Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation

For a Bose atom system whose energy operator is diagonal in the so-called number operators and its ground state has an internal two-level structure with negative energies, exact expressions for the limit free canonical energy and pressure are obtained. The existence of non-conventional Bose-Einstein...

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Published in:	Condensed Matter Physics
Date:	2010
Main Authors:	Corgini, M., Sankovich, D.P.
Format:	Article
Language:	English
Published:	Інститут фізики конденсованих систем НАН України 2010
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/32120
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Cite this:	Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation / M. Corgini, D.P. Sankovich // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43003:1-11. — Бібліогр.: 9 назв. — англ.

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spelling	Corgini, M. Sankovich, D.P. 2012-04-09T20:41:48Z 2012-04-09T20:41:48Z 2010 Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation / M. Corgini, D.P. Sankovich // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43003:1-11. — Бібліогр.: 9 назв. — англ. 1607-324X PACS: 05.30.Jp, 67.85.Jk https://nasplib.isofts.kiev.ua/handle/123456789/32120 For a Bose atom system whose energy operator is diagonal in the so-called number operators and its ground state has an internal two-level structure with negative energies, exact expressions for the limit free canonical energy and pressure are obtained. The existence of non-conventional Bose-Einstein condensation has been also proved. Для системи Бозе-атомів, чий оператор енергії є діагональним за так званим числом операторів і його основний стан має внутрішню дворівневу структуру з негативними енергіями, одержано точні вирази для граничних вільної канонічної енергії та тиску. Доведено існування нестандартної Бозе - Айнштайнівської конденсації. Partial financial support by PBCT–ACT13 (Stochastic Analysis Laboratory, Chile) and Programa de Magıster en Matematicas, Universidad de La Serena, Chile. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation Розв'язна модель Бозе-атомів з дворівневою внутрішньою структурою: нестандартна Бозе - Айнштайнівська конденсація Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation
spellingShingle	Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation Corgini, M. Sankovich, D.P.
title_short	Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation
title_full	Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation
title_fullStr	Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation
title_full_unstemmed	Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation
title_sort	soluble model of bose-atoms with two level internal structure: non-conventional bose-einstein condensation
author	Corgini, M. Sankovich, D.P.
author_facet	Corgini, M. Sankovich, D.P.
publishDate	2010
language	English
container_title	Condensed Matter Physics
publisher	Інститут фізики конденсованих систем НАН України
format	Article
title_alt	Розв'язна модель Бозе-атомів з дворівневою внутрішньою структурою: нестандартна Бозе - Айнштайнівська конденсація
description	For a Bose atom system whose energy operator is diagonal in the so-called number operators and its ground state has an internal two-level structure with negative energies, exact expressions for the limit free canonical energy and pressure are obtained. The existence of non-conventional Bose-Einstein condensation has been also proved. Для системи Бозе-атомів, чий оператор енергії є діагональним за так званим числом операторів і його основний стан має внутрішню дворівневу структуру з негативними енергіями, одержано точні вирази для граничних вільної канонічної енергії та тиску. Доведено існування нестандартної Бозе - Айнштайнівської конденсації.
issn	1607-324X
url	https://nasplib.isofts.kiev.ua/handle/123456789/32120
citation_txt	Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation / M. Corgini, D.P. Sankovich // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43003:1-11. — Бібліогр.: 9 назв. — англ.
