Devil's staircases without particle-hole symmetry
DDevil’s staircases without particle-hole symmetry
Zhihao Lan, Igor Lesanovsky, and Weibin Li
School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom andCentre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems,University of Nottingham, Nottingham NG7 2RD, United Kingdom
We present and analyze spin models with long-range interactions whose ground state features a so-calleddevil’s staircase and where plateaus of the staircase are accessed by varying two-body interactions. This isin contrast to the canonical devil’s staircase, for example occurring in the one-dimensional Ising model withlong-range interactions, where typically a single-body chemical potential is varied to scan through the plateaus.These systems, moreover, typically feature a particle-hole symmetry which trivially connects the hole part ofthe staircase (filling fraction f ≥ /
2) to its particle part ( f ≤ / I. INTRODUCTION
A devil’s staircase is a fractal structure that character-izes the ground state of a plethora of systems in physics .Examples include the Frenkel-Kontorowa model , theFalicov-Kimball model , Ising models , quantum dimermodels as well as certain discrete maps . In par-ticular, one-dimensional (1D) Ising models are paradigmaticsystems that may exhibit devil’s staircases both in the caseof long-range and short-range interactions. In Ising mod-els with long-range interactions it was shown rigorously thatthe permitted filling fractions (ratio of the number of parti-cles to that of lattice sites) of the ground state configurationsform a complete devil’s staircase when the chemical potentialis varied . This means when scanning the chemical poten-tial the filling fractions can assume all rational numbers. Forthe short-range interacting anisotropic next-nearest-neighbourIsing (ANNNI) model, such staircase appears only at finitetemperatures since there are solely two stable ground states(known as ferromagnetic phase and antiphase) at zero tem-perature. The staircase structure of ANNNI model has beenobserved in NaV O under high pressure .In recent years, controllable quantum systems haveemerged as platforms for exploring phenomena in condensed-matter and high-energy physics . This includes trappedions , cold polar molecules and strongly interactingRydberg atoms , ultracold atoms in bichromatic lattices ,synthetic dimensions and gauge fields , photons with engi-neered long-range interactions and optomechanical cavitysystems . These platforms allow not only to control single-body quantities (e.g. the trapping potential or the chemicalpotential), but also to tailor the shape of the underlying two-body interaction.In a recent study , we have identified a new mechanismunderlying the formation of a devil’s staircase within a spinmodel implemented by Rydberg atoms held in a 1D opti-cal lattice. By using a so-called double-dressing scheme ,we have shown how to create competing interactions withshort-range attraction and longer-range repulsion between twoatoms. In particular, we focused on a situation where thenearest-neighbour interaction is attractive and tunable whilethe interactions from next-nearest-neighbours onwards fol-low a repulsive van der Waals (vdW) potential. Such non- convex potential leads to the formation of a devil’s staircasein the ground state and its plateaus are accessed by varyingthe strength of the nearest-neighbour attraction.This situation is in contrast to that encountered, e.g. inabove-mentioned staircase of the Ising model, which is gov-erned by a single-body chemical potential term. The stair-case in the Ising model features a particle-hole symmetry, i.e.,the hole part of the staircase at filling fraction f ≥ / f ≤ /
2. In our previous work , we have shown that abroken particle-hole symmetry emerges for a staircase whoseplateaus are accessed by two-body attractive interactions. Inthis situation the staircase is a union of two sub-staircases thatare consisting of either dimer particles or dimer holes. Thisfinding opens up the possibility to study a plethora of hybridstaircases. For example, it is possible to encounter a situa-tion with a dimer particle sub-staircase in the particle sectorand a trimer hole sub-staircase in the hole sector. Finally, wewould like to note that the impact of different kinds of inter-actions on the devil’s staircase physics has been studied in