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Entropy for Colored Quark States at Finite Temperature

Article  in  Fizika B · September 2003

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 BI-TP 2003/18

ENTROPY FOR COLORED

QUARK STATES AT
FINITE TEMPERATURE
arXiv:hep-ph/0308192v3 9 Jan 2004

David E. Miller1,2 and Abdel-Nasser M. Tawfik1

1
Fakultät für Physik, Universität Bielefeld, Postfach 100131,
D-33501 Bielefeld, Germany ∗
2
Department of Physics, Pennsylvania State University, Hazleton Campus,
Hazleton, Pennsylvania 18201, USA †

Abstract
The quantum entropy at finite temperatures is analyzed by using models for colored
quarks making up the physical states of the hadrons. We explicitly work out some
special models for the structure of the states of SU(2) and SU(3) relating to the
effects of the temperature on the quantum entropy. We show that the entropy of
the singlet states monotonically decreases meaning that the mixing of these states
continually diminishes with the temperature. It has been found that the structure of
the octet states is more complex so that it can be best characterized by two parts.
One part is very similar to that of the singlet states. The other one reflects the
existence of strong correlations between two of the three color states. Furthermore,
we work out the entropy for the classical Ising and the quantum XY spin chains.
In Ising model the quantum (ground state) entropy does not directly enter into
the canonical partition function. It also does not depend on the number of spatial
dimensions, but only on the number of quantum states making up the ground state.
Whereas, the XY spin chain has a finite entropy at vanishing temperature. The
results from the spin models qualitatively analogous to our models for the states of
SU(2) and SU(3).

PACS: 05.30.-d Quantum Statistical Mechanics,

12.39.-x Phenomenological Quark Models,
03.75.Ss Degenerate Fermi Gases,
∗
email: dmiller@physik.uni-bielefeld.de; tawfik@physik.uni-bielefeld.de
†
email: om0@psu.edu
1 Introduction
In this paper we present a new method for evaluating the entropy of the colored
quark states at finite temperature. Clearly, the entropy which we mean here
is the quantum entropy which is to a great extent different in nature from the
classical (Boltzmann-Gibbs) entropy. From the classical point of view the entropy
characterizes an access to information between macroscopic and microscopic
physical quantities using statistical criteria. It can be considered to be the number
of possible microstates that the macrosystem can include. Whereas, the quantum
entropy1 can be calculated for very few degrees of freedom by using the density
matrix and should not vanish for zero temperature. It reflects the uncertainties in
the abundance of information about the quantum states in a system. The definition
of quantum entropy may date back to works of von Neumann [2] in the early
thirties on the mathematical foundations of quantum theory. Another milestone
has been set out by Stratonovich, when he introduced the reciprocative quantum
entropy for two coupled systems. In the Lie algebra the quantum fluctuation and
correspondingly the quantum entropy are given by non-zero commutative relations.
The most essential information we can retain from the physical systems is similar to
measuring the eigenvalues of certain observables. The faultless measurement is to be
actualized only if the degeneracy in the determined observable is entirely considered.

According to the Nernst’s heat theorem the entropy of a homogeneous

system at zero temperature is expected to be zero. However, it has been proven
that the mixing in ground states of subsystems with conserving the particle number,
the temperature and the volume results in a finite entropy [3], the so-called Gibbs
paradox. On the other hand, we know that the effects of quantization on the
fundamental laws of thermodynamics were generally well known to the founders of
quantum theory [4]. Planck, who had successfully predicted the measured intensity
distribution of different wavelengths by the postulation of a minimum energy size
for the light emitted from a dark cavity at a given temperature, had also realized
that the massive particles with non-zero spin also possess a finite entropy relating
directly to that of the spin in the limit of low temperatures. This realization
provided the chance for his students to resolve the Gibbs paradox, furthermore,
it also offered an exception to Nernst’s heat theorem, which is usually stated
under the name of the third law of thermodynamics that the entropy of a closed
system in equilibrium must vanish in the low temperature limit. A clear general
discussion of the entropy relating to Nernst’s heat theorem was given by Schrödinger
some years later in a lecture series [5]. In his selected example Schrödinger took
an N-particle-system, in which each one has two quantum states contributing
to the many particle ground state. Thus the ground state held 2N degenerate
configurations in its structure, which must then provide for it an entropy of N ln 2.
This value is obviously independent of all the thermodynamical quantities other
than the number of particles N itself.
1
The authors of [1] distinguished between different entropies: the classical one they called the
measurement entropy, and their relevant entropy refers to our quantum entropy.

