XXZ spin-1/2 representation of a finite-U Bose-Hubbard chain at half-integer filling

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by  Domenico Giuliano

XXZ spin-1/2 representation of a finite-U Bose-Hubbard chain at half-integer filling
Domenico Giuliano,1 Davide Rossini,2 Pasquale Sodano,3 and Andrea Trombettoni4
1Dipartimento di Fisica, Universita` della Calabria, Arcavacata di Rende I-87036, Cosenza, Italy &
INFN, Gruppo collegato di Cosenza, Arcavacata di Rende I-87036, Cosenza, Italy
2NEST, Scuola Normale Superiore & Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
3International Institute of Physics, Universidade Federal do Rio Grande do Norte, 59012-970, Natal, Brazil &
INFN, Sezione di Perugia, Via A. Pascoli, I-06123, Perugia, Italy
4CNR-IOM DEMOCRITOS Simulation Center and SISSA, Via Bonomea 265 I-34136 Trieste, Italy &
INFN, Sezione di Trieste, I-34127 Trieste, Italy
Using a similarity Hamiltonian renormalization procedure, we determine an effective spin-1/2
2
1 representation of the Bose-Hubbard model at half-integer filling and at a finite on-site interaction
0 energy U. By means of bosonization, we are able to recast the effective Hamiltonian as that of a
2 spin-1/2 XXZ magnetic chain with pertinently renormalized coupling and anisotropy parameters.
WeusethismappingtoprovideanalyticalestimatesofthecorrelationfunctionsoftheBose-Hubbard
t
c model.WethencomparesuchresultswiththosebasedonDMRGnumericalsimulationsoftheBose-
O HubbardmodelforvariousvaluesofU andforanumberLoflatticesitesaslowasL∼30.Wefind
an excellent agreement up to 10% between the output of analytical and numerical computations,

9 evenforrelativelysmallvaluesofU.Ouranalysisimpliesthat,alsoatfiniteU,the1DBose-Hubbard
model with suitably chosen parameters may be seen as a quantumsimulator of the XXZchain.
]
l
e PACSnumbers:75.10.Pq,75.10.Jm,67.85.-d
-
r
t
s I. INTRODUCTION phases (corresponding respectively to the checkerboard
.
t andthesupercounterfluidphases10)maybedetectedand
a
m studied.
Thestudyofmagneticsystemsisoneofthemoreactive
Akeytoolinthemanipulationofultracoldatomicsys-
- fields of researchin condensed matter physics1: the vari-
d temsisthepossibilitytosuperimposeandcontroloptical
ety of emerging ground-states, as well as the rich phase
n lattices11. The low-energy properties of ultracold bosons
diagram of magnetic lattices, makes these systems an
o in deep optical lattices are well captured by the Bose-
c optimal testbed where it is possible to study the com- Hubbard (BH) Hamiltonian12
[ petition between various orders and frustration effects2.

From this perspective, it would be very useful to be able
1 U
toengineersyntheticphysicalsystemseffectivelydescrib- H = t(b†b +b†b )+Vn n + n (n 1).
v BH − i j j i i j 2 i i−
5 ingmagneticmodelHamiltonians,withtunablegeometry hXi,jih i Xi
8 and parameters. (1)
6 A promising route is provided by cold atoms setups: InEq.(1), i, j denote the lattice sites,while i,j stands
2 forinstance,itinerantmagnetisminbulkultracoldFermi for any pair of nearestneighbouring sites. Thhe opierators
0. systems with repulsive interactions has been experimen- b†i (bi), with [bi,b†j] = δi,j and ni = b†ibi, create (annihi-
1 tallystudied3,whilesmallspinnetworkshavebeensimu- late) a boson in the site i. The parameter t denotes the
2
lated with ion chains4. Effective nearest-neighbour spin- hopping strength,and U (V) is the interactionenergyof
1
spin interactions for atoms in neighbour wells of an op- two particles atthe same site (at two nearestneighbour-
:
v tical lattice may result from super-exchange couplings: ing sites).
i
X the corresponding second-order tunneling has been ob- The use of optical lattices in ultracold atomic sys-
served in array of double wells5. Furthermore, using fast tems is also central in other proposals to simulate spin
r
a oscillationsoftheopticallattice,onecancontrolthesign Hamiltonians: in Ref. 13 it was suggested to use spinor
of the nearest-neighbour tunneling6, which has been re- Bosegasesinopticallatticestoimplementthequadratic-
cently used to simulate classical frustrated magnetism biquadratic spin Hamiltonian. The mapping of a Mott
in triangular lattices7. Finally, in ultracold atomic sys- insulator of one-component ultracold bosonic gases in a
tems one may use two-component gases: here, the two tilted1DopticallatticeontoaspinHamiltonianwasdis-
internal degrees of freedom correspond to the simulated cussed in Ref. 14. Based on this proposal, the quantum
(pseudo)spins.In opticallattices,in the regimeof strong simulationofan antiferromagneticspin chain(more pre-
interactions, by adjusting the external potential one can cisely,anIsingmodelinatransversefield15)hasbeenex-
control spin interactions between spin states of neigh- perimentally demonstrated16: varying the effective mag-
bouring atoms8. The recent experimental realization of netic field, a quantum phase transitionoccurs between a
controllableBose-Bosemixtures9 then paves the way to- paramagneticphaseandanantiferromagneticone.These
wardsthe experimentalsimulationof spinHamiltonians, phases(andthecorrespondingphasetransition)werede-
in whichthe atomic counterpartofmagnetic phases,like tectedbymeasuringtheprobabilitytohaveanoddoccu-
the antiferromagnetic N´eel and the XY ferromagnetic pation of sites, while the formation of magnetic domains
2
wasobservedusingin-situsite-resolvedimagingandnoise area of research32–41 and exact analytical results for the
correlation measurements16. correlationfunctions at small distance (both at zero and
Apart from the simulation of quantum magnetism, finite temperature) are by now available34. The asymp-
more generally the possibility of realizing BH models17 totic form of the ground-state correlation functions in
through the use of ultracold atoms in optical lattices12 the thermodynamic limit is power-law with an exponent
attracted a huge interest in the last decade for a num- thathasbeenobtainedbycomparingtheresultofabelian
ber of other reasons: the experimental detection of the bosonization with the Bethe ansatz solution42: for an
Mott-superfluidtransition18 motivatedathoroughstudy openchainintheregion 1 ∆ 1,thenumericalfind-
− ≤ ≤
ofthe quantumphasesofthe BHmodel.Fromthispoint ings for correlationfunctions obtainedwith DMRG were
of view, arraysof ultracold atoms are versatile synthetic compared with the results of a low-energy field theory,
systems for the study of strongly interacting lattice sys- showing a very good agreement and allowing for precise
tems19,20 in which it is possible to investigate a huge va- estimatesoftheamplitudesofthecorrelationfunctions43.
