Density matrix renormalization group for disordered bosons in one dimension

4 Pages·0.12 MB·English

by  S. Rapsch

Density matrix renormalization group for disordered bosons in one dimension
S. Rapsch∗, U. Schollw¨ock and W. Zwerger
Sektion Physik, Universit¨at Mu¨nchen, Theresienstr. 37/III, D-80333 Mu¨nchen, Germany
(February 1, 2008)
We calculate the zero-temperature phase diagram of the disordered Bose-Hubbard model in one
dimension using the density matrix renormalization group. For integer filling the Mott insulator
is always separated from the superfluid by a Bose glass phase. There is a reentrance of the Bose
glass both as a function of the repulsive interaction and of disorder. At half-filling where no Mott
insulator exists, thesuperfluiddensityhas amaximumwhere thekineticandrepulsiveenergies are
about the same. Superfluidityis suppressed both for small and very strong repulsion but is always
monotonic in disorder.
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9
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The interplay between disorder-induced localization integerfillingitisfoundthatthesuperfluidandMottin-
1
andinteractionshasattractedagreatdealofattentionin sulatingstatearealwaysseparatedbyaBoseglassphase

n recent years. The simplest model including both aspects as suggested by Fisher et al.2 The superfluid density is
a in a nutshell is a Hubbard model with random site en- nonmonotonic not only as a function of interaction but
J
ergies and a local repulsive interaction for either bosons also of disorder. Thus for strong repulsion increasing
1
or fermions with opposite spin. Unfortunately there are disorder drives a transition from a Bose glass to a su-
1
essentially no analytical results for this model if both perfluid. For half filling, where no Mott insulator exists,

disorderandinteractionsarepresent,notevenin onedi- the superfluid density is again a nonmonotonic function
1
v mension. For 1d bosons, however, there is a weak disor- oftherepulsiveinteraction,howeverdisordernowalways
0 der, perturbative calculation by Giamarchi and Schulz1, suppresses superfluidity as expected. The corresponding
8 who found that the superfluid ground state with quasi phase diagram is in agreement with that suggested by
0 long range order survives disorder up to a critical point, GiamarchiandSchulz1, howeverwe findno indicationof
1
where the effective exponent K in the decay of the one a qualitative difference between the glass phase at small
0
particle density matrix is equal to 2/3. More generally, orlargevaluesoftherepulsion(Andersonvs.Bose-glass).
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thequalitativephysicsoftheBose-Hubbardmodelinany The Bose-Hubbard model in 1d is defined by the
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/ dimension, and in particular the scaling behaviour near Hamiltonian1–3
at critical points has been elucidated by Fisher et al.2. For
t U
m quantitative results, however, it is necessary to resort Hˆ =−2X(b†i+1bi+h.c.)+ 2 Xni(ni−1)+Xǫini.
- to numerical simulations3,4. The latter were performed i i i
d using path integral (or “world line”) Monte Carlo cal-
(1)
n
culationswhichbecome increasinglydifficult inthe most
o
c interestinglimitofzerotemperature. Nowatleastinone Hereb†isthebosoncreationoperatoronsiteiofa1dlat-
i
v: diciamletnecshionniqtuheerfeorisinatnerinahcteirnegntqluyaznetruomtepmropberleamtusr,ethneumdeenr-- ticewithLsitesandni =b†ibithecorrespondinglocaloc-
i cupationnumberwitheigenvalues0,1,2,.... Thekinetic
X sity matrix renormalization group (DMRG) method de-
energy is described by a hopping matrix element t > 0,
velopedbyWhite5. Itisthereforenaturaltotryemploy-
r leading to a standard tight binding band ǫ(k)=−tcosk
a ing this method to the disordered Bose-Hubbard model
in the absence of interactions and randomness. The re-
in one dimension. This was first done by Pai et al.6,
pulsive interaction is described by a local, positive Hub-
who calculated the associated phase diagram for integer
bard U which increases the energy if more than one bo-
filling. As expected, it exhibits a Mott insulating, a su-
son occupies a given site. Finally the site energies ǫ are
perfluidandalsoaBoseglassphase,thelatterappearing i
assumed to be independent random variables with zero
onlyforsufficientlystrongdisorder. Quiterecently,how-
average and a box distribution in the interval from −∆
ever,theirresultswereseriouslyquestionedbyProkof’ev
to ∆. Throughout we work in the canonical ensemble
and Svistunov, who performed rather precise quantum
with a given (dimensionless) density n = N of bosons.
