Bose-Einstein Condensation in Quantum Glasses

Home , Superglass

arXiv:0909.2328v2 [cond-mat.dis-nn] 3 Nov 2009 xenlptnil xmlsaesi iud hssof phases liquids spin any are by Examples broken explicitly potential. not the external a which is is in symmetry phases frustration disordered translational Geometrical of feature recognised ingredient. well missing the is mechanism. other some by replaced be quantum of could description glasses, external the in the microscopic paradigm if current possible and disorder, the super-glassines to be leading could partic- mechanism In what wonder order. in we still ular, is interactions repulsive presence purely Bose- in of of and glasses physics quantum the in of condensation Einstein understanding general amorphous a and [9], superfluidity order of coexistence potential a random feature external and an by induced moment localization demonstrated the a recently for been calls has that it matter, Although of thought. phase of novel this emer- the of glasses”and gence “Bose insulating of picture current condensa- potential nal both superfluidity, features and solid tion amorphous where [8], metastable 7], evidence brought a [6, experimental phase has recent “superglass” a by Helium of corroborated of foresight theoretical phase the supersolid to a of discovery 5]. [2, phase insulating sharp Mott in the an excitations to contrast density of gapless existence and the compressibility in results This lating lowest the in state. particles inter- energy of repulsive introduce condensation [2], to prevent has to bosons one actions case and latter fermions the [1]. in for states but localized both ex- of happens disordered appearance Localization a the of is striking presence potential most ternal the the following from of experimental One resulting interest the 4]. features renewed by [3, cold-atoms a opened with with frontiers research exciting years, literature the vast past of from in subject 2] quan- the and [1, been disorder has between fluctuations tum competition phe- non-trivial The of plethora nomena. a exhibit environment dered nti etrw hwta emtia frustration geometrical that show we Letter this In the by stimulated research latest hand, other the On Introduction. attractively hs ald“oegas,caatrzdb finite a by characterized glass”, “Bose called phase 2 aoaor ePyiu herqed aMatière Condens´ la Théorique de Physique de Laboratoire 3 rneo etrfrTertclSine rneo Unive Princeton Science, Theoretical for Center Princeton h paetircniaiiy ewe the between irreconcilability, apparent The . ie ueia n nltclapoc.W eosrt th demonstrate We approach. analytical and numerical bined hs n h tews nuaig“oeglasses “Bose insulating otherwise the and spin-glass phase and a condensation Bose-Einstein both featuring h oeo emtia rsrto nsrnl interactin strongly in frustration geometrical of role The neatn atc ooscnovercome can bosons lattice interacting EORTS–Ntoa iuainCne,vaBiu -,I- 2-4, Beirut via Center, Simulation National – DEMOCRITOS unu atce oigi disor- a in moving particles Quantum 1 IS coaItrainl ueir iSuiAazt a Avanzati Studi di Superiore Internazionale Scuola – SISSA nasneo n admexter- random any of absence in M NS84,2 u hmn,721PrsCdx0,France. 05, Cedex Paris 75231 Lhomond, Rue 24 8549, CNRS UMR M NS70,4paeJsiu 55 ai ee 5 France. 05, Cedex Paris 75252 Jussieu, place 4 7600, CNRS UMR oeEnti odnaini unu glasses quantum in condensation Bose-Einstein ispeCarleo, Giuseppe 4 aoaor ePyiu herqe cl oml Supéri Normale Ecole Théorique, Physique de Laboratoire 1 ac Tarzia, Marco insu- ” ecneinl ecie ymaso h xeddHub- namely extended Hamiltonian, the bard of means by described conveniently be fti e hs ftematter. the of understanding phase better glass” new a “Bose this to to way of the leading paving effects and differ- localization insulators noticeably the systems mechanism from bosonic novel ent in a formation onto glass light frustration for a sheds Such superglass or- low- induced condensation. spin-glass Bose-Einstein a by and feature both der demonstrating characterised can question phase bosons this temperature interacting answer repulsively we that Hereby a in tion? phase quan- superglass can a disordered purposes: stabilise our phases fluctuations to relevant pertinently tum of Most nontrivial realisation matter. raises possible of interplay the on their geomet- questions meet, and of fluctuations frustration number fingerprint quantum large rical the When a constitutes of that spin-glasses. presence states the metastable manifes- is of prominent frustration Another of tation [12]. lattices and frustrated [11] on glasses valence-bond the [10], magnets frustrated nti e.W ildsusaeaepoete vrthis over properties average L discuss graph will each We to ensemble probability set. same this the in give and sites, other with lattice given a of vertices h indexes the where i ewl set will we ie,sc htec iei once to connected is site each that solid. of such the set sites, in the transparent consider frustration and therefore geometrical minimal We induce a to adopt we strategy exam, in problem of r elucidated. are 2 , Model. n n rnec Zamponi Francesco and ∞ → t nti ok ocpueteesnilpyiso the of physics essential the capture to work, this In . i order-by-disorder e nvri´ iree ai ui-ai 6, Curie-Paris Marie et Université Pierre ee, = eaiu.Tedffrne ewe such between differences The behaviour. ooi ytm ssuidwt com- a with studied is systems bosonic g st,Pictn e esy054 USA 08544, Jersey New Princeton, rsity, b i † h oiain o hscoc r h follow- the are choice this for motivations The . b H ( i † b xsec fanvlqatmphase quantum novel a of existence e b b frno rpsi h hroyai limit thermodynamic the in graphs random of i i t niomn nue ygoerclfrustra- geometrical by induced environment togyitrcigbsn naltiecan lattice a on bosons interacting Strongly rae dsry)ahr-oebsno site on boson hard-core a (destroys) creates ) = stest-est n h umtosover summations the and site-density the is ,ie ewl esr l nrisi units in energies all measure will we i.e. 1, = h ,j i, − i t h X r xeddt nearest-neighbouring to extended are i,j 41 ret,Italy. Trieste, 34014 dCNR-INFM nd i h ehns nuigsupersolidity inducing mechanism b i † eure, b j ,4 3, + b i b L j † i l osbegraphs possible all ie.I h following the In sites. + V h X i,j i exactly n i n j , z of self- 3 = (1) L 2

