Universal quantum behaviors of interacting fermions in 1D traps: from few particles to the trap thermodynamic limit

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by  Adriano Angelone

Universal quantum behaviors of interacting fermions in 1D traps:
from few particles to the trap thermodynamic limit
Adriano Angelone, Massimo Campostrini and Ettore Vicari
Dip. di Fisica dell’Universit`a di Pisa and INFN, Largo Pontecorvo 3, I-56127 Pisa, Italy
(Dated: February 18, 2014)
Weinvestigatetheground-statepropertiesoftrappedfermion systemsdescribedbytheHubbard
modelwithanexternalconfiningpotential. Wediscusstheuniversalbehaviorsofsystemsindifferent
4
regimes: from few particles, i.e. in diluteregime, to thetrap thermodynamic limit.
1
Theasymptotictrap-size(TS)dependenceinthediluteregime(increasingthetrapsizeℓkeeping
0
the particle number N fixed) is described by a universal TS scaling controlled by the dilute fixed
2
pointassociatedwiththemetal-to-vacuumquantumtransition. Thisscalingbehaviorisnumerically
b checked by DMRG simulations of the one-dimensional (1D) Hubbard model. In particular, the
e particledensity anditscorrelations show crossovers among differentregimes: for strongly repulsive
F
interactionstheyapproach thoseofaspinlessFermigas, forweak interactionsthoseofafreeFermi

5 gas,andforstronglyattractiveinteractionstheymatchthoseofagasofhard-corebosonicmolecules.
1 Thelarge-N limitkeepingtheratioN/ℓfixedcorrespondstoa1Dtrapthermodynamiclimit. We
addressissues related to theaccuracy of thelocal density approximation (LDA).Weshowthat the

] particledensityapproachesitsLDAinthelarge-ℓlimit. Whenthetrappedsystemisinthemetallic
s phase,correctionsatfiniteℓareO(ℓ−1)andoscillating aroundthecenterofthetrap. Theybecome
a
g significantly larger at the boundary of the fermion cloud, where they get suppressed as O(ℓ−1/3)
- only. This anomalous behaviorarises from thenontrivialscaling at themetal-to-vacuum transition
t
n occurring at theboundaries of thefermion cloud.
a
u PACSnumbers: 71.10.Fd,05.30.Fk,67.85.-d,05.10.Cc
q
.
t
a I. INTRODUCTION homogeneoussystem [5, 6]. Therefore, in experiments of
m trapped particle systems, a thorough understanding of
the quantum many-body dynamics calls for a quantita-
-
The progress in the experimental activity in atomic
d tiveanalysisofthetrapeffects. Thisissuehasbeenmuch
n physics, quantum optics and nanoscience has provided a
discussedbytheoreticalandexperimentalinvestigations,
o great opportunity to investigate the nature of the quan-
see e.g. Refs. 5–55.
c tum dynamics, and the interplay between quantum and
[ statistical behaviors in particle systems. In particular, Inthispaperweconsiderspin-1/2fermionsystemsde-
2 experiments with cold atoms realize systems which are scribed by the lattice Hubbard model, at zero tempera-
v accurately described by microscopic theoretical models ture, therefore in their ground state. We investigate the
4 such as dilute atomic Fermi and Bose gases, or Hubbard scaling behavior of unpolarized systems, i.e. with equal
1 and Bose-Hubbard models in optical lattices. See e.g. numberofspin-downandspin-upparticles,whenvarying
5
Refs. [1–4]. A peculiar feature of these experiments is the size of the trap confining the particles. We mostly
3
the confinement of the atoms, i.e. the presence of a in- considerone-dimensional(1D)systemswithbothattrac-
.
1 homogeneous space-dependent (usually harmonic) trap- tiveandrepulsiveinteractions,inthediluteregimewhen
0 ping potential. The tunability of the confining potential the trap size ℓ increases keeping the particle number N
4
allows the realization of quasi-two and quasi-one dimen- fixed, and in the trap thermodynamic limit defined as
1
sional systems, by tightly confining the particles along the large-N limit keeping the ratio N/ℓ fixed. Note that
:
v one or two transverse dimensions. 1D fermion systems are also experimentally interesting,
Xi The inhomogeneity induced by the trapping potential indeed their realizations, and evidences of pairing, have
been recently reported in Refs. [56, 57].
r gives rise to peculiar effects, such as the possibility of
a
simultaneously observing different phases, in particular The asymptotic trap-size dependence in the dilute
Mott incompressible (insulator) and superfluid (metal- regime for a system of N particles can be described in
lic) phases,depending onthe distancefromthe centerof theframeworkoftheTSStheory,whosescalingbehavior
the trap.[1–3]This is essentiallydue tothe factthatthe iscontrolledbythedilute fixedpointassociatedwiththe
effective local chemical potential decreases with increas- metal-to-vacuum quantum transition. This is controlled
ing the distance from the center of the trap center. At by the trap exponent θ related to the renormalization-
continuous quantum transitions the presence of the trap group (RG) perturbation arising from the trapping po-
does not allow the development of critical modes with tential, and requires a nontrivial rescaling of the on-site
diverging length scales. However, in the limit of large interactionin1Dsystems. TheTSSpredictionsarecom-
trap size ℓ the system develops peculiar critical modes, pared with numerical results based on DMRG simula-
whichgiverisetoauniversaltrap-sizescaling(TSS),con- tions. In particular, we address issues related to the
trolled by the universality class of the transition of the expected crossover from strongly repulsive interactions,
2
where the system assumes the properties of a free spin- II. THE HUBBARD MODEL
less Fermi gas, to strongly attractive interactions, where
the system is expected to be effectively constituted by The Hamiltonian of the Hubbard model reads
hard-core bosonic (spin zero) molecules of two fermions,
passing through a free Fermi gas in the absence of inter- H = t (c† c +h.c.)+U n n (1)
− σx σy ↑x ↓x
actions. σX,hxyi Xx
We also investigate the trap thermodynamic limit of where x are the sites of a cubic lattice, xy indicates
the Hubbard model, i.e. the large-ℓ and large-N limit nearest-neighborsites,c isafermionicopheraitor,σ =
σx
keeping the ratio N/ℓd fixed. In particular, we address labels the spin states, and n c† c . The par↑ti↓-
issuesrelatedtotheaccuracyofthelocal-densityapprox- cle number operators Nˆ = σx n≡ σaxreσxconserved, i.e.
