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Particle-hole symmetry and the dirty boson problem Peter B. Weichman1 and Ranjan Mukhopadhyay2 1BAE Systems, Advanced Information Technologies, 6 New England Executive Park, Burlington, MA 01803 2Department of Physics, Clark University, Worcester, MA 01610 (Dated: February 2, 2008) We study the role of particle-hole symmetry on the universality class of various quantum phase transitions correspondingtotheonset of superfluidityat zero temperatureofbosons in aquenched random medium. To obtain a model with an exact particle-hole symmetry it is necessary to use 8 the Josephson junction array, or quantum rotor, Hamiltonian, which may include disorder in both 0 the site energies and the Josephson couplings between wavefunction phase operators at different 0 sites. The functional integral formulation of this problem in d spatial dimensions yields a (d+1)- 2 dimensional classical XY-model with extended disorder, constant along the extra imaginary time n dimension—the so-called random rod problem. Particle-hole symmetry may then be broken by a adding nonzero site energies, which may be uniform or site-dependent. We may distinguish three J cases: (i)exactparticle-hole symmetry,in whichthesiteenergiesallvanish,(ii)statistical particle- 0 holesymmetryinwhichthesiteenergydistribution issymmetricaboutzero, vanishingonaverage, 1 and(iii)completeabsenceofparticle-holesymmetryinwhichthedistributionisgeneric. Weexplore ineachcasethenatureoftheexcitationsinthenon-superfluidMottinsulatingandBoseglassphases. ] Weshow, inparticular,that,sincetheboundaryoftheMott phasecanbederivedexactlyinterms n of that for thepure,non-disordered system, that therecan beno direct Mott-superfluidtransition. n - Recent Monte Carlo data to the contrary can be explained in terms of rare region effects that s are inaccessible to finite systems. We find also that the Bose glass compressibility, which has the i d interpretationofatemporal spinstiffnessorsuperfluiddensity,ispositiveincases(ii) and(iii),but . that it vanishes with an essential singularity as full particle-hole symmetry is restored. We then t a focus on the critical point and discuss the relevance of type (ii) particle-hole symmetry breaking m perturbationstotherandomrodcriticalbehavior,identifyinganontrivialcrossoverexponent. This exponent cannot be calculated exactly but is argued to be positive and the perturbation therefore - d relevant. Wearguenextthataperturbationoftype(iii)isirrelevanttotheresultingtype(ii)critical n behavior: thestatisticalsymmetryisrestored onlargescalesclosetothecriticalpoint,andcase(ii) o therefore describes the dirty boson fixed point. Using various duality transformations we verify all c of these ideas in one dimension. To study higher dimensions we attempt, with partial success, to [ generalize theDorogovtsev-Cardy-Boyanovskydoubleepsilon expansion techniquetothisproblem. 1 We find that when the dimension of time ǫτ < ǫcτ ≃ 289 is sufficiently small a type (ii) symmetry v breaking perturbation is irrelevant, but that for sufficiently large ǫτ > ǫcτ particle-hole asymmetry 6 isarelevant perturbationandanewstablefixedpointappears. Furthermore,forǫτ >ǫcτ2 ≈ 32,this 4 fixed point is stable also to perturbations of type (iii): at ǫ=ǫcτ2 the generic type (iii) fixed point 6 merges with the new fixed point. We speculate, therefore, that this new fixed point becomes the 1 dirtybosonfixedpointwhenǫτ =1. Wepointout,however,thatǫτ =1maybequitespecial. Thus, . althoughthequalitativerenormalization groupflowpicturethedoubleepsilon expansiontechnique 1 providesisquitecompelling,oneshouldremainwaryofapplyingitquantitativelytothedirtyboson 0 problem. 8 0 PACSnumbers: 64.60.Fr,67.40.-w,72.15.Rn,74.78.-w, : v i X I. INTRODUCTION glass (BG), and Mott insulating (MI) phases, motivated r mainly by the problem of superfluidity of 4He in porous a media. However,asisoftenthecaseinthestudyofcriti- Quantum phase transitions at zero temperature are calphenomena,theuniversalityclassofthistransition,or driven entirely by quantum fluctuations in the ground astraightforwardgeneralizationofit, alsoincludes other state wavefunction. In many cases a crucial require- physical phenomena, such as many aspects of quantum ment is the presence of quenched disorder. Examplesin- magnetism and superconductivity. clude randommagnets with various kinds ofsite, bond,1 or field2 disorder; the transitions between plateaux There have been a number of approaches to study- in the two-dimensional quantized Hall effects;3 metal- ing the SF–BG transition: mean field theories,7 strong insulator,4 metal-superconductor, and superconductor- coupling,8 large N,9 real space methods,10 quantum insulator5 transitions in disordered electronic systems; Monte Carlo calculations,11,12,13,14 double dimensional- and the onset of superfluidity of 4He in porous media.6 ityepsilon-expansions,15,16,17,realspacerenormalization In this article, we will study the quantum transi- groupin the limit of strong disorder in one dimension,18 tions between the superfluid (SF), the localized Bose renormalizationgroupin fixed dimension,19 and in 1+ǫ 2 dimensions.20 However, none of the above methods pro- lations are [a ,a†] = δ , and nˆ a†a is the number i j ij i ≡ i i videacontrolledanalyticalapproachtothecriticalpoint operator at site i. for d>1. The second model is the Josephson junction array A satisfactorydimensionality expansionaboutthe up- Hamiltonian, per critical dimension for the superfluid to Bose glass transition (analogous to the epsilon expansion about = J˜ cos(φˆ φˆ )+ (ε˜ µ˜)n˜ J ij i j i i H − − − d = 4 for classical spin problems) does