theliterature from different perspectives . For example, someaspects of the staircases discussed in the present work can belinked to studies of atoms adsorbed on a surface . How-ever, a systematic exploration of devil’s staircases from theperspective of particle-hole symmetry breaking has not beenconducted previously.In this work, we extend our previous study to spin mod-els that feature attractive interactions not only among nearestneighbours but over a longer range. The paper is structuredas follows: In section II, we present the model Hamiltonianand discuss the role played by the particle-hole symmetry. Insection III, we discuss analytical and numerical tools for ana-lyzing the ground state properties of the model Hamiltonian.In section IV, we benchmark our tools by applying them toa conventional staircase controlled by a chemical potential.In section V we investigate in detail the situation where two-body interactions drive staircases without particle-hole sym-metry, which constitutes the central part of this work. In sec-tion VI, we discuss the possibility of staircases controlled by n -body interactions ( n > a r X i v : . [ c ond - m a t . o t h e r] F e b (a) ⌦ ⌦ | R i| R i | i | i V / V (b) V ( r ) / V ( a ) V V V k V ( k ) / | W ( ) | R res Figure 1. (a) Level scheme. An electronically low-lying state | (cid:105) islaser-coupled to Rydberg states | R (cid:105) and | R (cid:105) with Rabi frequency Ω and Ω , and detuning ∆ and ∆ respectively. (b) Effective in-teraction potential between particles in dressed state | (cid:105) . This inter-action is attractive at short distances and repulsive at long distances.Here, we show a situation where the nearest neighbour and next-nearest neighbour are attractive, i.e. W ( ) < W ( ) < II. MODEL HAMILTONIAN
Staircases explored in this work rely on a non-convex long-range interaction which is attractive at short distances and re-pulsive at large distances. In our previous work , we pro-posed that a special form of this interaction, i.e., with vander Waals repulsive tail, can be engineered with the help ofRydberg atoms. Specifically the physical setting is a 1Dlattice with spacing a where each site can either be occu-pied by an atom in state | (cid:105) or | (cid:105) . For convenience, wedenote that a site is empty (occupied by a particle) whenthe atom of the site is in state | (cid:105) ( | (cid:105) ). We then employa double-dressing scheme , in which two blue- and red-detuned lasers are applied simultaneously to weakly couplethe | (cid:105) state with two Rydberg S -states | R (cid:105) and | R (cid:105) , as de-picted in Fig. 1(a). The Rabi frequency and detuning of theblue (red) detuned laser are Ω ( Ω ) and ∆ ( ∆ ), respec-tively. The vdW interaction of the Rydberg state | R j (cid:105) is C j / r with C j the corresponding dispersion constant ( j = , . The lasers induce longrange interactions between atoms in the Rydberg dressed | (cid:105) state . The blue-detuned laser induces an interaction po-tential U ( r ) = ˜ C / ( r − R ) , where ˜ C = R Ω / ∆ and R res = ( C / | ∆ | ) / determines the distance of the two-atomresonant excitation when 2 ∆ + C / R = . The result-ing interaction is attractive for r < R res and repulsive when r > R res . The red-detuned laser generates a long range soft-core interaction, U ( r ) = ˜ C / ( r + r ) where ˜ C = r Ω / ∆ and the core radius r = ( C / | ∆ | ) / . The overall dressedinteraction is given by the combined potential of V ( r ) = U ( r ) + U ( r ) , which is illustrated in Fig.1(b). By tuningthe laser parameters, strength and attractive range of the non-convex long-range two-body interactions can be varied (de-tails of the implementation is given in Ref. [35]).In this work, we will go beyond this special realization with Rydberg atoms and consider more general non-convex inter-actions, where the repulsive tail is not limited by the vdWtype, i. e. V ( r ) ∼ / r α with 1 < α , focusing more on thephysics rather than the experimental implementations. When α is taken as a parameter that can be freely tuned, many newfeatures are found in the respective staircase which is not re-vealed using the vdW interaction. Taking these considerationsinto account, we study a classical 1D spin chain governed bythe following Hamiltonian H = ∞ ∑ i = − ∞ ∞ ∑ r = R + V ( r ) n i n i + r + ∞ ∑ i = − ∞ R ∑ r = W ( r ) n i n i + r , (1)where W ( r ) ≤ r = , · · · , R ) parametrizes the strength of theattractive potential part, with R to be the range of the attractiveinteraction. The potential V ( r ) = ( R + ) α / r α ( r = R + , · · · )corresponds to the repulsive tail. Note that here and in thefollowing, the energy is expressed in units of V ( R + ) andlength in units of the lattice spacing a .A particle-hole symmetry is absent in Hamiltonian (1)which is explicitly seen by applying the particle-hole trans-formation, n i = − m i , where m i denotes the occupation of ahole at i -th site. This yields the Hamiltonian for the holes, H = ∞ ∑ i = − ∞ ∞ ∑ r = R + V ( r ) m i m i + r + ∞ ∑ i = − ∞ R ∑ r = W ( r ) m i m i + r − µ (cid:48) ∞ ∑ i = − ∞ m i + C , (2)where µ (cid:48) = ∑ ∞ r = R + V ( r ) + ∑ Rr = W ( r ) and C = ∑ ∞ i = − ∞ ∑ ∞ r = R + V ( r ) + ∑ ∞ i = − ∞ ∑ Rr = W ( r ) . The extra µ (cid:48) term, which is controlled by interactions V ( r ) and W ( r ) , istypically nonzero. In this case, the Hamiltonian of the hole isstructurally different from that of the particle. III. METHODS
To investigate the ground state of Hamiltonian (1), wewill use both analytical and numerical tools. The analyticalmethod is based on that by Bak and Bruinsma . It was origi-nally used to deal with repulsive and convex interactions andwe will adapt it to our system. The analytic treatment is ac-companied by “brute-force” numerical calculations to find theground state of (1). A. Analytical method
1. Stability regions of monomers
When studying the Ising model with convex interactions,Bak and Bruinsma showed that for any rational filing frac-tion f = qp ( p and q are nonnegative integers) of the particlesto the lattice sites, there will be a finite range, i.e., a stabil-ity region of chemical potential (with lower and upper bound µ − and µ + , respectively) such that the most homogeneousconfiguration with this filling fraction is the ground state con-figuration.The stability regions are determined by the following equa-tions (the derivation is for convenience given in Appendix A), µ − = ∞ ∑ n = , α n (cid:54) = [( r n + ) V ( r n ) − r n V ( r n + )]+ ∞ ∑ n = , α n = [( r n + ) V ( r n ) − r n V ( r n + )] (3)and µ + = ∞ ∑ n = , α n (cid:54) = [( r n + ) V ( r n ) − r n V ( r n + )]+ ∞ ∑ n = , α n = [( − r n + ) V ( r n ) + r n V ( r n − )] (4)where r n and α n are related to p and q through the relation np = r n q + α n with 0 ≤ α n < q . From these equations, weobtain the “width” of the stability region, ∆ µ = µ + − µ − = ∞ ∑ n = , α n = [ r n V ( r n − ) − r n V ( r n ) + r n V ( r n + )] . (5)
2. Effective interaction between two n-mer particles and holes
In our case particles tend to form clusters due to the short-range attraction. In general, if the first R nearest-neighbourinteractions are attractive, then R + ( i , i + , · · · , i + R ) . We will refer to such n-particle(hole) cluster as n-mer particle (hole). ! ! ! ! !! ! n-mer particle ! r .. n-mer hole .. Figure 2. Two n-mer particles (holes) separated by distance r willinteract according to the 2n binary interactions of their constituentparticles (holes), resulting in the interaction matrix given by Eq. (6). The method by Bak and Bruinsma is extended to capturethis case by treating an n-mer as an effective “monomer”. Tothis end one needs to know the effective chemical potentialfor a n-mer and the interaction between two n-mers. (Notethat, for an n-mer with filling fraction q / p , the correspondingfilling fraction of the actual monomers is nq / p .) The interac-tion between two n-mer particles (holes) separated by r lattice sites, as shown in Fig. 2, can be conveniently described by amatrix˜ V = V ( r ) V ( r + ) · · · V ( r + n − ) V ( r − ) V ( r ) · · · V ( r + n − ) ... ... . . . ... V ( r − n + ) V ( r − n + ) · · · V ( r ) , (6)where matrix element ˜ V i j describes the interaction betweenthe i -th particle of the first n -mer and the j -th particle of thesecond n -mer. The effective interaction between two n-mersis then given by ˜ V eff ( r ) = ∑ i j ˜ V i j . Under the condition thatthere is no overlap of two n-mers in the most homogeneousconfiguration, we can use the method by Bak and Bruinsma todescribe the n-mers, when we replace the interaction of Eqs.(3), (4), and (5) by ˜ V eff ( r ) . Furthermore, we need to replacethe monomer chemical potential by that of the n-mer particleor hole, which will be discussed in the following.
