Energy spectrum and phase diagrams of two-sublattice hard-core boson model

The energy spectrum, spectral density and phase diagrams have been obtained for two-sublattice hard-core boson model in frames of random phase approximation approach. Reconstruction of boson spectrum at the change of temperature, chemical potential and energy difference between local positions in su...

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Опубліковано в: :	Condensed Matter Physics
Дата:	2013
Автори:	Stasyuk, I.V., Vorobyov, O.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2013
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/120812
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Цитувати:	Energy spectrum and phase diagrams of two-sublattice hard-core boson model / I.V. Stasyuk, O. Vorobyov // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23005:1-9. — Бібліогр.: 19 назв. — англ.

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spelling	Stasyuk, I.V. Vorobyov, O. 2017-06-13T05:07:01Z 2017-06-13T05:07:01Z 2013 Energy spectrum and phase diagrams of two-sublattice hard-core boson model / I.V. Stasyuk, O. Vorobyov // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23005:1-9. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 03.75.Hh,03.75.Lm,71.35.Lk DOI:10.5488/CMP.16.23005 arXiv:1307.2005 https://nasplib.isofts.kiev.ua/handle/123456789/120812 The energy spectrum, spectral density and phase diagrams have been obtained for two-sublattice hard-core boson model in frames of random phase approximation approach. Reconstruction of boson spectrum at the change of temperature, chemical potential and energy difference between local positions in sublattices is studied. The phase diagrams illustrating the regions of existence of a normal phase which can be close to Mott-insulator (MI) or charge-density (CDW) phase diagrams as well as the phase with the Bose-Einstein condensate (SF phase) are built. Для двопiдґраткової моделi жорстких бозонiв в рамках наближення хаотичних фаз розраховано енергетичний спектр i спектральнi густини у рiзних фазах та побудовано фазовi дiаграми. Дослiджено перебудову бозонного спектру при змiнi температури, хiмiчного потенцiалу та рiзницi енергiй локальних позицiй у пiдґратках. Побудовано фазовi дiаграми, якi iлюструють областi iснування нормальної фази, що може бути подiбною до фази моттiвського дiелектрика (MI) чи зарядового впорядкування (CDW), а також фази з бозе-конденсатом (фази SF). en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Energy spectrum and phase diagrams of two-sublattice hard-core boson model Енергетичний спектр i фазовi дiаграми двопiдґраткової моделi жорстких бозонiв Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Energy spectrum and phase diagrams of two-sublattice hard-core boson model
spellingShingle	Energy spectrum and phase diagrams of two-sublattice hard-core boson model Stasyuk, I.V. Vorobyov, O.
title_short	Energy spectrum and phase diagrams of two-sublattice hard-core boson model
title_full	Energy spectrum and phase diagrams of two-sublattice hard-core boson model
title_fullStr	Energy spectrum and phase diagrams of two-sublattice hard-core boson model
title_full_unstemmed	Energy spectrum and phase diagrams of two-sublattice hard-core boson model
title_sort	energy spectrum and phase diagrams of two-sublattice hard-core boson model
author	Stasyuk, I.V. Vorobyov, O.
author_facet	Stasyuk, I.V. Vorobyov, O.
publishDate	2013
language	English
container_title	Condensed Matter Physics
publisher	Інститут фізики конденсованих систем НАН України
format	Article
title_alt	Енергетичний спектр i фазовi дiаграми двопiдґраткової моделi жорстких бозонiв
description	The energy spectrum, spectral density and phase diagrams have been obtained for two-sublattice hard-core boson model in frames of random phase approximation approach. Reconstruction of boson spectrum at the change of temperature, chemical potential and energy difference between local positions in sublattices is studied. The phase diagrams illustrating the regions of existence of a normal phase which can be close to Mott-insulator (MI) or charge-density (CDW) phase diagrams as well as the phase with the Bose-Einstein condensate (SF phase) are built. Для двопiдґраткової моделi жорстких бозонiв в рамках наближення хаотичних фаз розраховано енергетичний спектр i спектральнi густини у рiзних фазах та побудовано фазовi дiаграми. Дослiджено перебудову бозонного спектру при змiнi температури, хiмiчного потенцiалу та рiзницi енергiй локальних позицiй у пiдґратках. Побудовано фазовi дiаграми, якi iлюструють областi iснування нормальної фази, що може бути подiбною до фази моттiвського дiелектрика (MI) чи зарядового впорядкування (CDW), а також фази з бозе-конденсатом (фази SF).
