumericallyobtainedphasediagrambutalso Z h Xi
J
reveals low-energy excitations and interactions between −2 (Φ∗iΦj +c.c.) ,
them. This kind of effective field theory has been ob- hXi,ji i
tained and discussed for a granular superfluid etc by
using Hubbard-Stratonovichi transformation12. In this Zrotor = [dΦ]e−Arotor. (4.6)
sectionweshallemployaslightlydifferentmethod13. As Z
a result, we reveal some important point that has been The above derivation from Eq.(4.1) to Eq.(4.6) seems
overlookedso far. exactbecausetheintegrationofωi(τ)istheone-siteinte-
gralandessentiallyGaussianintegrationofthefreefields.
Infacttheabovemanipulationisexactaslongasthebo-
son field Φ does not Bose condense. On the other hand
A. Effective field theory i
forJ V ,the BosecondensationofΦ takesplace,i.e.,
0 i
≫
Φ = 0. One may wonder if the above manipulation
i
To illustrate the procedure to derive the effective field h i 6
is applicable even for this case because Eq.(4.4) seems
theory, we first consider a single rotor model as an ex- to indicate nonexistence of the long-range oder of Φ .
i
ample that describes superfluid phase transition. After
Furthermore in this case, the mass term of Φ becomes
i
considerationofthe simple model, weshallapplysimilar
negative, and the integration of Φ in Eq.(4.6) becomes
i
methods to the quantum XY model for the B-t-J model.
unstable, e.g., for the square lattice
We first rewrite the partition function of the rotor
J
model onthe squarelattice by introducingsource terms, 2 (Φ∗iΦi+µ+c.c.)−V0 |Φi|2
i,µ i
Zrotor = [dω]e−V1o RdτPω˙i2−RdτH(eiωi,e−iωi) =XJ Φ 2+ (XzJ V )Φ 2, (4.7)
Z −2 |∇µ i| − 0 | i|
= [dω]e−V1o RdτPω˙i2−RdτH(δηδi,δδη¯i) Xi,µ Xi
where is the lattice difference operator and z is the
Z µ
I(ηi,ωi)η=η¯=0, (4.1) number∇oflinksemanatingfromasinglesiteandz =4for
× |
I(η ,ω )=eRdτP(ηieiωi+η¯ie−iωi), (4.2) the square lattice. This instability comes from the fact
i i
that the order of the η-derivative and the ω-integration
is not interchangeable when the Bose condensation, i.e.,
where ω˙ =∂ ω and
i τ i
a phase transition to an ordered state, takes place. It is
H(eiωi,e−iωi)=−J cos(ωi−ωj). oabnvdiothuesrheΦfoirie=ithiseipωliaiu<sib1leintotehxepBecotsetecrmondlieknesλedΦst4atteo,
i
hXi,ji appear in the effective field theory to stabilize th|e i|nte-
gration of Φ for the Bose condensed state, though ex-
In Eq.(4.1), we evaluate the path integral of ω as i
i plicit calculation to determine the coefficient is difficult.
Simplespin-wavelikeapproximationfortherotormodel,
[dω]e−V10 RdτPσω˙i2I(ηi,ωi)
J
Z Jcos(ω ω ) J (ω ω )2,
=eRdτRdτ′η¯i(τ)e−V0|τ−τ′|ηi(τ′), (4.3) i− j ∼ − 2 i− j
where we have used gives the estimation like eiωi e−iωi e−VJ0, and then
h ih i ∼
thecoefficientofthe Φi 4-termisestimatedasλ JeVJ0
eiωi(τ)e−iωi(τ′) =e−V0|τ−τ′|, (4.4) for J V0. | | ∼
h i ≫
6
Hubbard-Stratonovichtransformationderivesasimilar
effective field theory to the above. But its straightfor-
wardapplicationgives a negative coefficientof the Φ 4-
i
| |
term indicating aninstability of the system. This means
