Quantum charge glasses of itinerant fermions with cavity-mediated long-range interactions

Transcription

1 Quantum charge glasses of tnerant fermons wth cavty-medated long-range nteractons (fxed) postons of the atoms can be chosen so that the effectve spn-spn nteractons become random and frustrated. It was shown that a quantum phase transton n the unversalty class of the (solvable) nfnte-range Isng quantum spn glass [3, 4] occurs, potentally enablng comparson between experment and the theory of spn glasses. In a related paper [5 7], Gopalakrshnan et al. provded expermental detals and detecton methods for the spn glass phase. In the present paper we explore quantum glassness n the charge or densty sector of tnerant fermonc atoms n mult-mode cavtes. An mportant dfference to the prevous study [] conssts n the ncluson of hoppng of atoms on the lattce; the resultng phase dagram now depends on the quantum statstcs of the tnerant partcles. Related bosonc versons (Bose Hubbard models coupled to cavty photons) were consdered n Ref. 9, wheren a superradant Mott nsulatng phase, dsplayng entanglement of the charge of the atoms wth a cavty mode, was found. Optomechancal applcatons of degenerate fermons n a cavty were consdered prevously n Ref.. Our calculatons provde evdence that mult-mode cavtes wth degenerate fermonc atoms can quantum-smulate varous phases and propertes of nfnte-range glasses that share several propertes wth Efros-Shklovsk Coulomb glasses n the quantum regme. For a sngle-mode cavty coupled to tnerant bosonc atoms [4], the onset of superradance s concomtant wth translatonal symmetry breakng and the formaton of a charge densty wave wth a perod of half the cavty photon wavelength. The glassy orderng of fermonc atoms n mult-mode cavtes can nstead be understood as the formaton of one out of many energetcally low-lyng amorphous charge densty patterns n a random lnear combnaton of cavty modes. Most of these amorphous densty orderngs are metastable and the ensung slow relaxaton dynamcs should n prncple be measurable as a response to probe laser felds. Glassy phases of fermons have long been predcted to exst n Coulomb frustrated semconductors, so-called electron- or Coulomb glasses []; n our case, the fermonc atoms play the role of the electrons n these earler studes. In these electronc systems frustraton results naturally from a competton between long range Coulomb repulsons and the random poarv:5.47v [cond-mat.quant-gas] 7 May Markus Müller, Phlpp Strack,, and Subr Sachdev, 3 The Abdus Salam Internatonal Center for Theoretcal Physcs, Strada Costera, 345 Treste, Italy Department of Physcs, Harvard Unversty, Cambrdge MA 38, USA 3 Insttuut-Lorentz for Theoretcal Physcs, Unverstet Leden, P.O. Box 956, 3 RA Leden, The etherlands (Dated: May, ) We study models of tnerant spnless fermons wth random long-range nteractons. We motvate such models from descrptons of fermonc atoms n mult-mode optcal cavtes. The soluton of an nfnte-range model yelds a metallc phase whch has glassy charge dynamcs, and a localzed glass phase wth suppressed densty of states at low energes. We compare these phases to the conventonal dsordered Ferm lqud, and the nsulatng electron glass of semconductors. Prospects for the realzaton of such glassy phases n cold atom systems are dscussed. PACS numbers: , 4.5.-p, 5.3.Rt, 7..Hf, Qr I. ITRODUCTIO There s much current nterest n experments wth ultracold atoms and photons that provde clean realzatons of models from condensed matter physcs. A varant of the antferromagnetc Isng model n one dmenson, for example, has recently been quantum-smulated wth bosons n optcal lattces [] an encouragng step toward quantum smulaton of more general, strongly correlated quantum magnets n low dmensons. The hope s that these quantum optcs experments eventually reach the parameter regmes and accuracy necessary to allow for predctons that can overcome the lmtatons of conventonal theoretcal approaches for strongly nteractng quantum many-body systems. On top of that, the tunablty of ultracold atoms allows one to explore new quantum many-body Hlbert spaces that have no drect condensed matter analog. In a seres of remarkable experments at ETH Zurch [ 4], Baumann et al., have begun the quantum-smulaton of strongly-nteractng quantum gases wth genune long-range nteractons [5]. In these many-body cavty QED systems, an atomc ensemble (a thermal cloud [6, 7] or Bose-Ensten condensate) s loaded nto an optcal cavty contanng quantzed photon modes. Because the photons are massless, they medate an nteratomc nteracton whch does not decay as a functon of dstance between atoms and therefore couples all the partcles n the ensemble to one another. Explotng ths property, Baumann et al. [4, 8] found a mappng of ther entangled atom-lght system to the classc Dcke model descrbng two-level atoms unformly coupled to a sngle quantzed photon mode [9 ]. The superradance transton of the Z Dcke model spontaneously breaks an Isng symmetry and may be vewed as a realzaton of an Isng ferromagnet whch s exactly solvable due to the nfnte range of the photon-medated spn-spn nteracton. Strack and Sachdev [] recently computed the phase dagram and spectral propertes for atomc spns n multmode cavtes assumng that the cavty mode functons and Electronc address: pstrack@physcs.harvard.edu

2 stons and energes of mpurty stes. Whle the former favor a regular densty pattern, such as n a Wgner crystal, the latter dsrupt ths order and provoke a non-crystallne densty order. The competton of these two ngredents leads to many longlved metastable confguratons wth very slow relaxaton dynamcs between them. Ths structure of phase space entals remarkable out-of-equlbrum propertes such as memory effects and logarthmc relaxaton persstng over many hours [ 4] whch are also present n quantum electron glasses approachng a metal nsulator transton [5]. An mportant hallmark of such nsulatng electron glasses s the Efros-Shklovsk gap n the sngle partcle densty of states n the glassy phase, whose elementary charge exctatons are strongly Anderson localzed. Such a pseudogap s requred by the stablty of metastable states n the presence of unscreened long range nteractons [6]. Remarkably, the amorphous orderng of the glass softens the hard gap that would exst n a regularly ordered system just to the maxmal extent whch s stll compatble wth stablty, leavng the glass n an nterestng state of crtcalty [7, 8]. Ths crtcalty results n a wdely dstrbuted response to a local exctaton and avalanches [3], features whch occur also n mean feld spn glasses [3, 3]. Ths crtcalty survves n the presence of weak quantum fluctuatons (non-zero tunnelng ampltude between stes n electron glasses or spontaneous spn-flps nduced by a transverse feld n spn glasses). It entals gapless collectve exctatons despte the absence of a broken contnuous symmetry. Eventually the glass order melts at a crtcal value of the hoppng or transverse feld [4, 45, 5, 5]. Upon further ncreasng the quantum fluctuatons, a common assumpton s that the delocalzaton of the fermonc quaspartcles,.e. the nsulator-to-metal transton, co-ncdes wth the dsappearance of glassy dynamcs, gvng way to a dsordered Ferm lqud. However, t s also possble that these two transtons are separated. The Anderson delocalzaton of the fermonc quaspartcles may precede the meltng of the glass, n whch case we obtan a metallc glass wth non-zero conductvty at zero temperature. Such a glassy state wth metallc conducton was obtaned n dynamcal mean feld theory by Dobrosavljevc and collaborators [33, 34]. Moreover, the fact that glassy phases may also exst n phases wth good transport propertes was recently shown n models of frustrated bosons [8, 4, 43] where a superglass phase wth mcroscopcally coexstng superflud and glassy densty order exsts. In the context of Coulomb frustrated systems n condensed matter (wthout dsorder), an ntermedate metallc phase wth perodc, strped densty order ( conductng crystal ) was dscussed by Spvak and Kvelson [35]. A. Overvew of key results and outlne of paper In the present paper, we argue that two types of glassy phases, a metallc glass and an nsulatng Anderson-Efros- Shklovsk glass also exst for fermons wth random nfnterange nteractons (see Fg. ). For low denstes, we fnd that the metallc glass s avoded; nstead, the lqud abruptly transtons to the localzed glass state (see Fg. ). Our metallc glass state should not be confused wth the metallc glass of metallurgy. In the latter materals, the glassy physcs s entrely due to classcal atoms freezng nto outof-equlbrum confguratons, and the metalllc conducton s due to conducton electrons whch move n the background of the frozen atoms. In contrast, n our system, the glassy dynamcs and metallc conducton are due to the same fermonc degrees of freedom, whch are electrons n the condensed matter realzatons, and fermonc atoms n quantum optcs realzatons. In Sec. II, startng from a Jaynes-Cummngs type Hamltonan for tnerant fermons coupled to cavty photons, we derve the fermonc model that we study n ths paper: H = t c c j + h.c., j (ε µ)n = V j n n j,, j= whch contans a short-range hoppng term t, dsordered, random onste energes ε, and long-range, random denstydensty nteractons V j medated by photons. In Fg., we show the phase dagram of ths model at moderate densty, n = O(), as a functon of effectve onste dsorder ( W defned n Eq. (6)) and photon-medated nteracton strength (J) n unts of hoppng t. For small effectve onste dsorder W, as the nteracton s ncreased, the dsordered Ferm lqud (FL) becomes unstable to the formaton of an rregular, glassy densty pattern, whch depends on the nteractons medated by the random cavty modes. However, the rregular densty waves do not gap the Ferm surface, but leave the fermons metallc, wth a fnte conductvty n the low temperature lmt. The glassy charge densty order s margnally stable, whch leads to soft collectve densty exctatons. The scatterng of fermons from collectve densty modes leads to some non-ferm lqud propertes (such as fnte-temperature transport) but the fermonc quaspartcles reman well-defned. The quantum glass transton and propertes of the metallc glass are analyzed wth effectve feld theory methods n Sec. V. Upon further reducton of the hoppng, or ncrease of nteractons, the random Hartree potental generated by the frozen densty pattern starts to localze the quaspartcles and nduces an Anderson nsulator. The latter has strongly suppressed dffuson, whch s expected to vansh at T =. At ths pont, the metallc background vanshes, leavng behnd an nsulatng charge glass wth a spatally strongly fluctuatng frozenn densty dstrbuton. Ths phase and an estmate for the transton pont (J/t) c,loc for a 3d cubc lattce s presented n Sec. III. We wll show there that for low fermon denstes, at equlbrum, the metallc glass s avoded and the Ferm lqud transtons dscontnuously to a localzed glass phase. evertheless, even at lower denstes, the metallc glass may be expermentally observable, as t s expected to exst as a long-lved metastable phase, whch eventually wll nucleate the energetcally more favorable localzed glass. In Fg., the metallc glass strp s predcted to separate the AES glass from the Ferm lqud even close to the Anderson transton at (J =, W loc ). In that regme, as we explan n ()

