Superfluid - Bose-Glass Transition in Weakly Disordered Commensurate One-Dimensional System
a r X i v :c o n d -m a t /9609243v 1 25 S e p 1996
SUPERFLUID -BOSE-GLASS TRANSITION IN WEAKLY DISORDERED
COMMENSURATE ONE-DIMENSIONAL SYSTEM
Boris V.Svistunov
Russian Research Center ”Kurchatov Institute”,123182Moscow,Russia
We study the e?ect of commensurability (integer ?lling factor)on the super?uid (SF)-Bose-glass (BG)transition in a one-dimensional disordered system in the limit of weak disorder,when the e?ect is most pronounced and,on the other hand,may be traced via the renormalization-group analysis.The equation for the SF-BG phase boundary demonstrating the e?ect of disorder-stimulated super?uidity implies that the strength of disorder su?cient to restore super?uidity from Mott insulator (MI)is much larger than that enough to turn MI into BG.Thus we provide an explicit proof of the fact that at arbitrarily small disorder the SF and MI phases are always separated by BG.
PACS numbers:05.30.Jp,64.60Ak,67.40.-w
The problem of super?uid -insulator transitions in commensurate and disordered bosonic systems has at-tracted a lot of interest in the past decade (for an intro-duction see Refs.[1–3]).A special part in the analysis of the problem is played by one-dimensional (1D)systems,where along with the exact ?nite-size numerical treat-ment (quantum Monte Carlo [4,5]and exact diagonal-ization [6,7])there exists macroscopic renormalization-group (RG)description [2].[Recently it was demon-strated that the two are in an excellent agreement with each other at least in the case of SF-MI transition [8].]Note also a remarkable success of density-matrix RG study of 1D bosonic Hubbard model [9].
Despite the fact that now the understanding of super-?uid -insulator transitions is close to being complete,especially in 1D,there is one point,however,which re-mains somewhat unclear up to now.That is the ques-tion of super?uid -insulator transition in a weakly dis-ordered commensurate system with integer ?lling factor.From general considerations one would expect here that SF and MI phases are always separated by BG phase [3,10],while a number of approximate and exact (but ?nite-size!)methods suggest that small disorder is not relevant and there occurs a direct transition from SF to MI [11,12,9].In this paper we consider this problem in 1D taking advantage of the asymptotically exact RG treat-ment.
Our main result is the demonstration of the existence of the BG phase at arbitrarily weak disorder.We provide the proof of this fact in a ’direct’and in an ’enhanced’forms.By ’direct’form we mean tracing the renormal-ization picture (in the super?uid phase,where the RG equations are asymptotically exact)with the observation that the commensurate scenario,taking place at smaller
scales of distance,is destroyed at larger distances by suf-?ciently strong disorder,in favor of the universal disor-dered scenario.[Incidentally,this accounts for the e?ect of disorder-stimulated super?uidity.]So that commensu-rability turns out to be irrelevant to any second-order-type transition arising with decreasing of the strength of disorder.The ’enhanced’proof does not rely on any as-sumption concerning the type of the transition leading to the destruction of super?uidity.We simply take the phase boundary for super?uid -insulator transition fol-lowing from the ’direct’treatment as a lower boundary for the strength of disorder su?cient to support super?u-idity,without specifying the phase adjacent to SF.Then we compare the result with that for the MI phase bound-ary,following from the exact relation between the mini-mal amount of disorder destroying MI and the insulating gap of the pure system [10].The comparison shows that,in relative units,the separation between the boundaries is ever increasing with vanishing disorder,thus revealing the presence of BG between MI and SF.