imation(LDA),whichapproximatesthespace-dependent σ x σx
[H,Nˆ ]=0. InthefollowingweconsiderbalancedFermi
particle density of a inhomogeneous trap by the particle σ P
systems, thus
density of the homogenous system at the corresponding
value of the effective chemical potential. LDA is usually
N =N =N/2 (2)
↑ ↓
used to determine the particle density in inhomogeneous
systems, providing an accurate approximation when the where N is the total number of particles. In this sym-
inhomogeneity is sufficiently smooth. Of course, LDA is metric case n = n and c† c = c† c .
not exact, in particular at finite trap sizes. Therefore, The presehnc↑exiof ah ↓trxaippinghp↑xote↑nytiialhca↓nx ↓byei taken
ananalysis of the deviations fromLDA, and thereforeof into account by adding a external-potential term in the
its accuracy, is required to get a robust confidence of its Hamiltonian of the Hubbard model,
results.
H =H +H , (3)
t e
LDA has been largely employed in studies of inho-
mogeneous interacting fermion systems, see Refs. [1– He = V(x)nx, nx nσx. (4)
≡
4] and references therein, an in particular of 1D sys- x σ
X X
tems [7, 9, 10, 13, 14, 16, 17, 19–27, 41–43, 48]. The
Forsimplicity,weassumearotationalinvariantpotential
comparison with numerical results for the particle den-
sityshowsthatitprovidesagoodapproximationinmany 1
V(x)= vprp, r x, (5)
cases. A more quantitative analysis of the deviations p ≡| |
fromLDAmaybeachievedbyestablishingwhetherLDA
where r is the distance from the center of the trap, and
becomes exact in the large trap-size limit, and how de-
p is a positive number. We set the origin x = 0 at the
viations get suppressed at large, but finite, ℓ. We in-
center of the trap. The trap size ℓ is defined as
vestigate this issue in the trapped 1D Hubbard model,
where LDA of the particle density can be exactly com-
(2t)1/p rp
puted by Bethe-Ansatz methods [58, 59], allowing us to ℓ , V(x)=2t (6)
≡ v pℓp
perform an accurate study of the deviations from LDA.
We show that LDA of the particle density tends to be-
The trapping potential is effectively harmonic in most
come exact in the large trap-size limit, with power-law
experiments, i.e. p = 2. [1] In the limit p we
suppressedcorrections. Inparticular,the correctionsap- → ∞
recover a homogenous spherical system of size L = 2ℓ
pear significantly larger at the boundary of the fermion
with hard-wall boundary conditions.
cloud, decreasing as O(ℓ−1/3), due to the critical modes
The above definition of trap size ℓ naturally arises
arisingfromthe metal-to-vacuumtransitionatthe edges
whenweconsiderathermodynamiclimitinatrap[1,60].
of the trap. Indeed, in the presence of trap, a trap thermodynamic
Thepaperisorganizedasfollows. InSec.IIwepresent limitcanbeconsistentlydefinedasthelarge-ℓlimitkeep-
the Hubbard model in the presence of an external po- ingtheratioN/ℓdfixed. Thisisequivalenttointroducing
tential coupled to the particle density; we provide the a chemical potential µ, by adding the term
definitions of the observables and correlations which are
H = µ n (7)
considered in the paper. In Sec. III we investigate the µ x
−
trap-size dependence at a fixed particle number, i.e. in Xx
thediluteregime,intheframeworkoftheTSStheory. In
to the Hubbard Hamiltonian (1).
Sec. IV we consider the thermodynamic limit in a trap, In the rest of the paper we set the kinetic constant
i.e. the large trap-size limit keeping the ratio N/ℓ fixed; t=1 and ℏ=1; their dependence can be easily inferred
in particular,we addressthe accuracyof the LDA of the by dimensional analyses.
particle density and the peculiar scaling behavior at the In our study we focus on the ground-state properties.
boundary of the cloud. Finally, in Sec. V we summarize We consider one-point observables such as the particle
ourmainresultsanddrawourconclusions. Someappen- density
dices report technical details of some results mentioned
in the paper. ρ(x)= n (8)
x
h i
3
and the double occupancy 6
do(x)= n↑xn↓x . (9) n=2 Mott
h i 4
insulator
Moreover, we analyze the behavior of correlation func- n=1 Mott
µ metallic
tions such as the one-particle correlation insulator
2
C(x,y)= c† c +h.c. , (10)
h σx σy i 0
σ
X superfluid/metallic
the connected density-density correlations
−2
G(x,y)= n n n n , (11) vacuum
x y x y
h i−h ih i
M(x,y)= n n n n , (12) −4 −2 0 2 4 6 8
↑x ↓y ↑x ↓y
h i−h ih i U
and the pair correlation
FIG. 1: (Color online) Phase diagram of the 1D unpolarized
P(x,y)= c† c† c c +h.c. . (13) Hubbard model, which shows the different quantum phases
h ↑x ↓x ↓y ↑y i relatedtothebehavioroftheparticledensity: vacuum,metal,
In the following we mostly consider 1D systems. The n=1 and n=2 Mott insulator phases.
homogeneous 1D Hubbard model has been extensively
studied,obtainingseveralexactresults,seee.g. Ref.[59].