2
 Furthermore, it is known that the more information present, the greater is the
reduction of the entropy. In other words the entropy is reduced by the amount
of information distinguishing the different degenerate states of the system. The
completely mixed state is that of minimal information. This is completely consistent
with the idea of confinement in the hadronic matter. If we were to consider a gas of
N hadrons in the sense that Schrödinger [5] considered a gas of two level atoms, we
should roughly expect an entropy of the form Nln 3 for the SU(3)c quark structure
in the color singlet ground state of this pure composite system [6]. In this article we
attempt to elucidate the physical meaning of the quantum entropy for the colored
quark states and investigate its behavior at finite temperature.
According to the third law of thermodynamics it is expected that the confined
hadron bags - as a pure state - have zero entropy at zero temperature. But
the hadron constituents are to be treated as subsystems composed of quantum
elementary particles, which in their ground states can exhibit a finite value of the
entropy. The quantum entropy of such subsystems reflects the degree of mixing
and entanglement [7] inside the hadron bag. In a recent letter [6] it was shown
that in a quantum system with SU(3)c internal color symmetry how the ground
state entropy arises in relation to the mixing of the quantum color states of that
system. So far we conclude that in a completely mixed ground state with N
internal components the entropy relates directly to the value ln N. Nevertheless,
one would expect some changes in this entropy in the presence of other states or
at finite temperatures. Such changes were mentioned in [6] in relation to the octet
hadronic states and the changes in the ground state entropy at finite temperature.

Understanding the thermal behavior of quantum subsystems [8] - in our

case the colored quark states - is very useful for different applications. It may
well bring about a device for the further understanding of the recent lattice
results with the heavy quark potential [9–11]. The lattice results for the entropy
difference in a quarks-antiquark singlet state lead to the value of 2 ln 3 at vanishing
temperature [10, 11]. Our evaluation for the quantum entropy of one quark as part
of the colorless singlet ground state [6] yields the value of ln 3. Furthermore, we
believe that the investigation of the quantum subsystems at finite temperature
might be useful for understanding the concept of confinement, which could exists
everywhere throughout T ∈ [0, ∞]. The cold dense quark matter in the interior of
stellar compact objects provides another application for this work. Also we think
that according to the existence of finite entropy inside the hadron bags, the confined
quarks get an additional heating and the bag constant should be correspondingly
modified. The quark distributions inside the hadron bags reflect themselves as
entanglement or - in our language - quantum entropy.

In this work we shall further investigate the implications of the quantum

entropy for the colored quarks relating to the known properties of the standard
model [12]. As previously mentioned [6] we shall not bring in other important
properties of the quarks in the standard model like flavor, spin, isospin, chirality,
electric charge and the spatial distribution of the quarks. From now on in this work

3
we shall assume that we are only looking at the color symmetry so that we shall
leave out any name distinctions between the states. Here we shall further look
into some specific properties of the other quark structures. Thereafter, we shall
investigate some specific properties of the other states in relation to the ground
state, which lead to some more results for the entropy. As a following investigation
we propose and solve for the quark quantum entropy some simple models of color
unmixing at finite temperatures, for which we check each model for the limiting
cases. Then we study the correlations of entangled spin systems. These spin
systems are used to compare our results for measurement entropy of a system of
colored quark states. Other degrees of freedom are not considered.

This paper is organized as follows: the next section is devoted to the formulation
of the ground state, from which we develop models for the entropy of mixing of
colored quark SU(2) and SU(3) states. Then we introduce our thermal models for
the quantum entropy of these states. Some spin models are investigated in relation
to the known exact solutions. The following section contains the discussion of our
results. Finally, we end with the conclusion and outlook.