issn	1607-324X
url	https://nasplib.isofts.kiev.ua/handle/123456789/120812
citation_txt	Energy spectrum and phase diagrams of two-sublattice hard-core boson model / I.V. Stasyuk, O. Vorobyov // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23005:1-9. — Бібліогр.: 19 назв. — англ.
work_keys_str_mv	AT stasyukiv energyspectrumandphasediagramsoftwosublatticehardcorebosonmodel AT vorobyovo energyspectrumandphasediagramsoftwosublatticehardcorebosonmodel AT stasyukiv energetičniispektrifazovidiagramidvopidgratkovoímodeližorstkihbozoniv AT vorobyovo energetičniispektrifazovidiagramidvopidgratkovoímodeližorstkihbozoniv
first_indexed	2025-11-26T13:13:19Z
last_indexed	2025-11-26T13:13:19Z
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fulltext	Condensed Matter Physics, 2013, Vol. 16, No 2, 23005: 1–9 DOI: 10.5488/CMP.16.23005 http://www.icmp.lviv.ua/journal Energy spectrum and phase diagrams of two-sublattice hard-core boson model I.V. Stasyuk, O. Vorobyov Institute for Condensed Matter Physics National academy of Sciences of Ukraine, 1 Sventsitskii St., 79011 Lviv, Ukraine Received March 14, 2013, in final form March 29, 2013 The energy spectrum, spectral density and phase diagrams have been obtained for two-sublattice hard-core boson model in frames of random phase approximation approach. Reconstruction of boson spectrum at the change of temperature, chemical potential and energy difference between local positions in sublattices is stud- ied. The phase diagrams illustrating the regions of existence of a normal phase which can be close to Mott- insulator (MI) or charge-density (CDW) phases as well as the phase with the Bose-Einstein condensate (SF phase) are built. Key words: hard-core bosons, spectral density, phase diagrams PACS: 03.75.Hh, 03.75.Lm, 71.35.Lk 1. Introduction Lattice Bose-gas model based on the hard-core bosons approach (the site occupancy ni = 0,1) has a wide range of possible applications starting from quantum effects in liquid He [1, 2]. This model was also applied to superconducting gas of Cooper electron pairs [3], physical properties of Josephson junctions [4], thermodynamics and energy spectrum of crystals with ionic conductivity [5, 6]. In recent years the hard-core boson approach has gained popularity in connection with investigations of ultra-cold atoms in optical lattices. At an arbitrary occupation of local particle positions optical lattices are usually described with Bose-Hubbardmodel (see [7] for review). In U →∞ limit of this model, when potential wells are ex- tremely deep, Bose-Hubbardmodel turns to hard-core boson model. In this paper we consider this model for the lattice with non-equivalent sites, particularly in the simplest case of two-sublattice structure. Such structures can be easily realized in optical lattices [8] and are also observed in the case of adsorption of hydrogen atoms on the surface of metals (the quantum surface diffusion of protons is described bymeans of Bose-Hubbard model [9, 10]). Crystal lattice is supposed to be centrosymmetrical of cubic type. Parti- cles have different local site energies on each of two sublattices (εA , εB , where A and B are sublattice indices). This model has been investigated in connection with thermodynamic properties of Bose atoms in complex optical lattices [11–13]. The main focus of our paper is to study the conditions of Bose-Einstein (BE) condensation and to construct the corresponding phase diagrams. Our goal is to investigate the energy spectrum and one- particle spectral densities as well as the changes of their shapes as the system enters