Asymptotically Exact Scenario of Strong-Disorder Criticality in One-Dimensional Superfluids
AAsymptotically Exact Scenario of Strong-Disorder Criticality in One-DimensionalSuperfluids
Lode Pollet
Department of Physics, Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience,University of Munich, Theresienstrasse 37, 80333 Munich, Germany
Nikolay V. Prokof’ev and Boris V. Svistunov
Department of Physics, University of Massachusetts, Amherst, MA 01003, USA andRussian Research Center “Kurchatov Institute”, 123182 Moscow, Russia (Dated: August 13, 2018)We present a controlled rare-weak-link theory of the superfluid-to-Bose/Mott glass transition inone-dimensional disordered systems. The transition has Kosterlitz-Thouless critical properties butmay occur at an arbitrary large value of the Luttinger parameter K . The hydrodynamic descriptionis valid under the correlation radius and defines criticality via mutual renormalization of the strengthof microscopic weak links and superfluid stiffness. The link strength renormalizes along the linesof Kane and Fisher [Phys. Rev. Lett. , 1220 (1992)], while the renormalization of superfluidstiffness follows the lines of classical-field flow. The hallmark of the theory is the relation K ( c ) = 1 /ζ between the critical value of the Luttinger parameter at macroscopic scales and the microscopic(irrenormalizable) exponent ζ describing the scaling ∝ /N − ζ for the strength of the weakest linkamong the N (cid:29) L disorder realizations in a system of fixed mesoscopic size L . PACS numbers: 03.75.Hh, 67.85.-d, 64.70.Tg, 05.30.Jp
I. INTRODUCTION
Scalar bosons with local interactions in one dimensionare generically described by the paradigm of Luttingerliquids (LL), which amounts to quantized superfluid hy-drodynamics augmented with instantons (aka “backscat-tering events” in the fermionic language) responsible forquantum slippages of the superfluid phase. It is via theinstantons that superfluid hydrodynamics is coupled toeither a commensurate external potential, or disorder, orboth (see, e.g., Refs. 1 and 2). The LL picture is typicallypreserved under the correlation length of the superfluid-to-insulator quantum phase transition, providing a natu-ral framework for an asymptotically exact description ofcriticality.Arguably, the most intriguing superfluid-to-insulatorquantum phase transition is the one that is induced bydisorder and leads to the formation of the Bose glass(BG), a compressible insulator.
In their seminal paperon localization in one-dimensional (1D) superfluids, Gi-amarchi and Schulz found—by means of a perturbativerenormalization group (RG) analysis—that the transi-tion to the BG is of the Kosterlitz-Thouless (KT) typeand is characterized by the universal value K ( c ) = 3 / K . Recently, this finding wasshown to hold at the two-loop level, in line with the ear-lier proof that K ( c ) = 3 / et al. conjectured thatpower-law distributed weak links can lead to a non-universal value of K ( c ) (see also recent Ref. 7). To cor-roborate their idea, the authors attempted a scenario inwhich they abandoned the usual hydrodynamic descrip- tion in favor of the “Coulomb blockade” nomenclatureallowing them to apply a real-space RG treatment. How-ever, the approach is not asymptotically exact and, as weshow below, inconsistent with hydrodynamics in the su-perfluid state (SF). Recently, we argued that the onlypossible effect of strong disorder is a prolonged classicalflow based on the vanishing fugacity of weak links. Wealso proved a theorem of critical self-averaging implyingthat the LL picture should hold at criticality. Based onthe classical-flow picture and the above-mentioned theo-rem, we claimed no alternative to the Giamarchi-Schultzuniversality class.In the present work, we observe that our Ref. 8 containsan arbitrary statement saying that applicability of hydro-dynamics necessarily implies Giamarchi-Schultz critical-ity, along with a major flaw: the quantum hydrodynamicrenormalization of weak links was overlooked. However,if the flaw is corrected, an asymptotically exact theory ofa new universality class of superfluid to Bose/Mott glasstransition in one dimension emerges, which we derive be-low.The paper is organized as follows. In Sec. II, we in-troduce the basic notation for the hydrodynamical de-scription and review the RG flow of a single weak linkin a homogeneous superfluid. In Sec. III, we renderthe RG description of the classical-field flow in the pres-ence of strong disorder, following Ref. 8. We then arguethat these two flows must be combined, which intuitivelyyields the central result of the paper, Eq. (9). Conse-quently, we rigorously prove Eq. (9) in Sec. IV, whereasymptotically exact semi-RG flow equations are derived.A technically involved—but quite important for justify-ing the theory—aspect, that the relevant weak links aremicroscopic and isolated from each other, is referred to a r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Sec. V. In the Conclusion (Sec. VI), we summarize themain results and argue that, despite the existence of twouniversality classes for the transition from a superfluidto Bose (Mott) glass, only one Bose (Mott) glass phaseexists.
