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Density matrix renormalization group for disordered bosons in one dimension

S. Rapsch? , U. Schollw¨ck and W. Zwerger o

Sektion Physik, Universit¨t M¨nchen, Theresienstr. 37/III, D-80333 M¨nchen, Germany a u u (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 ?lling the Mott insulator is always separated from the super?uid 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-?lling where no Mott insulator exists, the super?uid density has a maximum where the kinetic and repulsive energies are about the same. Super?uidity is suppressed both for small and very strong repulsion but is always monotonic in disorder.

arXiv:cond-mat/9901080v1 11 Jan 1999

The interplay between disorder-induced localization and interactions has attracted a great deal of attention in recent years. The simplest model including both aspects in a nutshell is a Hubbard model with random site energies and a local repulsive interaction for either bosons or fermions with opposite spin. Unfortunately there are essentially no analytical results for this model if both disorder and interactions are present, not even in one dimension. For 1d bosons, however, there is a weak disorder, perturbative calculation by Giamarchi and Schulz1 , who found that the super?uid ground state with quasi long range order survives disorder up to a critical point, where the e?ective exponent K in the decay of the one particle density matrix is equal to 2/3. More generally, the qualitative physics of the Bose-Hubbard model in any dimension, and in particular the scaling behaviour near critical points has been elucidated by Fisher et al.2 . For quantitative results, however, it is necessary to resort to numerical simulations3,4 . The latter were performed using path integral (or “world line”) Monte Carlo calculations which become increasingly di?cult in the most interesting limit of zero temperature. Now at least in one dimension there is an inherently zero temperature numerical technique for interacting quantum problems, the density matrix renormalization group (DMRG) method developed by White5 . It is therefore natural to try employing this method to the disordered Bose-Hubbard model in one dimension. This was ?rst done by Pai et al.6 , who calculated the associated phase diagram for integer ?lling. As expected, it exhibits a Mott insulating, a super?uid and also a Bose glass phase, the latter appearing only for su?ciently strong disorder. Quite recently, however, their results were seriously questioned by Prokof’ev and Svistunov, who performed rather precise quantum Monte Carlo calculations7 . Based on that, it was argued that the DMRG method is intrinsically unable to deal with disordered systems because randomness would invalidate building up a system in a block like fashion. Our aim in the present work is to show that a careful DMRG calculation can indeed be successfully applied in the presence of quenched disorder. In particular, we provide a quantitative phase diagram for the 1d disordered Bose-Hubbard model at both integer and half ?lling. For

integer ?lling it is found that the super?uid and Mott insulating state are always separated by a Bose glass phase as suggested by Fisher et al.2 The super?uid density is nonmonotonic not only as a function of interaction but also of disorder. Thus for strong repulsion increasing disorder drives a transition from a Bose glass to a super?uid. For half ?lling, where no Mott insulator exists, the super?uid density is again a nonmonotonic function of the repulsive interaction, however disorder now always suppresses super?uidity as expected. The corresponding phase diagram is in agreement with that suggested by Giamarchi and Schulz1 , however we ?nd no indication of a qualitative di?erence between the glass phase at small or large values of the repulsion (Anderson vs. Bose-glass). The Bose-Hubbard model in 1d is de?ned by the Hamiltonian1–3 t ? H =? 2 (b? bi + h.c.) + i+1

i

U 2

ni (ni ? 1) +

i i

? i ni . (1)

Here b? is the boson creation operator on site i of a 1d lati tice with L sites and ni = b? bi the corresponding local oci cupation number with eigenvalues 0, 1, 2, . . .. The kinetic energy is described by a hopping matrix element t > 0, leading to a standard tight binding band ?(k) = ?t cos k in the absence of interactions and randomness. The repulsive interaction is described by a local, positive Hubbard U which increases the energy if more than one boson occupies a given site. Finally the site energies ?i are assumed to be independent random variables with zero average and a box distribution in the interval from ?? to ?. Throughout we work in the canonical ensemble with a given (dimensionless) density n = N of bosons. L As usual we choose t = 1 as a unit of energy (note that t some authors have t instead of 2 in the hopping or 2?i in the site energies which leads to a trivial factor of two difference with our results). Apart from the density n, this leaves the two dimensionless parameters U and ? characterizing the interactions and disorder which completely specify the problem at zero temperature. In order to distinguish the various possible phases, we calculate both the energy gap Eg and the super?uid fraction ρs . The 1