0 2 . for the model defined on a random graph and on a square lattice [15, 16]. Hence, we believe that it is possible to 0.15 β = 1.5 find a model similar to Eq. (1), defined on a square lattice β = 2 β = 3 but with slightly more complicated interactions (proba- EA 0.1 q β = 4 bly involving many-body terms) that will show the same = 5 β qualitative behaviour of the model investigated here. 0.05 Methods. The stochastic sampling of the quantum b 0 partition function Z = Tr e−βH at finite temperature 012345 0.4 T = 1/β can be conveniently exploited to obtain numerically exact properties of a generic bosonic Hamilto- 0.3 nian such as (1). Quantum Monte Carlo (QMC) schemes based on the original Worm algorithm idea [19] have been

/ρ 0 2 recently extended to Canonical ensemble simulations [20,

c . ρ SGF 21]. These methods offer an efficient scheme based on the 0.1 sampling of the configuration space spanned by the ex- b b −(β−τ)H −τH tended partition function Zw(τ) = Tr e We , 0 where W is a suitable worm operator determining an 0 1 2 3 4 5 0.12 imaginary time discontinuity in the world-lines. Wec have L = 240 chosenc the worm operator introduced in [21], which is a = 160 0 09 L linear superposition of n-body Green functions, avoiding . L = 80 the complications arising in [20] where the commutability

