Chern-Simons-Fermion Vector Model with Chemical Potential

0.19 MB·English

by  Shuichi Yokoyama

TIFR/TH/12-34
Oct 2012
Chern-Simons-Fermion Vector Model
2
1 with Chemical Potential
0
2

t
c
O

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1
Shuichi Yokoyama

]
h
t
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p Department of Theoretical Physics, Tata Institute of Fundamental Research,
e
h Homi Bhabha Road, Mumbai 400005, India
[

1
v
9 E-mail: yokoyama(at)theory.tifr.res.in
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1
4
.
0
1
2 Abstract
1
:
v
We study SU(N) level k Chern-Simons theories coupling to a fundamental
i
X fermion with chemical potential for U(1) flavor symmetry. We solve this system
r exactly inthe’t Hooftlimit, N,k with λ = N/k fixed. The solutionexists
a
→ ∞
up to λ = 1 for a fixed value of chemical potential. We study the system
| |
beyond λ = 1 by considering U(N) gauge group and taking the diagonal U(1)
| |
holonomy into account. The result suggests the system becomes unstable and
a phase transition happens at λ = 1.
| |
1 Introduction
Vector (or sigma) models play important roles to understand not only phenomenology but
also theoretical highenergy physics. In the context of AdS/CFT duality they play a crucial
role for the duality proposed in [1]. The duality claims that the singlet sector of a three
dimensional O(N) (or U(N)) vector model with conformal symmetry and a higher spin
gravity theory on AdS [2–4] in the large N limit. This conjecture has been tested by
4
comparing spectra and correlation functions of both sides [5–8].
Motivated by this conjecture three dimensional vector models have been studied in
presence of CS interaction by gauging global symmetry [9, 10]. It was studied that confor-
mal symmetry and higher spin symmetry are still preserved up to 1/N correction and the
effect of CS interaction is reflected as a suitable change of boundary condition for a bulk
scalar field. This class of conformal field theories turned out to have strongly constrained
correlation functions [11–14]. The duality can be also studied by changing a base manifold
[15, 16] or in accordance with supersymmetry [5, 17–19]. A recent review is [20].
In this paper we consider a little generalization of [9] by adding chemical potential for
U(1) flavor symmetry into Chern-Simons-fermion vector model. One of the achievements
in [9] is solving the system by taking the light-cone gauge and the ’t Hooft limit. The
free energy is determined exactly as a function of λ. Interestingly the solution exists up
to λ = 1 and the free energy vanishes at that point. The motivation of this paper is
| |
to compute other various thermodynamic quantities such as the grand canonical potential
and the thermodynamic descendants thereof exactly as a function of λ and to address what
happens to the system beyond λ = 1.
| |
The rest of the paper is organized in the following manner. In Section 2 we present
the exact solution of Chern-Simons-fermion vector model with chemical potential in the
’t Hooft limit. In Section 3 we calculate various thermodynamic quantities exactly as a
function of λ. In Section 4 we study this system beyond λ = 1 by considering the gauge
| |
groupU(N)andtakingaccountofthediagonalU(1)holonomy. Thefinalsectionisdevoted
to summary and discussion.
2 Chern-Simons-fermion vector theory with chemical
potential
In this section we study SU(N) Chern-Simons gauge theory at level k with a fermion which
is in the fundamental representation of the gauge group in presence of chemical potential
µ for U(1) flavor symmetry. The action of this system is
k 2i
S = d3x iεµνρ Tr(A ∂ A A A A )+ψ¯γµD ψ µψ¯γ3ψ (2.1)
µ ν ρ µ ν ρ µ
4π − 3 −
Z (cid:20) (cid:21)
We use the same convention in [9, 19].
This sytem can be exactly solved in the ’t Hooft limit [9]
N,k with λ := N/k fixed. (2.2)
→ ∞
1
Themethodisthefollowing. Firstlyonefixesthegaugedegreesoffreedombythelight-cone
gauge in Euclidean space A := 1 (A +iA ) = 0 and integrate out the other components
− √2 1 2
of the gauge field. This generates four fermi interaction. Secondly one introduces a bilocal
singlet field Σ so that the four fermi interaction term cancels, which is known as Hubbard-
Stratonovich transformation. Σ turns out to coincide with the self energy of the fermionic
field onshell. The resulting action is quadratic with respect to the fermion and thus we can
integrate it out by gaussian integration. The remaining action is written in terms of the
singlet field and N can be factored out from the action. Finally one can obtain the exact
effective action in the leading of 1/N expansion by the steepest descent method under the
assumption of translational invariance.
