PHYSICAL REVIEW B 85, 104205 (2012)

Mean field theory of superglasses

Xiaoquan Yu International School for Advanced Studies (SISSA), via Bonomea 265, 34136 Trieste, Italy

Markus Muller¨ The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy (Received 5 December 2011; published 19 March 2012) We study the interplay of superfluidity and glassy ordering of hard core bosons with random, frustrating interactions. This is motivated by bosonic systems such as amorphous supersolid, disordered superconductors with preformed pairs, and helium in porous media. We analyze the fully connected mean field version of this problem, which exhibits three low-temperature phases, separated by two continuous phase transitions: an insulating, glassy phase with an amorphous frozen density pattern, a nonglassy superfluid phase, and an intermediate phase, in which both types of order coexist. We elucidate the nature of the phase transitions, highlighting in particular the role of glassy correlations across the superfluid-insulator transition. The latter suppress superfluidity down to T = 0, due to the depletion of the low-energy density of states, unlike in the standard BCS scenario. Further, we investigate the properties of the coexistence (superglass) phase. We find anticorrelations between the local order parameters and a nonmonotonous superfluid order parameter as a function of T . The latter arises due to the weakening of the glassy correlation gap with increasing temperature. Implications of the mean field phenomenology for finite dimensional bosonic glasses with frustrating Coulomb interactions are discussed.

DOI: 10.1103/PhysRevB.85.104205 PACS number(s): 61.43.−j, 67.25.dj

I. INTRODUCTION presence of slow relaxation and out-of-equilibrium phenomena due to frustrated interactions. However, if superconductivity Bosons in random environments occur in a variety of develops in a highly disordered environment, such frustration experimentally relevant systems, ranging from cold atomic may appear in the form of Coulomb interactions between the gases, superconductors, and quantum liquids. The superfluid- charged carriers, which may become important, as screening ity of helium-4 (4He) in porous media was one of the first is not very effective. It is well known that in more insulating phenomena observed in this type of system,1,2 featuring an regimes, strong disorder and Coulomb interactions may induce interesting competition between Bose-Einstein condensation 25–27 and localization by random potentials.3 a glassy state of electrons (the Coulomb glass). It is In more recent years, supersolidity in crystalline 4He4–6 therefore an interesting question whether such glassy effects has been reported. It soon became clear that defects of the can persist within the superconducting state of disordered crystalline order and amorphous solids sustain more robust films. A memory dip in resistance versus gate voltage (similar supersolidity, spurring the idea that disorder, or even glassy to the conductance dip in insulators, which is considered 28 order, may be a crucial element in understanding the superfluid a smoking gun of electron glassiness ), was reported as a part of those systems.7–11 possible indication of such a superglassy state in Ref. 29,even 30 A recent experiment12 reported indeed that the supersolidity though doubts about its intrinsic nature were raised later on. 31 in 4He is accompanied by the onset of very slow glassy The recent developments in ultracold atoms open new relaxation. This suggested that an amorphous glass with ways to studying bosonic atoms in the presence of both a superfluid component is forming, a state of matter that interactions and disorder. Those can exhibit superfluid or was dubbed a “superglass.” These experimental results have localized, and potentially also glassy phases, especially if the motivated several theoretical investigations into the possibility interactions are sufficiently long ranged and frustrated, as is and nature of such amorphously ordered and yet supersolid possibly the case for dipolar interactions. systems.13–18 Motivated by these experimental systems, we study a Similar questions as to the coexistence and interplay of solvable model of bosons subject to disorder and frustrating glassy density ordering and superfluidity arises in disordered interactions, as proposed previously in Ref. 15. This solvable superconducting films, which feature disorder- or field-driven case provides insight into the possibility of coexistence of superconductor-to-insulator quantum phase transitions.19 In superfluidity and glassy density order, as well as into the nature several experimental materials, this transition appears to be of the coexistence phase (the superglass). In particular, for driven by phase fluctuations of the order parameter rather than the considered mean field model we prove the existence of a by the depairing of electrons, suggesting that the transition superglass phase. This complements the numerical evidence can be described in terms of bosonic degrees of freedom for such phases provided by quantum Monte Carlo investiga- only.20 This had led to the dirty boson model21 and the notion tions in finite dimensions15 and on random graphs.16 Those of the Bose glass,22 in which disorder and interactions lead were, however, limited to finite temperature, and could thus to the localization of the bosons, while the system remains not elucidate the structure of the phases at T = 0. In contrast, compressible.23,24 In this context, the term “glass” refers the analytical approach allows us to understand the quantum mostly to the amorphous nature of the state rather than to the phase transition between glassy superfluid and insulator, and