work_keys_str_mv	AT corginim solublemodelofboseatomswithtwolevelinternalstructurenonconventionalboseeinsteincondensation AT sankovichdp solublemodelofboseatomswithtwolevelinternalstructurenonconventionalboseeinsteincondensation AT corginim rozvâznamodelʹbozeatomívzdvorívnevoûvnutríšnʹoûstrukturoûnestandartnabozeainštainívsʹkakondensacíâ AT sankovichdp rozvâznamodelʹbozeatomívzdvorívnevoûvnutríšnʹoûstrukturoûnestandartnabozeainštainívsʹkakondensacíâ
first_indexed	2025-11-24T16:57:52Z
last_indexed	2025-11-24T16:57:52Z
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fulltext	Condensed Matter Physics 2010, Vol. 13, No 4, 43003: 1–11 http://www.icmp.lviv.ua/journal Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation M. Corgini1,2 ∗, D.P. Sankovich3 † 1 Departamento de Matemáticas, Universidad de La Serena, Cisternas 1200, La Serena, Chile 2 Laboratorio de Análisis Estocástico, Chile 3 Steklov Mathematical Institute, Gubkin Str. 8, 119991, Moscow, Russia Received July 21, 2010 For a Bose atom system whose energy operator is diagonal in the so-called number operators and its ground state has an internal two-level structure with negative energies, exact expressions for the limit free canonical energy and pressure are obtained. The existence of non-conventional Bose-Einstein condensation has been also proved. Key words: approximating Hamiltonian method, non-conventional Bose-Einstein condensation PACS: 05.30.Jp, 67.85.Jk 1. Introduction We use an approach based on a suitable expression obtained for the limit free canonical energy in order to determine the limit pressure of a Bose-atom system with internal two-level structure. This enables us to recover some results, related to non-conventional Bose-Einstein condensation (BEC), obtained in [1] in the framework of the approximating Hamiltonians method ([2]). In section 2 we present a description of the main mathematical features associated with this model. In section 3 we obtain the limit free canonical energy of the model. It leads via Legendre transform to the limit pressure, recovering the previous results obtained in [1]. Finally in section 4 it is proved that the system undergoes non-conventional BEC (independent of temperature BEC). 2. The model The one-particle free Hamiltonian corresponds to the operator Sl = −△/2 defined on a dense subset of the Hilbert space Hl = L2(Λl), being Λl = [−l/2, l/2] d ⊂ Rd a cubic box of boundary ∂Λl and volume Vl = ld. In other words, the particles are confined to bounded regions. We assume periodic boundary conditions under which Sl becomes a self-adjoint operator. We consider a system of Bose atoms with an internal two-level structure analogous to the SU2 spin symmetry. In this case any one-particle wave function has the form φ ⊗ s where, φ ∈ L2(Λl) and s ∈ C 2 represents the internal state. Therefore, the vector space associated with this system is in fact, Hl s = L2(Λl)⊗ C2. We shall study a model of Bose particles whose Hamiltonian is given by: Ĥl = Ĥ0 l + a Vl ∑ p∈Λ∗ l ,σ (â†p,σ) 2â2p,σ + γ Vl n̂0,−n̂0,+ , (1) ∗E-mail: mcorgini@userena.cl †E-mail: sankovch@mi.ras.ru c© M. Corgini, D.P. Sankovich 43003-1 http://www.icmp.lviv.ua/journal M. Corgini, D.P. Sankovich