III. TRAP-SIZE SCALING IN THE DILUTE
The phase diagram of homogenous 1D unpolarized sys-
REGIME
tems presents various quantum phases related to the be-
havior of the particle density, such as vacuum, metallic
In this section we study the asymptotic trap-size de-
and Mott insulator (incompressible) phases, depending
pendence in the dilute regime, when increasing the trap
on the chemical potential µ and the on-site interaction
sizewhilekeepingtheparticlenumberN fixed. Thisissue
U. Their phase boundaries are known exactly, by com-
isbestaddressedinthe frameworkoftheTSStheory[6].
putations based on the Bethe Ansatz, see e.g. Ref. [59].
Its universal features are determined by the RG dimen-
In particular, the vacuum-to-metal transition occurs at
sionsoftherelevantperturbationsatthefixedpointcon-
µ = 2 for U 0, (14) trolling the vacuum-to-metal transition of the Hubbard
0
− ≥ model.
U2
µ = 2 1+ for U <0. (15) The scaling behavior of the Hubbard model in the di-
0
− r 16 lute regime can be inferred by a RG analysis of the cor-
responding quantum field theory, see e.g. Ref. [62],
Fig. 1 shows the phase diagram for the 1D unpolarized
Hubbard model, with the different phases characterized
1/T
by different behaviors of the particle density. In the Z = Dψ∗Dψ exp dτ ddx ,(16)
metallic phaseregionthe low-energypropertiesarechar- F Z σ σ −Z0 Z LF!
acterized by algebraically decaying correlations [59, 61]. ∂ψ 1
At small attractive interactions fermions form Cooper- LF = [ψσ∗ dτσ + 2m|∇ψσ|2−µ|ψσ|2]+
pair-like bound states resembling those of BCS super- σ
X
conductors, while in the limit of strong interactions the + uψ↑∗ψ↓∗ψ↓ψ↑.
pairs become tightly bounded within anextensionof the
The dynamic criticalexponentz andthe RG dimensions
latticespacing,givingeffectivelyrisetoasystemofhard-
y andy oftherelevantparameterµanduatthedilute
core bosons. µ u
fixed point (µ = 0 and u = 0) encode most important
We finally mention that the 1D Hubbard model, even
informationonthescalingpropertiesinthediluteregime.
in the presence of an external confining potential, can
Since the dilute fixed point is essentially Gaussian, they
be exactly mapped into a model of two interacting
can be obtained by simple dimensional analyses,
species(flavors)ofhard-corebosonsdescribedbyaBose-
Hubbard model, whose Hamiltonian is formally analo-
z =2, y =2, y =2 d. (17)
µ u
gous to that of the Hubbard model with the fermionic −
operators c replaced by hard-core bosonic operators In order to obtain the scaling behavior of the observ-
σx
b . Some details are reported in App. A. The mapping ables, such as the particle density and the correlations
σx
betweenfermionicandbosonicoperatorsisnonlocal,but introducedinSec. II, we alsoneedthe RGdimensions of
it directly maps the density operator of fermions into thefermionicfieldψ,densityoperatorn=ψ†ψ,andpair
that of bosons, i.e. c† c b† b . Thus the par- operator p=ψ ψ , which are respectively y =d/2 and
σx σx → σx σx ↑ ↓ ψ
ticle density and its correlations for the fermionic Hub- y =y =d.
n p
bard model are identical to those of the two-flavorBose- Thetrapeffectsinthediluteregimecanbeinferredby
Hubbard model. analyzing the RG perturbation arising from an external
4
confining potential V(x) such as that of Eq. (5), i.e. wherec isaspinlessfermionicoperator,n =c†c ,and
x x x x
the potential V(x) is given in Eq. (5). In the following
P (x)=V(x) ψ (x)2, V(x)=vp xp, (18) we report some of the results of Refs. [40, 65] which are
V σ
| | | | useful for the rest of the paper.
σ
X
In arbitrary dimensions the asymptotic trap-size de-
at the dilute fixed point. Proceeding analogously to the pendenceoftheparticledensityofafreegasofN spinless
case of spinless free-fermion systems [39, 40], the RG di- fermions (N = Nˆ and Nˆ = n ) is given by [40]
mension y of the potential coupling v can be obtained h i x x
v
from the RG relation ρ(x,ℓ,N) ℓ−dθPS (X,N), (25)
p
≈
py p=d+z y =y . (19) where θ is the same trap exponent (21), and X x/ℓθ
v− − n µ As shown in Refs. [6, 40], the TSS limit correspo≡nds to
ThisRGanalysistellsusthatthetrapinducesalength a continuum limit in the presence of the trap. Thus the
scale TSSfunctions providingthe asymptotic trap-sizedepen-
dence can be exactly derived from the ground state of
ξ ℓθ (20) a trapped spinless Fermi gas defined in the continuum.
∼
ThisisgivenbyaSlaterdeterminantofthelowestN one-
with particleeigenfunctionsϕ (x)oftheSchr¨odingerproblem
k
1 p p 1 1
θ = = . (21) Hϕ =ε ϕ , H = p2+ xp, (26)
≡ yv p+yµ p+2 k k k 2 p| |
This implies that the spatial coordinates x must be withunitmassandtrapsize. This allowsus towrite the
rescaled as X = x/ℓθ to get a nontrivial TSS limit. In TSS function Sp in Eq. (25) as
particular, θ = 1/2 for the harmonic p = 2 potential, in
N
any spatial dimension.