From the action (2.1) we can easily see that the result in our case can be obtained from
that in [9] by exchanging the momentum in the fermion propagator in such a way that
p p := p +iµδ . (2.3)
µ µ µ µ,3
→
After this repalacement we find the exact effective action
e
d3q 1 1
S = NV tr log[iγµq +Σ(q)]+ Σ(q) (2.4)
eff (2π)3 − µ 2 iγµq +Σ(q)
Z (cid:20) (cid:18) µ (cid:19)(cid:21)
where the trace is taken over a two-by-two matrix with spinor indices and V is the volume
e
of the Euclidean space. As shown in [9] this effective action corerectly reproduces the
connected vacuum diagram consisting of planar graphs perturbatively with respect to λ.
The exact propagator of fermionic field is
1
ψ¯( p)ψ(q) = − (2π)3δ3( p+q). (2.5)
− iγµp +Σ −
µ
(cid:10) (cid:11)
Spinor indices are suppressed here and hereafter. We also find the saddle point equation
e
d3q 1 1 2πλ
Σ(p) = γ+ γ3 γ3 γ+ . (2.6)
(2π)3 iγµq +Σ(q) − iγµq +Σ(q) i(p q)
Z (cid:18) µ µ (cid:19) − −
where p = 1 (p ip ), γ = 1 (γ iγ ). We can check that this equation precisely
agrees w±ith S√c2hw1in∓ger-D2 yso±n equ√a2etio1n±in t2he ’t Hooftelimit and also coincides with the
equation of 1PI self-energy diagram, which is now given only by the rainbow diagram [9].
Let us solve the saddle point equation (2.6). For this end we decompose Σ into
Σ = γµΣ +IΣ (2.7)
µ I
where I is a two by two unit matrix. Plugging this into (2.6) we find
4πλ d3q i(q iΣ) +γ+Σ 1
(γµΣµ +I2ΣI)(p) = i (2π)3 (q− iΣ−)2 +Σ21 (p q) (2.8)
Z (cid:18) µ − µ 1 (cid:19) − −
e
By comparing both sides we obtain
e
Σ = Σ = 0, (2.9)
3
−
4πλ d3q iq 1
ΣI(p) = − (2.10)
i (2π)3 (q iΣ (q))2 +Σ (q)2 (p q)
Z (cid:18) µ − µ 1 (cid:19) − −
4πλ d3q Σ (q) 1
I
Σ (p) = (2.11)
+ i (2π)3 (eq iΣ (q))2 +Σ (q)2 (p q)
Z (cid:18) µ − µ 1 (cid:19) − −
2
e
From the dimensional analysis and rotational covariance in two plane we can set
Σ (p) = f(p)p , Σ (p) = ig(p)p (2.12)
I ′ s + ′ +
where f and g are unknown functions of p = ps. Plugging these into the above equations
′ µ
we obtain
d3q q 1
f(p′)ps = 4πλ (2π)3 (1+g(q ))q2 +−q2 +(f(q )q )2 (p q) (2.13)
Z (cid:18) ′ s 3 ′ s (cid:19) − −
d3q f(q )q 1
′ s
g(p)p = 4πλ (2.14)
′ + − (2π)3 (1+g(q ))q2 +e q2 +(f(q )q )2 (p q)
Z (cid:18) ′ s 3 ′ s (cid:19) − −
By using ∂ 1 = 2πδ2(p q), we can show ∂ ((f(p)p )2 +g(p)p2) = 0, which leads
∂p+(p q) − ∂p+ e′ s ′ s
to − −
c
f(p)2 +g(p) = 0 , (2.15)
′ ′
p2
′
where c is a dimensionless constant which will depend on λ. g can bedetermined by (2.15)
0
so we focus on solving the equation of f, (2.13). Substituting (2.15) into (2.13) we obtain
d3q q 1
f(p′)ps = 4πλ (2π)3 q2 +q2−+c µ2 (p q) (2.16)
Z (cid:18) s 3 0 (cid:19) − −
Integrating the angular part of p we get
− e
q dq dq 1
∞ s s 3
f(p)p = 4πλ − (2.17)
′ s 2π 2π q2 +q2 +c µ2
Zps Z s 3 0
We regularize UV divergence by dimensional regularization, as done in [9, 19], which leads
e
to1
λ
f(p) = p2 +c θ( p2 +c 1) θ(1 p2 +c ) , (2.19)
′ ′ 0 ′ 0 ′ 0
p | | | |− − − | |
′(cid:18) (cid:19)
p p p
where θ(x) is a step function taking unit value when x > 0 otherwise zero.