3 3 II. MODEL Ferm lqud Metallc glass Anderson-Efros-Shklovsk glass (nsulator) J c,met J c,loc J/t FIG. : (Color onlne) Schematc phase dagram of model () for moderate denstes n the vcnty of half-fllng n /. J s the varance of the long-range dsorder V j, and W the varance of the effectve onste dsorder, Eq. (6). The black, dashed lne s the metalnsulator transton. The blue lne s the glass transton. Quanttatve computatons n ths paper are restrcted to the W = axs, whle the nature of the transton lnes around the Anderson transton (J = ) are nferred from scalng consderatons n Sec. VI. The localzaton transton from the metallc glass to the Anderson-Efros-Shklovsk glass and the sgnatures of ths nsulatng glass phase are dscussed n Sec. III. We descrbe the Ferm lqud to metallc glass transton and the propertes of the metallc glass phase n Sec. V. Ferm lqud Metallc glass Meta-stable Anderson-Efros- Shklovsk glass (nsulator) Meta-stable_ FIG. : (Color onlne) Schematc phase dagram of Eq. () for small effectve onste dsorder (W = ) as a functon of fermon densty n. Below the crossng pont (yellow dot) of the metal-nsulator transton (black-dashed lne) and the glass transton (blue), the transton becomes frst order. The Ferm lqud and the metallc glass may stll exst (as long-lved metastable states) to the rght of the dashed lne. Sec. VI, the quantum glass transton s senstve to the fractal nature of the fermonc wavefunctons close to the Anderson transton. In Sec. VII, we conclude wth a unfyng dscusson about the potental of many-body cavty QED as a quantum smulator of long-range quantum glasses exhbtng freezng n dfferent sectors (spn, fermonc charge, bosonc charge). We also outlne nterestng open questons for future research. J/t We consder spnless lattce fermons coupled to multple cavty photon modes. The absorpton of cavty photons (represented by canoncal bosonc creaton and annhlaton operators a, a) rases the nternal state of the fermons from a ground state (represented by canoncal fermon creaton and annhlaton operators c g, c g ) to an excted state (c e, c e ). In addton, a classcally-treated pump laser coherently drves transtons between the ground state and the excted state. The Hamltonan operator, H[c g, c g, a, c e, c e ] = H atom + H photon + H pump, () conssts of three terms. The frst one descrbes tnerant fermons on a d-dmensonal (optcal) lattce wth =,..., stes wth short-range, nearest-neghbor (, j) hoppng t (taken to be the same for ground and excted state): H atom = t c,g c j,g + c,e c j,e + h.c., j +,e µ + n,e +,g µ n,g. (3) = Here µ s the chemcal potental,,g,,e are the snglepartcle energes of the ground and excted state, respectvely, and n,g = c,g c,g are densty operators for the atoms n the ground state, and analogously for the excted state. Followng Maschler et al. [44], we wrte our equatons n a frame rotatng wth the frequency of the pump laser and descrbes the atom-pump detunng whch can generally be chosen large t,,e. We further have M cavty photon modes wth frequences ω and spatally non-unform atom-photon couplngs g that generally depend on the atom s ste and the characterstcs of the cavty photon mode. Fnally, we have a pump term wth ampltude h that does not nvolve photon operators: M M H photon = ω a a g c,e c,ga + c,g c,ea, H pump = = = = = = h c,e c,g + c,g c,e. (4) The excted state can be adabatcally elmnated. In the aforementoned regme of large atom-pump detunng, one can gnore the dependence of the effectve ground state nteractons on the dynamcs and spatal propagaton of the excted state [44]. Therefore, the,e do not appear n the parameters of the resultng effectve Hamltonan, whch reads: H[c g, c g, a] = t c,g c j,g + h.c. +, j M M + ω a a + + = M =,m= = =,g µ + h n,g = g h n,g a + a g g m n,ga a m. (5)

4 4 We drop the subscrpt g from now on. We nclude the expectaton value of the term n the last lne as a contrbuton to the sngle-partcle energes, ε = M,m= g g m a a m + + h. (6) In a fnal step, we ntegrate out the photons from Eq. (5) and get the expresson: H c, c = t c c j + h.c. +, j (ε µ) n = V j n n j., j= The long-range densty-densty nteracton wrtten n a path ntegral representaton V j (Ω) = M = g g j h h j ω Ω + ω makes the dependence on the bosonc Matsubara frequency Ω explct [7]. The magntude of V j s proportonal to the ampltude of the drvng laser h and can therefore be tuned flexbly. The sgn and spatal dependence of V j s determned by the choce of mode profles of the cavty modes and pump lasers as well as the orentaton of the lattce wthn the cavty. In ths paper, we are prmarly nterested n the case where the photon mode functons g g j n Eq. (8) can be realzed as randomly varyng n sgn and magntude n each dsorder realzaton and wth M suffcently large, we assume the V j (Ω) to be Gaussan-dstrbuted wth varance (7) (8) δv j (Ω)δV j (Ω ) = δ δ jj + δ j δ j V(Ω, Ω )/. (9) The overlne represents a dsorder average, and δv j s the varaton from the mean value. Such a mean value only shfts the chemcal potental and can be dropped. Further, we assume n ths paper that couplngs between dfferent stes are uncorrelated. We note that t should also be possble to generate random and frustrated nteractons of longer range by usng other means than random cavty modes. For example one mght employ a second fermon speces to generate RKKY-type nteractons among the prmary fermon speces. As n metallc spn glasses, such nteractons decay as a power-law wth dstance and oscllate wth perods of the Ferm wavelength of the second speces, whch nduces frustraton. In the calculatons below, the results only depend on the varance, whch respects tme-translaton nvarance V(Ω, Ω) J (Ω). () To capture the man effects, as n Ref., we may assume the smplfed form: J(Ω) = v ω / Ω + ω, () where ω s a prototypcal photon frequency representatve of the spectral range of photons that medate the nter-atomc nteracton and v the dsorder strength. For most of the paper we wll concentrate on frequences Ω ω and work wth the statc lmt of the couplngs δv j δv j = δ δ jj + δ j δ j J /, J J() = v ω. () The photon contrbuton to the sngle-partcle energes n Eq. (6) and the pump term generate random local potentals for the fermons, whch may addtonally be superposed by a random lattce potental. We summarze all these effects by assumng random (Gaussan-dstrbuted) onste energes wth ndependently tunable varance III. δε δε j = δ j W. (3) ISULATIG ADERSO-EFROS-SHKLOVSKII GLASS In the absence of hoppng, t =, the Hamltonan (7) reduces to the classcal Sherrngton-Krkpatrck (SK) model [56] of localzed spns (descrbng presence or absence of a partcle). Ths spns are subject to a random longtudnal feld ε, and kept at fxed magnetzaton M = n where n s the fermon densty. The low temperature glass phase of ths model s understood n great detal. As llustrated by the red graph n Fg. 3, a typcal confguraton of ths glass exhbts a lnear soft gap n the dstrbuton P(ϕ) = of local Hartree felds ϕ = δ(ϕ ϕ ) dh dn = ε µ α ϕ, ϕ J, (4) J V j n j, (5) wth coeffcent α 4.3 =.4 [57, 58] and Gaussan decay for ϕ J. The brackets... stand for the thermodynamc average. As compared to the canoncal SK model an extra factor of 4 arses because we consder Isng degrees of freedom wth magntude s z n / = ±/. Remarkably, the soft gap (4) at low felds s unversal, that s, ndependent of the strength of the random felds and the average magnetzaton (densty) [9]. A smlar soft gap, the Efros-Shklovsk Coulomb gap [6] s also known to exst n electron glasses wth Coulomb nteractons [9, 33]. A. Anderson-Efros-Shklovsk (AES) glass wth quantum fluctuatons/fnte hoppng (t ) Upon turnng on quantum fluctuatons va a fnte hoppng t, the fermons hop wthn the dsorder potental of the above dscussed Hartree felds. We refer to the egenenerges of the resultng sngle partcle problem as sngle partcle exctatons. In the lmt t the latter are completely j

