Another look at charge fractionalization at finite temperature

0.15 MB·English

by  Yeong-Chuan Kao

Another look at charge fractionalization at finite temperature
Yeong-Chuan Kao and Ming-Chiun Wu
Department of Physics, National Taiwan University, Taipei, Taiwan
Westudyagainthephenomenonofchargefractionalizationatfinitetemperature. Ourcalculations
aredoneinaframeworkinwhichtheconnectionbetweentheinducedfermionnumberandthechiral
anomaly is manifest. Wefind that thefractional fermion numberinduced on a soliton decreases as
temperaturerisesandvanishesatinfinitetemperatureatone-looplevel. Theseresultsareconsistent
with previous studies. As an application of our approach, we havealso studied the behavior of the
inducedChern-Simons term in the (2+1)-dimensional QED at finite temperature.
PACSnumbers: 11.10.Wx, 11.10.Kk,12.39.Fe
Keywords:
2
1
0 I. INTRODUCTION
2

n Twoofthe mostintriguingaspectsofquantumfieldtheoryarethechiralanomaly,whichreferstothe thequantum
a
mechanical breaking of the chiral symmetry[1, 2], and the charge fractionalization, which refers to the fact that
J
fractional fermion numbers can be induced by topologically nontrivial background configurations such as kinks or
7
solitons[3–6]. Since boththechiralanomalyandthechargefractionalizationcanbe understoodfromtheviewpointof
2
Dirac’snegativeenergysea[7],itisperhapsnotsurprisingthatatighterconnectionbetweenthesetwophenomenacan

] be made[8–10]. For example, Schaposnik[9] showed that, in a commonly used 1+1-dimensional model, the fermion
h current receives a contribution from the nontrivial Jacobian factor associated with a chiral rotation of the fermionic
t
- field. This contribution leads directly to charge fractionalization. Schaposnik’s analysis was inspired by Fujikawa’s
p
observation[11–14] that the chiral anomaly owes its existence to the nontrivial Jacobian associated with the chiral
e
transformation in the path integral. Therefore, the nontrivial chiral Jacobian links together the chiral anomaly and
h
[ the fractional fermion number.
This connection between the chiral anomaly and the induced fractional fermion number raise the question: How
2
canthechiralJacobianandhencethe chiralanomalybe temperatureindependent[15–23]while thefractionalfermion
v
number is affected by temperature[24–27]? This question is very much like the problem of how the amplitude for
3
π0 γγ, which is deeply related to the chiral anomaly, is affected by temperature in the face of the temperature
7
→
3 independence of the coefficient of the chiral anomaly[28–34]. Here, we present a study of this question. We show, in
4 the field theoretic model studied by Shaposnik, how the fractional fermion number acquires a negative temperature-
. dependentcontribution,whicheventuallycancelsexactlytheanomalycontributionatinfinitetemperatureatone-loop
4
0 order. We have been aided by a previous study[35] of the massive Schwinger model in obtaining this result. The
1 results presented here are consistent with previous studies on induced fermion numbers at finite temperature[25–27].
1 Our aim is to show that although the chiralanomaly and the fractional fermion number share a common origin they
: are ultimately very different due to their different finite temperature behaviors.
v
i With slightmodifications, ouranalysiscanbe immediately applied to the study ofthe induced Chern-Simonsterm
X
in 2+1-dimensional QED at finite temperature[36–43]. The reason is that the induced Chern-Simons term is related
r to the chiral anomaly in a way very similar to the relation between the induced fermion number and the chiral
a
anomaly[41, 44]. We will show in our approach that the coefficient of induced Chern-Simons term vanishes in the
so-called long wavelength limit at infinite temperature, consistent with previous studies.
II. INDUCED FERMION CURRENT AT FINITE TEMPERATURE
Wefirststudythefermionicmodeldefinedbythe followingactionintwo-dimensional(2D)Euclideanspace[3–6,9],
β
= dτ dx ψ¯iγµ∂ ψ+gψ¯e−γ5θψ , (1)
µ
S
Z0 Z
(cid:2) (cid:3)
whereβ isthe inversetemperature1/T,andthe EuclideanDiracmatricesγµ obeytherelations: γµ,γν = 2δµν =
{ } −
2ηµν, µ,ν =0 or 1, γ0 =iσ , γ1 =iσ , γ =σ , γµγ =iǫνµγν, ǫ01 = ǫ10 =1, γ = γν, and trγ γµγν = 2iǫµν.
