The stochastic limit in the analysis of the open BCS model 1

The stochastic limit in the analysis of the open BCS model

Fabio Bagarello

Bjalowieza, 2004

1J. Phys. A, 2004 and IDAQP, submitted

General problem

Suppose that

System

S

←→ Reservoir

R

How R affects the time evolution of S?

Description nr.1: S is not a conservative system (it changes energy with R) ⇒ its time evolution α^t_S is not a group but only a semigroup (of completely positive maps,...., Lindblad, Gorini,...);

Description nr.2: S + R is a conservative system, so that its time evolution α^t_S+R is a group whose generator is a self-adjoint operator, the hamiltonian H

H = H| _S {z+ H_R}

=:H₀

+λH_I Strategy nr.1: Master equation approach

Main reference: P.A. Martin, Mod`eles en M´ecanique Statis- tique des Processus Irr´eversibles, Lecture Notes in Physics, 103,

Springer-Verlag, Berlin, (1979)

Strategy nr.2: Stochastic limit approach

Main reference: L. Accardi, Y.G. Lu, I. Volovich, Quantum Theory and its Stochastic Limit, Springer (2002)

F.B., Many-body applications of the stochastic limit: a re- view, Reports in Math. Phys., submitted

the stochastic limit approach

Starting point: the interaction picture Let

H = H₀ + λH_I ⇒

Ψ(t) = e^{−i(H0+λHI)t}Ψ₀ (Schr¨odinger picture) and

Ψ_I(t) = e^itH0Ψ(t) = e^itH0e^{−i(H0+λHI)t}Ψ₀ (interaction picture)

∂_tΨ_I(t) = −iλH_I(t)Ψ_I(t), where H_I(t) = e^iH0tH_Ie^−iH0t and Ψ_I(t)|_t=t0 = Ψ_I(t₀). Its solution is

Ψ_I(t) = U(t, t₀)Ψ_I(t₀) where the wave operator U(t, t₀) is such that:

• ∂_tU(t, t₀) = −iλH_I(t)U(t, t₀) with U(t₀, t₀) = I;

• U(t, t₀) = U(t, t₁)U(t₁, t₀);

• U(t, t₀)^∗ = U(t₀, t).

We can also write

U(t, t₀) = I − iλ Z _t

0

H_I(t₁)U(t₁, t₀) dt₁ =

= I − iλ Z _t

0

H_I(t₁) dt₁+ (−iλ)² Z _t

0

H_I(t₁) dt₁ Z _t1

0

H_I(t₂) dt₂+ ....

Let now < . > be a state of S + R. We assume that:

• < H_I(t) >= 0, for all t. This is automatic, e.g., for all the dipole hamiltonians;

• < H_I(t₁+s)H_I(t₂+s)...H_I(t_n+s) >=< H_I(t₁)H_I(t₂)...H_I(t_n) >, which is a typical requirement for the n-points functions (Wight- man);

• R

R| < H_I(0)H_I(t) > | dt < ∞ (the two point functions are in L¹(R)).

Under these assumptions the first non trivial contribution in the van Hove limit (λ → 0, t → ∞ with λ²t = τ finite) of the iterated

series for < U (t, t₀) > is

< −λ² Z _t

0

H_I(t₁) dt₁ Z _t

1

0

H_I(t₂) dt₂ >−→ −τ Z ₀

−∞

< H_I(0)H_I(s) > ds

How to implement the van Hove limit:

• we rewrite ∂_tU(t, t₀) = −iλH_I(t)U(t, t₀) as

∂_tU_λ(t) = −iλH_I(t)U_λ(t);

• rescaling t → t/λ² this becomes

∂_tU_λ(t/λ²) = −i

λH_I(t/λ²)U_λ(t/λ²) or

U_λ(t/λ²) = I − i λ

Z _t

0

H_I(t₁/λ²)U_λ(t₁/λ²) dt₁ =

= I − i λ

Z _t

0

H_I(t₁/λ²) dt₁+ +

µ

−i λ

¶₂Z _t

0

H_I(t₁/λ²) dt₁ Z _t

1

0

H_I(t₂/λ²) dt₂ + ...