II. A SINGLE WEAK LINK IN A LUTTINGERLIQUID
The starting point for the theoretical analysis of liquidsin (1+1)d is Popov’s hydrodynamic action over the phasefield Φ( x, τ ), S [Φ] = (cid:90) dxdτ (cid:20) in ( x )Φ (cid:48) τ + Λ s (cid:48) x ) + κ (cid:48) τ ) (cid:21) . (1)Here, x stands for the spatial coordinate and τ for theimaginary time. The fields Φ (cid:48) τ and Φ (cid:48) x are the derivativesof the field Φ with respect to τ and x , respectively. Thequantity n ( x ) denotes the local number density wherethe spatial dependence originates from an external po-tential. The shape of the first term shows that it is oftopological nature and is able to destroy superfluidity.The Luttinger parameter K = π √ Λ s κ is directly relatedto the compressibility κ and the superfluid stiffness Λ s . FIG. 1. Renormalization of the effective link strength for K ≈ . /L to form a weak link. The plot is for2 J ( L ) /J (0) = [Λ − s ( L ) − Λ − s (0)] − . While our theory fordisordered systems can perfectly describe this Kane-Fisherrenormalization, the real-space RG approach can not. In their seminal work, Kane and Fisher addressed thequestion of renormalization of the strength of a weaklink in an otherwise homogeneous Luttinger liquid, find-ing, in particular, that, irrespectively of the microscopicstrength of the link, the superfluid transport gets com-pletely blocked at
K <
1. The hopping across a link ofstrength J is described by the term J (cid:90) dτ cos [Φ + ( τ ) − Φ − ( τ )] (2) added to the hydrodynamic action. Here, Φ ± are thevalues of the (1+1)-dimensional phase field Φ( x, τ ) rightbefore and after the weak link. The link can be consid-ered weak if its strength J satisfies the condition J (cid:96) (cid:28) Λ (0) s , (3)where (cid:96) is the length scale of the ultraviolet cutoff for thehydrodynamic description and Λ (0) s the superfluid stiff-ness at the scale (cid:96) . In what follows, we measure alllengths in units of (cid:96) and the stiffness in units of Λ (0) s .We note in passing that a large value of K = K ( (cid:96) ) (cid:29) κ .The full problem posed by Eq. (3) is non-trivial (butsolved long ago and well understood). Progress can bemade by using the renormalization group and noting thatsuch a weak link is perturbative with respect to the short-wave harmonics of Φ. Therefore, those harmonics canbe eliminated from the theory by averaging (cid:104) . . . (cid:105) λ overharmonics with wavelength shorter than λ and resultingin the corresponding renormalization of the link strength, J → J ( λ ), where λ is the new ultraviolet cutoff (in unitsof l , in accordance with the above-mentioned units).From the scaling dimension of the operators it followsthat this flow can be written as dJ ( λ ) d ln λ = − K J ( λ ) . (4)In our subsequent analysis for disordered systems, it willbe more economical to use an integral formulation of flowequations. Let us therefore re-express the Kane-Fisherrenormalization of weak links in this language. Averagingover the harmonics of Φ with wavelength shorter than λ leads to (cid:104) cos(Φ + − Φ − ) (cid:105) λ = λ − /K cos (cid:16) Φ ( λ )+ − Φ ( λ ) − (cid:17) . (5)Here, Φ ( λ ) is the phase field with the harmonics withwavelength shorter than Λ removed. In this standardKane-Fisher renormalization procedure, the Luttingerparameter is independent of λ because a single weak linkhas no effect on the superfluid far away from its location(i.e., dK/d ln λ = 0, or, in integral form, K ( λ ) = const).Equation (5) means that the effective strength of theweak link flows with λ as J ( λ ) = J λ − /K [ J ( λ ) λ (cid:28) , (6)which is Eq. (4) recast in integral form.We thus recover the well-known result forfermions/hard-bosons in 1D: In the thermodynamiclimit, a weak link becomes fully transparent for attrac-tive interactions ( K >
