On thermodynamic states of the Ising model on scale-free graphs
A new exactly solvable spatially one-dimensional quantum superradiance model describing a charged fermionic medium interacting with external electromagnetic field is proposed. The infinite hierarchy of quantuum conservation laws and many-particle Bethe eigenstates that model quantum solitonic impuls...
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irk-123456789-1208042017-06-14T03:03:25Z On thermodynamic states of the Ising model on scale-free graphs Blackmore, D. Prykarpatsky, A. A new exactly solvable spatially one-dimensional quantum superradiance model describing a charged fermionic medium interacting with external electromagnetic field is proposed. The infinite hierarchy of quantuum conservation laws and many-particle Bethe eigenstates that model quantum solitonic impulse structures are constructed. The Hamilton operator renormalization procedure subject to a physically stable vacuum is described, the quantum excitations and quantum solitons, related with the thermodynamical equilibrity of the model, are discussed. Запропоновано нову точно розв’язувану просторово-одновимiрну квантову супервипромiнювальну мо-дель, що описує заряджене фермiонне середовище, взаємодiюче iз зовнiшнiм електромагнiтним полем. Сконструйовано скiнченну iєрархiю законiв збереження та багаточастинковi власнi стани Бете, що моделюють квантовi солiтоннi iмпульснi структури. Описано процедуру ренормалiзацiї оператора Гамiльтона щодо стiйкого фiзичного вакуума, обговорюються квантовi збудження та квантовi солiтони, асоцiйованi iз термодинамiчною рiвновагою моделi. 2013 Article On thermodynamic states of the Ising model on scale-free graphs / D. Blackmore, A. Prykarpatsky // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23701:1-9. — Бібліогр.: 32 назв. — англ. 1607-324X PACS: 73.21.Fg, 73.63.Hs, 73.50.Fq, 78.67.De DOI:10.5488/CMP.16.23701 arXiv:1307.3866 http://dspace.nbuv.gov.ua/handle/123456789/120804 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A new exactly solvable spatially one-dimensional quantum superradiance model describing a charged fermionic medium interacting with external electromagnetic field is proposed. The infinite hierarchy of quantuum conservation laws and many-particle Bethe eigenstates that model quantum solitonic impulse structures are constructed. The Hamilton operator renormalization procedure subject to a physically stable vacuum is described, the quantum excitations and quantum solitons, related with the thermodynamical equilibrity of the model, are discussed. |
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Blackmore, D. Prykarpatsky, A. |
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Blackmore, D. Prykarpatsky, A. On thermodynamic states of the Ising model on scale-free graphs Condensed Matter Physics |
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Blackmore, D. Prykarpatsky, A. |
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Blackmore, D. |
| title |
On thermodynamic states of the Ising model on scale-free graphs |
| title_short |
On thermodynamic states of the Ising model on scale-free graphs |
| title_full |
On thermodynamic states of the Ising model on scale-free graphs |
| title_fullStr |
On thermodynamic states of the Ising model on scale-free graphs |
| title_full_unstemmed |
On thermodynamic states of the Ising model on scale-free graphs |
| title_sort |
on thermodynamic states of the ising model on scale-free graphs |
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Інститут фізики конденсованих систем НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/120804 |
| citation_txt |
On thermodynamic states of the Ising model on scale-free graphs / D. Blackmore, A. Prykarpatsky // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23701:1-9. — Бібліогр.: 32 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT blackmored onthermodynamicstatesoftheisingmodelonscalefreegraphs AT prykarpatskya onthermodynamicstatesoftheisingmodelonscalefreegraphs |
| first_indexed |
2025-07-08T18:36:15Z |
| last_indexed |
2025-07-08T18:36:15Z |
| _version_ |
1837104917430403072 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 2, 23701: 1–9
DOI: 10.5488/CMP.16.23701
http://www.icmp.lviv.ua/journal
A new exactly solvable spatially one-dimensional
quantum superradiance fermi-medium model
and its quantum solitonic states∗
D. Blackmore1, A. Prykarpatsky2,3
1 New Jersey Institute of Technology, University Heights, Newark, NJ 07-102, USA
2 AGH University of Science and Technology, 30-059 Krakow, Poland
3 Ivan Franko State Pedagogical University, Lviv region, Drohobych, Ukraine
Received October 23, 2012, in final form March 11, 2013
A new exactly solvable spatially one-dimensional quantum superradiance model describing a charged fermionic
medium interacting with an external electromagnetic field is proposed. The infinite hierarchy of quantum con-
servation laws and many-particle Bethe eigenstates that model quantum solitonic impulse structures are con-
structed. The Hamilton operator renormalization procedure subject to a physically stable vacuum is described,
the quantum excitations and quantum solitons, related to the thermodynamical equilibrity of the model, are
discussed.