that certain step of the derivation is invalid, e.g., intro-
duction of the Hubbard-Stratonovichi field and the ω-
integration is not interchangeable. This problem is un- 4
V2
der study and the result will be reported in a separate 0 1
-2
publication. 0
--11
Similar manipulation to the above can be applied 00 -1
to the present quantum extended XY model. From 11
Eqs.(3.1) and (3.2),
Φ1＝1.02771 Φ2＝1.11237
ZqXY = [dω]e−V1o Rdτω˙σ2i−A(eiΩσi,e−iΩσi) V= -2.05918
Z FIG. 4. (Color online) Potential V(Φσ) (4.14) and its mini-
= [dω]e−V1o RdτPω˙σ2i−A(δηδσi,δη¯δσi) m21,uλm3.=Sy−m1,mλe4tr=ic−ca1s,eanΦd2g==Φ31.. Parametersareλ1 =1,λ2=
Z
I(η ,Ω ) ,
σi σi η=η¯=0
× |
I(η ,Ω )=eRdτP(ησieiΩσi+η¯σie−iΩσi), (4.8) system. The finalformoftheeffectiveactionistherefore
σi σi
given by
whereω˙ =∂ ω andwehaveomittedthegaugefieldλ
σi τ σi i
as we consider the only gauge-invariant objects through A =A dτ (λ Φ 4)
eff 0 σ σi
− | |
Ωσi. Then the integrationoverωσi can be performed as, Z Xσ,i
[dω]e−V10 RdτPσω˙σ2iI(ησi,Ωσi)=I˜(ησi,η¯σi), (4.9) =Z dτLeff,
Z
Z = [dΦ]e−Aeff. (4.13)
and the Green function of ω (τ), G , is obtained as
σi ω Z
eiωσi(τ)e−iωσ′i(τ′) =δσσ′e−V0|τ−τ′|. (4.10)
h i B. Phase diagram and low-energy excitations:
Typical term of I˜(η ,η¯ ) is as follows, Square lattice
σi σi
eRdτRdτ′η¯σi(τ)e−2V0|τ−τ′|ησi(τ′), In this subsection, we shall apply the field-theoretical
approach explained in the previous subsection to the B-
t-Jmodelonthesquarelatticeasasimpleexample. Fur-
and this quantity is expressed by introducing complex
thermore we focus on the FM parameter region J < 0.
scalar fields Φ (τ) as
σi
In this case there exists no frustrationandtherefore it is
eRdτRdτ′η¯σi(τ)e−2V0|τ−τ′|ησi(τ′) ratherstraightforwardtoobtainthephasediagram. Nev-
ertheless study on this system reveals important aspect
1
=Z [dΦ]exph− 4V0 Z dτΦ∗σi(−∂τ2+4V02)Φσi oofftthheesNtaatmebwui-tGhomlduslttiopnleelboonsgo-nras.ngeordersandstructure
ThissystemwasstudiedinSec.III.A, andweobtained
+ dτ(η Φ +η¯ Φ ) . (4.11)
σi σi σi ∗σi the phase diagram by MC simulations. The potential
Z i V(Φ ) of the present system is given as follows from the
σ
ByinsertingI˜(ησi,η¯σi),whichisexpressedintermsofthe effective field theory in Sec.IV.A,
boson fields Φσi(τ) for eiΩσi, into Eq.(4.8), the action of
the effective field theory is obtained as follows, V(Φ )= (V 2za )Φ 2 g(Φ Φ Φ +c.c.)
σ 0− σ | σ| − ∗1 2 ∗3
σ
1 X
A0 =Z dτhσX,hi,ji(aσΦ∗σiΦσj)− 4V0 Xσi (|Φ˙σi|2+4V02|Φσi|2) +Xσ λσ|Φσ|4. (4.14)
+ g(Φ∗1iΦ2iΦ∗3i+c.c.) , (4.12)From the first term of Eq.(4.14), it is obvious that spon-
Xi i taneous symmetry breaking occurs for aσ > V0, but the
secondtermsthatrepresenttheinterplaybetweentheor-
where a = a , a = a = a and g = 3 . To describe
1 s 2 3 h 2V2 der parameters give nontrivial contribution to the phase
0
the Bose condensed state, we add the Φ4-terms in the diagram. In Fig.4, we show typical potential and its
| |
effectivefieldtheoryanddiscussthephasediagramofthe minimum obtained from Eq.(4.14), which derives qual-
7
itatively the same phase structure with that obtained in where
Sec.III.A.