S (X,N)= ϕ (X)2. (27)
TheknowledgeoftheabovecriticalexponentsandRG p k
k=1
dimensions allows us to write down the universal TSS X
ansatz for the observables introduced in Sec. II in the In particular, in the case of 1D systems in a harmonic
dilute regime, which provides the asymptotic behavior trap, it can be written as [10, 65]
of their trap-size dependence. For example, for the n-
point correlation of a generic local operator (x), we √N
expect [6, 39] O S2(x,N)= √2 ϕ′N+1(x)ϕN(x)−ϕ′N(x)ϕN+1(x) (28)
(cid:2) (cid:3)
F(x ,...,x ;ℓ,U,N) (x )... (x ) (22) where
1 n 1 n
≡ hO O i
≈ ℓ−εF(X1,...,Xn;Uℓθyu,N) ϕ (x)= Hk−1(x) e−x2/2,
k π1/42(k−1)/2(k 1)!1/2
where −
H are the Hermite polynomials, and the corresponding
k
ε=nθy , X =x /ℓθ, (23) eigenvalues are ε k 1/2. In the case of a 1D hard-
o i i k
∝ −
wall trap, corresponding to p ,
→∞
and y is the RG dimension of the operator (x) at the
o
O N 1 sin[π(N +1/2)(1+x)]
dilute fixed point. Corrections to the above asymptotic
S (x,N)= + (29)
∞
behavior are generally suppressed by further negative 2 4 − 4sin[π(1+x)/2]
powers of ℓ. TSS has some analogies with the standard
with x 1.
FSSforhomogeneoussystems[63,64]withtwomaindif- | |≤
The one particle correlationbehaves as
ferences: theinhomogeneityduetothespacedependence
of the external field, and the size L replaced by ℓθ. C(x ,x ,ℓ,N) ℓ−dθE (X ,X ,N), (30)
1 2 p 1 2
≈
where X =x /ℓθ and
i i
A. TSS in a lattice gas of free spinless fermions
N
E (X ,X ,N)= ϕ (X )ϕ (X ). (31)
p 1 2 k 1 k 2
The trap-size dependence of the simplest noninteract-
k=1
ingcaseU =0canbeeasilydeterminedusingtheresults X
obtained for trapped lattice gases of spinless fermions, The connected density correlation scales asymptotically
defined by as
H =−hXxyi(c†xcy+h.c.)+Xx V(x)nx, (24) YGp((xX1,1x,X2,2ℓ,,NN))=≈−ℓ−E2pdθ(XYp1(,XX12,,XN2),2N), ((3332))
5
for x x > 0, which can be derived from the free B. TSS in the dilute regime of the Hubbard model
1 2
| − |
fermion relation
G(x ,x )= C(x ,x )2+δ(x x )C(x ,x ). (34) 1. TSS for d>2
1 2 1 2 1 2 1 2
−| | −
Corrections to the above large-ℓ scaling behavior are
The RG analysis leading to Eqs. (17) shows that the
O(ℓ−2θ) relatively to the leading term [6].
U term is irrelevant at the dilute fixed point for d >
TheaboveTSSbehaviorsareuniversalwithrespectto
2, because its RG dimension y is negative. Therefore,
u
alargeclassofshort-rangeinteractions. Forexample,the
theasymptotictrap-sizedependence inthedilute regime
sameasymptoticbehaviorisalsoexpectedinthepresence
turns out to be the same as that of a free Fermi gases of
of a density-density nearest-neighbour interaction [39]
N particles with N = N = N/2, independently of U,
↑ ↓
at least for U >U∗ with U∗ <0 [62].
H =w n n . (35)
nn x y
TheasymptoticTSScanbeeasilydeterminedfromthe
hxyi
X results of Sec. IIIA, obtaining
Indeed the RG dimension of the coupling w is [62] y =
w
d, thus the interaction Hnn only induces O(ℓ−dθ) cor- ρ(x,ℓ,U,N)=ℓ−dθ 2Sp(X,N/2)+O(ℓ−κ) , (43)
−
rections to the asymptotic behaviors. do(x,ℓ,U,N)=ℓ−2d(cid:2)θ Sp(X,N/2)2+O(ℓ−κ(cid:3)) ,
TheTSSfunctions oftheparticledensityandthe one-
particle and density correlations, cf. Eqs. (25), (30) and C(x1,x2,ℓ,U,N)=ℓ−(cid:2)dθ 2Ep(X1,X2,N/2)+(cid:3)O(ℓ−κ) ,
(32), show also peculiar large-N scaling behaviors. In- G(x1,x2,ℓ,U,N)=ℓ−2dθ(cid:2) 2Yp(X1,X2,N/2)+O(ℓ−κ)(cid:3) ,
deed, [65]
M(x1,x2,ℓ,U,N)=O(ℓ−(cid:2)2dθ−κ), (cid:3)
Sp(x,N) Nθ p(N(θ−1)/dX), (36) P(x ,x ,ℓ,U,N)=ℓ−2dθ E (X ,X ,N/2)2+O(ℓ−κ) ,
≈ S 1 2 p 1 2
and
where the exponent of the le(cid:2)ading power law are deter- (cid:3)
Ep(X1,X2,N) Nθ p(Nθ/dX1,Nθ/dX2), (37) mined by the RGdimensions of the operatorsassociated
≈ E
Y (X ,X ,N) N2θ (Nθ/dX ,Nθ/dX ). (38) with the observables or the correlations, see Eq. (22).