where σ = + or − depending on the corresponding internal level. The second term at the right hand side of equation (1) represents the intrastate collisions (self-scattering term), the third term represents the interstate collisions (cross-scattering term). This model has been exhaustively studied in [1] by using the so-called method of approximating Hamiltonians developed in [2]. Here we shall obtain an analytical expression for the limit pressure of our model as the Legendre transform of the free canonical energy. The sum in (1) runs over the set Λ∗ l = {p = (p1, . . . , pd) ∈ Rd : pα = 2πnα/l, nα ∈ Z, α = 1, 2, . . . , d}. â†p,σ, âp,σ are the Bose operators of creation and annihilation of particles defined on the Bose Fock space FB and satisfying the usual commutation rules: [âq,σ1 , â † p,σ2 ] = âq,σ1 â † p,σ2 − â†p,σ2 âq,σ1 = δp,qδσ1,σ2 . n̂p,σ = â†p,σâp,σ is the number operator associated with mode p and internal level σ. In this case Ĥ0 l = ∑ p∈Λ∗ l ,σ λl(p, σ)n̂p,σ, a > 0, γ ∈ R. N̂ = ∑ p∈Λ∗ l ,σ â†p,σâp,σ is the total number operator, N̂ ′ = ∑ p∈Λ∗ l \{0},σ â†p,σâp,σ is the total number operator with exclusion of n̂0,−, n̂0,+ and λl(p, σ) = { λl(0, σ) < 0, p = 0, p2/2, p 6= 0, where λl(0,−) = −λ − O(V −s l ), λl(0,+) = −λ + O(V −s l ) with s > 0, λ > 0 and λl(0, σ) → λ(0, σ) = −λ as Vl → ∞. Note that the boson Fock space FB is isomorphic to the tensor product ⊗σ,p∈Λ∗ l FB p,σ where FB p,σ is the boson Fock space constructed on the one-dimensional Hilbert space Hp,σ = {γeip·x⊗eσ}γ∈C, where e− = (0, 1) and e+ = (1, 0). Let pl(β, µ) = 1 βVl lnTrFB exp ( −β(Ĥl − µN̂) ) (2) be the grand-canonical pressure, at finite volume, corresponding to Ĥl, where β = θ−1 is the inverse temperature. If Ĥl(µ) = Ĥl−µN̂ , the equilibrium Gibbs state (grand canonical ensemble) 〈−〉Ĥl(µ) is defined as 〈Â〉Ĥl(µ) = [ TrFB exp ( −βĤl(µ) )]−1 TrFB Â exp ( −βĤl(µ) ) , (3) for any operator Â acting on the symmetric Fock space. Finally, the total density of particles ρ(µ) for infinite volume is defined as lim Vl→∞ 〈 N̂ Vl 〉 Ĥl(µ) = lim Vl→∞ ρl(µ) = ρ(µ), (4) and the density of particles ρ0,σ(β, µ) associated with the energy λ(0, σ) = −λ is defined as lim Vl→∞ 〈 n̂0,σ Vl 〉 Ĥl(µ) = lim Vl→∞ ρ0,σ,l(β, µ) = ρ0,σ(β, µ). (5) We shall say that the system undergoes a macroscopic occupation of the single particle mode (0, σ) (BEC) if ρ0,σ(β, µ) > 0. 3. Pressure Let fl(β, ̺) be the free canonical energy at finite volume Vl, inverse temperature β and density ̺, corresponding to Hamiltonian given by equation (1). Let f̃l(β, ̺), f id l (β, ̺), f id ′ l (β, ̺) be the 43003-2 Model of Bose-atoms with internal structure finite free canonical energies associated with H̃l, Ĥ id l , and Ĥ id ′ l , respectively. These operators are given by H̃l = Ĥ id l + a Vl ∑ p∈Λ∗ l ,σ n̂2 p,σ + γ Vl n̂0,−n̂0,+ , where Ĥ id l = −λ(n̂0,− + n̂0,+) + ∑ p∈Λ∗ l \{0},σ λl(p, σ)n̂p,σ , Ĥ id ′ l = ∑ p∈Λ∗ l \{0},σ λl(p, σ)n̂p,σ . Let f(β, ̺), f̃(β, ̺), f id(β, ̺), f id ′ (β, ̺) be the corresponding limit free canonical energies. Let ̺l = N/Vl, ̺ = lim Vl,N→∞ ̺l = constant. We shall use the symbol ̺ when referring to ̺l or ̺ indistinctively, avoiding excessive notation. The strategy developed in [3] enables us to prove the