• if ϕ₀ is the ground vector for R (T=0) and ξ is an arbitrary vector for S we consider vector states like

< . >=< ϕ₀ ⊗ ξ, . ϕ₀ ⊗ ξ > .

The first non-trivial term in the expansion of U_λ(t/λ²) is there- fore

I_λ(t) :=

µ

−i λ

¶₂Z _t

0

dt₁ Z _t

1

0

< H_I(t₁/λ²)H_I(t₂/λ²) > dt₂. The first aim of the SLA consists in getting I(t) = lim_λ,0I_λ(t) and finding a new time-dependent operator H_I^(sl)(t), acting on a new Hilbert space H^(sl) with cyclic vector η₀ 6= ϕ₀ such that

I(t) = − Z _t

0

dt₁ Z _t

1

0

dt₂< H_I^(sl)(t₁)H_I^(sl)(t₂) >_η0⊗ξ

| {z }

nothing depends onλ

H_I^(sl)(t) is now used to deduce the expression of the generator, looking for the time evolution of operators of the algebra A = B(H_S) ⊗ I_R.

The Physicals Model

Our model consists of two main ingredient, the system, which is described by spin variables, and the reservoir, which is given in terms of bosonic operators. It lives in a box of volume V = L³, with N lattice sites.

The system

We define, following Martin and Buffet, [MB] (JSP, 1978 and P.A.Martin, Lecture Notes in Physics, 103),

H_N^(sys) =

int. with a mag. f ield

˜² XN

j=1

σ_j⁰ −

mean f ield interaction

g N

XN i,j=1

σ_i⁺σ_j⁻ ,

i, j −→ discrete values of the momentum that an electron in a fixed volume can have (lattice sites);

σ_j⁺ creates a Cooper pair with given momentum while σ_j⁻ an- nihilates the same pair;

˜² is the energy of a single electron and −g < 0 is the interaction close to the Fermi surface.

The algebra of the Pauli matrices is given by

[σ_i⁺, σ_j⁻] = δ_ijσ_i⁰, [σ^±_i , σ_j⁰] = ∓2δ_ijσ_i^±. We will use the following realization of these matrices:

σ⁰ ≡ σ^z =



 1 0 0 −1



 , σ⁺ =



 0 1 0 0



 , σ⁻ =



 0 0 1 0



 .

Introducing

S_N^α = 1 N

XN i=1

σ_i^α, R_N = S_N⁺S_N⁻ = R^†_N,

H_N^(sys) can be simply written as H_N^(sys) = N(˜²S_N⁰ − gR_N) and we have:

[S_N⁰ , R_N] = [H_N^(sys), R_N] = [H_N^(sys), S_N⁰ ] = 0, for any given N > 0.

The intensive operators S_N^α are all bounded by 1 in the operator norm, and the commutators [S_N^α, σ_j^β] go to zero in norm when N → ∞ as _N¹, for all j, α and β.

The reservoir

We consider here a realistic bosonic reservoir (in contrast with [MB]), so that some of the following formulas must be under- stood to be formal (....but they can be made rigorous using, e.g., [B,JMP1998]).

We introduce here as many bosonic modes a_~p,j as lattice sites are contained in V : j = 1, 2, ..., N . ~p is the value of the momentum of the j-th boson which, because of the periodic boundary condition, is ~p = ^2π_L~n, where ~n = (n₁, n₂, n₃), n_j ∈ Z. These operators satisfy the following CCR,

[a_~p,i, a_~q,j] = [a^†_~p,i, a_~q,j^† ] = 0, [a_~p,i, a^†_~q,j] = δ_ijδ_~p~q and their free dynamics is given by

H_N^(res) = XN

j=1

X

~p∈Λ_N

²_~pa^†_~p,ja_~p,j.