5 5 ΡE t > t loc t < t loc t =. FIG. 3: (Color onlne) Sketch of the evoluton of the dstrbuton of sngle partcle energes ρ(e) as the hoppng t ncreases. In the classcal Efros-Shklovsk lmt, t = (red sold curve), ρ(e) concdes wth the dstrbuton of Hartree felds, whch exhbts a lnear pseudogap. In the localzed AES glass at fnte hoppng (blue, dot-dashed curve) a pseudogap wth vanshng densty of states at the Ferm level ρ() = perssts. A fnte densty of states at E = emerges when the quaspartcles delocalze and the system turns metallc. The dp n the densty of states at low energy gradually weakens as t ncreases. localzed and ther energes concde wth the local Hartree felds. As long as the hoppng stays below a crtcal value (t < t loc ), the fermonc atoms reman Anderson-localzed. In that regme the sngle partcle densty of states n a gven local mnmum of the glass has to vansh at the Ferm level (down to energes of order /, at T = ), otherwse the state would be unstable wth respect to charge rearrangements. Ths follows from arguments analogous to those for quantum Coulomb glasses [6, 38 4]. As a consequence, n a typcal AES glass state, the lnear compressblty (the analogue of zero-feld-cooled susceptblty n spn glasses) vanshes, even though the full thermodynamc (feld-cooled) compressblty s fnte. Despte the vanshng lnear compressblty, there s no hard charge gap, as there are charge exctatons at any fnte energy. The qualtatve evoluton of the dstrbuton of sngle partcle energes wth ncreasng hoppng s sketched n Fg. 3. We now explan the key dfferences of the AES glass to other prevously dscussed glass phases and descrbe ts transport propertes. (Quantum) electron glass: The glassy electrons n strongly doped semconductors are localzed mostly due to quenched potental dsorder, whereas n the case of the AES glass the potental s mostly nteracton-nduced and thus self-generated. If W could be made to vansh, the localzaton would be entrely nduced by the random nteractons n a glassy frozen state. Both, the quantum electron glasses and the AES glass have lnearly suppressed densty of states at low energes. Anderson or Ferm glass: In these localzed nsulators, where quenched external dsorder domnates, the quaspartcles are localzed despte the presence of weak short range nteractons. The bosonc analogues of such nsulators are E often referred to as Bose glasses [36, 37]. Both fermonc and bosonc verson of these glasses onste-dsorder domnated and do not exhbt glassness n the sense of havng many metastable confguratons separated by hgh multpartcle barrers, except f strong nteractons are present n addton to the dsorder potental. Another dfference s that the strong nteractons n the AES glass cause gapless collectve modes, whch are absent n less strongly correlated Ferm and Bose glass nsulators. Mott glass: In fermonc lattce systems at commensurate fllng wth strong nearest-neghbor nteractons and dsorder, ths Mott glass arses as an ntermedate phase whch s ncompressble due to a hard charge gap, but has a fnte AC conductvty σ(ω) at any postve frequency [4]. The man contrbuton to σ(ω) comes from local partcle-hole exctatons, whch become gapless due to dsorder. The AES glass s very dfferent from ths Mott glass, n that t has no hard charge gap, exsts at any fllng and does not rely on external potental dsorder. Isng spn glass n transverse feld: If the occupaton of stes n =, s thought of as an Isng varable, the model (7) looks smlar to a mean feld Isng spn glass n a transverse feld Γ. The dfference wth the model consdered n ths paper conssts mostly n the way n whch quantum fluctuatons act on top of the classcal SK glass. Ths entals however a crucal dfference of the respectve phases to expect for weak long range nteractons, as we wll explan below. The transverse feld SK model s well known to undergo a quantum glass transton from a glass to a quantum dsordered state at a crtcal value Γ J. The spns undergo glassy freezng when ther nteracton strength s bgger than the nverse polarzablty of the softest two level systems. One can show that the thermodynamc glass transton perssts n the presence of random longtudnal felds (the equvalent of the dsorder potental ε n Eq. (7)), even though the latter spols the Isng symmetry of the system. The glass transton then has the character of an Almeda-Thouless transton, lke for mean feld spn glasses n an external homogeneous or random feld. The only symmetry whch s broken at ths transton s the replca symmetry, whch s rgorously establshed to occur n models wth nfnte-range nteractons, whle t remans controversal n fnte-dmensonal, short range nteractng systems. If the Isng states of a spn are thought of as two states of a localzed fermon (two-level systems), the transverse feld SK model descrbes nteractng, but fully localzed fermons. It follows from the above dscusson that such systems have a non-glassy nsulatng phase. However, for our model a regme of localzed but non-glassy fermons cannot exst. The reason s that there s no mechansm whch gaps sngle fermon exctatons away from zero energy, n contrast to the fnte tunnelng ampltude Γ of two level systems whch does gap the local exctatons. Therefore, n the model consdered here, the gapless low-energy fermonc states are always unstable to glass formaton for any small nfnte range nteractons V j (a smlar argument was put forward n Ref. 34). Ths s why the AES glass n Fg. above W loc extends all the way to the axs J =.

