New scenario for the emegence of non-conventional BEC. Beyond the notion of energy gap.

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NEW SCENARIO FOR THE EMERGENCE OF NON-CONVENTIONAL BOSE-EINSTEIN CONDENSATION BEYOND THE NOTION OF ENERGY GAP Marco Corgini Videla

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de

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NEW SCENARIO FOR THE EMERGENCE OF NON-CONVENTIONAL BOSE-EINSTEIN CONDENSATION BEYOND THE NOTION OF ENERGY GAP Marco Corgini Videla

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Abstract A stable non-ideal Bose system whose energy operator includes

a perturbations depending on the square root of the number operator associated to the zero mode energy is analyzed, demonstrating that, in presence or absence of a gap in the one particle energy spectrum, it undergoes the so-called non-conventional Bose-Einstein condensation.

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Contents 1.

Introduction

2

1.1.

Grand canonical ensemble

2

1.2.

Canonical ensemble

2

1.3.

Non-conventional Bose-Einstein condensation

3

2.

Perturbed system

5

2.1.

General framework

5

2.2.

Problem statement

5

3.

Emergence of non-conventional Bose-Einstein Condensation

4.

Remarks

11

5.

Final comments

13

REFERENCES

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1. Introduction

1.1. Grand canonical ensemble. Let

1 p` (β, µ) = log TrFB exp −β(Γ̂` − µN̂ ) , βV` be the grand-canonical pressure (grand canonical ensemble) at nite volume related to a quantum many particle Bose system whose Hamiltonian is represented by ical potential,

N̂

Γ̂` , being β = θ−1

µ the chem-

the inverse temperature,

the total number of particles operator and

N FB := ⊕∞ N =0 HB the Fock space for bosons constructed from the Hilbert space where

Λ` ⊂ R3

is a cubic box of volume

V` = `3 .

H = L2 (Λ` ),

In this case,

N −times

}| { z N HB := S H ⊗ H... ⊗ H represents the Hilbert space associated to a system composed by exactly

N

S the symmetrization operator. p(β, µ) = lim p` (β, µ) exists for all µ ∈ R

Bose-particles, being If the limit pressure

V` →∞ be said that it is a superstable system. If this holds only for

R, the system will be denominated stable".

Being Γ̂` (µ) = Γ̂` − µN̂ , the equilibrium Gibbs state ensemble) h−iΓ̂ (µ) is de ned as `

it will

µ in a proper

subset of

hÂiΓ̂` (µ) = TrFB exp −β Γ̂` (µ) for any operator

Â

acting on

FB .

−1

(grand canonical

TrFB Â exp −β Γ̂` (µ) ,

The total density of particles

ρ(µ)

for

in nite volume is de ned as

* lim

V` →∞

N̂ V`

+ = lim ρ` (µ) = ρ(µ) = constant. Γ̂` (µ)

V` →∞

1.2. Canonical ensemble. Let consider a sequence of nite systems enclosed in the cubic region

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Λ` .

In the canonical ensemble framework,

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the free energy of particles

N,

f` (β, %` )

3

Γ̂` at nite number temperature β and density %` , is

associated to Hamiltonian

V` ,

nite volume

inverse

de ned as:

f` (β, %) = − where

(N )

Γ`

1 (N ) log TrHBN exp −β Γ̂` , βV`

is the restriction of the energy operator to

%` =

n0

lim %0,` ,

V` ,N →∞

the number of particles associated to the zero mode.

In what follows, the symbols

%0,`

and

N , % = lim %` = constant, V` ,N →∞ V`

%0,` = n0 /V` , %0 = being

N HB ,

%, %0

will be used when referring to

%`

or

indistinctly, avoiding excessive notation.

Finally, being

Â

FB , Â :=

an arbitary operator de ned on the Fock space

the thermal average, in the canonical ensemble, of the operator

Â|HBN , at nite volume V` , inverse temperature β

and density

(N )

%, is de ned

as:

(N )

hÂ

iΓ̂(N ) (%) = TrHBN exp `

(N ) −β Γ̂`

−1

TrHBN Â

(N )

exp

(N ) −β Γ̂`

.

1.3. Non-conventional Bose-Einstein condensation. It will be said that a quantum many particle Bose system undergoes non-conventional Bose-Einstein condensation (NCBEC) if it shows an independent on temperature macroscopic occupation of the single particle zero mode, i.e.,

ρ0 (µ) > 0.