various phases that include the phase with BE condensate (also called superfluid or SF) and normal phase of the so-called Mott-insulator (MI) or charge-density wave (CDW) type. We use two-time Green’s function technique and random phase approximation (RPA). A similar approach has been used recently in [14]. © I.V. Stasyuk, O. Vorobyov, 2013 23005-1 http://dx.doi.org/10.5488/CMP.16.23005 http://www.icmp.lviv.ua/journal I.V. Stasyuk, O. Vorobyov 2. Boson Green’s functions and phase diagrams The Hamiltonian of noninteracting hard-core bosons on a lattice is as follows: Ĥ =− ∑ i j ti j b+ i b j + (ε0 −µ) ∑ i ni , (2.1) where ti j is the boson hopping parameter and bi , b+ i are Pauli operators. We proceed to pseudospins (bi = S+ i , b+ i = Si ) and generalize the model for two sublattices (i = n.α; α= A,B ; ε0 = εA, εB): Ĥ =− ∑ nα ∑ n′β J αβ nn′ ( Sx nαSx n′β+S y nαS y n′β ) − ∑ α hα ∑ n Sz nα . (2.2) The parameter of “transversal” interaction between pseudospins J αβ nn′ describes the transfer of par- ticles between nearest neighbours in the lattice; hα = εα−µ is the “field” acting on the pseudospin in α sublattice. To start with, we consider the mean-field Hamiltonian ĤMF =− ∑ nα ∑ n′β ( J αβ nn′ + J βα n′n ) 〈Sx β〉S x nα− ∑ α hα ∑ n Sz nα (2.3) which is diagonalized with the rotation transformation Sz nα = σz nα cosϑα+σx nα sinϑα , Sx nα = σx nα cosϑα−σz nα sinϑα (2.4) and takes the form ĤMF =− ∑ nα Eασ z nα. The following equations define the angles ϑα: hA sinϑA −〈σz B〉JA(0)cosϑA sinϑB = 0, hB sinϑB −〈σz A〉JB(0)cosϑB sinϑA = 0. (2.5) Here, Jα(0) = ∑ n′β ( J αβ nn′ + J βα n′n ) ; in the case of structurally equivalent sublattices JA(0) = JB(0) ≡ J (0). The trivial solution sinϑA = 0, sinϑB = 0 defines the normal phase (like MI or CDW), while at sinϑα , 0 the SF phase exists. For SF phase, the order parameter 〈Sx α〉 is not equal to zero (because 〈Sx α〉 = −〈σx α〉sinϑα). For nontrivial solution we have sin 2ϑα = 〈σz α〉 2〈σz β 〉2 J 4(0)−h2 αh2 β 〈σz α〉 2 J 2(0)[h2 α+〈σz β 〉2 J 2(0)] . (2.6) Here and below, β,α. In the mean-field approximation 〈σz α〉 = 1 2 tanh βEα 2 , (2.7) where Eα = hα cosϑα+〈σz β〉J (0)sinϑα sinϑβ = 〈σz α〉J (0) √ h2 α+〈σz β 〉2 J 2(0) √ h2 β +〈σz α〉 2 J 2(0) . (2.8) The set of equations (2.7) and (2.8) defines the pseudospin averages 〈σz A 〉, 〈σz B 〉 and internal fields EA, EB. On the other hand, in the case of normal phase Eα = hα , 〈σz α〉 = 1 2 tanh βhα 2 . (2.9) 23005-2 Energy spectrum and phase diagrams two-sublattice hard-core boson model The condition of transition to SF-phase is the divergence of boson Green’s function 〈〈S+\|S−〉〉q,ω at zero frequency and ~q = 0 (as we approach SF phase boundary from any of normal phases). To construct the equations for pseudospin Green’s functions, we use the linearized equations of mo- tion for ~σnα operators [ σx lα, Ĥ ] = Eαiσ y iα− 〈 σz α 〉 ∑ n′ ( J αβ ln′ + J βα n′l ) iσ y n′β , [ σ y lα , Ĥ ] =−Eαiσx iα+ 〈 σz α 〉 ∑ n′ ( J αβ ln′ + J βα n′l ) cosϑA cosϑBiσx n′β , [ σz lα, Ĥ ] = 0 (2.10) (these equations were written using RPA decoupling). It is taken into account that interaction J αβ nn′ (parti- cle hopping) takes place between lattice sites from different sublattices. As a result, we obtain the following set of equations for pseudospin Green’s functions ħω〈〈σx lα\|σ x l ′γ〉〉 = iEα〈〈σ y lα \|σx l ′γ〉〉− i 〈 σz α 〉 ∑ n′ ( J αβ ln′ + J βα n′l ) 〈〈σ y n′β \|σx l ′γ〉〉, ħω〈〈σ y lα \|σx l ′γ〉〉 = −i ħ 2π δl l ′δαγ 〈 σz α 〉 − iEα〈〈σ x lα\|σ x l ′γ〉〉+ i 〈 σz α 〉 ∑ n′ ( L αβ ln′ +L βα n′ l ) 〈〈σx n′β\|σ x l ′γ〉〉, (2.11) where LAB ln′ = J AB ln′ cosϑA cosϑB . (2.12) After Fourier transformation