1098-0121/2012/85(10)/104205(13)104205-1 ©2012 American Physical Society XIAOQUAN YU AND MARKUS MULLER¨ PHYSICAL REVIEW B 85, 104205 (2012) the nontrivial role played by glassy correlations. Indeed it is a function of T . The implications of this mean field analysis an interesting feature of the considered model that despite for realistic finite dimensional models, e.g., with frustrating its infinite connectivity, it features an insulating, Anderson long-range Coulomb interactions, will be discussed in Sec. VI. localized phase at sufficiently small but finite hopping strength. Some detailed derivations are collected in the Appendix. This is in contrast to nonglassy models where the localization and insulating behavior is lost in the large connectivity limit, II. MODEL unless the hopping is downscaled logarithmically with the connectivity.32 We consider the fully connected model of hard-core bosons The model studied here is an interesting type of a quantum with random pairwise interactions between all bosons, glass.33 Like other canonical quantum glass models of mean t † † 34 =− + − b + field type, such as the random exchange Heisenberg model, H Vij ni nj i ni (bi bj bj bi ). (1) 35,36 N the Ising spin glass in transverse field, and frustrated rotor iferromagnetism in is equivalent to the Sherrington-Kirkpatrick spin-glass (SK) a magnetic analogon. The microscopic coexistence of these model46 in a random field. It is well known that except for two types of order, which we demonstrate to exist in our lowering the transition temperature, the random fields do not model, is rather nontrivial, as one might instead expect a alter the low-temperature properties of the glass phase. We first-order transition between phases with either of the two thus restrict our attention mostly to a slightly simpler model orders. At zero temperature, the quantum glass transition proposed in Ref. 15, which corresponds to a special choice of is induced by frustrated interactions, which win over weak the i : quantum fluctuations. While the quantum phase transition of H =− V (n − 1/2)(n − 1/2) such models is understood relatively well,42 the deep glass ij i j isolid. Our model considers instead bosons from nonrandom spin-flip terms, on a predefined lattice, on which an inhomogeneous density t pattern establishes in the glass phase. H =− J szsz − sx sx + sy sy , (3) ij i j N i j i j The remainder of the paper is organized as follows: In i

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III. FREE-ENERGY AND SELF-CONSISTENT EQUATIONS δF = ⇒  = T z z  0  Qaa(τ,τ ) sa(τ)sa(τ ) eff A. General formalism δQaa(τ,τ )  The disorder average of the free energy of the model (3) ≡ R(τ,τ ), (12) can be obtained using the replica method,49 δF  n − = ⇒ x = x ≡ x Z J 1 0 Ma (τ) sa (τ) M , (13) lnZJ = lim , (5) δMx (τ) eff n→0 n a ··· where Z is the partition function and J indicates an δF = ⇒ y = y ≡ y average over the couplings Jij . 0 y Ma (τ) sa (τ) eff M , (14) Following a method introduced by Bray and Moore34 it is δMa (τ) useful to represent the partition function as an imaginary time where ···eff denotes the average with respect to the effective path integral: action Seff of a single site. We have used that, as usual, x,y 1 n the saddle-point values of Qa =b and Ma are independent n z z  Z = TrT exp β dτ Jij s (τ)s (τ) of imaginary time, while Qaa(τ,τ ) depends only on the ia ja 50 x,y 0 a=1 i

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⊥ where χ ≡ χxx(ω = 0) is the static transverse susceptibility. where the effective single replica action in a frozen field y These expressions must be calculated in the nonsuperfluid reads phase where M = 0. In this regime the effective action Seff 2 2 1 1 β J  z  z  is classical, which entails the further simplification R(τ) = 1. Seff(y) =− dτdτ s (τ)[R(τ − τ ) − qEA]s (τ ) 2 This feature is due to the suppression of quantum fluctuations 0 0 1 1 in the nonsuperfluid phase by factors of 1/N, due to the scaling −βtM dτsx (τ) − βy dτsz(τ). (23) of the transverse coupling. It allows us to find the superfluid- 0 0 insulator transition analytically, even at zero temperature, The Edwards-Anderson order parameter qEA characterizes the without solving a full quantum impurity problem. In particular, glassy freezing in a pure state of the glass and is given by we immediately find that the transition from the disordered high-temperature phase to a glassy phase is given by 2 qEA = q(x = 1) = dyP(y)m (y). (24)

Tg = J, (19) As first derived by Sommers and Dupont,52 the frozen field distribution P (y) ≡ P (y,x = 1) is obtained from a self- exactly as in the classical SK model. However, the glass consistent solution of the differential equations on the interval transition line will be modified if it is preceded by a superfluid x ∈ [0,1], transition at higher temperature. q˙(x) m˙ (y,x) =− [m(y,x) + 2xm(y,x)m(y,x)], (25) 2 B. Solution of the saddle-point equations q˙(x) P˙(y,x) = [P (y,x) − 2x(m(y,x)P (y,x))], (26) A full solution of the saddle-point equations involves the 2 solution of the problem of interacting replica as well as the evaluation of dynamical correlation functions with the with effective action Seff,ifM = 0 and the replica symmetry is m(y,x = 1) = m(y),P(y,x = 0) = δ(y), (27) broken as well. Here we describe what steps an exact solution involves, and then discuss the approximations we will use to where dots and primes denote derivatives with respect to x and study parts of the phase diagram, especially the bulk of the y, respectively. The solutions of these differential equations solve the saddle-point equations of the replica free energy superglass phase. 52 To describe a nonglassy superfluid phase, the replica (8). The overlap function q(x), which parametrizes the structure is trivial, and one needs to solve the self-consistency ultrametric matrix Qab by the distance x between replicas, must obey the self-consistency relation equations = 2 x z z q(x) P (y,x)m(y,x) . (28) M =s eff,R(τ) =T s (τ)s (0)eff, (20) Notice that Eqs. (25) and (26) are the same as in a classical spin with effective action glass. The influence of quantum fluctuations enters through m y,x = ≡ m y m y 2 2 1 1 the boundary condition ( 1) ( ), where ( )was β J  z  z  defined in Eq. (22) These differential equations provide an Seff =− dτdτ s (τ)R(τ − τ )s (τ ) 2 elegant way of integrating out all spins except for one.53 0 0 1 Once the above scheme has been solved self-consistently, x −βtM dτs (τ). (21) site-averaged observables such as the longitudinal magnetiza- 0 tion are given by These can be solved using techniques as used in dynamical 1 mean field theory.51 Mz ≡ sz = dyP(y)sz N i Seff (y) In a glassy phase the replica structure has to be taken into i account. Assuming the standard ultrametric structure of the = dyP(y)m(y). (29) saddle-point matrix Qab, the above single-replica scheme has to be generalized to include a self-consistent distribution of frozen longitudinal fields P (y) acting on a given replica. This The properties of the solution of these differential equations captures the distribution of random frozen fields y created are well understood in several classical models exhibiting i full replica symmetry breaking with continuous functions by the exchange of sites i with the frozen magnetization 54–57 pattern with a spin-glass state.52 In practice this requires the q(x). The full solution of mean field quantum glasses simultaneous solution of in the ergodicity broken has not been analyzed in the literature so far. However, an analysis of the transverse field SK model shows that most features of the low-temperature solution of m(y) =sz ,m(y) =sx  , Seff (y) x Seff (y) q(x) carry over rather naturally to the quantum case.43 A salient new feature in the quantum case is the fact that full replica M = dyP(y)mx (y), (22) symmetry breaking implies marginal stability of the whole glass phase, which in turn ensures the presence of gapless R(τ) = dyP(y)T sz(τ)sz(0) , Seff (y) collective excitations. The latter is very similar to what was