The RG descriptions of the super?uid-to-insulator transitions in 1D were initially constructed in terms of the e?ective action for the long-wave density deviations [2,3],employing Haldane’s representation of 1D bosons [13].For our purposes,however,it is more convenient to use a dual treatment in terms of the e?ective action for the phase ?eld [14].This language directly takes into account e?ects of the collective backscattering o?either commensurate or disordered potential,the corresponding instantons being just vortices in the phase ?eld in (1+1)dimensions.As a result,the descriptions for the com-mensurate and disordered cases turn out to be very close to each other.The only di?erence is that in the dis-ordered case one deals with ’vertical’vortex-antivortex pairs (the spatial coordinates of vortex,x 1,and antivor-tex,x 2,coincide up to some microscopic length),because contributions of non-vertical pairs are averaged out due to the random phase associated with the pair:
γ(x 1,x 2)=?2π
x 2
x 1
n 0(x )dx ,(1)where n 0(x )is the expectation of the density at the point
x [14].In the weak-disorder limit n 0(x )is related to the
disordered potential,?W (x ),perturbatively:n 0(x )=n (0)(x )+
dx ′S (x ′,x )?W (x ′),(2)where n (0)(x )and S (x ′,x )are the density expectation
and static susceptibility for the pure system.
1
S(x′,x)=i ∞0dt [n(x′,t),n(x,0)] ,(3)
n is the density operator.Substituting(2)into(1)and taking into account that in the integer-?lling case the contribution from n(0)(x)leads only to some irrelevant renormalizations[14],we get
γ(x1,x2)=?2π x2x1dx Q(x)?W(x),(4) where
Q(x)= dx′S(x,x′)(5)
is a periodic(commensurate)function.[In Eq.(4)| x2?x1|is meant to be much greater than the micro-scopic static correlation radius characterizing the decay of S(x′,x)as a function of|x?x′|.]
For de?niteness,we take?W(x)to be bounded,
?W/2≤?W(x)≤W/2,(6) and symmetric with respect to the change of the sign. Introducing the radius of a vortex pair
R=
R.(8) [From now on we treat W and R as dimensionless quan-tities,assuming them to be measured in units of char-acteristic interparticle energy and distance,respectively.] WhileγR?1,the disorder is irrelevant and the renor-malization picture follows the scenario of the pure com-mensurate case.At the scale
R?～1/W2,(9)
corresponding toγR
?～1,the renormalization picture crosses over to the disordered scenario which becomes
well-developed at R?R?,when only the vertical pairs (|y2?y1|?|x2?x1|～R?)contribute.The scale R?itself plays negligible role in the renormalization,since near the critical point in1D the total contribution comes from exponentially large range of distances.We thus can simply sew together the known RG descriptions[2](see also[3,14])for the purely commensurate and the purely disordered cases:
dK
dλ=(2?1/K)w(λ≤λ?);(10)
dK
dλ
=(3/2?1/K)w(λ≥λ?).(11)
Hereλ=ln R,λ?=ln R?;K= π√
2
+a tan[4aλ?π/2](|K?
1
p/p0?1.(14)
Here p is any parameter of the Hamiltonian,the value
p=p0corresponding to the point of SF-MI transition in
the pure system(p>p0in the Mott phase).For exam-
ple,one may choose p to be the strength of interparticle
interaction.
From Eq.(13)it is seen that K(λ)does not di?er essen-
tially from1/2untilλcomes closely to the pointπ/4a.
At this point K diverges and we thus conclude that to a
good accuracy
λ?=
π
only a rough order-of-magnitude estimate.This rough-ness,however,is essentially compensated by the sharp behavior of K(λ)in the vicinity of the pointλ=π/4a. So that the relative uncertainty of the value ofλ?is easily estimated to be on the order a?1.
Combining Eqs.(9),(14),and
(15)
we obtain the equa-
tion for the phase boundary between super?uid and Bose
glass[W=W SF-BG(p)]:
W SF-BG(p)～exp ?πp/p0?1 ,(16)
where b is some constant.Eq.(16)demonstrates the ef-
fect of disorder-stimulated super?uidity in the vicinity
of the SF-MI transition of the pure commensurate sys-
tem,which was observed previously in numerical studies
[11,9].Note a very sharp behavior of the critical value of
p for the SF-BG transition as a function of the strength
of disorder.