not appear to i,j i X X exist. In particular, we previously found17 that the ap- 1 + U n˜ n˜ , (1.2) proach based on a simultaneous expansion in both the ij i j 2 dimension, ǫτ, of imaginary time (physically equal to Xi,j unity), and the deviation, ǫ =4 D, of the total space- − with analogous parameters, but now the commutation time dimensionality, D = d+ǫ , from four15 does not τ relations [φˆ,n˜ ] = iδ . These two Hamiltonians are, in yield a perturbatively accessible renormalization group i j ij fact, verycloselyrelated. It is easyto checkthatif N is fixed point, and therefore does not produce a systematic 0 any positive integer then expansion for the dirty boson problem. Nevertheless it does give one a fairly detailed picture of the renormalization group fixed point and the flow structures, and a†i =(N0+n˜i)21eiφî produce an uncontrolled expansion for the dirty boson ai =e−iφî(N0+n˜i)12 (1.3) problem, though further work needs to be done in order to understand the analytic structure of the theory as a satisfy the correct Bose commutation relations, and we functionofǫ . Therefore,inspiteofitspoorconvergence, identify nˆ = N + n˜ . Note, however, that the com- τ i 0 i thedoubledimensionalityexpansionstillappearstopro- mutation relations between φˆ and n˜ permit n˜ to have i i i vide the most flexible analytic approach to the study of any integer eigenvalue, positive or negative, whereas the the SF–BGtransition,including insightinto the symme- eigenvalues of nˆ must be non-negative. Therefore it is i tries of the fixed point. As detailed below, attention to only when N is large, and the fluctuations in n˜ are 0 i the role ofparticle-hole symmetry,especially, is essential small compared to N , that and may be com- 0 J B H H to generating correct RG flow structures. In three (and pared quantitatively: inside the hopping term we may hisigthheer)ondliymmenestihoonds,tthhaetdhoausbbleeednimaebnlesi,ownaitlhitypaerxtpiaalnssuiocn- approximate a†i ≈ N021eiφî, ai ≈ N012e−iφî and make the identifications cess, to access the critical fixed point and obtain critical exponents.17 J˜ =N J ; U =V ; ε˜ =ε ; µ˜=µ N Vˆ + 1V , ij 0 ij ij ij i i 0 0 0 − 2 (1.4) where V = V and Vˆ = V , and there exists an A. Particle-hole symmetry 0 ii 0 j ij overall additive constant term E N = (1N Vˆ 1V P 0 2 0 0 − 2 0 − As stated, it will transpirethat anessentialingredient µ)N0N, where N is the number of lattice sites. that is necessary in order to correctly understand the Despite this asymptotic equivalence at large N0, the physics of the SF–BG transition is an extra “hidden” Josephson Hamiltonian (1.2) has an exact discrete sym- symmetry, which we call particle-hole symmetry, that is metrywhichthebosonHamiltonianlacks. Thusthecon- present at the critical point, but not necessarily away stant shift, n˜′i = n˜i +n0, where n0 is any integer, has from it. To make this notion precise, we compare the no effect on the commutation relations or the eigenvalue following two lattice models of superfluidity: the first is spectrum of the n˜i. The Hamiltonian correspondingly the usual lattice boson Hamiltonian, transforms as = 1 J [a†a +a†a ]+ (ε µ)nˆ HJ{n˜′i}=HJ{n˜i}+n0Uˆ0 n˜i+Nε0(n0,µ˜), (1.5) HB −2 ij i j j i i− i Xi i,j i X X 1 where ε0(n ,µ˜) = n (1n Uˆ µ˜), and Uˆ = U . + V nˆ (nˆ δ ), (1.1) 0 0 2 0 0 − 0 j ij 2 i,j ij i j − ij The free energy density, fJ = −β1N ln tre−βHJ P, where X β =1/k T, transforms as B (cid:2) (cid:3) where J = J is the hopping matrix element between ij ji sites i andj, which we will allowto havea randomcom- fJ(µ˜)=fJ(µ˜ n0Uˆ0)+ε0(n0,µ˜) (1.6) − ponent; µ is the chemical potential whose zero we fix by choosing the diagonal components, Jii, of Jij in such a independent of the J˜ij and ε˜i. This implies that the way that jJij = 0 for each i; εi is a random site en- only effect of a shift n0Uˆ0 in the chemical potential is a ergy with mean zero; V = V is the pair interaction trivial additive term in the free energy which is linear in ij ji potential,Passumed for simplicity to be nonrandom and µ˜. This term serves only to increase the overall density, translation invariant; the only nonzero commutation re- n=−∂∂fµ˜J, by n0 but otherwise has no effect whatsoever 3 on the phase diagram, which therefore will be precisely sufficiently small, short ranged J , there is a finite en- ij periodic in µ˜, with period Uˆ . ergygapforadditionofparticles,andtheoveralldensity 0 Considernextthetransformationn˜′i =−n˜i,φˆ′i =−φî. remains fixed at n0 for a finite range of µ˜. The Hamiltonian transforms as Consider first µ˜ = 0. Then only at a critical value 6 J (µ˜)ofJ doesthesystemfavoraddingextraparticles 0,c 0 HJ[n˜′i,φˆ′i,ε˜i−µ˜]=HJ[n˜i,φî,−(ε˜i−µ˜)], (1.7) (orholes). Equivalently,foragivenJ0thereisaninterval µ˜ (J ) < µ˜ < µ˜ (J ) of fixed density n . These extra − 0 + 0 0 so that particlesmaybethoughtofasadiluteBosefluidmoving atoptheessentiallyinertbackgrounddensity,n (seeFig. 