Here Λ_N = {~p = ^2π_L~n, ~n ∈ Z³} and ²_~p = _2m^~p2 = ^4π2(n_2mL²¹⁺ⁿ₂^22+n23).

The interaction

The form of the interaction between reservoir and system is assumed to be of the following form:

H_N^(I) = XN

j=1

(σ_j⁺a_j(f ) + h.c.), where a_j(f ) = P

~p∈Λ_N a_~p,jf (~p), f being a given test function.

Thefinite volumeopen system is now described by the following hamiltonian,

H_N = H_N⁰ + λH_N^(I), where H_N⁰ = H_N^(sys) + H_N^(res), and λ is the coupling constant.

The free evolution of the interaction hamiltonian is:

H_N^(I)(t) = e^{iHN0 t}H_N^(I)e^{−iHN0 t} =

= XN

j=1

(e^iHN(sys)tσ_j⁺e^−iHN(sys)te^iHN(res)ta_j(f )e^−iHN(res)t + h.c.).

The computation of the part of the reservoir is trivial and produces e^iHN(res)ta_j(f )e^−iHN(res)t = a_j(f e^−it²),

where a_j(f e^−it²) = P

~p∈Λ_N a_~p,jf (~p)e^−it²p.

Let us now compute the free evolution of the spin operators.

The differential equations of motion for the spin variables are





dσ⁺_j (t)

dt = 2i˜²σ_j⁺(t) + igS_N⁺(t)σ_j⁰(t)

dσ⁰_j(t)

dt = 2ig(σ_j⁺(t)S_N⁻(t) − σ_j⁻(t)S_N⁺(t)).

(1)

where we have called, with a little abuse of language useful to maintain the notation simple, σ_j^α(t) = e^iHN(sys)tσ_j^αe^−iHN(sys)t (instead of σ_j,N^{α,f ree}(t)).

Let us now call S^α = F − strong lim_N→∞S_N^α. The existence of this limit (and of all its powers) has been discussed, e.g., in [BM, JSP1993] and references therein. With this in mind we can take the sum over j = 1, 2, ..., N of (both sides of) the system (1), divide the result by N, and consider the F −strong lim_N→∞ of the equations obtained in this way. We find that ˙S⁰(t) = 0 and ˙S⁺(t) = i(2˜² + gS⁰(t))S⁺(t), which can be easily solved: S⁰(t) = S⁰ = (S⁰)^† and S⁺(t) = S⁺e^{i(2˜²+gS0)t}. Of course S⁻(t) = (S⁺(t))^†. System (1) is

now replaced by its semiclassical approximation,





dσ_j⁺(t)

dt = 2i˜²σ_j⁺(t) + igS⁺(t)σ_j⁰(t)

dσ_j⁰(t)

dt = 2ig(σ_j⁺(t)S⁻(t) − σ_j⁻(t)S⁺(t)),

which can be explicitly solved using, for instance, the Laplace transform technique, [MB]. It gives

σ_j⁺(t) = e^iνtρ^j₀ + e^i(ν+ω)tρ^j₊ + e^i(ν−ω)tρ^j₋, where we have defined the following operators













ρ^j₀ = ^g2_ω^S₂⁺ ¡

2S⁻σ_j⁺ + S⁰σ_j⁰ + 2S⁺σ_j⁻¢ ρ^j₊ = ^gS_ω2⁺

³

gS^{− ω−gS}_ω+gS⁰₀σ_j⁺ + ^ω−gS₂ ⁰σ_j⁰ − gS⁺σ_j⁻

´ ρ^j₋ = ^gS_ω2⁺

³

gS^{− ω+gS}_ω−gS⁰₀σ_j⁺ − ^ω+gS₂ ⁰σ_j⁰ − gS⁺σ_j⁻

´ , and the following quantities

ω = gp

(S⁰)² + 4S⁺S⁻, ν = 2˜² + gS⁰.

Defining further ν_α(~p) = ν − ²_~p + αω, α = 0, ±, we obtain H_N^(I)(t) =

XN j=1

X

α=0,±

¡ρ^j_αa_j(f e^itνα) + h.c¢ .