Key words: charged fermionic medium, quantum superradiance model, Bethe eigenstates, quantum solitons,
renormalization, conservation laws, quantum inverse spectral problem, Yang-Baxter identity
PACS: 73.21.Fg, 73.63.Hs, 73.50.Fq, 78.67.De
1. Fermionic medium and superradiance model description
We shall describe the quantum superradiance properties [1–9] of a model of a one-dimensional many
particle charged fermionic medium interacting with an external electromagnetic field. The Dirac type
N -particle Hamiltonian operator of the model is expressed as
HN := i
N
∑
j=1
σ
( j )
3
∂
∂x j
⊗1−iβ1⊗
∫
R
dxε+εx +α
N
∑
j=1
σ
( j )
1 ⊗E (x j ), (1.1)
where σ
( j )
3 ,σ
( j )
1 , j = 1, . . . N , are the usual Pauli matrices, α∈R+ is an interaction constant, 0 <β< 1 is the
light speed in the linearly polarized fermionic medium,
E (x) :=
(
ε(x) 0
0 ε+(x)
)
is the one-mode polarization matrix operator at particle location x ∈Rwith quantized electric field bose-
operators ε(x),ε+(x) : ΦB → ΦB acting in the corresponding Fock space ΦB and satisfying the commuta-
tion relationships:
[ε(x),ε+(y)]= δ(x − y), [ε(x),ε(y)] = 0 = [ε+(x),ε+(y)]
for all x, y ∈ R. We note that throughout the sequel we employ units for which the standard constants
×= 1 = c.
∗Authors with great pleasure devote their work to Professor Mykhaylo Kozlovskii in honor of his 60th Birthday Jubilee.
© D. Blackmore, A. Prykarpatsky, 2013 23701-1
http://dx.doi.org/10.5488/CMP.16.23701
http://www.icmp.lviv.ua/journal
D. Blackmore, A. Prykarpatsky
By construction, the N -particle Hamiltonian operator (1.1) acts in the Hilbert space L(as)
2 (RN ;C2)⊗
ΦB, where L(as)
2 (RN ;C2) denotes the square-integrable antisymmetric vector functions on R
N , N ∈ Z+.
Correspondingly, the Fock space ΦB allows for the standard representation as the direct sum
ΦB :=
⊕
n∈Z+
L(s)
2 (Rn ;C), (1.2)
where L(s)
2 (Rn ;C) denotes the space of symmetric square-integrable scalar functions on R
n ,n ∈Z+. Simi-
larly, the corresponding fermionic Fock space
ΦF :=
⊕
n∈Z+
L(as)
2 (Rn ;C2), (1.3)
can be used to represent [10–13] the Hamiltonian operator (1.1) in the second quantized form
H : = i
∫
R
dx
[
ψ+
1 ψ1,x −ψ+
2 ψ2,x −βε+εx + iα
(
εψ+
2 ψ1 +ε+ψ+
1 ψ2
)]
, (1.4)
which acts on the tensored Fock space Φ :=ΦF ⊗ΦB, where the spaces ΦF and ΦB, defined respectively,
by (1.2) and (1.3), can also be represented as
ΦB :=
⊕
n∈Z+
span
{
∫
Rn
dx1dx2 . . . dxnχn(x1, x2, . . . , xn )
×
n
∏
j=1
ε+(x j ) |0〉B ;χn ∈ L(s)
2 (Rn ;C)
}
,
ΦF :=
⊕
n∈Z+
span
{
∫
Rn
dx1dx2 . . . dxnϕ
(m)
n (x1, x2, . . . , xn )
×
n
∏
j=m+1
ψ+
1 (x j )
m
∏
k=1
ψ+
2 (xk )|0
〉
: 0 É m É n;ϕ(m)
n ∈ L(as)
2 (Rn ;C2)
}
, (1.5)
and |0〉B ∈ΦB, |0〉F ∈ΦF are the corresponding vacuum bose- and fermi-states, satisfying the determining
conditions
ψ1(x) |0〉F = 0 =ψ2(x) |0〉F , ε(x) |0〉B = 0 (1.6)
for all x ∈ R. The creation and annihilation operators ψ j (x),ψ+
k
(y) : ΦF → ΦF, j ,k = 1,2, satisfy the anti-
commuting
{ψ j (x),ψ+
k (y)} = δ j ,kδ(x − y),
{ψ j (x),ψk (y)} = 0 = {ψ+
j (x),ψ+
k (y)} (1.7)
and commuting
[ε(x),ψ j (y)]= 0 = [ε(x),ψ+
j (y)],
[ε+(x),ψ j (y)]= 0 = [ε+(x),ψ+
j (y)] (1.8)
relationships for all x, y ∈R.
