Superfluid-Insulator transition of quantum Hall domain walls in bilayer graphene
Victoria Mazo, Chia-Wei Huang, Efrat Shimshoni, Sam T. Carr, H. A. Fertig
SSuperfluid-Insulator transition of quantum Hall domain walls in bilayer graphene
Victoria Mazo, Chia-Wei Huang, Efrat Shimshoni, Sam T. Carr, and H. A. Fertig Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel School of Physical Sciences, University of Kent, Canterbury CT2 7NH, UK Department of Physics, Indiana University, Bloomington, IN 47405, USA (Dated: October 31, 2018)We consider the zero-filled quantum-Hall ferromagnetic state of bilayer graphene subject to akink-like perpendicular electric field, which generates domain walls in the electronic state and low-energy collective modes confined to move along them. In particular, it is shown that two pairs ofcollective helical modes are formed at opposite sides of the kink, each pair consisting of modes withidentical helicities. We derive an effective field-theoretical model of these modes in terms of twoweakly coupled anisotropic quantum spin-ladders, with parameters tunable through control of theelectric and magnetic fields. This yields a rich phase diagram, where due to the helical nature ofthe modes, distinct phases possess very different charge conduction properties. Most notably, thissystem can potentially exhibit a transition from a superfluid to an insulating phase.
PACS numbers: 73.21.-b, 73.22.Gk, 73.43.Lp, 72.80.Vp, 75.10.Pq, 75.10.Jm
Among the most intriguing electronic properties ofgraphene is the emergence of novel collective states in thequantum Hall (QH) regime [1]. In particular, the peculiar ν = 0 QH states in both monolayer graphene (MLG) [2–4] and bilayer graphene (BLG) [5] suggest that Coulombinteractions lift the degeneracies of the half-filled zero en-ergy Landau level. The multitude of discrete degrees offreedom (two valleys ( K , K (cid:48) ) and two spin states in MLG,and an additional layer index in BLG) dictates a rich vari-ety of possible exchange-induced broken symmetry states[6–8], as generalizations of the spontaneously polarizedferromagnetic state of an ordinary two-dimensional (2D)electron gas [9]. These can be controlled by externalfields: in MLG, primarily by tuning the Zeeman energyvia a strong parallel magnetic field [10]; in BLG, the or-bital isospin degeneracies can be lifted by applying a per-pendicular electric field [11].The unique features of the broken symmetry ν = 0QH states are most prominently manifested by the na-ture of their collective excitations. While in the standardQH ferromagnet the elementary charge excitations areSkyrmions [9], more complex forms of spin-textures havebeen predicted in graphene, e.g. charge-2 e Skyrmionsin BLG [12]. Yet more remarkably, the particle-holesymmetry of the bulk spectrum allows the formation ofcharge conducting edge modes associated with kinks inthe effective Zeeman field, where it changes sign acrossa line. These can be realized near physical edges of thegraphene ribbon [13–15], or in the interior of a BLG sheetsubject to non-uniform gating [16–18].The coherent domain wall (DW) forming in thespin/isospin configuration near such a kink supportsa gapless collective mode, which possesses a one-dimensional (1D) dynamics along the kink of a helical character. The latter arises from the constraint relatinga spin/isospin texture to the charge degree of freedom [9].This yields a mapping to a helical Luttinger liquid (HLL), with a single flavor encoding spin and charge related byduality, in analogy with the edge states of 2D topologi-cal insulators (TI) [19, 20]. However in distinction fromthe latter, DW modes are not topologically protected bytime-reversal symmetry and are therefore not immune tobackscattering due to perturbations which violate spinor isospin conservation. Their conduction properties de-pend crucially on the Luttinger parameter, which is sen-sitive to the ratio between the effective Zeeman energyand exchange interaction, and may be tuned by externalfields [13, 21]. Interestingly, this may trigger a transitionfrom an insulator to a conducting phase manifesting thequantum spin Hall effect (QSHE) [10, 22].