Remark:– This same free time evolution can be found also for a fermionic reservoir, since CCR and CAR produce the same free time evolution for both the annihilation and the creation operators.

The next step in the SLA consists in computing the following quantity

I_λ(t) = µ

−i λ

¶₂Z _t

0

dt₁ Z _t1

0

dt₂ω_tot µ

H_N^(I)(t₁

λ²)H_N^(I)(t₂ λ²)

¶ , and its limit for λ going to zero. Here ω_tot = ω_sysω_β, ω_sys being a state of the system, and ω_β a state of the reservoir, which we will take to be a KMS state corresponding to an inverse temperature β = _kT¹ . It is convenient here to use the so-called canonical rep- resentation of thermal states, [ALV,2002]. For that we introduce two sets of mutually commuting bosonic operators {c^(γ)_~p,j}, γ = a, b, as follows:

a_~p,j = p

m(~p)c^(a)_~p,j +p

n(~p)c^(b),†_~p,j ,

where

m(~p) = ω_β(a_~p,ja^†_~p,j) = 1

1 − e^−β²~p, n(~p) = ω_β(a^†_~p,ja_~p,j) = e^−β²~p 1 − e^−β²~p, and

[c^(α)_~p,j, c_~q,k^(γ)†] = δ_jkδ_{~p ~q}δ_αγ,

while all the other commutators are trivial. Then, if we define f_m(~p) = p

m(~p)f (~p) and f_n(~p) = p

n(~p)f (~p), we get a_j(f e^itνα) = c^(a)_j (f_me^itνα) + c^(b)_j ^†(f_ne^itνα),

(both c^(γ)_j (f ) and c^(γ)_j (f )^† are linear in their argument f). We conclude that

H_N^(I)(t) = XN

j=1

X

α=0,±

½ ρ^j_α

µ

c^(a)_j (f_me^itνα) + c^(b)_j ^†(f_ne^itνα)

¶

+ h.c

¾ . Let us now introduce the vacuum of the operators c^(α)_~p,j, Φ₀:

c^(α)_~p,jΦ₀ = 0, ∀~p ∈ Λ_N, j = 1, ..N, α = a, b.

The KMS state ω_β can be represented as the following vector state, as in a GNS-like representation:

ω_β(X_r) =< Φ₀, X_rΦ₀ >,

for X_r = a^]_~p,ja^]_~p,j, where a_~p,j^] = a_~p,j or a^]_~p,j = a^†_~p,j. Therefore ω_tot

µ

H_N^(I)(t₁

λ²)H_N^(I)(t₂ λ²)

¶

= XN

j=1

X

α,β=0,±

X

~p∈Λ_N

{ω_sys(ρ^j_αρ^j_β^†)|f_m(~p)|²·

·e^{iλ2t1να(~p)}e^{−iλ2t2νβ(~p)} + ω_sys(ρ^j_α^†ρ^j_β)|f_n(~p)|²e^{−iλ2t1να(~p)}e^{+iλ2t2νβ(~p)}}.

Since we are interested to lim_λ→0I_λ(t), we must require that the following integral exists finite:

Z ₀

−∞

dτ X

~p∈Λ_N

|f_r(p)|²e^{±iτ να(~p)} < ∞,

where f_r(p) is f_m(p) or f_n(p). Under this assumption we find:

I(t) = lim

λ→0I_λ(t) = −t XN

j=1

X

α=0,±

n

ω_sys(ρ^j_αρ^j_α^†)Γ^(a)_α + ω_sys(ρ^j_α^†ρ^j_α)Γ^(b)_α o

, where

Γ^(a)_α = Z ₀

−∞

dτ X

~p∈Λ_N

|f_m(p)|²e^{−iτ να(~p)}, Γ^(b)_α = Z ₀

−∞

dτ X

~p∈Λ_N

|f_n(p)|²e^{iτ να(~p)}. To this same result we could also arrive starting with the follow- ing stochastic limit hamiltonian