1 2 5 3 5 ν 5
The field θ represents the background soliton. To study the behavior o−f the fermionic−current ψ¯γµψ in the p−resence
2
of θ, we define the following generating functional:
Z[s]= ψ¯ ψeR0βdτRdx[ψ¯iγµ∂µψ+ψ¯γµsµψ+gψ¯e−γ5θψ], (2)
D D
Z
where a source term s is introduced and the expectation value of the fermion current is given by
µ
1 δZ
Jµ = ψ¯γµψ = . (3)
h i Z δs (x)
µ (cid:12)sµ=0
(cid:12)
(cid:12)
(cid:12)
Following Schaposnik[9], we now perform a chiral rotation ψ = eγ5θ2tχ (t is a parameter varying from 0 to 1) and
turn the Lagrangianinto
=χ¯D χ, (4)
t
L
where
D iγµ∂ +γµs +tγµa +geγ5θ(t−1), (5)
t µ µ µ
≡
with a = 1ǫµν∂ θ. The point of performing the chiral rotation is to eliminate the γ exponential in (1) for t=1
µ −2 ν 5
Again,followingSchaposnik(andFujikawa),wecalculatetheJacobian (θ)associatedwiththeabovechiralrotation
J
for t=1 at finite temperature
ψ¯ ψ = (θ) χ¯ χ, (6)
D D J D D
1 θ
ln (θ) = 2 lim dtTr x,τ γ5 eDt2/M2 x,τ (7)
J − M→∞Z0 h | 2 | i
1 β ∞ dk 1
= lim dt tr dτ dxγ5θ e−ikx−iωnτeDt2/M2eikx+iωnτ (8)
−M→∞Z0 Z0 Z Z−∞ 2πβ Xωn
1 β iγµγν(∂ s )+itγµγν∂ a +g2γ sinh(2θ(t 1))
µ ν µ ν 5
= lim dt tr dτ dxγ θexp −
−M→∞Z0 Z0 Z 5 (cid:26) M2 (cid:27)
∞ dk −k2 1 ∞ −1((2n+1)π)2
eM2 eM2 β (9)
× 2π β
Z−∞ n=−∞
X
1 β iγµγν(∂ s )+itγµγν∂ a +g2γ sinh(2θ(t 1))
µ ν µ ν 5
= lim dt tr dτ dxγ θexp −
−M→∞Z0 Z0 Z 5 (cid:26) M2 (cid:27)
∞ dk −k2 ∞ dk′ −k′2
eM2 e M2 (10)
× 2π 2π
Z−∞ Z−∞
1 1 β
= dt dτ dx tr γ θ iγµγν(∂ s )+itγµγν∂ a +g2γ sinh(2θ(t 1)) (11)
5 µ ν µ ν 5
−4π −
Z0 Z0 Z
1 β (cid:8) 1(cid:2) 1 (cid:3)(cid:9)
= dτ dx ǫνµs ∂ θ+ (∂ θ)2+ g2(cosh2θ 1) , (12)
ν µ µ
2π − 4 2 −
Z0 Z (cid:20) (cid:21)
where in Eq. (9) we have kept only terms which are not zero in D2 after taking the Dirac trace; in Eq. (10) we turn
t
the sum over Matusbara modes into an integral [45],
2π ∞ e−((2nM+β1)π)2 = ∞ dk e−Mk22. (13)
β
n=−∞ Z−∞
X
In the end, we have
3
1 β 1 1
Z =exp dτ dx[ ǫνµsν∂µθ+ (∂µθ)2+ g2(cosh2θ 1)] χ¯ χeR0βdτRdxχ¯(iγµ∂µ+γµsµ+γµaµ+g)χ.(14)
"2π Z0 Z − 4 2 − #Z D D
Substituting Eq. (14) into Eq. (3), we obtain the induced fermion current
1 a
Jµ = ǫµν∂ θ+jµ = µ +jµ, (15)
ν
−2π π
where
δ
jµ = lndet(iγµ∂ +γµa +g), (16)
µ µ
δa
µ
is the induced current in the massive Schwinger model[46] with fermion mass g at finite temperature. Just like the
temperatureindependence ofthe chiralanomaly[15–23],the coefficientofthe aµ terminEq. (15)is alsoindependent
π
of temperature as expected. Because of the aµ term, when we integrate J0 over x, we find that the induced fermion
π
number is dictated by by the asymptotic values of θ at x= , and hence the fermion number can be fractional in
the presence ofasoliton. The currentjµ inthe massiveSchw±in∞germodelwill onlycontainderivativeterms ofa and
µ
thereforedoesnotcontributetotheinducedfermionnumber. Butjµ willplayanimportantroleatfinitetemperature
as is shown in the next section.
