Dipolar Molecules in Optical Lattices Can [10] K

arXiv:1109.4782v3 [cond-mat.quant-gas] 5 Apr 2012 eedn Nhpigpoess efidta ihin- with that find We processes. hopping NN occupation- novel dependent the including model Hubbard one-dimensional extended the solve renor- we entanglement (MERA), ansatz multiscale malization unit-filling di- and exact and Using (ED) the half- states. agonalization In at insulating dipo- states (CB) sufficient checkerboard ground at are 18]. the interaction, [17, strength, NN only lar supersolidity on with stable focusing case usual model, of litera- Hubbard presence the one- extended in the and the discussed square within been in ture have bosons dipolar lattices of Soft-core dimensional physics that the bosons. show change we soft-core Letter, considerably discussing this terms interaction- when In these Usually, neglected molecules. are bosonic band. dipolar terms Bloch hopping interac- lowest induced dipolar the long-range in addi- from tions includes arising which processes model ping hop- (NN) BH nearest-neighbor occupation-dependent dipoles. modified tional the a of harmonic derive direction a We polarization with the lattice optical along square trapping 2D a in molecules [9–15]. etc. or- pair-supersolids, system, long-range supersoli crystals, states, the with Wigner insulating states of structured various of behavior like appearance der, collective the the to in leading role play interactions crucial long-range a molecules, dipo polar electric strong of the in moment of systems Because such [4–7]. model lattices optical can which models extended in Bose-Hubbard interest renewed have ul- molecules on polar long- Experiments tracold including particles. from between interactions comes range model Bose–Hubbard the of B)mdl hc a w aaees tunneling a interaction parameters: on-site two an has and sys- which Bose-Hubbard these model, the case, by pre- (BH) simplest described been theoretically the be has In can (MI) tems [1]. insulator observed Mott a and a dicted from to transition via (SF) phase interacting superfluid quantum atoms a For interaction, environment. contact controlled physics many-body a of in realization the allowed has lattices nti etr esuytegon-tt fdipolar of ground-state the study we Letter, this In rpigadmnpltn lrcl ae noptical in gases ultracold manipulating and Trapping 2 CO–h nttt fPooi cecs v alFriedric Carl Av. Sciences, Photonic of Institute –The ICFO 3 CE nttcóCtln eRcraiEtdsAaçt,L Avançats, Estudis i Recerca de Catalana Institució – ICREA oazSowiński Tomasz 1 nttt fPyiso h oihAaeyo cecs l L Al. Sciences, of Academy Polish the of Physics of Institute ob ae noacuti poigeprmnswt dipola 37.10Jk,67.85.Hj,75.40.Cx with numbers: experiments PACS considerable upcoming in These account phases. into taken quantum novel be to t to ne show lead and we and on-site terms. phases ansatz, all tunneling renormalization account entanglement pair into multiscale and taking tunneling lattice, occupation-dependent optical an in esuyteetne oeHbadmdldsrbn nultr an describing model Bose–Hubbard extended the study We U 1 , 2 2 ] aua extension natural A 3]. [2, myt Dutta Omjyoti , ioa oeue nOtclLattices Optical in Molecules Dipolar 2 hlp Hauke Philipp , Dtd uy2,2018) 22, July (Dated: ds, le J sa xenlltieptnilo atc depth lattice of potential lattice external an is nasur pia atc.TecrepnigHamilto- corresponding The reads lattice. nian optical square a in rtdb ae edo wave-length of field laser a by erated xenleeti edaogthe along field electric external coexists con- at them sign of alternating sites. has both secutive parameter where order SF region the a su- with Par- find one-particle properties. a we (PSF) has ticularly pair-superfluid which and state (SF) novel the perfluid a from to enters phases system CB the interaction, dipolar creasing Ψ operators. eiigteetra amncptnilin potential harmonic external the terizing h ioedpl neato sdntdby denoted is interaction dipole–dipole The where site h tnadtneigcoefficient, tunneling standard the etitn usle oo-ieadN em,w arrive we model BH terms, extended NN the and at on-site to ourselves restricting nlws lc-adWannier-functions Bloch-band lowest in adn h edoperator field the panding in and tion, a neatosla otonvltrsi q 1:The (1): Eq. in terms to novel proportional two term to Dipo- lead term. interactions dominating the lar to interactions dipolar the of u fdplrand dipolar of sum Npi unln [19–21]. tunneling pair NN ie n h empootoa to that of proportional occupation term the the by and induced site, site neighboring a to 1 2 as,nm ,080Cselees(acln) Spain (Barcelona), Castelldefels 08860 3, num. 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150 0 0.5 1 1.5 2 2.5 3 0 0 0.5 1 1.5 2 2.5 3 70 (c) 8 12 50 -1 1 16 TD /J 40 P /J (d) -50 -2 D χ 0.5 UD /J 10