SG 0.06 of the worm operator with the non-diagonal part of the χ Hamiltonian is required. Full details of the Stochastic 0.03 Green Function (SGF) method are described in Ref. [21], 2.5 2.75 3 we only stress here that access to exact equal-time ther- 0 mal averages of n-body Green functions is granted as 0 0.5 1 1.5 2 2.5 3 3.5 well as to thermal averages of imaginary time correla- V tion functions of local, i.e. diagonal in the occupation Figure 1: Edwards-Anderson order parameter (top) and con- numbers representation, quantum operators. densate fraction ρc/ρ (middle) as functions of V at half- filling, computed via the cavity method at different values A different and complementary approach to models de- of β. In the middle panel ρc/ρ as obtained by SGF at fined on random lattices consists in solving them exactly β = 5 is reported. (Bottom) Scaled spin-glass susceptibility in the thermodynamic limit L → ∞, by means of the cav- 5/6 χSG = χSG/L reported as a function of V ; standard finite- ity method [14]. Since local observables are self-averaging size-scaling arguments [22] show that the different curves must in this limit, this results in automatically taking into ac- intersect at the spin-glass transition. count the average over the different realisations of the random graphs. For bosonic systems, the cavity method allows to reduce the solution of the model to the prob- ing: i) On a square lattice, model (1) is known to produce lem of finding the fixed point of a functional equation for a solid insulating phase at high enough density, where the local effective action, in a similar spirit to bosonic the particles are arranged in a checkerboard pattern [13]. DMFT. All the details of the computation have been dis- This is due to the fact that all loops have even length. cussed in [18], where it has been shown that the method On the contrary, typical random graphs are character- allows to compute the average of all the relevant observ- ized by loops of even or odd length; in the classical case ables. However, in the simplest version discussed in [18], t = 0, this frustrates the solid phase enough to produce a the cavity method can only describe homogeneous pure thermodynamically stable glass phase at high density [14]. phases such as the low-density liquid. In order to de- ii) Typical random graphs have the important property scribe exactly the high density glassy phase, where many that they are locally isomorphic to trees, since the size different inhomogeneous states coexist, one has to intro- of the loops scales as ln L for large L: indeed, this is a duce a generalization of the simplest cavity method which consistent way of defining Bethe lattices without bound- goes under the name of replica symmetry breaking (RSB). ary [14]. This locally tree-like structure allows to solve Unfortunately, this is already a difficult task for classical the model exactly, at least in the liquid phase, by means models, in particular in spin-glass like phases [14]. Hence, of the cavity method [17, 18]. iii) These lattices are quite in this paper we describe the glassy phase using the sim- different from square lattices. Yet, it has been shown in plest version of the method, the so-called replica sym- the classical case, and for some more complicated interac- metric (RS) one. This yields an approximate description tions, that the phase diagram is qualitatively very similar of the glassy phase which we expect to be qualitatively 3

0.8 which can be easily computed by the cavity method, or 0.7 Liquid by the divergence of the spin-glass susceptibility 0.6 β 1 2 χSG = dτ hδni(0)δnj (τ)i , (4) 0.5 L Z0 i,j T X 0.4 Superﬂuid Glass which is more easily accessible in SGF. It is possible to 0.3 show [22] that χSG is the susceptibility naturally associ- 0.2 ated to the order parameter qEA, because it can be de-