λ =V 2za , (4.19)
It is interesting to see how gapless low-energy excita- 4 0− s
tions,i.e.,Nambu-Goldstone(NG)bosons,appearinthe λ5 =V0 2zah. (4.20)
−
present system. In the phase with the FM spin order
Next, we introduce three phase fluctuation fields that
Φ = v > 0, we set Φ = v+ψ +iχ , and then it is
hobv3iious that the field χ3iidescribesia NGiboson that ap- 0re,pnres=ent0,NG bosons in the FM+2FS phase with n0 6=
pears as a result of the spontaneous symmetry breaking 1 6
of the U(1) pseudo-spin symmetry. Φ (x)=√n +iφ(x),
1 0
Interesting point is that how many NG bosons appear
Φ (x)=√n +ib (x),
2 1 1
inthephasewiththeFM+2SF.Ifg =0inEq.(4.12),itis
Φ (x)=√n +ib (x). (4.21)
obviousthatthereexistthreeNGbosons. Howeverinthe 3 1 2
originalB-t-Jmodel,thesymmetryisU(1) U(1)forthe
By substituting Eq.(4.21) into V(Φ ) in Eq.(4.16), the
× σ
globalphaserotationofaandb-bosonoperators. Inorder
quadratic terms of the fluctuating fields are obtained as
to study the low-energy excitations in the effective field
theory, we shall take the continuum description instead 2λ1n0+λ4 g√n1 g√n1
−
of the lattice one though it is not essential. V =(φ,b ,b ) g√n 2λ n +λ g√n
2 1 2 1 2 1 5 0
 − − 
For the FM B-t-J model on the square lattice, it is g√n1 g√n0 2λ2n1+λ5
−
straightforwardto deriveeffective LagrangianLeff inthe  φ 
continuum spacetime from the effective field theory on
b .
the lattice. For example in Eq.(4.12), we put × 1 
b
2
Φ∗iΦi+µ+ΦiΦ∗i+µ By using Eq.(4.18),  
i,µ
X
=−Xi,µ |∇µΦi|2+Xi 2z|Φi|2 V2 =(φ,b1,b2)−gg√√nn1n01 −gg√√nn01 −gg√√nn10 bφ1 
∂ Φ 2+ 2z Φ 2. g√n1 −g√n0 g√n0 b2
⇒− | µ i| | i|  φ  
i,µ i
X X =(φ,b ,b )gK b . (4.22)
Then Leff in the continuum is derived as follows from 1 2 b1 
Eq.(4.13) 2
 
ItisstraightforwardtodiagonalizethematrixKbyusing
Lsq = a Φ (x)2+a Φ (x)2+a Φ (x)2
eff s|∇ 1 | h|∇ 2 | h|∇ 3 | a unitary matrix U and obtain the mass gap,
(cid:18) (cid:19)
1 1 0 0 0
∂ Φ (x)2+ ∂ Φ (x)2
−(cid:18)4V0| τ 1 | 4V0| τ 2 | U−1KU =00 00 2n00+n1. (4.23)
+ 1 ∂ Φ (x)2 +V(Φ ), (4.15) √n0
4V | τ 3 | σ  
0 (cid:19) Thereforetherearetwogaplessmodesthatcorrespondto
where the potential V(Φ ) is given as theNGbosons,thoughonemayexpectthreeNGbosons
σ
V(Φ )=(V 2za )Φ (x)2+(V 2za )Φ (x)2 as the U(1) pseudo-spinsymmetry is spontaneously bro-
σ 0 s 1 0 h 2
− | | − | | ken and also both the a and b-atoms Bose condense.
+(V 2za )Φ (x)2
0 h 3 From the above derivation of the mass gap, it is obvi-
− | |
−g(Φ∗1(x)Φ2(x)Φ∗3(x)+c.c.) (4.16) ousthatthecubiccouplinggΦ∗1Φ2Φ∗3 playsanessentially
+λ Φ (x)4+λ Φ (x)4+λ Φ (x)4. important role.
1 1 2 2 3 3
| | | | | |
We first substitute
Φ (x)=√n , Φ (x)=√n , Φ (x)=√n C. Phase diagram and low-energy excitations:
1 0 2 1 3 1
Triangular lattice
into V(Φ ),
σ
V(n0,n1)=λ1n20+2λ2n21+λ4n0+2λ5n1 In this subsection we shall focus on the system on the
2g√n n , (4.17) triangularlatticeandstudythephasestructureandlow-
0 1
−
energy excitations. The obtained results will be com-
and derive the equations that determine the value of
pared with those by the numerical study on the quan-
n , n as,
0 1
tumXYmodelgivenintheprevioussectionandthoseof
∂V n
=2λ n +λ g 1 =0, the 3D B-t-J model in the following section. To study
1 0 4
∂n0 − √n0 the system, we assume a homogeneous state and put
∂V ρ = ρ = ρ (ρ = 1 ρ ρ ). The field Φ repre-
1i 2i 3i 1i 2i 1
=4λ n +2λ 2g√n =0, (4.18) − −
∂n 2 1 5− 0 sents pseudo-spin degrees of freedom, and it is plausible
1
8
22
toassumethatitscondensationhasauniformamplitude --22 00
Φ =constant as holes are distributed homogeneously.