riety of phenomena, ranging from quantum phase tran- In turn, the obtained amplitudes were found in agree-
sitions11 to superfluid dynamics in lattices21 and non- ment with the analytical expressions given by Lukyanov
equilibrium dynamics of quantum systems22. and Zamolodchikov44,45. Finally, exact results obtained
For very large values of U, i.e. for t/U 1, the BH for the XXZ chain in a special scaling limit were used
≪
modelmaybemappedintotheHeisenbergXXZspin-1/2 to compute the local correlations of a continuous Lieb-
Hamiltonian Liniger 1D Bose-gas46.
H = J sxsx+sysy ∆szsz , (2) In this paper we determine a correspondence between
XXZ − i j i j − i j the BH chain at half-integer filling for finite U and a 1D
hXi,ji(cid:0) (cid:1) XXZ spin-1/2 model. This enables us to provide analyt-
where ~s = ~σ /2 = (sx,sy,sz) are the S = 1/2 spin ical expressions for the BH correlation functions, which
i i i i i
wecomparewithnumericalresultsobtainedwithDMRG,
operators, ~σ being the Pauli matrices, J is the nearest-
i
showingthatthereisaverygoodagreementbothatlarge
neighbour coupling, and ∆ is the anisotropy parameter
and small distances and also for U/J as low as 2 and
(∆ = 1 respectively correspond to the antiferromag-
∼
± for a number of sites L 30. As a consequence, the nu-
neticandtheferromagneticisotropicHeisenbergmodel).
≥
mericaldeterminationofthephasetransitionssuperfluid-
The use of lattice spin systems for interacting bosons
charge density wave and superfluid-Mott insulator (re-
traces back to the classical papers by Matsubara and
Matsuda in the 50’s23. In those papers the properties of spectively corresponding, in the effective XXZ chain, to
∆ = 1 and ∆ = 1) well agrees with the analytical
helium II are studied assuming that each atom in liquid eff eff
−
resultsforthe XXZ chain.Besidesprovidingawaytoan
heliumcanoccupyoneofthelatticepoints:thisassump-
analyticalstudy of the quantum phase transitions of the
tion naturally leads to a (BH-like) lattice Hamiltonian.
BH model, our analysis may reveal very useful in deter-
The further assumption that two atoms cannot simulta-
miningthe structurefactoraswellasthe responseofthe
neously occupy the same lattice site (due to the hard-
BH chain to external perturbations. As a result, we are
corepartofthe interparticleinteractionbetweenHelium
atoms24 or, in BH language, by a an infinite U) leads to ablenotonlytoprovideanalyticalexpressionsfor1DBH
a XXZ spin-1/2 model in a magnetic field23. To quali- correlation functions, but also to show that by realizing
the 1D BH chain at half-integer filling one canprovide a
tatively understand the emergence of a spin representa-
quantum simulation of the XXZ chain.
tion of the one-component BH model one may say that,
when U and if two states per site give a dominant In the following we shall derive an effective spin-1/2
→∞
contribution to the energy, an XXZ Hamiltonian is re- Hamiltonian for the BH chain at half-integer filling as
trieved: this is exactly what happens when the filling f, a power series of t/U. Following Refs. 47,48, we per-
defined as the average number of bosons per lattice site, formacontinuousunitarytransformationSwhichblock-
is half-integer. Indeed, when f = n¯ +1/2, with n¯ inte- diagonalizestheBHHamiltonianinthebasisoftheeigen-
ger,therelevantstatesintheFockspacearegivenby n¯ vectorsofH witht=0.Thelatterrequirementsyields
BH
and n¯ +1 (deviations from half-integer fillings wou|ldi an equation for S which we solve perturbatively to the
| i
result in a magnetic term in the XXZ Hamiltonian). For order (t/U)2. We finally show that, using bosonization,
half-integer f, at the leading order in t/U 0 one has this Hamiltonian can be recasted in the XXZ form with
→
J = 2t(f + 1/2) and ∆ = V/J (see the discussion in pertinent coupling and anisotropy parameters. We ob-
Section III). serve that, while to the first order in t/U one finds a
The XXZ model is a paradigmatic spin Hamilto- XXZ model with J = 2t(f +1/2) and ∆ = V/J, to the
nian which has been the object of many investiga- next order in t/U one gets an effective spin Hamiltonian
tions (see e.g. Ref. 25) and that in 1D is exactly solv- whichisnotoftheXXZform,sinceitcontainsalsonext-
able by Bethe ansatz26–28; this provides an ideal arena nearest neighbours and 3-spin terms (this is the bosonic
to test different analytical and numerical techniques, counterpart of a similar computation done for the 2D
frombosonization29,30 todensitymatrixrenormalization Fermi-Hubbard model49,50, where 4-spin terms appear).
group (DMRG)31. The study of (static and dynamical) However, in 1D it is possible to further proceed using
correlationfunctions in this model is currently an active bosonization: we do this for the BH chain, where one
3
canobtain analyticalresults introducing a Luttinger liq- liesonthefactthatthismodelexhibitsaquantumphase
uid description of the effective Hamiltonian enabling to transition between a superfluid phase (for t/U 1) and
≫
incorporate the long-wavelength behaviour of non-XXZ a Mott insulator (for t/U 1)17. A finite V gener-
≪
terms in the effective coupling and anisotropy parame- ally favours charge density wave phases: e.g., for half-
ters, J and∆ ,whicharenow functionof t, V, f and integer filling f = 1/2, a large V t,U will result in a
eff eff
≫
U. ground-state of the type 1,0,1,0, (where in general
| ···i
Theplanofthe paperisthe following:inSectionII we n ,n ,n , is an eigenfunction of H with t = 0).