Parameters -3 -150 V /J D (d) π/8 N = 16 π contribution (a) -4 0 /4 Checkerboard 0.2 3π/8 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 1 π/2 κ κ (b) 5π/8 Lattice flattening Lattice flattening 3π/4 0.5 7π/8 π S(q) 0.1 0 functions

FIG. 1. Dependence of the dipolar part (subscript D) of Correlation U, V , T , and P on the lattice flattening κ for lattice depth 0 0.5 1 1.5 2 2.5 3 0 0 0.5 1 1.5 2 2.5 3 Electric dipole d (D) V0 = 6ER and γ = 52. Electric dipole d (D)

FIG. 2. We plot various properties of the exact ground state of ′ 2 (3) ′ 1 (z−z ) a half-filled system as a function of dipole moment d. Fig. (a) gδ (r − r )+ γ ′ 3 − 3 ′ 5 . We measure all h  |r−r | |r−r | i shows the contribution of the CB states to the ground state lengths in units of the laser wave length λ and all en- of the system for N = 8. In Fig. (b) we plot the one-particle 2 2 2 ergies in recoil energies ER = 2π ~ /(mλ ), where m and two-particle correlation functions φi and Φi. The dotted is the bosonic mass. Additionally, we define the lattice line shows φi when we neglect the terms T and P . When T, P =6 0, the solid line and dash-dotted line shows φi and Φi flattening κ = ~Ωz/2ER as well as the dimensionless cou- pling constants describing contact and dipolar interac- respectively as a function of dipole moment d. In Fig. (c) we plot the fidelity susceptibility χ(d) for the half-filled system tion, g = 16π2a /λ and γ = md2/(~2ε λ) (where a is s 0 s for different system sizes. In Fig. (d) we have shown the the s-wave scattering length, ε0 is the vacuum permittiv- structure-factor S at different ordering wave vectors for the ity, and d is the electric dipole moment of the bosons). half-filled system with 16 sites. For concreteness, we consider an ultracold gas of dipo- lar molecules confined in a optical lattice with lattice depth V0 = 6ER, mass m = 220a.m.u and λ = 790 nm tion to an insulating state. In the half-filled system, the [22]. We also assume that the s-wave scattering length transition occurs because for large enough V the parti- of the molecules, as ≈ 100a0. For these parameters, cles can decrease their energy by avoiding every second g ≈ 1.06 is approximately constant. We consider dipole site. If we neglect T and P , the situation will not change moments d up to ∼ 3 D (γ up to ∼ 470), which can be by further increasing d (dotted lines in Fig. 2), since achievable for molecules like bosonic RbCs, KLi [8] etc. this only increases V even more. However, the situa- To illustrate the relative strengths of different parame- tion changes significantly when we take into account the ters, in Fig. 1, we compare for γ = 52 the tunneling J density-induced tunneling T and the pair tunneling P . In with the dipolar contribution (subscript D) to the pa- this case, for d ≈ 1.1 D, a second phase transition occurs, rameters U, V , T and P . For the parameters chosen, TD destroying the CB order [solid lines in Fig. 2(a)]. Pre- and PD are 1 orders of magnitude smaller than VD where vious studies have completely neglected such a possible as UD/TD can be tuned by changing κ. On the other destruction of CB order at large d. At the transition, the hand, TD can dominate over J for large γ. In addition, contribution of the CB state to the ground state decreases T and J can have opposite sign as seen in Fig. 1. For rapidly, and the one-particle as well as the two-particle † † concreteness, we choose the lattice parameter κ ≈ 1.95, NN correlation function Φi = ha a aiaii [dashed- P{j} j j making (additionally to J) the on-site interaction U al- dotted line in Fig. 2(b)] attain finite positive values, in- most independent of the dipole moment (UD ≈ 0). In this dicating that the new phase shows single-particle as well case, for large enough γ, we expect that with increasing d as pair superfluidity. In this