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

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).


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).

The model
The one-particle free Hamiltonian corresponds to the operator S l = −△/2 defined on a dense subset of the Hilbert space H l = L 2 (Λ l ), being Λ l = [−l/2, l/2] d ⊂ R d a cubic box of boundary ∂Λ l and volume V l = l d .In other words, the particles are confined to bounded regions.We assume periodic boundary conditions under which S l 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 ∈ L 2 (Λ l ) and s ∈ C 2 represents the internal state.Therefore, the vector space associated with this system is in fact, H l s = L 2 (Λ l ) ⊗ C 2 .We shall study a model of Bose particles whose Hamiltonian is given by: 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.

Pressure
Let f l (β, ̺) be the free canonical energy at finite volume V l , inverse temperature β and density ̺, corresponding to Hamiltonian given by equation (1).Let fl (β, ̺), f id l (β, ̺), f id ′ l (β, ̺) be the finite free canonical energies associated with Hl , Ĥid l , and Ĥid ′ l , respectively.These operators are given by 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.
Proof.Being n p,σ = 0, 1, 2, . . ., ̺ 0,− = n 0,− /V l , ̺ 0,+ = n 0,+ /V l , the finite canonical free energies f id l (β, ̺), fl (β, ̺), can be written in the following form, where and On the other hand, being n 0 = n 0,+ + n 0,− , we have, Thus, we obtain the inequalities inf Therefore, in the thermodynamic limit it follows that, f (β, ̺) = lim h l (̺, ̺ 0,− , ̺ 0,+ ) can be rewritten as being − Ĥid ′ l (β,̺−̺0) the canonical Gibbs state associated with Ĥid ′ 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 being p id ′ (β, α) the limit grand canonical pressure associated with Ĥid ′ l .From the Jensen inequality we get exp For p and V l fixed and r 1, r ∈ Z + , the moments nr in the canonical ensemble, are monotonously increasing functions of ̺ (see [4,5]).Therefore, where is the Gibbs state associated with Ĥid ′ l in the grand-canonical ensemble given by equation ( 3) and KV l (̺ − ̺ 0 , dx) is the so-called Kac measure of the perfect Bose gas at finite volume V l ( [4,6]) given by, KV l (̺, dx) = δ(x − ̺)dx , the density and the critical density of the perfect Bose gas (the case of atoms with internal structure), respectively, and Using the latter inequality and taking into account that = 0, we get lim Then, equations ( 15) and ( 17) imply that, lim On the other hand, since exp 1 from equation (15) we get, lim Equations ( 18), (19) imply lim Proof.Let Ĥ(N) l and H(N) l be the restrictions of the self-adjoint operators Ĥl and Hl defined on D ⊂ F B to the N-particles symmetrized Bose-space.In this case the following well-known Bogolyubov inequalities for free energies, hold, where ∆H , respectively.Being ∆n 0 = n0,+ − n0,− , this leads to the following inequalities, Noting that, lim and lim we obtain lim 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 Let q(x, y) : R 2 → R be the symmetric quadratic form defined by, q(x, y) = (µ + λ)(x + y) − a(x 2 + y 2 ) − γxy. (28) Definition 1.The domain of stability D(p) of p(β, µ) is defined as, Corollary 1.For (β, µ) ∈ D(p), Proof.Since f (β, ̺) is a convex function of ̺, its Legendre transform coincides with the grand canonical limit pressure p(β, µ), i.e., Therefore This corollary implies that, the derivation of the limit pressure and demonstration of the occurrence 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).
, µ ∈ (−λ, 0], Proof.Let us introduce some basic notions concerning minimization and maximization of convex and concave quadratic functions.Let f : R n → R be the quadratic form given by: where Q is a symmetric n × n-matrix of real entries and c ∈ R n .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. xQx T 0, (xQx T 0, respectively) for all x ∈ R n .Then, being f a convex (concave, respectively) function it attains its global minimum (global maximum, respectively) at x * if and only if x * solves the equations system ∇f (x) = Qx T + 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 Therefore, q is a strictly concave function (xQx T < 0) if a, γ satisfy the following condition: ), and it is a strictly convex function (xQx The same results can be obtained by using the standard approach based on second derivatives to study the functions of two variables.
The above results completely determine D(p), and lead to the following theorem.

Non-conventional BEC
From theorem 3 one easily deduces the following corollary on non-conventional BEC.
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, where, 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 nonconventional 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 crossscattering term (for example, the case of a cross-scattering term with a negative coupling parameter γ close to −2a).

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].

Розв'язна модель
̺) are the Gibbs states in the canonical ensemble associated with the Hamiltonians Ĥ(N)