On the other hand, the kinetic energy of a single particle in a Bose system is described by the free Laplacian operator

4`

de ned on

Λ`

endowed with appropriate boundary conditions and a numerable set of eigenvalues

E,

obtained from the equation

1 − 4` φ = Eφ 2

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(φ's are the free Lapacian eigenfunctions).

The non-conventional condensation can emerge, among other reasons, as consequence, either from the existence in the thermodynamic limit of an isolated and negative eigenvalue (the smallest one) of the laplacian or by shifting the zero energy level of the kinetic energy operator to a negative value, generating a gap in its spectrum.

In the rst case, the so-called attractive boundary conditions (see ref. [5]), given by the condition:

âˆ‚Ď† + ĎƒĎ† = 0, ∂n play a fundamental role. boundary and

Ď&#x192;<0

â&#x2C6;&#x201A;Ď&#x2020; is the outward normal derivative on the â&#x2C6;&#x201A;n

is the parameter governing the attractivity of the

boundaries. In this context, the lower eigenvalue is an isolated negative point of the spectrum.

In the second case (periodic boundary conditions), the mentioned nite gap in the one-particle excitations spectrum can be obtained by adding to the corresponding energy operator -in the formalism of second quantization- the term

â&#x2C6;&#x2020;nĚ&#x201A;0 ,

where

â&#x2C6;&#x2020; < 0.

As an example, a homoge-

neous Bose gas with periodic boundary conditions and a two-body interaction enclosed in a region of volume

V`

has been exhaustively studied

in refs. [1, 2]. In these works the emergence of NCBEC has been proven.

Unlike the described situations, the goal of this work is to demonstrate that non-conventional macroscopic occupation of the fundamental state is possible, from a purely mathematical point of view, without resorting to the above mentioned arguments. Indeed, in ref. [14] it has been shown that a kind of small perturbations of the energy operator of the ideal Bose gas -represented by the square root of the number operator associated to the zero mode energy in absence of any gap in the one particle energy spectrum of the system, leads to the emergence of non-conventional BoseEinstein condensation. This fact opens new scenarios for the ocurrence of this type of critical behavior.

This paper is organized as follows. Section 2 is devoted to introduce the perturbed noninteracting Bose particle system to be studied. In section 3 the e ects of the before mentioned disturbance on its critical behavior,

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in presence or absence of gaps, is analyzed. Finally, sections

4 and

5

consider some remarks and the main conclusions, respectively.

2. Perturbed system

2.1. General framework. Let consider a particle system enclosed within the cubic box

Λ` ⊂ R3

of volume

V` . The free energy operator Ĥ`0

is given

as

X

Ĥ`0 =

λ` (p)n̂p ,

p∈Λ∗` where,

N̂ =

X

â†p âp

is the total number operator.

The sum runs

p∈Λ∗`

Λ∗` = {p = (p1 , . . . , pd ) ∈ Rd : pα = 2πnα /`, nα ∈ Z, α = 1, 2, . . . , d}. â†p , âp are the Bose operators of creation and annihilation of particles de ned on the Bose Fock space FB , satisfying the usual commu† † † tation rules: [âq , âp ] = âq âp − âp âq = δp,q I (I is the identity operator), † and n̂p = âp âp is the number operator associated to mode p. On the 0 00 other hand, the operators Ĥ` and N̂ are de ned as:

over the set

Ĥ`00 =

X

λ` (p)n̂p , N̂ 0 =

p∈Λ∗` \{0}

X

â†p âp .

p∈Λ∗` \{0}

Assuming periodic boundary conditions, the energies

λ` (p)

associated

to the single modes are given by,

( λ` (p) = where

∆ < 0,

p=0

p2 /2,

p 6= 0,

∆ has been arbitrarily introduced for producing a gap in the spec-

trum.

2.2. Problem statement. Let

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Ĥ`

be the following Hamiltonian:

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HĚ&#x201A;` = HĚ&#x201A;`0 +

(1)

a â&#x20AC; 2 2 aĚ&#x201A; aĚ&#x201A; , a > 0.

V` 0 0

The second term on the right hand side of (1) can be physically understood as the simultaneous creation of 2-single zero mode particles after the disappearance of an equivalent amount of single particles associated to the same mode. It is a well-known fact that the particle system with energy operator (1) -extensively studied in refs. [6, 7, 10, 11, 13] by using di erent mathematical techniques- undergoes NCBEC for

Âľ â&#x2C6;&#x2C6; (â&#x2C6;&#x2020;, 0].