0 fJ(µ˜, ε˜i )=fJ( µ˜, ε˜i ). (1.8) 2). The physics is identical to that of a dilute Bose gas { } − {− } in the continuum, and is well described by the Bogoli- Combining the two symmetries (1.6) and (1.8) we see ubov model.22 From this one concludes that the system thatifallε˜ =0,thenforµ˜ =µ˜ 1kUˆ ,wherek isany immediately becomes superfluid (recall that we assume i k ≡ 2 0 integer,the Hamiltonian possessesa special particle-hole T =0)withasuperfluiddensityρ n n J J , s 0 0 0,c ∼ − ∼ − symmetry, namely invariance under the transformation and an order parameter ψ0 eiφî (n n0)21 an˜21d′ikd=pitekiro−nsitan˜eni,d,aφrˆne′dm=toh−veaφˆtl.hoefArptmaµ˜rotd=icylneµãskm(ttihhceesridseemsnyosmivtaymlieostfrppicraeructniiscdeleelyrs (ξJ0 =−JJ0021,/c[)µ˜21−. µ˜T±h(eJ0c)h]a21r∼act(enr−istn≡ic0)hl−en21g∼tih(∼Jin0−thJi0−s,cp)−ha12saen∼ids representsthedistancebetween“uncondensed”particles: being synonymous with the addition of holes). If the ε i n n ψ 2 ξ−d. This zero-temperature superfluid are nonzero, but have a symmetric probability distribu- − 0−| 0| ∼ 0 onset transition is therefore trivial, in the sense that all tion, p ε˜ = p ε˜ , then the exact particle-hole sym- i i { } {− } exponents are mean-field-like. In fact, historically this metry is lost, but there still exists a statistical particle- onset was never really viewed as an example of a phase hole symmetry at the same special values µ˜ of µ˜: self k transition. averaging will ensure that f (µ˜ , ε˜ ) = f (µ˜ , ε˜ ). J k i J k i The lattice boson Hamiltonian ({1.1)} clearly can{−nev}er Furthermore,althoughallquantitiesvarycontinuously possess either form of particle-hole symmetry since the as J0 decreases towardJ0,c, the actual onset point is en- hopping termmixesthe number andphase inaninextri- tirely noncritical. Thus, for a given value of J0 within cable fashion. a Mott lobe, the interval µ˜−(J0) < µ˜ < µ˜+(J0) represents a single unique (incompressible) thermodynamic state. Incompressibility implies that for the given (integer) density, the value of the chemical potential is am- B. Phase diagrams biguous. One might just as well set µ˜ = n U , its value 0 0 atthecenterofthelobe. Thecorrelationlength,ξ(µ˜,J ), 0 In Fig. 1 we sketch zero temperature phase diagrams, is independent of µ˜, and remains perfectly finite in the with and without various types of disorder, in the µ˜-J 0 Mott phase at J (µ˜). In this sense the transition has 0,c plane,whereJ isameasureoftheoverallstrengthofthe 0 some elements of a first order phase transition. hopping matrix (e.g., J = 1 J˜ ), in the simplest 0 N i6=j ij The more important transition is the one occurring case of onsite repulsion only:21 U = U δ . This phase diagram has been discussed inPdiejtail p0reivjiously6,11 for at fixed density, n = n0, at µ˜ = n0U0 through the tip of the Mott lobe at J (0). At this transition ξ(J ) 0,c 0 the lattice boson Hamiltonian, HB. Here we emphasize (J0,c J0)−νpure divergescontinuouslywithacharacteri∼s- the features unique to , namely the periodicity in µ˜, − HJ ticexponent,νpure. Onemayshow(seeRef.6andbelow) andthe specialpoints µ˜ correspondingtolocalextrema k thatthe transitionispreciselyinthe universalityclassof in the phase boundaries. the classical (d+1)-dimensional XY-model. What dis- tinguishes this transition from the previous ones is precisely particle-hole symmetry: superfluidity is achieved 1. Phase diagram for the pure system not by adding a small density of particles or holes atop the inert background, but by the buildup of superfluid In Fig. 1(a) we show the phase diagram in the ab- fluctuations within the background, to the point where sence of all disorder. For J 0 the site occupancies particlesandholessimultaneously overcomethepotential ij ≡ are good quantum numbers and each site has precisely barrier U0 and hop coherently without resistance. The n0 particles for n0 − 12 < µ˜/U0 < n0 + 21. The points exponent νpure exhibits itself in the phase diagram as µ˜/U = k+ 1 for integer k are 2N fold degenerate with well: ascalingargumentshowsthattheshapeoftheMott eithe0r k or k2+1 particles placed independently on each lobe is singular near its tip,6 µ±(J0) (J0,c J0)νpure. site. For Jij > 0 communication between sites occurs In d=2, νpure ≃ 32, while in d≥ 3, ν∼pu±re = 21.−This will and the effective wavefunction for each particle spreads be discussed in a more general scaling context in Sec. to neighboring sites (see Fig. 2). We denote by ξ(J ) (to IVB. 0 be defined carefully later) the range of this spread. One WehavealreadyobservedthatthelatticebosonHamil- can show within perturbation theory,6 however, that for tonian (1.1) never has an exact particle-hole symmetry. 4 (a) (b) (c) µ~/U ~µ/U µ~/U 3_ 0 _3 0 _3 0 SF SF SF 2 2 2 MI <n>=1 MI <n>=1 MI <n>=1 1 1 1 n=1 1_+δ n=1 n=1 1_2 _J_0U,_c_(µ~_)_ 2 12_ BG _µ~_+− U _(_ J 0_ _)−+δ 12_ BG RRG 0 1_−δ 0 MI <n>=0 2 MI <n>=0 MI <n>=0 0 0 0 J /U n=0 J /U n=0 J /U n=0 0 0 0 0 0 0 ~ −1_ SF −1_ BG SF −1_ BG _J_0,_c_(µ_)___ 2 2 2 U (1+δ ) 0 + MI <n>=−1 MI <n>=−1 MI <n>=−1 −1 −1 −1 n=−1 n=−1 n=−1 SF 3_ _3 3_ − − − 2 2 2 FIG. 1: (Color online) Schematic zero temperature phase diagram for the Josephson junction model. (a) The model without disorder, showing the periodic sequence of Mott lobes (MI) and the superfluid phase (SF). The transitions to superfluidity at the tips of the Mott lobes, J = J0, are special and are in the universality class of the (d+1)-dimensional XY-model. The c points µ /U = k/2 have an exact particle-hole symmetry. As described in the text, there is a corresponding singularity, k 0 determined by the correlation length exponent νXY, in the shape of the lobe near its tip. (b) The model with site energy d+1 disorder whose distribution is supported on the interval −∆ε ≤ ε˜i ≤ ∆ε, showing the now shrunken (or even absent, if the disorder is sufficiently strong, δ ≡ ∆ε/U0 ≥ 12) Mott lobes, and the new Bose glass phase that now intervenes between them and the superfluid phase. As indicated, the boundaries of the Mott lobes are determined entirely by δ and their pure system counterparts. The transition to superfluidity now takes place only from the Bose glass phase (BG), and it is believed that the natureof this transition is independent of where it occurs, even at thespecial points µ which now have only a statistical k particle-hole symmetry. (c) Phase diagram for themodel with only nearest