of pseudospin interaction matrix J ( ~q ) = ∑ n−n′ ( J AB nn′ + J B A n′n ) e i~q ( ~RnA−~Rn′B ) (2.13) as well as Green’s functions 〈〈σα\|σβ〉〉we obtain, in particular, the following equations ħωGxx AA = iEAG y x AA − i 〈 σz A 〉 J ( ~q ) G y x BA , ħωG y x AA = −i ħ 2π 〈 σz A 〉 − iEAGxx AA + i 〈 σz A 〉 L ( ~q ) Gxx BA , ħωGxx BA = iEBG y x BA − i 〈 σz B 〉 J ( ~q ) G y x AA , ħωG y x BA = −iEBGxx BA + i 〈 σz B 〉 L ( ~q ) Gxx AA . (2.14) The system of equations (2.14) can be easily solved to obtain the expressions for matrix Green’s func- tions 〈〈σ µ α\|σ ν γ〉〉q,w and 〈〈S µ α\|S ν γ〉〉q,w , (we can calculate the latter using relations (2.4)). Here, µ and ν indices denote +,−,z components. 3. Boson spectrum in normal phase and phase diagrams Let us consider the one-particle boson Green‘s function 〈〈bα\|b + β 〉〉q,w = 〈〈S+ α\|S − β 〉〉q,w . In the normal phase case 〈〈S+ α\|S − β 〉〉q,w = 〈〈σ+ α\|σ − β 〉〉q,w . For α=β, we have the following result G+− αα (~q, w) ≡ 〈〈σ+ α\|σ − α〉〉q,w = ħ π 〈 σz α 〉 ħω−Eβ (ħω−Eα)(ħω−Eβ)−Φq , (3.1) derived from equations (2.14). Here, Φq = 〈 σz A 〉〈 σz B 〉 J 2 ( ~q ) . The boson excitation spectrum is defined from the poles of the G+− αα function ε(NO) 1,2 (~q) = h± √ δ2 + 〈 σz A 〉〈 σz B 〉 J 2(~q) . (3.2) We have introduced the general notations h = EA+EB 2 ; δ= EA−EB 2 . In normal phases h = hA+hB 2 ; δ= hA−hB 2 . 23005-3 I.V. Stasyuk, O. Vorobyov The features of the obtained spectrum may vary depending on the values of the model parameters: For δ= 0 (A and B positions are equivalent; crystal is not split to sublattices and the unit cell is two times smaller): ε(NO) 1,2 (~q) = h± ∣ ∣ 〈 σz 〉∣ ∣ J (~q), 〈 σz 〉 = 1 2 tanh βh 2 . (3.3) There is only one band ε(~q) = h−〈σz〉 J (~q) inside the two times bigger Brillouin zone. For δ, 0; δ> 0. There are two bands in this case. The edges of the bands are defined by the inequalities which depend on the sign of 〈 σz A 〉〈 σz B 〉 = 1 4 tanh β 2 (h+δ) tanh β 2 (h−δ) expression: h+δ< ε1(~q) < h+ √ δ2 + 〈 σz A 〉〈 σz B 〉 J 2(0) h− √ δ2 + 〈 σz A 〉〈 σz B 〉 J 2(0) < ε2(~q)< h−δ        〈 σz A 〉〈 σz B 〉 > 0 and h+ √ δ2 + 〈 σz A 〉〈 σz B 〉 J 2(0) < ε1(~q) < h+δ h−δ< ε2(~q) < h− √ δ2 + 〈 σz A 〉〈 σz B 〉 J 2(0)        〈 σz A 〉〈 σz B 〉 < 0. -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 3 1 h=1 q a a MI -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 q a 4 3 2 hA/2+hB/2=0.7 b SF 1 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -0.5 0.0 0.5 1.0 3 1 h=0.3 q a c CDW -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -2.0 -1.5 -1.0 -0.5 0.0 3 1 h=-1 q a d MI Figure 1. Dispersion laws ε(q) for different phases. Dashed line denotes the chemical potential level. The Fourier transform J (~q) = 1 z J (0) z ∑ α=1 cos qαa is used with the aim of illustration. J (0) is chosen as the energy unit. T ≡ 1/β= 0.05,δ = 0.8. The numbers indicate the corresponding branches. For SF phase: 1 – ε1, 2 – ε2 , 3 – ε3, 4 – ε4. For MI and CDW phases: 1 – ε1, 3 – ε2. 23005-4 Energy spectrum and phase diagrams two-sublattice hard-core boson model In the first case (〈σz A 〉〈σz B 〉 > 0), which holds for h −δ > 0, two different bands always exist; the gap between these bands disappears as δ→ 0. The chemical potential (which is located on the energy scale at ε = 0 point) is placed either higher or lower than the bands ε1(~q) and ε2(~q) [figures 1 (a), 1 (d)]. In the second case (〈σz A 〉〈σz B 〉 < 0), which corresponds to the inequalities h −δ < 0; h +δ > 0, two different bands exist only at δ > √ \|〈σz A 〉〈σz B 〉\|J (0). The gap disappears when this condition is violated [at T = 0 this happens at δ = δc ≡ 1 2 J (0)]. When the bands are separated in normal phase, the chemical potential is located between the bands [figure 1 (c)]. The instability connected with SF transition takes place when the level of chemical potential touches the edge of one of the bands that may be driven either by the temperature, chemical potential or energy difference δ change. At J (0) > 0 (ti j > 0), this always happens in the ~q = 0 point. The condition for this is as follows: h2 = δ2 + 〈 σz A 〉〈 σz B 〉 J 2 (0). (3.4) Two equations derived from this relation allow us to construct the phase diagrams in (J (0),h) and (T,h) planes that show the areas of SF and normal (MI, CDW) phases. Diagram in figure 2 illustrates the change of the shape of phase boundary curve on (J (0),h) plane as the temperature increases (at T = 0, the phase boundary curve corresponds to the one obtained in [11, 12]). The definitive boundary between MI and CDW regions exists only at zero temperature. In this case, MI and CDW states can be interpreted as different phases. When one departs from T = 0 limit, this boundary disappears and one may observe a single normal phase. However, this normal phase is close to either MI or CDWphases in different regions of phase diagram (also see below). -2 -1 0 1 2 0 1 2 3 4 SF MIMI 5 4 3 J( 0) / h/ 1 2 CDW -1.0 -0.5 0.0 0.5 1.0 0.00 0.05 0.10 0.15 0.20 0.25 MI MI CDW SF SF MI MI 1 2 3 4 5 6 T/ J( 0) h/J(0) SF Figure 2. Phase diagram of two-sublattice model of hard-core bosons for different temperatures: 1. T = 0.00005, 2. T = 0.05, 3. T = 0.15, 4. T = 0.2, 5. T = 0.5. Energy quantities aremeasured in units of δ= (εA −εB)/2. Figure 3. Phase diagram (T,h) at various values of δ: 1. δ = 0.1, 2. δ = 0.3, 3. δ = 0.48, 4. δ = 0.499, 5. δ= 0.501, 6. δ= 0.8. Energy quantities are mea- sured in units of J (0). If the existing critical value of the difference of sublattice local energies (δ= δc) is exceeded, it leads to the splitting of the SF-phase area on (T,h) plain (figure 3). This result is in agreement with the papers mentioned above, where all calculations were performed only at T = 0. Therefore, at δ > δc, there are two critical points for T , 0. For intermediate values of chemical potential, the normal phase is similar to the charge ordered phase (CDW) while at large positive (or negative) values of h this phase is of Mott-insulator (MI) type. This conclusion is confirmed by one-particle spectral density ρα(ω) calculations. We use the relation ρα(ω)=− 1 N ∑ q 2Im〈〈S+ α\|S − α〉〉q,ω+iε = 2 N ∑ q 〈 σz α 〉 { Aα 1 (~q)δ [ ω− ε1(~q) ħ ] + Aα 2 (~q)δ [ ω− ε2(~q) ħ ]} , which follows from the decomposition into partial fractions. 23005-5 I.V. Stasyuk, O. Vorobyov Here, AA 1,2(~q) = 1 2 ± δ 2 √ δ2 +Φq , while expression for AB 1,2 (~q) is derived from AA 1,2 (~q) by A⇄B (δ→−δ) substitution. Using non-perturbative density of states ρ0(z)= 1 N ∑ q δ [ z − J (~q) ] , we can rewrite the expression (3.5) for α= A ρA(ω) = 2〈σz a〉 J(0) ∫ −J(0) dzρ0(z)         1 2 + δ 2 √ δ2+ 〈 σz A 〉〈 σz B 〉 z2    δ [ ω− 1 ħ ( h+ √ δ2+ 〈 σz A 〉〈 σz B 〉 z2 ) ] +    1 2 − δ 2 √ δ2+ 〈 σz A 〉〈 σz B 〉 z2   δ [ ω− 1 ħ ( h− √ δ2+ 〈 σz A 〉〈 σz B 〉 z2 ) ]      . (3.5) When performing numerical calculations, we use the semi-elliptical function ρ0(z)= 1 πJ2(0) √ J 2(0)− z2. -0.