104205-4 MEAN FIELD THEORY OF SUPERGLASSES PHYSICAL REVIEW B 85, 104205 (2012) found, e.g., in the threshold states of quantum p-spin models,58 D. Static and one-step approximation or in the quantum dynamics of elastic manifolds, approximated In order to avoid solving numerically a full self-consistent 59 with a replica symmetry-breaking variational approach. quantum problem as outlined in Eqs. (22) above, we will resort to the widely used static approximation. The latter consists in C. Alternative derivation by a cavity approach seeking a minimum of the free energy not with respect to the full function space R(τ) but, instead with respect to a constant The replica-diagonal part of the above scheme will become → 60 value R(τ) R. easier to understand, if we derive it in a cavity framework A further approximation, which we will use in the study of similarly to the derivation of the quantum analog of Thouless- 58 the quantum glassy phases, is the one-step approximation for Anderson-Palmer equations by Biroli and Cugliandolo. the structure of replica symmetry breaking. It is equivalent to From a cumulant expansion in the couplings involving site assuming a step form of q(x), o it is easy to obtain the following effective action for the site o: q(x) = (x − x1)Q1 + (x1 − x)Q0, (35) eff So and optimizing the free energy over x1,Q1,Q0.Thisis β2 1 1 =−  z 2 z z  o z  expected to give qualitatively good results, especially at dτdτ so(τ) Joi si (τ)si (τ ) c so(τ ) 2 0 0 intermediate temperatures and close to the glass transition. i Combined with the static approximation for the replica diago- 1 − z z + x x nal, short-time part one obtains the free-energy functional per β dτ ho(τ)so(τ) h (τ)so (τ) . (30) 0 spin:

Here 2 2 β J 2 2 2 βt 2 βf = (x1 − 1)Q − x1Q + R + M o o 1 0 hz (τ) = J sz(τ) = J sz (31) 4 2 o oi i oi i 1 i i − Dy log Dy x 0 1 1 is the site-dependent longitudinal field, which does not depend x1 × 2 + 2 2 on time, however. The index o denotes a “cavity average,” i.e. DyR2 cosh β hy t M , (36) an average over the action of the system, in which the site o has been removed. The subscript c indicates a connected where hy = y0 + y1 + yR. Dy0, Dy1, and DyR are correlator. The effective transverse field, −y2/ Q J 2 exp√( 0 2 0 ) Gaussian measures: Dy0 = dy0, Dy1 = 2 t o t o 2πQ0J x = x = x − 2 − 2 − 2 − 2 h (τ) si (τ) si exp[√y1 /2(Q1 Q0)J ] exp[√yR /2(R Q1)J ] N N dy1, and DyR = dyR. i i 2π(Q −Q )J 2 2π(R−Q )J 2 1 0 1 t Note that Q1 = qEA is the Edwards Anderson order = sx = tM, (32) N i parameter in the one-step approximation, while Q0 is the i overlap between different spin-glass states. We point out that does not fluctuate from site to site, and is independent of τ if the above free energy differs from the expression given in we neglect subleading terms, which scale as inverse powers of Ref. 15, where the static approximation was not carried out N. Note that for large N correctly. This error was at the origin of several strange features of the phase diagram reported there, such as a o J 2 sz(τ)sz(τ ) → J 2[R(τ − τ ) − q ], (33) T -independent transition between superfluid and superglass oi i i c EA and a J -independent superfluid transition. i

z independently of the site o. The distribution of hi over the sites i is the frozen field distribution, E. 1RSB free-energy and self-consistent equations Here we rewrite the one-step self-consistency equations N with the help of the local-field distribution. = −1 − z P (y) N δ y hi , (34) The effective partition function of a single spin is i=1 Z (y) = TrT exp[−S (y)] computed in the replica formalism. Thus we precisely recover eff eff the self-consistency problem for the replica diagonal, while β2J 2 1 1 = TrT exp dτdτ the solution of the replica off-diagonal part furnishes the 2 distribution P (y). 0 0 × sz(τ)[R(τ − τ ) − q ]sz(τ ) For the study of the phase transition from the insulating EA glass phase into the superfluid, it will prove crucial to use the 1 1 + βtM dτsx (τ) + βy dτsz(τ) . (37) full low-temperature solution of the SK model. However, in 0 0 order to analyze properties of the mixed superglass phase we will restrict ourselves to a one-step approximation, which we In the case of one-step replica symmetry breaking (RSB), discuss in the next section. the frozen field distribution within one pure state can be