neighbor bond disorder, J˜ij =J0(1+∆Jij) whose distribution has support −δ ≤ ∆Jij ≤ δ+, with δ ≤ 1, showing again shrunken Mott lobes (which will vanish entirely if − − ∆Jij is unbounded from above). The boundaries of the Mott lobes are now determined entirely by δ+ and their pure system counterparts. At integer filling the Bose Glass phase turns into an incompressible random rod glass (RRG). The transition is intheuniversalityclass ofthe(d+1)-dimensionalclassical random rod problem. Awayfrom integerfillingthecompressibility turnson continuously,and the Bose glass phase reappears. The transition is once more in theuniversality class of thegeneric Bose glass to superfluid transition. The random rod temporal correlation length exponent ντ,0=z0ν0 (expected to be greater than unit in d = 315) now describes the shape of critical line near the commensurate point. In a neighborhood of half filling (half-integer µ/U0),whose size must dependon theprecise distribution of ∆Jij, for d>1,25 thesystem remains superfluidall the way down to J = 0: in the absence of site disorder the the exact particle-hole symmetry that is restored at half filling 0 guarantees delocalized excitations. Nevertheless, one still has Mott lobes (now asymmetric parameter and the chemical potential. anddecreasinginsizewithincreasingn )foreachinteger This phenomenon of “asymptotic symmetry restora- 0 density, and a unique extremal point, [J (n ),µ (n )], tion” at a critical point is actually fairly common (and 0,c 0 c 0 atwhichoneexitstheMottlobeatfixeddensityn=n . we shallencounter it againbelow). For example, though 0 One may show6 that the transition through these ex- theusualIsingmodelofmagnetismhasanup-downspin tremal points is still in the (d+1)-dimensional XY uni- symmetry, the usual liquid–vapor or binary liquid crit- versality class, and that particle-hole symmetry must ical points do not. However the Ising model correctly thereforebeasymptoticallyrestoredat thecriticalpoint. describesthe universalityclassofthe transition,andone The difference now is that there is a nontrivial balance concludes that the up-down symmetry must be restored betweenthedensitiesofparticleandholeexcitations,and nearthecriticalpoint. Similarly,thep-stateclockmodel the interactionsbetweenthem. The positionofthe criti- with Hamiltonian cal point is no longer fixed by an explicit symmetry, but 2π must be located by carefully tuning both the hopping = J cos (q q ) , q =1,2,...,p, (1.9) i j i H − p − Xhiji (cid:20) (cid:21) 5 J 0 < J0,c ξ n = 1 J < J ξ n = 1 0,c MI MI n < 1 µ~ < µ~ ( )−∆ Hole + J0 ε n > 1 SF BG n > 1 Particle µ~ ( ) > µ~ > µ~ ( )−∆ SF sf J0 +J0 ε n > 1 FIG.2: (Coloronline)Schematicillustrationoflatticebosons SF nearunitfillingn=1atsomevalueofthehoppingamplitude ~µ > µ~ (J ) in the interval 0 < J0 < J0,c. In the Mott insulating phase sf 0 (n = 1), the effective wavefunction of each particle spreads only a finite distance ξ(J ), and the state is insulating. For 0 FIG.3: (Coloronline)Schematicillustrationoflatticebosons n>1 (n<1), theextra particles (holes) travel freely within near unit filling in the presence of bounded disorder. If the the essentially inert background, and the state is superfluid for arbitrarily small |n−1|. disorder is not too strong (∆ε < U0/2), there is still a Mott phase with a finite energy gap for adding or removing particles. Unlike in the pure case (Fig. 2), superfluidity is not generated immediately with the addition of particles. For whichmaybethoughtofasakindofdiscreteXY-model, sufficiently small n−1, the additional particles are Ander- has for sufficiently large p (specifically, p > 4 in d = 2; son localized by the residual random background potential clearly p = 2 corresponds to the Ising model and p = 3 of the effectively inert layer. The finite compressibility dis- to the three-state Potts model) a transition precisely in tinguishes this Bose glass phase from the Mott phase. The the XY-model universality class. Note, however,that in superfluid critical point µ˜sf(J0) occurs only once the added particleshavesufficientlysmoothedthebackgroundpotential theordered phase,correspondingtothezerotemperature that its lowest lying states become extended. fixed point, the order parameter will (for d>2) sponta- neouslyalignalongone ofthep equivalentdirections,q , i breaking the XY-symmetry and generating a mass for totically restored at the critical point.24 Only if µ˜ = µ˜ the spin-wave spectrum (more interestingly, in d = 2 a k and ε˜ 0 does the disorder fully respect particle-hole power-laworderedKosterlitz-Thoulessphase exists for a i ≡ symmetry. Weshallseethatinthiscasethetransitionis finite temperature interval below the transition, with a entirely different, lying in the same universality class as second transition to a long-range ordered phase taking place only at a lower temperature23). This latter prop- the classical (d+1)-dimensional XY-model with colum- nar bonddisorder,preciselythekindofsystemaddressed erty is not relevant in the present case since breaking in Ref. 15. particle-hole symmetry does not break the symmetry of In Fig. 1(b) we sketch the phase diagram in the pres- the order parameter: the nature of the superfluid phase ence of site disorder, whose distribution is supported on is unaffected. the finite interval ∆ ε ∆ . We see thatthe Mott ε i ε − ≤ ≤ lobes have shrunk (and may in fact disappear altogether for sufficiently strong disorder), and a new Bose glass 2. Phase diagrams with disorder phaseseparatestheselobesfromthesuperfluidphase.6In thisnewphasethecompressibilityisfinite,buttheparti- Wewillconsidertwotypesofdisorder: (i)onsitedisor- clesdonothoplargedistancesduetolocalizationeffects: derintheε˜,and(ii)disorderinthehoppingparameters, particles on top of the inert backgroundstill see a resid- i J˜ . If µ˜ = µ˜ for any k, that is if particle-hole symme- ual randompotential, whose lowestenergy states will be ij k 6 try is broken, we expect the two types of disorder to localized (Fig. 3). As particles are added to the system, yield the same type of phase transition. In renormaliza- bosonswilltendtofillthesestatesuntiltheresidualran- tion group language, each in isolation will generate the dompotentialhasbeensmoothedoutsufficientlythatex- other under renormalization. If µ˜ = µ˜ and the ε˜ have tended states can form, finally producing superfluidity.6 k i a symmetric distribution about zero so that the Hamil- As argued above, the nature of the superfluid transition tonianpossessesastatisticalparticle-holesymmetry,the is the same everywhere along transition line. obvious question is whether or not the transition in this As indicated in the figure, the boundaries of the case is different from the one in the presence of generic Mott lobes are determined entirely by the pure system, nonsymmetric disorder. We shall argue below that it is together with ∆ .8 The upper half of the boundary, ε not, i.e., that breaking particle-hole symmetry locally is µ˜ (J ,∆ ) = µ˜ (J ,0) ∆ , is pushed down by ∆ , + 0 ε + 0 ε ε − not substantially different from breaking it globally, and while the lower half, µ˜ (J ,∆ ) = µ˜ (J ,0) + ∆ , is + 0 ε − 0 ε that in fact statistical particle-hole symmetry is asymp- pushed up by ∆ . This result, which relies on the exis- ε 6 τ tence of large, rare regions of nearly uniform superfluid, (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) will be derived in Sec. III. 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discussed in Sec. IVB, the superfluid transition line has baosrihnogoudlaoriftyth,eµp±a(rJt0i)cle∼-h±ol(eJscRyRm−meJt0r)icz0pν0o,initnst(hneotneetihgaht- FlikIGe.st4r:u(cCtuorloergoennleinraet)eSdchbeymqautiecniclhluesdtrdatisioonrdoerf rianndqoumanrtoudm- z = 1). One expects ν > 1 in d = 3, yielding the models. The cylinders represent, for example, a picture of pure τ,0 random regions of enhanced or suppressed hopping strength. pictured cusps. The boundaries of the Mott lobes are again deter- minedentirelybythepuresystem,togetherwithδ . The + boundary, J (µ˜,δ ) = J (µ˜,0)/(1+δ ), is this time are τ-independent generates rod-like disorder (Fig. 4). 0,c + 0,c + scaled to the left by ∆+. The result again relies on the These models are intended to be used in the vicinity of J existence of large, rare regions of nearly uniform super- the n = 0 Mott lobe, hence g in the neighborhood of 0 fluid, and will also be derived in Sec. III. zero, since fluctuations in the amplitude ψ destroy any | | At half filling, where µ˜ = µ˜k = (k + 21)U0 is a half- translation symmetry in g0, analogous to (1.6), and the integer,anexactparticle-holesymmetryisrestored,and, nonzero lobes are no longer symmetric. for d > 1,25 the superfluid phase can penetrate all the The Lagrangian = (g = 0) generates the clas- 0 1 0 waytozerohopping. Ingeneralthesuperfluidphasewill sical (d+1)-dimensLionalLXY critical behavior along the survive on a finite interval in density (which maps into line µ˜ = 0 in Fig. 1(a): the superfluid transition occurs the single point µ˜ = µ˜k at J0 = 0) around half filling, fordecreasingr0 atacriticalvaluer0,c. Nonzerog0 in 1 whose size depends on the precise distribution of J˜ij.26 generatestheremainderoftheMottlobe,r0,c(g0),corrLe- sponding to the remainder of the phase diagram in Fig. 1(a). The transition at nonzero g is in the universality 0 C. Criticality and restoration of statistical class of the dilute Bose gas superfluid onset transition particle-hole symmetry (described by the Bogoliubov model) in d dimensions.27 The usual coherent state representation of the (contin- In Sec. II various functional integral representations uum versionof the) boson Hamiltonian (1.1) is obtained of the Hamiltonians (1.1) and (1.2) will be introduced. by dropping the ∂τψ 2 term in 1 and setting g0 = 1. | | L In order to discuss, in the most transparent fashion, the This demonstrates explicitly the lack of a simple inter- role of various symmetries at the superfluid transition, polation between the particle-hole symmetric and asym- we summarize in Table I the basic classical continuum metric models in the original boson Hamiltonian. ψ4 models that may be abstracted from these represen- The Lagrangian = (g = 0) incorporates hop- 2 3 0 L L tations. The phase of ψ = ψ eiφ represents the Joseph- pingdisorderintheformofτ-independentdisorderδr in | | son phases in (1.2). The controlparameter r represents r , and represents the previously analyzed (particle-hole 0 0 the hopping strength J , while g represents the chemi- symmetric) random rod model.15 The model describes 0 0 cal potential µ˜. Quenched hopping and site energy dis- the transitions along the line g =0 in Fig. 1(c), includ- 0 order are correspondingly represented, respectively, by ing the incompressible randomrodglass(RRG) separat- the random fields δr(x) and δg(x). The fact that they ing the tip of the Mott lobe from the superfluid phase. 