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 ~ h=1 A =0.5 B =0.48 A a MI h- -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -1.0 -0.5 0.0 0.5 1.0 ~ A ha/2+hb/2=0.7 A =0.5 B =0.449 b SF h- -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -1.0 -0.5 0.0 0.5 1.0 ~ h=0.3 A =0.5 B =-0.5 A c CDW h- -2.0 -1.5 -1.0 -0.5 0.0 0.5 -2.0 -1.5 -1.0 -0.5 0.0 h=-1 A =-0.482 B =-0.5 A d MI ~ h- Figure 4. Spectral density of A-sublattice for different phases. T = 0.05, δ= 0.8. All energy quantities are measured in units of J (0). ρ̃A = ρA/ħ is the spectral density as function of energy ħω. Figure 4 illustrates spectral density for all phases. For CDW region of normal phase [figure 4 (c)], the chemical potential is located within the gap between the bands ρα(ω); the sign of ρα(ω) function is different in each band [ρα(ω) < 0 at ħω< µ and ρα(ω) > 0 at ħω> µ]. For MI region [figures 4 (a), 4 (d)] the chemical potential is at the same side of both bands ρα(ω). 23005-6 Energy spectrum and phase diagrams two-sublattice hard-core boson model The values of σz A and σz B averages presented in figures 4 (a)–4 (d) are very close to those at T = 0. We observe themodulated occupancy nA = 1 2 −σz A = 0,nB = 1 2 −σz B = 1 in CDW-like case [figure 4 (c)]. Contrary to this, in MI-like cases, this occupancy is either close to zero or unity depending on the chemical potential value. The latter two possibilities are illustrated in figure 4 (a) (nA ≈ nB ≈ 0 when µ is positioned below the energy bands) and figure 4 (d) (nA ≈ nB ≈ 1 when µ is placed above the bands). 4. Excitation spectrum in SF phase In the case of a phase with BE-condensate (SF phase), when sinϑA , 0, sinϑB , 0, 〈〈S+ A \|S − A 〉〉q,w = ħ 2π 〈σz A〉 P A q (ħ2ω2 −E 2 A )(ħ2ω2 −E 2 B )−2Mqħ 2ω2 −2Nq EAEB +M2 q , (4.1) where P A q (ħω) = [ EA ( cos 2ϑA +1 ) +2ħωcosϑA ]( ħ 2ω2 −E 2 A ) −2ħωMq cosϑA + Φ̃ A q EB , (4.2) and the following notations are introduced: Mq =Φq cosϑA cosϑB , Nq = 1 2 Φq ( 1+cos 2ϑA cos 2 ϑB ) , Φ̃ A q =Φq cos 2ϑA ( 1+cos 2ϑB ) (4.3) (the replacement A⇄B gives an expression for the 〈〈S+ B \|S− B 〉〉q,w function). The boson spectrum consists now of four branches ε(SF) 1,2 (~q) =± ( Pq +Qq )1/2 , ε(SF) 3,4 (~q) =± ( Pq −Qq )1/2 . (4.4) Here, Pq = 1 2 ( E 2 A +E 2 B ) +Mq , Qq = [ 1 4 ( E 2 A −E 2 B )2 +2Nq EAEB +Mq ( E 2 A −E 2 B ) ]1/2 . (4.5) Energies EA and EB, as well as averages 〈σz A 〉 and 〈σz B 〉 are determined now as solutions of equations (2.7) and (2.13). Regions of existence of SF phase are shown in figures 2, 3. The dispersion curves ε(SF) 1..4 (~q) are present in figure 1 (b) for certain values of h and δ parameters. The presence of branches with linear dispersion at small values of q [ε3(~q) and ε4(~q) in the case presented in figure 1 (b)] is the specific feature of SF phase; their energy goes to zero in the point of the location of chemical potential. This peculiarity of spectrum is well known from investigations of the simple hard-core boson model [3]. However, in our case, at εA , εB, the additional gapped branch [ε2(~q) in figure 1 (b)] appears in the negative energy region. Similarly to the normal phase case, one can perform calculations of the boson spectral density ρα(ω). Using decomposition of expression (4.1) into partial fractions, we obtain ρα(ω)= 2 N ∑ q 〈σz α〉 4 ∑ i=1 Aα i (~q)δ ( ω− εi (~q) ħ ) , (4.6) where Aα i (~q) = Pα q ( ħω= εi (~q) ) 4Qqεi (~q) . (4.7) It is easy to obtain an expression like (3.5) passing to integration with the ρ0(z) density of states. The contributions from all four bands are present in the total spectral density. The plots of the ρA(ω) functions in the case of SF phase are presented in figure 4 (b). For branches with linear dispersion [ε3,4(~q)], the spectral density changes its sign in the point ħω= 0 (at that point the chemical potential is located). The change of the spectral density shape at MI → SF transition, when we observe the appearance