following theorem. Theorem 1. f̃(β, ̺) = − inf ̺0,−,̺0,+∈[0,̺] { − λ(̺0,− + ̺0+) + a̺20,− + a̺20,+ + γ̺0−̺0+ + f id ′ (β, ̺− ̺0) } . (6) Proof. Being np,σ = 0, 1, 2, . . . , ̺0,− = n0,−/Vl, ̺0,+ = n0,+/Vl, the finite canonical free energies f id l (β, ̺), f̃l(β, ̺), can be written in the following form, f id l (β, ̺) = − 1 βVl ln ∑ np,σ=0,1,2,...,p∈Λ∗ l ,σ exp ( −β ∑ λl(p, σ)np,σ ) δ∑ p∈Λ∗ l ,σ np,σ=[̺Vl] , (7) f̃l(β, ̺) = − 1 βVl ln   ∑ ···+np,σ+···=[̺Vl] e−βVlhl(̺,̺0,−,̺0,+)   , (8) where hl(̺, ̺0,−, ̺0,+) = −λ(̺0,− + ̺0,+) + ̺20,+ + a̺20,+ + γ̺0,−̺0,+ − 1 βVl ln ∑ np,σ=0,1,...,p∈Λ∗ l \{0},σ exp [ −β ( ∑ λl(p, σ)np,σ + a Vl n2 p,σ )] δN ′=[̺Vl]−[̺0Vl] (9) and N ′ = ∑ p∈Λ∗ l \{0},σ np,σ. The following inequality − f̃l(β, ̺) = 1 βVl ln   ∑ ..+np,σ+..=[̺Vl] e−βVlhl(̺,̺0,− ,̺0,+)   > 1 βVl ln ( e−βVlhl(̺,̺0,−,̺0,+) ) = −hl(̺, ̺0,−, ̺0,+), (10) holds for ̺0,−, ̺0,+ ∈ [0, ̺], being [b] the integer part of b. Equation (10) implies that, f̃l(β, ̺) 6 inf ̺0,−,̺0,+∈[0,̺] hl(̺, ̺0,−, ̺0,+). (11) 43003-3 M. Corgini, D.P. Sankovich On the other hand, being n0 = n0,+ + n0,−, we have, − f̃l(β, ̺) 6 1 βVl ln   [̺Vl] ∑ n0=0,N ′=0 exp ( −βVl inf ̺0,−,̺0,+∈[0,̺] hl(̺, ̺0,−, ̺0,+) )   6 1 βVl ln  exp ( −βVl inf ̺0,−,̺0,+∈[0,̺] hl(̺, ̺0,−, ̺0,+) ) [̺Vl] ∑ n0=0,N ′=0 1   6 − inf ̺0,−,̺0,+∈[0,̺] hl(̺, ̺0,−, ̺0,+) + 2 βVl ln ( 1 + [̺Vl](1 + [̺Vl]) 2 ) 6 − inf ̺0,−,̺0,+∈[0,̺] hl(̺, ̺0,−, ̺0,+) + 4 βVl ln([̺Vl] + 1). (12) Thus, we obtain the inequalities inf ̺0,−,̺0,+∈[0,̺] hl(̺, ̺0,−, ̺0,+) − 4 βVl ln([̺Vl] + 1) 6 f̃l(β, ̺) 6 inf ̺0,−,̺0,+∈[0,̺] hl(̺, ̺0,−, ̺0,+). (13) Therefore, in the thermodynamic limit it follows that, f̃(β, ̺) = lim Vl→∞ inf ̺0,−,̺0,+∈[0,̺] hl(̺, ̺0,−, ̺0,+). (14) hl(̺, ̺0,−, ̺0,+) can be rewritten as hl(̺, ̺0,−, ̺0,+) = −λ(̺0,− + ̺0,+) + a̺20,− + a̺20,+ + γ̺0,−̺0,+ − 1 Vl ln 〈 exp  − βa Vl ∑ p∈Λ∗ l \{0},σ n̂2 p,σ   〉 Ĥid ′ l (β,̺−̺0) + f id ′ l (β, ̺− ̺0), (15) being 〈−〉 Ĥid ′ l (β,̺−̺0) the canonical Gibbs state associated with Ĥ id ′ l (β, ̺−̺0). Since the limit free canonical energy of the free Bose gas is the Legendre transform of the corresponding pressure, we get, lim Vl→∞ f id ′ l (β, ̺) = f id ′ (β, ̺) = sup α60 {α̺− pid ′ (β, α)}, (16) being pid ′ (β, α) the limit grand canonical pressure associated with Ĥ id ′ l . From the Jensen inequality we get 〈 exp  − βa Vl ∑ p∈Λ∗ l \{0},σ n̂2 p,σ   〉 Ĥid ′ l (β,̺−̺0) > exp  −βaVl ∑ p∈Λ∗ l \{0},σ 〈 n̂2 p,σ V 2 l 〉 Ĥid ′ l (β,̺−̺0)   . (17) For p and Vl fixed and r > 1, r ∈ Z+, the moments 〈 n̂r p,σ 〉 Ĥid ′ l (β,̺−̺0) in the canonical ensemble, are monotonously increasing functions of ̺ (see [4, 5]). Therefore, 〈 n̂2 p,σ V 2 l 〉 Ĥid ′ l (µ(̺−̺0)) = ∫ [0,∞) 〈 n̂2 p,σ V 2 l 〉 Ĥid ′ l (β,x) K̂Vl (̺− ̺0, dx) > ∫ [̺−̺0,∞) 〈 n̂2 p,σ V 2 l 〉 Ĥid ′ l (β,x) K̂Vl (̺− ̺0, dx) > 〈 n̂2 p,σ V 2 l 〉 Ĥid ′ l (β,̺−̺0) K̂Vl (̺− ̺0, [̺− ̺0,∞)), 43003-4 Model of Bose-atoms with internal structure where 〈·〉 Ĥid ′ l (µ) is the Gibbs state associated with Ĥ id ′ l in the grand-canonical ensemble given by equation (3) and K̂Vl (̺ − ̺0, dx) is the so-called Kac measure of the perfect Bose gas at finite volume Vl ([4, 6]) given by, K̂Vl (̺, dx) = δ(x− ̺)dx for ̺ 6 ̺c(β), being ̺, ̺c(β) = 2(2π)−d ∫ Rd(e βp2/2 − 1)−1dp, the density and the