| 1| 6
On the other hand, its phase degrees of freedom has a
nontrivial behavior as a result of the frustration. More 4
E(k)
general cases will be studied in a future publication. k2
2
We firststudy the effective actionbya mean-fieldthe-
orylikeapproximationassumingthe√3 √3symmetry. 0
× --22
The triangular lattice is divided into three sublattice la- 00
22
beledA,B andC sublattices. Eachfieldonthesublattice k1
A is denoted as Φ etc. It is not so difficult to search
σA
thegroundstateofthepotential. Forsmalla (i.e.,small
h
t), Φ and Φ do not Bose condensate. On the other
2 3 3
hand,anontriviallong-rangeoderofthepseudo-spinap-
2
pears for a > V . For constant Φ , one can show that
s 0 | 1| 1
the state with the three sublattice coplanar order like
Φ = Φ , Φ = Φ ei2π/3, Φ = Φ e i2π/3 ap- k2 0
1A 1 1B 1 1C 1 −
| | | | | | -1
pears as the lowest-energy state. This state obviously
corresponds to the phase of the 120o spin order in Fig.2. -2
-3
To obtain the expectation value of Φ and to study
1
| | -3 -2 -1 0 1 2 3
low-energy excitations, we relabel the lattice site by k1
dividing it into three sublattices as i (s,o) where
FIG.5. (Color online) Dispersion relation E(k) inEq.(4.33).
→
o = A,B and C. Then we parameterize the field Φ1 Parameters are as=1,V0 =1,γ =5 and β=1.
as
Φ (s)=ρ
1A i To study the low-energy excitations, we introduce a
Φ1B(s)=ρiei23π (4.24) complex scalar field ηi as follows,
Φ1C(s)=ρie−i23π. ρi =ρcl+ηi, (4.28)
Effective Hamiltonian of the spin part Hspin is readily
eff and then from Eq.(4.25), we obtain the quadratic terms
obtained from L in (4.13) (please notice a < 0 in the
eff s of η in the Hamiltonian as follows
i
AF case),
Hesffpin = |as|Φ†1iΦ1i+µ+c.c Hesffpi(n2) ∼ k (cid:20)χ(k)ηk†ηk+β(ηk†η−†k+ηkη−k)(cid:21)(4.29)
i,µ(cid:18) (cid:19) X
X
where
+ V Φ 2+λ Φ 4 , (4.25)
0 1i 3 1i
| | | |
Xi (cid:18) (cid:19) χ(k)= 2a cos k1a cos √3k2a
and substituting Eq.(4.24) into Hspin, − | s| 2 2
eff (cid:18) (cid:19) (cid:18) (cid:19)
a cosk a+γ, (4.30)
Hesffpin = cos23π |as|ρiρi+µ+c.c β =−λ1|ρ2csl|, γ =14λ1ρ2cl+V0. (4.31)
i,µ (cid:18) (cid:19)
X
We diagonalize Eq.(4.29) by the Bogoliubovtransforma-
+ V0ρ2i +λ1ρ4i tion and obtain
i (cid:18) (cid:19)
= |asX| ˜ ρ 2 Heff = E(k)b†kbk, (4.32)
µ i k
2 |∇ | X
i,µ
X
where
+ λ ρ 4 (3a V )ρ 2 , (4.26)
1 i s 0 i
| | − | |− | |
i (cid:18) (cid:19) E(k)= χ(k)2 4β2, (4.33)
X −
where ˜ denotes the difference operator on the trian- q
∇µ anditstypicalbehaviorisshowninFig.5. Asinthelimit
gular lattice. k 0,
From Eq.(4.26), Bose condensation of ρ takes place →
i
for3as >V0, andthe classicalexpectationvalueρcl ofρi E(k) 0, (4.34)
is easily obtained as →
thenbkrepresentsaNGbosoncorrespondingtothespon-
3as V0 taneoussymmetrybreakingoftheU(1)pseudo-spinsym-
ρ = | |− . (4.27)
cl
s 2λ1 metry.