As we are interested in describing the so-called super-resonance processes in our fermionic medium
induced by an external electromagnetic field, in particular, a possibility of generating strong localized
photonic impulses, it is first necessary to investigate the bound photonic medium states and their spectral
energy characteristics. Toward this aim, we make an important note that it has been observed that the
spectral properties of the Hamiltonian operator (1.4) can be analyzed in great detail owing to the fact that
the related Heisenberg nonlinear dynamical system
ψ1,t = i[H,ψ1] =ψ1,x + iαε+ψ2,
ψ2,t = i[H,ψ2] =−ψ2,x + iαεψ1 ,
εt = i[H,ε] =−βεx + iαψ+
1 ψ2 (1.9)
23701-2
An exactly solvable quantum superradiance model
is a quantum exactly solvable Hamiltonian flow on the quantum operator manifold M :=
{(ψ1,ψ2,ε;ε+ ,ψ+
2 ,ψ+
1 ) ∈ EndΦ6}, where the notation EndΦ6 denotes the space of all endomorphisms from
the linear operator space Φ
6 to itself. The system (1.9) can be linearized by means of the quantum Lax
type spectral problem
d f
dx
= l(x;λ) f , (1.10)
where the generalized eigenfunction f ∈Φ
3, and the operator matrix l(x;λ) ∈ EndΦ3 equals
l(x;λ) :=
− iλ
3−β iξ1ψ1 iξ2ψ2
iξ1ψ
+
1 − iλ
2β iξ3ε
iξ2ψ
+
2 iξ3ε
+ iλ(3+β)
2β(3−β)
(1.11)
for all x ∈R, with λ ∈C being an arbitrary time-independent spectral parameter, and
ξ1 := ξ1(α,β) =−18α
[
(9−3β)(β+1)
β+3
]1/2 (
12β
β+3
)1/2 β+3
2β2 +3β+3
,
ξ2 := ξ2(α,β) = 6α(3−3β)1/2
(
12β
β+3
)1/2 (9−3β)(β+1)
(β−1)(2β2 +3β+3)
,
ξ3 := ξ3(α,β) = 72αβ
(3(1−β))1/2
(β−1)(2β2 +3β+3)
[
(9−3β)(β+1)
β+3
]1/2
(1.12)
being constants, depending on the interaction parameter α ∈ R+ and the light speed β in the polarized
fermionic medium 0 <β< 1.
Remark 1. Concerning the quantum Lax type spectral problem (1.10) and its determination, one can consult
[10, 14–17], where the corresponding analytical tools are developed and described in detail.
The quantum dynamical system (1.4) may be also regarded as an exactly solvable approximation of
the three-level quantum model studied in [18] subject to its superradiance properties. Concerning the
studies of such superradiance Dicke type one-dimensional models, it is necessary to mention the work
[19] in which it was shown that the well-known quantum Bloch-Maxwell dynamical system
ψ1,t = i[Ĥ,ψ1]= iαε+ψ2 ,
ψ2,t = i[Ĥ,ψ2]= iαεψ1 ,
εt = i[Ĥ,ε]=−βεx + iαψ+
1 ψ2 (1.13)
generated by the the reduced quantum Hamiltonian operator
Ĥ :=−i
∫
R
dx
[
βε+εx − iα
(
εψ+
2 ψ1 +ε+ψ+
1 ψ2
)]
(1.14)
in the strongly degenerate Fock space Φ is also exactly solvable. Moreover, it has the corresponding Lax
type operator spectral problem [10, 15, 20] in the space Φ
3. However, the important problem of con-
structing the stable physical vacuum for the Hamiltonian (1.14) was on the whole not discussed in [19],
and neither was the problem of studying the related thermodynamics of quantum excitations over it.