c can be determined as follows. First let us plug (2.19) into (2.15) and expand g
0
around p 0:
′
∼
2
λ2 c θ( c 1) θ(1 c )
c 0 0 0
0 | | | |− − − | |
g(p) . (2.20)
′
∼ p2 − (cid:16)p p p2 p (cid:17)
′ ′
From (2.14) and (2.19), on the other hand, we can show g(p) 0 around p 0 in the
′ ′
∼ ∼
same way as in [9, 19]. Therefore c has to satisfy
0
2
c λ2 c θ( c 1) θ(1 c = 0 (2.21)
0 0 0 0
− | | | |− − − | |
(cid:16)p p p (cid:17)
1We note a usuful formula to derive this equation
dq π
3 = θ(a µ). (2.18)
R (q3+iµ)2+a2 a | |−| |
Z | |
3
A unique solution exists only when λ < 1 as c = λ2 and no solution when λ > 1.
0
| | | |
As a result we obtain the solution of the saddle point equation
λ
f(p) = p2 +λ2θ( p2 +λ2 1) θ(1 p2 +λ2) , (2.22)
′ ′ ′ ′
p − − −
′(cid:18) (cid:19)
λ2 p p p
g(p) = (1 p2 λ2)θ( p2 +λ2 1), (2.23)
′ p2 − ′ − ′ −
′
p
where λ < 1. Σ finally becomes
| |
Σ(p) = f(p)p I +ig(p)γ+p . (2.24)
′ s ′ +
Remark that the effective mass becomes λµ .
| |
On the solution (2.24) the effective action (2.4) can be evaluated by using the dimen-
sional regularization as
λ2(1 λ )
S /V = N −| | µ 3. (2.25)
eff
− 6π | |
We can also evaluate momentum distribution of U(1) flavor charge density
d3q dp
n := 3 ψ¯(p)γ3ψ(q) ψ¯(p)γ3ψ(q) , (2.26)
h psi (2π)3 (2π) − |µ=0
Z Z
(cid:0)(cid:10) (cid:11) (cid:10) (cid:11) (cid:1)
where the normalization is such that it vanishes when the chemical potential does. By
using (2.5) and (2.24) this becomes
n = N 1 θ( p2 +(λµ)2 µ) θ( p2 +(λµ)2 µ) . (2.27)
h psi − s − − − s −
(cid:16) p p (cid:17)
This momentum distribution indicates thissystem is described by anideal Fermi gaswhose
Fermi surface is p = µ √1 λ2.
s
| | −
To summarize, the Chern-Simons-fermion vector model in presence of chemical poten-
tial µ is effectively described by a free fermion system with the mass squared (λµ)2 under
the ’t Hooft limit.2
3 Finite temperature analysis
In this section we study the Chern-Simons-fermion system in nonzero temperature T by
placing the system on R2 S1 with circumfirence β = 1/T. We impose anti-periodic
×
boundary condition for S1 direction on the fermionic field in order to study the (grand)
canonical ensemble thereof. Since Fourier mode in the thermal direction is quantized by
2π(n+1)
p = 2 , wecanobtaintheresult innonzerotemperaturefromthepreviouscalculation
3 β
by doing replacement such that
dp 1 1 2π(n+ 1)
3F(p ) F(p ) := F( 2 ), (3.1)
3 3
2π → β β β
Z Xp3:F Xn∈Z
2 Inthisspecificmodelweanalyticallyseeanimportantassumptioninstatisticalphysicsthataweakly
interacting many body system reduces to a free ideal one with appropriate parametrization in the ther-
modynamic limit [21].
4
where F(p ) is an arbitrary function of the momentum in the thermal direction.