K 'K d d - d E z X - + + -V +V x=0 x z B w (a) (b) FIG. 1: (color online) (a) BLG in a split-double-gated setup;the y -axis is perpendicular to the page. (b) The non-interacting energy levels crossing at (cid:15) = 0 vs. guiding center X . Arrows denote spin ( S z along B ), full red lines correspondto valley K and dashed blue lines to valley K (cid:48) . The blow-uppresents a typical S z -configuration of a coupled DW-pair. a r X i v : . [ c ond - m a t . s t r- e l ] S e p In this paper, we propose a realization of QH DWstates in BLG subject to a kink in the perpendicular elec-tric field (see Fig. 1), where pairs of HLL with identicalhelicities couple to form strongly correlated 1D collec-tive modes. Depending on the external fields manipulat-ing their Luttinger parameter K and coupling strengths,these correlated modes exhibit a charge-density wave(CDW) or superfluid (SF) character, marked by distincttransport properties. We suggest a measurement of the“antisymmetric conductance” as a particularly revealingprobe of transitions between the two phases.We consider a split-double-gated geometry as discussedin [16–18], where the inter-layer bias V ( x ) imposed on aBLG changes from + V to − V over a distance w alongthe x direction [Fig. 1(a)]. In addition, a strong tiltedmagnetic field B enforces Landau quantization, as wellas a Zeeman energy E z ∼ | B | . The single-particle ze-roth Landau levels separate into eight different levelswhich cross at four distinct guiding center coordinates X n ± ( n = 1 , V ( x ) = 0[see Fig. 1(b)]. This results in two pairs of parallel 1Dchannels propagating along the y -axis, marked by dis-tinct helicities: h = sgn( X n h ).Exchange interactions modify this picture, forming aspin-valley DW structure as depicted in Fig. 1(b) andreplacing the single-particle helical states by mutuallycoupled collective modes. As explained in earlier work[13, 21, 23], the quantum dynamics of each DW-mode(along the y -direction) can be described in terms of aneffectively 1D spin-1 / S n h ,y ), encoding both spinand charge degree of freedom: S x and S y are associatedwith electric charge (a property which descends from thespin-charge coupling inherent in quantum Hall ferromag-nets [9]), and the S z operator coincides with the electric current . Note that the latter correspondence depends onhelicity, S zn h ∼ hj e . The coupling between adjacent DWsdepends on their spacing [ d, d in Fig. 1(b)]: d ≈ w E z eV , d ≈ d ω c γ − ω c , where γ is the interlayer hopping and ω c = √ (cid:126) v F /(cid:96) (with (cid:96) = (cid:112) c (cid:126) /eB ⊥ the magnetic length)[18], so that typically d (cid:28) d . Hence modes with thesame helicity are more strongly coupled. The system isthen modeled by the effective Hamiltonian H = (cid:88) h = ± H h + H + − , (1)where the weak coupling H + − will be specified later on,and H h describe anisotropic spin-1 / H h = (cid:88) n =1 , H n h + H ( h ) ⊥ , (2) H n h = (cid:88) y (cid:20) J xyn (cid:0) S + n h ,y S − n h ,y +1 + h.c. (cid:1) + J zn S zn h ,y S zn h ,y +1 (cid:21) H ( h ) ⊥ = (cid:88) y (cid:20) J xy ⊥ (cid:0) S +1 h ,y S − h ,y + h.c. (cid:1) + J z ⊥ S z h ,y S z h ,y (cid:21) . The dependence of J αn , J α ⊥ on the original system pa-rameters and external fields is complicated; however, theanisotropy factors ∆ n ( ⊥ ) ≡ J zn ( ⊥ ) J xyn ( ⊥ ) qualitatively reflect theratio of kinetic energy ( ∝ V ) to exchange interaction( ∼ e /(cid:96) ). Note that the signs of J α ⊥ depend on the spatialoverlap of DWs, and are hereon assumed arbitrary.We next employ standard Bosonization to express thespin operators in terms of Bosonic fields φ n h ( y ) and theirdual θ n h ( y ) [24]: S + ∼ e − iθ [( − ) y + cos(2 φ )], S z ∼ [ − ∂ y φ + ( − ) y Λ cos(2 φ )] with Λ ∼ /(cid:96) a short distancecutoff. Defining symmetric and antisymmetric modes ineach ladder [ φ s h = ( φ h + φ h ) / θ s h = ( θ h + θ h ) and φ a h = ( φ h − φ h ), θ a h = ( θ h − θ h ) /