104205-5 XIAOQUAN YU AND MARKUS MULLER¨ PHYSICAL REVIEW B 85, 104205 (2012) obtained by stepwise integration of the flow equations (25) capture equilibrium states, we should further extremize with ∂f and (26), yielding [cf. Eq. (49)] respect to the Parisi parameter x1, i.e., = 0, which yields ∂x1 − + x1 + the further condition Dy1δ[y (y0 y1)]Zeff(y0 y1) P (y) = Dy0 , (38) 2 2 x1 + β J Dy1Zeff(y0 y1) − 2 − 2 2 Q1 Q0 m 4 where Dy1 is a Gaussian measure like Dy1 with variance (Q − Q )J 2. = x1 + 1 0 Dy0ln Dy1Zeff(y0 y1) Under the static approximation, Eq. (37) becomes x1 + + Dy1Zeff(y0 y1)lnZeff(y0 y1) − x1 Dy0 . (49) Zeff(y) = DyRZstat(y + yR), (39) x1 + Dy1Zeff(y0 y1) where It is a useful check that upon imposing Q1 = Q0,the saddle-point equations for Q and Q reduce to the same Z y = β y2 + M2t2 . 0 1 stat( ) 2 cosh( ) (40) replica symmetric constraint. When M = 0, the local-field One can interpret yR as a random field, which is generated distribution, the free energy and the saddle-point equations by the thermal fluctuations of the nonfrozen part of the reduce to those of the classical SK model, as it should be. magnetization. The longitudinal and transverse magnetizations of a spin in IV. PHASE DIAGRAM a frozen field y introduced in Eqs. (22) are easily seen to be given by Let us now study the phase diagram of our model (3). The gross features of the phase diagram we find are similar 1 ∂ = z = to the ones found in Refs. 15 and 16: The low-temperature m(y) s Seff (y) ln[Zeff(y)], (41) β ∂y phase exhibits three phases: a nonglassy superfluid at small J/t, an insulating (nonsuperfluid) glass phase at large J/t, 1 ∂ = x  = and, most interestingly, a phase in between with both glassy mx(y) s Seff (y) ln[Zeff(y)]. (42) β ∂(tM) order and superfluidity. However, as mentioned before, we The saddle-point equations for the Edwards-Anderson param- find a distinctly different behavior of the phase boundaries eter Q1 and the superfluid order parameter M can now be than Ref. 15. expressed as Moreover, we are able to analyze the limit T → 0, whose properties were inaccessible in previous works.15,16 The latter 1 Q = sz 2 = dyP(y)m2(y), (43) is of particular interest in the context of the superfluid-insulator 1 N i i transition. The findings of the mean field analysis are in qualitative 1 agreement with Monte Carlo studies in finite dimensions at M = sx = dyP(y)m (y). (44) N i x low but finite temperatures. The analytical approach allows i for a detailed analysis of the properties of the mixed phase, The saddle-point equation for the parameter R reads and of the glass-to-superglass transition.

β(R − Q1) = P (y)χ (y)dy, (45) loc A. High-temperature phase which relates the static approximation of the connected sz The high-temperature phase is simple to describe. Since correlator, R − Q1, to the average local susceptibility M = 0, the system behaves identically to the paramagnetic phase of the classical SK model, and R = 1 holds exactly. In ∂m(y) χ (y) = . (46) this regime the static approximation is of course exact. loc ∂y At large enough J/t, the leading instability upon lowering The saddle-point equation for the Q0 can be written in a similar the temperature is the classical glass transition at Tg = J , way: as mentioned earlier. However, at small values of J/t,the tendency to form a superfluid wins. The instability condition 2 Q0 = dy0P (y0; x1)m (y0; x1), (47) toward XY symmetry breaking, = where ⊥ ∂mx(y 0) tχ = t = 1, (50) 2 ∂h 1 y x hx =0 P (y ; x ) = exp − 0 , 0 1 2 2 2Q0J can be evaluated exactly. In this expression h is a 2πQ0J x x1 + + uniform transverse field. The transverse susceptibility is Dy1Zeff(y0 y1)m(y0 y1) m(y0; x1) = , (48) easily calculated for the replicated Hamiltonian with a DyZx1 (y + y) 1 eff 0 1 Hubbard-Stratonovich transformation of the quadratic term are discrete versions of the continuous functions R dτdτsz(τ)sz(τ ). This results in the instability criterion P (y,x),m(y,x) introduced above. 1 dhe−h2/2J 2 sinh(βh)/βh Optimizing the one-step free energy with respect to Q1, M, = ⊥ = χ β − 2 2 . (51) R, and Q0 yields the saddle-point equations Eqs. (43)–(47). To t dhe h /2J cosh βh

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1.2 in the paramagnetic phase. Note that the instantaneous field distribution is not a simple Gaussian, but small fields are 1.0 under-represented. This phenomenon is closely related to the PM suppression of small fields encountered in the cavity approach 0.8 to Ising systems,60 and is a precursor effect of the opening of the pseudogap in glassy phases at low temperatures.26,56,57 T/t 0.6

