Phase diagrams of bosonic A B n chains
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Phase diagrams of bosonic AB n chains G. J. Cruz , R. Franco , and J. Silva-Valencia Departamento de Ciencias B´asicas, Universidad Santo Tomas, Bogot´a, Colombia. Departamento de F´ısica, Universidad Nacional de Colombia, Bogot´a, Colombia.Received: date / Revised version: date
Abstract.
The AB N − chain is a system that consists of repeating a unit cell with N sites where betweenthe A and B sites there is an energy difference of λ . We considered bosons in these special lattices andtook into account the kinetic energy, the local two-body interaction, and the inhomogenous local energyin the Hamiltonian. We found the charge density wave (CDW) and superfluid and Mott insulator phases,and constructed the phase diagram for N = 2 and 3 at the thermodynamic limit. The system exhibitedinsulator phases for densities ρ = α/N , with α being an integer. We obtained that superfluid regionsseparate the insulator phases for densities larger than one. For any N value, we found that for integerdensities ρ , the system exhibits ρ + 1 insulator phases, a Mott insulator phase, and ρ CDW phases. Fornon-integer densities larger than one, several CDW phases appear.
PACS.
The study of the ground state of bosonic systems is aninteresting area within the study of cold atoms due to thecapability of emulating them in optical lattices, with theadvantage of having absolute control over the parameters(kinetic energy, interactions, and density) and access tovarious dimensions [1,2].Experimental progress on this topic has allowed theprediction of the superfluid-Mott insulator transition inultracold bosonic atoms [3] and the observation of quan-tum phase transitions [4] in an optical lattice, which hasopened the way for potential studies that have revealedthe fundamental physics of these systems.Recently, the implementation of optical lattices hasmotivated the exploration of bosonic systems in super-lattices whose arrangement is characterized by a periodicpotential [5,6], for instance an energy difference of λ inthe unit cell between sites called A and N − B in theunit cell (notation AB N − ). Bidimensional systems with AB configurations have been created confining Rb [7]and K [8] atoms in optical lattices. This experimentalprogress allows us to believe that experimental study ofthe one-dimensional AB N − chains will be carried out inthe coming years.In 2004, Buonsante and Vezzani [9] elaborated an an-alytical description of the physics of cold bosonic atomstrapped in superlattices. Within the study of the finite-temperature phase diagram, they included the results forzero temperature, and insulator domains for fractionaldensities were demonstrated. The phase diagrams due tothe interplay between the on-site repulsive interaction, su- Fig. 1. (color online). Schematic representation of the super-lattices considered in this paper. The chain AB (a) is formedby the superposition of two waves, where the first has doublethe frequency of the other, while the chain AB in (b) is formedby the superposition of two waves where the first has triple thefrequency of the other. We consider that the difference of thehopping parameter between neighboring states is very small. perlattice potential strength, and filling were found byRousseau et al. [10].Later, Dhar and colleagues [11,12,13] studied the quan-tum phases that emerge when the AB chain is considered(see Fig. 1(a)). Such studies proved that a new insulatorphase results at density ρ = 1 /
2, and the Mott insulatorphase corresponding to ρ = 1 undergoes a phase transi-tion to another insulator when the parameter λ is near G. J. Cruz et al.: Phase diagrams of bosonic AB n chains = (a) (b) µ /t 1/L ρ =1 λ /t=20 ρ =3/2 /t=20 λ µ p µ h p µµ h Fig. 2.