2, respectively], theleading continuum limit of H h becomes ( (cid:126) = 1) H h = (cid:88) ν = s,a [ H ( ν h )0 + H ( ν h ) int ] + H ( h ) as (3)where under the assumption J α ⊥ , | J α − J α | (cid:28) J αn , H ( ν h )0 = v ν π (cid:90) dy (cid:104) K ν ( ∂ y θ ν h ) + 1 K ν ( ∂ y φ ν h ) (cid:105) (4) H ( s h ) int = ˜ g z Λ (cid:90) dy cos(4 φ s h ) H ( a h ) int = g xy Λ (cid:90) dy cos(2 θ a h ) + g z Λ (cid:90) dy cos(2 φ a h ) H ( h ) as = (cid:90) dy π (cid:2) g xyas ( ∂ y θ a h )( ∂ y θ s h ) + g zas ( ∂ y φ a h )( ∂ y φ s h ) (cid:3) ;here v ν ∼ J xy / Λ with J α ≡ J α + J α , g α , ˜ g α ∼ J α ⊥ / Λ, K s ≈ K (cid:18) − KJ z ⊥ πv Λ (cid:19) , K a ≈ K (cid:18) KJ z ⊥ πv Λ (cid:19) (5)where K = π , ∆ ≡ − J z J xy . Since g αas ∼ ( J α − J α ) / Λ (cid:28) v , the marginal last term in Eqs. (3),(4) can betreated perturbatively [25] and is henceforth neglected.Under the above approximation, the s and a modesare decoupled: H h = H ( h ) s + H ( h ) a , with H ( h ) ν = H ( ν h )0 + H ( ν h ) int , as in a standard spin-1 / H ( h ) s , H ( h ) a de-scribe the conduction properties of the charge degree offreedom as well: ∂ y φ ν h denote spin-density fluctuations,and also encode the total and relative electric currentoperators through channels 1 h , h : J h + J h = − evhπK ∂ y φ s h , J h − J h = − evhπK ∂ y φ a h . (6)The dual fields ∂ y θ ν h encode the corresponding chargedensity operators.The behavior of the symmetric mode is controlled by asine-Gordon model H ( h ) s , where the cosine term is irrele-vant for K s > / K = 1 and arbitrary J z ⊥ < H ( h ) a contains two competing cosineterms; for arbitrary K a , at least one of them is relevant.In particular, both terms are relevant for 1 / < K a < K a = 1, wherethe model was shown to exhibit an Ising ( Z ) quantumphase transition [26–28] from a phase with ordered φ a h to an ordered θ a h . Below we argue that this behaviorpersists throughout the entire range 1 / < K a <
2, anddiscuss the interpretation of the two phases as SF andCDW, respectively [29].For K a = 1, the model H ( h ) a can be exactly mapped tomassive free Fermions [26, 27]. In terms of two species ofright and left moving Majorana fields ξ ± R , ξ ± L , one obtains H ( h ) a = (cid:88) τ = ± (cid:90) dy (cid:26) iv a ξ τL ∂ y ξ τL − ξ τR ∂ y ξ τR ) − im τ ξ τR ξ τL (cid:27) , (7) m ± = Λ( g xy ± g z ). This represents two independent Isingchains ( τ = ± ) in a transverse field, which possess quan-tum critical points at g xy = ± g z (depending on the rela-tive sign of g xy , g z ) where one of the masses m τ vanishes.The phases separated by this critical point are related byduality: for | g xy | > | g z | , the original field θ a h acquires afixed value ( π or 0 for g xy > g xy < φ a h is disordered; for | g xy | < | g z | , the roles of θ a h , φ a h and g xy , g z are interchanged.We next consider a deviation from the self-dual point, K a = 1 + g ( − < g < τ = ± : H g = − g Λ (cid:90) dyξ + R ξ + L ξ − R ξ − L . (8)However, one of the sectors is always more massive, anddoes not undergo a transition. This justifies a mean-fieldapproximation for the other sector (denoted τ c ), wherethe operator ξ − τ c R ξ − τ c L is replaced by its expectation value[26]. The resulting approximation for H g merely yieldsa shift of m τ c by δm = ig Λ (cid:104) ξ − τ c R ξ − τ c L (cid:105) . Consequently,the critical point determined by m τ c = 0 is shifted butmaintains its Z character. This yields a transition line ,which can be derived by equating the effective massesof the two competing order fields. The scaling of thesemasses with K a [24] implies that this occurs at | g xy | − /Ka ∼ | g z | − Ka . (9)The resulting phase diagram is depicted in Fig. 2. TheCDW phase is characterized by a gap ∆ c ∼ m τ c to fluc-tuations in the relative charge fields θ a h , while the SFphase exhibits a gap ∆ s to fluctuations in φ a h .To derive the conduction properties characterizing thedistinct phases, we first introduce local coupling termsbetween the channels 1 h , h which break translation in-variance in the y -direction, and are necessary to inducenon-trivial transport coefficients. As a minimal choice ofsuch terms [30], we consider defects at y = 0 which add alocal correction J to J xy ⊥ [Eq. (2)], and a spin-flip term K a g z (cid:72) a (cid:76) SF CDW K a g xy (cid:72) b (cid:76) SF CDW
FIG. 2: Phase diagram of the model H ( h ) a ; the phase boundaryis derived from Eq. (9), (a) for g xy = 0 .