3. Effective chemical potential of n-mer particle and hole
In case of the n-mer particle, we find that n particles willcluster together to lower their energy when the range of theattractive interactions is R = n −
1. According to Hamilto-nian (1), the interaction energy within the n-mer particle thenserves as effective chemical potential given by µ np = W ( n − ) + W ( n − ) + · · · + ( n − ) W ( ) . (7)Note, that if W ( i ) = W for all i , then µ pn = n ( n − ) W / µ ( R ) mh = − E ( R ) mh = − R ∑ j = ( m − j ) W ( j ) − m − ∑ k = R + ( m − k ) V ( k )+ m ∞ ∑ r = R + V ( r ) + m R ∑ r = W ( r ) . (8)When m ≥ R + m depends on both R and the long-range repulsive tail. This is a manifestation of the particle-hole symmetry breaking, i.e., the size of the hole cluster is notnecessarily the same as the particle cluster which would give m = R +
1. In the following, we list explicitly the effectivechemical potentials of particle and hole clusters of differentsizes for R = R =
2, which are relevant for our discus-sions below. • R =
1: the chemical potentials of particle and holedimers, and hole trimers are given by, µ p = W ( ) (9) µ ( ) h = ∞ ∑ r = V ( r ) + W ( ) (10) µ ( ) h = ∞ ∑ r = V ( r ) − V ( ) + W ( ) (11) • R =
2: the chemical potentials of particle and holetrimers, and hole tetramer are given by, µ p = W ( ) + W ( ) (12) µ ( ) h = ∞ ∑ r = V ( r ) + W ( ) + W ( ) (13) µ ( ) h = ∞ ∑ r = V ( r ) − V ( ) + W ( ) + W ( ) (14)In order to obtain the stability regions of n-mer particles andholes [for Hamiltonians (1) and (2)], we can now use Eqs. (3-5) with the effective interaction ˜ V eff ( r ) , the effective chemicalpotentials µ np and µ ( R ) nh as well as the true monomer fillingfraction nq / p (associated with a n-mer particle) or hole fillingfraction q / p .The above effective theory for n-mer particles and holesonly works when the staircase contains no mixtures of n-mersof different kinds. The aim of this work is to understand whenthe staircase can be described by a union of two pure sub-staircases in the particle and hole sectors, respectively. B. Numerical method
The filling fraction associated with the ground state con-figuration of Hamiltonian (1) as functions of the attractive in-teraction W ( r ) ( r = , , · · · , R ) can be calculated by a bruteforce method. In this numerical method , we check all pos-sible periodic configurations of an infinite chain with period p up to a certain limit ( p =
23 in this study, due to the limitationof computational resources). The ground state configurationis determined by the one that has the lowest energy density(energy of a single period divided by the length of the period-icity). This captures the coarse structure of the staircase as thephases with large p usually have very small stability regions. IV. PARTICLE-HOLE SYMMETRY IN TRADITIONALDEVIL’S STAIRCASES
To provide some context, we review here briefly the resultsby Bak and Bruinsma , which are based on an Ising modelwith long-range interactions, H = ∞ ∑ i = − ∞ ∞ ∑ r = V ( r ) n i n i + r − µ ∞ ∑ i = − ∞ n i , (15)where n i = , i is empty or occupied by a par-ticle, respectively. Here, V ( r ) describes a long-range repul-sive interaction between two particles separated by r sites and µ is the chemical potential for the particle. For any rationalfilling fraction f of the particles, the ground state configura-tion will assume a distribution in space as uniform as possi-ble if the infinite-range interaction V ( r ) is strictly convex .In this case the ground state configuration is independent ofthe actual details of the interaction potential and features so-called generalized Wigner crystals. The filling fractions f of the ground state configurations form a complete devil’s stair-case as a function of the chemical potential µ .For power-law interactions V ( r ) = / r α , the Hamiltonian(15) is invariant (apart from an irrelevant constant term) underthe particle-hole transformation, n i = − m i and the corre-sponding hole Hamiltonian reads H = ∞ ∑ i = − ∞ ∞ ∑ r = V ( r ) m i m i + r − µ (cid:48) ∞ ∑ i = − ∞ m i + C , (16)where µ (cid:48) = ∑ ∞ r = V ( r ) − µ and C = ∑ ∞ i = − ∞ ∑ ∞ r = V ( r ) − µ ∑ ∞ i = − ∞ . One can find the transition point to the state withoutholes (or a state where the lattice is fully occupied by parti-cles) by setting µ (cid:48) =
0, i.e., µ = ∑ ∞ r = V ( r ) . For power-lawinteractions, the corresponding critical chemical potential µ c is determined by µ c = ζ ( α ) with ζ ( α ) = ∑ ∞ n = / n α beingthe Riemann zeta function. α µ f Figure 3. Ground state filling fraction f of Hamiltonian (15) as func-tions of the chemical potential µ and the power α of the power-lawinteraction potential. The large plateau at half-filling, f = /
2, cor-responds to the configuration of 101010 ··· . The red solid lines areanalytical results obtained from Eqs. (3) and (4) at α = , , ,
5, and6. The black line is the critical chemical potential µ c ( α ) = ζ ( α ) atwhich the ground states of Hamiltonian (15) turn into the fully-filledparticle states with f = We numerically obtain the staircase structure by varyingboth the power α of the repulsive power-law interaction andthe chemical potential. The result is shown in Fig. 3, whichhas a “devil’s terrace” structure. The big plateau at filling frac-tion f = / · · · . Itswidth increases as α increases since the large commensuratephases (with large p ) occupy negligible parameter space of µ due to the fast decaying property of 1 / r α at large α . At small α , the large commensurate phases play important roles andoccupy a large portion of the parameter space of µ .When f ≤ /
2, particles in the lattice are all separated fromeach other by empty sites and there is no cluster behaviour ofthe particles. However, when f > /
2, particles will clustertogether to form different kinds of n-mers ( n ≥ f ≥ / f ≤ / f = / µ direction . When the chemical potential of the particlesis zero, the ground state configuration would have no parti-cles in it and similarly, if the chemical potential of the holesis zero, one would have no holes in the lattice, i.e., the tran-sition point to a fully-filled particle state with f = µ c ( α ) = ζ ( α ) . The analytical result (marked by the blackcurve in Fig. 3) agrees with the numerical calculation. In thesame figure, we also present the analytical result from Eqs.(3) and (4) at α = , , ,
5, and 6 on top of the numerical datain red lines, which agree with each other very well.