The total transverse susceptibility is obtained as an average of the local susceptibility χ(h) over Pinst(h): 0.4 SF G ⊥ ⊥ χ = dhPinst(h)χ (h) 0.2 SuG −h2/2J 2 −β2J 2/2 e sinh βh 0.0 = e dh√ , (55) 0.00.20.40.60.81.01.2 2πJ2 h J/t which indeed coincides with the replica result (51). The static approximation for superfluid phases has a FIG. 1. (Color online) Phase diagram of glassy hard-core bosons. = completely analogous effect. The approximation replaces the At high temperature, the straight blue line T J indicates the dynamically fluctuating exchange fields on the various sites classical SK glass transition line. The red solid line shows the by a random distribution of quasistatic fields. The latter differs superfluid phase boundary which is given by the instability condition from the distribution of frozen fields (which is Gaussian at (51). The two lines cross at the tricritical point (T/t) = (J/t) = T T 2 − 0.7248. At low temperature, the blue line shows the phase boundary high T ) by a random Gaussian smearing with variance J (R of the glass within the superfluid phase, as evaluated within the static qEA), and a reweighing factor proportional to cosh(βh), which approximation, cf. Eq. (16). The glass transition at T = 0 occurs at accounts for the fact that a small instantaneous field is less likely to be observed on a given site, as it implies a positive (J/t)g,stat = 1/2. The red solid line indicates the location of the onset of superfluidity within the glass phase, as evaluated within the full free-energy fluctuation in the environment. breaking of the replica symmetry to the instability condition (63). = = The superfluid transition at T 0 takes place at (J/t)s 1.00. B. Onset of glassy order within the superfluid = The glass transition and the superfluid transition line cross at The instability toward forming a glass occurs when Jχ  −  = the tricritical point, βJ dτdτ R(τ τ ) 1. Within the superfluid phase it is difficult to calculate this susceptibility exactly, and we thus −z2/2 −  → dze sinh(z)/z first resort to the static approximation, R(τ τ ) R.The TT = JT = tT = 0.7248 tT . (52) dze−z2/2 cosh z instability of the statically approximated free energy occurs when JβR = 1, βR being the static approximation for the The result (51) does not have the familiar looking form of an longitudinal susceptibility χ . Within the nonglassy superfluid average local transverse susceptibility. However, it can indeed phase there are no frozen fields, P (y) = δ(y). Thus, from be recast in such a way. This furnishes us a better understanding Eqs. (44) and (45), the two relevant saddle-point equations of the interaction effects in the high-temperature phase, and at read the same time illuminates the nature of the static approximation = = in the superfluid phases. M mx (y 0), (56) Let us rederive the above result directly from the nonrepli- = = βR χloc(y 0), (57) cated Hamiltonian:

where mx (y = 0) and χ (y = 0) are to be evaluated from 1 z z t x x y y x loc H =− Jij s s − s s + s s − hx s , Eqs. (39)–(42) and (46). 2 i j 2N i j i j i ij ij i They have a relatively simple low-temperature limit. One (53) verifies that it is self-consistent to assume that r T where hx is an infinitesimal field. In a given classical Ising βR = χ → ,M→ 1 − m , (58) configuration, the spin i sees an “instantaneous” local field t t z = z → hi j =i Jij sj , while the transverse coupling is negligible with finite numbers r,m,asT 0. −1  x = = = Injecting this into the above self-consistency equations, and in the paramagnetic phase where N j sj M 0 −1  y  evaluating the Gaussian integral over yR in Eq. (39) around the N j sj . Thus the transverse susceptibility can be calcu- lated as a site and configuration average of the susceptibility stationary point, the equations simplify to ⊥ = of a single spin sitting in an instantaneous field h, χ (h) J 2r β dτsx τ sx = βh /h r = 1 + + O(T/t), (59) 0 ( ) (0) tanh( ) . t2 − J 2r The thermal distribution of instantaneous local fields of the 61 J 2r SK model has been well studied, and takes the rather simple m = + O(T/t). (60) form 2(t2 − rJ2) 2 2 2 exp − h − β J This yields the solution for the susceptibility Jχ = Jr/t = = √2J 2 2 √ Pinst(h) cosh(βh) (54) t + − J 2 2πJ2 2J (1 1 4( t ) ).

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The static approximation predicts the quantum glass tran- We recall that in the nonsuperfluid glass phase the static sition to occur at the critical point, approximation is exact with R = 1, so that the instability criterion can be expressed in the form J = 1 = , (T 0), (61) + 1 t g,stat 2 DyR sinh(β(y yR)) + t dyP y y yR = , ( ) + 1 (63) where Jχ = 1. DyR cosh[β(y yR)] It is difficult to predict whether we over- or underestimate y2 = √ 1 − R where DyR exp( 2 ). This condition 2 2(1−qEA)J the phase boundary with the static approximation in the 2π(1−qEA)J superfluid phase. This is because the approximation has two can be expressed in terms of the instantaneous field distribution competing effects with respect to the onset of glassy order. as On one hand, we approximate the dynamic longitudinal tanh(βh) susceptibility by the static one. Since the latter is bigger, t dhPinst(h) = 1, (64) we tend to overestimate the stability of the glassy ordering h z of s . This effect is well known from the SK model in a where (constant) transverse field .50,62,63 On the other hand, the β(h−y)2 βh static approximation underestimates quantum fluctuations of cosh βh exp − − O P (h) = P (y)dy √ 2hO 2 (65) sx ,atleastatlowT . Indeed we see above that at T = 0, the inst cosh βy 2πhO /β static approximation predicts maximal transverse order, M = 1, independently of the value of J/t, while it is easy to show is the instantaneous field distribution, which was first derived in 2 that quantum fluctuations around the transverse ferromagnetic Ref. 61.ThetermhO = βJ (1 − qEA) is known as Onsager’s state decrease the magnetization as M = 1 − O[(J/t)2]. The back reaction. Equation (64) can be recognized as a BCS overestimate of M leads to an underestimate of the longitudinal equation, where the instantaneous field distribution Pinst(h) susceptibility, and thus of the tendency to glassy order. In view takes the role of the density of states. of these competing tendencies, it is hard to predict on which The temperature-dependent local-field distribution can be side with respect to Eq. (61) the exact glass instability will be obtained from a numerical solution of the self-consistent located. set of full RSB equations (25)–(28), from which the phase However, there is a simple way to obtain an upper bound for boundary of the insulator-to-superfluid transition is deduced. the quantum critical point. In the superfluid phase our model This yields the solid (red) line in Fig. 1. For comparison we also (3) is very similar to the SK model in a constant transverse field evaluate the phase boundary within a one-step approximation, ,63 with the difference that the effective transverse field Mt which works well at moderate temperatures. However, it is self-generated and has to be determined self-consistently. fails badly at low T where a nonphysical reentrance of the However, it is clear that the effective transverse field is always superfluid instability would be predicted, and the quantum smaller than t. From quantum Monte Carlo results for the phase transition at T → 0 is completely missed. transverse field SK model, one knows that a quantum glass We note in passing that the thermodynamics of the phase is obtained for J/  0.76.50 This implies that the insulating phase is essentially classical because of the scaling model studied in the present work must certainly be in a glassy of the transverse√ coupling as t/N. If instead t were random and phase if J/t  0.76. The latter value is thus an upper bound scaled as 1/ N, the glass phase would also exhibit quantum for (J/t)g. Approaching from large values of J/T we will find fluctuations and would not reduce to the purely classical below in Eq. (66) that the nonsuperfluid glass phase becomes SK model. In that case, the analysis of the transition would unstable toward superfluidity already at (J/t)s = 1.00. Hence become much more complicated. However, even though the we conclude that a phase with both superfluid and glassy thermodynamics can be obtained by a purely classical saddle- order parameters exists for a substantial range of parameters point computation, one should not conclude that excitations covering at least the interval 0.76  J/t  1.00. do not have any quantum dynamics.

C. Superfluid instability within the insulating glass phase 2. Low temperatures and quantum phase transition 1. Instability criterion At low temperatures, the most prominent feature of the local-field distribution P (y) is a linear pseudogap which Our discussion of the phase boundaries will be complete, opens at small fields. The latter is required to assure the once we have addressed the superfluid instability with in the stability of the glass phase,64,65 in a very similar manner as the glass phase at large J/t. The instability condition reads Efros-Shklovskii Coulomb gap arises in electron glasses with unscreened, long-range 1/r interactions.27,57 More precisely, ∂mx (y) = t dyP(y) 1, (62) it is known that P (y) = α|y|+O(T ) with α = 0.301 for ∂hx = hx 0 fields in the range T |y|J , while the distribution where P (y) is the nontrivial distribution of frozen local fields decays like a Gaussian for |y|J . At zero temperature in the classical glass phase of the SK model. The properties of the pseudogap extends down to y = 0 (i.e., the chemical P (y) are well studied, and turn out to be crucial to understand potential in the terminology of hard-core bosons), while at the low-temperature behavior of the phase boundary and the finite but low temperatures T  J , P (y)assumesascaling physics of the glassy superfluid-to-insulator quantum phase form P (y) = Tp(y/T ) with P (0) = const and p(x  1) = transition. α|x|+const.56,57

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This scaling form asserts that only a fraction of (T/J)2 is Recalling the dictionary (4), the mean field estimate of the thermally active. Therefore the Edwards-Anderson parameter superglass-to-glassy insulator quantum phase transition is tends to 1 as 1 − q ∼ (T/J)2. Accordingly, as T → 0 √ √ EA V  MF V/ z J/ z there is no difference between the distribution of frozen and = = 4  (68) instantaneous fields, P (h), since no thermal fluctuations are t s tb/z s 2t/z s inst left. In this limit the instability condition (63) for onset of √  QMC = J  ≈ V  superfluidity then takes the form 2 z 4.9  5, t s t s which comes close to the extrapolation of QMC results to P (y; T = 0) t dy = . T = 0. The transition point between superglass and nonglassy s | | 1 (66) y superfluid is estimated from the static approximation as  MF √ V = J  Using the above-mentioned features of P (y)atlowT one  2 z 2.45 can easily obtain a rough estimate for the superfluid-insulator t g t g,stat  ±  transition point as (J/t)s 1.05 0.1. However, since the V QMC precise value is also sensitive to the part of P (y) at high fields, ≈  3.2. (69) t y  J , a full numerical evaluation of the condition (66)is g necessary to obtain the exact location of the quantum critical This indicates that the static approximation overestimates the point. Using high precision data for P (y; T )atlowT from stability of the superglass phase, similarly as what is known Ref. 66, we find (J/t)s  1.00 ± 0.01. from the mean field version of the transverse field Ising spin We emphasize an important difference between the quan- glass. tum phase transition we have found here and a standard The mean field prediction (with static approximation) for   BCS transition. The latter, in the presence of a constant the interaction-to-hopping ratio (V /t )T at the tricritical point low-energy density of states always yields a finite Tc,even is rather good, too, though it becomes exponentially small in 1/t for small t. V  MF √ J In our glassy system the situation is fundamentally different = 2 z  3.55 in that the frustrated interactions suppress the density of t t T T,stat states around the chemical potential with P (y → 0) → 0.  QMC This quenches the tendency for superfluidity and allows for ≈ V   3.8. (70) a superfluid-to-insulator transition at a finite value of t,even t T →∞ in the mean field limit of N , which we consider here. While the tricritical ordering temperature is overestimated by This has important consequences for the nature of excita- a factor of 2 (similarly as in the classical Ising spin glass),69 tions and transport properties across the superfluid-insulator transition. In particular, the transition to the Bose insulator T MF z T =  . , is accompanied by the Anderson localization of lowest  2 2 (71) t T 2 t T energy excitations, whereas higher energy excitations remain 67 delocalized relatively far into the insulator. We believe that QMC T  the physics revealed by this mean field model is relevant  1.1. (72) for Coulomb frustrated bosonic systems which undergo a t T transition from a superfluid to a Bose glass state in finite dimensions. This will be discussed in detail elsewhere.68 D. Robustness of the phase diagram to random-field disorder It is interesting to compare our mean field predictions for the phase diagram with the three-dimensional (3D) quantum In the previous sections we have seen that the model (3) Monte Carlo (QMC) simulation results reported in Ref. 15.The possesses an intermediate phase which is simultaneously su- mean field predictions for the quantum critical points actually perfluid and glassy. We have determined the phase boundaries match the numerical results surprisingly well. The latter were as instability lines, assuming second-order phase transitions. done for the Hamiltonian Indeed it seems unlikely that any of the instabilities could be preempted by a first-order transition. Since the superfluid- to-insulator transition at (J/t)s is of particular interest, we =−  − − H Vij (ni 1/2)(nj 1/2) provide further arguments in this section that the parts of i,j the phase diagram related to the phase boundary of the −  † + nonsuperfluid glass remain robust when disorder potentials, t (bi bj H.c.), (67) 2 i.e., random fields i of variance W , are restituted to the i,j model. In particular we will show that glass and superfluid transition lines meet at a tricritical point at finite temperature  =±  with binary disorder, Vij V with equal probability. TT /J and (J/t)T . Further, we determine the superfluid Contact with the mean field model (2) is made by replacing instability of the glass phase at T = 0 and show that it the coordination number with N → z = 6 for the 3D cubic always occurs at a larger ratio (J/t) than the tricritical point, lattice, and taking a Gaussian disorder with the same variance, (J/t)s > (J/t)T . This suggests that for any W the transition 2 2  V /z = V , as well as a hopping tb/z = t . line between nonsuperfluid and superfluid glass is not reentrant