critical density of the perfect Bose gas (the case of atoms with internal structure), respectively, and K̂Vl (̺, dx) = { 0, for x 6 ̺c(β), (̺− ̺c(β)) −1 exp ( −x−̺c(β) ̺−̺c(β) ) dx, for x > ̺c(β) for ̺ > ̺c(β). Using the latter inequality and taking into account that lim Vl→∞ K̂Vl (̺− ̺0, [̺− ̺0,∞)) 6= 0, lim Vl→∞ ∑ p∈Λ∗ l \{0},σ 〈 n̂2 p,σ V 2 l 〉 Ĥid ′ l (µ(̺−̺0)) = 0, we get lim Vl→∞ ∑ p∈Λ∗ l \{0},σ 〈 n̂2 p,σ V 2 l 〉 Ĥid ′ l (β,̺−̺0) = 0. Then, equations (15) and (17) imply that, lim Vl→∞ inf ̺0,−,̺0,+∈[0,̺] hl(̺, ̺0,−, ̺0,+) 6 inf ̺0−,̺0+∈[0,̺] {−λ(̺0,− + ̺0,+) + a̺20,+ + a̺20,+ + γ̺0,−̺0,+ + f id ′ (β, ̺− ̺0)}. (18) On the other hand, since exp { − βa/Vl ∑ p∈Λ∗ l \{0},σ n2 p,σ } 6 1 from equation (15) we get, lim Vl→∞ inf ̺0,−,̺0,+∈[0,̺] hl(̺, ̺0,−, ̺0,+) > lim Vl→∞ inf ̺0,−,̺0,+∈[0,̺] {−λ(̺0,− + ̺0,+) + a̺20,− + a̺20,+ + γ̺0,−̺0,+ + f id ′ l (β, ̺− ̺0)}. (19) Equations (18), (19) imply lim Vl→∞ inf ̺0,−,̺0,+∈[0,̺] hl(̺, ̺0,−, ̺0,+) = lim Vl→∞ inf ̺0,−,̺0,+∈[0,̺] {−λ(̺0,− + ̺0,+) + a̺20,− + a̺20,+ + γ̺0,−̺0,+ + f id ′ l (β, ̺− ̺0)}. (20) Theorem 2. f(β, ̺) = f̃(β, ̺). (21) Proof. Let Ĥ (N) l and H̃ (N) l be the restrictions of the self-adjoint operators Ĥl and H̃l defined on D ⊂ FB to the N-particles symmetrized Bose-space. In this case the following well-known Bogolyubov inequalities for free energies, 〈 ∆H (N) l Vl 〉 Ĥ (N) l (̺) 6 fl(β, ̺) − f̃l(β, ̺) 6 〈 ∆H (N) l Vl 〉 H̃ (N) l (̺) , (22) 43003-5 M. Corgini, D.P. Sankovich hold, where ∆H (N) l = Ĥ (N) l − H̃ (N) l , 〈−〉 Ĥ (N) l (̺) , 〈−〉 H̃ (N) l (̺) are the Gibbs states in the canonical ensemble associated with the Hamiltonians Ĥ (N) l , H̃ (N) l , respectively. Being ∆n̂0 = n̂0,+ − n̂0,−, this leads to the following inequalities, fl(β, ̺) > f̃l(β, ̺)− a V 2 l ∑ p,σ 〈n̂p,σ〉Ĥ(N) l (̺) + O(V −s l ) Vl 〈∆n̂0〉Ĥ(N) l (̺) , (23) fl(β, ̺) 6 f̃l(β, ̺)− a V 2 l ∑ p,σ 〈n̂p,σ〉H̃(N) l (̺) + O(V −s l ) Vl 〈∆n̂0〉H̃(N) l (̺) . (24) Noting that, lim Vl→∞ 1 V 2 l ∑ p,σ 〈n̂p,σ〉Ĥ(N) l (̺) = lim Vl→∞ 1 V 2 l ∑ p,σ 〈n̂p,σ〉Ĥ(N) l (̺) = 0, (25) and lim Vl→∞ O(V −s l ) Vl 〈∆n̂0〉Ĥ(N) l (̺) = lim Vl→∞ O(V −s l ) Vl 〈∆n̂0〉H̃(N) l (̺) = 0, (26) we obtain lim Vl→∞ fl(β, ̺) = lim Vl→∞ f̃l(β, ̺) = lim Vl→∞ inf ̺0,−,̺0,+∈[0,̺] hl(̺, ̺0,−, ̺0,+). (27) This completes the proof. This result enables us to derive an explicit expression for the limit pressure p(β, µ) given by p(β, µ) = lim Vl→∞ pl(β, µ). Let q(x, y) : R2 → R be the symmetric quadratic form defined by, q(x, y) = (µ+ λ)(x + y)− a(x2 + y2)− γxy. (28) Let A = { µ ∈ R : q∗ = sup x,y∈[0,∞) q(x, y) < ∞ } . Definition 1. The domain of stability D(p) of p(β, µ) is defined as, D(p) = { (β, µ) ∈ R 2 : β > 0, µ ∈ A ∩ (−∞, 0] } . (29) Corollary 1. For (β, µ) ∈ D(p), p(β, µ) = sup ̺0,−,̺0,+∈[0,∞) {(µ+ λ)(̺0,− + ̺0,+)− a(̺20,− + ̺20,+)− γ̺0,−̺0,+}+ pid ′ (β, µ). (30) Proof. Since f(β, ̺) is a convex function of ̺, its Legendre transform coincides with the grand canonical limit pressure p(β, µ), i.e., p(β, µ) = sup ̺>0 {µ̺− f(β, ̺)} = sup ̺>0 {µ̺− f̃(β, ̺)}. (31) Therefore p(β, µ) = sup ̺>0 { µ̺− inf ̺0,−,̺0,+∈[0,̺] { − λ(̺0,− + ̺0,+) + a(̺20,− + ̺20,+) + γ̺0,−̺0,+ + f id ′ (β, ̺− ̺0) } = sup ̺>0 sup ̺0,−,̺0,+∈[0,̺] { (µ+ λ)(̺0,− + ̺0,+)− a̺20,− − a̺20,+ − γ̺0,−̺0,+ − f id ′ (β, ̺− ̺0) + µ(̺− ̺0 } } = sup ̺0,−,̺0,+∈[0,∞) { (µ+ λ)(̺0,− + ̺0,+)− a(̺20,+ + ̺20,+)− γ̺0,−̺0,+ } + pid ′ (β, µ). (32) 43003-6 Model of Bose-atoms with internal structure This corollary implies that, the derivation of the limit pressure and demonstration of the oc- currence of non-conventional Bose-Einstein condensation (independent of temperature) can be reduced to the study of the occurence of extreme values of the symmetric quadratic form q given in equation (28). Proposition 1. q∗ = sup x,y∈[0,∞) q(x, y) satisfies q∗ = +∞, for γ ∈ (−∞,−2a) , µ ∈ (−λ, 0]. q∗ = { 0, µ ∈ (−∞,−λ], +∞, µ ∈ (−λ, 0], for γ = −2a. q∗ = { 0, µ ∈ (−∞,−λ], (µ+λ)2 2a+γ , µ ∈ (−λ, 0], for γ ∈ (−2a, 2a). q∗ = { 0, µ ∈ (−∞,−λ], (µ+λ)2 4a , µ ∈ (−λ, 0], for γ ∈ [2a,∞). Proof. Let us introduce some basic notions concerning minimization and maximization of convex and concave quadratic functions. Let f : Rn → R be the quadratic form given by: f(x) = 1 2 xQxT + cxT, where Q is a symmetric n × n- matrix of real entries and c ∈ Rn. The function f is a convex (concave, respectively) function if and only if it is a symmetric and positive (negative,respectively) semidefinite function, i.e. xQxT > 0, (xQxT 6 0, respectively) for all x ∈ Rn. Then, being f a convex (concave, respectively) function it attains its global minimum (global maximum, respec- tively) at x∗ if and only if x∗ solves the equations system ∇f(x) = QxT + c = 0. In this case the Hessian matrix H(x) satisfies H(x) = Q. For the quadratic form q(x, y) given by equation (28) we have Q = − ( 2a γ γ 2a ) , c = (µ+ λ)(1, 1). Therefore, q is a strictly concave function (xQxT < 0) if a, γ satisfy the following condition: detQ = ∣ ∣ ∣ ∣ 2a γ γ 2a ∣ ∣ ∣ ∣ = −4a2 + γ2 < 0, i.e., γ ∈ (−2a, 2a), and it is a strictly convex function (xQxT > 0) if detQ = −4a2 + γ2 > 0, i.e., γ ∈ (−∞,−2a) ∪ (2a,∞). The same results can be obtained by using the standard approach based on second derivatives to study the functions of two variables. We define the auxiliary quadratic forms q1, q2, q3 by q1(x, y) =(µ+ λ)(x + y)− a(x− y)2, q2(x, y) =(µ+ λ)(x + y)− a(x+ y)2, q3(x, y) =(µ+ λ)(x + y)− γ 2 (x+ y)2. 43003-7 M. Corgini, D.P. Sankovich a) Case µ ∈ (−∞,−λ] (µ+ λ 6 0). For µ satisfying the condition above we have, q1(x, y) 6 0, q2(x, y) 6 0 for all x, y ∈ [0,∞) and sup x,y∈[0,∞) q1(x, y) = sup x,y∈[0,∞) q2(x, y) = 0. On the other hand, for γ > 0, q3(x, y) 6 0 for all x, y ∈ [0,∞) and sup x,y∈[0,∞) q3(x, y) = 0. If γ ∈ (−∞,−2a) and x ∈ [0,∞), the following inequality holds q∗ > q(x, x) = (µ+ λ)2x− (2a+ γ)x2. Since −(2a+ γ) > 0, we get lim x→+∞ q(x, x) = +∞. Then, q∗ = +∞. If γ = −2a and x, y ∈ [0,∞), we have q(x, y) = q1(x, y). Then q∗ = 0. If γ ∈ (−2a, 2a) and x, y ∈ [0,∞), the following inequalities, q2(x, y) 6 q(x, y) 6 q1(x, y) hold, leading to the conclusion that q∗ = 0. For γ ∈ [2a,+∞) and x, y ∈ [0,∞) we obtain, q3(x, y) 6 q(x, y) 6 q2(x, y), which finally implies that q∗ = 0. b) Case µ ∈ (−λ, 0] (µ+ λ > 0). In this case, for γ ∈ (−2a, 2a), q is a concave function, taking a global maximum at x∗ = y∗ = (µ+λ)/(2a+γ), for (x, y) ∈ [0,∞)× [0,∞). Then, using these results and equation (28) we get q∗ = (µ+ λ)2/(2a+ γ). For γ ∈ (−∞,−2a], x ∈ [0,∞), taking into account