2 Formulation for the Ground State Entropy

In order to understand the ground state structure, we recall some common properties
of spin and color quantum systems which relate to the ideas of superposition and
entanglement [13]. The orthonormal basis usually taken [8] for spin SU(2) can be
written as |0i and |1i for the two states. For such two-state-system we have four
combinations of |iji a useful linear orthonormal combination thereof. When we use
the Pauli matrices σ x , σ y , σ z together with the two dimensional identity matrix 12 ,
we may easily write down the singlet and triplet structure for the structure of the
states of SU(2). The usual symmetric triplet and antisymmetric singlet states also
provide a proper basis. After we have written down the density matrices ρt and ρs
for each of the states, we find that after projecting out the second state the single
quark reduced density matrices are both in the form

p i 12
X
ρq,2 = (1)
i

where pi is the probability of i-th state. We can calculate the entropy S of the
quantum states [2, 4], which makes direct use of the density matrix ρ

S = −Tr (ρ ln ρ), (2)

Nevertheless, this equation could be found in text books, we apply it here where the
trace is taken over all the quantum states (Eq. 1). For quantum operators the trace is
independent on the representation, therefore the quantum states might well be used
to write down the either quantum canonical or grand canonical partition function.
The density matrix can be illustrated as mixing of subsystems of a closed system.

4
According to the Nernst’s heat theorem this enclosed system has zero entropy at
zero temperature [3].
When, as is presently the case, the eigenvectors are known for ρ, we may directly
write this form of the entropy in terms of the eigenvalues λi as follows:
X
S=− λi ln λi (3)
i

It is obviously important to have positive eigenvalues. For the special case of a

zero eigenvalue we use the fact that xln x vanishes in the small x limit. Then for
the density matrix ρ we may interpret λi as the probability pi of i-th state. This
meaning demands that 0 < pi ≤ 1. Thus the orthonormality condition for the given
states results in the condition This is a very important condition for the entropy.
Thus we can easily see that for SU(2) the value of probabilities pi is always 1/2
yielding the same total entropy for both the singlet and triplet states

Sq,2 = ln 2 (4)
We get the same results if we apply the so-called Schmidt decomposition on the
pure state |ψ >, which consists of substates, |0 > and |1 >, then there is sets of
orthonormal states, {|0 >i } and {|1 >i }, so that
X
|Ψ > = Ni |0 >i ⊗|1 >i (5)
i

where N are real position numbers, known as the Schmidt numbers, which satisfy
X
Ni2 = 1 (6)
i

One important consequence of last equation, is that the reduced density matrices
of substates, |0 > and |1 > should have identical eigenvalues. Plugging Ni ≡ λi in
Eq. 3, Sq,2 = ln 2. Eq. 5 can be generalized for decomposition into n subsystems.
The pure state |ψ > can be expanded in a number of factorizable n-states. The
number of coefficients is minimal and Eq. 6 still valid.

In the case of SU(3)c the state structures for the singlet and octet are very
different. We recall that we have found for the single quark density matrix ρq the
form [6]

pi 13
X
ρq,3 = (7)
i

where 13 is the three dimensional identity matrix.

We now apply the above definitions of the entropy to the SU(3)c quark states – as
was done in [6]. It is clear that the original hadron states are pure colorless states
which posses zero entropy as in the third law of thermodynamics. For the meson
it is immediately obvious since each colored quark state has the opposing colored

5
antiquark state for the resulting colorless singlet state. The sum of all the cycles
determine the colorlessness of the baryon singlet state thereby giving no entropy.
However, the reduced density matrix for the individual quarks (antiquarks) ρq or ρq̄
has a finite entropy. In this context, the reduced density matrix can be illustrated
as a certain mixing inside the closed system [3]. Meanwhile the whole system have
zero entropy at zero temperature, the subsystems - as shown above - are expected
to have finite entropy at vanishing temperature.

For SU(3)c all the eigenvalue λi in Eq. 3 have the same value 1/3. Thus we
find for all quark (antiquark) in singlet states

Sq,3 = ln 3 (8)
As a further exercise we may compare this result with those of the quark octet
states. The octet density matrices ρo,i may be constructed from the eight Gell-Mann
matrices (λ)i with i = 1, 2, · · · , 8. The density matrix for each state is constructed
by using the properties of Ψ(λ)i Ψ∗ . The first seven matrices all give the same value
for the entropy ln 2, since all of these states are constructed only from the Pauli
matrices. This result comes from the fact that these first seven octet states each
involve only two of the three color states – that is these octet states are not pure
states in all the colors. Although the mixing of the two states is equal, it is not
complete since the third color is absent. However, the eighth diagonal Gell-Mann
matrix involves all three colors, but the mixing is unequal. It yields an entropy
1
So,8 = ln 3 − ln 2 (9)
3
Thus we can clearly state that the entropy of any of the quark octet states is always
smaller than the quark color singlet state. This means that the colorless quark
singlet state is the most probable individual state for the hadrons.