energy gap which is only nonzero in a Mott insulating phase, can either be evaluated directly from a numerical calculation of the energy of the ground and ?rst excited state. Alternatively the gap can be obtained as the di?erence Eg = ?p ? ?n between the chemical potential for particle (?p = EN +1 ? EN ) or hole excitations2 (?n = EN ? EN ?1 ). We have employed both methods in order to check our results. For the super?uid fraction ρs , we use the thermodynamic de?nition proposed by Fisher, Barber and Jasnow8 . It is based on de?ning ρs via the sensitivity to a change in the boundary conditions between periodic (pbc) and antiperiodic (apbc) ones. In one dimension, at a given density n = N , the super?uid L fraction ρs is thus given by (t = 1) 2L L apbc pbc ρs = 2 · [E0 (L) ? E0 (L)] π N (2)

antiperiodic boundary condition has been implemented by replacing the hopping energy t at one of the bonds by ?t thus enforcing a localized twist in the phase by π.

n=1

4

? max Bose glass Bose glass

?

2

superfluid Mott insulator

where E0 (L) are the ground state energies for the speci?c boundary condition. In the absence of interactions and disorder it is straightforward to show that ρs = 1 for arbitrary densities n as it should be. It is important to note that it is precisely a nonvanishing value of ρs (in the limit L → ∞) which is the relevant criterion for super?uidity despite the fact that the one particle density matrix b? b0 decays to zero algebraically, i.e. only i exhibits quasi-long range order. For the numerical calculations it is obviously necessary to limit the number of bosons which can occupy a given site. In order to be able to cover also small values of U , where many bosons tend to cluster at locally favorable sites, we have truncated our basis to m = 7 states for each site i which allows up to 6 bosons occupying the same place. We have checked carefully that our results do not depend on m, which was the case at least down to U ≈ 0.5. In the DMRG calculation we studied system lengths up to L = 50 and included up to M = 190 states. For the truncation error, which is one minus the density matrix eigenvalues λα of all M states kept in the decimation,

M

0

2

4

U

FIG. 1. Phase diagram for commensurate ?lling n = 1. Error bars are mainly due to the dependence of results on the realisations of disorder. Above a disorder strength ?max = 4 it is always energetically advantageous to destroy super?uidity in favor of a glass phase.

ρ = 1?

α=1

λα ,

(3)

we ?nd values of the order of 10?10 . A very important point which turns out to be absolutely crucial for applying the DMRG to disordered systems is to apply the ?nite size (“sweeping”) algorithm5 . After the system has been grown to its full length, renormalization group transformations have not yet been able to take into account the full structure of disorder while working on shorter systems. The ?nite size algorithm then works on the complete system, and improves results essentially in a variational fashion. We ?nd good convergence of both the gap and the super?uid fraction after several sweeps. The dependence on the number of states kept was comparatively weak (also compared to the scattering of results in various realisations of disorder) such that we preferred to invest computational resources rather in sweeps. The 2

For the discussion of our results we ?rst concentrate on a commensurable density n = 1, where a Mott insulating phase is expected2 at su?ciently large U . In the limit of vanishing disorder ? = 0 the system is super?uid at small values of U with a super?uid fraction ρs which monotonically decreases from one at U = 0 to zero at U = Uc . Since the transition to the Mott insulator is driven by phase ?uctuations at a given density, it is a KosterlitzThouless like transition2 very similar to the one present in a chain of Josephson junctions with a local charging energy9 . Our numerical result for the critical value of U is Uc (? = 0) = 1.92 ± 0.04 which is surprisingly close to that found in mean ?eld theory10 . It also agrees with a very recent DMRG calculation of the Bose-Hubbard model without disorder by K¨ hner und Monien11 . They u have used the condition that the exponent K characterizing the decay of the o?-diagonal density matrix b+ b0 ? |i|?K/2 i (4)

in the super?uid phase takes on the value Kc = 1/2 at the transition12 . Note that K scales like U/t at least in a Josephson junction array description which is equivalent to the Bose-Hubbard model at large integer densities. At ?nite disorder the Mott insulating phase is suppressed because the energy gap is reduced. For vanishing hopping, i.e. U → ∞ e?ectively, the reduction2 is just 2?. Thus in the limit of large U the Mott-insulator disappears of ? > U/2. This is in fact the asymptotic behaviour of