This work deals with a modi ed version of the Hamiltonian given in (1), which consists in including in it the disturbance

â&#x2C6;&#x161; â&#x2C6;&#x2019;2ν V` nĚ&#x201A;0 , being ν > 0.

In other words, the perturbed energy operator has the following form:

HĚ&#x201A;`per = HĚ&#x201A;`0 +

(2)

Since

p a â&#x20AC; 2 2 aĚ&#x201A;0 aĚ&#x201A;0 â&#x2C6;&#x2019; 2ν V` nĚ&#x201A;0 .

V`

per 2 aĚ&#x201A;â&#x20AC; 2 0 aĚ&#x201A;0 = nĚ&#x201A;0 (nĚ&#x201A;0 â&#x2C6;&#x2019; I), HĚ&#x201A;`

HĚ&#x201A;`per

(3)

=

HĚ&#x201A;`00

+ V`

can be rewritten as:

a â&#x2C6;&#x2020;â&#x2C6;&#x2019; V`

nĚ&#x201A;0 +

p a 2 nĚ&#x201A;0 â&#x2C6;&#x2019; 2ν V` nĚ&#x201A;0 .

V`

The main goal of this article is to prove that such a perturbation leads to the emergence of non-conventional Boseâ

Âż Einstein condensation in

presence or absence of a gap in the kinetic energy spectrum.

3. Emergence of non-conventional Bose-Einstein

Condensation

Let Âľ â&#x2030;¤ 0, ν > 0. Then, the limit pressure pper (β, Âľ, ν, â&#x2C6;&#x2020;) associated to operator (2) is given as: Theorem 3.1.

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0

pper (β, µ, ν, ∆) = g(z0 ) + pid (β, µ),

(4)

where g : R+ ∪ {0} → R is the function de ned as: √ g(z) = −az 2 + (µ − ∆)z + 2ν z,

and z0 (µ, ν, ∆) is the unique solution of the equation z+

(5)

∆−µ ν = √ 2a 2a z

for z > 0.

Proof.

The demonstration of this theorem is a simple variation of an

approach based on the use of the canonical ensemble, already applied by other authors with similar purposes (see for example [16] and references therein). per Let f` (β, %),

f`id (β, %) be the free canonical energies at nite volume V` , inverse temperature β and density %` , corresponding to the perturbed system and the free Bose gas, respectively.

0

Let

f`id (β, % − %0 )

be the nite free canonical energy associated to

00

Ĥ . np , N ∈ N ∪ {0} for all p ∈ Λ∗` , the nite canonical f`id (β, %), f`per (β, %) can be written in the following form, Being

 f`id (β, %) = −

1 log βV` n

X

exp −β

∗ p =0,1,2,..., p∈Λ`

1 log  βV`

 X

λ` (p)np  δPp∈Λ∗ np =[%V` ] , `

p∈Λ∗`

 f`per (β, %) = −

free energies

 X

e−βV` h` (%,%0 ,%0 )  ,

···+np +···=[%V` ]

where

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√ h` (%, %0 ) = ∆%0 + %20 − 2ν %0 +

 −

1 log βV`

and

X

exp −β 

np =0,1,2,...,p∈Λ∗l \{0}

X

N0 =

np .



 X

λ` (p)np  δN 0 =[%V` ]−[%0 V` ] ,

p∈Λ∗` \0

The following inequality

p∈Λ∗` \{0}

 −f`per (β, %) =

1 log  βV`

 X

e−βV` h` (%,%0 )  ≥

..+np +..=[%V` ]

1 log e−βV` h` (%,%0 ) = −h` (%, %0 ), βV` holds for

%0 ∈ [0, %], being [b] the integer part of b.