Graphs of chemical potential versus 1 /L for λ/t = 20in a) ρ = 1 and b) ρ = 3 /
2. The points are DMRG results, andthe lines represent the regression to the thermodynamic limit. the strength of the interaction between the particles ( U ),leading to a different boson arrangement.Although the results of these new phases indicate theconsequences produced by the superlattice on a bosonchain, we believe that there are new aspects to be dis-covered. For instance, we can ask: is it possible that theinsulator phases at ρ > λ increases?What happens in the phase diagram if we increase thenumber of sites in the unit cell? To answer these ques-tions, we used the density matrix renormalization groupmethod [14] and the von Neumann entropy [15] to studythe quantum phases of the model. We found that for frac-tional densities lower than one there is a unique CDWphase for any value of λ . For densities larger than one,we obtained a finite number of insulating phases when λ increases, two CDW phases for fractional densities, and ρ + 1 for integer ones. The von Neumann entropy showsthat the insulator phases are separated by superfluid re-gions.This paper is organized in the following way: In sec-tion 2, we explain the Hamiltonian for AB N − chains. Thethermodynamic limit phase diagrams and the von Neu-mann entropy results for AB and AB chains are shownin sections 3 and 4. Finally, the conclusions are presentedin section 5. The ground state of ultracold bosonic atoms in homoge-neous systems is described by the Bose-Hubbard model.When a system of bosons in an inhomogeneous lattice isconsidered, the Hamiltonian is given by: H = − t X (ˆ a † i ˆ a j + H.c. ) + U X i ˆ n i (ˆ n i − Fig. 3. (color online). Phases and transitions in the AB chain. + X i λ i ˆ n i − µ X i ˆ n i . (1) t being the hopping parameter, < i, j > denotes a pairof nearest-neighbor sites i , and j , µ is the chemical po-tential, a † i ( a i ) creates (annihilates) a boson at site i . U represents the local interaction in the second term of theHamiltonian, where ˆ n i = a † i a i is the number operator,and λ i denotes the shift in the energy levels of the sites ineach unit cell. We set our energy scale taking t = 1 andthe interaction parameter U/t = 10.Particle chains in a superlattice are denoted as AB N − ,where the site A has a difference of energy of λ from the N − B per unit cell. In the present paper,we consider two types of superlattices. The first superlat-tice has a potential with periodicity 2 ( AB chains), mean-ing two sites per unit cell with a potential difference λ ,schematically represented in Fig. 1(a). The second one hasa periodicity of 3 and is shown in Fig. 1(b) ( AB chains).As is well known, in a homogeneous environment theone-dimensional bosonic system exhibits a phase transi-tion between a Mott insulator phase, characterized by in-teger filling, and the superfluid phase, which is compress-ible [16,17,18]. In a superlattice type AB , the behavior ofthe system in the ground state exhibits additional phasesfor half-filling and density equal to one, where insulatingphases due to the superlattice potential are exhibited [11].The current investigation involved calculating the en-ergies E ( N, L ) for lattices with different lengths L and N , N + 1 and N − µ p ) and decrease ( µ h ) thenumber of particles to one, where the general expressionsare: µ p = E ( N + 1 , L ) − E ( N, L ) (2)and µ h = E ( N, L ) − E ( N − , L ) . (3)To obtain the energies, we used the density matrix renor-malization group method for lattices from 24 to 84 sites,obtaining an error of around 10 − with open bound-ary conditions. Then we extrapolated our results at thethermodynamic limit and repeated the process for a wide . J. Cruz et al.: Phase diagrams of bosonic AB n chains 3 λ =0 λ =6 λ =10 λ =16 λ =20 λ =24 ρ µ (a) (b) (c) (d)(e) (f)/t /t/t /t/t /t /t Fig. 4. (color online). Density profile ρ versus chemical poten-tial µ for bosons in superlattice type AB and various values of λ , with U/t = 10. range of values of the parameter λ . For example, all in-sulator phases are characterized by a gap. The analysisof this quantity at the limit L → ∞ gives us informationabout the phase in which the system exists. The gap forany boson system is given by: ∆ = µ p − µ h . (4)If ∆ = 0, the system is in an insulator phase; if it is not,the system is in a superfluid phase. This is shown in Fig.2 for the AB chain for two cases: in Fig. 2(a), we considera density ρ = 1 and λ/t = 20 and we observe that the sys-tem has a large gap at the thermodynamic limit, whichindicates that the system is in a CDW phase, and specifi-cally the charge distribution generated by the superlatticestructure is { , , , , , ... } , as reported before (see Fig.3). If we maintain the λ parameter the same and changethe global density to ρ = 3 /