104205-9 XIAOQUAN YU AND MARKUS MULLER¨ PHYSICAL REVIEW B 85, 104205 (2012) as a function of temperature. The absence of reentrance in turn 0.00 suggests that the quantum phase transition out of the insulating glass remains second order, independent of the strength of the disorder potential. -0.05 The Hamiltonian with a disorder potentials reads C M,qEA z z t x x y y z H =− Jij s s − s s + s s + i s . (73) i j N i j i j i -0.10 i

The disorder potential breaks the Z2 symmetry, therefore = Qa =b 0 already in the high-temperature phase, where it -0.15 assumes a constant replica symmetric value Q0. The glass phase occurs at the Almeida-Thouless instability, which is 0.1 0.2 0.3 0.4 0.5 0.6 0.7 given by57 T/t P (y) 2 2 W = β J dy 4 1, (74) FIG. 2. (Color online) Cross correlation between the local order cosh (βy) parameters for superfluidity and glassy order, respectively. The where correlations are evaluated from Eq. (85) in the temperature range 0.05 0, establishing that (J/t) > (J/t) In the limit W/J  1, Pinst(h; T = 0) is known to have a s T simple structure: even in the presence of strong disorder. We point out that Eq. (84) overestimates Eq. (82), but this overestimation should α|h|/J 2, |h|h , be much smaller than c = 0.231. − − 2 2 Pinst(h; T = 0) = exp (h γJ /W) (80) √ 2W2 , |h|h , 2πW2 V. PROPERTIES OF THE SUPERGLASS PHASE with a smooth crossover between the two limiting forms 2 Having established the phase diagram of the model, we around h = √1 J + O( J ). The value of the constant J α 2π W W 2 now focus on the properties of the bulk of the superglass γ = O(1) can be estimated by the normalization condition phase. There the interplay between temperature, glassy order, dhPinst(h; T = 0) = 1, but will be irrelevant below. and superfluid order induce several interesting phenomena, For W/J  1, (J/t)s and (J/t)T both behave as which potentially survive also in finite dimensional models √4 ln(W/J) to leading order. Their difference scales like cJ/W. 2π W/J of frustrated bosons. In the following, we investigate how the

104205-10 MEAN FIELD THEORY OF SUPERGLASSES PHYSICAL REVIEW B 85, 104205 (2012) glassy and superfluid orders evolve with temperature, and how 1.0 they are locally correlated. R M 0.8 Q1 A. Competition between glassy and superfluid order x = While the effective transverse field hi Mt is uniform 0.6 z for every site, the frozen longitudinal field hi depends on the site (and on the pure state in which the system is frozen). 0.4 Therefore the magnetization of the local spin s due to the local −→ i = x z field hi (hi ,hi ) fluctuates from site to site. It is interesting 0.2 to study the correlation of the local magnetization, whose components are the local order parameters of the glassy and the 0.0 superfluid order, respectively. More precisely, we investigate 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 the following correlation function: T/t 1 x z 2 − 1 x 1 z 2 N i si si N i si N i si FIG. 3. (Color online) The order parameters in the superglass CM,q ≡ EA 1 x 1 z 2 phase as a function of temperature 0.05 sj . We note that also the local order-parameter correlations

The maximal amplitude of the normalized correlation CM,qEA CM,qEA exhibit a nonmonotonous behavior within our static is only of order ≈ 0.1, suggesting that in the superfluid one-step approximation, as shown in Fig. 2. The absolute value phase the nonuniformity of the two local order-parameter of CM,qEA increases with temperature at very low temperatures, fields is actually not very strong. It may be that the one-step and decreases at higher temperatures. This can be seen again approximation underestimates these correlations a bit. The as a consequence of the nonmonotonicity of the superfluid relative weakness of the anticorrelations might be the reason order. At fixed T , the larger hx the stronger the normalized x = they have not been noticed in the quantum Monte Carlo studies anticorrelation CM,qEA . Since h tM initially increases with of Refs. 15 and 16. T , it is natural to expect an increasing CM,qEA until eventually

