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Bose-Einstein Condensation in Quantum Glasses

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Bose-Einstein Condensation in Quantum Glasses Bose-Einstein condensation in quantum glasses Giuseppe Carleo,1 Marco Tarzia,2 and Francesco Zamponi3, 4 1SISSA – Scuola Internazionale Superiore di Studi Avanzati and CNR-INFM DEMOCRITOS – National Simulation Center, via Beirut 2-4, I-34014 Trieste, Italy. 2Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, Universit´ePierre et Marie Curie-Paris 6, UMR CNRS 7600, 4 place Jussieu, 75252 Paris Cedex 05, France. 3Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA 4Laboratoire de Physique Th´eorique, Ecole Normale Sup´erieure, UMR CNRS 8549, 24 Rue Lhomond, 75231 Paris Cedex 05, France. The role of geometrical frustration in strongly interacting bosonic systems is studied with a com- bined numerical and analytical approach. We demonstrate the existence of a novel quantum phase featuring both Bose-Einstein condensation and spin-glass behaviour. The differences between such a phase and the otherwise insulating “Bose glasses” are elucidated. Introduction. Quantum particles moving in a disor- frustrated magnets [10], valence-bond glasses [11] and dered environment exhibit a plethora of non-trivial phe- the order-by-disorder mechanism inducing supersolidity nomena. The competition between disorder and quan- on frustrated lattices [12]. Another prominent manifes- tum fluctuations has been the subject of vast literature tation of frustration is the presence of a large number [1, 2] in past years, with a renewed interest following of metastable states that constitutes the fingerprint of from the exciting frontiers opened by the experimental spin-glasses. When quantum fluctuations and geomet- research with cold-atoms [3, 4]. One of the most striking rical frustration meet, their interplay raises nontrivial features resulting from the presence of a disordered ex- questions on the possible realisation of relevant phases ternal potential is the appearance of localized states [1]. of matter. Most pertinently to our purposes: can quan- Localization happens both for fermions and bosons [2], tum fluctuations stabilise a superglass phase in a self- but in the latter case one has to introduce repulsive inter- disordered environment induced by geometrical frustra- actions to prevent condensation of particles in the lowest tion? Hereby we answer this question demonstrating energy state. This results in the existence of an insu- that repulsively interacting bosons can feature a low- lating phase called “Bose glass”, characterized by a finite temperature phase characterised both by spin-glass or- compressibility and gapless density excitations in sharp der and Bose-Einstein condensation. Such a frustration contrast to the Mott insulating phase [2, 5]. induced superglass sheds light onto a novel mechanism On the other hand, latest research stimulated by the for glass formation in bosonic systems noticeably differ- discovery of a supersolid phase of Helium has brought ent from the localization effects leading to “Bose glass” to the theoretical foresight of a “superglass”phase [6, 7], insulators and paving the way to a better understanding corroborated by recent experimental evidence [8], where of this new phase of the matter. a metastable amorphous solid features both condensa- Model. Strongly interacting bosons on a lattice can tion and superfluidity, in absence of any random exter- be conveniently described by means of the extended Hub- nal potential. The apparent irreconcilability, between the bard Hamiltonian, namely current picture of insulating “Bose glasses”and the emer- † † gence of this novel phase of matter, calls for a moment H = −t bi bj + bibj + V ninj , (1) of thought. Although it has been recently demonstrated hi,ji hi,ji X h i X that attractively interacting lattice bosons can overcome b † the localization induced by an external random potential where bi (bi) creates (destroys) a hard-core boson on site arXiv:0909.2328v2 [cond-mat.dis-nn] 3 Nov 2009 † and feature a coexistence of superfluidity and amorphous i, ni = bi bi is the site-density and the summations over order [9], a general understanding of the physics of Bose- the indexes hi, ji are extended to nearest-neighbouring Einstein condensation in quantum glasses and in presence vertices of a given lattice with L sites. In the following of purely repulsive interactions is still in order. In partic- we will set t = 1, i.e. we will measure all energies in units ular, we wonder what could be the possible microscopic of t. In this work, to capture the essential physics of the mechanism leading to super-glassines and if the external problem in exam, we adopt a minimal and transparent disorder, current paradigm in the description of quantum strategy to induce geometrical frustration in the solid. glasses, could be replaced by some other mechanism. We therefore consider the set of all possible graphs of L In this Letter we show that geometrical frustration sites, such that each site is connected to exactly z = 3 is the missing ingredient. Geometrical frustration is a other sites, and give the same probability to each graph well recognised feature of disordered phases in which the in this set. We will discuss average properties over this translational symmetry is not explicitly broken by any ensemble of random graphs in the thermodynamic limit external potential. Examples are spin liquids phases of L → ∞. The motivations for this choice are the follow- 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 nu- merically 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].
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