1, (b) for g z = 0 . allowing backscattering between the closest channels ofopposite helicities [ H + − in Eq. (1)]: δH ( h ) = J (cid:2) S +1 h , S − h , + h.c. (cid:3) , H + − = J (cid:2) S +1 − , S − + , + h.c. (cid:3) . (10)In terms of Bosonic fields, these yield δH = (cid:88) h = ± J Λ cos[ θ h (0) − θ h (0)] (11)+ (cid:88) n,n (cid:48) =1 , J n,n (cid:48) Λ cos[ θ n + (0) − θ n (cid:48)− (0)]where J n,n (cid:48) with n, n (cid:48) = 2 are generated to second orderin the perturbations Eq. (10). a V a I y J J J FIG. 3: Schematic transport measurement geometry, illus-trated for antisymmetric conductance. Blue solid lines repre-sent DW’s. J , J are defined in Eq. (10). We now consider a multi-terminal contact to an ex-ternal circuitry where current can be driven along the y -direction of the BLG sample (Fig. 3). We particu-larly focus on two observables: the total two-terminalconductance G , and the “antisymmetric conductance” G a = I a /V a where I a is a counter-propagating currentin the channels 1 h , h , short-circuited at one edge (seeFig. 3). From Kubo’s formula, G is given by the re-tarded correlation function of the fully symmetric cur-rent J s = (cid:80) h = ± ,n =1 , J n h ; similarly, G a is dictated bythe correlation of relative current operators [second termof Eq. (6)].We first consider the behavior of G ( T ) at a finite tem-perature T . The main contribution to the scattering ofthe current J s arises from the second term in Eq. (11),which couples the a and s modes via the operators O ± = cos (cid:20) ( θ s + (0) − θ s − (0))2 (cid:21) cos (cid:2) θ a + (0) ± θ a − (0) (cid:3) . (12)To leading order in δH , the conductance G (in units of e / π (cid:126) ) is then given by [31] G = 4 − δG , δG ∼ (cid:90) ∞ dt t (cid:104) [ F ± ( t ) , F ± (0)] (cid:105) where F ± ≡ i [ J s , O ± ] . (13)In the CDW phase, since θ a h are ordered, the secondcosine in Eq. (12) can be replaced by its finite expecta-tion value and O ± ∼ cos θ , θ ≡ ( θ s + (0) − θ s − (0))2 . Eq. (13)with J s related to φ s h via Eq. (6) hence yields [24, 32] δG ∼ T Ks − . (14)For accessible values of K s , this typically diverges at low T implying a breakdown of the weak backscattering ap-proximation. The system therefore exhibits an insulatingbehavior, G ( T →
0) = 0. The finite low T dependence of G can be evaluated perturbatively in the dual tunnelingoperator cos(4 φ ) [32–34], resulting in G ∼ T K s − . (15)In the SF phase, θ a h are disordered and the correlationsof e ± iθ ah yield an exponential decay of δG evaluated fromEq. (13) for T (cid:28) ∆ s . The leading backscattering istherefore governed by second order terms generated by δH , which decouple the a -mode [29, 34, 35], of the formcos(2 θ ). One obtains δG ∼ T Ks − , which under ourassumption K s > / insulating behaviorat T →