III. FINITE TEMPERATURE EFFECT ON INDUCED FERMION NUMBER
To see how the induced fermion number obtained in the previous section changes with temperature, we need to
calculate vacuum polarization tensor of the massive Schwinger model at finite temperature. To one loop-order, the
vacuum polarization tensor at finite temperature is given by
1 ∞ dk
Πµν(p ,p )= 1tr[γµS (p+k)γνS (k)], (17)
0 1 β β
β 2π
n Z−∞
X
which can be written as[35]
Πµν(p ,p )= p2ηµν pµpν Π(p ,p ), (18)
0 1 0 1
−
with (cid:0) (cid:1)
Π(p ,p )=Π0(p ,p )+δΠ(p ,p ), (19)
0 1 0 1 0 1
where Π0 is the zero-temperature part, and δΠ is the temperature-dependent part. As there is a fermion mass gap,
the zero-temperaturepartΠ0 does notcontaina 1/p2-pole termandtherefore couldnot contribute to the a term as
µ
mentioned above. If we want to know how jµ contributes to the fermion number at finite temperature, we only need
to look at (the real part of) δΠ. Borrowing the result from [35], we know that
2g2 ∞ dk n (E ) 1 1
1 F +
δΠ(p ,p )= +
0 1 p2 p2 2π E (E p )2 E2 (E +p )2 E2
− 0− 1 Z−∞ (cid:26) + (cid:20) +− 0 − − + 0 − −(cid:21)
n (E ) 1 1
F −
+ + , (20)
E (E p )2 E2 (E +p )2 E2
− (cid:20) −− 0 − + − 0 − +(cid:21)(cid:27)
with E = (k +p /2)2+g2 , and E = (k p /2)2+g2.
+ 1 1 − 1 1
−
We write δΠ as
p p
1 1
δΠ(p ,p )= I(p ,p ;βg)= I(p ,p ;βg), (21)
0 1 p2 p2 0 1 p2 0 1
− 0− 1
4
where I(p ,p ;βg) is the integral in Eq. (20). Straightforward calculations give us the following expression for the
0 1
static limit of I(p ,p ;βg):
0 1
∞ dy 2 2βgeβg√y2+1
I(p =0,p 0;βg)= − + − . (22)
0 1 → Z−∞ 2π  eβg√y2+1+1 (y2+1)32 eβg√y2+1+1 2(y2+1)
(cid:16) (cid:17) (cid:16) (cid:17)
 
If I(p = 0,p 0;βg) is not zero, δΠ will contain a 1/p2-pole term and j0 will contain a term proportional to a
0 1 0
→
which can contribute to the induced fermion number when integrated over x. If we only keep terms proportional to
a , the fermion number density J0 becomes
0
a a c(βg)
J0 = 0 +j0 = 0 +Π00a = a , (23)
0 0
π π π
where
c(βg)=1+πI(p =0,p 0;βg). (24)
0 1
→
Numericalintegrationsallowustoplotc(βg)inFig. 1. Wecanseethatc(βg)decreasesfrom1atzerotemperatureto
zeroatinfinite temperature. Therefore,theinducedfermionnumber(obtainedbyintegratingJ0 overx)willdecrease
when temperature increases and vanish completely at infinite temperature.
In comparison with earlier results, we give the following low- and high-temperature limits of Eq. (22),
2 βge−βg+ , βg 1
I(p0 =0,p1 0;βg) − 2π ··· ≫ . (25)
→ ≈(−π1q+2c2β2g2+O((βg)3) , βg ≪1
with
∞
1 1
c = . (26)
2 π3 (2n+1)3
n=0
X
Given the above results, the relevant terms in j0 will be
2 βge−βg+ a , βg 1
j0 =I(p0 =0,p1 →0;βg)a0 ∼ (cid:18)−1q+22πc β2g2+···(cid:19)a0 , βg ≫1 . (27)
 −π 2 ··· 0 ≪
and  (cid:0) (cid:1)
1 √2πβge−βg+ , βg 1
c(βg) − ··· ≫ . (28)
∼ 2πc2β2g2+ , βg 1
(cid:26) ··· ≪
These results are consistent with previous studies[25–27].