Monte Carlo calculations7. Basedonthat, it was argued L
As usual we choose t = 1 as a unit of energy (note that
that the DMRG method is intrinsically unable to deal
someauthorshavetinsteadof t inthe hoppingor2ǫ in
with disordered systems because randomness would in- 2 i
thesiteenergieswhichleadstoatrivialfactoroftwodif-
validate building up a system in a block like fashion.
ference with our results). Apart from the density n, this
Our aim in the present work is to show that a careful
leaves the two dimensionless parameters U and ∆ char-
DMRG calculation can indeed be successfully applied in
acterizingtheinteractionsanddisorderwhichcompletely
the presenceofquencheddisorder. Inparticular,wepro-
specifytheproblematzerotemperature. Inordertodis-
vide a quantitative phase diagram for the 1d disordered
tinguish the various possible phases, we calculate both
Bose-Hubbardmodelatbothintegerandhalffilling. For
the energy gap E and the superfluid fraction ρ . The
g s
1
energy gap which is only nonzero in a Mott insulating antiperiodic boundary condition has been implemented
phase, can either be evaluated directly from a numeri- byreplacingthe hoppingenergytatoneofthe bondsby
cal calculation of the energy of the ground and first ex- −t thus enforcing a localized twist in the phase by π.
citedstate. Alternativelythe gapcanbe obtainedasthe
difference E = µ − µ between the chemical poten-
g p n
tial for particle (µ = E −E ) or hole excitations2 n = 1
p N+1 N
(µ =E −E ). We haveemployedboth methods in
n N N−1
ordertocheckourresults. Forthesuperfluidfractionρ , ∆
s
weusethethermodynamicdefinitionproposedbyFisher, 4 max
Barber and Jasnow8. It is based on defining ρ via the
s
sensitivity to a change in the boundary conditions be-
Bose glass
tween periodic (pbc) and antiperiodic (apbc) ones. In
∆
one dimension, at a given density n= N, the superfluid Bose
L glass
fraction ρ is thus given by (t=1)
s
2 super-
2L L
ρ = · [Eapbc(L)−Epbc(L)] (2) fluid
s π2 N 0 0
where E (L) are the ground state energies for the spe-
0 Mott insulator
cific boundary condition. In the absence of interactions
and disorder it is straightforward to show that ρ = 1
s
for arbitrary densities n as it should be. It is important 0 2 4
to note that it is precisely a nonvanishing value of ρ U
s
(in the limit L → ∞) which is the relevant criterion for
superfluidity despite the fact that the one particle den-
FIG.1. Phasediagramforcommensuratefillingn=1. Er-
sity matrix hb†ib0i decays to zero algebraically, i.e. only ror bars are mainly due to the dependence of results on the
exhibits quasi-long range order. realisations of disorder. Abovea disorderstrength ∆max =4
Forthenumericalcalculationsitisobviouslynecessary it is always energetically advantageous to destroy superfluid-
to limit the number of bosons which can occupy a given ity in favor of a glass phase.