3
We first solve the Schwinger-Dyson equation, which is now given by
1 d2q 1 1 2πλ
Σ(p) = γ+ γ3 γ3 γ+ . (3.2)
β (2π)2 iγµq +Σ(q) − iγµq +Σ(q) i(p q)
Xq3:F Z (cid:18) µ µ (cid:19) − −
Repeating the same calculation we obtain
e e
Σ = Σ = 0, (3.3)
3
−
4πN 1 d2q iq 1
ΣI(p) = − (3.4)
ik β (2π)2 (q iΣ(q))2 +Σ (q)2 (p q)
Xp3:F Z (cid:18) − 1 (cid:19) − −
4πN 1 d2q Σ (q) 1
1
Σ+(p) = ik β (2π)2 (eq iΣ(q))2 +Σ (q)2 (p q) (3.5)
Xp3:F Z (cid:18) − 1 (cid:19) − −
From the dimensional ground the ansatz is such that
e
Σ (p) = f(pˆ,µˆ)p , Σ (p) = ig(pˆ,µˆ)p (3.6)
I s + +
where f and g are unknown functions of pˆ= ps and µˆ = µ. Hereafter we will suppress µˆ
T T
in the argument. Then the above equations become
1 d2q q 1
f(pˆ)ps = 4πλβ (2π)2 (1+g(qˆ))q2 +−q2 +(f(qˆ)q )2 (p q) ,
pX3:F Z (cid:18) s 3 s (cid:19) − −
1 d2q f(qˆ)q 1
s
g(pˆ)p+ = −4πλβ (2π)2 (1+g(qˆ))q2 +eq2 +(f(qˆ)q )2 (p q) . (3.7)
Xp3:F Z (cid:18) s 3 s (cid:19) − −
The same argument in the previous section leads to
c
f(pˆ)2 +g(pˆ) = , (3.8)
pˆ2
where c is a dimensionless constant depending on λ and µˆ. Substituting (3.8) into (3.7)
we obtain
1 d2q q 1
f(pˆ)ps =4πλβ (2π)2 q2 +q2−+cT2 (p q)
Xp3:F Z (cid:18) s 3 (cid:19) − −
q dq 1 1
∞ s s
=4πλ e− . (3.9)
2π β q2 +q2 +cT2
Zps Xp3:F s 3
Regularizing UV divergence by dimensional regularization we find3
e
λ
f(pˆ) = log 2(cosh pˆ2 +c +coshµˆ) . (3.11)
pˆ | |
(cid:16) p (cid:17)
3 A useful formula to derive this is
1 1 tanhβ(h+µ) +tanhβ(h−µ)
= 2 2 . (3.10)
β nX∈Z(2π(nβ+21) +iµ)2+h2 4h
5
c can be determined in the same way as in the previous section. Substituting (3.11)
into (3.8) and expanding g around pˆ 0 we find
∼
2
λ2 log 2(cosh c +coshµˆ)
c
| |
g(pˆ) . (3.12)
∼ pˆ2 − (cid:16) (cid:16) ppˆ2 (cid:17)(cid:17)
We can show g(pˆ) 0 around p 0 from (3.7) and (3.11). Therefore (3.12) implies c 0
′
∼ ∼ ≥
and
√c = λ log 2(cosh√c+coshµˆ) . (3.13)
| |
A solution exists when λ < 1 and does n(cid:0)ot when λ > 1. Th(cid:1)is fact can be easily seen by
| | | |
rewriting (3.13) as
√c
e λ = e√c +e √c +eµˆ +e µˆ, (3.14)
| | − −
√c
which suggests e λ > e√c and thus λ < 1.
| | | |
Let us summarize the solution of the Schwinger-Dyson equation
λ c
f(pˆ) = log 2(cosh pˆ2 +c+coshµˆ) , g(pˆ) = f(pˆ)2, (3.15)
pˆ pˆ2 −
(cid:16) p (cid:17)
where √c is determined by (3.13) or (3.14). Note that √c physically means the thermal
mass of the fermion divided by T.
As in the previous section we can compute the momentum distribution of U(1) charge
density in finite temperature
1 d2q
n = ψ¯(p)γ3ψ(q) ψ¯(p)γ3ψ(q) (3.16)
h psi β (2π)2 − |µ=0
p3X,q3:F Z (cid:0)(cid:10) (cid:11) (cid:10) (cid:11) (cid:1)
where we did replacement (3.1) in (2.26). Substituting the above solution and simplifying
it we find
√pˆ2+c µˆ √pˆ2+c+µˆ
tanh − +tanh
n = N− 2 2 . (3.17)
h psi 2
This is the momentum distribution of the free fermion gas in nonzero temperature with
thermal mass √cT. The fermi surface clearly seen in (2.27) is now smeared by finite
temperature effect.