that µ+ λ > 0, −(2a+ γ) > 0, we have, q∗ > q(x, x) = (µ+ λ2)2x− (2a+ γ)x2 > 0. Then, noting that lim x→+∞ q(x, x) = +∞, we obtain q∗ = +∞. For γ ∈ [2a,∞) and x, y ∈ [0,∞), the following inequality holds, q(x, y) 6 q2(x, y). It is not hard to see that, sup x,y∈[0,∞) q2(x, y) = (µ+ λ)2/4a, i.e., q∗ 6 (µ+ λ)2 4a . Since for x∗ = (µ+ λ)/2a, y∗ = 0 the quadratic form q satisfies q(x∗, 0) = (µ+ λ)2/4a, we conclude that q∗ = (µ+ λ)2/4a. The above results completely determine D(p), and lead to the following theorem. Theorem 3. For (β, µ) ∈ D(p) the limit pressure is given by p(β, µ) = q∗ + pid ′ (β, µ). (33) 43003-8 Model of Bose-atoms with internal structure 4. Non-conventional BEC From theorem 3 one easily deduces the following corollary on non-conventional BEC. Corollary 2. Let ρ0(µ) = ρ0,−(µ) + ρ0,+(µ) be the total amount of condensate. i.) For µ ∈ (−∞,−λ], ρ0(µ) = 0 if γ ∈ [−2a,∞). ii.) For µ ∈ (−λ, 0], ρ0(µ) = { 2(µ+λ) 2a+γ , γ ∈ (−2a, 2a), 2(µ+λ) 4a , γ ∈ [2a,∞). Non-conventional BEC takes place only for µ ∈ (−λ, 0] provided that (β, µ) ∈ D(p). Moreover, the term representing interstate collisions satisfies: lim Vl→∞ 〈 n̂0,−n̂0,+ V 2 l 〉 Ĥl(µ) =    ( µ+λ 2a+γ )2 , γ ∈ (−2a, 2a), 0, γ ∈ [2a,∞). For µ, λ fixed, with µ ∈ (−λ, 0], the total amount of condensate ρ0 : (−2a,+∞) → [ µ+ λ 2a ,+∞ ) as function of γ is a decreasing function, satisfying lim γ↓−2a ρ0 = +∞, lim γ↑2a ρ0 = (µ+ λ)/2a and taking the constant value (µ+ λ)/2a when γ ∈ [2a,+∞). This behavior is a direct consequence of the fact that the symmetric quadratic form q is a strictly concave function for γ ∈ (−2a, 2a) and becomes a convex function for γ ∈ (−∞,−2a) ∪ (2a,+∞). Hamiltonian (1) can be rewritten in the following form, Ĥl = ∑ p∈Λ∗ l ,σ λl(p, σ)n̂p,σ + ( 2a+ γ 2Vl ) (n̂2 0,− + n̂2 0,+) − γ 2Vl (n̂0,− − n̂0,+) 2 − a Vl (n̂0,− + n̂0,+) + a Vl ∑ p∈Λ∗ l \{0},σ (â†p,σ) 2â2p,σ . (34) For µ ∈ (−λ, 0] and γ ∈ (−2a, 2a), in the thermodynamic limit, only the term ( 2a+ γ 2Vl ) (n̂2 0,− + n̂2 0,+) contributes to the emergence of non-conventional BEC. In this sense, under those restrictions, our model is thermodynamically equivalent to the system whose energy is represented by the Hamiltonian, Ĥ (1) l = ∑ p∈Λ∗ l ,σ λl(p, σ)n̂p,σ + ( 2a+ γ 2Vl ) (n̂2 0,− + n̂2 0,+) + a Vl ∑ p∈Λ∗ l \{0},σ (â†p,σ) 2â2p,σ . (35) On the other hand, for µ ∈ (−λ, 0] and γ ∈ [2a,+∞), the model under study is thermodynam- ically equivalent to the model given by the following Hamiltonian, Ĥ (2) l = ∑ p∈Λ∗ l ,σ λl(p, σ)n̂p,σ + a Vl (n̂2 0,− + n̂2 0,+) + a Vl ∑ p∈Λ∗ l \{0},σ (â†p,σ) 2â2p,σ . (36) 43003-9 M. Corgini, D.P. Sankovich Quantum many-particle systems associated with energy operators given by equations (35) and (36) have been extensively studied in [1, 7–9] and references therein. Adapting results in [6], it is easy to verify that for d > 2, the models of the type given by equations (35) and (36) undergo generalized BEC in the following sense, lim δ→0+ lim Vl→∞ ∑ {p∈Λ∗ l :0<\|\|p\|\|<δ,σ=±} 〈 n̂0,σ Vl 〉 Ĥl(µ) = { 0, ρ 6 ρPc (β), ρ− ρPc (β), ρ > ρPc (β) where, ρPc (β) = 2 (2π)d ∫ Rd (eβ p2 2 − 1)−1dp is the critical density of the perfect Bose gas (Bose-atoms with internal two-level structure). In this sense, for the model under