1 2 3 BH
| ···i
introduce the BH and the XXZ models, recalling some The ground-state of the BH model has been studied in
usefulpropertiesandresults.InSectionIIIweemploythe the seminal paper in Ref. 17 using the grand-canonical
continuous unitary transformationintroduced by Glazek ensemble, where the chemical potential µ is introduced
andWilson47toapproximatetheBHchainathalf-integer toenforcethe constraintonthe numberofparticles.The
filling with an effective spin-1/2 effective Hamiltonian. phasediagramintheU µplaneshowsthecharacteristic
−
In Section IV we use bosonizationto recastthis effective lobes17:forapertinently fixedvalue ofµ the half-integer
HamiltonianasanXXZ Hamiltonian,with effective cou- fillings may be made correspond to the “basis” of the
pling J and anisotropy ∆ . In Section V we establish lobes(i.e.wherethelobestouch)and,forV =0,onehas
eff eff
the correspondence between the correlation functions of a superfluid for each finite value of t, while a finite and
theBHmodelandtheonesoftheXXZchain.Theresults positive value of V gives rise to a charge density wave
of Section III-V are then used to compare the analytical region among the Mott lobes.
results obtained for the BH chain correlation functions The Mott-insulator/superfluid transition was first ob-
with the numerical findings obtained by DMRG numeri- served in 3D18 and subsequently in 1D54 and 2D55. The
calsimulations:theresultsofthiscomparisonaredetailed effect of a superimposed external potential (typically a
in Section VI, both for the correlationfunctions and the parabolic one) has been also considered: the so-called
phase transition points. Section VII is devoted to our wedding-cake-like density has been studied both theo-
conclusions, while more technical details are contained retically56,57 and experimentally58,59. The phase coher-
in the Appendices. encepropertiesofultracoldbosonsinopticallatticeshave
been studied, as well, showing that phase coherence on
short length scales still persists deep in the insulating
II. MODEL HAMILTONIANS phase60. The BH model in a 1D geometry can be ob-
tained either by tightly confining the bosonic cloud in
In this Section we review the basic properties of the two radial directions in presence of a periodic potential
BHandofthe spin-1/2XXZ Hamiltonians,inparticular in the transverse direction, or by creating many (even-
focusingonknownanalyticalresultsabouttherealspace tually uncoupled) tubes with a 2D optical lattice. The
spin correlations in the XXZ chain. Bragg spectroscopy of interacting one-dimensional Bose
gases loaded in an optical lattice across the superfluid
to Mott-insulator phase transition has been performed
A. Bose-Hubbard model and the properties of the strongly correlated phases in-
vestigated looking at the Bragg spectra61. The excita-
The low-energy properties of interacting bosons in a tionspectruminthestronglyinteractingregimehasbeen
one-dimensional deep optical lattice are in general well studiedalsoinpresenceofatunable disorder,createdby
described by the Bose-Hubbard Hamiltonian (1), which, abichromaticopticallattice,showingabroadeningofthe
in 1D and with open boundaries, reads: Mott-insulator resonances62.
The finite-V 1D BH model has been studied with a
L−1 L
U number of analytical and numerical techniques: in par-
H = t b†b +b† b + n (n 1)
BH − i i+1 i+1 i 2 i i− ticular inRef. 63 the phase boundaries ofthe Mottinsu-
Xi=1 (cid:16) (cid:17) Xi=1 lators and charge density wave phases were determined
L−1
byDMRG.Thezero-temperaturephasediagrambothof
+ V n n . (3)
i i+1 theBHmodelandofaspin-SHeisenbergmodelwascon-
Xi=1 structed and their relation investigated64. The role of V
We denote with N the total number of particles in the in inducing supersolid phases in the 1D BH model was
L-site chain, so that the filling f, that is, the average alsostudied65–68.Bosonizationtechniquesquitegenerally
number of particles per site, is given by f = N. For al- provide in 1D a very effective way to compute the corre-
L
kali atoms usually V U, but with dipolar gases (or lation functions and their decay at large distance: thus,
polar molecules) V cou≪ld be comparable with U: exper- they have been applied as well to 1D BH chains (see the
iments with dipolar gases51 and long-lived ground-state review 69).
polar molecules52 in optical lattices have been already Finally, we mention that the effect of intersite inter-
performed (see also the review 53). actions was considered since the 90’s in a related model,
A large amount of experiments investigated the prop- the quantum phase model, describing Josephson junc-
erties of the BH model: the main reason for this interest tionarrays70:the quantumphase modelcanbe obtained
4
from the BH model for large filling per site when the The global minus sign in the couplings has been intro-
numberfluctuationsarenegligibleinthe kineticterm.In duced in order to more easily perform the comparison
the correspondence between the two models the chemi- with the BH model, and it can be readily gauged away
calpotential µ termin the BH modelcorrespondsto the by implementing the canonical mapping to the spin-1/2
so-called “offset charge” q, which are external charges operatorsτa definedasτx,y =( 1)jsx,y,τz =sz.There-
present in the superconducting network70: the lobes in forethe chajinis antiferrojmagne−tic(fejrromjagnetjic)for ∆
the quantum phase model are equal, since there is an positive (negative).
invariance for q q + 2e (2e being the charge of the Following Ref. 23, one can derive the Hamiltonian in
→
Cooper pairs), and an half-integer value of the filling f Eq. (4) from the BH Hamiltonian (3) at half-integer fill-
corresponds to half-integer values of the offset charges ingf andforU .To doso,letus define sz n f
→∞ j ≡ j−
q/2e. The study of intersite interactions is relevant in (sothattheeigenvaluesofsz are 1).Sincefort=0the
Josephson junction arrays since the interaction term de- j ±2
energyperparticleis(forL )ε=Uf(f 1)/2+Vf2,
pends on the capacitance matrix C , which is in gen- →∞ −
ij it follows that H t→0 Vszsz , i.e.