p 1 2 p 1 2
≈ Y The presence of the on-site interaction induces scaling
Note the different scaling of the space variable between corrections with
the density and the one-particle and connected density
correlations. κ=(d 2)θ. (44)
−
Inthecaseof1Dsystemsinaharmonictrap,thelarge-
N scalingbehaviorcanbederivedfromEq.(28),obtain- Ford=3theydominatethescalingcorrectionsexpected
ing within the lattice model of free spinless fermion, i.e. the
Hubbard model with U = 0, which are relatively sup-
S2(X,N)≈N1/2S2(X/N1/2), (39) pressed as O(ℓ−2θ). [40]
(z)= 1 2 z2 for z z =√2, (40) The on-site interaction becomes marginal in 2D, thus
2 b
S π − ≤ a residualweakdependence onU is expected in the TSS
and (z)=0forz pz . Thecorrectionstothislarge-N limit, with at most logarithmic rescalings of the onsite
2 b
behaSvior are known≥[40]; they are O(N−1) (relatively to interaction.
the leading term) for z <z . In particular,
b
√2 ( 1)N+1 2. TSS in 1D systems
S (0,N)= N1/2 1+ − +... (41)
2
π 4N
(cid:20) (cid:21)
The relevance of the U term in 1D gives rise to non-
We finally mention that around the boundary of the
trap, i.e. at the spatial point z = z = √2 where the trivial TSS limits, requiring an appropriate rescaling of
b
± the parameter U. Indeed, we expect the large-ℓ scaling
function vanishes,adifferentlarge-N scalingbehavior
2
S behavior
sets in: [40, 66]
limN→∞N−1/6S2(N1/2zb+N−1/6z,N)=F(z), ρ(x) ≈ ℓ−θR(X,Ur,N), (45)
F(z)=21/2 Ai′(21/2z)2 2z Ai(21/2z)2. (42) do(x) ℓ−2θ (X,Ur,N), (46)
| | − | | ≈ D
C(x ,x ) ℓ−θ (X ,X ,U ,N), (47)
This implies that at the boundary zb of the cloud the 1 2 ≈ C 1 2 r
TSS function of the particle density increases as N1/6 G(x1,x2) ℓ−2θ (X1,X2,Ur,N), (48)
≈ G
only, instead of the O(N1/2) behavior for |z| < zb. This M(x1,x2) ≈ ℓ−2θM(X1,X2,Ur,N), (49)
isrelatedtothefactthatthe z →zb limitoftheO(1/N) P(x1,x2) ℓ−2θ (X1,X2,Ur,N), (50)
corrections is singular, see also below. ≈ P
InthefollowingweextendtheseresultstotheHubbard where
model,andingeneraltolatticefermiongaseswithshort-
ranged interactions. X =x /ℓθ, U =Uℓθ. (51)
i i r
6
4
N=10 U=-100 N=20 U=-10
r r
3
l=200
l=1000
l=5000 3
) l=10000 )
x U = 0 x
ρ( 2 Ur → − ∞ ρ(
θl r θl
2
1 1 l=50
l=100
l=200
U=0
U → − ∞
r
0 0
0 1 2 3 0 1 2 3 4 5
θ θ
x/l x/l
3
N=10 U=100 N=20 U=10
r r
2 l=10
l=30
l=100
) l=200 )
x x 2
( U = 0 (
ρ U → ∞ ρ
θl r θl
1
1
l=50
l=100
l=200
U=0
U → ∞
r
0 0
0 1 2 3 4 5 0 1 2 3 4 5 6 7
θ θ
x/l x/l
FIG.2: (Coloronline)TSSoftheparticledensityforN =10 FIG.3: (Coloronline)TSSoftheparticledensityforN =20
particles, at fixed U ≡ Ulθ = −100 (top) and U = 100 particles, U = −10 (top) and U = 10 (bottom). The lines
r r r r
(bottom). In both cases the data appear to converge to a show thecurves for U =0, U →∞ and U →−∞.
r r r
nontrivialcurvewithincreasingℓ,whicharequiteclosetothe
U →∞ and U →−∞ limits respectively, cf. Eqs. (54) and
r r
(56), shown by the full lines. The dashed line is the U = 0
r which gives a maximum discarded weight below 10−9. 1
curve for a free Fermi gas. Due to the reflection symmetry
Weusewavefunctionpredictionandexploitfullythecon-
with respect the center of the trap, we show only data for
servationofthe totalnumber ofparticlesofeachspecies.
x≥0.
The implicitly restartedArnoldi method is used to diag-
onalize the Hamiltonian.
We note that, for large negative U, the strong on-site
attraction tends to bunch together the particles in the
middle ofthe trap: this effectreducesthe chainlengthL
θ is the same exponent of Eq. (21). These TSS behav-
neededtohostalltheparticles,evendowntoO(N)inthe
iors are expected to be approached with power-law sup-
limit U . Moreover,it also reduces the number of
pressed corrections. Of course, for Ur = 0, i.e. for a →−∞
stateskeptintheDMRGtruncationforagivenaccuracy.
strictly vanishing U, we can derive the scaling functions
This fact allows us to get data for larger and larger trap
fromtheresultsofSec.IIIA,takingintoaccountthatan
size as the attractive interaction increases.
unpolarized free Fermi gases of N particles is equivalent
totwoindependentspinlessFermigasesofN/2particles. InFigs.2and3weshowresultsfortheparticledensity
of systems with N = 10 and N = 20 particles, for some
positive andnegativevalues ofthe rescaledon-site inter-
In order to check our predictions for the TSS behav-
actionU Uℓθ. TheyclearlyconfirmtheTSSpredicted
iors,we presentnumericalDMRG [67] results for the 1D by Eq. (r4≡5), indeed the data for ℓθρ(x) plotted versus
Hubbard model in the presence of a harmonic trap, for
various values of U, N, and ℓ. DMRG simulations are
performedforachainofLsiteswithopenboundarycon-
ditions. The size L is chosen sufficiently large to make
1 InafewcomputationsatlargeN,themaximumdiscardedweight
finite-size effects negligible; in practice, we set L large is slightly larger than 10−9; however the quality of the results
enoughto haveρ<10−15 atthe edges ofthe chain. The isadequate forourneeds: weestimate(byvaryingM)thatthe
number of states M kept in the truncation is M 1120, truncationerrorisnegligibleinallfigurespresented.