2, we expect that the systemwill be in a CDW state due to the superlattice, accordingto the results of Buonsante and Vezzani[9], with a den-sity profile { , , , , , ... } (see Fig. 3). However, in Fig.2(b), we observe a gap equal to zero at the thermodynamiclimit, which indicates that the system is in a superfluidstate. In conclusion, for λ/t = 2 U , the ground state isCDW for ρ = 1 and is superfluid for ρ = 3 / AB phase diagram Dhar et al. [11] have shown that considering bosons in asuperlattice with periodicity equal to two, a CDW phasewith one particle per unit cell appears for ρ = 1 / λ , while for ρ = 1, theyobtained a Mott insulator phase and a CDW phase, be-cause when the hopping and the local interaction parame-ters are fixed, for values of λ < U the ground state will bea Mott insulator with one particle per site. However, forvalues of λ ≈ U , the local repulsion can be compensatedby the superlattice term, and the quantum fluctuations
20 30 40
20 30 40 (cid:7) (cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13)
20 30 40 (cid:14) < n ( i ) > i ρ =3/2 λ =10 ρ =2 λ =24 ρ =3/2 λ =2 (a) ρ =2 λ =24 (b)(d)(c) /t /t/t/t Fig. 5. (color online). On-site number density plotted againstlattice site index with L = 50 sites at density ρ = 3 / ρ = 2. Open boundary conditions were used. The lines arevisual guides. increase, delocalizing the particles, which leads to a su-perfluid state. As λ is increased further, the bosons preferto be in the wells, and the ground state is a CDW (seeFig. 3). They showed that the superfluid region betweenthe Mott insulator and the CDW phases is in the range { . , . } . In this section, we extend these results by con-sidering densities larger than ρ = 1 in the AB chain.In Fig. 4, we show the density ρ versus the chemicalpotential µ/t for some specific values of λ/t , considering U/t = 10, because the previous phase diagram was con-structed for this value. The case of a homogeneous lattice,with λ/t = 0, is displayed in Fig. 4(a). In this figure, itis possible to clearly observe plateaus at integer densitiesof ρ = 1 and ρ = 2. The width of the plateaus indicatesthe size of the energy gap at the thermodynamic limit foreach density. Between these values of µ/t , a Mott insula-tor phase is found. For different values of µ/t , the systemis compressible and is in a superfluid phase.When an inhomogeneous lattice is considered, for ex-ample with λ = 6, shown in Fig. 4(b), the number ofplateaus increases. It is crucial to emphasize that the plateausfor ρ = 1 and ρ = 2 are smaller than for λ = 0, where theplateau for ρ = 1 is larger than the plateau for ρ = 2.Also, the other plateaus for half-integer ρ are shown aspredicted. These plateaus are caused by the superlatticestructure type that characterizes the boson chain, whilethe plateaus for integer densities are caused by the inter-action between the particles of each site. It is important tohighlight the fact that between each plateau at a specificvalue of λ/t , we always obtain a superfluid region.In Fig. 4(c), for λ/t = 10, plateaus for densities ρ =1 / ρ = 3 / ρ = 1 and ρ = 2the width of the plateaus is very small, and around λ ≈ U we found that the Mott insulator phase disappears forall integer densities and a superfluid region anticipates anew insulator phase. On the basis of Fig. 3 for ρ = 2,we can explain why the Mott insulator phase disappears.A Mott insulator phase means ρ particles per site, which G. J. Cruz et al.: Phase diagrams of bosonic AB n chains λ /U µ /U Fig. 6. (color online). Phase diagram of one-dimensionalbosonic system for the AB chain. The points (open circles)represent the boundaries and are DMRG results. implies an energy U ρ ( ρ −