eralnotdiagonal,resulting in terms of the formVijninj, BH −→ j j+1
where V (C )−1: as a mean-field analysis shows71,
ij ∝ ij J∆ V. (5)
for a diagonal capacitance matrix one has that at T =0 ≡
the superconducting phase is obtained for each value of
Similarly, for f 1, one gets J 2tf as one can see by
the Josephson energy EJ (∝t in the mapping) and that putting bi √f≫eiφi and mappin≈g the obtained result in
at q = e one has a finite critical temperature for the the XXZ s∼pin-1/2 language72: for finite values of f one
Mott-insulator/superfluidtransition for eachfinite value
gets (see Section III)
of E (unlike q = 0, where a critical value of E is re-
J J
quired). Non-diagonal terms of the capacitance matrix 1
favour charge density waves70: the role of the intersite J 2t f + . (6)
≡ 2
termswasconsideredforsuperconductingchainsandthe (cid:18) (cid:19)
correspondingphase diagraminvestigated72,73, revealing Eqs. (5) and (6) provide the desired mapping between
thatin1Da(superconducting)repulsiveLuttingerliquid the BH modelandthe XXZ Hamiltonianto lowestorder
phase exists. in t/U. However, as we are going to see in Section V,
To conclude this Section let us mention that, in the to geta quantitativeagreementbetween the BH andthe
rest of the paper, we will mostly deal with half-integer XXZ correlation functions even for t/U relatively small
fillings, f n¯ + 1, with n¯ = 0,1,2, . The reason for (aslowas0.1forf =1/2)onehastogotothenextorder
≡ 2 ···
such a choice is that in this case the relevant states for in t/U: the corresponding Hamiltonian is determined in
the description of system for U are just n¯ and Section III and recasted in XXZ form via a Luttinger
→ ∞ | i
n¯ + 1 . Simple arguments - reviewed in Section IIB - representation in Section IV. We remark that, since our
| i
then show that, to first order in t/U, the BH Hamilto- resultareobtainedathalf-integerfilling,wemayomitthe
nianismappedintoanXXZspin-1/2Hamiltonianwhich addition of a magnetic field term of the form L sz
∝ i=1 i
is integrable in 1D. Within the XXZ-model framework, to Eq. (4). Indeed such a term is proportional to the
it is also possible to consider small deviations from the totalspinSz = L sz inthe z directionand,sPince the
T i=1 i
half-filled regime, which mainly give rise to a uniform systemis half-filled,onlyeigenstatesofH withSz =
XXZ T
magneticfieldinthez-direction.Eventhoughwewillnot 0 arephysicallyPmeaningful - notice that in the following
considerlargefluctuationsinf (oforder1),itis possible analyticalresults basedonthe XXZ Hamiltonian (4) are
to take them into account,by keeping, as relevant states compared with numerical DMRG simulations of the BH
for U , n¯ , n¯ 1 , n¯+1 .Inthis case,aneffective chaininthecanonicalensemble,where n isconserved
→∞ | i | − i | i i i
spin-1 XXZ effective model (in general not integrable) and equal to N.
is expected74.Spin-1 models exhibit agapped(Haldane) TheHamiltonianH isexactlysolPvablebymeansof
XXZ
insulator phase75,76, which has been investigated in the standard Bethe ansatz techniques26–28: however, explic-
context of the 1D BH model77–81. itly computing the real space spin-spin correlation func-
tions is quite a difficult task.Exact analyticalresults for
short-range correlators in a range of up to seven lattice
sites were reported for the isotropic Heisenberg model
B. XXZ chain in Ref. 40, in the thermodynamic limit (L ) and
→ ∞
at arbitrary finite temperature, and for finite chains of
For a chain with L sites and open boundaries, the arbitrarylengthLinthe ground-state.Resultsforshort-
Hamiltonian of a spin-1/2 XXZ model given in Eq. (2) rangecorrelationfunctionsarealsoavailablefortheXXZ
particularizes to: chain34.Forlargedistances,usingthestandardbosoniza-
tion approach29,30 to spin-1/2 XXZ model82, one may
findoutallthespin-spincorrelationfunctionsintermsof
L−1
H = J sxsx +sysy ∆szsz . (4) two-pointcorrelatorsofpertinentconformaloperators43:
XXZ − i i+1 i i+1− i i+1 in the thermodynamic limit one finds the asymptotic
i=1
X(cid:0) (cid:1)
5
∞
forms dt cosh(2ηt)e−2t 1
˜ = −
x
ψ szsz ψ = ( 1)i−j Az 1 , I Z0 t 2sinh(ηt)sinh(t)cosh[(1−η)t]
hψ0|sxisxj|ψ0i = (−1)i−j|i−Axj|1/η − 4πA2˜ηx(i−j,)2 (7) + sinh1(ηt) − η2η+1e−2t!,
h 0| i j| 0i − i j η − i j η+1/η
| − | | − |
where ψ is the ground-state of H and we set42
0 XXZ
| i
1 ∞
η =1− π arccos∆. (8) z = dt sinh[(2η−1)t] 2η−1e−2t
I t sinh(ηt)cosh[(1 η)t] − η !
Z −
0
Analytical expressions for the correlation amplitudes
A , A˜ and A entering Eqs. (7)-(7) were presented in
x x z
Refs. 44,45 and further discussed in Ref. 83 (see also the and
discussion in Section V of Ref. 41):
η Γ η
Ax = 8(1A−η)2 e−Ix; (9) A= 2√π(cid:16)Γ2(1−η1)(cid:17) , (12)
η+1/η 2(1−η)
A˜ = A e−I˜x; (10) (cid:16) (cid:17)
x
2η(1 η)
−
2 1/η and Γ(x) being the Euler’s Gamma function.
A = A eIz (11)
z π2 Analytical expressions (in the large-L limit) for the
subsequentprefactorsofthe correlationfunctions arere-
with
ported in Refs. 36,41.
∞ For chains of finite size L with open boundary condi-
dt sinh(ηt)
= ηe−2t tions, one obtains43:
x
I t sinh(t)cosh[(1 η)t] − !
Z −
0
( 1)i−ja2 f1(i+j) f1(i j) 1 1 1
ψ szsz ψ = − η η − +
h 0| i j| 0i 2f21η(2i)f21η(2j) fη1(i−j) − fη1(i+j)!− 4π2η (cid:18)f2(i−j) f2(i+j)(cid:19)
a ( 1)i ( 1)j
− [g(i j)+g(i+j)] − [g(i j) g(i+j)] (13)
−2πη(f 1 (2i) − − f 1 (2j) − − )
2η 2η
and
fη(2i)fη(2j) c2 b2 f1(i+j) f1(i j)
ψ sxsx ψ = 2 2 ( 1)i−j η + η −
h 0| i j| 0i fη(i j)fη(i+j) ( − 2 − 4f 1 (2i)f 1 (2j)"f1(i j) f1(i+j)#
− 2η 2η η − η
bc ( 1)i ( 1)j
sgn(i j) − − , (14)
− 2 − "f 1 (2j) − f 1 (2i)#)
2η 2η
where sgn(x) is the sign function and (here and in the following all the distances are in units
of the lattice constant).