≤
7
N=10 U=-100
2.6 r
5
0)2.4 N=10 0,x)
θρ(l2.2 UUr == 1-100 θ2lG( 0
r l=200
l=1000
2.0 −5 l=2000
l=5000
l=10000
U=0
r
1.8 U → − ∞
−10 r
0.00 0.01 0.02 0 1 2
-2θ θ
l x/l
FIG.4: (Coloronline)DMRGdataofℓθρ(0)vsℓ−2θ forN = N=10 Ur=-10
10andU =−10, 10. Thedottedlinesshowlinearfits,which 0
r
support the O(ℓ−2θ) behavior of the corrections analogously
)
to thefree U =0 case. 0,x
(
G
3 U → ∞ θ2l
r
Ur = 100 l=200 −2 l=100
Ur=10 l=200 l=200
U = 2 l=100 l=300
r
U = 0 l=500
Ur = -2 l=100 U=0
x) 2 Ur = -10 l=500 U → − ∞
ρ( Ur = -100 l=10000 r
θl Ur → − ∞
r −4
0 1 2 3
θ
x/l
1
1
N=10 U=100
r
N=10
00 1 2 3 4 5 x) 0
θ 0,
x/l (
G
θ
2
l
FIG. 5: (Color online) TSS functions of the particle density −1
of N =10 particles, for various values of the scaling variable l=10
U . Wereport theexact curvesfor a free Fermi gas (U =0) l=30
r r l=100
and the limits Ur → ±∞, cf. Eqs. (54) and (56). For the l=200
othervaluesofUr weplottheDMRGresultsforasufficiently −2 UU = → 0 ∞
large trap size, providingalready theasymptoticcurveswith r
high precision, as shown in Figs. 2 and 3 for some values of 0 1 2 3
θ
Ur. x/l
FIG.6: (Color online) TSSof thedensity-densitycorrelation
x/ℓθ appeartoapproachalimitingcurverepresentingthe G(0,x),i.e. with a point fixedat thetrap center, forN =10
NTS.STfhuencntuiomnerRic(aXl,rUesru,lNts)sahtowthtehgaitvethnevaasluyemopftoUtircabned- pxa/rℓtθi.clTesh,ealnindesUrsh=ow−t1h0e0U, −=100,c1u0r0v.eWanedptlhoet Uℓ2θG→(0∞,x)anvds
r
havior is generally approached with O(ℓ−2θ) corrections Ur →−∞ limits, cf. Eqs. (55) and (57).
relativetotheleadingbehavior,asinthecaseofnoninter-
actingfermionsystems,seeSec.IIIA.SomeDMRGdata
for the approach of the particle density at the origin to clearly depend on the scaling variable U . In Fig. 5 we
r
its asymptotic behavior are reported in Fig. 4. The am- show them for N = 10 and several values of U . As ex-
r
plitude of these corrections is significantly larger for at- pected, they extend to largerregions when we pass from
tractive interactions, increasing with increasing U , re- attractive to repulsive interactions. We also note that
r
| |
quiringlargerandlargertrapsizestoobservetheasymp- the scaling density shows N/2 peaks for U . 10, while
r
totic behavior. For example the ratio between the am- they become N for large U , see in particular the data
r
plitudes of the leading O(ℓ−1) corrections at U = 10 for U = 100. As we shall discuss below, when varying
r r
−
and U =10 is approximately 10. U from the strongly attractive to the strongly repul-
r r
The asymptotic TSS curves of the particle density sive regimes, the system experiences a crossover from a
8
l=2000
0 U=-100
r l=10000
2 U = 0
U=100 l=30
x)−1 N=20 x) r l=100
0, 0, 1 U → −∞
G( M( r
θ2l−2 Ur = -10 ll==120000 θ2l 0
U = 10 l=100
r l=200
−3 U = 0
Ur → ∞ −1
r
U → − ∞
r
−4
−2 N=10
0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
θ θ
x/l x/l
FIG.7: (Color online)ℓ2θG(0,x)versusx/ℓθ forN =20and FIG. 8: (Color online) TSS of the correlation M(0,x) ≡
Ur = −10, 10. The lines show the Ur = 0 and Ur → ±∞ hn0↑nx↓ic for N = 10 and Ur = −100, 100. We plot
limits. ℓ2θM(0,x) vs x/ℓθ. The full line represents the U → −∞
r
limit given byEq. (58).
hard-corebosonic gasofN/2 molecules to a free spinless
fermion gas of N particles.
We also present results for the correlation functions 3 N=10
introduced in Sec. II, in particular we consider corre- U=-100 l=2000
r l=10000
lation functions with a point fixed at the trap center U=-10 l=100
r l=200
xne=cte0d. Fdiegnss.i6tyancodr7reslhatoiwonthfuenscctailoinngGb(e0h,axv)i,orfoorftNhe=co1n0- (0,x) 2 UUr==11000 lll===5250000
and N = 20 respectively, and various values of Ur. The θlC r lU=r2=000
data nicely support the TSS Ansatz (48). In Fig. 8 we
1
show results for the connected correlation M(0,x) be-
tween up and down density. TSS is also confirmed by
the data of the one-particle correlation, see Fig. 9, and
0
the pair correlationin Fig. 10. In particular, the quanti-
ties which are more sensitive to the correlation between
0 1 2 3
up and downfermions, such as the up-down density cor- θ
x/l
relationM(x,y)andthepaircorrelationP(x,y),tendto
become more and more significant with decreasing U ,
r FIG. 9: (Color online) TSS of the one-particle correlation:
asexpectedbecauseupanddownfermionsbecomemore
lθC(0,x) versusx/lθ for N =10 and various values of U .
and more tightly correlated for large negative values of r
the on-site interaction. This is also shownby the behav-
ior of the double occupancy shown in Fig. 11.