thermal fluctuations become dominant and diminish CM,qEA . B. Nonmonotonicity of the superfluid order In the superglass phase, the glass order parameter q = EA VI. DISCUSSION Q1 monotonously decreases with increasing temperature, as one should expect. However, surprisingly, the superfluid In this paper we have analyzed a fully connected mean order parameter M exhibits nonmonotonic behavior with a field model. The full connectivity is not a real limitation, maximum at an intermediate crossover temperature Tm,as however. Indeed, one can generalize the model to a highly shown in Fig. 3.BelowTm, the superfluid order parameter connected Cayley tree. While this does not affect the ther- M decreases, anomalously, when lowering the temperature. modynamics of the model, this generalization allows for Above Tm, M decreases with increasing temperature as usual the study of localization and delocalization of excitations, in a standard superfluid. since this model is endowed with a notion of distance. The This phenomenon is related to the anticorrelation between analysis of localization properties is of particular interest in the glassy and superfluid order discussed in the previous section. vicinity of the superglass-to-insulating-glass quantum phase While on one hand, thermal fluctuations tend to diminish transition, where the boson system collectively delocalizes both glassy and superfluid order, there appears to be a low- into a superfluid at low energies. The nature of higher energy temperature regime T

104205-11 XIAOQUAN YU AND MARKUS MULLER¨ PHYSICAL REVIEW B 85, 104205 (2012) suppression of superfluidity is the linear pseudogap within the ACKNOWLEDGMENTS glass phase. A very similar pseudogap is known to occur in We thank L. Leuzzi for providing us the high precision data disordered Coulomb interacting systems, where it is due to on the local-field distribution P (y) of the SK model at finite unscreened 1/r interactions between charged particles. This temperature. We thank L. Foini, M. Gingras, F. Zamponi, and Coulomb gap may well be of importance in strongly disordered A. W. Sandvik for useful discussions. superconductors and play a significant role in the competition between glassy insulating behavior and superconductivity. In particular, in materials with strong negative U centers, one APPENDIX: DERIVATION OF RS FREE ENERGY may think of preformed electron pairs constituting hard core With standard replica trick49 we can get the RS free energy: bosons which interact with Coulomb repulsions.70 The power J 2β2 tβ J 2β2 1 1 law suppression of the low-energy density of states makes it βf =− Q2 + M2 + dτdτR2(τ,τ) likely that the superfluid condensate is entirely destroyed once 4 2 4 0 0 the hopping becomes too small. For short-range interactions 2 1 J 2β2 1 the density of states is merely reduced at low energy, but does − T z lim lnTr exp Q dtsa(τ) n→0 n 2 not tend to zero. On a Cayley tree of very large connectivity a 0 this will always lead to delocalization, unless the hopping t J 2β2 1 1 is scaled down logarithmically with the connectivity. In finite + dτ dτR(τ,τ)sz(τ)sz(τ ) 2 a a dimensions, however, sufficiently strong disorder is known a 0 0 23 to suppress superfluidity, and thus one may expect that 2 J 2β2 1 at sufficiently large ratios J/t, the disordered boson model − Q dtsz(τ) 2 a will localize due to spontaneously created, frozen-in local a 0 fields. Such a conclusion may be suggestive from a straight 1 extrapolation of the quantum Monte Carlo results of Ref. 15 + x x tβ dτMa sa (τ) . (A1) to T = 0, but it seems difficult to exclude a scenario in which a 0 T becomes merely exponentially small with J/t.Amore c Under static approximation: R(τ,τ) = R,wehave careful analysis will be necessary to settle this question in finite dimensional, short-range interacting glasses. J 2β2 tβ J 2β2 βf =− Q2 + M2 + R2 As for the coexistence phase, the superglass, the numerical 4 2 4 15,16 data provide evidence that it exists also in finite dimen- 2 1 J 2β2 1 sions. It would be interesting to confirm and quantify the local − T z lim lnTr exp Q dtsa(τ) n→0 n 2 anticorrelation of order parameters in such simulations. From a 0 our mean field analysis one expects that the anticorrelation J 2β2 1 2 is in fact relatively weak. A further nontrivial prediction + (R − Q) dtsz(τ) 2 a with measurable consequences is the nonmonotonicity of the a 0 superfluid order parameter, which should translate into an 1 equivalent nonmonotonicity of the superfluid stiffness as a + tβ dtMx sx (τ) . (A2) function of temperature. This nonmonotonicity has its origin a a a 0 in the softening of the glassy order at low T , a feature which may potentially survive in finite dimensions, especially when According to Hubbard-Stratonovich transformations, we lin- 1 z 2 1 z 2 the lattice connectivity is large, or the interactions are not earize the quadratic terms ( a 0 dtsa(τ)) and ( 0 dtsa(τ)) too short ranged. We should caution though that we obtained by introducing extra fields y0 and yR: this effect by employing a static approximation and a replica J 2β2 J 2β2 tβ symmetry breaking at the one-step level only. However, we βf =− Q2 + R2 + M2 4 4 2 believe that it is a real feature of the model. z z x As discussed earlier, a number of experiments have already − Dy0ln DyRTr exp[β(y0s + yRs + tMs )] shown promising indications of possible coexistence of glassy J 2β2 J 2β2 tβ order with superfluidity. We hope that our analysis will =− 2 + 2 + 2 help to unambiguously identify such phases in experiments. Q R M 4 4 2 Note that finding an experimental system exhibiting a glassy − + 2 + 2 2 superfluid-insulator transition might also be of great interest to Dy0ln DyR cosh(β (y0 yR) t M ). study the intricate interplay of interactions and disorder with (A3) respect to glassy ergodicity breaking, and quantum ergodicity breaking, i.e., Anderson localization. One can get Eq. (36) following the similar steps above.

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