H_N^(sl)(t) = XN

j=1

X

α=0,±

½ ρ^j_α

µ

c^(a)_αj(t) + c^(b)_αj^†(t)

¶

+ h.c

¾ ,

where the operators c^(γ)_αj(t) are assumed to satisfy the following commutation rule,

[c^(γ)_αj(t), c^(µ)_βk^†(t⁰)] = δ_jk δ_αβ δ_γµδ(t − t⁰)Γ^(γ)_α , for t > t⁰. It is now a simple exercise to check that

J(t) = (−i)² Z _t

0

dt₁ Z _t

1

0

dt₂Ω_tot(H_N^(sl)(t₁)H_N^(sl)(t₂))

coincides with I(t), where Ω_tot = ω_sysΩ = ω_sys < Ψ₀, Ψ₀ >.

Here Ψ₀ is the vacuum of the operators c^(γ)_αj(t): c^(γ)_αj (t)Ψ₀ = 0 for all α, j, γ and t.

Following the SLA we now use H_N^(sl)(t) to compute thegenerator of the theory. Let X be an observable of the system and 11_r the identity of the reservoir. Its time evolution (after limit λ → 0 is taken) is j_t(X ⊗11_r) = U_t^†(X ⊗11_r)U_t, where U_t is the wave operator satisfying the following differential equation ∂_tU_t = −iH_N^(sl)(t)U_t (then ∂_tU_t^† = iU_t^†H_N^(sl)(t)).

Then we find

∂_tj_t(X ⊗ 11_r) = iU_t^†[H_N^(sl)(t), X ⊗ 11_r]U_t =

= iU_t^† XN

j=1

X

α=0,±

½

[ρ^j_α, X](c^(a)_αj(t) + c^(b)_αj^†(t)) + [ρ^j_α^†, X](c^(a)_αj^†(t) + c^(b)_αj(t))

¾ U_t To order normally this formula above, i.e. to move to the right all

the annihilation operators c^(γ)_αj and to the left the creation operators c^(γ)_αj ^†, we need the following commutators

[c^(a)_αj(t), U_t] = −iρ^j_α^†Γ^(a)_α U_t, [c^(b)_αj(t), U_t] = −iρ^j_αΓ^(b)_α U_t, and, taking their adjoint,

[U_t^†, c^(a)_αj^†(t)] = iU_t^†ρ^j_αΓ^(a)_α and [U_t^†, c^(b)_αj^†(t)] = iU_t^†ρ^j_α^†Γ^(b)_α . (These results are easily found using the time consecutive prin- ciple, [ALV,2002]).

Going back to ∂_tj_t(X ⊗ 11_r) and recalling that Ω_tot(∂_tj_t(X ⊗ 11_r)) = Ω_tot(j_t(L(X))), we get

L(X) = XN

j=1

X

α=0±

{[ρ^j_α, X]ρ^j_α^†Γ^(a)_α + [ρ^j_α^†, X]ρ^j_αΓ^(b)_α −

−ρ^j_α[ρ^j_α^†, X]Γ^(a)α − ρ^j_α^†[ρ^j_α, X]Γ^(b)α } If X is self-adjoint (X = X^†) we have

L(X) = L₁(X) + L₂(X), where

L₁(X) = XN

j=1

X

α=0±

n

[ρ^j_α, X]ρ^j_α^†Γ^(a)_α + h.c.

o ,

L₂(X) = XN

j=1

X

α=0±

n

[ρ^j_α^†, X]ρ^j_αΓ^(b)_α + h.c.

o

The phase transition

As discussed in [MB], the phase structure of the model is given by the dynamical behavior of S_N⁰ and R_N := S_N⁺S_N⁻. These inten- sive operators are both self-adjoint, so that we can use equations above.