IV. INDUCED CHERN-SIMONS TERM AT FINITE TEMPERATURE
As an immediate extension of the above analysis, in this section we shall recalculate the much studied coefficient
of the induced Chern-Simons term in the (2+1)-dimensional QED at finite temperature[36–43]. It is known that
nonanalyticityatpµ =0 appearsinthe parity-oddpartofthe gaugefieldself-energyΠµν(p) atfinite temperature[39,
43]. Itturnsoutthatourapproachallowsthe use ofasetofbackgroundgaugefieldconfigurationsthatwillnaturally
leadustothe longwavelengthlimit(p 0,p~=0)oftheparity-oddpartofthe self-energy[39,40,43], insteadofthe
0
→
static limit (p =0,~p 0) commonly obtained in [36–38, 41, 42]. Therefore, while the generalstrategy we employ is
0
→
the same as that of Fosco, Rossini, and Schaposnik[41, 42], our result also nicely complements theirs.
The theory we study is given by the following action in three-dimensional (3D) Euclidean space:
5
β L1 L2
= dτ dx dx ψ¯(τ,x ,x ) i∂/+ea/+M ψ(τ,x ,x ), (29)
1 2 1 2 1 2
S
Z0 Z0 Z0
(cid:2) (cid:3)
withthe3DEuclideanDiracmatricesγµchosentobeγ0 =iσ ,γ1 =iσ ,γ2 =iσ . Forconvenience,weletx [0,L ]
1 3 2 1 1
∈
and x [0,L ], with L and L serving as large infrared cutoffs. The goal is to integrate out the fermionic fields to
2 2 1 2
∈
find the effective action as was done in previous sections. We choose to have a spatially uniform but time-dependent
electric field as the backgroundelectromagnetic field. Therefore, the class of gauge field configurationscorresponding
to this particular kind of electric field can be set as a (τ)=(a (τ),a (τ),a (τ)). For comparisons,we point out that
µ 0 1 2
a time-independent magnetic field is employed in [41, 42] as the background electromagnetic field.
The first step of the analysis is to Fourier transform the fermionic field over the x coordinate as
1
1 (2n+1)π
ψ(τ,x ,x )= e−iknx1ψ (τ,x ) ; with k = , (30)
1 2 L kn 2 n L
1 1
n
X
then the action becomes
1 β L2
S = dτ dx ψ¯ (τ,x ) /d+γ1(k +ea )+M ψ (τ,x ), (31)
L 2 kn 2 n 1 kn 2
1 n Z0 Z0
X (cid:2) (cid:3)
where
d/=γ0(i∂ +ea )+γ2(i∂ +ea ).