1) + ρλ for two sites. For λ ≈ U , this energy can be compensated by a particle jumpinto wells, the energy for two sites is U ( ρ − ρ + 1) +( ρ − λ , and the fluctuations lead to a superfluid statefor any density ρ . Note that the hopping term is small( U/t = 10). This contributes to the fluctuations, but isnot the most important factor. This fact should be takeninto consideration throughout this paper.For λ/t = 16, the plateaus reappear, and we get insu-lator phases at all commensurate densities ρ = 1 / , , / ρ = 2. Also, we can observe that the plateau for ρ = 1 / λ/t increases, which is shownin Fig. 4(d). In Fig. 4(e), for λ/t = 20 the plateau at ρ = 3 / λ ≈ U . From Fig. 3, wesee that the CDW phase for this density consists of twobosons in the wells and one in the barriers. Note that ifthe barrier particle jumps to the well, the energy for twosites 3 U compensates for the energy of the CDW arrange-ment U + λ , when λ ≈ U , which leads the system to asuperfluid state.Finally, for λ/t = 24, it is possible to observe thatagain we have plateaus for all the densities, and a newinsulator phase for ρ = 3 / ρ ≥
1, a new CDW insulator phase occurs forlarger values of λ/t . For integer densities, this CDW phaseis achieved by first passing through a Mott insulator phaseand a superfluid one, while for ρ = 3 /
2, we obtained twoCDW phases separated by a superfluid phase.Each of the above insulator phases that have beenmentioned is characterized by a particular charge distri-bution in the system. In Fig. 5, the lattice density profileis shown for densities ρ = 3 / ρ = 2, with some pa-rameters taken before and after the transition region. InFig. 5(a), for ρ = 3 / λ/t = 10, the distribution oftwo bosons at one site and one boson at the other site isobserved, while for λ/t = 24, in Fig. 5(b), the charge dis- λ =6 /t λ =9.1 /t λ =15 /t λ =19 /t ρ µ /t (a) (b)(c) (d) Fig. 7. (color online). Density versus the chemical potentialfor some values of λ/t , for a system of bosons in a superlatticetype AB . tribution is close to { , , , , , ... } . Clearly, the systemexperiences a change of state caused by the spatial struc-ture of the lattice. We emphasize that this possibility wasnot taken into account by Dhar et al. In Fig. 5(c) for λ/t = 2, a Mott insulator phase withtwo particles at each lattice site is shown, but as λ/t in-creases to λ/t = 24 in Fig. 5(d), the charge distributionin the system indicates that the system is organized as { , , , , , ... } .As was reported, the insulator phase at density ρ = 1 / λ/t = 0 is always present, and for ρ = 1, a phase tran-sition is located near λ/t ≈ U (see Fig. 6). Also, this figureshows that for all commensurate densities with ρ ≥
1, thesystem exhibits a superfluid region between two insula-tor regions, and the parameter λ can generate a quan-tum transition between these states due to the superlat-tice structure. Within the values of λ/t considered, we seethat there are two insulator phases for densities ρ ≥
1, andthe gap of the CDW phase for the densities ρ = 1 / ρ = 1 saturates, because for these densities, the barriersare unoccupied (see Fig. 3), and this implies that no statechange may occur. AB phase diagram Now the case of bosons in a superlattice type AB willbe studied. In this superlattice, each unit cell is formedby three sites. Two B sites differ by a potential of λ withrespect to A (see Fig. 1(b)). Note that real systems withthis structure in one or three dimensions have been createdand studied [19,20,21].Again, we consider a local repulsion interaction be-tween the bosons of U/t = 10, and the chemical potentialis determined at the thermodynamic limit.The evolution of the plot of density versus chemicalpotential as a function of the superlattice parameter isshown in Fig. 7. For λ/t = 6 (Fig. 7(a)), the number ofplateaus increases, in contrast with the superlattice type . J. Cruz et al.: Phase diagrams of bosonic AB n chains 5 AB Fig. 8. (color online). Phases and transitions in the AB chain. AB . The first two plateaus are related to the possibility ofhaving one or two bosons in the wells of each unit cell (seeFig. 8), which means CDW phases with densities ρ = 1 / ρ = 2 /
3. As in the AB case, we expected that theseplateaus would increase with λ until they saturate. Also,we obtained other plateaus at commensurate densities ofmultiples of 1 /
3, as was predicted.Fig. 7(b) shows the results for λ/t = 9 .