7 Pure PH-sym [(d+1)-dimensional XY model]: L0 = − ddx dτ 21|∇ψ|2 + 21|∂τψ|2 + 12r0|ψ|2 + 14u0|ψ|4 R R (cid:8) (cid:9) Pure PH-asym [d-dimensional dilute Bose gas]: L1 = − ddx dτ 21|∇ψ|2 − 12ψ∗(∂τ −g0)2ψ + 12r0|ψ|2 + 41u0|ψ|4 R R (cid:8) (cid:9) PH-sym RR [(d+1)-dimensional classical random rod model]: L2 = − ddx dτ 21|∇ψ|2 + 21|∂τψ|2 + 12[r0 +δr(x)]|ψ|2 + 41u0|ψ|4 R R (cid:8) (cid:9) PH-asym RR [(d+1)-dimensional incommensurate random rod model]: L3 = − ddx dτ 21|∇ψ|2 − 12ψ∗(∂τ −g0)2ψ + 12[r0 +δr(x)]|ψ|2 + 41u0|ψ|4 R R (cid:8) (cid:9) Statistical PH-sym [commensurate dirty boson problem]: L4 = − ddx dτ 21|∇ψ|2 − 12ψ∗[∂τ −δg(x)]2ψ + 21[r0 +δr(x)]|ψ|2 + 41u0|ψ|4 R R (cid:8) (cid:9) Generic PH-asym [incommensurate dirty boson problem]: L5 = − ddx dτ 21|∇ψ|2 − 12ψ∗[∂τ −g0 −δg(x)]2ψ + 21[r0 +δr(x)]|ψ|2+ 14u0|ψ|4 TABLE I: ψ4 represeRntationRof m(cid:8)odels with various types of disorder and various degrees of particle-hole symm(cid:9)etry. The coefficients of |∇ψ|2 and |∂τψ|2 have been normalized to 21. The control parameters r0 and g0 are analogous to J0 and µ˜, respectively. Disorder in thehopping strengths is represented by δr, while that in thesite energies is represented by δg. Both are independent of τ. In field theoretic treatments, both are taken as quenched Gaussian random fields with zero mean and delta-functioncorrelationscharacterizedbyvariances∆r and∆g,respectively. Disorderintheotherparameters(includingthe unit gradient-squared coefficients) may also be introduced, butproduces no new critical behavior. Nonzero g in generates the remainder of the phase ricproblemsmaybetreatedwithinthesamemodel. The 0 3 L diagram in Fig. 1(c). We will show in Sec. III that the JosephsonjunctionarrayHamiltonian,(1.2),fromwhich breakingofparticle-holesymmetryimpliesthattheRRG all of the Lagrangians listed in Table I may be derived, becomesthecompressibleBoseglassexistsatallnonzero satisfies this requirement. g —in the renormalization group sense, nonzero g , to- 0 0 Using and , we shall see in Sec. VI that, in- 4 5 gether with nonzero δr, generates a finite δg even if it L L deed, a new statistically particle-hole symmetric dirty vanishes to begin with. bosonfixedpointcanbeidentified,andthatalltechnical Undertheconditionthatthedistributionofδg issym- problems encountered in Ref. 16 may then be avoided. metric, he Lagrangian 4 = 5(g0 =0) maintains a sta- Specifically, certain diagrams that were ignored in Ref. L L tistical particle-hole symmetry. It is believed, however, 16areinfactimportantandaccomplishthe requireden- that the superfluid transition at zero and nonzero g0 are largement of the parameter space. There is, however, a in the same universality class. In renormalization group price: thisnewfixedpointisnotperturbativelyaccessible language,g0 mustflowtozeroatlargescales,sothat 4, within the double epsilon expansion. Thus, the random L nnioctaLl 5d,edfiensictrioibnesofthtehecrriteisctaolrfiaxtieodnpoofinstt.atTishtiiscaisltphaerttiecclhe-- rdoimdepnrsoibolnesmifmtahyebdeimanenaslyiozned, ǫfo,rosmf tailmlǫeiinsdal=so4s−mǫa−ll.ǫ1τ5 τ hole symmetry. We will confirm this picture within the We find, however, that for small ǫ and ǫ , site disor- τ double-epsilon expansion in Sec. VI. der is irrelevant atthe randomrod fixed point, and that The significance of this symmetry restoration at the full particle-holesymmetry is therefore restoredonlarge critical point is the following. In Ref. 16 only the boson scales close to criticality. This remains true for suffi- Hamiltonian (1.1), and correspondingcoherentstate La- ciently small ǫ < ǫc(D), with ǫc(D) = O(1). Only for τ τ τ grangian, was considered, and a fixed point sought only ǫ > ǫc(D) does a new fixed point appear which breaks τ τ inthespaceofparametersaccessibletothisHamiltonian. full particle-holesymmetry. Extrapolationof the critical Suchafixedpointcanthereforeneverpossessastatistical behaviorassociatedwiththisnewfixedpoint,thoughun- particle-hole symmetry. However if the true fixed point controlled,showssignificantsimilaritiesto some features does possess this symmetry, it is clear that it must then of the known behavior at ǫ = 1. To lowest nontrivial τ lie outside the space of boson Hamiltonians of the form order in ǫ we find ǫc(D =4) 8 (D =4, hence ǫ=0, (1.1) (or Lagrangianswithout a ∂ ψ 2 term). Accessing corresponτding to d =τ 3 at ǫ ≃=219). This value of ǫc is | τ | τ τ thisfixedpointrequiresanenlargementoftheparameter significantlylessthanunity,andleadsustohopethates- spacesothatbothparticle-holesymmetricandasymmet- timates based on these extrapolations from small ǫ are τ 8 not too unreasonable. Kac-Hubbard-Stratanovich transformation to decouple , generate effective classical actions with an infinite 1 H numberofterms,whichmustbetruncatedatsomefinite D. Outline order.6 Inaddition,suchrepresentationsworkonlywhen µ˜ lieswithinaMottphasewhenJ =0,andhencebreak 0 downfor unbounded, e.g.,Gaussian,distributions ofsite Theoutlineoftheremainderofthispaperisasfollows. energies. We turn instead to representations obtained In Sec. II we introduce various useful functional integral from the Trotter decomposition (see App. A). formulations for the thermodynamics of the Hamiltoni- ans (1.1) and (1.2). We begin in Sec. III by considering the role of particle-hole symmetry in the nature of the excitation spectra of the glassy phases. Using a phe- A. Lattice boson model nomenological model in which we view the structure of the random rod and Bose glass phases as a set of ran- For the lattice boson model, the coherent state repre- domsized,randomlyplaced,isolatedsuperfluiddroplets, sentation is most appropriate, and yields a classical La- we focus on the density and compressibility and exam- grangian ine how they vanish as full particle-hole symmetry is restored. The droplet model also confirms the relation be- β tween the boundaries of the pure and disordered Mott = dτ ψ∗(τ)∂ ψ (τ) ψ∗(τ),ψ (τ) , lobes in Figs. 1. In Sec. IV we begin focusing on the LB Z0 " i i τ i −HB{ i i }# X critical point through various phenomenological scaling (2.1) arguments. In