As a first step we compute L(S_N⁰ ) = L₁(S_N⁰ )+L₂(S_N⁰ ). We have L₁(S_N⁰ ) = 1

N XN

j=1

L₁(σ_j⁰) = 1 N

XN j=1

X

α=0,±

n

[ρ^j_α, σ_j⁰]ρ^j_α^†Γ^(a)_α + h.c.

o

which can be written as L₁(S_N⁰ ) = X

α=0,±

n

(b_α+S_N⁺ + b_α−S_N⁻ + b_α0S_N⁰ + b_α111_N)Γ^(a)_α + h.c.

o , where 11_N = _N¹ P_N

j=111_j and the various coefficients {b_{α γ}} have been introduced here only to stress the fact that L₁(S_N⁰ ) is linear in the intensive operators. We have already discussed the fact that the limit of the right hand side of this formula exists in the strong topology restricted to a certain family F of states, since all the operators S_N^α (and their powers) converge in this topology, [BM, JSP1993]. Therefore, also the limit of the left hand side does exist in the same topology. After some non trivial algebra we find

L₁(S⁰) := F − strong lim

N→∞L₁(S_N⁰ ) =

= −8g⁴S⁰(S⁺S⁻)² ω³

½

<Γ^(a)₊ ω − g

(ω + gS⁰)² + <Γ^(a)₋ ω + g (ω − gS⁰)²

¾ , where <Γ^(a)_± indicates the real part of Γ^(a)_± .

The computation of L₂(S⁰) := F − strong lim_N→∞L₂(S_N⁰ )

follows essentially the same steps and produces L₂(S⁰) = −8g⁴S⁰(S⁺S⁻)²

ω³

½

<Γ^(b)₊ ω + g

(ω + gS⁰)² + <Γ^(b)₋ ω − g (ω − gS⁰)²

¾ , so that

L(S⁰) = −8g⁴S⁰(S⁺S⁻)²

ω³ h(S⁰, S⁺S⁻), where

h(S⁰, S⁺S⁻) = {<Γ^(a)₊ ω − g

(ω + gS⁰)² + <Γ^(a)₋ ω + g (ω − gS⁰)²+ +<Γ^(b)₊ ω + g

(ω + gS⁰)² + <Γ^(b)₋ ω − g

(ω − gS⁰)²},

and we have written explicitly the dependence of h on S⁺S⁻ = F −strong lim_N→∞S_N⁺S_N⁻ (via the pulsation ω). It is interesting to observe that the same function h(S⁰, S⁺S⁻) appear also in the computation of L(S⁺S⁻) := F − strong lim_N→∞L(S_N⁺S_N⁻). Here it is convenient to use the fact that lim_N→∞[S_N^α, σ_j^β] = 0, for all α, β and j. We get

L(S⁺S⁻) = −16g⁴(S⁺S⁻)³

ω³ h(S⁰, S⁺S⁻).

The phase structure of the model is now given by the right-hand sides of equations for L(S⁺S⁻) and L(S⁰), see [MB], that is, from the zeros of the functions

f₁(x, y) = −8g⁴xy²

ω³ h(x, y), f₂(x, y) = −16g⁴y³

ω³ h(x, y), where x = S⁰, y = S⁺S⁻ ( ⇒ ω = gp

x² + 4y and ν = 2˜²+ gx).

In particular, the existence of a super-conducting phase corre- sponds to the existence of a non trivial zero of f₁ and f₂, [MB], or, equivalently, to the existence of a non trivial zero of h.

To find these zeros, we need the explicit expression for the coef- ficients <Γ^(γ)_± :

<Γ^(a)_± = Z _∞

−∞

X

~p∈Λ_N

|f_m(~p)|²e^{−iτ ν±(~p)} dτ = 2π X

~p∈Λ_N

|f_m(~p)|²δ(ν_±(~p)), and

<Γ^(b)_± = 2π X

~p∈Λ_N

|f_n(~p)|²δ(ν_±(~p)).