0 0 2 2
Hence,theproblemofobtainingthedeterminantofthe 3DDiracoperatordet[i∂/+ea/+M]becomesthatofobtaining
the determinant of the reduced 2D Dirac operators, which we write as
det[d/+γ1(kn+ea1)+M]=det[d/+ρkn(τ)eiγ5φkn(τ)], (32)
where γ σ and
5 3
≡
ρ (τ)= M2+(k +ea (τ))2; (33)
kn n 1
k +ea (τ)
φ (τ)=ptan−1 n 1 . (34)
kn M
(cid:18) (cid:19)
Following the strategy explained in section II and in [41, 42], we now make a chiral rotation to transform away the
factor eiγ5φkn in Eq. (32) so that the determinant becomes
det[d/+ρkn(τ)eiγ5φkn(τ)]=Jkndet[D/ +ρkn(τ)], (35)
where D/ =γ0(i∂ +ea +b )+γ2(i∂ +ea +b ), b = i∂ φ =0, b = i∂ φ . Note that the 2DDirac operator
0 0 0 2 2 2 0 2 2 kn 2 −2 0 kn
D/ +ρ (τ) is that of the massive Schwinger model with fermion mass ρ (τ) . We easily find the Fujikawa Jacobian
kn kn
to be
Jkn
e β i 1
ln = i L dτ φ ∂ a (∂ φ )2+ ρ2 (cos2φ 1) , (36)
Jkn − 2π 2 − kn 0 2− 4 0 kn 2 kn kn −
Z0 (cid:20) (cid:21)
by adapting Eq. (12) to the present case. In the end we have
lndet[∂/+iea/+M] = ln det[D/ +ρ (τ)] (37)
Jkn kn
n
X (cid:8) (cid:9)
ln + lndet[D/ +ρ (τ)], (38)
≡ J kn
n
X
6
with
ln = ln (39)
J Jkn
n
X
e β k +ea
= i L dτ tan−1 n 1 ∂ a (40)
2 0 2
2π M
n Z0 (cid:20) (cid:18) (cid:19) (cid:21)
X
M e2 β
= i L L dτa ∂ a (41)
1 2 1 0 2
M 4π
| | Z0
1 M e2 β
= i L L dτ(a ∂ a a ∂ a ). (42)
1 2 1 0 2 2 0 1
2 M 4π −
| | Z0
It is seen that the Fujikawa Jacobianleads to the following induced Chern-Simons term in the effective action in our
chosen backgroundfield,
1 M e2 β ∞ ∞
S = i dτ d2x ǫµνα a ∂ a . (43)
eff µ ν α
− 2 M 4π
| | Z0 Z−∞Z−∞
Next, we consider the contributions from lndet[D/ +ρ (τ)] in Eq. (38). At zero temperature, these contributions
kn
vanish,buttheybecomenonvanishingatfinitetemperaturesimilartothecontributiontotheinducedfermionnumber
from the jµ term in Eq. (16) . The one-loop contribution from lndet[D/ +ρ ] to the local Chern-Simons term is
kn
again easily obtained from the vacuum polarization tensor Πij(p ,p ) = p2ηij pipj Π(p ,p ) (with i,j = 0 or 2,
0 2 0 2
−
and η00 = 1=η22) of the massive Schwinger model given in Sec. III in the long wavelength limit x [0,L ]
− (cid:0) (cid:1) 2 ∈ 2
S lndet[D/ +ρ (τ,x )] (44)
kn ≡ kn 2
L β
= 2 dτ(ea +b ) [Πij(p 0,p =0)] (ea +b )+ (45)
i i 0 2 j j
2 · → · ···
Z0
L β
= 2 dτI(p 0,p =0;βρ )ηij(ea +b )(ea +b )+ , (46)
2 0 → 2 kn i i j j ···
Z0
where I(p ,p ;βρ ) is defined by Eq. (21) with p replaced by p and βg replaced by βρ .
0 2 kn 1 2 kn
Again by using the result in [35], we have the following long wavelength limit
∞ dq 2
I(p 0,p =0;βρ )= − . (47)
0 → 2 kn Z−∞ 2π eβρkn√q2+1+1 (q2+1)32
(cid:16) (cid:17)
Substituting Eq. (47) into Eq. (46), we see that S contains a term of the form
kn
e β
i L dτI(p 0,p =0;βρ )a ∂ φ (48)
− 2 2 0 → 2 kn 2 0 kn
Z0
M e β ∞ dq 2φ (M )
=i L dτ − kn | | ∂ a . (49)
2 0 2
|M|2 Z0 Z−∞ 2π eβρkn√q2+1+1 (q2+1)23
(cid:16) (cid:17)
Hence, the contributions from S to the effective action contains the following term
n kn
P
M e β ∞ 2φ (M )
i L dτ dq − kn | | ∂ a (50)
2 0 2
|M|4π n Z0 Z−∞ eβρkn√q2+1+1 (q2+1)32
X
M e L L ∞ β (cid:16) ∞ (cid:17) 2φ (M )
1 2 k
i dk dτ dq − | | ∂ a , (51)
0 2
→ |M|4π 2π Z−∞ Z0 Z−∞ eβρk√q2+1+1 (q2+1)32
(cid:16) (cid:17)