region we also find that the the parameters V , T and P determine the system prop- † long-ranged correlation function haj aii for |i − j| ≤ 6 erties. For clarity, we restrict ourselves to a 1D chain of decays slowly with alternation sign for consecutive sites. N lattice sites with periodic boundary conditions. This suggests appearance of antiferromagnetic like order To get a first understanding of the system, we find due to the positive hopping T resulting in the conden- the ground state |ψ0(d)i as a function of d by exact di- sation of bosons at the edge of the Brillouin zone. We agonalization (ED) of a half-filled system with N = 8 also looked into the relative effect of T and P on the PSF sites. We also present results for N = 12 and 16 to check state. We found that PSF is generated due to the cor- for dependence on system size. Without the occupation- related tunneling term T (in interplay with the nearest- dependent tunneling terms T and P , we observe the usual neighbor interaction V ). scenario with only two phases, a single-particle SF and For even larger electric moments, a third phase transi- a CB phase. The transition happens at d ≈ 0.4 D. It is tion happens, where φi changes sign. Another signature marked by an increase of the contribution of the checker- of this transition is a rapid growth of Φi. Since this board states to the ground state to almost 100% [inset quantity measures fluctuations of bosonic pairs, this is of Fig. 2(a)]. Also, the one-particle correlation function a signature of a novel pair-superfluid (PSF) phase. The † φi = ha aii almost vanishes, indicating the transi- appearance of pair superfluidity has previously been pre- P{j} j 3 T = P = 0 T ≠ 0, P ≠ 0 dicted in bilayer dipolar systems where the particles are 2 2 1.5 bound by an attractive interaction between the layers CB2 CB2 1 1 1

[14–16]. Though in bilayer systems, the state is a true /U CB CB µ molecular superfluid as Φi 6= 0, whereas φi = 0 iden- 0.5 tically. In the present system, in spite of the particles 0 SF 0 SF PSF 0 interacting repulsively, the pairs are created due to the 0 1 2 3 0 1 2 3 occupation-dependent tunneling terms in Eq. (1) (similar Electric dipole d (D) Electric dipole d (D) to [23]). To confirm that all these transitions are indeed quan- FIG. 3. ED phase diagram without (left) and with (right) tum phase transitions, we calculated – for different taking into account T and P . The color denotes the super- chain lengths N – the ground-state fidelity suscepti- fluidity fractions, φi and Φi. Neglecting T and P , for large 2 ∂ F(d,δ) enough d and µ the system is always in an insulating phase bility [24–26] χ(d) = − ∂δ2 , where F(d, δ) = and the average number of particles is a multiple of 1/2. CB δ=0 |hψ0(d)|ψ0(d + δ)i|. Peaks in χ are efficient indicators of (CB2) denotes a checkerboard phase where sites with 0 and quantum phase transitions. In Fig. 2(c), we present χ(d) 1 (2) particles alternate. Including the new terms, the insu- for different chain sizes. There are three clear peaks at lating phases vanish for large enough d, and a PSF appears. We truncate the Hilbert space at a maximal occupation num- the quantum phase transitions found from the correlation ber of 4 particles per site. We exclude data points where the functions [as presented in Fig. 2(b)]. The positions of the occupation number becomes too high (white region). transition points (TPs) do not significantly depend on the number of sites, especially for the 1st and 3rd TP. The middle peak in Fig. 2(c) refers to the transitions from checkerboard to antiferromagnetic superfluidity. More- over, the magnitude of the fidelity susceptibility at all TPs increases with chain length, which suggests that the transitions will survive in the thermodynamic limit. More insight into the properties of the observed phase comes from the static structure factor, which is defined 1 N iq(j−k) as S(q) = N 2 j,k=1 e (hnˆj nˆki − hnˆj ihnˆki), with q =2πm/N, 0 ≤Pm ≤ N − 1 integer.A peak in the struc- ture factor at finite momentum points towards presence of periodic density modulation in the systems. In Fig. FIG. 4. Results for the BH Hamiltonian (1) in a chain with 2(d), we present S(q) for a half-filled system with N = 16 N = 128 using MERA with m = 8, revealing checkerboard (CB and CB2) order, as well