α
2(L+1) π x
f (x) = sin | | , (15) Theagreementbetweenexactnumericalcalculationsof
α
π 2(L+1)
(cid:20) (cid:18) (cid:19)(cid:21) theXXZcorrelationfunctionsandanalyticalexpressions
π πx in (13)-(14) is very good, and it becomes excellent with
g(x) = 2(L+1)cot 2(L+1) (16) L 100 for 0.8 < ∆ < 0.843. Thus one may readily
(cid:18) (cid:19) ass∼umethatE−qs.(1∼3)-(14∼)providequite anaccuratean-
with
alyticalexpressionforthe spin-spincorrelationfunctions
c2 b2 a2 in the XXZ model84. As a consequence, constructing a
Ax, A˜x, Az (17) rigorousmappingbetweentheBHandtheXXZspin-1/2
2 ≡ 4 ≡ 2 ≡
6
Hamiltonian and expressing correlation functions in one states at each site is in one-to-one correspondence with
model in terms of the ones of the other gives an efficient the Hilbert space of states of a quantum spin-1/2degree
andstraightforwardwaytoprovideaccurateanalyticex- of freedom; we shall see that, at half-integer filling, even
pressions for real space correlation functions in the BH for finite U the BH model may be replaced by an effec-
model at half-integer filling. tive spin-1/2 Hamiltonian, with pertinently determined
Wefinallyobservethattheonlysystem-dependentpa- parameters. The method amounts to an iterative block-
rameter determining the spin-spin correlation functions diagonalization of the BH Hamiltonian on the space of
is the coefficient η: thus, in tracing out the mapping be- eigenfunctions of HBH with t=0.
tweenthetwomodels,thisisthekeyquantitytobecalcu- To illustrate the procedure, we start from the explicit
lated as a function of the BH parameters. In particular, construction of the “low-energy” Hilbert space of physi-
one may distinguish between the regions in parameter cally relevant states, in the large-U limit. Neglecting ex-
space with η > 1/2 and η < 1/2: while the former one citations with energy U amounts to truncating the
∼
correspondstoanantiferromagneticspinchain,thelatter Hilbert space to a subspace , defined as
F
one (which may be realizedfor pertinently chosenvalues
=Span n¯+µ ,...,n¯+µ , (18)
of the parameters of H , as we shall show below) cor- 1 L
BH F {| i}
responds to a ferromagnetic chain.
with µ taking the values µ = 0,1 and L µ = L.
i i i=1 i 2
In Eq. (18) n , ,n labels the state in the Hilbert
1 L
| ··· i P
space with n particles on site i. To implement the GW
i
III. EFFECTIVE SPIN-1/2 HAMILTONIAN FOR
approach,onesplitsthe Hamiltonian(3)asH =H +
THE BOSE-HUBBARD MODEL AT BH 0
H , with
HALF-INTEGER FILLING I
U
H = n (n 1)+V n n (19)
As reviewed in the previous Section, for U , the 0 2 i i− i i+1
BH Hamiltonian maps onto the XXZ model i→n E∞q. (4), Xi Xi
with the parameters J,∆ given in Eqs. (5, 6). This may H = t b†b +b† b . (20)
I − i i+1 i+1 i
be seen as a first-order term in an expansion (in powers Xi (cid:16) (cid:17)
of t/U) aimed at computing the effective Hamiltonian:
From Eqs. (19)-(20) one sees that H is diagonal with
in this Section we compute this effective Hamiltonian to 0
respect to the partition of the Hilbert space into plus
the next order. As we shall show in the following, this is
F
its orthogonalcomplement, since
enough to fit quite well the numerical data for the cor-
relation functions of the BH model using the analytical
H n ,...,n =E [n ,...,n ]n ,...,n (21)
results obtained for the correlators of the XXZ chain. 0| 1 Li 0 1 L | 1 Li
To approachthe large-U limit one may either proceed withE [n ,...,n ]=(U/2) n (n 1)+V n n ,
byperformingastrongcouplingexpansiontothesecond- while H0 1exhibitsLoff-diagonali(wiithi−respect toithei pi+ar1-
I
, or higher-order of perturbation theory, or by deriv- tition of the Hilbert space)Pmatrix elementsPwhich are
ing effective Hamiltonians using alternative techniques, (tn¯). In order to block-diagonalize H , one needs to
BH
based on canonical transformations or continuous uni- pOerform a similarity transformation47
tary transformations85. Strong coupling techniques were
extensively applied to the BH model at integer filling: H ˜ =S†H S, (22)
BH BH BH
→H
as a result, one may, for instance, evaluate the energy of
the Mott insulator and of the superfluid state in higher- with S unitary. Upon setting S=I+T, the unitarity of
orderperturbationtheoryanddeterminethephasetran- S implies the optical theorem
sitionpointandthe phase diagramin the U µplane86.
− T+T†+T†T=0. (23)
Since we are rather interested to the BH at half-integer
filling, i.e., in the region of the phase diagram where Setting T h+a, with
the lobes touch and the superfluid phase persists also ≡
at very small U (with V = 0), we found it convenient 1 1
h= T+T† , a= T T† (24)
to use an approach based on continuous unitary trans- 2 2 −
formations47,48. We follow the notation and the method
(cid:0) (cid:1) (cid:0) (cid:1)
presented in the paper by Glazek and Wilson (GW)47: one finds that Eq. (23) yields
systematicallyusing the GW renormalizationprocedure,
1
we work out an effective description of the dynamics of h= (a2 h2). (25)
2 −
the BH model, restricted to the low-energy subspace de-
termined by the constraint on the total number of par- Eq. (25) shows that h is always “higher order” than a.
ticles and by the large-U assumption. As a result, the FollowingRef.47,itismostconvenienttodefinethenew
low-energysubspaceisspannedbystateswitheithern¯ or interaction Hamiltonian ˜ as
I
n¯+1particlespersite,withthetotalnumberofparticles H
being fixed to N. Thus, the space of physically relevant ˜ = ˜ H (26)
I BH 0
H H −
7
so that the new “free” Hamiltonian is the same as the The first term in the right hand side of Eq. (33) yields
old one (H ). a spin-1/2 Hamiltonian which is actually the spin-1/2
0
TofurtherproceedanddetermineS,onehastorequire XXZ chain introduced in Section IIB and having the
that the matrix elements of ˜ between states with en- anisotropy and the coupling given by Eqs. (5)-(6):
I
ergydifference>U areequalHtozero,amountingtostate
that ˜ isblock∼-diagonalwithrespecttothepartitionof
I
H V
theHilbertspaceinto plusitsorthogonalcomplement, H(0) H = J sxsx +sysy szsz
i.e. F XXZ ≡P BHP − i i+1 i i+1− J i i+1
i (cid:18) (cid:19)
X
(34)
˜I +(I ) ˜I(I )= ˜I, (27) withJ =2t f + 1 (constanttermshavebeenomitted).