1, where theplateaus at ρ = 1, ρ = 4 / ρ = 2 disappear, whilethe size of the gaps for the other commensurate densitiesremains finite. The reappearance of these plateaus hap-pens for larger values of λ , as can be seen in Fig. 7(c).This means that there is a superfluid region separatingtwo insulator regions for λ around U for each density. Forinteger densities, the explanation is similar to the previouscase ( AB ); fixing the density ρ (each site in the well has ρ bosons) and a barrier strength λ , the energy of two sites(the barrier and one of the well) is U ρ ( ρ − ρλ (see Fig.8). If one boson jumps into the well, previous energy canbe compensated when λ ≈ U . Now the energy of the twosites is U ( ρ − ρ + 1) + ( ρ − λ . We can conclude that forany AB N chain, there is always a superfluid region around U separating a Mott insulator phase and a CDW one forany density ρ .Finally, in Fig. 7(d), with λ/t = 19, the calculationsshow that the gaps for ρ = 5 / ρ = 2 are very small,which indicates that for these densities a superfluid regionwill appear around 2 U . Note that in the AB chain, a su-perfluid region around 2 U appears for the commensuratedensity ρ = 3 /
2, but this does not occur for the integerdensity ρ = 2 (see Fig. 6). For λ = 2 U and ρ = 2, thecharge distribution of the ground state is three particlesin the well and one in the barrier (see Fig. 3), so the quan-tum transition happens when the particle in the barrierjumps to the well. The above happens for λ ≈ U . Beforethe transition, the energy for two sites is 3 U + λ , whichcan be compensated with the energy 6 U once the particlehas jumped into the well. The location of this transitionmarks one of our new findings due to the increase of thenumber of sites of the wells.
10 20 30 (cid:16)(cid:17)(cid:18)
20 30 (cid:19)(cid:20) (cid:21)(cid:22) (cid:23)(cid:24) (cid:25)(cid:26) (cid:27)(cid:28) (a)(c) ((cid:29)(cid:30) (b)(d) (cid:31) ! λ =2 /t λ =15 /t λ =10 /t /t λ =24 /t λ =24 λ =15 < n ( i ) > i ρ =4/3, ρ =4/3, /t ρ =5/3, ρ =5/3, ρ =2, ρ =2, Fig. 9. (color online). On-site number density plotted againstlattice site index at densities ρ = 4 / , / ρ = 2 andvalues smaller or larger than the critical point. Open boundaryconditions were used. The lines are visual guides. In order to distinguish between insulators, we calcu-late the arrangement of the particles in the superlatticeaccording to the increase of the parameter λ/t ; i. e. thedensity profiles for values of λ/t lower and higher than thetransition region are shown in Fig. 9, where the densities ρ = 4 / ρ = 5 / ρ = 2 are considered. For den-sity ρ = 4 / λ/t = 2, the superlatticeinduces fluctuations of charge such that there is one par-ticle in the barrier and three in the well (see Fig. 8), whileFig. 9(b), with λ/t = 15, shows the particles organizedsuch that the occupancy is { , , , , , , ... } . The quan-tum phase transition takes place around U , which can beexplained by taking into account that the smallest num-ber of particles in a site within the well is one. The energyof the two sites, the barrier, and one site of the well withone boson, is λ . This energy can be compensated if thebarrier particle jumps to the single occupied site, so theenergy of the two sites is U , and the transition happensfor λ ≈ U . Note that other possibilities are more expen-sive, which is corroborated by the phase diagram of the AB chain shown in Fig. 10, where the gap tends to sat-urate for larger values of λ , banning quantum transitionsin this region. A quantum phase transition around U fornon-integer densities is a newly discovered fact, due to the AB chain.In Figures 9(c) and 9(d), we see that for the density ρ = 5 / λ/t varies from 10 to 24, respectively. Thissituation is similar to the AB chain case for the density ρ = 3 /