0; the same procedure leading to Eq. (15) yields G ∼ T K s − (16)(which approaches a non-universal constant for K s ∼ / G ( T ) as a jump in the power-law G ∼ T κ , from κ = 4 K s − κ = 16 K s − T -dependence of G a , which probes the response to a pure antisymmetric current I a . Backscat-tering in this channel is solely due to the first term in Eq.(11), which can be cast as O a = J Λ (cid:88) h = ± cos [2 θ a h (0)] . (17)In the SF phase, we evaluate the deviation δG a fromperfect conductance ( G a = 1 − δG a for each ladder h = ± ) from Eq. (13) with O ± replaced by O a , associatedwith the disordered operators in this phase. For T (cid:28) ∆ s , δG a ∼ exp (cid:18) − ∆ s T (cid:19) (18)which implies an exponentially small voltage drop V a ∼ δG a in the setup depicted in Fig. 3. A similar calculation,with O a replaced by its dual (cid:80) h cos [2 φ a h (0)], yields anexponentially small conductance in the CDW phase: G a ∼ exp (cid:18) − ∆ c T (cid:19) , (19)with ∆ c a charge gap characterizing this phase. We thuspredict that G a would exhibit a true “superconductor-insulator” transition, indicated by a jump of G ( T → K a (which monotonicallyincreases with the physical parameter e/(cid:96)V ∼ √ B ⊥ /V )through the phase boundaries of Fig. 2.In summary, we have shown that pairs of DW’s form-ing in the ν = 0 QH state of BLG subject to a kink-likegate potential V ( x ) provide a unique realization of spin-1 / B where E z is appreciable [10], the tuningof V or the size and tilt-angle of B can induce a SF-CDWtransition, clearly observable in the low- T conductance.As a final remark, note that the paired DW’s discussedin our case are apparently analogous to a “helical ladder”formed by coupling two parallel edge states of TI, with acrucial distinction: in the latter, a coupling in the form ofthe first term in H ( h ) ⊥ [Eq. (2)] is forbidden. The secondterm of H ( h ) ⊥ , analogous to a Josephson coupling resultingfrom electron-pair tunneling between the HLL’s, likewisecompetes with a term generating a CDW order. How-ever, the latter has a different scaling dimension. Hence,the resulting SF-CDW transition is of a different nature.This will be discussed in more detail elsewhere [36].We thank E. Berg, T. Pereg-Barnea and A. Young foruseful discussions. E. S. is grateful to the hospitality ofthe Aspen Center for Physics (NSF Grant No. 1066293)and to the Simons Foundation. This work was supportedby the US-Israel Binational Science Foundation (BSF)grant 2008256, the Israel Science Foundation (ISF) grant599/10, and by NSF Grant No. DMR-1005035. [1] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov and A. K. Geim, Rev. Mod. Phys. , 109(2009), and references therein.[2] D. A. Abanin, K. S. Novoselov, U. Zeitler, P. A. Lee, A.K. Geim and L. S. Levitov, Phys. Rev. Lett. , 196806(2007).[3] J. G. Checkelsky, L. Li and N. P. Ong, Phys. Rev. Lett. , 206801 (2008); J. G. Checkelsky, L. Li and N. P.Ong, Phys. Rev. B , 115434 (2009).[4] Xu Du, I. Skachko, F. Duerr, A. Luican, and E. Y. An-drei, Nature , 192 (2009).[5] B. E. Feldman, J. Martin and A. Yacoby, Nature Phys. , 889 (2009).[6] J. Jung and A.H. MacDonald, Phys. Rev. B , 235417(2009).[7] M. Kharitonov, Phys. Rev. Lett. , 046803 (2012); M.Kharitonov, Phys. Rev. B , 155439 (2012).[8] Y. Zhang, Z. Jiang, J. P. Small, M. S. Purewal, Y.-W.Tan, M. Fazlollahi, J. D. Chudow, J. A. Jaszczak, H. L.Stormer and P. Kim, Phys. Rev. Lett. , 136806 (2006);Y. Zhao, P. Cadden-Zimansky, F. Ghahari and P. Kim,arXive:1201.4434.