particular we identify a new crossoverex- where ψ∗(τ),ψ (τ) is obtained by substituting the H{ i i } ponent that describes the relevance ofparticle-hole sym- classical complex variable ψ (τ) for the boson site anni- i metry breaking perturbations to the random rod critical hilation operator a [and ψ∗(τ) for the creationoperator i i behavior. We revisit the original arguments6,28 for the a†] wherever it appears in the (normally ordered form i dynamicalexponentscalingequalityz =d, showingthat of the) quantum Hamiltonian. The partition function they can be violated, and hence that z may remain an is given by Z = trψ[eL], where trψ[] is an unrestricted independent exponent.29 This is consistent with recent integral over all complex fields, ψ (τ·). Notice that the i quantum Monte Carlo simulations in d = 2 that find only term that couples different time slices is the “Berry z =1.40 0.02.14Wealsodiscusstheasymptoticrestora- phase” ψ∗∂ ψ term which arises from the overlapof two ± τ tionofstatisticalparticle-holesymmetryatthedirtybo- coherent states at neighboring times. This should be son critical point. In Sec. V we illustrate all of these contrastedwith the spatialcoupling, 1 J ψ∗ψ (es- 2 i,j ij i j ideas using an exactly soluble one-dimensional model. sentially a discrete version of ψ∗ 2ψ), which appears in The analysis is very similar to that of the Kosterlitz- ∇ P . The imaginary time dimension is therefore highly B Thoulesstransitionintheclassicaltwo-dimensionalXY- H anisotropic. This anisotropyis increasedfurther ifdisor- model. In Sec. VI we generalizethe previous analyses to derispresentsincetheε andJ areτ-independent: the i ij general ǫ = 1, observing along the way some apparent τ 6 disorder appears in perfectly correlated columns, rather pathologies that make ǫ = 1 very special, leading one τ than as point-like defects, in (d+1)-dimensional space- to question how smooth the limit ǫ 1 might be. For τ time. → example, the Bose glass phase has finite compressibility If the ψ∗∂ ψ term were replaced by ψ∗∂2ψ, only the for ǫ =1, but is incompressible for all ǫ <1. It is dis- τ τ τ τ disorderwouldcontributetotheanisotropy(thefactthat tinguished from the Mott phase only by having a diver- the coefficients of ψ∗∂2ψ and ψ∗ 2ψ are different is not gent order parameter susceptibility, χ . We then intro- τ ∇ s important,andmaybecuredbyasimplerescaling). The duce the Dorogovtsev-Cardy-Boyanovskydouble epsilon model becomes precisely a special case in the family of expansion formalism15 and derive the results outlined in classicalmodelswithrod-likedisordertreatedinRef.15. the previous subsection. Finally, two appendices outline The linear time derivative term in (2.1) is more singular the derivations of variouspath integralformulations and than a term with a second derivative in time. It should duality transformations used in the body of the paper. therefore not be too surprising that its presence leads to new critical behavior. Yetanothercrucialpropertyoftheψ∗∂ ψ termisthat II. FUNCTIONAL INTEGRAL τ it is purely imaginary: FORMULATIONS ∗ β β In order to obtain a formulation of the problem more dτψ∗∂ ψ = dτψ∗∂ ψ, (2.2) τ τ amenable to analytic treatment, we turn to functional "Z0 # −Z0 integral representations of the partition function. It will turn out to be important to have an exact representa- where integration by parts and periodic boundary con- tion. Representations which involve dividing the Hamil- ditions have been used. Therefore the statistical factor, tonian into two pieces, = 0 + 1, then using the eLB, used to compute the thermodynamics is in general H H H 9 acomplexnumber,andleadstointerferencebetweendif- and takes precisely the form of a classical XY-model in ferent configurations of the ψ (τ). Unlike that with cou- (d+1) dimensions. i pling ψ∗∂τ2ψ, the resulting model therefore does not cor- The periodicity, (1.6), of the phase diagram is a con- respond to any classical model with a well defined real sequence of the periodicity of the φ in τ: substituting i Hamiltonian in one higher dimension. In fact, it is pre- µ˜ n Uˆ for µ˜ and multiplying out the (U−1) term 0 0 ij cisely this property that reflects the particle-hole asym- yie−lds metry in the model. Interchanging particles and holes is equivalenttointerchangingψ∗(τ)andψ (τ). Theψ∗∂ ψ i i τ β term changes sign under this operation, while HB[ψ∗,ψ] LJ[µ˜−n0Uˆ0] = LJ[µ˜]−in0 dτ φ˙i(τ) is unaffected. Thus although B is invariant under the Z0 i L X combination (known as time reversal) of complex con- + βNε0(n ,µ˜). (2.5) 0 jugation and τ τ, the boson model always violates → − each separately. However 0βdτφ˙i(τ) = 2πmi and ei2πmin0 = 1, so the second term simply drops out of the statistical factor, B. Josephson model eLJ,andwRerecoverthefreeenergyidentity(1.6). Notice that if µ˜ ǫ˜ 1Uˆ , we obtain a statistical factor − i ≡ 2 0 Consider now the canonicalcoordinate Lagrangianfor theJosephsonarraymodel(seeApp.Aforaderivation): eLJ[µ˜−ǫ˜i=12Uˆ0] =( 1)PimieLJ[µ˜−ǫ˜i=0]+βNε0(21,0), (2.6) − β = dτ J˜ cos[φ (τ) φ (τ)] which,thoughreal,is notalwayspositive. Althoughthis J ij i j L − Z0 (cid:26) i,j Lagrangian is also particle-hole symmetric, it too does X notcorrespondtotheHamiltonianofanyclassicalmodel. 