More interesting quantum one-dimensional models with the Hamiltonian similar to (1.4) describing the
quantum interaction of just fermionic particles and only bosonic particles with an external electromag-
netic field were studied, respectively, in [21] and [22]. In these investigations, the quantum localized Bethe
states were constructed and analyzed in detail. The corresponding classical version of the quantum dy-
namical system (1.9), called the three-wave model, was studied in [20, 23, 24] and elsewhere.
It is also worthy to mention here that the spectral operator problem (1.11) makes sense [5, 6, 8] only
if the light speed inside the polarized fermionic medium is less than the light speed in a vacuum. This is
ensured by the dynamical stability of the quantumHamiltonian system (1.9) following from the existence
of an additional infinite hierarchy of conservation laws, suitably determined on the quantum operator
23701-3
D. Blackmore, A. Prykarpatsky
phase space M. Consequently, one can expect that the quantum dynamical system (1.9) also possesses
many-particle localized photonic states in the Fock space Φ, which are called [9, 19, 25–29] quantum soli-
tons, whose spatial range is inverse to the number of interior particles, and which can be interpreted as
special Dicke type superradiance laser impulses. In particular, the quantum stability, solitonic formation
aspects and construction of the physical ground state [10, 14, 17] related to the unbounded a priori from
below Hamiltonian operator (1.4) will be the main focus of the succeeding sections.
2. Bethe eigenstates and the energy localization
In this section we construct some of the finite-particle Bethe eigenstates [10, 14, 16, 17, 30] for the
quantumHamiltonian operator (1.4) and discuss their energy localization property. The localizationman-
ifests itself in the fact that the energy of a many-particle cluster appears to be less [10, 14, 30] than that of
the corresponding system of free particles, giving rise to the formation of the so-called quantum solitonic
states localized in the space.
The following number operators
NF :=
∫
R
dx
(
ψ+
1 ψ1 +ψ+
2 ψ2
)
,NB :=
∫
R
dx
(
ε+ε+ψ+
2 ψ2
)
(2.1)
commute with each other and with the Hamiltonian operator (1.4):
[NF,NB]= 0, [H,NF] = 0 = [H,NB]. (2.2)
Hence, the Bethe eigenstates of the Hamiltonian operator (1.4) can be indexed by two integers N , M ∈Z+ :
the state |(N , M)〉 ∈Φ satisfies the determining equation
H |(N , M)〉 = E |(N , M)〉 , (2.3)
where E ∈R is the energy and
NF |(N , M)〉 = N |(N , M)〉 ,
NB |(N , M)〉 = M |(N , M)〉 . (2.4)
Owing to (2.4), the state
|(N , M)〉 =
N
⊕
n=0
∫
RN+M
dx1dx2 . . . dxn dy1dy2 . . . dyN−n dz1dz2 . . . dzM−N+n
×ϕ(n)
(M ,N)
(x1, x2, . . . , xn ; y1, y2, y3, . . . , yN−n ; z1, z2, . . . , zM−N+n )
×
n
∏
j=1
ψ+
1 (x j )
N−n
∏
k=1
ψ+
2 (yk )
M−N+n
∏
l=1
ε+(zl ) |0〉 (2.5)
for any M , N ∈ Z+, where for bounded states ϕ(n)
(M ,N)
∈ L(as)
2 (RN ;C)×L(s)
2 (RM−N+n ;C),0 É n É N , one can
easily construct [16, 21, 30] the Bethe state
|(1,1)〉 =
∫
R
dy1ϕ(y1)ψ+
2 (y1) |0〉+