V. TWO-BODY INTERACTION DRIVEN STAIRCASESWITHOUT PARTICLE-HOLE SYMMETRY
In this section we turn to the discussion of devil’s staircasescorresponding to the ground state of Hamiltonian (1). We willmainly focus on two aspects of the problem. First, we wouldlike to understand how the range of the attraction R changesthe structure of the devil’s staircases. Second, we investigatethe effect of the power α of the interaction potential V ( r ) =( R + ) α / r α . For simplicity, we will consider the case wherethe short-range attraction W ( i ) ( i = , · · · , R ) are the same andequal to W .In particular, we find that the feature of the staircase non-trivially depends on α . For certain α the staircase can bedescribed by a union of two pure sub-staircases, i.e., a pure ( R + ) -mer particle sub-staircase and a pure ( R + ) -mer or ( R + ) -mer hole sub-staircase. For other values of α , thestaircases consist of more than two kinds of basic buildingblocks.We use an “complexity parameter” P to describe this ef-fect. When the value of α is such that the whole staircase canbe described by two pure sub-staircases (a pure n -mer particlesub-staircase and a pure m -mer hole sub-staircase), i.e., a sin-gle pair of integers ( n , m ) is sufficient to describe the emergentstaircase, then P ( α ) =
1, otherwise P ( α ) = R = R =
2. A general description of R ≥ R = , ,
5, and 6 will also be presented. A. R = When the range of the attraction is R =
1, only the nearest-neighbour interaction is attractive and the interactions fromnext nearest neighbour onwards ( r = , · · · ) are repulsive andfollow the form V ( r ) = α / r α . The case of α =
6, corre-sponding to the van der Waals interaction, has been studiedin detail by us in a recent work based on a concrete systemof Rydberg atoms. We found that the staircase structure hasa dimer particle sub-staircase with f ≤ / f ≥ /
2, as shown in Fig. 4(a). The brokenparticle-hole symmetry in this case does not manifest in the α W f α ( a ) ( b ) narrow region ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ
Figure 4. (a) Ground state filling fraction f of Hamiltonian (1) at R = W and the power α of the long-range tail of the potential. (b) The complexity parameter P ( α ) . Only when P ( α ) =
1, the staircase is described by a unionof two pure sub-staircases. Dots are numerical data and the line isused to guide the eye. different sizes of the clusters in the two sectors, but rather inthe asymmetric shape of the staircase along the W -direction inthe vicinity of f = /
2. There is no symmetry around f = / α . For example, the plateau correspond-ing to f = / f = /
5) becomes very narrow (wide) around α =
4, as can be seen in Fig. 4. Such feature is not found inthe Ising model studied by Bak and Bruinsma.Our next goal is to understand the dependence of the stair-cases on the interaction exponent α . The complexity parame-ter P ( α ) shows two regions where the staircases can be de-scribed by two pure sub-staircases. One region is α > . α = α > . f ≤ / f ≥ / α = f = / · · · . Decreasing α in this regimenarrows this central plateau (at f = /
2) up to α ≈ . α =
3, we find the staircase consists of a dimer par-ticle sub-staircase in the sector of f ≤ / f ≥ / α = α in this regime the central plateau at f = / ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ - 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- - - - ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ - - - ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ - - - f numerics dimer particle theory f =
12 : ! ! α = dimer hole theory numerics dimer particle theory α = dimer hole theory W f =
25 : ! ! numerics dimer particle theory α = trimer hole theory W numerics dimer particle theory α = f ( a ) ( b ) ( c ) ( d ) trimer hole theory dimer hole theory trimer hole theory tetramer hole theory Figure 5. Comparison between analytically and numerically calcu-lated staircases at α = , , , α = , P =
1. The case of α = where we considered α = α =
3, we find a dimer particle sub-staircase and a trimer holesub-staircase which meet at f = / ··· . At α = ,
4, the staircases at the hole sectorcontain n-mer holes of different kinds. can be described by two pure sub-staircases.When the hole sector can be described by a single kind ofcluster hole a simple analytic calculation of the phase bound-ary is possible. For example, in the regime of α > .