1) and reflects any (even dc)current for repulsive interactions (
K < mesoscopic scales. Specifically, a decisive role in ouranalysis will be played by a simple fact that, at
K > λ ∗ such that J ( λ ∗ ) λ ∗ ∼
1. We refer to this wavelength as theclutch scale—to underline that the phase field becomescontinuous across the link. At length scales λ (cid:29) λ ∗ , thequantum phase slippages are suppressed, and the linkbehaves as a classical-field link of strength J ∗ ≡ J ( λ ∗ ) ∼ J KK − . (7)The Kane-Fisher renormalization of the weak link andits progressively stronger suppression of the superfluidstiffness is demonstrated numerically in Fig. 1. It was ob-tained by a Monte Carlo simulation of the Bose-Hubbardmodel with parameters corresponding to the superfluidstate with K ≈ . /L to form a weaklink. It illustrates the most important new ingredient inour theory of the strong-disorder critical point that iso-lated links are renormalized by hydrodynamic phonons.This undeniable physics of leak links is missed in the real-space RG treatment, which relies on Coulomb blockadephenomena even in the SF phase. III. PROLONGED CLASSICAL FLOW
Before entering the strong-disorder critical regime, su-perfluid systems with K (0) (cid:29) . In Ref. it was argued that the strength of theweakest link in a system of size L scales as a certain powerof L , conveniently parameterized as J ∼ /L − ζ , where ζ is determined by the microscopic parameters. However,according to Eq. (7), the quantum-renormalized classicaltheory corresponds to J ∗ ∼ /L − ˜ ζ , with˜ ζ = ζK − K − . (8)Thus, the effective “classical-field” exponent ˜ ζ turns outto be a function of K . This fact—central for the scenariorevealed below—was overlooked in our Ref. 8.The microscopic exponent ζ itself can be measuredexperimentally/numerically directly by examining thesuperfluid response of an appropriately large number N of mesoscopic systems of a fixed size L . It is ex-pected that the weakest link that can be found un-der these circumstances has J ∝ /N − ζ because for λ ∗ > L the quantum renormalization amounts to a con-stant N -independent L − /K factor, see Eq. (6). Forexponentially-rare-exponentially-weak distributions theweakest links are composed of stronger links placed rightnext to each other implying that the length of the link ∝ ln J . This leads to the following requirement on themeasurements of ζ : ln N (cid:28) L (cid:28) N .On the basis of Eq. (8) we see that the classical-fieldapproach of Ref. 8 only applies in the following two limits:(i) at ζK, K (cid:29)
1, when the quantum renormalization of˜ ζ is negligible, and (ii) in the superfluid phase beyond thecorrelation radius corresponding to the saturation of the superfluid stiffness (and thus K ) to its infinite-size value.Most importantly, at the special point ζK = 1 the quan-tum renormalized ˜ ζ changes sign . This means that thesystem can only remain superfluid at ζK ≥
1. Moreover,as we show below, the equality indeed corresponds to thecritical point, K ( c ) = 1 /ζ . (9)As long as ζ < /
3, the transition to the Bose glassfollows a novel strong-disorder scenario with K ( c ) > / inevitably happens if K (cid:29)