3 Thermodynamical Models for Mixed Colored

Quark States
The structure of the color singlet hadronic ground state for SU(3) was shown in [6]
to have a complete uniform mixing of all the colors of the quarks and antiquarks for
both the mesons and the baryons. We have seen above that this situation does not
happen for the single quark (antiquark) entropy Sq or Sq̄ in the presence of octet
states, where the mixture is either partial or unequal. We now want to return to the
single quark (antiquark) reduced density matrices ρq,2 or ρq̄,2 , which we shall assume
to be the same for the fundamental and antifundamental representations since we
are not considering the differences between flavors. In this section we will extend
these calculations of the structural entropy of the ground states for the SU(2) and
SU(3) quarks in color singlet states to models for color mixing at finite temperature.
In these models the Boltzmann weighting for the finite temperature states of the
single colored quark states will contain the single particle relativistic energies ε(p),

6
 0.7 0.7
(a) (b)
0.6 0.6
10 MeV 0.5 GeV
5 MeV 0.1 GeV
0.5 0.5
50 MeV
1 MeV 10 MeV
0.4 0.4
Sq,2

Sq,2
0.3 0.3

0.2 0.2

0.1 0.1

0 0
0 10 20 30 40 50 0 100 200 300 400 500
T [MeV] T [MeV]

Figure 1: Single quark entropy for various quark masses as a function of the temperature T for
SU (2), Eq. 13. The left panel shows the results for the stated light quark masses. The right panel
depicts the results for masses up to 500 MeV. The dotted lines at the top represent the value of the
ground state entropy at zero temperature in Eq. 4.

p
which is given by m2 + p2 for the relativistic quark momentum p and mass m.
Since the biggest effect at the given temperature T comes with the lowest value of
ε(p), we may well assume that ε(p) is just determined by the lowest quark mass
threshold. Furthermore, ε(p) is also color independent.

3.1 SU (2) Thermodynamical Model

We start our consideration of thermodynamics with a very simple model for quarks
with an internal SU(2) symmetry at a finite temperature T . We postulate that the
single quark reduced density matrix with two colors have the following form:
1
1 − e−ε(p)/T |0 >< 0| + 1 + e−ε(p)/T |1 >< 1|



ρq,2 (T ) =
2
1
12 − σ z e−ε(p)/T

= (10)
2
which shifts the weighting of the eigenstates due to the temperature T . Thus we
note that the total probability is still one, so that ρq,2 (T ) will still have the correct
probabilistic interpretation. Furthermore, we note that if we had taken all the Pauli
matrices with equal weighting in front of the Boltzmann factor, we would still be
able to diagonalize the full SU(2) reduced density matrix into this form. Then the
eigenvalues from the T -depending reduced density matrix, Eq. 10, read

1 e−ε(p)/T 1 e−ε(p)/T e−2ε(p)/T

    
λ
det < i|ρq |j > − = − √ −λ + √ −λ − =0
2 2 2 3 2 2 3 6
1
1 − σ z e−ε(p)/T


λi (T ) = (11)
2
i = ±1 refers to the states, |0 > and |1 >, which are included in σ z in Eq. 10.
As it was above carried out for the ground state, we are now able to calculate the
entropy Sq,2 (T ) at finite temperature [2,4] from the eigenvalues λi (T ) and under the

7
assumption that the color at finite temperature have the same energy eigenstates,
we find
X
Sq,2 (T ) = − λi (T ) ln λi (T ) (12)
i=±1
 