as superfluid (SF and SF’) and sites. In the CB phase (between the 1st and 2nd TP), pair-superfluid phases (PSF). (a) Mean occupation, (b) mean the dominant peak of S(q) is at q = π, and its magnitude SF order parameter, (c) mean PSF order parameter, and (d) is almost independent of system size. Above the 3rd TP, mean NN density correlations. the system is in a phase where φi has an inverted sign and Φi is large. This means that states where bosons oc- cur in pairs dominate (their contribution to the ground tions of 4 sites with occupation truncated at 4 particles state is about 95%). Since, due to the dipolar interac- per site. When the additional terms T and P are large, tions, boson pairs do not occupy neighboring sites, the they destroy the CB phase, making place for a PSF. system has some local structure, leading to a predomi- To get a more detailed analysis of larger systems π nant structure-factor peak at q = 2 . The intermediate than tractable in ED, we have performed a Multi-Scale- phase (between the 2nd and 3rd TP) has interesting prop- Entanglement-Renormalization-Ansatz (MERA) [27–29] erties: the ground state of the finite system deforms its computation of the phase diagram The MERA is a quasi- structure stepwise, changing the dominant q from π to exact variational method that consists in postulating π/2 by one quantum ∆q =2π/N at a time. For N = 16, a tensor-network structure for the low-energy states of this leads to three changes in the dominant q. Since in Hamiltonian (1), which in particular yields especially an infinite system q can take every value between 0 and good results in critical phases, where other methods such 2π, we expect in large chains a continuous change from as DMRG are very costly [27, 28]. the CB with q = π to the two-particle SF with q = π/2. The results are presented in Figs. 4(a-d), where we Finally, we analyze the influence of the additional show, averaged over the chain, the occupation hnii, the terms T and P on the grand-canonical phase diagram, SF order parameter haii, the PSF order parameter haiaii, where the particle number is not conserved. For this, we and NN density–density correlations hnini+1i. The phase add a chemical potential term −µ i nˆi to Hamiltonian diagram extracted from these observables is sketched in (1). In Fig. 3, we present the phaseP diagram as well as Fig. 4(a). At low d, there is a single-particle SF, which the average number of particles per site for ED calcula- gives way to CB phases for d ≥ µ. Increasing d, the sys- 4 tem undergoes a transition to a SF phase, where initially T=P=0 a) T≠0, P≠0 b) for a range of ≈ 0.2D one-particle superfluidity domi- 0.8 0.8 nates (similar to the ED results), and afterwards pair 0.4 0.4 superfluidity. At low µ, we find a phase (SF’) which has 0 0 additionally to SF order (i.e., a finite haii) small nearest- neighbor density–density correlations. Hence, it has a Correlation functions -0.4 -0.4 0 0.4 0.8 1.2 0 0.4 0.8 1.2 local structure where sites with high and low occupation d (D) d (D) alternate. We checked that this phase is not due to phase separation. The novel aspect of this is that in the usual FIG. 5. The one-particle and two-particle correlation func- extended BH model with soft-core interactions stable su- tions φi (solid line) and Φi (dotted line) as a functions of persolidity appears only at the particle-doped region of dipole moment d when the full dipolar interactions are taken the CB phase [17, 18]. For higher µ and d ∼ 1, we get into account corresponds qualitatively to the calculations truncated at NNs (Fig. 2). The large Φi and negative φi when a CB of two particles(CB2 phase) in the filled site. This behavior is a result of having low U so that it is ener- terms T and P are taken into account indicate the break down of the CB phase to a PSF. Calculations for ED at half-filling getically favorable than having one particle at each site. with N = 16. As already indicated by ED, the new terms T and P destroy CB order in favor of PSF phases, meaning that these terms cannot be neglected. We also checked at few occupation-dependent tunneling and pair tunneling (in- points in the phase space of the PSF region to look for duced by long-range dipolar interactions) destroy insu- the sign of the SF order parameter as a function of lat- lating