where we approximate the sum over n by an integral over k for very large L . Expanding Eq. (34) to first order in
1
7
ea , we find
1
∞ 2φ ∞ 2
k
dk − = dk −
Z−∞ eβρk√q2+1+1 (q2+1)23 Z−∞ ( eβ|M|√(k2+1)(q2+1)+1 (1+k2)(q2+1)32
(cid:16) (cid:17) (cid:16) 2β M (tan−1k)keβ(cid:17)|M|√(k2+1)(q2+1)
+ | | ea + , (52)
eβ|M|√(k2+1)(q2+1)+1 2(q2+1)√k2+1) 1 ···
(cid:16) (cid:17)
Atthe end,wefindthatinadditiontoEq. (39), theeffectiveactioncontainsanextraChern-Simonstermofthe form
M e2 β
ic (β M ) L L dτa ∂ a (53)
3 1 2 1 0 2
− | | M 4π
| | Z0
c (β M ) M e2 β ∞ ∞
= i 3 | | dτ d2x (a ∂ a a ∂ a ), (54)
1 0 2 2 0 1
− 2 M 4π −
| | Z0 Z−∞Z−∞
where
∞ ∞ dq 2
c (β M ) dk
3
| | ≡ Z−∞ Z−∞ 2π( eβ|M|√(k2+1)(q2+1)+1 (1+k2)(q2+1)32
(cid:16) 2β M (tan−1k)ke(cid:17)β|M|√(k2+1)(q2+1)
| | . (55)
− eβ|M|√(k2+1)(q2+1)+1 2(q2+1)√k2+1)
(cid:16) (cid:17)
Combining Eq. (42) and Eq. (54), we finally have the full-induced Chern-Simons term in the effective action at
finite temperature,
(1 c (β M )) M e2 β ∞ ∞
Sβ = i − 3 | | dτ d2x (a ∂ a a ∂ a ) (56)
eff 2 M 4π 1 0 2− 2 0 1
| | Z0 Z−∞Z−∞
κ(β M ) M e2 β ∞ ∞
i | | dτ d2x (a ∂ a a ∂ a ) (57)
1 0 2 2 0 1
≡ 2 M 4π −
| | Z0 Z−∞Z−∞
κ(β M ) M e2 β ∞ ∞
= i | | dτ d2x ǫµνα a ∂ a . (58)
µ ν α
− 2 M 4π
| | Z0 Z−∞Z−∞
We obtainthe coefficientofthe inducedChern-Simonstermκ(β M )asafunctionofβ M bynumericalintegrations.
| | | |
The result is plotted in Fig.(2). We see that κ(β M ) vanishes at infinite temperature in agreement with a previous
| |
study in the long wavelength limit[39].
V. DISCUSSIONS
Schaposnik has shown that the induced fermion number on a soliton can be seen as arising from the Fujikawa
jacobian associated with a chiral rotation. We have shown in Schaposnik’s approach that the induced fractional
fermion number on a soliton will indeed reduce smoothly to zero at one-loop order when temperature increases to
infinity. (As a byproduct of our analysis, we have also shown that the coefficient of the induced Chern-Simons
term in 2+1 QED vanishes in the long wavelength limit at infinite temperature.) This behavior is in contrast
with the well-known temperature independence of the chiral anomaly. Therefore, while the phenomena of charge
fractionalizationand chiralanomaly are related to eachother, they are in the end very different, as seen mostclearly
at finite temperature. Besides the analogy to the temperature dependence of the amplitude for π0 γγ mentioned
→
in the introduction, we also want to point out that the fate of the induced fermion number is not unlike that of
the chiral condensate ψ¯ψ in the massless Schwinger model[47, 48]. It is known that without the chiral anomaly to
spoil the chiral symmehtry,ithere will be no nonvanishing ψ¯ψ in the massless Schwinger model because the Coleman
theorem[49] forbids spontaneous breakdown of continuouhs syimmetries in 2D. In a way, we may say that ψ¯ψ owes
h i
8
its existence to the chiral anomaly. Yet, ψ¯ψ still decreases smoothly to zero at infinite temperature[50] even if the
h i
chiral anomaly is not affected by temperature.
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9
c(βg)
1
0.8
0.6
0.4
0.2
1/(βg)
0
0 2 4 6 8 10
FIG. 1: Plot of c(βg) definedby Eq. (25)
κ(β|M|)
1
0.8
0.6
0.4
0.2
1/(β|M|)
0
0 1 2 3 4 5(X103)
FIG. 2: Plot of κ(β|M|)