1, because the initial(classical) flow requires ζ (cid:28) and the Kane-Fisher renormalization of a single weak link. The keyquantity in real-space RG is the exponent α governingthe distribution of renormalized weak links, vanishing atcriticality and taking a small positive value in the SFphase. Since real-space RG does not account for renor-malization of isolated links due to long-range zero-pointhydrodynamic fluctuations (phonons), we can relate α to ζ as ζ = α/ (1 + α ). Equation (9) implies then that astate with small enough but finite α inevitably becomesincompatible with superfluidity, while the real-space RGputs the critical point at α = 0.We can go even a step further. Consider a hypothet-ical theory, possibly in combination with numerics, ca-pable of producing values α ( L ) and K ( L ) up to somefinite system size L for the distribution of weak linksand the Luttinger parameter, respectively. Renormal-ization effects by phonons with wavelength larger than L have not been included yet. Consequently, if the criterion α ( L ) / [1 + α ( L )] < /K ( L ) is satisfied, the Kane-Fisherrenormalization of weak links by long-wave phonons willinevitably result in the insulating state. (Note that α and K can only decrease with the system size.) IV. SEMI-RG FLOW
A controlled description of the weak-link quantum crit-icality is achieved by combining the RG treatment ofRef. 8 with the Kane-Fisher renormalization of the linkstrength at a given length scale. The applicability of hy-drodynamics below the correlation radius is guaranteedby the theorem of critical self-averaging proven in Ref. 8and stating that the superfluid stiffness is well definedfor the critical flow as long as it does not vanish in thethermodynamic limit. If governed by single weak links,the flow of superfluid stiffness Λ s has to obey the equa-tion d Λ − s /d ln λ ∝ [ J ∗ ( λ ) λ ] − (see Ref. 8). We cast thisequation in the form [below z = ln( λ/(cid:96) )] d Λ − s dz ∝ r ( z ) , r ( z ) ≡ λ ∗ λ , (10)by recalling that J ∗ ( λ ) is the strength of the typical weak-est link in a system of size λ , and λ ∗ = 1 /J ∗ ( λ ) ≡ λ ∗ ( λ ).It is instructive to observe that for Λ s to stay finite inthe z → ∞ limit, the ratio r ( z ) has to obey the limitingrelation lim z →∞ z r ( z ) = 0 . (11)This relation implies lim λ →∞ λ ∗ /λ →
0, which is a nec-essary condition (but potentially not a sufficient one be-cause of the occurrence of composite weak links, see be-low) for single weak links with the same λ ∗ to be treatedas independent. Below we will see that our semi-RG flowsatisfies the condition (11).We now proceed with constructing a self-consistentquantitative description in which weak links result in aslow flow of K ( λ ) while the flow of K ( λ ) enhances therenormalization of microscopic weak links. To this end,we recall that the factor λ − /K in (6) is, in fact, a prod-uct of factors ( λ i /λ i +1 ) /K associated with renormal-ization coming from the wavelength intervals [ λ i , λ i +1 ].For a slowly flowing K ( λ ), each term in the productcan be written as, ( λ i /λ i +1 ) /K ( λ i ) , provided the inter-vals [ λ i , λ i +1 ] are small enough to guarantee K ( λ i ) ≈ K ( λ i +1 ). This leads to the integral analog of (6) J ( λ ) = J exp (cid:34) − (cid:90) ln λ d ln λ (cid:48) K ( λ (cid:48) ) (cid:35) [ J ( λ ) λ (cid:28) . (12)Next, we have to generalize Eqs. (7) and (8) for thetypical weakest link at the length scale λ , or J ( λ ) ∝ /λ − ζ . The clutch condition now reads λ ∗ λ − ζ exp (cid:34) − (cid:90) ln λ ∗ d ln λ (cid:48) K ( λ (cid:48) ) (cid:35) = const , (13)which can conveniently