1 −ε(p)/T

1 −ε(p)/T


= − 1−e ln 1−e
2 2
 
1 −ε(p)/T

1 −ε(p)/T


− 1+e ln 1+e (13)
2 2

In the low temperature limit we can immediately find that the ground state quark
entropy Sq,2 (0) has again, as in Eq. 4, the value of ln 2, a completely equally mixed
quark color state. However in the high temperature limit the contribution to the
entropy of the first state vanishes from deoccupation, while the second state becomes
a pure state with a probability of one, which also contributes a vanishing value to
the entropy. In Fig. 1 we look at some particular cases for Sq,2 (T ) both at lower
and higher temperatures. In Fig. 1a we see how the single quark entropy varies in a
range of temperature up to 50 MeV with quark masses 1, 5 and 10 MeV. We notice
that in all cases the entropies monotonically decline in this region meaning that the
mixing of the states continually decreases. For the sake of comparison in Fig. 1b we
look at some larger quark masses in a much larger temperature range. We see the
same tendency for the separation of the states. We note that with increasing quark
masses the range of temperature within which the entropy is entirely determined
by the ground state value becomes wider and the entropy remains larger for higher
temperatures. This reflects the importance of the ground state entropy for the
massive quark systems. Thus in this model for SU(2) colored quark states we
find in the high temperature limit a pure single colored quark state in the classical
sense. Whereupon, we would expect a pure quark phase in which all the correlations
between the different colors have vanished - free quarks!

3.2 SU (3) Thermodynamical Model

Now we construct a similar model for SU(3) with the right quark mixing in the
ground state, which also providesP the proper probabilities for each of the states
from the trace condition Trρ = i pi = 1 (section 2). Again we demand that the
energy eigenstates remain the same for each of the color states |ii. We think that the
eighth (diagonal) Gell-Mann matrix λ8 is not a very suitable choice as a weighting
matrix for the thermal states even though it maintains the trace condition for the
probability. The eighth Gell-Mann matrix weights the third color twice as much as
the other two taken individually. Therefrom, we should look for another weighting
matrix for the thermal states of SU(3). A good possibility, which we shall choose,
is that we take the three complex roots of −1, which are R0 = −1, R1 = exp iπ/3
and R2 = exp −iπ/3.
1
13 + (R0 |0 >< 0| + R1 |1 >< 1| + R2 |2 >< 2|) e−ε(p)/T

ρq,3 = (14)
3

8
 1.00
1.10
(a) (b)

0.80
1.05

0.60 5 MeV
1.00 50 MeV
0.5 GeV
Sq,3

Sq,3
0.1 GeV 500 MeV
50 MeV
0.95 10 MeV 0.40
5 MeV
1 MeV

0.90 0.20

0.85
0.00
0.0 100.0 200.0 300.0 400.0 500.0 0.0 100.0 200.0 300.0 400.0 500.0
T [MeV] T [MeV]

Figure 2: The left panel shows the single quark entropy change for various quark masses as a function
of temperature T for SU (3), Eq. 20. The right panel gives details on this behavior. We plot separately
the two contributing terms of Eq. 20 for masses, 1, 50 and 500 MeV. The curves in bottom part of the
figure represent only the first term.

Then we get three eigenvalues:

1
1 − e−ε(p)/T ,


λ0 =
3
1
1 + eiπ/3 e−ε(p)/T ,


λ1 = (15)
3
1
1 + e−iπ/3 e−ε(p)/T


λ2 =
3
We may write the three roots as the weights wi for the states |ii with the Boltzmann
factor. Thus the reduced quark density matrix, Eq. 14 for SU(3) reads
1
13 + |ii wi e−ε(p)/T hi| .


ρq,3 (T ) = (16)
3
Clearly, in the low temperature limit we get back the completely mixed state with
a probability of 1/3 for each color state |ii. We are able to calculate the entropy
Sq,3 (T ), Eq. 12 by carefully using the proper definitions for the complex logarithms.
 
1 −ε(p)/T

1 −ε(p)/T


Sq,3 (T ) = − 1 − e ln 1−e
3 3
 
1 iπ/3 −ε(p)/T

1 iπ/3 −ε(p)/T


− 1+e e ln 1+e e
3 3
 
1 −iπ/3 −ε(p)/T

1 −iπ/3 −ε(p)/T


− 1+e e ln 1+e e (17)
3 3

Using the properties of the logarithms of complex variable2 we can write the real
part as
1 1/2
z = 1 + e−ε(p)/T + e−2 ε(p)/T (18)
3
2
For the phase −π < θ ≤ +π we define the complex variable as Z = ReZ eiθ .