This inequality implies

that,

f`per (β, %) ≤ inf h` (%, %0 ). %0 ∈[0,%]

On the other hand we have,



[%V` ]

−βV`

inf h` (%, %0 )



1 %0 ∈[0,%]  log  e βV` n0 =0,N 0 =0   [%V` ] −βV` inf h` (%, %0 ) X 1 %0 ∈[0,%] ≤ log e 1 βV` n0 =0,N 0 =0

[%V` ](1 + [%V` ]) 2 log 1 + ≤ − inf h` (%, %0 ) + %0 ∈[0,%] βV` 2 4 ≤ − inf h` (%, %0 ) + ln([%V` ] + 1). %0 ∈[0,%] βV`

−f`per (β, %) ≤

X

Thus, we obtain the inequalities,

inf h` (%, %0 ) −

%0 ∈[0,%]

4 ln([%V` ] + 1) ≤ f`per (β, %) βV` ≤ inf h` (%, %0 ). %0 ∈[0,%]

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Therefore, in the thermodynamic limit it follows that,

f per (β, %) = lim

inf h` (%, %0 ).

V` →∞ %0 ∈[0,%]

In other words,

f per (β, %) =

0 √ inf {∆%0 + a%20 − 2ν %0 + f id (β, % − %0 )},

%0 ∈[0,%]

where

f per (β, %) = Since

0

0

lim f`per (β, %), f id (β, % − %0 ) =

lim f`id (β, % − %0 )

V` ,N →∞

f per (β, %)

is a convex function of

%,

V` ,N →∞

its Legendre transform coin-

cides with the grand canonical limit pressure

pper (β, µ, ν, ∆),

i.e.,

pper (β, µ, ν, ∆) = sup{µ% − f per (β, %)}. %≥0

µ ≤ 0, β > 0,

Hence, for

0 √ sup {(µ − ∆)%0 − a%20 + 2ν %0 } + pid (β, µ).

pper (β, µ, ν, ∆) =

%0 ∈[0,∞)

The continuous function

g : [0, ∞) → R

de ned as

√ g(%0 ) = −a%20 + (µ − ∆)%0 + 2ν %0 , satis es

g(0) = 0 and lim g(%0 ) = −∞. From this it follows that g %0 →±∞

a global maximum for Taking

z = %0

has

%0 ≥ 0.

we get 0

pper (β, µ, ν, ∆) = sup{f (z)} + pid (β, µ). z≥0

Therefore, under the hypotheses of the theorem ximum at

z0 (µ),

f (z)

attains its ma-

positive root of the equation:

ν g 0 (z) = −2az + (µ − ∆) + √ = 0, z > 0. z

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On the other hand,

10

0

pid (β, µ)

is nite for

µ ≤ 0.

Thus,

0

pper (β, µ, ν, ∆) = g(z0 ) + pid (β, µ).

Under the hypotheses of Theorem 3.1 the Bose system with energy operator Ĥ`per undergoes non-conventional macroscopic occupation of the ground state, being Corollary 3.2.

2 ρper 0 (µ) = z0 (µ, ν, ∆)

(6)

the density of particles in the condensate.

Proof.

Convexity arguments yield, in the thermodynamic limit, to the

following inequalities:

lim

V` →∞

ρ̂20,`

Ĥ` (µ)

per = − lim ∂a pper (β, µ) ` (β, µ) = ∂a p V` →∞

2 = z04 (µ, ν, ∆) ≥ lim hρ̂0,` i2Ĥ` (µ) = (ρper 0 (µ)) V` →∞

and,

lim

V` →∞

1 1 per lim ∂ν pper (β, µ) ` (β, µ) = ∂ν p 2 V` →∞ 2 Ĥ` (µ) q p = z0 (µ, ν, ∆) ≤ lim hρ̂0,` iĤ per (µ) = ρper 0 (µ).

Dp

ρ̂`,0

E

=

V` →∞

The bounds for the thermal averages

u(x) = x2 for x ≥ 0.

from the fact that concave function,

ρ̂20,`

Ĥ`

, (µ)

is a convex function

p ρ̂`,0 Ĥ (µ) , follow ` √ and v(x) = x is a

These inequalities lead to

2 ρper 0 (µ) = z0 (µ, ν, ∆).

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As a straightforward consequence of Theorem 3.1 and Corollary 3.2 it follows that in absence of energy gap, i.e., for limit pressure

(β, 0, ν, µ)

∆ = 0, µ ≤ 0, ν > 0,

the

is given as:

√ 0 pper (β, 0, ν, µ) = −az02 + µz0 + 2ν z0 + pid (β, µ),

(7)

where

p

per

z02 (µ, ν, 0)

represents, simultaneously, the solution of the equation

ν − 2az + µ + √ = 0, z > 0, z

(8)

and the fraction of condensed particles.