study, conventional condensate coexists at µ = 0 with the non-conventional condensate (see [6]). From a physical point of view the above facts imply that for γ ∈ [2a,+∞), the interstate collisions term does not play any role in the thermodynamic behavior of the system and non- conventional BEC is only the consequence of the presence of intrastate collisions (self-scattering term). However, for γ ∈ (−2a, 2a), non-conventional BEC is enhanced by the presence of a cross- scattering term (for example, the case of a cross-scattering term with a negative coupling parameter γ close to −2a). 5. Conclusion We have made use of a strategy based on the derivation of the limit free canonical energy (see [3]) to obtain an analytic expression for the limit pressure of a system of Bose atoms whose ground state has two internal levels. We have proved that negative ground state energies, for a range of values of the chemical potential, leads to non-conventional BEC, being the amount of condensate as a function depending on the variables γ, a associated with the interstate collisions and intrastate collisions. In this way we recover the previous results obtained in the framework of the approximating Hamiltonians method [1]. Acknowledgements Partial financial support by PBCT–ACT13 (Stochastic Analysis Laboratory, Chile) and Pro- grama de Maǵıster en Matemáticas, Universidad de La Serena, Chile. References 1. Corgini M., Rojas-Molina C., Sankovich D.P., Int. J. Mod. Phys. B, 2008, 22, 4799–4815. 2. Bogolyubov N.N, Lectures on Quantum Statistics: Quasiaverages, Vol. 2, Gordon and Beach, New York, 1970. 3. Lewis J.T., Mark Kac Seminar on Probability and Physics: The Large Deviation Principle in Statistical Mechanics, syllabus 17, Centrum voor Wiskunde en Informatica (1985–1987) (Amsterdam: Centrum voor Wiskunde en Informatica CWI). 4. Pulé J.V., Zagrebnov V.A., J. Math. Phys., 2004, 268, 3565–3583. 5. Buffet E., Pulé J.V., J. Math. Phys., 1983, 24, 1608–1616. 6. Bru J.V., Nachtergaele B., Zagrebnov V.A., J. Stat. Phys., 2002, 109, 143–176. 7. Zagrebnov V.A., Condens. Matter Phys., 2000, 3, 265–275. 8. Corgini M., Sankovich D.P., Phys. Lett. A, 2007, 360, 419–422. 9. Bru J.B., Zagrebnov V.A., J. Phys. A: Math. Gen., 2000, 33, 449–464. 43003-10 Model of Bose-atoms with internal structure Розв’язна модель Бозе-атомiв з дворiвневою внутрiшньою структурою: нестандартна Бозе-Айнштайнiвська конденсацiя М. Коргiнi1,2, Д.П. Санковiч3 1 Факультет математики, Унiверситет де Ла Серена, Ла Серена, Чилi 2 Лабораторiя стохастичного аналiзу, Чилi 3 Математичний iнститут iм. В.А. Стєклова, Москва, Росiя Для системи Бозе-атомiв, чий оператор енергiї є дiагональним по так званому числу операторiв i його основний стан має внутрiшню дворiвневу структуру з негативними енергiями, отримано точнi вирази для граничних вiльної канонiчної енергiї та тиску. Також доведено iснування нестандартної Бозе-Айнштайнiвська конденсацiї. Ключовi слова: метод апроксимуючого гамiльтонiана, нестандартна Бозе-Айнштайнiвська конденсацiя 43003-11 Introduction The model Pressure Non-conventional BEC Conclusion

Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation

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