of the negative branch of ρα(ω) [figure 4 (b)], corresponds to the results obtained 23005-7 I.V. Stasyuk, O. Vorobyov in [15, 16] as well as to the ones obtained for generalized hard-core boson model with excited states trans- fer [17]. Additional branch ε2(~q) that appears in SF phase is characterized by a negative spectral density. Its intensity (at the chosen values of h and δ parameters) is small. Qualitatively, this shape of the ρα(ω) function is specific for the Bose-Hubbardmodel [18]. However, contrary to the standard case, where addi- tional branches separated by gaps exist due to the local energy splitting (caused by the Hubbard repulsion of bosons), in our two-sublattice model such an effect is a consequence of the energy non-equivalence of sublattices. The behaviour of ρA(ω) function is in agreement with the results of numerical calculations performed in [19] with exact diagonalization technique for one-dimensional (d = 1) chain structures. In [19], the authors take into account the two-particle interaction between nearest neighbouring sites. This inter- action forms the effective internal field which is similar to the field δ considered here, and both fields are responsible for the appearance of CDW-like phase. The shape of spectral densities in various phases, obtained here, lets one identify the equilibrium states on phase diagrams (diagrams of state) obtained numerically for d = 1. 5. Conclusions Within the randomphase approximation, we have calculated the spectral densities of a two-sublattice model of hard-core bosons and analyzed the features of the boson single-particle spectrum in various phases. These features are connected with the position of the chemical potential level. It is placed: • within the gap between two boson bands in the case when normal phase is similar to the charge- ordered (CDW) phase; • above (or below) both bands in the case when normal phase is similar to the Mott insulator (MI) phase; • within a certain boson band, for SF phase (the phase with BE condensate); the additional boson bands appear in this case. We have obtained the equation that describes the transition to the SF phase and have built the corresponding phase diagrams at various temperatures and at different values of energy difference δ= 1 2 (εA −εB). The temperature increase leads to the gradual vanishing of the difference between CDW- like and MI-like modifications of normal phase; there are no border lines separating them. SF-phase re- gion also decreases with the temperature increase;at the same time, two regions of the SF phase, which exist at T = 0 and at a fixed value of δ, join together. On the other hand, a similar effect takes place for fixed temperature at the decrease of δ. At high values of δ, there are two critical points in which the SF phase disappears at an increase of temperature. When δ decreases, only one central critical point remains. At the same time, it should be mentioned that nonzero value of δ is the main reason for the appear- ance of the CDW-like state in our system. We have not included direct intersite interactions between particles into consideration. This kind of interaction may induce the phase transition into “true” CDW phase. More elaborate study of the boson spectrum reconstruction at the transitions between different re- gions in phase diagrams and the change of their topology remains an interesting task. It is worthy of special attention. 23005-8 Energy spectrum and phase diagrams two-sublattice hard-core boson model References 1. Matsuda H., Tsuneto T., Sup. Prog. Theor. Phys., 1970, 39, 411; doi:10.1143/PTPS.46.411. 