∫
R
dx1
∫
R
dz1χ(x1; z1)ψ+
1 (x1)ε+(z1) |0〉 . (2.6)
Here, the functions ϕ : R→C and χ :R2 →C satisfy the generalized differential equations
1
i
∂ϕ(x1)
∂x1
−αχ(x1; x1) = Eϕ(x1),
−
1
i
∂χ(x1; x2)
∂x1
+
β
i
∂χ(x1; x2)
∂x2
−αϕ(x1)δ(x1 − x2) = Eχ(x1; x2), (2.7)
23701-4
An exactly solvable quantum superradiance model
having solutions
ϕ(x1) = S+(λ,µ)exp
[
i
(
p1 +q1
)
x
]
,
χ(x1, x2) =
[
ϑ (x1 − x2)+S1(λ,µ)ϑ(x2 − x1)
]
exp
[
i
(
p1x1 +q1x2
)]
, (2.8)
where the momenta λ,µ ∈R and
p1 = (β−1)λ, q1 = 2µ, E = p1 −βq1,
S1(λ,µ) =
λ−µ− iα2/[4(1−β2)]
λ−µ+ iα2/[4(1−β2)]
, S+(λ,µ) =
α
2(1−β)λ−µ+ iα2/[4(1−β2)]
. (2.9)
In the same way, one can represent the other quantum Bethe state as
|(2,1)〉 =
∫
R2
dx1dx2ϕ(x1, x2)ψ+
1 (x1)ψ+
2 (x2) |0〉+
∫
R3
dx1dx2dx3χ(x1, x2; x3)ψ+
1 (x1)ε+(x2) |0〉 , (2.10)
where the functions ϕ :R2→C and χ : R3 →C satisfy the generalized differential equations:
−
1
i
[
∂ϕ(x1, x2)
∂x1
−
∂ϕ(x1, x2)
∂x2
]
+αχ(x2, x1; x2)+αχ(x1, x2; x1) = Eϕ(x1, x2),
−
1
i
[
∂χ(x1, x2; x3)
∂x1
+
∂χ(x1, x2; x3)
∂x2
]
+
β
i
∂χ(x1, x2; x3)
∂x3
+
α
2
ϕ(x1, x2)δ(x2 − x3)+
α
2
ϕ(x2, x1)δ(x1 − x3) = Eχ(x1, x2; x1), (2.11)
with solutions similar to those of (2.8) and (2.9).
It is important to mention here that the eigenstates (2.6) and (2.10) become degenerate as β → 1,
meaning that the corresponding bound quantum soliton states cannot be formed. The same statement
is also true for an arbitrary eigenstate (2.5). To demonstrate this, we shall in the next section make use
of the quantum spectral problem (1.11) to prove that the quantum dynamical system (1.9) possesses an
infinite hierarchy of commuting conservation laws, thereby ensuring its complete quantum integrability
and the formation of quantum solitons.
3. The quantum solitons
We now consider the following quantum operator Cauchy problem for the spectral equation (1.10)
subject to the periodic conditions l(x +2π;λ) = l(x;λ) ∈ EndΦ3 for all x ∈ R and λ ∈C :
dF (x, y ;λ)
dx
=
...l(x;λ)F (x, y ;λ)
... , (3.1)
where F (x, y ;λ) ∈EndΦ3 is the corresponding fundamental transition operator matrix satisfying
F (x, y ;λ)
∣
∣
y=x = 1, (3.2)
and the operation
... ·
... arranges operators ψ j ,ψ
+
j
, j = 1,2, ε and ε+, via the standard normal ordering
[11, 31] that does not change the position of any other operators; for instance,
...Aψ+
1 ψ2ε
+B
... =ψ+
1 ε
+ABψ2
for any A,B ∈ EndΦ.
Construct now the operator products
F̂ (x, y |λ,µ) := F̃ (x, y ;λ)
≈
F (x, y ;µ),
F̌ (x, y |λ,µ) :=
≈
F (x, y ;µ)F̃ (x, y ;λ), (3.3)
23701-5
D. Blackmore, A. Prykarpatsky
where
F̃ (x, y ;λ) := F (x, y ;λ)⊗1,
≈
F (x, y ;µ) := 1⊗F (x, y ;µ) (3.4)
are for all x, y ∈ R , λ,µ ∈ C, the corresponding tensor products of operators acting in the space Φ
3 ⊗
Φ
3. The following proposition is crucial [10, 17, 31] for further analysis of integrability of the quantum
dynamical system (1.9) and is proved by a direct computation.
Proposition 1. The operator expressions (3.3) satisfy the following differential relationships:
∂
∂x
F̂ (x, y |λ,µ) =
...L̂ (x;λ,µ)F̂ (x, y |λ,µ)
... ,
∂
∂x
F̌ (x, y |λ,µ) =
...
