Casimir Energy in a Bounded Gross-Neveu model

0.42 MB·

by  F. Escalante

Casimir Energy in a Bounded Gross-Neveu model
F. Escalante∗ and J.C. Rojas†
Departamento de F´ısica, Universidad Cato´lica del Norte, Angamos 0610, Antofagasta, Chile
InthisletterwestudysomerelevantphysicalparametersofthemasslessGross-Neveu(GN)model
in a finite spatial dimension for different boundary conditions. It is considered the standard homo-
geneousHartreeFocksolutionusingzetafunctionregularizationforthestudythemassdynamically
generated and its respective beta function. It is found that the beta function does not depend on
the boundary conditions. On the other hand, it was considered the Casimir effect of the resulting
effective theory. There appears a complex picture where the sign of the generated forces depends
on the parameters used in the study.
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∗ [email protected]
† [email protected]
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I. INTRODUCTION
The Gross-Neveu (GN) model was born as a toy model of Quantum Chromodynamics (QCD) [1]. Despite its
simplicity, it keeps many interesting features, such as asymptotic freedom, dynamical mass generation and discrete
chiral symmetry. Later, it was used in the study of baryons with explicit symmetry breaking by a mass term [2].
Curiously, this model has also application in condensed matter physics, where it describes the conductivity in
certainpolymers. Inparticular, itcanbementionedthecaseoftrans-polyacetylene, which, inasimplifiedcontinuous
model,isdescribedbythesymmetricGNmodel[3],besides,themassiveGNmodelhasacondensedmatteranalogue;
which are polymers with non-degenerate ground states [4].
TheoriginaltreatmentoftheGNmodelwasundertheassumptionoftheunbrokentranslationalinvariance,itmeans
an standard treatment based on the large N approximation, where the use of the Hartree Fock (HF) approximation
is well founded, that leads a condensate independent of the space coordinates. Later, it was realized that there are
crystal solutions of the model i.e. an spatial realization solution which have a rich interpretation in the realm of
condensed matter physics [5].
In our study, we shall concentrate on the homogeneous solutions of the GN model for a finite space of fixed size L.
We are interested in the behaviour of physical parameters for different boundary conditions (BC’s).
The spatial BC’s considered are the periodic , anti periodic conditions. There are also considered the situation of
no current transmission on the borders, there we consider two cases where such condition is fulfilled (see appendix
C).
The HF approximation, implies the use of a large momentum cutoff. Since we shall deal with systems of spacial
finite size, the momentum integrals must be replaced by summation on discrete modes, meaning that the natural
regularization to be used is the zeta regularization technique [6].
In this work, we first ask about the ultraviolet dependence of the physical parameters on the BC’s, considering the
GNmodelatzerobaremass(m =0)wheretemperatureandchemicalpotentialarenotconsidered. Weassumethat
0
the spatial length L is a fixed parameter, so, if the physical mass is independent of the cutoff, it implies that the beta
function does not depend on the BC’s. There appears an arbitrary mass scale and the functional dependency of the
dynamical mass clearly depends on the BC’s.
A second step in our work is to study the Casimir energy and force due to the quantum fluctuation of the effective
free system that arises from the HF approximation. We consider the non dimensional parameter µ = mL, since the
value of m is fixed by ultraviolet considerations, the variation of µ is equivalent to the variation of L. We find that
the value of energy and Force are sensitive to the BC’s. In particular, the signature of the energy clearly differs in
the small size limit, but it is universally negative for infinite size limit. On the other hand, the force is also sensitive
to the BC’s, implying situations where the forces are such that they compress or expand our space depending on the
BC’s used. There is also a universal metastable point where the force becomes zero independently of the BC’s used.
For the large L limit the force becomes positive for any BC’s considered.
THE GROSS-NEVEU MODEL
The Gross-Neveu Lagrangian is given by
L =ψ¯iiγµ∂ ψi+ 1g2(cid:8)(ψ¯iψi)2−λ(ψ¯iγ ψi)2(cid:9)−m ψ¯iψi. (1)
GN µ 2 5 0
Where i runs from 1 to N, it was introduce a finite mass in order to consider a general expression and we use the
convention
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
0 −1 i 0 0 i
γ0 = , γ1 = , γ5 = .