PH P −P H −P H 2
The effective spin-1/2 operators are defined as
where is the projector onto and I the projec- (cid:0) (cid:1)
P F −P
tor onto its complementary subspace87. One sees that
Eq. (27) implies that 1
sx = b +b† ,
˜I(I )=(I ) ˜I =0. (28) i 2 f + 12P(cid:16) i i(cid:17)P
PH −P −P H P
qi
Using Eqs. (22)-(24)-(26), one may write H˜I as syi = 2 f + 1P −bi+b†i P,
2 (cid:16) (cid:17)
H˜I =(I+h−a)(H0+HI)(I+h+a)−H0 (29) sz = qb†b f ; (35)
i P i i− P
and Eq. (28) then becomes (cid:16) (cid:17)
H + H ,h +[H ,a]+T†H +H T the boson number eigenstates at site i correspond to
I 0 0 I I
P{ { } (30) the eigenstates of sz according to n¯ , and
+T†H T (I )=0. i | ii ↔ | ↓ii
I } −P n¯ +1 i i. Therefore, the result in Eq. (34) cor-
| i ↔ | ↑i
responds to the “naive” large-U limit for the BH model
Eq. (30), together with the identity
at half-integer filling discussed in Section IIB, in which
a= a(I )+(I )a (31) off-diagonal matrix elements of relevant operators (in-
P −P −P P cluding the Hamiltonian itself) are set to zero from the
and with Eq. (25), is all what one needs in principle to very beginning.
fullydetermineaandh(and,therefore,theoperatorT). Corrections to H(0) arising from virtual transitions
However, except for some simple cases48, an explicit XXZ
involvingstates outside of may be properly accounted
solution for T cannot be exhibited. For this reason we F
for within GW procedure, allowing to get the effective
proceed by writing the solution for T iteratively, in a
spin-1/2Hamiltoniantothenextorderint/U.Summing
series in H : in particular, we use Eq. (30) to determine
I over all virtual transitions outside of induced by H ,
a to first order (a1) in HI. We provide the details in one finds F I
Appendix A and the result for a1 in Eq. (A4). Using
Eq.(A4)andsettingT a1,wefindthatEq.(22)reads
≈
as H(1) ≡P[HI,a1]P =−t2 P b†jbj+1+b†j+1bj ×
S†HBHS=HBH+[H0,a1]+[HI,a1]. (32) Xj,ℓ (cid:16) (cid:17)
(I )(H )−1(I ) b†b +b† b . (36)
TheGWproceduremaybereadilyiteratedtodetermine, × −P BH −P ℓ ℓ+1 ℓ+1 ℓ P
inprinciple,TtoanydesiredorderinH .However,since (cid:16) (cid:17)
I
keeping only second-order contributions in H provides
I In particular,when computing H(1), one has to consider
alreadyquiteanexcellentestimateforthereal-spacecor-
intermediate states with either one of the µ in Eq. (18)
relation functions of operators in the BH model (as ex- j
being equal to 2, or to 1 (all these states have en-
plicitly shown by the numerical calculations we report −
ergy U, with respect to states in the subspace ), or
in Section VI), setting T a1 already provides quite a ∼ F
good approximation to the≈exact T. states with one of the µj equal to 2 (−1), and the other
equal to 1 (2) (all these states have energy 2U, with
Since the approach we are implementing is pertur- − ∼
respect to states in the subspace ). Thus, one even-
bative in HI, one should enforce Eq. (30), as well tually finds out that H(1) can beFwritten as the sum
as Eq. (31), to each order in H ; moreover, since
[H0,a1] = 0, one may neglect thIe term [H0,a1] in of two terms: H(1) = Hd(1ia)g + Ho(ff1)d, with Hd(1ia)g being
EPq. (32) aPnd approximate the effective Hamiltonian act- the part of H(1) having 1- and 2-nearest-neighbour spin
ing within as terms,whileH(1) contains2-next-nearest-neighbourand
F offd
3-spin terms. Omitting constant terms, their expression
Heff =P{HBH+[HI,a1]}P ≡HX(0X)Z+H(1). (33) are given by:
8
4(n¯+1)t2 t2
H(1) = sz 3n¯2+6n¯+4 szsz (37)
diag − U i − U i i+1
i i
X (cid:0) (cid:1)X
t2(n¯+1)2 2t2(n¯+1)
H(1) = s+ s− +s− s+ s+ s− +s− s+ sz. (38)
offd − U i+1 i−1 i+1 i−1 − U i+1 i−1 i+1 i−1 i
i i
X(cid:0) (cid:1) X(cid:0) (cid:1)
As we shall in the next Section, using a Luttinger liquid fined by ǫ(k )=0, are given by
F
representation, H(1) may be recasted in the XXZ form
with coupling and anisotropy coefficients depending on U(n¯+1) U(n¯+1) 2
U. coskF = +n¯+2. (42)
2J −s 2J
(cid:18) (cid:19)
Uponlinearizingthedispersionrelationaround k and
F
IV. EFFECTIVE XXZ PARAMETERS VIA A setting k =k +p, one gets ±
F
LUTTINGER LIQUID REPRESENTATION
2J
ǫ( k +p) Jsink 1 cosk p. (43)
F F F
The effective spin Hamiltonian in Eq. (33) is not in ± ≈± − U(n¯+1)
(cid:20) (cid:21)
the XXZ form: in this Section we show how the con-
tribution coming from H(1) may be accounted for by a FromEq.(43)oneseesthat,sincecoskF 6=0,H2 implies
a nonzero effective magnetic field B 88, as well as a re-
pertinent redefinition of the parameters of the spin-1/2 eff
definitionoftheFermivelocityv .Thisyieldsaredefined
XXZ-Hamiltonian H(0) . F
XXZ coupling given by B /J = cosk . Since
ThefirstcontributiontoH(1) intheright-handsideof eff eff − F
diag
Eq. (37) describes aneffective magnetic field in the z di- 2J
B = Jcosk 1 cosk ,
rection88, while the second term simply shifts the value eff − F − U F