2, the holesector can be solely described by dimer holes and one can findthe exact transition point to the fully filled f = W c = − ∞ ∑ r = V ( r ) = − [ ζ ( α ) − ] α . (17)Similarly, for the regime around α =
3, the hole sector can bedescribed by trimer holes. From Eq. (11) we obtain then thetransition point, W c = − ∑ ∞ r = V ( r ) − V ( ) = − [ ζ ( α ) − ] α − . (18)For other values of α , the staircase has more complicatedstructures. Examples with α = α = W , W = − . → · · · f = , (19) W = − . → · · · f = , (20)which clearly shows that the staircase consists of dimer parti-cles, trimer particles and dimer holes, and trimer holes. B. R = We will now consider the case R =
2, i.e. where the at-tractive range of the interaction potential spans two sites. Forsimplicity, we will focus on the case where W ( ) = W ( ) = W <
0. The long-range repulsive interaction tail now becomes V ( r ) = α / r α ( r = , · · · ). For such interactions, three parti-cles tend to cluster together on neighbouring lattice sites, toform a trimer particle serving as the basic building block ofthe staircase at the particle sector. The ground state fillingfraction f of Hamiltonian (1) at R = P ( α ) in Fig. 6(b). When α > .
8, the staircase can be de-scribed by a union of trimer particle sub-staircase in the par-ticle sector and a trimer hole sub-staircase in the hole sector,where the two sub-staircases meet at f = / · · · . Around α = .
2, we obtain a trimerparticle sub-staircase in the particle sector and a tetramer holesub-staircase in the hole sector, where the two sub-staircasesmeet at f = / · · · .Apart from these two regimes, the staircases cannot be de-scribed as a union of two pure sub-staircases. The detailedresults of the staircases at α = , . ,
5, and 6 are presented inFig. 7.We can also find the exact transition points to the unit filling f = W c = − ∞ ∑ r = V ( r ) = − [ ζ ( α ) − − α ] α , (21)and W c = − ∑ ∞ r = V ( r ) − V ( ) = − [ ζ ( α ) − − α ] α . (22) C. R > When R >
2, the qualitative feature of the physics is largelysimilar to the case R = R =
2. The sub-staircase in α W f α ( a ) ( b ) ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ
Figure 6. (a) Ground state filling fraction f for R = W and the power α . (b) Thecomplexity parameter P ( α ) . See also Fig. 4. ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ - - - ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ - - - - - - ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ - - - - - ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ - - - - - f numerics trimer particle theory f =
12 : ! ! α = trimer hole theory numerics trimer particle theory α = trimer hole theory W f =
37 : ! ! numerics trimer particle theory α = tetramer hole theory W numerics trimer particle theory α = f ( a ) ( b ) ( c ) ( d ) tetramer hole theory trimer hole theory tetramer hole theory pentamer hole theory Figure 7. Analytically calculated staircases compared to the numer-ically obtained staircases at α = , , .
2, and 3 in panels (a-d) re-spectively. The other parameters are R = W ( ) = W ( ) = W .See also Fig. 5. the particle sector is built up from ( R + ) -mer particles. Inthe hole sector, there are two regimes where the staircase canbe described by a union of two pure sub-staircases in boththe particle and the hole sectors. In one region, we havea ( R + ) -mer particle sub-staircase and a ( R + ) -mer holesub-staircase, where the two sub-staircases meet at f = / · · · · · · · · · (with both R + R + α , wherethe staircase is made of a ( R + ) -mer particle sub-staircaseand a ( R + ) -mer hole sub-staircase, where the two sub-staircases meet at f = ( R + ) / ( R + ) with a configurationof 1 · · · · · · · · · (i.e., R + R + f and order parameter P ( α ) of Hamiltonian (1) with R = , ,
5, and 6 as shown inFig. 8. Moreover, the critical transition points of the ( R + ) -mer hole sub-staircase and the ( R + ) -mer hole sub-staircaseto the full-filled f = VI. N ( ≥ ) -BODY INTERACTION DRIVEN STAIRCASE The above results indicate that there should be R -mer stair-case when the range of the attractive interactions is R − R -mer staircase can beinduced by R -body interactions directly? To answer thisquestion, we investigate the following model Hamiltonian,which contains a two-body long-range repulsive interactiondescribed by V ( r ) , and a N ≥ U N , H = ∞ ∑ i = − ∞ ∞ ∑ r = V ( r ) n i n i + r − U N ∞ ∑ i = − ∞ n i n i + · · · n i + N − . (23)Numerical calculations of the above Hamiltonian show thatin the ground state, there is always a direct transition from theempty state of · · · · · · to the fully filled state of · · · · · · .The energies of the two states are 0 and ∑ r ≥ V ( r ) − U N .Hence the transition happens when ( ∑ r ≥ V ( r ) − U N ) <