−1 0 0 −i −i 0
The Euler Lagrange equation from (1) is given by
iγµ∂ ψ+g2(cid:8)ψ¯ψi−λ(ψ¯jγ ψj)γ (cid:9)ψi−m ψi =0. (2)
µ 5 5 0
For the sake of simplicity, from now, we suppress the index i. In the framework of Hartree-Fock relativistic
approximation, it is assumed the expectation value (cid:104)ψ¯γ ψ(cid:105)=0 and (cid:104)ψ¯ψ(cid:105)=Nρ. We end up with the expression
5
3
(iγµ∂ −m)ψ(x)=0, (3)
µ
where m=m −g2Nρ and ρ=(cid:104)ψ¯ψ(cid:105)/N.
0
From (3) we obtain a free Dirac equation
(cid:18) (cid:19) (cid:18) (cid:19)
∂ψ 0 −i 0 1
i =Hψ = (−i∂ )ψ−m ψ.
∂t i 0 x 1 0
In order to obtain a stationary solution, we use the usual decomposition
(cid:18) (cid:19)
φ(x)
ψ(x)=e−iλt .
χ(x)
We obtain
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
φ 0 i ∂ φ 0 −m φ
λ =−i x + , (4)
χ −i 0 ∂ χ −m 0 χ
x
giving a system of coupled equations
d
χ(x)−mχ(x)=λφ(x), (5)
dx
d
− φ(x)−mφ(x)=λχ(x). (6)
dx
By making the redefinition of the fields
f =χ+φ, g =χ−φ, (7)
we obtain a general solution
α β
f(x)= √ cos(Ωx)− √ sin(Ωx), (8)
λ+m λ+m
α β
g(x)= √ sin(Ωx)+ √ cos(Ωx), (9)
λ−m λ−m
√
where Ω = λ2−m2 ande the constants α and β are not independent since they are determined by the boundary
conditions.
II. HARTREE FOCK FOR DIFFERENT BOUNDARY CONDITIONS
Following the standard procedure [7], it is possible to compute the negative energy in an infinite space taking the
value of m as a parameter to be determined
E (cid:90) dk(cid:112) m2
=−2 m2+k2 + , (10)
N 2π n 2G
|k|≤Λ
where G=Ng2 and Λ is a momentum cut off.
Sincewehaveafinitespatialsize,thewavenumberkisdiscretizedk =(2πn+φ)/rL,implying(cid:82) dk/(2π)→ 1 (cid:80),
n rL
Where r is a number which depends on boundary conditions. So, we have
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E =− 2 (cid:88)(cid:0)m2+k2(cid:1)1/2+ m2. (11)
N rL n 2G
n
Thesummationtermcanbeexpressedasgeneralizedzetafunctionregularizationanditsresultisdescribedinappendix
A. Since the power 1/2 in the summation is replaced by a term 1/2−(cid:15), it appears a mass scale η. We have the
following momentum decomposition for the BC to be considered (section (IV)):
Periodic k =2πn/L n∈(−∞,∞),
n
Antiperiodic k =(2n+1)π/L n∈(−∞,∞),
n
Zerocurrenti) k =(2n)π/2L n∈(−∞,∞),
n
Zerocurrentii) k =(2n+1)π/2L n∈(−∞,∞).
n
ForthefourconsideredBC,weobtainedthefollowingexpressionsfortheenergydensity, whereitwasintroducedthe
non dimensional variables µ=mL and η˜=ηL (see appendix B):
• Periodic BC
EP =− µ2 + µ2 − µ2 ln(cid:18)2η˜(cid:19)+ 4µ (cid:88)∞ K1(µn) + µ2 . (12)
N 2π(cid:15)L2 2πL2 πL2 µ πL2 n 2GL2
n=1
• Anti periodic BC
EAP =− µ2 + µ2 − µ2 ln(cid:18)2η˜(cid:19)+ 4µ (cid:40)(cid:88)∞ K1(2µn) −(cid:88)∞ K1(µn)(cid:41)+ µ2 . (13)
N 2π(cid:15)L2 2πL2 πL2 µ πL2 n n 2GL2
n=1 n=1
• Zero current BC
Ei =− µ2 + µ2 − µ2 ln(cid:18)2η˜(cid:19)+ 2µ (cid:88)∞ K1(2µn) + µ2 , (14)
N 2π(cid:15)L2 2πL2 πL2 µ πL2 n 2GL2
n=1
Eii =− µ2 + µ2 − µ2 ln(cid:18)2η˜(cid:19)+ 2µ (cid:88)∞ (−1)nK1(2µn) + µ2 . (15)
N 2π(cid:15)L2 2πL2 πL2 µ πL2 n 2GL2
n=1
In the following step, we minimize the energy densities with respect to µ. Then, we use (B5) and obtain for each BC
an expression for G
(cid:40) (cid:18) (cid:19) ∞ (cid:41)−1
1 2η˜ (cid:88)
GP =π −2+2ln +4 K (nµ) , (16)
(cid:15) µ 0
n=1
(cid:40) (cid:18) (cid:19) ∞ (cid:41)−1
1 2η˜ (cid:88)
GAP =π −2+2ln −4 (K (nµ)−2K (2nµ)) , (17)
(cid:15) µ 0 0
n=1
(cid:40) (cid:18) (cid:19) ∞ (cid:41)−1
1 2η˜ (cid:88)
Gi =π −2+2ln +4 K (2nµ) , (18)
(cid:15) µ 0
n=1
(cid:40) (cid:18) (cid:19) ∞ (cid:41)−1
1 2η˜ (cid:88)
Gii =π −2+2ln +4 (−1)nK (2µn) , (19)
(cid:15) µ 0
n=1
where (cid:15)=s+1/2 goes to zero and must be considered as the ultraviolet cut-off.