(cid:18) (cid:19)
of the XXZ anisotropy. At variance, the term H(1) in
offd one obtains
Eq. (38) contains 3-spin, as well as non-nearest neigh-
bour, couplings. To show how these terms can be ac- 2J
J =J 1 cosk . (44)
counted for via a redefinition of H(0) , it is most con- eff − U F
XXZ (cid:18) (cid:19)
venient to rewrite H(1) in terms of the Jordan-Wigner
offd The quartic term H4 canbe dealt with by noticing that,
(JW) fermions aj,a†j, as outlined in Appendix B. As a in the low-energy,long-wavelengthlimit, one can write
result, one writes H(1) as a sum of a bilinear (H ), plus
offd 2
a quartic (H4) term, that is a†j−1aj+1+a†j+1aj−1 −→− ρR(xj)+ρL(xj)
(cid:26)
H(1) H +H ,
offd ≡ 2 4 −(−1)j[ψR†(xj)ψL(xj)+ψL†(xj)ψR(xj)] , (45)
with (cid:27)
where the chiral fermion fields ψ (x ), ψ (x ) are de-
t2(n¯+1) R j L j
H2 = U a†i−1ai+1+a†i+1ai−1 (39) fiAnpepdenfrdoixmBt)he long-wavelength expansion of aj as (see
Xi (cid:16) (cid:17)
H4 = 2t2(n¯U+1)2 :a†iai: a†i−1ai+1+a†i+1ai−1 (4.0) aj ≈eikFxjψR(xj)+e−ikFxjψL(xj), (46)
Xi (cid:16) (cid:17) with x = aj, and the chiral fermion densities given by
j
Since H2 is bilinear in the JW fermions, it merely mod- ρR(xj)=ψR†(xj)ψR(xj)andρL(xj)=ψL†(xj)ψL(xj).As
ifies the single-fermion dispersion relation, yielding the a result, H may be written as
4
quadratic Hamiltonian in the JW fermions reading
L
4t2(n¯+1)2
HX(0X)Z+H2 = k (cid:26)−2Jcosk+ t2(n¯U+1)cos(2k) H4 = − U Z0 dxn(ρR(x))2
X
B a†a , (41) +(ρL(x))2+4ρR(x)ρL(x) . (47)
− k k
(cid:27) o
ComparingEq.(47)toEq.(B13),oneseesthatH takes
4
with B = 4(n¯ + 1)t2/U. Setting ǫ(k) = 2Jcosk + the same form as the term J szsz in the spin-1/2
t2(n¯+1)cos(2k) B, one finds that the Ferm−i points, de- XXZ Hamiltonian in Eq. (4). j j j+1
U − P
9
0.75 From Fig. 1 one also sees that ∆ may be tuned by
eff
1.5 n = 0 η varying the ratio J/U: in particular ∆eff can be differ-
n = 10 ent from 0 even if V = 0 (as it is typical for alkali
n = +∞ 0.5 atoms). Fig. 1 also suggests the possibility of describing
1 thewholephasediagramoftheXXZspin-1/2chainusing
BH model for a single species of bosons with pertinently
f 0.25
ef 0 0.2 J/U 0.4 chosen parameters, see also Section VI89.
∆ 0.5
Finally we notice that, since the sign of ∆ may be
eff
changed by a pertinent choice of J/U and V, the Lut-
0 tinger liquid effectively describing the XXZ-Hamiltonian
maybe repulsiveor attractive.As noticedin the context
of 1D Josephson junction arrays72,73, the transition be-
-0.5
tweenthe repulsiveandtheattractivesidemaybe moni-
toredbyinsertingaweaklink(i.e.,anonmagneticimpu-
rity82):itwouldbetheninterestingtoanalyzetheeffects
-1
0 0.1 0.2 0.3 0.4
of a weak link introduced in a bosonic system described
J/U
by the BH Hamiltonian.
FIG. 1: ∆eff vs. J/U for different values of V/J and n¯. The
top (bottom) dotted line corresponds to thevalueof ∆eff for
U/J →∞ and V/J =0.5 (V/J =0). The other lines are for V. CORRELATION FUNCTIONS
V/J = 0.5 (top) and V/J = 0 (bottom), with n¯ = 0 (solid
black lines), n¯ = 10 (dashed red lines) and n¯ → ∞ (dot-
The mapping between H and Heff derived in Sec-
dashedbluebluelines).Inset:sameasinthemainpanel,but BH XXZ
tion IV enables to select the ground-states on which to
for η vs. J/U.
compute the pertinent vacuum expectation values. In-
deed if Φ is the ground-state of the BH Hamiltonian
0
Collectingtogetheralltheaboveresultsallowstowrite given in|Eqi. (3), and Ψ0 S† Φ0 is the ground-state
an effective XXZ Hamiltonian, describing the BH model of Heff =S†HBHS, the| GWi ≡appr|oacih requires
to the order (t/U)2, as:
Φ b,b† Φ = Ψ S† b,b† SΨ
0 BH 0 0 BH 0
h |O i h | O | i
HXeffXZ =−Jeff sxjsxj+1+syjsyj+1−∆effszjszj+1 , (cid:2)(cid:8) (cid:9)(cid:3) ≡ hΨ0|OXXZ[{(cid:2)s(cid:8)a}]|Ψ(cid:9)0(cid:3)i, (52)
j
X(cid:0) (cid:1)
(48) where b,b† ( [ sa ]) denotes a generic BH
BH XXZ
with Jeff defined in Eq. (44) and (XXZ)Ooperator. Of coOurse, E{q.}(52) is exact only if S is
(cid:2)(cid:8) (cid:9)(cid:3)
the exact solution of the GW equation (31): by comput-
∆¯
∆ = (49) ing it perturbatively at a given order, one recovers the
eff 1− 2UJ cos(kF) correspondencebetweenground-stateexpectationvalues
of BH and spin-1/2 operators at the chosen order.
with
In the restofthe paper, we will be interestedin corre-
V t2(3n¯2+6n¯+4) 4t2(n¯+1)2 lation functions of the following BH operators:
∆¯ = . (50)
J − JU − JU
z (n f)(n f), (53)
Since J acts just as an effective over-all scale of Mi,j ≡ i− j −
eff
HXeffXZ, then ∆eff is the only parameter determining the M⊥i,j ≡ b†ibj. (54)
behaviorofspin-spincorrelationsintheXXZmodel.Sub-
stituting Eq. (49) in Eq. (8) one gets Using the results of Appendix C one has S† z S =
Mi,j
η =1 1 arccos∆ , (51) Mzi,j 1+O tU2n¯22 , so that
eff
− π h (cid:16) (cid:17)i
t2n¯2
which provides an explicit formula for the effective Lut- Φ (n f)(n f) Φ = Ψ szsz Ψ + .