9
Finally let us study the low-energy excitations in the
SF correlations C3=1.0 C1=1.5
phase with the 120o spin order plus 2SF’s that corre- 1
sponds to the phase B in Fig.2. From Eq.(4.13), the ef-
fective Lagrangian Ltri of low-energy excitations in this 0.5
pLhterffais=e is obt−ai|naesd||∇asΦe1ff,i|2+ah|∇Φ2,i|2+ah|∇Φ3,i|2 SF correlations-0 .05 Effective
i (cid:18) (cid:19) MC
X
-1
1 1 1 0 1 2 3 4 5 6
∂ Φ 2+ ∂ Φ 2+ ∂ Φ 2 r
τ 1,i τ 2,i τ 3,i
− 4V | | 4V | | 4V | |
Xi (cid:18) 0 0 0 (cid:19) FIG. 6. (Color online) Boson correlation functions GΦ(r) =
+V(Φσ), (4.35) hΦ∗2,iΦ2,i+ri = hΦ∗3,iΦ3,i+ri in the effective field theory, and
Ga(r)=Gb(r),whichisnumericallyobtainedinthequantum
XY model. Weset n0 =n1 =0.5, g=6, as =1 and ah =1.
V(Φ )= V 2z a Φ 2+ V 2za Φ 2
σ 0 s 1,i 0 h 2,i
− | | | | − | |
i (cid:20)(cid:18) (cid:19) (cid:18) (cid:19)
X Φ =√n +ib ,
3,A 1 2
+ V0−2zah |Φ3,i|2 −g Φ∗1,iΦ2,iΦ∗3,i+c.c. Φ3,B =(√n1+ib2)e−iβ,
(cid:18) (cid:19) (cid:19) (cid:18) (cid:19)
Φ =(√n +ib )eiβ. (4.41)
3,C 1 2
+λ Φ 4+λ Φ 4+λ Φ 4 , (4.36)
1 1,i 2 2,i 2 3,i
| | | | | | Substitutingtheaboveequations(4.39),(4.40)and(4.41)
(cid:19)(cid:21)
where z is again the number of links emanating from a into Lterffi in (4.35), we obtain the mass term of the fluc-
single site. tuating fields as
AnTsoatzobfotaritnhethveargiorouusncdonsdteantesa,twioensa,dwohpitchthreespfoelclotswtinhge gδ√nn10 −gδ√n1 gδ√n1 φ
three-sublattice symmetry, V2′ =(φ,b1,b2)−gδ√n1 gδ√n0 −gδ√n0b1 
gδ√n1 gδ√n0 gδ√n0 b2
Φ1,A =√n0,Φ1,B =√n0ei23π,Φ1,C =√n0e−i23π,  φ −  
Φ2,A =√n1,Φ2,B =√n1eiβ,Φ2,C =√n1e−iβ, =(φ,b1,b2)gδK′ b1 , (4.42)
 
Φ3,A =√n1,Φ3,B =√n1e−iβ,Φ3,C =√n1eiβ. (4.37) b2
 
The values of n , n and β are determined by substitut- wherewehaveusedEq.(4.38). We caneasilydiagonalize
0 1
ing Eq.(4.37) into (4.35) and imposing stationary condi- K′ and get eigenvalues,
tion,
0 0 0
∂Ltri 2π U 1KU = 0 0 0 . (4.43)
∂βeff →12ahn1sinβ−4g√n0n1sin 3 −2β =0, − ′ 0 0 2n0+n1
(cid:18) (cid:19) √n0
∂Ltri n  
eff 2λ n +λ gδ 1 =0, The above result indicates that there are only two NG
1 0 4
∂n0 → − √n0 bosons, though the pseudo-spin U(1) symmetry is spon-
∂Ltri taneously broken and both the a and b-atoms Bose con-
eff 4λ2n1+2λ5 2gδ√n0 =0, (4.38) dense.