2. Fisher M., Weichman P., Grinstein G., Fisher D., Phys. Rev. B, 1989, 40, 546; doi:10.1103/PhysRevB.40.546. 3. Micnas R., Ranninger J., Robaszkiewicz S., Rev. Mod. Phys., 1990, 62, 113; doi:10.1103/RevModPhys.62.113. 4. Czathy G.A., Reppy J.D., Chan M.H.W., Phys. Rev. Lett., 2003 91, 235301; doi:10.1103/PhysRevLett.91.235301. 5. Mahan G.D., Phys. Rev. B, 1976, 14, 780; doi:10.1103/PhysRevB.14.780. 6. Stasyuk I.V., Dulepa I.R., Condens. Matter Phys., 2007, 10, 259; doi:10.5488/CMP.10.2.259. 7. Bloch I., Dalibard J., Zwerger W., Rev. Mod. Phys., 2008, 80, 885; doi:10.1103/RevModPhys.80.885. 8. Aizenman M., Lieb E.H., Seiringer R., Solovej J.P., Yugvason I., Phys. Rev. A, 2004, 70, 023612; doi:10.1103/PhysRevA.70.023612. 9. Puska M.J., Niemenen R.M., Surf. Sci., 1985, 157, No. 2–3, 413; doi:10.1016/0039-6028(85)90683-1. 10. Brenig W., Surf. Sci., 1993, 291, No. 1–2, 207; doi:10.1016/0039-6028(93)91492-8. 11. Iskin M., Eur. Phys. J. B, 2012, 85, 76; doi:10.1140/epjb/e2012-20852-5. 12. Hen I., Iskin M., Rigol M., Phys. Rev. B, 2010, 81, 064503; doi:10.1103/PhysRevB.81.064503. 13. Pich C., Frey E., Phys. Rev. B, 1998, 57, 13712; doi:10.1103/PhysRevB.57.13712. 14. Stasyuk I.V., Dulepa I.R., J. Phys. Studies, 2009, 13, 2701. 15. Ohashi Y., Kitaura M., Matsumoto H., Phys. Rev. A, 2006, 73, 033617; doi:10.1103/PhysRevA.73.033617. 16. Menotti C., Trivedi N., Phys. Rev. B, 2008, 77, 235120; doi:10.1103/PhysRevB.77.235120. 17. Stasyuk I.V., Velychko O.V., Theor. Math. Phys., 2011, 168, 1347; doi:10.1007/s11232-011-0110-2. 18. Dupuis N., Sengupta K., Physica B, 2009, 404, 517; doi:10.1016/j.physb.2008.11.084. 19. Stasyuk I.V., Vorobyov O., Stetsiv R.Ya., Ferroelectrics, 2012, 426, 6; doi:10.1080/00150193.2012.671087. Енергетичний спектр i фазовi дiаграми двопiдґраткової моделi жорстких бозонiв I.В. Стасюк, О. Воробйов Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна Для двопiдґраткової моделi жорстких бозонiв в рамках наближення хаотичних фаз розраховано енерге- тичний спектр i спектральнi густини у рiзних фазах та побудовано фазовi дiаграми. Дослiджено перебудо- ву бозонного спектру при змiнi температури, хiмiчного потенцiалу та рiзницi енергiй локальних позицiй у пiдґратках. Побудовано фазовi дiаграми, якi iлюструють областi iснування нормальної фази, що може бути подiбною до фази моттiвського дiелектрика (MI) чи зарядового впорядкування (CDW), а також фази з бозе-конденсатом (фази SF). Ключовi слова: жорсткi бозони, густина станiв, фазовi дiаграми 23005-9 http://dx.doi.org/10.1143/PTPS.46.411 http://dx.doi.org/10.1103/PhysRevB.40.546 http://dx.doi.org/10.1103/RevModPhys.62.113 http://dx.doi.org/10.1103/PhysRevLett.91.235301 http://dx.doi.org/10.1103/PhysRevB.14.780 http://dx.doi.org/10.5488/CMP.10.2.259 http://dx.doi.org/10.1103/RevModPhys.80.885 http://dx.doi.org/10.1103/PhysRevA.70.023612 http://dx.doi.org/10.1016/0039-6028(85)90683-1 http://dx.doi.org/10.1016/0039-6028(93)91492-8 http://dx.doi.org/10.1140/epjb/e2012-20852-5 http://dx.doi.org/10.1103/PhysRevB.81.064503 http://dx.doi.org/10.1103/PhysRevB.57.13712 http://dx.doi.org/10.1103/PhysRevA.73.033617 http://dx.doi.org/10.1103/PhysRevB.77.235120 http://dx.doi.org/10.1007/s11232-011-0110-2 http://dx.doi.org/10.1016/j.physb.2008.11.084 http://dx.doi.org/10.1080/00150193.2012.671087 Introduction Boson Green's functions and phase diagrams Boson spectrum in normal phase and phase diagrams Excitation spectrum in SF phase Conclusions

Energy spectrum and phase diagrams of two-sublattice hard-core boson model

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