[9] S. M. Girvin and A. H. MacDonald in Perspectives inQuantum Hall Effects , S. Das Sarma and A. Pinczuk,eds. (John Wiley & Sons, 1997); S.L. Sondhi, A.Karlhede,S.A. Kivelson and E. H. Rezayi, Phys. Rev. B , 16419(1993); H.A. Fertig, L Brey, R. Cˆot´ e , A.H. MacDonald,Phys. Rev. B , 11018 (1994); K. Moon, H. Mori, K.Yang, S. M. Girvin, A. H. MacDonald, L. Zheng, D.Yoshioka and S.-C. Zhang, Phys. Rev. B , 5138 (1995).[10] A. F. Young, J. D. Sanchez-Yamagishi, B. Hunt, S. H.Choi, K. Watanabe, T. Taniguchi, R. C. Ashoori, P.Jarillo-Herrero, arXiv:1307.5104.[11] E. McCann and V. I. Fal’ko, Phys. Rev. Lett. , 086805(2006); E. McCann, Phys. Rev. B , 161403 (2006).[12] D. A. Abanin, S. A. Parameswaran and S. L. Sondhi,Phys. Rev. Lett. , 076802 (2009).[13] L. Brey and H. A. Fertig, Phys. Rev. B , 195408 (2006).[14] D. A. Abanin, P. A. Lee, and L. S. Levitov, Phys. Rev.Lett. , 176803 (2006).[15] V. Mazo, E. Shimshoni and H. A. Fertig, Phys. Rev. B , 045405 (2011).[16] I. Martin, Y. M. Blanter and A. F. Morpurgo, Phys. Rev.Lett. , 036804 (2008).[17] M. Killi, T.-C. Wei, I. Affleck and A. Paramekanti, Phys.Rev. Lett. , 216406 (2010); S. Wu, M. Killi and A.Paramekanti, Phys. Rev. B , 195404 (2012). [18] C.-W. Huang, E. Shimshoni and H. A. Fertig, Phys. Rev.B , 205114 (2012).[19] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010); X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. ,1057 (2011).[20] J. C. Y. Teo and C. L. Kane, Phys. Rev. B , 235321(2009); C.-Y. Hou, E.-A. Kim, and C. Chamon, Phys.Rev. Lett. , 076602 (2009); A. Str¨om and H. Johan-nesson, Phys. Rev. Lett. , 096806 (2009); C.-X. Liu,J. C. Budich, P. Recher, and B. Trauzettel, Phys. Rev.B , 035407 (2011).[21] E. Shimshoni, H. A. Fertig and G. V. Pai, Phys. Rev.Lett. , 206408 (2009).[22] M. Kharitonov, Phys. Rev. B , 075450 (2012).[23] V. Mazo, H. A. Fertig and E. Shimshoni, Phys. Rev. B , 125404 (2012).[24] T. Giamarchi, Quantum Physics in One Dimension , (Ox-ford, New York, 2004).[25] M. Fabrizio, Phys. Rev. B , 15838 (1993).[26] A. O. Gogolin, A. A. Nersesyan and A. M. Tsvelik, Bosonization and Strongly Correlated Systems (Cam-bridge University Press, 1998).[27] D. G. Shelton, A. A. Nersesyan and A. M. Tsvelik, Phys.Rev. B , 8521 (1996).[28] P. Lecheminant, A. O. Gogolin and A. A. Nersesyan,Nucl. Phys. B , 502 (2002).[29] See also Y. Atzmon and E. Shimshoni, Phys. Rev. B ,220518(R) (2011); Phys. Rev. B , 134523 (2012).[30] Our analysis can be straightforwardly modified to acountfor a more general type of disorder, e.g. a finite densityof random impurities [24], yielding qualitatively similarresults.[31] See, e.g., Chap. 7 in Ref. 24.[32] C. L. Kane and M. P. A. Fisher, Phys. Rev. B , 15233(1992).[33] H. Saleur, “Course 6: Lectures on Non-perturbative FieldTheory and Quantum Impurity Problems” in TopologicalAspects of Low Dimensional Systems , Eds. A. Comtet,T. Jolicoeur, S. Ouvry and F. David, p. 473 (1998);arXiv:cond-mat/9812110.[34] S. T. Carr, B. N. Narozhny and A. A. Nersesyan,Phys. Rev. Lett. , 126805 (2011); S. T. Carr, B. N.Narozhny and A. A. Nersesyan, Ann. of Phys. , 22(2013).[35] E. Orignac and T. Giamarchi, Phys. Rev. B57