2; therefore the quantum phase transition takesplace around 2 U .Finally, for λ/t = 15 and ρ = 2, the configuration forthe first CDW phase is shown in Fig. 9(e), with five parti-cles in the wells and one in the barriers, but at λ/t = 24,Fig. 9(f) shows six particles located at two sites with thebarrier site empty. The values of λ indicate that the quan- G. J. Cruz et al.: Phase diagrams of bosonic AB n chains µ /U λ /U Fig. 10. (color online). Phase diagram for bosons in a su-perlattice type AB at the thermodynamic limit. The points(open circles) represent the boundaries in the density matrixrenormalization group method. tum transition happens around 2 U , a value for which thereis no quantum transition in the AB chain case. On the ba-sis of Fig. 9(e), note that the smallest number of particlesin a site of the wells is two, and the barrier is always occu-pied. The energy for two sites, the barrier, and one site ofthe well with two bosons, is U + λ , while 3 U is the energyof the two sites after the boson at the barrier jumps to thesite with two bosons. The fluctuations will increase when λ ≈ U , and a superfluid region appears between the twoCDW phases. The above discussion allows us to concludethat for a AB N − chain with a integer density ρ , thereare ρ + 1 insulator phases separated by superfluid regions; ρ insulator phases are CDW, and one is Mott insulator.The values of λ around which the transitions occur willbe determined by the values of N and ρ .We show the phase diagram at the thermodynamiclimit for bosons in a AB chain in Fig. 10. For commen-surate densities lower than one, the system exhibits twoinsulator phases separated by a superfluid one; the sizeof these CDW regions increases as a function of λ , butquickly saturates. We believe that there is a non-zero fi-nite value of λ for which the CDW phase appears for thedensities ρ = 1 / ρ = 2 /
3. Note that these phasesremain stable for all values of λ considered in this paper;i. e., for these densities the arrangement of the particles isalways the same regardless of the depth of the potential.The phase diagram shows us that for non-integer den-sities larger than one, there are two CDW phases, similarto the AB chain case, but for the AB chain, we obtainedthat the transition regions depend on the density. How-ever, the transitions happen around multiples of U . When λ = 0, the ground state is superfluid for the densities ρ = 4 / ρ = 5 /
3; however, we expect a quantumphase transition to a CDW phase for larger values of λ .For integer densities, we observe that the ground stateis of the Mott insulator type at λ = 0 for any density ρ .When λ increases, we see that CDW phases can appear,
10 11 12 " ’)*+, -. /0 ρ =4/3 ρ =5/3 ε λ /t AB AB L=80 L=80* (b) (c)
19 19.5 20 20.5 211.1401.1451.1501.1551.160 λ ε ρ =3/2 /t AB L=80 * (a) Fig. 11.
Average entropy versus λ for a AB chain with density ρ = 3 / AB chain with ρ = 4 / AB chain with ρ = 5 / specifically ρ CDW phases for a fixed global density ρ (seeFig. 10).In the present paper, we consider bosons in an inho-mogeneous lattice. Information about the system can beobtained using the tools of information theory, which havebeen shown to be useful for determining critical points ofsystems with a harmonic potential, showing that there isa one-to-one correspondence between the local von Neu-mann entropy and the charge fluctuations [22]. Fran¸ca andCapelle [15] showed that the average of local von Neu-mann entropy is able to indicate the critical points wherethe phase changes happen in an inhomogeneous system,which is given by: ǫ ∗ = 1 L X i ǫ νN ( i ) , (5)where ǫ νN ( i ) = − T rσ i log σ i is the local von Neumannentropy, and σ i = T r B σ is the density matrix of a singlesite located at i , where B represents the environment with L − σ is the density matrix of the whole system.In Fig. 11, we show the average von Neumann entropyversus λ/t . In (a) we consider the AB chain, and the AB chain in (b) and (c). In all cases, we consider densities be-tween 1 and 2, and we observe the same behavior: the vonNeumann entropy increases, reaching a maximum value, . J. Cruz et al.: Phase diagrams of bosonic AB n chains 7 λ =: λ-λ /2 λ λ i = λ cos(2 π i/T+δ) AB Fig. 12.