6 6 Transport propertes: The man contrbuton to the fnte temperature transport n the AES glass comes from nelastc scatterng of the quaspartcles, wth an nelastc rate whch s expected to decrease as a power law of T. At low T, where ths rate becomes smaller than the level spacng n the localzaton volume of the fermons, the charge transport s expected to proceed frst by power law hoppng, and at lowest T by varable range hoppng, asssted by nelastc processes among the fermons. Ths type of transport s drastcally less effcent than n the metallc glass of Secton V. In the AES glass, the resstvty that should dverge wth vanshng T. At the same tme thermal transport may stll be rather good due to collectve densty modes, whch reman delocalzed. It s temptng to speculate that, upon advancng the technques of Ref. 6 to many-body cavty QED, the next generaton of experments could measure also the transport propertes of neutral, glassy fermons. IV. DELOCALIZATIO TRASITIO OUT OF THE ADERSO-EFROS-SHKLOVSKII GLASS AT W As mentoned above, n the localzed, nsulatng phase the Efros-Shklovsk stablty argument assures a vanshng densty of sngle partcle states at the Ferm level. However, the latter becomes fnte when the quaspartcles delocalze at the nsulator-to-metal transton, where dffusve transport behavor sets n. Ths takes place when the hoppng becomes comparable to the typcal potental dfference between a ste and ts neghbors. A precse calculaton of ths transton would requre the soluton of the glass problem ncludng quantum fluctuatons, to obtan the dstrbuton of effectve onste dsorder, and the subsequent analyss of a delocalzaton transton of quaspartcles. However, an estmate for the crtcal value J t, below whch the system behaves metallc, can c,loc be obtaned as follows. A. Moderate densty: n O() Even f W of Eq. (6) s neglgble, the soft gap n the Hartree potentals acts lke a rather strong onste dsorder for the fermons. To dscuss delocalzaton at the Ferm level we need to consder stes wth Hartree potentals of the order of the hoppng t. At such energes, the densty of states n the soft gap can be roughly approxmated by the constant densty of states of a box-dstrbuted onste potental of wdth W J gven by W J P(ϕ t) J αt. (6) For a box dstrbuton of onste dsorder, Anderson delocalzaton of quaspartcles on a cubc lattce n d = 3 dmensons s known to occur at a crtcal dsorder strength [53] Wc t 3d = Z U (7) From ths we may nfer an estmate for the delocalzaton transton out of the nsulatng glass phase at J t c,loc αz U 4.6. (8) In ths rough estmate we have neglected quantum fluctuatons of the densty order n the localzed phase whch are expected to ncrease the value of (J/t) c,loc further. Ths s because they weaken the densty nhomogenety and thus the onste dsorder generated by the nteractons. B. Low densty: n At low fermon denstes, n, the pseudogap s rrelevant for the delocalzaton transton because the gap s restrcted to very small energes, far below the hoppng strength needed for delocalzaton. The frozen felds (5) have a typcal magntude W J nj. If ths s the domnant contrbuton to dsorder (.e., for weak external dsorder, W W J ), the delocalzaton nstablty arses n the glass phase when t W J,.e., when the nteractons are reduced below the value (J/t) loc n /. (9) At larger external dsorder, delocalzaton happens when t W, ndependently of the nteracton strength J. However, the nteracton strength (9) s parametrcally smaller than the nstablty (J/t) met n 5/6 of the Ferm lqud toward the metallc glass, whch we wll derve below n Eq. (44). Ths mples that at low denstes, beyond the yellow, tr-crtcal ponts of Fg., the localzed glass state jumps dscontnuously nto a non-glassy Ferm lqud va a frst order transton. The locaton of that transton can be estmated by consderng the competton of knetc energy cost t and potental energy gan nj of transformng the Ferm lqud nto a fully localzed glass state. Ths yelds (J/t) st n /, () whch scales n the same way wth densty as the delocalzaton nstablty of the localzed phase, but s physcally dstnct from t. However, snce the nteractons are essentally nfnte ranged, the system cannot smply nucleate a localzed glass droplet locally, but has to undergo the localzaton transton more or less at once n the whole system. Ths suggests that the Ferm lqud phase s a very long lved metastable state, even far beyond the equlbrum transton pont (J/t) st. V. METALLIC GLASS AT W In ths secton, we develop an effectve feld theory approach for the Ferm lqud to metallc glass transton (the blue lne for W n Fg. ) and compute key propertes of the metallc glass. We can wrte the path ntegral pendant to Eq. (7) n terms of collectve Hubbard-Stratonovch felds for charge fluctuatons ρ (τ) and Lagrange multplers α whch

7 7 enforce ρ (τ) = c (τ)c (τ), where c, c are Grassmann varables representng the fermon felds and T wll denote temperature. Wth ths feld content, the partton functon correspondng to the model of Eq. (7), Z = DαDρD cdc e S +S nt, s determned by the acton S = S nt = + j= dτ c (τ) ( τ µ) c (τ) = dτ t c (τ)c j (τ) + c j (τ)c (τ),, j, j= dτ dτ V j (τ τ )ρ (τ)ρ j (τ ) dτ α j c j (τ)c j (τ) ρ j (τ) + ρ j (τ)ε j. () In order to streamlne notaton, we wll use sgns for all lattce ste summatons and ntegratons over magnary tme. We now ntegrate out the fermons, performng a cumulant expanson n the terms α to obtan an effectve acton for the densty felds Z = DαDρ e S[α,ρ] wth S[α, ρ] = V j (τ τ )ρ (τ)ρ j (τ ) + ρ j (τ)ε j, j,τ,τ j,τ α j (τ) n j (τ) ρ j (τ) j,τ + α (τ)α j (τ )n (τ)n j (τ ) +..., (), j,τ,τ where... denotes the quantum and thermal average wth respect to the non-dsordered free fermon acton S. We ntroduce the auxlary Hubbard-Stratonovch densty felds ρ as an ntermedate step to make the physcs more transparent and to make contact wth our prevous work []. As n Ref., we wll later ntegrate them out agan to obtan an acton for the order parameter felds Q ab. In App. A, we present an alternatve route to derve the fnal self-consstency equatons (38-4), whch offers nsght nto the nature of the approxmatons made below. We proceed by ntegratng over α, wthout keepng explct track of the hgher-order terms n α denoted by the dots n Eq. (). Below we drop those correctons. However, the alternatve approach n App. A can n prncple resum them, at the expense of replacng the bare densty-densty correlator Eq. (4) wth the full proper polarzablty [3]. By droppng the nteracton correctons to the latter, we operate at a level equvalent to the approxmaton used by Mller and Huse for the closely related nfnte-range quantum spn glass problem [3]. Those authors obtaned a good estmate for the quantum glass transton pont when compared wth more elaborate studes [54, 55]. It s convenent to express the resultng acton n terms of local fluctuatons n the onste densty δρ (τ) = ρ (τ) n (τ) ρ (τ) n: S[ρ] = δρ (τ) Π () δρ j(τ ) (3) τ,τ,, j, j,τ,τ δρ (τ)v j (τ τ )δρ j (τ ) + δρ (τ) ε +...,, j,τ,τ τ, wth the bare densty-densty correlator denoted by Π (), j,τ,τ = n (τ)n j (τ ). (4) The effectve onste dsorder conssts of two terms ε = ε n j (τ ) V j (τ τ ). (5) j,τ Focusng on low frequences we neglect the retardaton, and fnd the dsorder varance from Eqs. (9, 3) as: δ ε δ ε j = δ j W + J ()n k δ j W. (6) The dots n (3) stand for nteractons between more than two δρ. We proceed by employng standard replca methods [46] to average over the dsorder confguratons of the V j (Ω) and ε. The resultng nter-replca nteracton term δρ 4 proportonal to the varance of V j s non-local n both, magnary tme and poston space. We decouple ths terms wth an nter-replca matrx-valued Hubbard-Stratonovch feld, that depends on two frequences, Q ab (Ω, Ω ) δρ a (Ω)δρb (Ω ), whch s blocal n magnary tme, but local n poston space. Fnally, we wrte the n-tmes replcated, dsorder-averaged partton functon Z n = DQDρ e S wth the acton S =, j,τ,τ,a + T δρ a (τ) Π (), j,ω,ω,a,b τ,τ,, j δρa j (τ ),τ,τ,a,b W δρa (τ)δρb (τ ) V(Ω, Ω ) 4 Qab (Ω, Ω )Q ab j ( Ω, Ω ) Qab ( Ω, Ω )δρ a j (Ω)δρb j (Ω ). (7) In the end we wll take the replca lmt n to extract quenched averages. The dots stand for further replcadagonal nteractons between several δρ, whch are generated by the hgher order cumulants n Eq. (). Incorporatng local dsorder nto the polarzablty, whch replaces Π (), j,τ,τ n a complete soluton wll actually generate replca off-dagonal terms n Π that self-consstently depend on W and J ()n. Ths sgnfcantly complcates the analyss. As announced above, we restrct here formally to the W = slce of Fg., and defer the quanttatve analyss of onste-dsorder effects to future work [7]. A. Saddle-pont free energy (W = ) Wthout any truncaton n the cumulant expanson (), the functonal ntegraton would yeld results for the correlators