An important issue concerns the universality of the
N=10 l=2000
TSS reported in Eqs. (45-50). They are expected to be U=-100 l=5000
r
l=10000
universal apart from a global multiplicative normaliza- 60 U=-10 l=100
tion, and normalizations of the arguments of TSS func- r l=200
x) U=10 l=50
tions. More precisely, we expect that they are universal 0, r l=200
with respect to a large class of further short-ranged in- θ2P(40 Ur=100 llU==52=0000
teraction terms, such as l r
Hnn = wσσ′ nσxnσ′y. (52) 20
Xσ,σ′ hXxyi
Indeed, H may only give rise to a change of the effec-
nn 0
tive quartic coupling U (when adding H to the Hub- 0 1 2 3
nn θ
bardHamiltonian,the effectiverelevantquarticcoupling x/l
becomes U +2w ), and to further O(l−θ) corrections,
↑↓
due to the fact that they introduce other irrelevant RG FIG.10: (Coloronline)TSSofthepaircorrelation: l2θP(0,x)
perturbations of RG dimension yw = d at the dilute versusx/lθ for N =10 and various values of Ur.
−
fixed point.
9
U = 0 The relation between the TSS of the trapped Hub-
r
Ur = -2 l=100 bard model and the continuum GY model can be ex-
U = -10 l=500
30 Ur = -100 l=10000 ploited to determine the the TSS functions of the par-
r
ticle density and its correlation, i.e. (X,U ,N) and
x) R r
( (X ,X ,U ,N) respectively, in the strongly repulsive
o 1 2 r
θ2ld20 N=10 aGnd attractive limits, i.e. Ur →∞ and Ur →−∞.
We know that in the g limit the particle density
→∞
andits correlationsofthe GYmodelbecomeidenticalto
10 those of a gas of N spinless fermions [4, 72, 73]. This
wouldimplythattheU limitoftheTSSfunctions
r
→∞
is
0
0 1 2 3 (X,U ,N)=S (X,N), (54)
θ R r →∞ p
x/l
(X ,X ,U ,N)=Y (X ,X ,N), (55)
1 2 r p 1 2
G →∞
FIG. 11: (Color online) TSS functions of the double occu-
whereS andY arethesamefunctionsenteringthespin-
pancy for N = 10 and some values of U , obtained plotting p p
l2θd (x) versus x/lθ. Note that the dourble occupancy van- less free-fermion TSS, cf. Eqs. (27) and (33). Moreover,
o
the TSS functions of the double occupancy and the pair
ishes in thelimit of strong repulsive interactions.
correlation, i.e. (X,U ,N) and (X ,X ,U ,N), de-
r 1 2 r
D P
fined in Eq. (46) and (50) respectively, trivially vanishes
3. Crossover behavior as a function of the on-site in the Ur limit.
→∞
interaction In the g limit the density properties of the GY
→−∞
model is expected to match that ofan ensemble of hard-
It is importantto note thatthe TSSlimit corresponds core N/2 bosonic molecules constituted by up and down
to a continuum limit in the presence of the trap, i.e. it fermions. Indeed, with increasing attraction, the pair-
generally realizes a continuum quantum field theory in ing becomes increasingly localized in space, and eventu-
the presence of an inhomogeneous external field. In the ally the paired fermions form a tightly bound bosonic
caseoftheHubbardmodelinthediluteregime,thiscon- molecule. Actually, the results of Ref. [16] for harmonic
tinuum limit is given by the quantum field theory (16), traps, obtained by LDA, show that these bound states
replacing the constant µ with a space-dependent poten- get trapped in a smaller region, with an effective trap
tial µ V(x). This implies that the TSS of the observ- size ℓb = ℓ/2 in the strongly attractive limit. Thus, we
ables o−f the 1D trapped Hubbard model must approach expect that in the g limit the particle density
→ −∞
the solutions of the continuous problemof fermions with of the unpolarized GY model (53) with a harmonic trap
contact interactions, which is also the so-called Gaudin- matches that of N/2 hard-core doubly-charged bosons
Yang (GY) model [68, 69], with equal number of up and withaneffective trapsize ℓb =ℓ/2,whichin turncanbe
down particles, N = N = N/2. The GY Hamiltonian mapped into a free gas of N/2 spinless doubly-charged
↑ ↓
of a trapped fermion gas can be written as fermions in a harmonic trap of size ℓb. On the basis of
these arguments and using the results of Sec. IIIA, we
H = N p2i +V(x ) +g δ(x x ). (53) conjecturethefollowingUr →−∞limitoftheTSSfunc-
GY i i j tions for harmonic traps:
2m −
i=1(cid:20) (cid:21) i6=j
X X
(X,U ,N)=23/2S (√2X,N/2), (56)
We expect that the TSS limit of the 1D Hubbard model r 2
R →−∞
at fixed N is related to the GY model with g ∼ Ur. G(X1,X2,Ur →−∞,N)=8Y2(√2X1,√2X2,N/2).(57)
More precisely, the TSS functions entering formulas (45-
50) are exactly given by corresponding quantities of the Moreover,since fermion pairs are tightly bounded in the
GY problem with a trap of unit size. strongly attractive limit, we also predict
The equation of state of the homogenous GY model
is exacty known for both repulsive and attractive zero- (X ,X ,U ,N)=2Y (√2X ,√2X ,N/2).(58)
1 2 r 2 1 2
M →−∞
rangeinteraction[68,69]. It is characterizedbydifferent
asymptotic regimes with respect to the effective dimen- Using analogous arguments we may also expect that in
sionlesscouplingγ g/ρ,whereρistheparticledensity. this U limit the TSS function of the double oc-
r
≡ → −∞
At weak coupling γ 1 it behaves as a perfect Fermi cupancy becomes proportional to (X,U ,N).