Harmonic potential used by Rosseau et al. [10] andLi et al. [23], and the AB potential considered here. and then decreases. We note that it is impossible to de-termine the critical points that delimit the superfluid re-gions; however we can obtain information. In all cases, forvalues of λ/t on the left side, the system has a characteris-tic distribution of particles, for instance { , , , , , ..... } for ρ = 3 / AB chain. The average von Neumannentropy has a finite value associated with the number ofdegrees of freedom. When λ/t increases, fluctuations alsoincrease, and we expect that the average von Neumannentropy will grow. This tendency continues until the sys-tem reaches its maximum number of degrees of freedom,which would be associated with a coherent behavior of theparticles, i. e. with a superfluid state. Note that the posi-tions of the maximum values of the average von Neumannentropy agree with the middle point of the superfluid re-gions in the phase diagram for each density. Further in-crease in λ/t lets the particles localize, and a differentarrangement of the particles begins. Therefore, the num-ber of degrees of freedom decreases and the average vonNeumann entropy is smaller. An important fact is that theslopes before and after the maximum are always different,which indicates that the insulator regions surrounding thesuperfluid phase are different. For instance, for ρ = 3 / { , , , , , ..... } . This behavior was also obtained for in-teger densities.During the final editing of this paper, we became awareof a Li et al. manuscript [23], who studied a boson systemthat undergo an external potential λ i = λ cos(2( i + 1) π/ AB phase diagram reported here, i. e. they show insulatorphases for commensurate densities, multiples of 1 /
3, andobserve that for ρ = 1 and ρ = 4 / ρ ≥
1, which results will be reported soon [24].It is important to observe that in the phase diagramsfound by Rosseau et al. [10] and Li et al. [23], who usedthe external potential λ i = λ cos(2 iπ/T + δ ) ( T is the pe-riod and δ a phase), the phase boundaries for the lowerinsulating phases have a negative slope, whereas in ourphase diagrams, the slopes are positive and the borders Fig. 13. (Color online) The phase diagram of bosons on an AB chain in the atomic limit. Here ◦ represents a empty siteand • a boson. tend to be constant as λ increases. To understand whatis happening, we must first note that the potentials aredifferent (see Fig. 12 ) and then consider the atomic limit t = 0 of Hamiltonian (1). The ground state energy at site i , with n i bosons, is E ( n i ) = 0 . U n i ( n i −
1) + λ i n i − µn i .The energy difference between the energy with n i + 1 and n i bosons is ∆E = U n i + λ i − µ . When the result is zero,the border between insulating phases with different den-sity is determined. If we consider the unit cell of the AB chain (Fig. 1(b)) to be BAB , the occupation at site A isdetermined by µ/U = λ/U + n i . This relation generatesparallel lines in the ( µ/U, λ/U ) plane that separate re-gions with different numbers of particles. However at siteB, λ = 0, and the occupation is given by µ/U = n i , whichgenerates horizontal lines for each filling of site B. Theselines are independent of λ , a fact that points out the maindifference between our phase diagrams and the ones foundby Rosseau et al. [10] and Li et al. [23]. The phase diagramof bosons in an AB chain at the atomic limit is shownin Fig. 13. Here it can be seen that for larger values of λ the borders between regions will be horizontal lines, in amanner similar to what we show in our DMRG calcula-tions, which confirm the continuity of the phase diagramfrom the atomic limit to the delocalized case. Using the density matrix renormalization group method,we determined the chemical potential of AB N − chainsat the thermodynamic limit and found the phase diagramfor the N = 2 and 3 cases. For a small energy differencebetween the A and B sites ( λ ), we observe that insulatorregions with peculiar charge distribution (CDW) appearfor densities ρ = α/N ( α an integer), except at integerdensities, for which the Mott insulator phase still appears. G. J. Cruz et al.: Phase diagrams of bosonic AB n chains The size of the new CDW phases for densities less thanone grows with λ , but then it stabilizes, and these phasesremain in the phase diagram. On the other hand, for den-sities ρ ≥
1, we always found superfluid regions that sep-arate two insulator phases, a result that was confirmedusing the von Neumann entropy. Regardless of the valueof N , we found that for integer densities ρ , there are ρ + 1insulator phases, these being ρ CDW phases and one Mottinsulator phase with ρ particles per site.For AB N − chains, we observed that there are twoCDW phases for any non-integer densities larger than one.This result can be generalized by saying that for non-integer densities larger than one, there are ρ + 1 CDWphases, where ρ corresponds to the previous integer den-sity. For instance, for commensurate densities larger thantwo, we expect three CDW phases.The superfluid regions that separate the CDW phasesoccur around multiples of the local repulsion U , but thespecific values and the size of these regions depend on theglobal density ρ and N . Acknowledgments
The authors are thankful for the support of DIB- Universi-dad Nacional de Colombia and COLCIENCIAS (grant No.FP44842-057-2015). Silva-Valencia and Franco are grate-ful for the hospitality of the ICTP, where part of this workwas done. G. J. Cruz programmed the DMRG code andcarried out the calculations. R. Franco contributed to thediscussions. J. Silva-Valencia planned and designed thestudy. All contributed to writing the paper.
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