8 8 of δρ that automatcally obey the Paul prncple. However, after a truncaton the latter mght be volated. In order to proceed wth a saddle-pont analyss, we ntroduce nto the acton a global Lagrange multpler λ a conjugate to densty fluctuatons, enforcng that correlaton functons stll obey the Paul prncple on average. amely, λ a s determned selfconsstently so that the equalty δρ a (τ)δρa (τ) = n( n) (8) = s satsfed. The lefthand sde can be vewed as the varance of the bnomal probablty dstrbuton of lattce stes beng ether occuped (ρ = ) wth probablty n or empty (ρ = ) wth probablty n. We recall that n s the average densty. n = corresponds to an empty band, n = / to half-fllng, and at n = the band s flled. Ths global constrant s analogous to the sphercal approxmaton n spn glasses. It can be shown that relaxng the (exact) local constrants to a global constrant does not affect the quantum crtcal behavor [4, 47 49]. otcng that the acton n Eq. (7) s quadratc n the ρ- varable, we can ntegrate t out exactly. ote that the bare densty correlator Π () τ,τ,, j only depends on the dfferences n tme ( τ τ ) and space ( j ). The resultng acton has a prefactor, the number of atoms, and thus, a saddle-pont evaluaton becomes exact. The free energy T F = lm lm n n ln Zn, (9) becomes a functonal of the order parameter feld Q ab (τ, τ ) Q ab (τ, τ ) = δρ a (τ)δρb (τ ) (3) and the Lagrange multplers λ a. Assumng that on the saddle pont the latter take a replca-symmetrc value λ, the functonal takes the form F = lm n n F n wth F n (Q ab,λ) = T 4 J (Ω) Q ab ( Ω, Ω) λ ( n) n + T a,b,ω ω,q tr ab ln Π () (Ω, q) + λ δ ab J (Ω)Q ab ( Ω, Ω), (3) where tr ab denotes the trace over replca ndces and the dots stand for the neglected terms mentoned above. We have mposed tme-translatonal nvarance of Q ab : Q ab (Ω, Ω ) Q ab (Ω, Ω )δ Ω+Ω,/T. As long as we are prmarly nterested n the crtcal behavor and the propertes of a sngle typcal state close to the glass transton, we may proceed wth a replca symmetrc calculaton, even though the replca-symmetrc saddle pont s strctly speakng unstable towards the breakng of replca symmetry. The latter sgnals the emergence of many pure states n the glass phase, the breakdown of full thermalzaton,.e., ergodcty breakng and the assocated out-of-equlbrum phenomena at long tme scales. As we show n App. A, the glass nstablty condtons obtaned below (Eq. 38,43) ndeed sgnal the nstablty of a replca symmetrc saddle-pont toward replca symmetry breakng. For the Q ab felds, the followng ansatz s natural: Q ab (Ω, Ω) = D(Ω)δ ab + δ Ω, T q EA. (3) Here, the Edwards-Anderson order parameter q EA shows up both n dagonal and off-dagonal entres of Q ab. Ths ansatz n terms of q EA and the (ste- and dsorder-averaged) dynamc densty response D(Ω) s the most general one, respectng replca symmetry and tme-translatonal nvarance. A non-zero value of q EA sgnals a frozen-n densty dstrbuton of the atoms: q EA = lm δρ (t)δρ (). t In the glass phase, the spatally non-unform on-ste denstes dffer randomly from ther average value (dependng on the state nto whch the glass freezes) and retan that value for nfntely long tmes. As n any glass, these frozen densty patterns are expected to depend senstvely on the detals of the quench protocol or the preparaton hstory n general. ote however, that only n the case of vanshng effectve dsorder, W =, q EA serves as an order parameter for the glass transton. In the more realstc case of fnte onste dsorder, W, the system stll exhbts a thermodynamc glass transton, as do mean feld spn glasses n random felds, but the only symmetry to be broken n that case s the replca symmetry, snce densty nhomogenetes already exst n the Ferm lqud phase (cf. App. A). The average dynamc densty response, D(Ω) = δρ (Ω)δρ ( Ω), (33) characterzes the response of the fermons to local, tmeperodc modulatons of the densty. In terms of the parametrzaton (3), the free energy (3) obtans as F = T J (Ω) D(Ω) + 4 J ()D()q EA λ ( n) n Ω + T d d q Π Ω (π) ln () (Ω, q) + λ J (Ω)D(Ω) d d d q J ()q EA (π) d Π () (Ω, q). + λ J ()D() (34) Ths free energy depends on the mcroscopc parameters of the orgnal fermonc theory va the bare densty response d n (τ)n j (τ d q )+q(x ) = T x j) (π) d e Ω(τ τ Π () (Ω, q), Ω (35)

9 9 where Π () s the real part of the partcle-hole bubble,.e. the convoluton of two bare fermon propagators: d Π () d k (Ω, q) = Re T ω (π) G (ω +Ω, k + q)g d (ω, k) d d k f (ξ k ) f (ξ k+q) = Re (π) d Ω, (36) ξ k+q ξ k wth f (x) = /(exp[x/t] + ) the Ferm functon and ξ k the fermon lattce dsperson. For a 3d cubc lattce, for nstance ξ k = t cos k x + cos k y + cos k z µ, (37) wth t the nearest-neghbor hoppng matrx element and µ the chemcal potental whch fxes the lattce fllng (µ = for half-fllng, that s, n = /). From the free energy functonal (34) we wll extract the phase boundary between the Ferm lqud and the metallc quantum charge glass and the assocated quantum-crtcal dynamcs of the densty correlatons. Mnmzaton of Eq. (34) wth respect to q EA, λ and D(Ω) for each Ω, yelds a set of coupled saddle-pont equatons. The dervatve wth respect to D(Ω) for Ω > together wth the dervatve wth respect to q EA requres the densty response to obey the selfconsstency relaton D(Ω) = d d q (π) d Π () (Ω, q) + λ J (Ω)D(Ω). (38) The lefthand sde, when wrtten as a geometrc seres, can be seen to express the self-consstent resummaton of all cactus dagrams n the nteractons V j. Mnmzaton of Eq. (34) wth respect to λ gves back the global constrant on densty fluctuatons: T D(Ω) + q EA = n( n). (39) Ω Fnally, the mnmzaton wth respect to D() determnes the Edwards-Anderson parameter self-consstently: d d q J ()q EA q EA = (π) d Π () (, q) + λ J ()D(). (4) For vanshng nteractons, J(Ω) = J() =, we have q EA = and Eqs. (38,39) have the free fermon soluton: d D () d q (Ω) = Π () (Ω, q) (π) d λ =. (4) Indeed, for free fermons, the constrant T Ω D () (Ω) = n ( n) s automatcally fulflled, and thus λ(j = ) =. ote that Eq. (4) s a varant of the general glass nstablty condton: J ˆχ j (Ω=) = (4) j where ˆχ s the full densty-densty correlator. In Appendx A, we present an alternatve route to derve Eqs. (38-4,4). B. umercal results for metallc glass: phase dagram and densty response We now compute the glass transton lne (Fg. 4) and the assocated densty response (Fg. 5), assumng no effectve onste dsorder W = on a 3d-cubc lattce. We smultaneously solve Eqs. (38, 39) together wth the crtcalty condton derved from Eq. (4) (where t permts a soluton wth nfntesmal q EA ): = d d q J () (π) d Π () (, q) + λ c,glass J ()D(). (43) Ths crtcalty has an mportant consequence for the dynamc response. By wrtng D(Ω) = D() δd Ω, and expandng Eq. (38) around Ω=, we fnd that the condton (43) entals a more sngular low frequency behavor δd Ω Ω, as compared to the behavor of the bare densty response δd () Ω Ω. Ths s llustrated by the explct soluton of D(Ω) on a frequency grd n Fg. 5. Eq. (39) wth fnte q EA, and more general arguments presented below ensure that ths crtcalty extends nto the glass phase. We wll present an approxmate analytcal calculaton of D(Ω) at the glass transton and n the metallc glass phase n the followng subsecton V C. For the numercal soluton of these equatons we dscretzed D(Ω) on a frequency grd wth varyng step sze up to a hundred grd ponts explotng D(Ω) = D( Ω). We employed a modfed verson of Powell s Hybrd method algorthm for mult-dmensonal root-solvng [5]. For the 3dnumercal momentum ntegratons of the rght-hand-sdes, we employed the VEGAS Monte Carlo algorthm [5]. To avod 6-dmensonal ntegratons at each step of the multdmensonal root solver, the momentum-ntegrated partclehole bubble was catalogued as 4-dmensonal array and then quadru-lnearly nterpolated n the ntegrands for the 3- dmensonal q-ntegraton (see App. B for some excerpts of Π () (Ω, q x, q y, q z )). The resultng phase boundary between Ferm lqud and metallc glass s plotted n Fg. 4 for varous fermon denstes. Along the phase boundary the glass transton s of contnuous nature,.e. wth a contnuous onset of the Edwards- Anderson parameter q EA (n the case W = ). Lke n quantum spn glasses, but n contrast to structural glasses (supercooled lquds), the dynamc freezng and the thermodynamc glass transton are thus expected to concde. Accordngly one expects the replca symmetry breakng pattern wthn the glass phase to be contnuous (full replca symmetry breakng) whch ensures crtcalty and the exstence of gapless collectve modes [5]. One can see from Eq. (43) that as a functon of densty, the glass nstablty of the Ferm lqud occurs when J (Π (n)n / ). Usng that Π (n) n/e F (n) n /d /t, we nfer that n dmensons d = 3 the glass nstablty of the Ferm lqud scales as (J/t) met n 3/+/d = n 5/6 (44)