0, i.e., U N > ∑ r ≥ V ( r ) = ζ ( α ) .In the following, we provide a simple explanation of thisresult based on energy arguments. The energy of a N -mer is E N = ( N − ) V ( ) + ( N − ) V ( ) + · · · V ( N − ) − U N . One can readily show that the energy of two separate N -mersare 2 E N + E int with E int the interaction energy of the two N -mers, which is larger than the energy of a ( N + ) -mer,2 E N + E int − E N + =( N − ) V ( ) + ( N − ) V ( ) + · · · + ( − ) V ( N ) + E int > , when N ≥
3. This result excludes the possibility of havingexotic staircases driven solely by N-body attraction when N ≥ α α α α W W W W f f f f α α α α R = R = R = R = ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ
ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ
ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ
ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ
Figure 8. (a) Ground state filling fraction f of Hamiltonian (1) for R = , , , W ( i ) = W ( i = , , ··· , R ) and the power α of the long-range power law repulsion V ( r ) = ( R + ) α / r α with ( r = R + , ··· ). Shown in (b) is P ( α ) similar to Figs. 4(b), 6(b) at R = R = VII. DISCUSSION AND OUTLOOK
We have explored a new class of devil’s staircases that ex-hibit a broken particle-hole symmetry. The symmetry break-ing is purely induced by the interplay between short-rangeattraction and long-range repulsion. When the staircase canbe described by a union of two pure sub-staircases in boththe particle and hole sectors, the value of the critical attrac-tive strength W c and the “width” of the stability region canbe found analytically. These confirm that the resulting stair-case is complete . However, when the staircase containsmixtures of n-mers of different kinds, it is an open questionwhether an analytic understanding of the staircase structurecan be obtained. Another possible way to understand theproblem-which may lead to an answer-is to consider periodicconfigurations as consisting of segments of different phasesseparated by interfaces , where the nature of interface inter-actions determine the detailed structure of the phase diagram.One interesting question is why for attractive interactions withrange R , there can only be ( R + ) -mer hole and ( R + ) -merhole sub-staircases but not ( R + ) hole sub-staircase in thehole sector for power law repulsion. The answer might be thatwithout particle-hole symmetry, the staircase structure willdepend on the specific form of the repulsive tail itself. Thisalso suggests an interesting way to manipulate the hole partof the staircase by controlling the form of the repulsive tail.We expect that for example with exponential interactions for the repulsive part, the hole part may indeed display a dif-ferent structure. Furthermore, it is known that some 2D latticegas and adsorption models can have devil’s staircase ofphase transitions in the ground state. So it would be interest-ing to extend the current work to 2D by coupling 1D chainstransversely.A further interesting problem for future studies is the ex-ploration of the role of thermal and quantum fluctuations. For example, for the ANNNI model, it is the thermal fluctuationsthat stabilize the staircase. Quantum fluctuations, however,can destroy the staircase at zero temperature , i.e., thestability regions shrink and at most a finite number of com-mensurate phases survives. So it would be interesting to un-derstand how quantum fluctuations will melt the emerginghybrid staircases, such as the dimer-particle and trimer-holestaircase [see Fig. 5(c)], studied in this paper. One might beable to address these questions experimentally for examplewith a recently established quantum simulator platform basedon Rydberg atoms . The preparation of the ground stateof our model on a Rydberg atom quantum simulator requiresan adiabatic sweep protocol. A detailed discussion of this pro-cedure can be found in the recent review . VIII. ACKNOWLEDGMENTS
We thank Emanuele Levi and Jiˇr´ı Min´aˇr for their contri-bution at the early stage of this work. The research leadingto these results has received funding from the European Re-search Council under the European Union’s Seventh Frame-work Programme (FP/2007-2013) / ERC Grant AgreementNo. 335266 (ESCQUMA), the EU-FET Grant No. 512862(HAIRS), the H2020-FETPROACT-2014 Grant No. 640378(RYSQ), EPSRC Grant No. EP/M014266/1, and the UKIERI-UGC Thematic Partnership No. IND/CONT/G/16-17/73. I.L.gratefully acknowledges funding through the Royal SocietyWolfson Research Merit Award.