site. In order to be able to cover also small values of U,
where many bosons tend to cluster at locally favorable Forthediscussionofourresultswefirstconcentrateon
sites,wehavetruncatedourbasistom=7statesforeach acommensurabledensityn=1,whereaMottinsulating
site i which allows up to 6 bosons occupying the same phase is expected2 at sufficiently large U. In the limit of
place. We have checked carefully that our results do not vanishingdisorder∆=0thesystemissuperfluidatsmall
depend on m, which was the case at least down to U ≈ valuesofU withasuperfluidfractionρ whichmonoton-
s
0.5. IntheDMRGcalculationwestudiedsystemlengths ically decreases from one at U = 0 to zero at U = U .
c
uptoL=50andincludeduptoM =190states. Forthe Since the transition to the Mott insulator is driven by
truncation error, which is one minus the density matrix phase fluctuations at a given density, it is a Kosterlitz-
eigenvalues λ of all M states kept in the decimation, Thouless like transition2 very similar to the one present
α
in a chain of Josephson junctions with a local charging
M energy9. Our numerical result for the critical value of U
ρ=1−Xλα, (3) is U (∆ = 0) = 1.92±0.04 which is surprisingly close
c
α=1 to that found in mean field theory10. It also agrees with
we find values of the order of 10−10. A very important a very recent DMRG calculation of the Bose-Hubbard
model without disorder by Ku¨hner und Monien11. They
point which turns out to be absolutely crucial for apply-
have used the condition that the exponent K character-
ingtheDMRGtodisorderedsystemsistoapplythefinite
izing the decay of the off-diagonal density matrix
size (“sweeping”)algorithm5. After the system has been
grown to its full length, renormalization group transfor- hb+b i∼|i|−K/2 (4)
i 0
mations have not yet been able to take into account the
full structure of disorder while working on shorter sys- inthesuperfluidphasetakesonthevalueKc =1/2atthe
tems. The finite size algorithm then works on the com- transition12. Note that K scales like pU/t at least in a
plete system, and improves results essentially in a vari- Josephsonjunction arraydescription which is equivalent
ational fashion. We find good convergence of both the to the Bose-Hubbard model at large integer densities.
gapandthesuperfluidfractionafterseveralsweeps. The AtfinitedisordertheMottinsulatingphaseissuppressed
dependence on the number of states kept was compar- because the energy gap is reduced. For vanishing hop-
atively weak (also compared to the scattering of results ping, i.e. U → ∞ effectively, the reduction2 is just 2∆.
invariousrealisationsofdisorder)suchthatwepreferred ThusinthelimitoflargeU theMott-insulatordisappears
to invest computational resources rather in sweeps. The of ∆>U/2. This is in fact the asymptotic behaviour of
2
the transition line shown in Fig. 1. For nonzero t, i.e.
finite U the transitionappears earlier,until the Mott in- n = 0.5
sulator completely disappears at U <U (∆=0)=1.92. 1.0 ∆=0.95
c
OutsidetheMott-insulatingphasethegapvanishes,how-
everat finite disorderthe systemneed notbe superfluid. 0.8
Indeedcalculatingthesuperfluidfractionρ ,wefindthat
s
ρ is nonvanishing only in the superfluid regime in Fig. ∆
s
1, which bends down to ∆ = 0 both near U = 0 and 0.6 Bose glass
U = U (∆ = 0). As a consequence, at finite disorder,
c
there is no direct transition from a Mott insulator to
a superfluid in agreement with the arguments given by 0.4 superfluid
Fisheretal.2 andFreericksandMonien13. Thecomplete
phase diagram is shown in Fig. 1. It agrees well with
that found by Prokof’ev and Svistunov7 using a rather 0.2
differentmethodandalsowiththequalitativepictureput
forwardbyHerbut14. Bycontrast,therearestrong,even
qualitative differences with the phase diagram found by
0 1 2 3 4
Pai et al.6. Their failure to see the intervening Bose-
U
glass between the superfluid and the Mott-insulator is
probably related to the fact that without the sweeping
FIG.2. Phasediagram forincommensuratefillingn=0.5.
algorithm the treatment of a disordered problem by the
Errors due to the dependence of results on the realisations
DMRGisnotreliable. Regardingthegeneralstructureof
ofdisorder arenow muchstronger(with theexception ofthe
thephasediagramshowninFig.1,weexpectthatitwill
point at ∆ = 0). Above a disorder strength of 0.95 we find
notbequalitativelydifferentforthetwodimensionalcase
that ρS =0 within ournumerical accuracy.