`
L (x;λ,µ)F̌ (x, y |λ,µ)
... , (3.5)
where the matrices
L̂ (x;λ,µ) = l̃ ( x;λ)+
≈
l (x;µ)−α△̂(x;λ,µ) ,
`
L (x;λ,µ) = l̃ ( x;λ)+
≈
l (x;µ)−α△̌(x;λ,µ) , (3.6)
and △̂(x;λ,µ),△̌(x;λ,µ) satisfy the algebraic relationship P△̂(x;λ,µ)P = △̌(x;λ,µ) for all x ∈ R, λ,µ ∈C ,
where P ∈ EndΦ3⊗Φ
3 is the standard transmutation operator in the spaceΦ3⊗Φ
3, that is P (a⊗b) := b⊗a
for any vectors a,b ∈Φ
3.
Using proposition 1, one can easily verify that there exists a scalar R-matrix R(λ,µ), R ∈ EndC9,
such that
R(λ,µ)L̂ (x;λ,µ) =
`
L (x;λ,µ)R(λ,µ) (3.7)
holds for all λ,µ ∈C and x ∈R. This, owing to the equations (3.5), implies the main functional Yang-Baxter
type [14, 17, 31] operator relationship
R(λ,µ)F̂ (x, y |λ,µ) = F̌ (x, y |λ,µ)R(λ,µ) (3.8)
is satisfied for any x, y ∈R and λ,µ ∈C, where, as a result of [22],
R(λ,µ) = (λ−µ)P − iα1. (3.9)
Recalling now the periodicity condition, from (3.8), one easily deduces by means of the trace-operation
that the monodromy operator matrix T (x;λ) := F (x +2π, x;λ) satisfies the following commutation rela-
tionship for all x ∈ R and λ,µ ∈ C :
[
trT (x;λ), trT (x;µ)
]
= 0. (3.10)
Actually, it follows from (3.8) that
tr
{
T (x;λ)⊗T (x;µ)
}
= tr
{
R
−1T (x;µ)⊗T (x;λ)
}
= tr
{
T (x;µ)⊗T (x;λ)
}
. (3.11)
Taking into account that tr(A ⊗B) = trA · trB for any operators A,B ∈ EndΦ3, one easily obtains (3.10)
from (3.11). Consequently, the λ-dependent operator functional
γ(λ) := trT (x,λ)�Σ j∈Z+
γ jλ
− j , (3.12)
as |λ|→∞ generates an infinite hierarchy of commuting conservation laws γ j :Φ→Φ, j ∈Z+ :
[
γ j ,γk
]
= 0 (3.13)
23701-6
An exactly solvable quantum superradiance model
for all j ,k ∈Z+, where, in particular,
γ1 = NF =
∫
R
dx(ψ+
1 ψ1 +ψ+
2 ψ2), γ2 = NB =
∫
R
dx(ε+ε+ψ+
2 ψ2),
γ3 = P = i
∫
R
dx(ψ+
1 ψ1,x +ψ+
2 ψ2,x +ε+εx ),
γ4 = H = i
∫
R
dx
[
ψ+
1 ψ1,x −ψ+
2 ψ2,x −ε+εx + iα(εψ+
2 ψ1 +ψ+
1 ψ2ε
+)
]
. (3.14)
Since the operator functional γ4 = H is the Hamiltonian operator for the dynamical system (1.9), from
(3.13) one obtains
[
H,γ j
]
= 0 (3.15)
for all j ∈ Z+; that is, all of functionals γ j :Φ→Φ, j ∈Z+, are conservation laws.