be written as z ∗ + ( ζ − z − (cid:90) z ∗ x ( z (cid:48) ) dz (cid:48) = const , (14)with z ∗ = ln λ ∗ and x ( z ) ≡ /K ( z ). Differentiating withrespect to z , we find dz ∗ dz = 1 − ζ − x ( z ∗ ) . (15)Given that K ( λ ) = π (cid:112) Λ s ( λ ) κ with the λ -independentcompressibility κ we see that Eqs. (10) and (15) com-pletely define the semi-RG flow of K ( λ ). In terms of x and y = − ln r ≡ z − z ∗ , we get [below x ≡ x ( z = 0)]1 x dx dz = e − y , (16) dydz = ζ − x − x . (17)By Eq. (15), x in the r.h.s. of (17) should be understoodas x ≡ x ( z − y ). However, we are allowed to substitute x ( z − y ) → x ( z ) because by Eq. (16) the correction issupposed to be small, ( ydx/dz ) /x (cid:28)
1. [And on thecritical line, y/z → z → ∞ , as we will see soon.]As can readily be verified, the classical flow equationsfrom Ref. 8 are recovered by ignoring the x dependencein Eq. (17). The Kane-Fisher equation is recovered byignoring the ( ζ − x ) dependence in Eq. (17), which iseasiest seen by ignoring the second term in Eq. (14).Note that ζ is a bare parameter not subject to theRG flow (hence the name semi-RG flow) because we aredealing with single links and ignore pairs of links (seebelow). Moreover, the Luttinger parameter is the keyquantity governing the flow toward an insulating state.Both facts are in sharp contrast with the real-space RGtreatment, where Coulomb blockade physics plays thekey role instead of hydrodynamics and where single weaklinks cannot be treated along the lines of Kane-Fisherbecause the real-space RG phenomenology is such thatsingle weak links surrounded by a Luttinger cannot occurin a context of a strong randomness.The first integral of equations (16)–(17) can easily befound in a closed form,(1 − ζ )[ x + ln(1 − x )] + x / − ( x / e − y + C . (18)Since y ( λ → ∞ ) → ∞ , the value of C can be expressedas C = (1 − ζ )[ x ∞ + ln(1 − x ∞ )] + x ∞ / . (19)The critical point for the SF-BG transition is locatedat x ∞ = 1 /K ( ∞ ) = ζ and corresponds to the strong-disorder scenario if x ∞ < /
3. At criticality, C = ζ − ζ / − ζ ) ln(1 − ζ ). One can further see that x ( z ) = ζ − − ζ ) /z (at criticality) , (20)implying, in particular, the critical behavior r ( z ) ∝ /z ,consistent with Eq. (11).Finally, the dependence of x ∞ on the external param-eters is of the KT-type, x ∞ ( g ) = ζ − D √ g − g c , imply-ing the standard KT-type exponential divergence of thecorrelation length on approach to the critical point. Theeasiest way to derive these results is to approach the crit-ical point along the C = const trajectory and consider ζ − ζ C and ζ − x as small parameters to simplify theequations.In the limit K ( (cid:96) ) (cid:29) x (cid:28)
1) and ζ (cid:28)
1, but ζ > x , the solution is particularly simple: ζ x − x − x e − y + C , C = ζ x ∞ − x ∞ . (21)At the SF-BG criticality, when x ∞ = ζ , we have C = ζ /
6. We further notice that the case ( x , x ) (cid:28) ζ (cid:28) x can be ne-glected in the r.h.s. of dy/dz . Then x ( L ) = x (cid:113) e − y (1 − e − ζz ) /ζ , (22) FIG. 2. Flow of the Luttinger parameter with system sizein the vicinity of the critical point when we have K (cid:29) (cid:96) . The initial behavior of the flowpertains up to the length scale 1 /ζ [see Eq. (22)], when K will level off to a constant in the superfluid phase (SF). If theflow reaches the condition K ( L ) = 1 /ζ ∼ K / , then K willfurther decrease to 0 and the thermodynamic phase is a Boseglass (BG). with x ∞ = x (cid:112) e − y /ζ ∝ x √ ζ . (23)From this estimate we see that in the classical field limitthe SF-BG transition corresponding to x ∞ = ζ occurswhen x ∞ ∝ x / . The overall picture is illustrated inFig. 2. V. IRRELEVANCE OF COULOMB BLOCKADEPHYSICS