∂n → −
1
where δ = 1(1+2cos(2π 2β)). It is verified that non-
3 3 −
trivialsolutionsofn0 andn1existforsufficientlylargeah V. PHASE DIAGRAM OF THE BOSONIC t-J
and a (i.e., negative λ and λ ). Correlation functions MODEL ON STACKED TRIANGULAR LATTICE:
s 4 5
obtainedfromatypicalsolutiontoEq.(4.38)areshowin NUMERICAL STUDY
Fig.6, which have similar behavior to those obtained by
the previous MC simulations of the quantum XY model. InSec.III.BandIV.C,westudiedthelow-energyeffec-
As in the square lattice case, we introduce fields that tive theories of the B-t-J model on the triangular lattice
describe low-energy excitations, at T = 0 and obtained phase diagram and low-energy
excitations. As we argued previously, if fluctuations of
Φ =√n +iφ,
1,A 0
particle density at each site is not so large, the model at
Φ1,B =(√n0+iφ)ei23π, T = 0 reduces to the system with “Lorentz invariance”,
Φ1,C =(√n0+iφ)e−i23π, (4.39) i.e., a linear-time derivative term change to a quadratic-
time derivative term8. In this case, the imaginary time
Φ =√n +ib , τ plays a similar role to the inter-layer dimension of the
2,A 1 1
stacked 3D lattice.
Φ =(√n +ib )eiβ,
2,B 1 1 In the present section, we shall study the B-t-J model
Φ2,C =(√n1+ib1)e−iβ, (4.40) onthestackedtriangularlatticebynumericalsimulations
10
25
20
phase separation
15
10 120°+SF
5 solid
2.5 5 7.5 10 12.5 15 17.5 20
FIG. 7. Phase diagram of thebosonic t-Jmodel in a stacked
triangular lattice. ρa =ρb=0.3, and c′1 =10.0.
FIG. 9. (Color online) Spin and boson correlation functions
obtained by MC simulation of the B-t-J model in a stacked
triangular lattice. c′τ = 15.0 and c′3 = 5.0 in Fig.7. GSS =
N4 PihS~i · S~i+ri, GB(r) = N1 Piha†iai+ri = N1 Pihb†ibi+ri,
where sites i and i+r are located in thesame 2D triangular
lattice. The result indicates the phase with 120o spin order
and 2SF’s.
previous studies in this paper, we employ the canonical
ensemblewiththeparticlenumberofeachatomfixed. To
impose the localconstraint,we employ the slave-particle
representation (2.2). Then the partition function Z is
given as
Z = [DφDϕ1Dϕ2]e−βH3DtJ. (5.2)
Z
The path integral in (5.2) is performed by the MC sim-
ulation with local update keeping each particle number
fixed. Wecallthecalculation(5.2)thequasi-classicalap-
proximation as we ignore the Berry phase in the action
of the path integral. Some detailed discussiononthe va-
FIG. 8. (Color online) Snapshots of pseudo-spin (upper lidity and physical meanings of this approximation was
panel), density of a-atom and b-atom (lower panel). Sx−Sy given in the previous papers14.
istheeasyplaneofthepseudo-spin(seeFig.9),andlengthof Wefirstshowthephasediagramforρ =ρ =0.3and
a b
arrowsindicatesmagnitudeofthepseudo-spin. Astripeoder
theholedensity=0.4inFig.7,wherec =t/(k T),c =
forms and also there are hole-rich regions. ′τ ′ B ′1
J/(k T)andc =t/(k T). Forsmallhoppingamplitude
B ′3 B
t, the system forms a solid with voids whose snapshot is
withaquasi-classicalapproximation. Hamiltonianonthe shown in Fig.8. As c is increased, phase transition to
′3
stacked triangular lattice is given as follows, a state with the 120o spin order and 2SF’s takes place.
In Fig.9, we show the spin and particle correlationfunc-
H3DtJ =− t(a†rar′ +b†rbr′ +h.c.) tionsthatverifythisconclusion. Thepreviousnumerical
hXr,r′i studyofthequantumXYmodelandtheanalyticalstudy
by the effective field theory predict the existence of this
− t′(a†rar+ˆ3+b†rbr+ˆ3+h.c.) phase. This result again indicates a strong resemblance
r
X of the phase diagram of 2D system at T =0 and that of
+J (SxSx +SySy) (5.1)
r r′ r r′ the corresponding model in stacked 3D lattice. As c′3 is
hXr,r′i increased further, phase transition to a phase-separated
state takes place. In this phase, the system is divided
where r,r denotestheNNsiteofthe2Dtriangularlat-
′
tice, anhd ˆ3iis the unit vector in the inter-layer direction. into a-atom rich region and b-atom rich region, and in
each region a SF of single atom forms.
There are (at least) two waysof the MC simulation, i.e.,
the grand-canonical and canonical ensemble. As in the