checkerboard phases for large enough electric mo- tice sites and we found the alternating sign as seen in ED ments d, leading to a novel pair-SF phase. MERA re- calculations. sults suggest also that a supersolid phase could appear To make better contact with experiment, we examine for 1/2 filling even in the hole-doped case. Any presence the disappearance of the CB pattern when the long-range of additional weak trapping potential can result in shell part of the full dipolar interactions is taken into account, like structures seen in usual BH model as long as local- i.e., we replace the NN term in Hamiltionian (1) with density approximation is valid. We note that, as our nu- V 3 nˆinˆj . Using ED at half-filling for N = 16, merical calculations in carried out in one dimension, the P{ij},i6=j |i−j| we find that qualitatively the phase diagram does not various superfluid correlations decay in a power law with change much with respect to our previous calculations distance. In this sense, the superfluid phases mention with the simplified Hamiltonian (1) [compare Fig. 2(b)]: here will show quasi-long-range order in infinite systems. When the occupation-induced tunneling terms T and P Our calculations are done for parameters experimentally are neglected, the CB phase remains stable for arbitrarily achievable in the near future, and the changes to the large d [Fig. 5(a)]. In contrast, when taking into account phase diagram have to be taken into account in the inter- the tunneling terms T and P it disappears, making way pretation of future experiments with dipolar molecules. for a PSF phase [Fig. 5(b)]. This happens even at smaller This Letter was supported by the EU STREP NAME- d than when truncating the interactions at NNs. Namely, QUAM, IP AQUTE, ERC Grant QUAGATUA, Spanish the PSF phase appears for d ≥ 0.7D. We also note that MICINN (FIS2008-00784 and Consolider QOIT), Caixa in Fig. 5(b), there is a kink in φi around d ∼ 0.5D. Manresa, and Marie Curie project FP7-PEOPLE-2010- This kink corresponds to the appearance of a crystal IIF “ENGAGES” 273524, and AAII-Hubbard. T.S. ac- like phase with modulation |....200100200100.... >. Ade- knowledges hospitality from ICFO. tailed discussion of this phase is outside the scope of this paper. We have further checked that counter-intuitively PSF arises predominantly due to correlated tunneling T . Without this term PSF phase can not be reached for rea- sonable electric moments. We also note that for very low [1] M. Greiner et. al., Nature 415, 39-44 (2002). [2] M.P.A. Fisher, P.B. Weichman, G. Grinstein, D.S. dipolar strength Φ has a small nonzero value. As seen i Fisher, Phys. Rev. B 40, 546 (1989). in Figs. 2(a), (b) and 5(a), (b), a small but finite Φi is [3] D. Jaksch et. al., Phys. Rev. Lett. 81, 3108 (1998). present as d → 0 irrespective of the presence of T and P . [4] K.-K. Ni et. al., Science 322, 231 (2008). This can be traced back to second-order processes due to [5] S. Ospelkaus et. al., Faraday Discuss. 142, 351 (2009). J which can also give rise to pair correlations with small [6] K. 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Gorshkov et. al, Nature Physics 6, 289 (2010). (2009). [20] G. Mazzarella, S. M. Giampaolo, and F. Illuminati, Phys. [29] See supplementary material for technical details regard- Rev. A 73, 013625 (2006). ing implementation of MERA. [21] P. Bader and U. R. Fischer, Phys. Rev. Lett. 103, 060402 arXiv:1109.4782v3 [cond-mat.quant-gas] 5 Apr 2012 h lattice the lr nieagvnlyraecoe ob h ae the same, the be to chosen are layer factor given a inside glers opeeteE rnfrainfo h lattice the to from required transformation as ER same times the many the complete layer as en- disentangler each and inside to isometry use order we in periodi conditions, with example, the boundary chains For of in states invariant encoded translational tensors. DMRG. code be the e.g., can of as, Hamiltonian structure such the chains of 1D Symmetries the traditional studying more is for to This respect methods with critical MERA resources. infinite the of computational describe advantage finite to with ansatz states MERA the of ability spacing ahlayer each lattice a rmeeetr esr eogn otodffrn fami- different two isometries to lies, belonging tensors elementary from n h oetBohbn o igepril mov- we single-particle