10 n Ferm lqud Metallc glass class of nfnte-range, metallc quantum Isng spn glasses. Those are characterzed by the nteracton- and dsordernduced freezng of Isng degrees of freedom n the presence of a metallc charge sector wth gapless fermonc exctatons, whch damp the order parameter fluctuatons. Ths was orgnally put forward n the context of metallc spn glasses by Sachdev, Read, and Oppermann [5], as well as by Sengupta and Georges [47]. Later on ths unversalty class was further analyzed n the form of a Landau theory for a mean feld verson of the electron-glass transton out of the Ferm lqud to by Daldovch and Dobrosavljevć [45] Jt FIG. 4: (Color onlne) umercally computed phase boundary of the metallc glass for varous denstes and W =. The crosses correspond to data ponts computed from a smultaneous soluton of the saddle-pont equatons (38,39,43); they are connected as a gude to the eye. ote that the phase boundary to the nsulatng glass s not plotted (see Fg. for an llustraton of both phase boundares). D D c,glass () D xxxx FIG. 5: (Color onlne) Comparson of the collectve densty response as a functon of magnary frequences at the glass transton (blue crosses on the frequency grd computed from saddle-pont equatons (38,39,43), and ts bare pendant from Eq. (4). The slope of the low energy response n the glass phase s sngular, wth a Ω behavor for small frequences. at low denstes. Ths scalng wth the fermon densty s confrmed by the numercal results of Fg. 4. However, as we argued n Sec. IV B, at lower denstes (beyond the yellow trcrtcal ponts n Fg. ), ths nstablty s pre-empted n equlbrum by a frst order transton whch takes place at (J/t) st n / (J/t) met. evertheless, the Ferm lqud phase should reman expermentally relevant even at these low denstes, snce t s very dffcult to nucleate the localzed nsulator out of the metastable Ferm lqud phase, owng to the hgh nucleaton barrer mposed by the long range nteractons. The glass transton, Fg. 4, and the assocated emergence of a sngular densty response, Fg. 5, are that of the unversalty C. Dynamc densty response n the metallc glass It s nstructve to derve an approxmate analytcal soluton for D(Ω) neglectng the q-dependence of the partclehole bubble n Eq. (38). To ths end we take Π () (Ω, q) Π () (Ω, Q) wth a fxed, prototypcal Q dfferent from any potental nestng vectors, Q < k F. Then, Eq. (38) becomes a quadratc equaton for the (approxmate) densty response, whch we call D Q (Ω). It can be solved n closed form: Π () (Ω, Q) + λ D Q (Ω) = J (Ω) Π () (Ω, Q) + λ J (Ω) J (Ω). (45) where we chose the physcal soluton wth mnus-sgn n front of the square-root, ensurng that D Q (Ω) decays to zero for large frequences. Recall that the partclehole bubble becomes zero for large external frequences: lm Ω Π () (Ω, Q). The crtcalty condton, Eq. (43) for fxed momentum Q, appled to Eq. (45), yelds: λ c,glass = J() Π () (, Q), and DQ () c,glass = J(). At the glass transton D Q (Ω) develops a sngular response to small frequency pertubatons of the local fermon densty. The approxmate energy scale Π () (, Q) = λ + J () (46) controls the dstance to the transton and separates two qualtatvely dstnct regmes n the dynamc densty response of the Ferm lqud. Contnung Eq. (45) to real frequences, Ω Ω+ +, we obtan: Ω, Ω, Im D Q (Ω) (47) Ω, Ω. The self-consstency condton (39) pns = n the entre glass phase, and the densty response remans sngular at zero frequency. Ths property holds true ndependently of the above approxmatons, as can be formally derved assumng full replca symmetry breakng n the glass phase [5].

11 The key ngredent for the unusual square-root behavor for both, D Q (Ω) and the full D(Ω) of Fg. 5, s the couplng of the collectve charge fluctuatons to the metallc partcle-hole contnuum whch coexsts wth glassy, amorphous densty order. The low frequency behavor of the partcle-hole bubble (see Fg. 6 n the appendx) at fnte momentum transfer Q: Π () (Ω, Q) Π () (, Q) Ω, (48) underles Eq. (47) and also enters nto the numercal computaton of Fg. 5. D. Fermonc quaspartcles n the metallc glass The sngle-partcle propertes of the underlyng fermons n systems belongng to the unversalty class of nfnte-range, metallc quantum Isng glasses have been worked out by Sengupta and Georges [47]. At T =, at the crtcal pont and n the glass phase, the fermons reman well-defned quaspartcles. Indeed, the leadng self-energy correcton due to low frequency charge fluctuatons scale as Σ f (ω) ω 3/, (49) wth a frequency exponent >. evertheless, at fnte temperatures ths translates nto non-analytc correctons n the transport propertes. The fnte-temperature resstvty, for example, scales as δρ(t) T 3/ n the quantum-crtcal regme above the QCP [47] and n the entre metallc quantum glass phase. Deeper n the glass phase, the metallc dffuson eventually breaks down when the localzaton transton to the Anderson-Efros-Shklovsk glass of Sec. III s reached. snce stronger hoppng s necessary to compensate for the extra dsorder. A. The role of fractalty for the glass transton On the other hand, we have to analyze the glass nstablty (4) wthn the metallc phase. In the lmt J, t holds that ˆχ χ(j = ), whch reduces to the (exact) susceptblty of non-nteractng fermons. As noted n Ref. 34, the sum over susceptblty squares, χ j χ j (Ω=) (5) dverges at the Anderson transton. Ths mples a glass nstablty already wthn the metallc phase, even for very weak nteractons J t, W. To the best of our knowledge, the precse dvergence of χ, or equvalently, of χ j, as W t W () t, loc s not known. However, the dsorder-averaged densty-densty correlator has been well studed, snce t reveals nterestng propertes of the fractal nature of the electronc wavefunctons and the anomalous dffuson at the Anderson transton [6]. In partcular, the spatal Fourer transform of χ j (Ω=) behaves as χ q (Ω ) D(q, Ω ) δ ξ f ( q ξ), (5) where ξ s the correlaton length, whch dverges at the Anderson transton as ξ ( t/t c ) ν ; δ ξ = /νξ d s the sngle partcle level spacng n the correlaton volume and ν s the densty of states. The scalng functon f (x) behaves as [6, 63] VI. PHASE BOUDARIES OF THE METALLIC GLASS AT W > J f (x) x d, x, (53) f (x) x, x. (54) In the precedng secton we have argued for a metallc glass phase at moderate denstes and weak external dsorder. However, also when the effectve onste dsorder W s larger than the nfnte-range nteracton J, we expect an ntermedate metallc glass phase strp between the Ferm lqud and the AES glass, as shown n Fg.. Below we wll present scalng arguments to justfy ths scenaro, wth both transtons at the borders of the metallc glass strp (the metal-nsulator transton on top and the glass transton on the bottom ) beng contnuous. For vanshng nteractons J =, the two nstablty lnes must jon at the Anderson transton of the dsordered free fermons, where W loc /t = W loc /t =.9 (for Gaussan dsorder at half fllng [53]). In the presence of weak nteractons, the frozen felds from the glassy densty order tend to ncrease the dsorder varance by δ(w ) J, whch has a smlar effect as a weak ncrease of dsorder δw J /W. Accordngly one expects the crtcal value W/t for delocalzaton to decrease by a relatve amount W W () W (J) δ t loc t t loc J, (5) W Here, d s the fractal dmenson assocated wth the dstrbuton of ψ 4 (x) over all space, ψ(x) beng a crtcal sngle partcle wavefuncton at the delocalzaton transton. From ths one obtans that ˆχ χ j (Ω=) = j d d q (π) d χ q(ω =) ξ (d d ) ( t/t c ) ˆα. (55) The exponent results as ˆα = ν(d d ) 4, usng the known values d [d = 3].3 [6, 64] and ν =.57. [53] It s reasonable to expect that χ ( t/t c ) α needed n Eq. (5) dverges at least as fast as ˆχ, and thus α ˆα. Hence we expect that the glass nstablty s dsplaced from the nonnteractng Anderson transton by an amount W W () W (J) δ t met t t met J /α. (56) W Comparng wth Eq. (5) and notng that α >, one sees that as J the glass nstablty lne approaches the nonnteractng Anderson transton from smaller values of (W/t)