Superglass fined as the derivative of qEA with respect to an external 0.1 0 0.5 1 1.5 2 2.5 3 3.5 4 field coupled to the order parameter itself (as in standard critical phenomena). V At half-filling factor ρ =1/2, the condensate fraction, Figure 2: Finite temperature phase diagram at half-filling. the Edwards-Anderson order parameter, and the scaled spin-glass susceptibility are shown in Fig. 1. In middle panel we compare the values of condensate fraction ob- correct. To summarize, in the low-density liquid phase tained via the cavity method and via SGF in a linear we can compute averages numerically with SGF and ana- extrapolation to L → ∞. The very good coincidence lytically with the cavity method, and we obtain a perfect of these results supports our conjecture that the approxi- agreement between the two results. In the glassy phase, mate RS description of the glass phase we adopted here is the RS cavity method is only approximate, an exact so- quantitatively and qualitatively accurate. At the lowest lution for L → ∞ requiring the introduction of RSB. temperature, the system becomes a glass around V ∼ 2.7 On the other hand, SGF is limited for large L by the while it still displays BEC; the condensate fraction only unavoidable divergence of equilibration times due to the vanishes at V ∼ 3.5 inside the glass phase. This clearly glassy nature of the system. Still, we find a good agree- establishes the existence of a zero-temperature superglass ment between the result of SGF for fairly large L, where < < phase in the region 2.7 ∼ V ∼ 3.5. Note additionally that the system can still be equilibrated, and the RS cavity both transitions are of second order, hence the conden- method for L → ∞, making us confident that the qual- sate fraction is a continuous function; since the latter itative and quantitative picture of the glassy phase we stays finite on approaching the spin-glass transition from obtained here is fully consistent. Moreover, we solved the liquid side (where the cavity method gives the exact the model at the simplest (one-step) RSB level in some solution), it must also be finite on the glass side just after selected state points and we found a very small quanti- the transition. In Fig. 2 we report the finite temperature tative difference with the RS solution. phase diagram of the model at half-filling. It is defined by Results. The presence of off-diagonal long range or- two lines: the first separates the non-condensed (hbi=0) der can be conveniently detected by considering the large from the BEC (hbi= 6 0) phase, the second separates the separation limit of the one-body density matrix, i.e. the glassy (qEA 6= 0) from the liquid (qEA = 0) phase. The condensate density reads intersection between these two lines determines the existence of four different phases (normal liquid, superfluid, † 2 ρc = lim bi bj = |hbii | , (2) normal glass, superglass). |i−j|→∞ Ground-state degeneracy. Geometrical frustration in- D E duces the existence of a highly degenerate set of ground- where the square brackets indicate a quantum and ther- states, each of them characterized by a different average mal average and the bar indicates averages over inde- on-site density, which is absent in glassy phases induced pendent realizations of the random graphs. The cav- by localization in disordered external potentials such as ity method works in the grand-canonical ensemble and the Bose glass. To demonstrate this peculiar feature, it is gives direct access to the average of b, while canonical instructive to consider a variational wave-function explic- ensemble simulations done with SGF give easy access to itly breaking the translational symmetry of the lattice the one-body density matrix. On the other hand, spin- glass order is signaled by the breaking of translational −1 L n invariance, namely hnii 6= L i=1hnii = ρ. Introduc- h |Ψαi ∝ exp αini , (5) ing δn = (n − ρ), the on-site deviation from the av- " i # i i X erage density, spin-glass orderP can be quantified by the where the variational parameters α are explicitly site- Edwards-Anderson order parameter i dependent and tend to (dis)-favour the occupation of a L given site. In the spin-glass phase of the bosons, the opti- 1 2 mal set of the variational parameters is highly dependent qEA = hδn i , (3) L i i=1 on the initial conditions associated with the αi, whereas X 4

0.8 acterization of its properties. 0.6 Acknowledgements. We wish to warmly thank G. Se- 0.4 merjian who participated to an early stage of this work 0.2 and gave us many important suggestions. We also thank S. Baroni, F. Becca, G. Biroli, S. Moroni, and S. Sorella /ρ

i 0 for many precious discussions. G.C. acknowledges the δn −0.2 allocation of computer resources at the CINECA super- −0.4 computing center from the CNR-INFM Iniziativa Calcolo −0.6 per la Fisica della Materia program. −0.8 Note added. After this work was completed, we be- 0 10 20 30 40 50 60 70 80 came aware of the paper [25] where related results have been obtained. i Figure 3: Variational expectation values of the site density for diﬀerent sets of the optimized parameters at half-ﬁlling density for L = 80 and V = 4.