It is now almost straightforward to recover the results of [MB]. We look for solutions corresponding to ν = 0. Then x = S⁰ = −2˜²/g

and ν₊(~p) = ω − ²_~p, which is zero if and only if ω = ²_~p. Also, ν₋(~p) = −ω − ²_~p, which is never zero. Therefore <Γ^(γ)₋ = 0, γ = a, b, while the sums for <Γ^(γ)₊ are restricted to the smaller set, E_N ⊂ Λ_N, of values of ~p such that, if ~q ∈ E_N then ²_~q = ω. Finally, recalling the expression of m(~p) and n(~p), we get

<Γ^(a)₊ = 2π e^βω e^βω − 1

X

~p∈E_N

|f (~p)|², <Γ^(b)₊ = 2π 1 e^βω − 1

X

~p∈E_N

|f (~p)|².

Equation h(x, y) = 0 becomes now 2π e^βω

e^βω − 1 X

~p∈E_N

|f (~p)|² ω − g

(ω + gx)²+2π 1 e^βω − 1

X

~p∈E_N

|f (~p)|² ω + g

(ω + gx)² = 0,

=⇒ e^βω = g + ω g − ω.

This equation replaces the equation obtained in [MB],g tanh

³βω 2

´

= ω. The conclusion is now straightforward:

1) let ξ = ^ω_g. Then equation above has a non-trivial solution if and only if the function g(ξ) = e^βgξ − ^1+ξ_1−ξ has a zero ξ 6= 0. This solution does exist only if the first derivative of g(ξ), computed in

ξ = 0 is positive, i.e. when βg − 2 > 0, that is if T < T_c = _2k^g . We recover therefore the first result of [MB].

2) it is also possible to find numerically the value of y = S⁺S⁻ from our equation. However, it is more convenient to observe that:

g tanh

µβω 2

¶

= ge^βω2 − e^−βω2

e^βω2 + e^−βω2 = ge^βω − 1 e^βω + 1 = g

g+ω g−ω − 1

g+ω

g−ω + 1 = ω, which is the same result found in [MB].

Remarks:– (1) These results are independent of the cutoff N.

This suggests that, at least for what concerns the critical temper- ature and the related value of S⁺S⁻, nothing changes after the thermodynamical limit is considered (i.e., when N → ∞).

(2) Concerning a comparison with [MB], the SLA allows to avoid the use of a perturbative expansion on the master equation, which is very hard to control and which is also based on the assumption that the observables whose time evolution we are interested in com- mute with the free hamiltonian. This makes the SLA more flexible and powerful wrt the standard techniques.

Further results and future projects

(1) In a recent paper (IDAQP, submitted) we have analyzed the role of a second reservoir interacting with R in the attempt of increasing the value of the critical temperature. This approach has produced surprising (to me!!) results: superconducting phases exist also for T > T_c under very special assumptions on the pa- rameters of the model, also for the original [BM] model.

(2) We are now considering, again in the same perspective, a double stochastic limit.

Indeed: suppose that the free evolution of the annihilation op- erator of the reservoir, a_~p,i(t) = a_~p,ie^−i²~pt, is replaced, for some reason, by a_~p,i(t) = a_~p,ie^−iγ²~pt, with γ < 1. Then the function ν_α(~p) will be replaced by ν_α(~p) = ν − γ²_~p + αω. All the other formulas are left unchanged, at least formally. It is easy to check now that equation h(x, y) = 0 produces now, taking ν = 0 as

before, the following equation:

e^βω/γ = g + ω g − ω,

which admits a non trivial solution in ]0, g[ if gβ/γ − 2 > 0, that is under a new critical temperature T_c^(γ) = _2kγ^g = ^T_γ^c, which is larger than T_c since γ is smaller than 1 by construction. There- fore, the critical temperature increases leaving invariant all the parameters! It is worth stressing that a similar conclusion was by no means evident in Martin’s papers.

How to get this change in the free evolution of the bosons?

No-go results:

1) reservoir with different statistic;

2) exact solution of two reservoirs mutually interacting.

The working idea: a nested stochastic limit.

We introduce a second reservoir R₂ interacting only with R with an O(λ) term and the R interacts with S with an O(λ²) hamiltonian.