12 than the delocalzaton nstablty, as ndcated n Fg.. These arguments confrm the exstence of an ntermedate phase between dsordered Ferm lqud and Anderson localzed glass, whch exhbts both metallc dffuson and glassy densty order. VII. COCLUSIO Based on the calculatons and arguments of ths paper and a prevous work [], we beleve that many-body cavty QED could evolve nto a new platform to explore the physcs of long-range quantum glasses. The long range of the photonmedated nteractons smplfes the theoretcal analyss for these systems; ths may allow for quanttatve comparson between experment and theory. Our work complements prevous proposals on glassy many-body systems n quantum optcs [69, 7] whch have focussed onto the creaton of onste dsorder or random short range exchanges by usng random optcal lattces and speces admxture. In Ref., we predcted a quantum spn glass transton of fxed, statonary atoms where the source of quantum fluctuatons was spontaneous tunnelng between sutable chosen nternal states of the atoms. In the present paper, we consdered tnerant fermons where the source of quantum fluctuatons s tunnelng between adjacent lattce stes. We found a metallc glass phase wth a gapless Ferm sea n the presence of random densty order and, for stronger nteractons, a localzed state wth strongly random charge dstrbutons and vanshng conductvty at T =. Varous aspects of the phase transtons nto and out of these glass phases are worth further studes: One s to quantfy further the mpact of the fractalty of wavefunctons close to the Anderson transton and compute the crtcal exponent α wth whch the glass nstablty lne approaches the non-nteractng Anderson transton pont. Expermentally, repeated ramps from the Ferm lqud phase nto the glass phase should produce metastable states wth a dfferent densty pattern at each run. The resultng varatons n absorpton mages could be partcularly strong, and therefore perhaps easer to detect, n the low densty regme where the glass and localzaton transton s of frst order. What happens when one consders models of type Eq. () wth bosonc atoms? umercal smulatons [4, 43] and replca calculatons [8] for smlar models agree on the exstence of a stable superglass phase wth superflud phase coherence n the presence of glassy random densty order. It s temptng to dentfy the superglass as the bosonc pendant to the metallc fermon glass. However, at fnte temperatures, the doman of the superglass (cf. Fg. of Ref. 8) s reduced because the man effect of thermal fluctuatons s to weaken the superflud phase coherence of the bosons. For fermons, on the other hand, the doman of the metallc glass ncreases at fnte T because the man effect of thermal fluctuatons s to enhance the nelastc scatterng rate and to weaken the localzng dsorder potental. We descrbed the dsordered atom-lght quantum phases usng an effectve equlbrum ground-state descrpton. In actual optcal cavtes, one deals wth pumped, steady-state phases certan features of whch may not be captured n an effectve equlbrum descrpton [, 66, 67]. We hope to address the non-equlbrum propertes of open, dsordered Dcke models n the near future n a forthcomng paper. We also want to analyze how the ntrgung propertes of classcal and quantum glasses such as agng (see Ref. 68 and references theren), or out of equlbrum dynamcs and avalanches [3] are modfed n the open, drven, steady-states n many-body cavty QED. Fnally, t wll be nterestng to nvestgate how the results of ths paper and Ref. scale wth the number of cavty modes and the choce of cavty mode profles. Acknowledgments We thank Alexe Andreanov, Vladmr Kravtsov, Govann Modugno, Francesco Pazza and Gacomo Roat for useful dscussons and Manuel Schmdt for brngng Ref. 48 to our attenton. Ths research was supported by the DFG under grant Str 76/-, by the SF under Grant DMR-386, and by a MURI grant from AFOSR. Appendx A: Alternatve dervaton of the glass nstablty equatons (38,43,4). Replca approach Here we formally derve the glass nstablty for arbtrary onste dsorder usng the replca approach. We start from model (7). Replcatng n tmes and takng the dsorder average and ntroducng Hubbard Stratonovch felds to decouple the quartc densty nteractons, one obtans the replcated partton functon Z n = D c a (τ)dca (τ) DQ ab (τ, τ ) exp[ S Q S c ] a,,τ a b,τ,τ (A) wth the acton S c = + n a= n a= S Q = 4 dτ c a (τ) ( τ µ) c a (τ) = dτ t c a (τ)ca j (τ) + ca j (τ)ca (τ), j n a,b= n a,b= dτ dτ W + J (τ τ )Q ab (τ, τ ) n a (τ)nb (τ ), dτ dτ J (τ τ )Q ab (τ, τ ). (A) In the thermodynamc lmt ( ), the nfnte range of the nteractons allows us to take the saddle pont wth respect

13 3 to Q ab, whch satsfy the saddle pont equatons Q ab (τ, τ ) = n a (τ)nb (τ ). (A3) (ote the slghtly dfferent defnton of Q as compared to Q ab defned n Eq. (3).) From here on we neglect the retardaton n the medated couplng J(τ τ ) and replace t smply wth J. As usual, the saddle pont values of Q ab are ndependent of τ, τ for a b, and depend only on τ τ for the replca dagonal Q aa. In the dsordered, replca symmetrc phase the term W W + J Q ab can be recognzed as the varance of the self-consstent effectve dsorder (5). ote that snce Q ab n, the effectve dsorder never really vanshes, unless and V j have specal correlatons. To detect a glass nstablty one has to solve frst the saddle pont equatons of the dsordered phase. Ths yelds a replca symmetrc soluton Q RS ab (τ, τ ) whch encodes the Edwards Anderson overlap q EA Q RS ab = and the average local susceptblty, Q RS aa (τ τ ) q EA = n, n (τ)n (τ ) c χ loc (τ τ ). (A4) (A5) The glass nstablty s found by wrtng Q = Q RS + δq and expandng the free energy n δq. A standard cumulant expanson yelds Z n [Q RS + δq] = exp[ (βf RS + δ(βf))], (A6) whch by vrtue of the extremalty of Q RS starts wth a quadratc term. The glass nstablty s sgnalled by the vanshng of the coeffcent of the term δq ab (the mass of the replcon mode), = (βj) J4 dτdτ n,a (τ)n j,a (τ ), j (βj) J, j ˆχ j. Hereby the correlator ˆχ j = dτ n,a (τ)n j,a (), c c (A7) (A8) has to be evaluated wth the replca symmetrc acton. Eq. (A7) s to be compared wth Eqs. (4,43).. Cavty approach: local selfconsstent acton In ths subsecton we derve the glass nstablty condtons from a selfconsstent local acton derved wthn a cavty approach. We start from the acton (), and splt t nto sngle partcle and nteracton part S = S + S V, S = S V = dτ c (τ) ( τ + ε µ) c (τ) = dτ t c (τ)c j (τ) + c j (τ)c (τ),, j=, j dτ dτ V j (τ τ ) (n (τ) n )(n j (τ ) n j ), (A9) where n (τ) c (τ)c (τ), and ε = ε V j (Ω=) n j. j (A) We now take advantage of the nfnte range nature of the nteractons, to transform the above problem exactly nto a selfconsstent sngle-ste problem wth retarded densty-densty nteractons. The extra contrbuton to the dsorder (A) can be treated as an addtonal Gaussan dsorder wth varance J n = J n + n c. Ths type of dsorder s essentally unavodable n the system. ote that f t does not vansh, ε, there are densty nhomogenetes already n the dsordered phase,.e., n n j for j. Unfortunately ths renders the exact evaluaton of the self-consstency problem very hard, and one has to resort to approxmatons n order to obtan quanttatve results. Upon average over V j a subsequent Hubbard Stratonovch transformaton and a saddle pont approxmaton, one fnds that the nteracton term can be resummed as S V = (n (τ) n )R(τ τ )(n j (τ ) n j ), (A) wth the kernel R(Ω) = J (Ω) n j (Ω)n j ( Ω) S +S J (Ω)χ V loc (Ω), j (A) where the average local susceptblty χ loc must be found selfconsstently. When computng the densty-densty correlator ˆχ j (Ω) n (Ω)n j ( Ω) S +S V, (A3) wth the self-consstent acton, one should be aware, however, that the above resummaton does not nclude terms whch contan a gven couplng V j only once wthn the expanson n nteractons. Those are ndeed rrelevant upon ste or dsorder averagng. However, they cannot be neglected when the glass susceptblty s computed, χ glass = χ j (Ω=). j (A4)