Consider pairs of weak links separated by a large dis-tance d —called d -pairs for brevity, with the numberof particles on the d -interval being well defined (theCoulomb blockade regime). Let us show that renormal-ization of Λ s by such complexes can be neglected. To thisend, we first establish the functional form of the Kane-Fisher factor [see Eq. (12)] in the asymptotic λ → ∞ limit by using Eq. (20) f ( λ ) = exp (cid:34) − (cid:90) ln λ x ( z ) dz (cid:35) ∝ λ − ζ ln − ζ ) ( λ ) . (24)Consider now some length scale λ and account for the d -pairs which have a probability of the order of unity tooccur at this scale (pairs with higher density are absorbedinto the renormalized value of Λ s ( λ ); unlikely events willbe accounted for at larger scales). For clarity, we startwith the case of two weak links having similar values of J separated by a distance scale d . The requirement forsuch pairs to occur with a probability of order unity, (cid:104) J / (1 − ζ )0 (cid:105) d = 1 /λ (for d -pairs) , (25) translates into the J = (1 /dλ ) − ζ relation for thestrength of weak links in the pair. As long as J ( d ) = J f ( d ) remains smaller than the “charging” energy of thesystem’s interval between the links, κ/d , one can use theresult of second-order perturbation theory to estimatethe strength of the composite link as J ( d ) / ( κ/d ). Thecontribution of the pair to the renormalization of Λ − s isgiven, as before, by the ratio λ ∗ /λ where λ ∗ is defined by (cid:2) J f ( d ) d (cid:3) f ( λ ∗ ) f ( d ) λ ∗ ∼ d -pairs) . (26)The second factor accounts for the composite link renor-malization between the d − and λ ∗ -scales. By substitut-ing here Eqs. (25) and (24) we readily find (cid:20) ln ( d ) ln ( λ ∗ ) λ ∗ λ (cid:21) − ζ ∼ d -pairs) , (27)or, by replacing λ ∗ with λ up to logarithmic precision, λ ∗ λ ∼ ( d ) ln ( λ ) (for d -pairs) . (28)We immediately see that the contribution of large d -pairs is suppressed by a factor ln − ( d ). Most im-portantly, the integral over the pair scales (cid:82) d ln( d ) isconverging at the lower limit where microscopic pairs(and other multi-link complexes) are part of the origi-nal exponentially-rare-exponentially-weak distribution ofsingle links. The same final conclusion (28) is reachedfor a pair of links with different strength, J = J δ , J = J /δ . Since J J = J , all equations in the analysispresented above remain identically the same. VI. CONCLUSION AND OUTLOOK
We presented an asymptotically exact theory for theSF-BG transition in the presence of appropriately strongdisorder by combining the classical field flow of Ref. 8with the Kane-Fisher renormalization, originally derivedfor single weak links. This constitutes the crucial dif-ference with the real-space RG treatment introduced byAltman and coworkers, as the latter is unable to addressthe Kane-Fisher renormalization—since it does not treatproperly the phonon degrees of freedom in the system.The hallmark of our theory is the relation K ( c ) = 1 /ζ stating that there are no superfluids with a Luttingerparameter smaller than 1 /ζ ; all future work has to dealwith this microscopic quantity.We have checked that the available data from Refs. 7and 8 are compatible with the present scenario (and, inparticular, with a Kosterlitz-Thouless-type transition ata non-universal value of K c ), but the data are