[1] P. W. Anderson, Phys. Rev. 109, 1492 (1958). all the variational states, even with different parameters, [2] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40, 546 (1989). have almost degenerate variational energies. Each set of [3] G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, optimized variational parameters is then representative M. Zaccanti, G. Modugno, M. Modugno, and M. Ingus- of one of the many degenerate ground-states of H. As cio, Nature 453, 895 (2008). an example, we show in Fig. 3 the variational expecta- [4] J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, tion values of the site densities for two differentb solu- P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, tions resulting from the minimization of the variational and A. Aspect, Nature 453, 891 (2008). energy with the SRH method [23], a robust stochastic [5] W. Krauth, N. Trivedi, and D. Ceperley, Phys. Rev. Lett. 67, 2307 (1991). variant of the Newton Method. We further checked, us- [6] M. Boninsegni, N. Prokof’ev, and B. Svistunov, Phys. ing the zero-temperature Green Function Monte Carlo Rev. Lett. 96, 105301 (2006). method [24], that if one applies the imaginary-time evo- [7] G. Biroli, C. Chamon, and F. Zamponi, Phys. Rev. B 78, lution |φαi = exp(−τH)|Ψαi to one of these states, the 224306 (2008). density profile remains amorphous for a time τ that is [8] B. Hunt, E. Pratt, V. Gadagkar, M. Yamashita, A. V. 324 divergent with the sizeb of the system. Balatsky, and J. C. Davis, Science , 632 (2009). [9] L. Dang, M. Boninsegni, and L. Pollet, Phys. Rev. B 79, Conclusions. The aim of this Letter was to show the 214529 (2009). existence of a stable superglass phase in a lattice model of [10] L. Capriotti, F. Becca, A. Parola, and S. Sorella, Phys. geometrically frustrated bosons, in absence of quenched Rev. Lett. 87, 097201 (2001); M. J. Lawler, H.-Y. Kee, disorder in the Hamiltonian. This has been done by Y. B. Kim, and A. Vishwanath, Phys. Rev. Lett. 100, combining the analytical solution of the model via the 227201 (2008). quantum cavity method and numerical simulations via [11] M. Tarzia and G. Biroli, Europhys. Lett. 82, 67008 Quantum Monte Carlo. The glass phase we found is (2008). [12] S. Wessel and M. Troyer, Phys. Rev. Lett. 95, 127205 very different from the usual Bose glass, since the latter (2005). is driven by localization effects in presence of an exter- [13] P. Sengupta, L. P. Pryadko, F. Alet, M. Troyer, and nal disorder and is then insulating, while the former is G. Schmid, Phys. Rev. Lett. 94, 207202 (2005). driven by self-induced frustration on a disordered lattice [14] M. Mezard and G. Parisi, Eur. Phys. J. B 20, 217 (2001). and displays Bose-Einstein condensation. This results [15] G. Biroli and M. Mézard, Phys. Rev. Lett. 88, 025501 in a coexistence of a large number of degenerate amor- (2001). phous ground states, whose existence we showed by a [16] M. P. Ciamarra, M. Tarzia, A. de Candia, and A. Coniglio, Phys. Rev. E 67, 057105 (2003). variational argument corroborated by QMC. We expect, [17] C. Laumann, A. Scardicchio, and S. L. Sondhi, Phys. by analogy with the classical case [15], that the glassy Rev. B 78, 134424 (2008). phase found here will exist also on regular finite dimen- [18] G. Semerjian, M. Tarzia, and F. Zamponi, Phys. Rev. B sional lattices, provided the interactions are modified to 80, 014524 (2009). induce sufficient geometrical frustration. In that case, its [19] N. V. Prokof’ev, B. V. Svistunov, and I. S. Tupitsyn, properties should be very similar to the one showed by Phys. Lett. A 238, 253 (1998). metastable superglasses observed both in numerical sim- [20] S. M. A. Rombouts, K. Van Houcke, and L. Pollet, Phys. Rev. Lett. 96, 180603 (2006). ulations [6] and experiments [8] on Helium 4. The main [21] V. G. Rousseau, Phys. Rev. E 77, 056705 (2008); V. G. difference is that, due to the randomness of the under- Rousseau, Phys. Rev. E 78, 056707 (2008). lying lattice, the superglass studied here is a truly stable [22] M. Guo, R. N. Bhatt, and D. A. Huse, Phys. Rev. Lett. equilibrium state, allowing for a much more precise char- 72, 4137 (1994). 5

[23] S. Sorella, Phys. Rev. B 71, 241103(R) (2005). R. G. Melko, arXiv:0909.1845 (2009). [24] S. Sorella and L. Capriotti, Phys. Rev. B 61, 2599 (2000). [25] K.-M. Tam, S. Geraedts, S. Inglis, M. J. P. Gingras, and