(thoughthecorrespondingpathintegralMonteCarlocal-
culationsofKrauth,TrivediandCeperley4andalsomore
For the study of a noncommensurate density, we have
recentones15failedtoseetheintermediateBose-glassbe-
chosen n = 0.5 where the Mott-insulating phase is com-
tween the superfluid and the Mott-insulator). Assuming
pletelyabsent2. Theresultingphasediagramisshownin
thatthe phasediagramofFig.1isindeedgenericforthe
Fig. 2. It exhibits a superfluid phase in a finite regime
disorderedBose-Hubbardmodelatcommensuratedensi-
U (∆) < U < U (∆) of the repulsion provided the dis-
ties, one can draw two general conclusions: − +
order is below a critical value ∆ ≈ 0.95. The er-
(i) Since the superfluid fraction is a nonmonotonic max
ror bars in the determination of the phase boundary are
function of U for a given disorder, repulsive interactions
largerthaninthe casen=1becausethesuperfluidfrac-
have a delocalizing tendency at small U but enhance lo-
tion exhibits rather strong realization dependent fluctu-
calization at large U. This is in fact a general prop-
ations. This problem becomes particularly relevant in
erty, valid also at incommensurate densities as verified
the limit of small U. In fact noninteracting bosons are
by Scalettar, Batrouni and Zimanyi3 for n = 0.625 and
a singular limit of the disordered Bose-Hubbard model
our own results at n=0.5.
since particles will collapse into the single level with the
(ii) More surprisingly, for fixed repulsion in the range
lowest ǫ , which will vary between different realizations.
i
U > U (∆ = 0) but not too large, increasing disorder
c For smallbut finite U the groundstate densities are still
drives a Bose-glass to superfluid transition. Thus in-
rathernonuniform. Nowonthebasisofthat,ithasbeen
creasing disorder may in fact favour superfluidity (see
conjectured by Scalettar et al.3 that there are two quali-
dash-dotted line in Fig. 1). The associated superfluid
tatively different localized states, a suggestion originally
fraction is finite only for ∆ > ∆ (U). It first increases
− due to Giamarchi and Schulz1. The two phases would
with ∆ but eventually decreases to zeroagainat the up-
be separated by a line ∆ (U) above ∆ which meets
per boundary ∆ (U) This effect may be understood by c max
+ the phase boundary to the superfluid at a multicritical
observing that with increasing distance from the Mott
point. Inordertolookforsignaturesofthisboundaryat
insulator the density of mobile particle-hole excitations
∆ > ∆ , we have calculated the expectation value of
increases, thus enhancing ρ . At larger values of ∆ the max
s the dimensionless disorder energy per particle
disorder induced localization takes over and ρ goes to
s
zero again at the upper phase boundary ∆ (U). 1
+ S = Xǫihnii, (5)
∆N
i
which is finite for a localized state4. Although S be-
comesincreasinglynegativeasU islowered,approaching
the limit S = −1 at U ≪ 1, we have found no indica-
tion of any abrupt changes. This suggests that there is
no quantitative distinction betweenan “Andersonglass”
3
for small U and a Bose-glass for larger repulsion. Verly 1T. Giamarchi and H.J. Schulz, Europhys. Lett. 3, 1287
likely it is only the line U = 0 which is singular. This (1987).
point of view is supported further by the fact that the 2M.P.A. Fisher, P.B. Weichmann, G. Grinstein and D.S.
phasediagramfoundbyProkov’evandSvistunov7onthe Fisher, Phys. Rev. B 40, 546 (1989).
basis of the Giamarchi and Schulz criterion1 K = 2/3 3R.T. Scalettar, G.G. Batrouni and G.T. Zimanyi, Phys.
c
for the renormalized exponent in the decay of the off- Rev.Lett. 66, 3144 (1991).
diagonaldensitymatrix(4)essentiallycoincideswithour 4W. Krauth, N. Trivedi and D. Ceperley, Phys. Rev. Lett.
results. Thus for any point on the phase boundary be- 67, 2307 (1991).