tingerparameterforthe BHmodelathalf-integerfilling. h 0| i− j − | 0i h 0| i j| 0i O U2
(cid:18) (cid:19)
In Fig. 1 we plot both ∆ and η versus J/U, for dif- (55)
eff
ferent values of V/J and n¯. One sees that n¯ = 10 and More generally, if the operator BH satis-
n¯ are almost indistinguishable, and that the limit fies (I ) BH = BH(I ) O= 0, then
of→the∞quantum phase model for Josephson junction ar- Φ0 BH−ΦP0 O PΨ0 XXPZOΨ0 , w−ithP XXZ obtained
h |O | i ≈ h |O | i O
rays(n¯ 1) at offset chargeq =e is practically reached from by substituting b , b† and n f respectively
at n¯ 1≫0. Furthermore,one sees that the dependence of with sO−B,Hs+ and sz. At variaincie, for i⊥− one obtains a
∼ i i j Mi,j
η upon n¯ is rather small. more involved expression (see Appendix C for details):
10
t(n¯+2)(n¯+1)
Φ b†b Φ (n¯+1) Ψ s−s+ Ψ + Ψ [s− +s− ]s++ s+ +s+ s− Ψ
h 0| i j| 0i ≈ h 0| i j | 0i 2U h 0| i+1 i−1 j j+1 j−1 i | 0i
tn¯(n¯+1) (cid:2) 1 (cid:3) 1
+ Ψ s−[s+ +s+ ]+s+ s− +s− Ψ +δ Ψ sz +sz Ψ ,
2U h 0| i j−1 j+1 j i−1 i+1 | 0i |i−j|,1h 0| 2 − i+1 2 i | 0i
(cid:18) (cid:19)(cid:18) (cid:19)
(cid:2) (cid:3)
where again we neglected contributions arising to tions szsz in the ground-state of the XXZ chain with
t2n¯2 . effectihveianjiisotropy∆eff givenby(49)andthea,b,ccon-
O U2 stants numerically determined from DMRG simulations
(cid:16) (cid:17)
of the XXZ chain, while the green diamonds (line) cor-
respondto a, b,c analyticallydeterminedfromEqs.(17)
VI. RESULTS
and (9)-(10)-(11). One sees that, up to numerical accu-
racy < 10−5, the results obtained analytically for the
In this Section we compare the numerical results ob- XXZ∼effective model are in excellent agreement with re-
tained through DMRG for the correlation functions and sults of the density-density BH model even at small dis-
the phase diagram of the BH model with the analytical tance.Bluetriangles(line)displaytheXXZHamiltonian
predictions for the correlators from the effective Hamil- results in the U limit, with anisotropy ∆=V/J –
tonian Heff given by Eq. (48). one sees that the→re∞lative error is noticeably larger.
XXZ
In Fig. 3 we plot the off-diagonal correlations b†b
h i ji
forthesamesetofvaluesofthe BHparameters(namely,
A. Correlation functions U = 10t, V = 0.5t, f = 0.5, L = 150). The meaning of
dots and lines is the same as in Fig. 2: also here one sees
Let us focus on the BH correlation functions. Since thattheresultsobtainedfromtheGWeffectiveHamilto-
DMRGsimulationsaremadeonafinitenumberofsitesL nian HXeffXZ are in much better agreement than the ones
andfor openboundary conditions,we mayuse Eqs.(13) obtained using HXXZ with ∆ = V/J, this happens even
and (14) yielding the zz and xy correlation functions though J/U is as low as 0.2.
of the XXZ model. We evaluate the values of the non- To quantify the agreement between BH and XXZ re-
universal constants a, b, c defined in Eq. (17) both nu- sults, we consider the absolute value of the relative error
merically,by DMRG simulations ofthe XXZ model, and done in evaluating a correlator (r) as the ground-state
C
using the analytical expressions presented in Refs. 44,45 averageof the corresponding operators in the BH model
andreportedinSectionIIB.AsconfirmedinRef.43,the [ BH(r)], and in the XXZ model [ XXZ(r)]. More pre-
C C
values ofa, b, c obtainedinthe twowaysareinexcellent
agreement. We show that the analytical expressions for
the XXZ correlations are well confirmed by the numer-
ical BH correlations also for small L (e.g., for L = 30)
andforJ/U relativelylarge(aslargeas∼0.5).Itshould 〉) |10-2
be stressed that, at variance, the agreement is not very f

ngoiaondHby(0)setotbintgain∆eedfffo=rUV/J, i.e.nebgyleucstiinnggctohnetrHibaumtiilotnos- (n -j infinite-U limit
arisingXfrXoZm the GW proce→du∞re. f)10-4
-
The correlators hΦ0|(ni−f)(nj −f)|Φ0i and n i
Φ b†b Φ are evaluated from the corresponding XXZ (
h 0| i j| 0i 〈
quantities using respectively Eqs. (55) and (56). They |
10-6
are plottedas a function ofr = i j ,with i andj such
that43 i = (L r+1)/2, j = (|L−+r| +1)/2 for odd r,
andi=(L r)−/2, j =(L+r)/2 forevenr (for instance, 0 50 r 100 150
−
for L = 100 sites, r = 1 corresponds to i = 50, j = 51;
r = 2 corresponds to i = 49, j = 51; r = 3 corresponds
FIG. 2: Density-density correlations |h(n −f)(n −f)i| vs.
i j
to i=49, j =52, and so on).
r=|i−j|forU =10t,V =0.5tandf =0.5,withnumberof
In Fig. 2 we plot our results for the density-density sitesL=150.Blackdotsandline:BHresults;redsquaresand
correlations (n f)(n f) for a typical set of val- line: XXZ result with a, b, c numerically determined; green
i j
h − − i
ues,i.e. for U =10t,V =0.5t, f =0.5,correspondingto diamonds and line: XXZ result with a, b, c analytically de-
J/U = 0.2. Black dots (joint by a black line as a guide termined; bluetriangles and dotted line: U →∞ XXZ result
for eye) are the density-density correlations evaluated in (indicatedbythelabel“infinite-U limit”–seetextfurfurther
details).
theBHmodel,redsquares(line)arethecorrelationfunc-