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If we Consider µ and η˜ constants, so the running of G should depends on the BC’s. But, fixing the value of a
common G for certain scale, implying different values of µ for each BC. If we take the limit (cid:15) → 0, we observe an
universal behaviour for G, independent of the BC’s
We observe from the general relation (B5), that there is dependency of the constants µ and η˜for each BC through
the transcendental equation:
(cid:18) (cid:19) ∞
2η˜ 4 (cid:88)
2ln + cos(φn)K (µrn)=C,
µ π 0
n=1
being C an arbitrary constant.
Considering the traditional point of view where the physical µ must be independent of the cutt-off (cid:15) we have a
renormalization group equation
dµ ∂µ ∂G ∂µ ∂η˜∂µ
(cid:15)2 =(cid:15)2 +(cid:15)2 +(cid:15)2 =0.
d(cid:15) ∂(cid:15) ∂(cid:15) ∂G ∂(cid:15) ∂η˜
For (17)-(19), it is computed the beta function
dG
β =(cid:15)2 ,
d(cid:15)
we obtain
G2
βP =βAP =βi =βii = , (20)
π
meaning an universal behaviour of G((cid:15)) as it is shown if figure (1).
FIG. 1. The running of G, for different BC’s fixing the parameters in order to have G=1 for (cid:15)=1.
III. CASIMIR ENERGY FOR GLOBAL BOUNDARY CONDITIONS
Imposing BC’s of the form
(cid:18) (cid:19) (cid:18) (cid:19)
φ(x+L) φ(x)
=eiα .
χ(x+L) χ(x)
It is equivalent to study
(cid:18) (cid:19) (cid:18) (cid:19)
f(x+L) f(x)
=eiα ,
g(x+L) g(x)
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because of the linear relation (7).
In equations (8) and (9) we have solutions of the form
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
f(x) f (x) f (x) α
= 1 2 , (21)
g(x) g1(x) g2(x) β
where the values of α,β depend on the imposed BC on the problem. We can define the matrix
(cid:18) (cid:19)
f (x) f (x)
H(x)= 1 2 . (22)
g (x) g (x)
1 2
Assuming that it is invertible, i.e. det(H(x))(cid:54)=0, we can isolate the constants
(cid:18) (cid:19) (cid:18) (cid:19)
α f(0)
=H−1(0) , (23)
β g(0)
meaning that
(cid:18) (cid:19) (cid:18) (cid:19)
f(x) f(0)
=H(x)H−1(0) . (24)
g(x) g(0)
On the other side, the BC can be expressed in the following way
(cid:18) (cid:19) (cid:18) (cid:19)
f(L) f(0)
=M , (25)
g(L) g(0)
so, evaluating (24) in x=L and comparing with (25), we have
(cid:18) (cid:19)
(cid:2)M−H(L)H−1(0)(cid:3) f(0) ⇒det(cid:2)M−H(L)H−1(0)(cid:3)=0. (26)
g(0)
Which is the condition for the eigenvalues of the problem.
We can include the periodic and anti periodic case by the parametrization
(cid:18) (cid:19)
exp(iα) 0
M= .