14 4 A glass transton manfests tself by a dvergence of ths susceptblty, whch n mean feld systems sgnals the emergence of ergodcty breakng n the form of many pure states, and replca symmetry breakng. In ths formula χ j s the full densty-densty correlator. To the relevant order n V j t can be obtaned from ˆχ j by the summaton of the geometrc seres, χ j = ˆχ j + ˆχ k V k ˆχ j (A5) k, Upon takng the square and summng over all pars of stes one fnds χ glass = (A6) J j ˆχ j (Ω=), whch dverges when the denomnator vanshes, reproducng Eq. (A7). In mean feld glasses, the contnuous breakng of replca symmetry ensures usually that the condton (A6) remans fulflled, even wthn the glass phase, where ˆχ j s to be nterpreted as the densty-densty correlator wthn one metastable state (and contrbutons from sngle couplngs dropped). Ths was explctly shown n the case of the nfnte range transverse feld Isng spn glass [5]. Ths phenomenon of mantaned margnal stablty s at the bass of the permanent gaplessness of the quantum glass phase. 3. Generalzed Mller-Huse type analyss Followng a reasonng smlar to the one by Mller and Huse [3] for the transverse feld spn glass, we formally compute the local densty-densty correlator ˆχ n a perturbaton seres n the nteracton acton S V. The perturbaton seres for the correlator ˆχ can be formally summed up as a geometrc seres of local nteractons R(τ τ ) lnkng proper polarzablty blobs Π j (τ τ ) that cannot be reduced nto two separate peces by cuttng a sngle R-lne, ˆχ j (Ω) =Π j (Ω) + Π k (Ω)R(Ω)Π kj (Ω) +... k =. (A7) Π (Ω) R(Ω) j The proper polarzablty has tself a power seres expanson n R. To lowest order one has smply Π j (Ω) = χ () j (Ω) + O(R), (A8) where χ () denotes the non-nteractng densty-densty correlator, n (Ω)n j ( Ω ) J= =: χ () j (Ω)δ Ω,Ω. (A9) As derved above n Eqs. (A7,A6), the glass transton occurs when = J() Tr Π (Ω=) R(Ω =). (A) In the approxmaton where we neglect the effectve dsorder, ε =, n the quantum dsordered phase, one has translatonal nvarance, whch allows one to work n Fourer space. The average local densty-densty correlator, D(Ω) = ˆχ (Ω) = Tr Π (Ω) R(Ω), must satsfy the selfconsstency condton R(Ω) = J (Ω)D(Ω). It must also obey the constrant (A) (A) dω π D(Ω) = (n (τ) n ) (A3) = n ( n ) = n( n) [n n ]. In the absence of onste dsorder, the last term n brackets vanshes. The constrant (A3) s fulflled automatcally by an exact soluton. However, f D s evaluated wthn an approxmate scheme, e.g. wth the help of Eqs. (A7,A8), one should mpose ths constrant to obtan a better approxmaton. In partcular, we can satsfy the short tme constrant (A3) by addng an adjustable short-tme component λ to the relaton J (Ω)(D(Ω) λ) = R(Ω), to correct for the errors at hgh frequences ntroduced by the approxmatons nvolved n computng D. The better the approxmaton, the smaller wll be the λ requred to enforce the short tme constrant. Ths recpe turns out to be essentally equvalent to the global constrant we ntroduced n Sec. V A. We thus have to solve smultaneously and dω π D(Ω) = n( n) [n n ], (A4) D(Ω) = Tr. (A5) Π (Ω) J (Ω)[D(Ω) λ] The glass transton arses when = J() Tr Π () J ()[D() λ] = [D()] Tr Π () J ()[D() λ]. (A6) The latter relaton leads to a sngular behavor of D(Ω) around Ω, as one may see by expandng Eq. (A5) around Ω=, very smlarly as n quantum spn glasses [3, 5]. Ths sngularty ensures the presence of spectral weght ImD(Ω ω + δ) at all fnte real frequences ω. The three last equatons are to be compared wth Eqs. (38-4) to whch they reduce n the translatonally nvarant case.

15 5 Q Π 4, Π 4, Π 4 Appendx B: Partcle-hole bubble as functon of external frequency and momenta,q,q In Fg. 6, we dsplay the partcle-hole bubble as a functon of external frequency for fxed momentum transfer. The lowfrequency part behaves as Ω as alluded to n Eq. (48). In Fg. 7, we plot an exemplary Ω = bubble as functon of momenta as occurrng n the saddle-pont equatons (38,39,43). As expected, the domnant contrbutons come from momenta n the vcnty of the nestng condton q Qnest = (π, π, π). Although logarthmcally dvergent at q = Qnest, the rght-hand-sde of the saddle-pont equatons remans regular as t nvolves an addtonal 3-dmensonal ntegraton over q. FIG. 6: (Color onlne) Ω behavor of the partcle-hole bubble, Eq. (36), as a functon of external frequency for fxed momentum transfer Q. FIG. 7: (Color onlne) Exemplary momentum behavor of the statc (Ω = ) partcle-hole bubble, Eq. (36), at half-fllng n 3 dmensons. [] J. Smon, W. S. Bakr, R. Ma, M. E. Ta, P. M. Press, and M. Grener, ature 47, 37 (); S. Sachdev, K. Sengupta, and S. M. Grvn, Phys. Rev. B 66, 758 (). [] R. Mottl, F. Brennecke, K. Baumann, R. Landg, T. Donner, and T. Esslnger, arv:3.3 (). [3] K. Baumann, R. Mottl, F. Brennecke, and T. Esslnger, Phys. Rev. Lett. 7, 44 (). [4] K. Baumann, C. Guerln, F. Brennecke, and T. Esslnger, ature 464, 3 (). [5] C. Maschler and H. Rtsch, Phys. Rev. Lett (5). [6] P. Domokos and H. Rtsch, Phys. Rev. Lett. 89, 533 (). [7] A. T. Black, H. W. Chan, and V. Vuletc, Phys. Rev. Lett. 9, 3 (3). [8] D. agy, G. Ko nya, G. Szrma, and P. Domokos, Phys. Rev. Lett. 4, 34 (). [9] Y. K. Wang and F. T. Hoe, Phys. Rev. A 7, 83 (973). [] C. Emary and T. Brandes, Phys. Rev. E 67, 663 (3). [] F. Dmer, B. Estenne, A. S. Parkns, and H. J. Carmchael, Phys. Rev. A 75, 384 (7). [] P. Strack and S. Sachdev, Phys. Rev. Lett. 7, 77 (). [3] J. Mller and D. A. Huse, Phys. Rev. Lett. 7, 347 (993). [4] J. Ye, S. Sachdev, and. Read, Phys. Rev. Lett. 7, 4 (993). [5] S. Gopalakrshnan, B. L. Lev, and P. M Goldbart, at. Phys. 5, 845 (9). [6] S. Gopalakrshnan, B. L. Lev, and P. M Goldbart, Phys. Rev. A 8, 436 (). [7] S. Gopalakrshnan, B. L. Lev, and P. M. Goldbart, Phys. Rev. Lett. 7, 77 (). [8]. Yu, and M. Mu ller, Phys. Rev. B 85, 45 (). [9] A. O. Slver, M. Hohenadler, M. J. Bhaseen, and B. D. Smons, Phys. Rev. A 8, 367 ().