1 + 2 (U−1)ij iφ˙i(τ)+µ˜−ǫ˜i iφ˙j(τ)+µ˜−ǫ˜j , Thismodelisverydifferentfromthatwithµ˜−ǫ˜i ≡0. For Xi,j h ih i(cid:27) example, as shown in Fig. 1(c), it always has superfluid (2.3) order at T = 0, for arbitrarily small Jij, as opposed to (2.4) which orders only for sufficiently large J .6 ij with partition function Z = trφeLJ. Notice that the linear time derivative, φ˙ , now appears in a much more i symmetric looking fashion. If µ˜ ǫ˜ 0, is particle- i J − ≡ L hole symmetric and real: C. Josephson model for general ǫτ β [µ˜+ǫ˜ 0] = dτ J˜ cos[φ (τ) φ (τ)] For later reference, note that, in contrast to (2.1), the J i ij i j L ≡ Z0 (cid:20) i,j − Lagrangian(2.3)has anobviousgeneralizationto nonin- X tegerdimensionsoftime.30 Ifǫ isthedimensionoftime, 1 τ (U−1) φ˙ (τ)φ˙ (τ) , (2.4) we simply write ij i j − 2 i,j (cid:21) X β 1 (ǫτ) = dǫττ J˜ cos[φ (τ) φ (τ)]+ (U−1) [i φ (τ)+µ ǫ ] [i φ (τ)+µ ǫ ] , (2.7) LJ  ij i − j 2 ij ∇τ i − i · ∇τ j − j  Z0 Xi,j Xi,j  where τ, µ and ǫ are ǫ -dimensional vectors: τ =(τ ,...,τ ), etc.  i τ 1 ǫτ Renormalization group calculations are performed most conveniently on Lagrangians, such as (2.1), which are polynomials in unbounded, continuous fields and their gradients. Therefore we would like to convert (2.7) to such a model, while retaining the essential physics. If we write ψi(τ)=eiφi(τ), then (2.7) may be written β 1 (ǫτ) = dǫττ J˜ [ψ∗(τ)ψ (τ)+c.c.]+ (U−1) ψ∗(τ)( +µ ǫ )ψ (τ) ψ∗(τ)( +µ ǫ )ψ (τ) . LJ  ij i j 2 ij i ∇τ − i i · j ∇τ − j j  Z0 Xi,j Xi,j  (2.8)   Foronsiteinteractionsonly,U =U δ ,thesecondterm which is conveniently quadratic in ψ. We now relax ij 0 ij simplifies to the assumption ψ = 1, employing instead the usual i | | 1 dǫττ ψ∗(τ)( +µ ǫ )2ψ (τ), (2.9) 2U i ∇τ − i i Z 0 i X 10 rψ 2+v ψ 4 Landau-Ginzburg-Wilsonweighting factor, compressibility and an excitation spectrum with an ex- o|bt|aining|fi|nally ponentially small density of states, ρ(ε) e−ε0/ε, i.e. a ∼ “softgap.” We will show that the compressibility is pre- (ǫτ) = dǫττ 1 J˜ [ψ∗(τ)ψ (τ)+c.c.] ciselythe spin-wavestiffness inthe time direction,which Lψ 2 ij i j therefore vanishes in the “symmetric glass,” but is finite Z (cid:26) i,j X in the Bose glass. This yields an upper bound z d 1 0 ≤ + ψ∗(τ)( +µ ǫ )2ψ (τ) for the dynamicalexponentat the particle-holesymmet- 2U0 i ∇τ − i i ric transition. An effective lower bound on z may be i 0 X obtained by demanding that particle-hole asymmetry be rψi(τ)2+v ψi(τ)4 . (2.10) a relevant operator at the symmetric transition. This − | | | | i (cid:27) is a necessary condition in order that the particle-hole X(cid:2) (cid:3) asymmetric transition be in a different universality class This model retains the exact particle-hole symmetry at from the symmetric one. We will obtain estimates for µ+ǫ 0, but loses the precise periodicity of the phase i ≡ this lower bound within the double ǫ-expansion in Sec. diagramwhen ǫ =1: thus the secondterm in (2.5) now τ VI. becomes β n0 dτ ψi∗∂τψi, (2.11) A. Superfluid densities or helicity modulii Z0 i X We begin by defining the superfluid density—the “he- [compare the first term in (2.1)] which reduces to the licitymodulus,”or“spinwavestiffness,”inclassicalspin previous form if ψ = 1. However if ψ fluctuates, as i i | | | | models. This quantity is computed from the change in in (2.10), this term is no longera perfect time derivative freeenergyunderachangeinboundaryconditions. Con- and will not integrate to a simple integer result. We will sider a box-shaped system with sides L , α = 1,...,D. therefore only use (2.10) near µ˜ = 0 when we study the α WeareinterestedinD =d+1andL =β. Wesaythat role of particle-hole symmetry near the phase transition. D ψ obeys θ -boundary conditions if α ψ(x ,...,x +L ,...,x ) = eiθψ(x ,...,x ,...,x ) 1 α α D 1 α D D. Continuum models ψ(x ,...,x +L ,...,x ) = ψ(x ,...,x ,...,x ), 1 β β D 1 β D β =α (3.1) ThefieldtheoreticLGW-typeLagrangianslistedinTa- 6 ble I follow (for ǫτ =1, but with obvious generalizations i.e., a twist angle θ is imposed in the α direction, while to general ǫτ) from the continuum limit of (2.10) (and periodic boundaryconditionsaremaintainedinallother its various special cases), in which the nearest neigh- directions. Let bor hopping term maps to ψ 2. Space and time are aanlsdo resψca2letdermtos.pArosdaucreesuunl|ti∇,ttch|oeehffiocpiepnintsg odfisothrdeer∂τnψo|w2 fθα =−V1 lntr eLθα , VD ≡ D Lβ, (3.2) maps|∇to|disorder in the ψ 2 coefficient. As is standard, D (cid:16) (cid:17) βY=1 | | themappingisnotexact,butisintendedonlytoproduce where θα istheLagrangian,bethefreeenergyobtained a minimal model that preserves the basic symmetries of using θL-boundary conditions. We may define α theoriginal,sothatitsphasetransitionslieinthecorrect universalityclass. Thecontrolparametersr0 andg0 have ψ˜(x1,...,xD)=e−iθxα/Lαψ(x1,...,xD), (3.3) only a rough correspondence with J and µ˜, but never- 0 which obeys periodic boundary conditions in all direc- theless generatephase diagramswith the same topology. tions. In all cases of interest one may write θα[ψ]= 0[ψ˜]+δ [ψ˜;θ/L ] (3.4) α III. PARTICLE-HOLE SYMMETRY AND THE L L L EXCITATION SPECTRUM OF THE GLASSY where δ may be expanded as a Taylor series in powers L PHASES of θ/L , and superscript “0” denotes periodic boundary α conditions. Thus In this section we will consider the nature of the non- 1 1 superfluid phases in the presence of the two types of dis- δfθα fθα f0 = δ + δ 2 c+... , (3.5) ≡ − −V h Li 2h L i order, ǫ and J (to simplify the notation we henceforth D (cid:20) (cid:21) i ij drop the tildes on the Josephson junction model param- where δ 2 = δ 2 δ 2,andthe averagesarewith c h L i h L i−h Li eters). Recall that for the boson problem the ǫi produce respect to 0[ψ˜]. Equation (3.5) yields a series of terms a Bose glass phase,6 with a finite compressibility and a in powers oLf θ/L , and we use the notation α finitedensityofexcitationstatesatzeroenergy. Weshall contrastthiswiththecaseoftheparticle-holesymmetric, iθ 1 θ 2 δfθα = ρ + Υ +... (3.6) random Jij model, which we will show has a vanishing −Lα α 2(cid:18)Lα(cid:19) α