the transition line shown in Fig. 1. For nonzero t, i.e. ?nite U the transition appears earlier, until the Mott insulator completely disappears at U < Uc (? = 0) = 1.92. Outside the Mott-insulating phase the gap vanishes, however at ?nite disorder the system need not be super?uid. Indeed calculating the super?uid fraction ρs , we ?nd that ρs is nonvanishing only in the super?uid regime in Fig. 1, which bends down to ? = 0 both near U = 0 and U = Uc (? = 0). As a consequence, at ?nite disorder, there is no direct transition from a Mott insulator to a super?uid in agreement with the arguments given by Fisher et al.2 and Freericks and Monien13 . The complete phase diagram is shown in Fig. 1. It agrees well with that found by Prokof’ev and Svistunov7 using a rather di?erent method and also with the qualitative picture put forward by Herbut14 . By contrast, there are strong, even qualitative di?erences with the phase diagram found by Pai et al.6 . Their failure to see the intervening Boseglass between the super?uid and the Mott-insulator is probably related to the fact that without the sweeping algorithm the treatment of a disordered problem by the DMRG is not reliable. Regarding the general structure of the phase diagram shown in Fig. 1, we expect that it will not be qualitatively di?erent for the two dimensional case (though the corresponding path integral Monte Carlo calculations of Krauth, Trivedi and Ceperley4 and also more recent ones15 failed to see the intermediate Bose-glass between the super?uid and the Mott-insulator). Assuming that the phase diagram of Fig. 1 is indeed generic for the disordered Bose-Hubbard model at commensurate densities, one can draw two general conclusions: (i) Since the super?uid fraction is a nonmonotonic function of U for a given disorder, repulsive interactions have a delocalizing tendency at small U but enhance localization at large U . This is in fact a general property, valid also at incommensurate densities as veri?ed by Scalettar, Batrouni and Zimanyi3 for n = 0.625 and our own results at n = 0.5. (ii) More surprisingly, for ?xed repulsion in the range U > Uc (? = 0) but not too large, increasing disorder drives a Bose-glass to super?uid transition. Thus increasing disorder may in fact favour super?uidity (see dash-dotted line in Fig. 1). The associated super?uid fraction is ?nite only for ? > ?? (U ). It ?rst increases with ? but eventually decreases to zero again at the upper boundary ?+ (U ) This e?ect may be understood by observing that with increasing distance from the Mott insulator the density of mobile particle-hole excitations increases, thus enhancing ρs . At larger values of ? the disorder induced localization takes over and ρs goes to zero again at the upper phase boundary ?+ (U ).

1.0

?=0.95

n = 0.5

0.8

?

0.6

Bose glass superfluid

0.4

0.2

0

1

2

3

4

U

FIG. 2. Phase diagram for incommensurate ?lling n = 0.5. Errors due to the dependence of results on the realisations of disorder are now much stronger (with the exception of the point at ? = 0). Above a disorder strength of 0.95 we ?nd that ρS = 0 within our numerical accuracy.

For the study of a noncommensurate density, we have chosen n = 0.5 where the Mott-insulating phase is completely absent2 . The resulting phase diagram is shown in Fig. 2. It exhibits a super?uid phase in a ?nite regime U? (?) < U < U+ (?) of the repulsion provided the disorder is below a critical value ?max ≈ 0.95. The error bars in the determination of the phase boundary are larger than in the case n = 1 because the super?uid fraction exhibits rather strong realization dependent ?uctuations. This problem becomes particularly relevant in the limit of small U . In fact noninteracting bosons are a singular limit of the disordered Bose-Hubbard model since particles will collapse into the single level with the lowest ?i , which will vary between di?erent realizations. For small but ?nite U the ground state densities are still rather nonuniform. Now on the basis of that, it has been conjectured by Scalettar et al.3 that there are two qualitatively di?erent localized states, a suggestion originally due to Giamarchi and Schulz1 . The two phases would be separated by a line ?c (U ) above ?max which meets the phase boundary to the super?uid at a multicritical point. In order to look for signatures of this boundary at ? > ?max , we have calculated the expectation value of the dimensionless disorder energy per particle S= 1 ?N ? i ni ,

i

(5)

which is ?nite for a localized state4 . Although S becomes increasingly negative as U is lowered, approaching the limit S = ?1 at U ? 1, we have found no indication of any abrupt changes. This suggests that there is no quantitative distinction between an “Anderson glass” 3