5S.R.White, Phys. Rev.Lett. 69, 2863 (1992).
tween the superfluid and the Bose-glass, scaling is to-
6R.V.Pai, R.Pandit,H.R.Krishnamurthyand S.Ramase-
wards ∆=0,K =2/3 evenfor very small U. Finally we
sha, Phys. Rev.Lett. 76, 2937 (1996).
havealsodeterminedthesuperfluidfractionasafunction
7N.V. Prokof’ev and B.V. Svistunov, Phys. Rev. Lett. 80,
of U, which exhibits a maximum at U ≈ 1−1.5. Un-
4355 (1998).
like the case for n = 1 this maximum does not scale to 8M.E. Fisher, M.N. Barber and D. Jasnow, Phys. Rev. A
larger U if ∆ is increased. For very small ∆ the critical
8, 1111 (1973).
value Uc(∆ = 0+) = 3.2(2) beyond which ρs vanishes in 9R.H. Bradley and S. Doniach, Phys. Rev. B 30, 1138
the presence of even a very small randomness has been (1984),seealsoW.Zwerger,Europhys.Lett.9,421(1989).
determined by calculating the exponent K in the pure 10L.AmicoandV.Penna,Phys.Rev.Lett.80,2189(1998).
system and using the criterion Kc = 2/3. Quite gener- 11T.D. Ku¨hner and H. Monien, Phys. Rev. B 58, R14741
ally, however, the numerical calculation of ρs becomes (1998).
rather difficult for small disorder. This is probably re- 12T. Giamarchi and A.J. Millis, Phys. Rev. B 46, 9325
lated to the strong divergence ξ ∼ 1 1/(3−2/K) of the (1992).
localization length in the limit ∆ →(cid:0)∆0(cid:1)near the critical 13J.K. Freericks and H. Monien, Phys. Rev. B 53, 2691
point K = 2/3, which follows from the integration of (1996).
the Giamcarchi and Schulz flow equations. For vanishing 14I.Herbut,Phys. Rev.B 57, 13729 (1998).
disorder ρ is finite for arbitrary values of U, approach- 15J.Kiskerand H.Rieger, Phys.Rev.B 55,R11981 (1997);
s F. Pazmandi and G.T. Zimanyi, Phys. Rev. B 57, 5044
ing ρ =2/π as U →∞ where the Bose-Hubbard model
s
at n = 1 is equivalent to the exactly soluble quantum (1998).
2 16E. Lieb, T. Schultz and D.C. Mattis, Ann. Phys. 16, 407
XY-model16 in zero magnetic field. Since K = ∞ in
(1961).
this limit, the localisation length in the XY-model with 17A. van Oudenaarden, S.J.K. Vardy and J.E. Mooji, Phys.
a random local field is expected to diverge like (cid:0)∆1(cid:1)1/3. Rev.Lett. 77, 4257 (1996).
In conclusion we have demonstrated that the DMRG 18D.Jaksch et al. ,Phys. Rev.Lett. 81, 3108 (1998).
method can be successfully applied to systems with
quenched disorder. The phase diagram of the 1d Bose-
Hubbardmodelhasbeendeterminedbothatintegerand
at halffilling. It exhibits significantdifferences with ear-
lierDMRGresults6 butessentiallyagreeswithaveryre-
centquantumMonteCarlocalculation7. Ourconclusions
quantitatively support the general picture for the disor-
deredBose-Hubbardmodeldevelopedby Giamarchiand
Schulz1 and by Fisher et al.2. The model studied here is
probably the simplest example for the interplay between
interactionsanddisorderandassuchisclearlyofinterest
in itself. Experimental realisations e.g. in terms of vor-
ticesina1darrayofJosephsonjunctionswithdisorder17
orthe recentsuggestionthat Bose-Hubbardphysicsmay
be relevant for cold atoms in optical lattices18, will cer-
tainly further the interest in this model.
Acknowledgements: Useful comments by D.S. Fisher
are gratefully acknowledged.
∗
Present address: Department of Theoretical Physics, Uni-
versity of Oxford, 1 Keble Road, OX13NP Oxford, UK.
4