To provide the’enhanced’proof of the existence of the
BG in the weak disorder limit we compare Eq.(16)with
the equation for the phase boundary between MI and
a gapless phase(which subsequently will be identi?ed
with BG;until this is done the notations MI-BG and SF-
BG should be understood as a convention).To obtain
the latter equation one can take advantage of the exact
relation taking place at the MI phase boundary[10]:
?=W,(17)
where?is the insulating gap of the pure system.This
relation follows from the fact that the in?nitesimal de-
struction of the MI is due to the most favorable exchange
of particles between exponentially rare Lifshitz’s regions,
where local con?gurations of disorder mimic uniform po-
tential of the value W/2or?W/2[3,10].An estimate for
?one can obtain from the relation?～1/R?[3].So we
have
W MI-BG(p)～exp ?πp/p0?1 (18)
and see that
W SF-BG(p)
8b
R?.
Let us now discuss recent results of numerical density-
matrix RG study of1D commensurate disordered bosonic
Hubbard model[9].In the weakly disordered case the
authors observe a direct transition from SF to MI,in
contrast to the case of a substantial disorder where BG
intervenes between SF and MI,and to our asymptotically
exact(at W→0)analytical results.
It is known,however,that in the limit of weak disorder
it is di?cult to distinguish BG from MI by a?nite-size
scaling,because the di?erence is associated with expo-
nentially rare Lifshitz’s regions[3,10].If the size of the
system is not large enough,these regions are simply ab-
sent with probability close to unity,and the scaling anal-
ysis reveals nothing.[Maximal size of the system in Ref.
[9]is of the order of100sites.]
Therefore,one may suspect that the transition ob-
served in Ref.[9]at a small disorder is actually the SF-
to-BG one.In this connection it is interesting to com-
pare the numerical data with Eq.(16).Unfortunately,the
phase diagram in Ref.[9]contains only few points in the
weak-disorder region,too little to make possible an un-
ambiguous?tting.However,one may check a non-trivial
qualitative prediction of Eq.(16),the requirement that
the strength of disorder on the SF-BG phase boundary
be much larger than the Mott gap in the absence of dis-
order.The data clearly demonstrate this feature.At any
of the three presented points on the super?uid-insulator
phase boundary in the weak-disorder region the strength
of disorder is at least an order of magnitude larger than
the insulating gap.
We can allow also for the e?ect of a small deviation
from commensurability.To this end we should add to
the right-hand side of Eq.(4)the term2πδn(0)(x1?x2),
whereδn(0)is the deviation of the density from the com-
mensurate value.This term is negligible for a vortex pair
of a radius much less than
R c～|δν|?1(20)
(δνis the deviation of the?lling factor),and leads to
the cancellation of contributions of pairs with radii R c<
R
plays no role at all.Hence,at R c replaceλ?by ln R c in Eq.(10)and set K(λ?)=K(ln R c) as the initial condition for Eq.(11).Consequently,for the SF-BG phase boundary we obtain the equation exp ?πp/p0?1 ～max W2,|δν| .(21) Note that in the variables p andδν(at?xed W)the phase boundary has an independent of W cusp-like shape except for the tip of the cusp cut o?by disorder. A remark is in order here concerning fractional?lling factors.The fact that the point of SF-MI transition of a pure commensurate system turns out to be the limit- ing point for both SF-BG and BG-MI phase boundaries 3 of the disordered system is characteristic of only integer ?lling factors,because only in this case the critical value of the parameter K for the pure system,K=1/2,is less than that of disordered system,K=2/3.In the vicin-ity of SF-MI transition at a fractional?lling,where the transition occurs at K=q2/2(q is the denominator of the?lling factor)[15,13],an in?nitesimal disorder takes the system away from the super?uid region. Summarizing,we have presented a description of the super?uid-Bose-glass phase boundary in1D weakly disordered system near the point of super?uid-Mott-insulator transition of the pure system with integer?ll-ing factor.The results demonstrate the e?ect of stim-ulation of super?uidity by disorder,previously observed numerically.In the limit of weak disorder the strength of disorder su?cient to restore super?uidity from Mott insulator turns out to be much larger than that enough to turn Mott insulator into Bose glass.This provides an explicit proof of the fact that at arbitrarily small disorder the Bose glass always intervene between super?uid and Mott phases. The author is grateful to V.Kashurnikov,N.Prokof’ev, and J.Freericks for valuable discussions. This work was supported by the Russian Foundation for Basic Research(under Grant No.95-02-06191a)and by the Grant INTAS-93-2834-ext[of the European Com-munity].