First a for (1). potential band Eq. the Bloch in in ing lowest model the Hubbard find modified the in n i a aee structure layered a has ii) and ttso aitna.Tetno network low-energy tensor The the Hamiltonian. for of structure states tensor-network a tulating m tions aaeesfrteHbadmdl npriua,the term term particular, pair-hopping the hopping the In correlated derive and the we calculate model. to basis, used Hubbard integrals Wannier the the for in parameters operator field the Ω eew ecietepoeuet aclt h terms the calculate to procedure the describe we Here EAi aitoa ehdta osssi pos- in consists that method variational a is MERA z 2 z W 2 n / P T aclto fhpigtrsTadP and T terms hopping of Calculation i 2 2 b o nycaatrz h lcigfco fthe of factor blocking the characterize only not L ( rmta,w osrc h ane func- Wannier the construct we that, From . i V × = W × = +1 ,y x, i T L Z Z ihltiespacing lattice with i 2 1 i = ( )e efrsa Rtasomto 2 ]from 3] [2, transformation ER an performs +1 Z Z oazSwńk,OjoiDta hlp ak,Lc Taglia Luca Hauke, Philipp Dutta, Omjyoti Sowiński, Tomasz i r ( I − x nb − hnalteioere n disenta- and isometries the all When . i ecito fMERA of Description κz ′ I d n disentanglers and y , i r i 3 2 I rpryi)i tteoii fthe of origin the at is ii) Property . d r ′ ′ V ) i † 3 ) oaie tsite at localized d W latt W r 3 = d r i j 3 ( ′ ( I ( W x r r ; x ′ ′ = ) ′ y , W i P y , 2 ( D ′ ,y x, i ′ ) are: ( )e i V W ,y x, D 0 − b )e i † j ioa oeue nOtclLattices Optical in Molecules Dipolar  T i κz ( sin ) = − oalattice a to x D W ′ ′ 2 κz = y , i i 2 I j 2 htaeisometric, are that 1.B expanding By [1]. ( 2 ; ′ λ V Q ,y x, π )e ( x i − r upeetr Material Supplementary T sin + κz )e − i T uhthat such , − ′ 2 r κz L si built i) is ′ ) 2 i 2 +1 2 λ π L with y (S2) (S1) i  to + T c I.S.Tensors S1. FIG. 1 .Kh,Py.Rev. Phys. Kohn, W. [1] uha oflfilteioer osrit endi q (S2 Eq. in layer defined chos constraints A are isometry tensors the ii) con fullfill The to tensor i) as represent indices. such involved tensors the two over connecting tractions representing Lines legs trailing with indices. circles by represented are asoeaosadsae endo lattice a on defined states and operators maps 2 .Vdl hs e.Lett. Rev. Phys. Vidal, G. [2] spacing 4 .Vdl hs e.Lett. Rev. Phys. Vidal, G. [4] Transitions”, Phase Quantum “Understanding in Vidal G. [3] ihltiespacing lattice with Rpoeue(etl about talk (we procedure ER 5 .EebyadG ia,Py.Rv B Rev. Phys. Vidal, G. and Evenbly G. [5] al noea es ntcl ftosts hscnbe can This sites. two of a cell natu- unit by can a that accomplished extracted least ansatz at diagram an encode need phase we rally the that suggests of ED parts from some in state. patterns the of cell unit the of size the defines ue h opttoa oto the re- of and phase cost CB computational a the of cell duces unit two-site the accomodates n oetvle of values modest and hnew hoea choose we whence 1 to 3 the 1 However, than to expensive procedure. more ER computationally the is of MERA step this each at one into sites enmn parameter refinement r and ory ic h opeiyo h loih is algorithm time, the simulation of larger complexity imply the structure since but TN results accurate the more of layer a show we the ii), for S1(a) Fig. In ogtacretqaiaiepcueo h model. the of picture qualitative correct a get to ntemdlw r osdrn h rsneo CB of presence the considering are we model the In dtdb icl .Cr Tyo rni,Bc Raton, Boca Francis, & (Taylor 2010) Carr D. Lincoln by Edited EA notntl,tetasainlyinvariant translationally the Unfortunately, MERA. EAde o aiyacmdt Bpattern, CB a accomodate easily not does MERA o1 to 4 b O i ooeaosadsae endo lattice a on defined states and operators to ( T m i 8 fthe of ) EAta ehv sd EAhsa has MERA used. have we that MERA nnme foeain e trto [5], iteration per operations of number in oz,Mce Lewenstein Maciej cozzo, 4 o1 to 2 b o1 to 4 I i o1 to 4 i m [4]. m ioere)and (isometries) 115 uhas such agrvle fwihprovide which of values larger EA .. ybokn two blocking by i.e., MERA, 0 (1959). 809 , EAtno network tensor MERA 99 101 EAta ohnaturally both that MERA o1 to n 245(2007). 220405 , 151(2008). 110501 , m 8 = EA u talso it but MERA) o1 to 2 79 D r fe enough often are O i 418(2009). 144108 , ( (disentanlgers) L m i EA[5]. MERA 5 ihlattice with ) nmem- in T their L that i +1 en ). 3 -