Moreover, making use of the exact operator relationships (3.8) one can construct the physically stable
quantum states |(N , M)〉 ∈ Φ for all N , M ∈ Z+ upon redefining the Fock vacuum |0〉 ∈ Φ, which is non-
physical for the dynamical system (1.9), governed by the unbounded from below Hamiltonian operator
(1.4). Following a renormalization scheme similar to those developed in [14, 17, 32], one can construct a
new physically stable vacuum
|0〉phys :=
∏
qɵ j ÉQ
B+(µ j ) |0〉 (3.16)
by means of the new, commuting to each other, “creation” operators B+(µ) : Φ→Φ,µ ∈ C, generated by
suitable components of the monodromy operator matrix T (x;µ) : Φ3 → Φ
3, x ∈ R, whose commutation
relationships with the Hamiltonian operator (1.4)
[H,B+(µ)]= S(µ;α,β)B+(µ) (3.17)
are parameterized by the two-particle scalar scattering factor S(µ;α,β),µ ∈ C, and where the values q <
Q ∈R are to be determined from the condition that quantum excitations over the physical vacuum (3.16)
have positive energy. Since the physical vacuum (3.16) is an eigenstate of the Hamiltonian operator (1.4),
the corresponding quantum eigenstates of the excitations can be represented as
|(µ)
〉
:= B+(µ) |0〉phys (3.18)
for some µ ∈ R and the new energy level can be taken into account in the renormalized Hamiltonian
operator (1.4) by means of the chemical potentials aF, aB ∈R :
Ha := H−aFNF −aBNB , (3.19)
which should be determined from the conditions
Ha |0〉phys = 0,
〈
(µ) |Ha |(µ)
〉
> 0 (3.20)
for any µ ∈ R. The physical vacuum state and quantum Hamiltonian renormalization construction de-
scribed above make it possible to study the properties of superradiance quantum photonic impulse struc-
tures generated by interaction of the charged fermionic medium with an external electromagnetic field.
Owing to the existence of quantum periodic eigenstates over the physically stable vacuum, one can also
investigate the related thermodynamic properties of the model and analyze the generated superradiance
photonic structures, which are important for explaining many [1] existing experiments.
4. Conclusion
The exact solvability of our model describing a one-dimensional many-particle charged fermionic
medium interacting with an external electromagnetic field allows one to calculate diverse superradi-
ance effects, which are closely related to the formation of the bound quantum solitonic states and their
23701-7
D. Blackmore, A. Prykarpatsky
stability. The existence of the bound states is established by suitably applying the physical vacuum renor-
malization subject to which all quantum excitations are of positive energy. This procedure, based on the
determining operator relationships (3.8), (3.16) and (3.17) enables one to describe the thermodynamic
properties of the quantum dynamical system over a stable physical vacuum. In addition, it facilitates
the analysis of the corresponding thermodynamic states of the resulting quantum photonic system and
its superradiance properties. Our results indicate that a more detailed investigation of these and related
topics is in order, which we plan to undertake elsewhere.
Acknowledgements
A.P. is cordially thankful to Prof. J. Slawianowski (IPPT of PAN, Warsaw) for his invitation to deliver a
report for his Seminar at IPPT of PAN, his gracious hospitality, fruitful discussions and valuable remarks.
D.B. was partially supported by NSF Grant CMMI–1029809. Last but not least, cordial thanks belong to
the referees, whose professional reading of the original version of the manuscript was instrumental in
making the exposition of the results both clearer, correct and readable.
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Нова точно розв’язувана просторово-одновимiрна квантова
модель супервипромiнювального фермi-середовища та
квантовi солiтоннi стани
Д. Блекмор1, А.К. Прикарпатський2,3
1 Iнститут науки та технологiй, Унiверситетськi висоти, Ньюарк, 07-102 Нью Джерсi, США
2 Академiя гiрництва та металургiї, 30-59 Кракiв, Польща
3 Державний педагогiчний унiверситет iм. I. Франка, Дрогобич, Україна
Запропоновано нову точно розв’язувану просторово-одновимiрну квантову супервипромiнювальну мо-
дель, що описує заряджене фермiонне середовище, взаємодiюче iз зовнiшнiм електромагнiтним полем.
Сконструйовано скiнченну iєрархiю законiв збереження та багаточастинковi власнi стани Бете, що моде-
люють квантовi солiтоннi iмпульснi структури. Описано процедуру ренормалiзацiї оператора Гамiльтона
щодо стiйкого фiзичного вакуума, обговорюються квантовi збудження та квантовi солiтони, асоцiйованi
iз термодинамiчною рiвновагою моделi.
Ключовi слова: заряджене фермiонне середовище, квантова обернена спектральна проблема, власнi
стани Бете, ренормалiзацiя вакуума, квантова супервипрмiнювальна модель, закони збереження,
тотожнiсть Янга-Бакстера
23701-9
http://arxiv.org/abs/cond-mat/9706063v1
Fermionic medium and superradiance model description
Bethe eigenstates and the energy localization
The quantum solitons
Conclusion
|