Let us compute the grand canonical potential density in this system, which is defined
by
S S
eff eff T,µ=0
G(T,µ) = − | . (3.18)
V β
2
where V is the volume of two plane given by V = V/β and S is the effective action on
2 2 eff
R2 S1 with chemical potential
×
d2q 1 1
S = NV tr log[iγµq +Σ(q)]+ Σ(q) (3.19)
eff 2 (2π)2 − µ 2 iγµq +Σ(q)
Xq3:F Z (cid:20) (cid:18) µ (cid:19)(cid:21)
e
6 e
G
-
NT3
c 2.5
6
2.0
5
4 1.5
3
1.0
2
0.5
1
Λ Λ
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
(a) (b)
Figure 1: The graphs of √c and G(T,µ) are shown as a function of λ varying µ/T =
− NT3
0,1,2,3,4fromthe bottom in Fig.1(a), Fig.1(b), respectively. They are taking finite values
in the region 0 λ < 1.
≤
and we normalize the grand canonical potential as zero when T = µ = 0. By employing
the dimensional regularization the grand canonical potential density can be computed as4
NT3 √c3
G(T,µ) = √c3 +3 ∞dy yln 1+e y µˆ +3 ∞dy yln 1+e y+µˆ .
− − −
− 6π λ −
(cid:18) | | Z√c Z√c (cid:19)
(cid:0) (cid:1) (cid:0) (3(cid:1).21)
We plot √c and G(T,µ) as a function of λ varying µˆ in Fig.1(a), Fig.1(b), respectively. We
− NT3
have assumed λ 0 here. Those figures indicate that as λ goes larger the thermal mass
≥ | |
becomes bigger andthegratitudeofthegrandcanonical potential smaller, andfinally when
λ reachesonethethermalmassblowsupandthe (N)grandcanonicalpotentialvanishes.
| | O
Note that this grand canonical potential (3.21) reduces to the free energy density obtained
in [9] when µ 0 and to the effective action (2.25) when T 0 since √c λµ /T from
→ → → | |
(3.13).
The behaviors of √c and G(T,µ) around µˆ , which means low temperature or
∼ ∞
small chemical potential, with λ fixed is the following.
√c λµˆ+λe(λ 1)µˆ + , (3.22)
−
∼ ···
NT3 1 π2
G(T,µ) (1 λ2)µˆ3 + µˆ+ . (3.23)
∼ − 6π 2 − 2 ···
(cid:18) (cid:19)
4 We note a useful formula
1 2π(n+ 1)
log ( 2 +iµ)2+h2 =log(1+e−β(h+µ))+log(1+e−β(h−µ))+h. (3.20)
β β
n Z (cid:18) (cid:19)
X∈
7
Q S C
NT2 NT2 NT2
1.4 2.0 2.0
1.2
1.0 1.5 1.5
00..68 1.0 1.0
0.4 0.5 0.5
0.2
0.2 0.4 0.6 0.8 1.0Λ 0.2 0.4 0.6 0.8 1.0Λ 0.2 0.4 0.6 0.8 1.0Λ
(a) (b) (c)
Figure 2: The U(1) flavor charge density, entropy density and heat capacitance normalized
by NT2 are plotted as a function of λ varying µ/T = 0,1,2,3,4 from the bottom in
Fig.2(a), Fig.2(b), Fig.2(c), respectively.
Their behaviors around λ 1 with µˆ fixed are
∼
2coshµˆ 2coshµˆ
√c log loglog + , (3.24)
∼ 1 λ − 1 λ ···
(cid:18) − (cid:19) (cid:18) − (cid:19)
NT3(1 λ) 3
3 2
G(T,µ) − √c + √c + . (3.25)
∼ − 6π coshµˆ ···
(cid:18) (cid:19)
From the grand canonical potential one can obtain its thermodynamical descendants
such as the pressure p, the mean U(1) flavor charge density Q, the average internal energy
U, the entropy density S and heat capacitance C with fixed two dimensional volume.