insufficientfor studying the transition accurately and/or extracting ζ . Our treatment applies to both Bose and Mott glasses.In the latter case, the system remains compressible atcriticality even though it is incompressible on the in-sulating side (the renormalization of κ starts when K drops to values close to 2). The semi-RG equations thatwe derived here can straightforwardly be upgraded toa system of three equations describing both the strong-disorder and Giamarchi-Schultz (or Mott-glass) critical-ities, as well as the competition between the two. Thisis achieved by accounting for the standard (for KT the-ory) instanton–anti-instanton renormalization terms inthe flows of Λ s (and κ , if necessary), and introduc-ing the RG equation for the flow of concentration ofthe instanton–anti-instanton pairs (for the details, seeRef. 1).Furthermore, in the Mott glass case (where instantonshave no phase factors), the ground-state 1D quantumsystem is directly mapped onto a 2D finite-temperature“scratched-disordered” superfluid film. The film is sup-posed to have a peculiar disorder in the form of straightparallel scratches cutting through the film. If the dis-order is strong enough to guarantee ζ < /
2, the filmexperiences the superfluid-to-normal phase transition ofthe above-discussed strong-disorder universality class,happening at the critical temperature T c = πζ ¯ h n s /m (with n s the superfluid density and m the mass of the atoms), thus preempting the usual Berezinskii-Kosterlitz-Thouless transition. Despite a different tran-sition scenario, the state at T > T c is just a normalfilm. This is seen from the fact that the vortex pairsare dangerously irrelevant with respect to the scratches:When n s is suppressed to zero at T > T c , the vortex-antivortex pairs inevitably proliferate, rendering the fi-nal phase indistinguishable from the standard normalstate. Likewise, there are no two different Bose glassphases because the rare weak links and the instanton–anti-instanton pairs are dangerously irrelevant with re-spect to each other. Hence, while only one of the twois responsible for criticality, the other one also becomesimportant on the insulating side (at distances at whichthe value of K becomes appropriately small), removingthe potential qualitative difference between the glasses.We are grateful to Ehud Altman, Thierry Giamarchi,Susanne Pielawa, and Anatoli Polkovnikov for valuablediscussions and S. Pielawa for sharing her numericaldata for the Luttinger parameter with us. This workwas supported by the National Science Foundation un-der the grant PHY-1314735, FP7/Marie-Curie Grant No.321918 (“FDIAGMC”), FP7/ERC Starting Grant No.306897 (“QUSIMGAS”) and by a grant from the ArmyResearch Office with funding from DARPA. V. A. Kashurnikov, A. I. Podlivaev, N. V. Prokof’ev, andB. V. Svistunov, Phys. Rev. B , 13091 (1996). B. V. Svistunov, Phys. Rev. B, T. Giamarchi and H.J. Schulz, Europhys. Lett. , 1287(1987); Phys. Rev. B , 325 (1988). M.P.A. Fisher, P.B. Weichman, G. Grinstein, and D. S.Fisher, Phys. Rev. B , 546 (1989). Z. Ristivojevic, A. Petkovi´c, P. Le Doussal, and T. Gia-marchi, Phys. Rev. Lett. , 026402 (2012). E. Altman, Y. Kafri, A. Polkovnikov, and G. Refael, Phys. Rev. Lett. , 150402 (2004); ibid Phys. Rev. B , 174528(2010). S. Pielawa and E. Altman, Phys. Rev. B , 224201 (2013). L. Pollet, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev.B , 144203 (2013). C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. , 1220(1992). F. Hrahsheh and T. Vojta, Phys. Rev. Lett.109