4. Remarks

Ĥ` and Ĥ`per are full diagonal self-adjoint operators in momentum representation with respect to the number operators n̂p , therefore the U (1) transformations Note 4.1.

â†p → eiφ â†p , âp → e−iφ âp , φ ∈ R

leave the Bose commutation rules and the Hamiltonians invariant (the number of particles is conserved). This gauge symmetry can be broken by adding the perturbation −2ν

p V` (â†0 + â0 )

to Ĥ`, i.e., by de ning a new operator Ĥ`sbr as Ĥ`sbr := Ĥ` − 2ν

(9)

p V` (â†0 + â0 )

From a result due to A. Süt® in ref. * lim

V →∞

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â†0 √ V

+

= lim

Ĥ`sbr (µ)

V →∞

[12]

â √0 V

it follows that:

=ν

q

ρsbr 0 (µ)

Ĥ`sbr (µ)

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On the other hand, * lim

V →∞

↠√0 V

+

= lim

V →∞

Ĥ`per (µ)

â √0 V

=0 Ĥ`per (µ)

since Ĥ`per is U (1)− invariant.

Thus, from the above identities and Theorem 4.3 in ref. [14] it follows that for xed parameters µ ≤ 0, ν > 0, the limit pressures of the systems whose energy operators are given by (1) and (9) coincide, i.e.,

(10)

pper (β, µ, ν, ∆) = psbr (β, µ, ν, ∆).

Theorem 3.1 can be easily extended to the case of energy operators of the type: Note 4.2.

(11)

H̄` = Ĥ`0 +

p a X †2 2 âp âp − 2ν V` n̂0 , V` p∈Λ∗ `

by using the same technique applied in its demonstration. However, in this case, unlike Theorem 3.1, it is necessary to prove additionally that (12)

lim

V` →∞

X p∈Λ∗` \{0}

n̂2p V`2

= 0.

0

Ĥ`id (µ(%−%0 ))

which is a straightforward consequence of the de nition of the so-called Kac measure of the ideal Bose gas at nite volume and the fact that for p, V` xed and r ≥ 1, r ∈ Z + , in the canonical ensemble, the moments (13)

n̂rp

0

Ĥlid (%−%0 )

are monotonic increasing functions of % (see refs.

[8, 9]

).

All the results related to NCBEC obtained in ref. [14] for a system whose energy operator is the sum of the free gas Hamiltonian plus √ √ the perturbation −2ν V n̂0 + 1, ν > 0, can be recovered by applying

Note 4.3.

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the technique used for proving Theorem 3.1. In such a case, for strictly negative values of ¾ and ν > 0, the density of particles in the condensate 2 is ¾ν 2 .

5. Final comments

In section 3, it has been demonstrated that the inclusion of the disturbance

â&#x2C6;&#x161; â&#x2C6;&#x2019;2ν V` nĚ&#x201A;0

in the Hamitonian given by (1) -corresponding to

a well-known particle system displaying NCBEC- a ects the density of condensed particles. Moreover, in absence of the energy shift induced by the introduction of the parameter

â&#x2C6;&#x2020;

in the single particle spectrum, it

has been also proven that the studied system undergoes NCBEC. These results de ne a quite di erent scenario from those represented by the inclusion of energy gaps. Furthermore, it has to be stressed that the limit pressures associated to the models represented by Hamiltonians (2) and (9) coincide with each other despite the fact that one of them preserves the

U (1)

symmetry while the other one does not. In some sense,

each system mimics the thermodynamic behavior of its counterpart.

However, although this new context looks promising, it is not less true that the introduction of this type of disturbances, from the point of view of physics, must have a reasonable justi cation, an issue that is far from the purpose of this work.

On the other hand, independent on physi-

cal consistency and even though these results could seem at rst glance counterintuitive, the fact that the systems whose energy operators are of the kind given by (2) admit the emergence of NCBEC, beyond the traditional approach (energy gaps), makes this matter per se mathematically interesting. In this context, it is necessary to point out that possibly even small disorders (of a random type, for example) could produce similar e ects to those caused by the introduced disturbances.

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http://www.dmatuls.cl/portal/

?page_id=1070

Soluble model of Bose-atoms with two level internal structure: non-conventional Bose-Einstein condensation Condensed Matter Physics 2010,

[16] Corgini, M. Sankovich, D. P.

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