0 exp(iα)
From (8) and (9), we have
(cid:32)c√os(Ωx) −s√in(Ωx)(cid:33)
H(x)= λ+m λ+m .
sin(Ωx) cos(Ωx)
√ √
λ−m λ−m
Eq. (26) leads to the condition
(2πn+φ)2
cosΩL=cosφ→Ω2 = , n∈Z.
r2L2
√
Since Ω= λ2−m2, we have
(2πn+φ)2 4π2 (cid:34)m2L2r2 (cid:18) φ (cid:19)2(cid:35)
λ2 =m2+ = + n+ . (27)
n r2L2 r2L2 4π2 2π
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We define µ = mL and r a parameter which depends of the boundary conditions, so, the general expression for the
Casimir energy is given by
E =(cid:104)Hˆ(cid:105)= 1(cid:88)λ = lim 1(cid:18)2π(cid:19)−2s (cid:88)∞ (cid:34)µ2r2 +(cid:18)n+ φ (cid:19)2(cid:35)−s.
Cas 2 n s→−1/22 rL 4π2 2π
n n=−∞
From appendix A, we have
1 (cid:18) 4π2 (cid:19)1/2−(cid:15)(cid:18) 1 (cid:19)−(cid:15)(cid:34)µ2r2 µ2r2 µ2r2 µr µ (cid:88)∞ cosφn (cid:35)
E = FP − − ln − K (µrn) , (28)
Cas 2 r2L2 η2 8π2(cid:15) 8π2 4π2 4π π2 n 1
n=1
π (cid:18)ηLr(cid:19)2(cid:15)(cid:34)µ2r2 µ2r2 µ2r2 µr µ (cid:88)∞ cosφn (cid:35)
= FP − − ln − K (µrn) , (29)
rL 2π 8π2(cid:15) 8π2 4π2 4π π2 n 1
n=1
π (cid:34)µ2r2 µ2r2 µ2r2 2ηL µr (cid:88)∞ cosφn (cid:35)
= FP − + ln − K (µrn) . (30)
rL 8π2(cid:15) 8π2 4π2 µ π2 n 1
n=1
Ending with
µ2r µ2r 2ηL µ (cid:88)∞ cosφn
ξ ≡LE =− + ln − K (µrn). (31)
Cas Cas 8π 4π µ π n 1
n=1
The Casimir Force
dE ξ 1 ∂ξ m∂ξ
F =− Cas = Cas − Cas − Cas,
Cas dL L2 L ∂L L ∂µ
µ2r ξ µ ∂ξ
=− + Cas − Cas,
4πL2 L2 L2 ∂µ
µ2r µ2r 2ηL µ2r (cid:88)∞ µ (cid:88)∞ cos(φn)
= − ln − cos(φn)K (µrn)− K (µrn). (32)
8πL2 4πL2 µ πL2 0 πL2 n 1
n=1 n=1
We define η˜≡ηL, and F ≡F/m2, so
∞ ∞
F ≡ FCas = r − r ln2η˜ − r (cid:88)cos(φn)K (µrn)− 1 (cid:88) cos(φn)K (µrn). (33)
Cas m2 8π 4π µ π 0 µπ n 1
n=1 n=1
IV. SPECIFIC BOUNDARY CONDITIONS
Anti Periodic BC
We first, assume anti periodic BC for our spinor solution
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
f(x+L) f(x) −1 0 f(x)
=− = .
g(x+L) g(x) 0 −1 g(x)
So, we have φ=π in (27), so
(2πn+π)2
λ2 =m2+ →r =1. (34)
n L2
8
Taking non-dimensional parameter µ=mL, using (31) we have the Casimir energy
ξAP =−µ2 + µ2 ln(cid:18)2η˜(cid:19)+ µ (cid:88)∞ (cid:18)K1µn) − K1(2µn)(cid:19), (35)
Cas 8π 4π µ π n n
n=1
and the Casimir force
(cid:18) (cid:19) ∞ ∞ (cid:18) (cid:19)
FAP = 1 − 1 ln 2η˜ + 1 (cid:88)(K (µn)−2K (2µn))+ 1 (cid:88) K1(µn) − K1(2µn) . (36)
Cas 8π 4π µ π 0 0 µπ n n
n=1 n=1
FIG. 2. Behaviour of ξ for different BC’s and η˜=1. We FIG. 3. Behaviour of ξ for different BC’s and η˜=4. We
observethatξP andξi neverreachthezeropointenergy. obtain that that ξP and ξi crosses the zero point energy
WehaveanasymptoticbehaviourcoincidingξP withξAP for a certain region of the parameter µ.
and ξi with ξii.