for small U and a Bose-glass for larger repulsion. Verly likely it is only the line U = 0 which is singular. This point of view is supported further by the fact that the phase diagram found by Prokov’ev and Svistunov7 on the basis of the Giamarchi and Schulz criterion1 Kc = 2/3 for the renormalized exponent in the decay of the o?diagonal density matrix (4) essentially coincides with our results. Thus for any point on the phase boundary between the super?uid and the Bose-glass, scaling is towards ? = 0, K = 2/3 even for very small U . Finally we have also determined the super?uid fraction as a function of U , which exhibits a maximum at U ≈ 1 ? 1.5. Unlike the case for n = 1 this maximum does not scale to larger U if ? is increased. For very small ? the critical value Uc (? = 0+ ) = 3.2(2) beyond which ρs vanishes in the presence of even a very small randomness has been determined by calculating the exponent K in the pure system and using the criterion Kc = 2/3. Quite generally, however, the numerical calculation of ρs becomes rather di?cult for small disorder. This is probably re1 1/(3?2/K) of the lated to the strong divergence ξ ? ? localization length in the limit ? → 0 near the critical point Kc = 2/3, which follows from the integration of the Giamarchi and Schulz ?ow equations. For vanishing disorder ρs is ?nite for arbitrary values of U , approaching ρs = 2/π as U → ∞ where the Bose-Hubbard model at n = 1 is equivalent to the exactly soluble quantum 2 XY-model16 in zero magnetic ?eld. Since K = ∞ in this limit, the localisation length in the XY-model with 1 1/3 . a random local ?eld is expected to diverge like ? In conclusion we have demonstrated that the DMRG method can be successfully applied to systems with quenched disorder. The phase diagram of the 1d BoseHubbard model has been determined both at integer and at half ?lling. It exhibits signi?cant di?erences with earlier DMRG results6 but essentially agrees with a very recent quantum Monte Carlo calculation7 . Our conclusions quantitatively support the general picture for the disordered Bose-Hubbard model developed by Giamarchi and Schulz1 and by Fisher et al.2 . The model studied here is probably the simplest example for the interplay between interactions and disorder and as such is clearly of interest in itself. Experimental realisations e.g. in terms of vortices in a 1d array of Josephson junctions with disorder17 or the recent suggestion that Bose-Hubbard physics may be relevant for cold atoms in optical lattices18 , will certainly further the interest in this model. Acknowledgements: Useful comments by D.S. Fisher are gratefully acknowledged.

T. Giamarchi and H.J. Schulz, Europhys. Lett. 3, 1287 (1987). 2 M.P.A. Fisher, P.B. Weichmann, G. Grinstein and D.S. Fisher, Phys. Rev. B 40, 546 (1989). 3 R.T. Scalettar, G.G. Batrouni and G.T. Zimanyi, Phys. Rev. Lett. 66, 3144 (1991). 4 W. Krauth, N. Trivedi and D. Ceperley, Phys. Rev. Lett. 67, 2307 (1991). 5 S.R. White, Phys. Rev. Lett. 69, 2863 (1992). 6 R.V. Pai, R. Pandit, H.R. Krishnamurthy and S. Ramasesha, Phys. Rev. Lett. 76, 2937 (1996). 7 N.V. Prokof’ev and B.V. Svistunov, Phys. Rev. Lett. 80, 4355 (1998). 8 M.E. Fisher, M.N. Barber and D. Jasnow, Phys. Rev. A 8, 1111 (1973). 9 R.H. Bradley and S. Doniach, Phys. Rev. B 30, 1138 (1984), see also W. Zwerger, Europhys. Lett. 9, 421 (1989). 10 L. Amico and V. Penna, Phys. Rev. Lett. 80, 2189 (1998). 11 T.D. K¨hner and H. Monien, Phys. Rev. B 58, R14741 u (1998). 12 T. Giamarchi and A.J. Millis, Phys. Rev. B 46, 9325 (1992). 13 J.K. Freericks and H. Monien, Phys. Rev. B 53, 2691 (1996). 14 I. Herbut, Phys. Rev. B 57, 13729 (1998). 15 J. Kisker and H. Rieger, Phys. Rev. B 55, R11981 (1997); F. Pazmandi and G.T. Zimanyi, Phys. Rev. B 57, 5044 (1998). 16 E. Lieb, T. Schultz and D.C. Mattis, Ann. Phys. 16, 407 (1961). 17 A. van Oudenaarden, S.J.K. Vardy and J.E. Mooji, Phys. Rev. Lett. 77, 4257 (1996). 18 D. Jaksch et al. , Phys. Rev. Lett. 81, 3108 (1998).

1

?

Present address: Department of Theoretical Physics, University of Oxford, 1 Keble Road, OX1 3NP Oxford, UK.

4

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