∂G ∂2G
p = G, Q = (∂ G) , U = ∂ (βG) , S = , C = T , (3.26)
− − µ T β µˆ − ∂T − ∂T2
(cid:18) (cid:19)µ (cid:18) (cid:19)µ
where subscripts are such that partial derivatives are taken with them fixed. From a
straight forward calculation we obtain
NT2 NT2 µˆ NT2 2 µˆ2
Q = g , U = 2G, S = (g g ), C = (g µˆg + g ), (3.27)
′ ′ ′ ′′
6π − 2π − 3 π − 3 6
where we set g = 6π G(T,µ) = g(µˆ) and g = dg,g = d2g.5 In particular we can show
−NT3 ′ dµˆ ′′ dµˆ2
U and Q are related in large µˆ region by U 16π Q23.
∼ 9N(1 λ2)
−
q
1
4 Comment on instability in λ >
| |
In the previous section we solved the Chern-Simons-fermion vector theory with chemical
potential in finite temperature in the ’t Hooft limit. The solution exists when λ < 1 and
| |
5 The explicit forms of g (µˆ),g (µˆ) are calculated as
′ ′′
cosh√c−µˆ √c+µˆ y
g′(µˆ)=3 √clog 2 + dylog(2cosh ) , (3.28)
(cid:20) cosh√c2+µˆ! Z√c−µˆ 2 (cid:21)
√c √c µˆ √c+µˆ √c
g′′(µˆ)=3 2 (∂µˆ√c−1) tanh 2− −tanh 2 + λ . (3.29)
(cid:20) (cid:18) (cid:19) (cid:21)
8
ceases to exist at λ = 1. It is natural to ask what happens when λ > 1. It is likely
| | | |
that at λ = 1 the system undergoes the phase transition from conformally symmetric
| |
(deconfining) phase in λ < 1 to broken (confining) one in λ > 1. Let us briefly discuss
| | | |
the possibility of this phase transition by employing the results obtained above.
For this purpose we consider the same system in the U(N) (not SU(N)) gauge group
and take the diagonal U(1) holonomy in the thermal direction into account, which has the
leading contribution in 1/N expansion. The self-energy and free energy density including
the holonomy contribution can be respectively obtained directly from the self-energy (3.6)
with (3.15), (3.14) and the grand canonical potential density (3.21) by performing an
analytic continuation for chemical potential µ iα, where α represents the diagonal U(1)
→
holonomy. After this manipulation the solution of the self energy becomes
λ c
f(pˆ) = log 2(cosh pˆ2 +c+cosαˆ) , g(pˆ) = f(pˆ)2, (4.1)
pˆ pˆ2 −
√c (cid:16) p (cid:17)
e λ = e√c +e √c +2cosαˆ, (4.2)
| | −
where αˆ = α. There exists a solution in 0 αˆ π when λ < 1. In the region λ > 1
T ≤ ≤ | | | |
there also exists a solution in αˆ αˆ π, where α is a function of λ taking values between
c c
≤ ≤
π/2 and 2π/3.6 The free energy density is
NT3 √c3
F(T,α) = √c3 +3 ∞dy yln 1+2cosαˆe y +e 2y . (4.3)
− −
− 6π λ −
(cid:18) | | Z√c (cid:19)
(cid:0) (cid:1)
Since the holonomy is dynamical, the large N leading free energy is determined by varying
the holonomy satisfying (4.2) so that the free energy is minimized. It turns out by simple
numerics that the value of the holonomy which minimizes the free energy is trivial when
λ < 1,7 but becomes non-trivial, α = α , when λ > 1.
c
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This result suggests that the first order phase transition may happen at λ = 1. How-
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ever it also turns out that the minimized free energy (4.3) at α = α in λ > 1 is positive,
c
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which means the pressure is negative and thus the system is no longer stable. This is
presumably a signal of break-down of our assumption for the system to have conformal in-
variance. Therefore operators neglected in this analysis, which break conformal symmetry,
would play an important role to investigate the phase in λ > 1. We leave detailed study
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in the future.
5 Summary and discussion
We have studied SU(N) Chern-Simons theory coupled to a fermion in the fundamental
representation with chemical potential for U(1) flavor symmetry and temperature in the
’t Hooft limit. We have calculated the self energy, the momentum distribution and several
thermodynamic quantities exactly as a function of λ, which suggests that the solution
6 Precisely speaking, α is given by α (λ)=arccosMax 1exp(√c) cosh√c , where Max A(x) is
c c √c 2 λ − x
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the maximum value of A(x) with respect to x. (cid:16) (cid:17)
7 This result was assumed in [9] to determine the free energy.
9