r
≪ R → −∞
gas; in the strongly repulsive regime, γ 1 the equa- This is supported by the DMRG data by comparing the
≫
tion of state approaches that of spinless Fermi gas; in dataofthedoubleoccupancyandtheparticledensityfor
the stronglyattractiveregime γ and forunpolar- N = 10 and U = 100, shown in Fig. 11 and the top
r
→−∞ −
ized gases it matches that of a 1D gas of impenetrable Fig. 2 respectively. Moreover,in the same limit the TSS
bosons [70], more precisely hard-core bosonic molecules function of the pair correlation should get proportional
of fermion pairs [16, 71]. to the one-particle correlation (density matrix) of a gas
10
of N/2 hard-core bosons. This can be checked by com- 0.5
paring the U = 100 data of Fig. 10 with the results U=0
r
−
of Ref. [40] for the one-particle correlation of hard-core 0.4
Bose gases.
0.3
The curves corresponding to the U limits are
r
→ ±∞
shown in Figs. 2, 3, 5, 6, and 7 for both the particle
0.2 N = 20
density and its connected correlations. They are clearly N = 40
tahpeprroeasuchltesdfobryNthe=d1a0tainfoFrilgasr.g2e |aUnrd|,6s.eeNiontepathrtaitcutlhaer 0.1 NNN === 81102600
N = 200
limit Ur of the connected density-density corre- N → ∞
lationsG→an−d∞M is apparentlyapproachednonuniformly 0.00.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
at small distance. Indeed the data for U = 100 shown X/N1/2
r
−
in the top Fig. 6 appear to follow the asymptotic curve
(57) for x/ℓθ &0.1, while atsmaller distances they show FIG. 12: (Color online) Large-N scaling of the TSS function
a sudden departure where the correlation increases sig- of the particle density of free Fermi gases: N−1/2R(X,Ur =
nificantly. The comparison with the data for U = 10, 0,N) versus X/N1/2 for several values of N. The finite-N
r
− curves,obtainedusingEq.(28),clearlyapproachthelarge-N
seethemiddleFig.6,suggeststhatthisoccursatsmaller
andsmallerdistanceswithincreasing U ,beinglikelyre- limit R∞(z), cf. Eq. (60), with oscillations that get sup-
| r| pressed as 1/N, in agreement with Eq.(60).
latedwiththesizeofthemoleculesformedbythefermion
pairs.
0.8
These data show that the dilute TSS of quantities re-
lated to the particle density of the trapped Hubbard
model experience a smooth crossover from an effective 0.6
hard-core bosonic gas of N/2 molecules (U ) to
r
→ −∞
an effective free gas of N spinless fermions (U ),
r
passingthroughtwononinteractingfermiongases→ofN∞/2 0.4
particles. In these 1D systems the formation of a gas N=10
of molecules of pair fermions has some analogies to the 0.2 NN==2300 Ur / N1/2=-2
formation of a 3D molecule BEC and the BCS-BEC N=40
U=0
crossoverfor 3D Fermi systems, which has been recently U / N1/2 → − ∞
r
observed in experiments with ultracold Fermi gases, see 0.0
0.0 0.5 1.0
e.g. Refs. [3, 74–80]. 1/2
X/N
Let us note that the Ur limit of the TSS cannot 0.6
→∞
be obtainedby firsttaking the U limit ofthe Hub-
→∞
bardmodelandthenthelargetrap-sizelimit. Indeed,on
the one hand, the U limit of the TSS reproduces
r
→ ∞ 0.4
the g limit of the continuous GY model, essen-
→ ∞
tially given by a gas of free spinless fermions. On the
other hand, the ground state of the Hubbard model in
tthraelUN→sit∞es alirmouitnsdhothueldcbeneteorbtoafinteradpbuypfiltloingxth.e Nce/n2- 0.2 NNN===123000 Ur / N1/2 = 2
| | N=40
(actuallyweexpectsomedegeneration),withoutanypar- U=0
ticular scaling with respect to the trap size. U / N1/2 → ∞
r
0.0
0.0 0.5 1.0
1/2
AnalogousconsiderationsapplytothelimitU , X/N
r
→−∞
whichcorrespondstoagasofN/2bosonicmoleculeswith
hard-core interactions. Indeed, the ground state of the FIG. 13: (Color online) Large-N scaling of the TSS function
trapped Hubbard model in the U limit of the oftheparticledensity: N−1/2R(X,Ur,N)versusX/N1/2 for
Hubbardmodel(atanyfinitetrapsi→zeℓ−)i∞sjustobtained Ur/N1/2 = −2 (top) and Ur/N1/2 = 2 (bottom). The TSS
by completely filling the central N/2 sites around the functions are derived from simulations with ℓ = 100, 200,
center of trap (up to x (N/2 1)/2 if N/2 is an odd which are expected to be sufficiently large to effectively re-
| |≤ − produce the ℓ→∞ limit. For comparison, we also show the
number,inthecaseofevenN/2therearetwodegenerate
asymptotic curve for U = 0, cf. Eq. (60), and the expected
ground states filled for N/4 x < N/4 and N/4 <
− ≤ − asymptotic curves for U /N1/2 → ∞ and U /N1/2 → −∞,
x N/4), without any particular scaling property with r r
≤ given by Eq. (62) and (63) respectively.
respect to the trap size.