Appendix A: Stability regions
In this part of the appendix, we give a brief derivation ofthe stability regions of Eqs. (3) and (4) used in the main text f f f f µ µ µ µ α α α α R = R = R = R = Figure 9. Ground state filling fraction f of Hamiltonian (2) for R = , , , µ and the power α of thelong-range power-law repulsion V ( r ) = R α / r α ( r = R + , ··· ), where the short-range attractions W ( i ) ( i = , , ··· , R ) have been set to zero.This chemical potential µ driven staircase has the particle-hole symmetry, so the hole sector is trivially related to the particle sector, which isvery different from our two-body attraction W ( i ) driven staircase, where the hole sector contains very rich physics as studied in the main text. (see Refs. for the original literature). The energy of theground state configuration can be written as E = E + E + · · · + E n + · · · + E ∞ + E µ , where E , , ··· , ∞ is the interaction energy with nearest-neighbour and next-neast-neighbour and so on and E µ is theenergy with the chemical potential term. For any filling frac-tion of f = qp , the n -nearest-neighbour interaction energy ofthe most homogeneous configuration requires np = r n q + α n ,where 0 ≤ α n < q . To make this relation clear, we rewriteit as np = r n x + ( r n + )( q − x ) by introducing a new integer x = ( r n + ) q − np . It means that there are x particles sepa-rated from each other by r n lattice sites while q − x particlesseparated from each other by r n + E n with L periods (a very large number) is then E n = [ xV ( r n ) + ( q − x ) V ( r n + )] L . Now if we have one more particle in the above configu-ration, the interaction will reorganise the particle distributionsuch that npL = r n y +( qL + − y )( r n + ) , i.e., compared withthe above case, we have qL + y = ( qL + )( r n + ) − npL , so the energy of E + n with onemore particle is E + n = yV ( r n ) + ( qL + − y ) V ( r n + ) . In the same way for one less particle in the configuration, npL = r n z + ( qL − − z )( r n + ) and we get z = ( qL − )( r n + ) − npL and E − n = zV ( r n ) + ( qL + − z ) V ( r n + ) . However, if α n =
0, i.e, x = q , we have slightly differentsituations, E n = [ qV ( r n )] LE + n = yV ( r n ) + ( qL + − y ) V ( r n − ) E − n = zV ( r n ) + ( qL − − z ) V ( r n + ) r n y + ( qL + − y )( r n − ) = npL ⇒ y = npL − ( qL + )( r n − ) r n z + ( qL − − z )( r n + ) = npL ⇒ z = ( qL − )( r n + ) − npL In summary, we get µ + = ∞ ∑ n = , α n (cid:54) = ( E + n − E n ) = ∞ ∑ n = , α n (cid:54) = [( r n + ) V ( r n ) − r n V ( r n + )]+ ∞ ∑ n , α n = [( − r n + ) V ( r n ) + r n V ( r n − )] and µ − = ∞ ∑ n = , α n (cid:54) = ( E n − E − n ) = ∞ ∑ n = , α n (cid:54) = [( r n + ) V ( r n ) − r n V ( r n + )]+ ∞ ∑ n , α n = [( r n + ) V ( r n ) − r n V ( r n + )] which are used in the main text. Appendix B: Polymer staircases with particle-hole symmetry
In the main text, we study the devil’s staircase physics de-scribed by Hamiltonian (1) where the short-range two-bodyattraction is the main driving force for the emergence ofdevil’s staircase behaviour. The emergent staircases couldbe termed as polymer staircases, which consist of some ba-sic clusters with different sizes. Our main motivation of themain text is to study the effect of broken particle-hole sym-metry on the staircase structure where the polymer behaviourof the staircases is a byproduct. However, we note that in theliterature there have been studies where the motivation was tolook for mechanism to form a polymer staircase, whereas theparticle-hole symmetry is not the focus. Actually, one can stillpreserve the particle-hole symmetry of the polymer staircasesby using the single-body chemical potential as the drivenmechanism, e.g., the papers by Jedrzejewski and Miekisz fit to this category.Here we would like to briefly discuss the connection of ourwork with those of Jedrzejewski and Miekisz . These au-thors have proven rigorously the existence of the dimer stair-cases in 1D lattice gas models with certain nonconvex long-range interactions, where the particle density versus the chem-ical potential, ρ ( µ ) , exhibits the complete devil’s staircase0structure. The authors also speculated that for interactionswith values near zero for distances up to R and strictly con-vex from distance R + R + using the numerical tool described in ourmain text, where we set W ( ) = · · · W ( R ) =
0, and from dis-tance R + ( R + ) α / r α . From the numerical results, we verify that given R , the staircase is a ( R + ) -mer staircase with exact particle- hole symmetry (see Fig. 9, where the mesa is symmetric withrespect to the f = / ( · · · · · · ) · · · with both R + R + P. Bak,
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