Periodic BC
Now we have the BC’s
f(x+L)=f(x),
g(x+L)=g(x). (37)
Proceeding as before, we obtain
4π2n2
λ2 =m2+ →φ=0, r =1. (38)
n L2
The Casimir energy is given by
µ2 µ2 (cid:18)2η˜(cid:19) µ (cid:88)∞ 1
ξP =− + ln − K (nµ), (39)
Cas 8π 4π µ π n 1
n=1
and the Casimir force
9
(cid:18) (cid:19) ∞ ∞
1 1 2η˜ 1 (cid:88) 1 (cid:88)
FP = − ln − K (µn)− K (µn). (40)
Cas 8π 4π µ π 0 µπ 1
n=1 n=1
Zero current BC
The confining condition is imposing the zero current condition at the borders
inµΨ¯γµΨ=0(cid:12)(cid:12)x=0, nµΨ¯γµΨ=0(cid:12)(cid:12)x=L. (41)
And the eigenvalues are
n2π2
λi,2 =m2+ →r =2andφ=0,
n L2
(2n+1)2π2
λii,2 =m2+ →r =2andφ=π. (42)
n (2L)2
According to ec.(31) with r =2, the Casimir energy and the Casimir force for this eigenvalues are given by
ξi =−µ2 + µ2 ln(cid:18)2η˜(cid:19)− µ (cid:88)∞ K1(2µn),
Cas 4π 2π µ π n
n=1
ξii =−µ2 + µ2 ln(cid:18)2η˜(cid:19)− µ (cid:88)∞ (−1)nK1(2µn). (43)
Cas 4π 2π µ π n
n=1
(cid:18) (cid:19) ∞ ∞
Fi = 1 − 1 ln 2η˜ − 2 (cid:88)K (2µn)− 1 (cid:88) K1(2µn),
Cas 4π 2π µ π 0 µπ n
n=1 n=1
(cid:18) (cid:19) ∞ ∞
Fii = 1 − 1 ln 2η˜ − 2 (cid:88)(−1)nK (2µn)− 1 (cid:88)(−1)nK1(2µn). (44)
Cas 4π 2π µ π 0 µπ n
n=1 n=1
Limiting values
As can be seen from figures (2) and (3), the behaviour for small µ depends on the BC’s. In fact, the parameter φ
determines the sign of the force as µ goes to zero. We are interested in the sign of the force for µ∼0, where the force
clearly goes to ±∞. Keeping the leading terms for µ≈0:
∞
1 (cid:88) cos(φn)
F ≈− +constants. (45)
µ2π n2
n=1
It is more clear to take the derivative to leading order
dF ≈ 1 (cid:2)Li (eiφ)+Li (e−iφ)(cid:3), (46)
dµ µ3π 2 2
µ→0
where Li (x) are Polylogarithm functions (see, for example [9]). Since the positive derivative means a negative force
n
and vice versa. The regime changes for the non physical value of φ = φ∗ ≈ 1.328, as it is shown in the figure (6),
notice that φ∗ does not depend on η˜.
Another curious feature happen with FAP and Fii. When η goes beyond a given value η˜∗ = η˜∗(φ), the force
becomes negative, having an equilibrium points A(A(cid:48)) and a metastable point B, as it is clear from figure (7).
10
FIG.4. BehaviourofFfordifferentBC’sandηL=η˜=1. FIG. 5. Behaviour of F for different BC’s and η˜ = 4.
We can see that FAP and Fii are always positive. It is There it happens that FAP and Fii acquire a negative
also seen an asymptotic behaviour coinciding FP with value in some limited region of µ.
FAP and Fi with Fii.
FIG. 6. Behaviour of the numerator in (46). which indi- FIG. 7. Behaviour of FAP and Fii for η˜ = 4. There it
cates the slope of the force when µ≈0 happens that FAP and Fii acquire a negative value in
region A(A(cid:48)) and becomes zero in the point B.
V. CONCLUSIONS AND DISCUSSION
The first part of this letter was aware of the ultraviolet behaviour of he GN model for different BC’s, in the
framework of mean field theory assuming homogeneous solution and using zeta function regularization. We found
that the beta function is independent of the type of boundary condition used, and that there appears a mass scale
of arbitrary value. The generated dynamical mass should depend on the BC’s, if we have no prescription on the
arbitrary mass scale.
Later, assuming, an homogeneous solution, we studied the Casimir energy and forces for different BC’s, if we