9
and the phase is given by
√ !
3 e−ε(p)/T
θ = arctan (19)
2 + e−ε(p)/T

After a little algebra we find that the single quark quantum entropy at finite tem-
perature for SU(3) reads
 
1 −ε(p)/T

1 −ε(p)/T


Sq,3 (T ) = − 1 − e ln 1−e
3 3
−2z [ln(z) cos(θ) − θ sin(θ)] , (20)

In the low temperature limit we have z just equal to 1/3 and θ is exactly zero.
Then the quark entropy is √ clearly again just ln 3. However, in the limit of very high
temperature z becomes 1/ 3 and θ is just π/6. The contribution of the first term
of Eq. 20 to Sq,3 (T ) simply vanishes as was the case for Sq,2 (T ) (review Fig. 1).
However, the second term still remains at high temperatures leaving a limiting
entropy of 0.8516. We see this effect in Fig. 2 where we have plotted Sq,3(T ) for
various values of the quark masses. We note that the first term behaves qualitatively
very similarly to the single quark for SU(2) in Fig. 1. The entropy begins from
ln 3/3 and exponentially decreases with the temperature T . The second term has a
remarkable dependence on T . For T → 0, it has the value of 2/3 (ln 3) analogously
to SU(2). With
√ increasing temperature it rapidly goes to the asymptotic value
(ln 3)/2 + π( 3/18). Furthermore, we note that by decreasing the mass it limits the
range of temperature to reach this asymptotic region. Since the asymptotic value
at high temperatures has remained more than three quarters of its ground state
value of ln 3, there are still considerable correlations between two of the three color
states. This observation points to an important fact about the structure of SU(3)
in these statistical models: the root structure forbids a complete cancellation of the
real solutions which is needed to maintain the trace condition on the reduced density
matrix. Thus two states are always matched against one. Hence for this model in
the high temperature limit one color vanishes while the other two remain mixed and
thereby correlated. The ground state favors the color singlet state with complete
mixing. In this model the high temperature limit favors the octet states involving
mostly two colors. This situation for SU(3) can be contrasted with SU(2) where
the triplet and the singlet states have the same reduced density matrices except for
the pure triplet states.

4 Spin Models with Strong Correlations

Our objective in this section is to further investigate the structure of the entropy
for some known spin models at finite temperature in which strong correlations exist.
The general category of all spin models goes under the name of the Heisenberg model,
which is the central to the theory of magnetism [14]. In general, the Hamiltonian for

10
 0.7 0.7

0.6 0.6

0.5 0.5

log 2 - SIsing
0.4 10.0 0.4 10.0
SIsing 5.0 5.0
0.3 1.0 0.3 1.0
0.2 0.2
0.0 0.0
0.2 0.2

0.1 0.1

0 0

0 5 10 15 20 0 5 10 15 20
T T

Figure 3: The entropy for 1D Ising model without external field calculated from the canonical
Hamiltonian, Eq. 22, depicted in dependence upon the temperature T and for different values of J, the
exchange coupling. Two asymptotic regions exist here. They are S → 0 and S → ln 2 for temperatures
T → 0 and T → ∞, respectively. In the right panel we depict the entropy difference from the SU (2)
ground state entropy, ln 2, (see text). This entropy difference (Eq. 26) represents the ground state
entropy for the 1D Ising model at finite temperature.

these models involves the vector of spin matrices s(ri ) for an electron at the position
ri coupled to another electron with the vector of spin matrices s(rj ) at position rj
X
H = −2 Jij si · sj , (21)
i<j

where Jij is the exchange integral arising from the integration over the interaction
containing the overlap of the spatial wavefunctions of the two electrons located at
the two points ri and rj . The relation of this exchange integral Jij to the different
spin directions si and sj determines the local structure of the interaction.

4.1 Classical Ising spin chain

The simplest special case of the electron-electron interaction is the Ising nearest
neighbor chain interaction, for which only the spin matrices in the z-direction
szi (≡ σ z Pauli spin matrix) and szi+1 are present. Then the interaction Hamilto-
nian is simply
X X
HIsing = −J szi szi+1 − h szi (22)
i i

where J is the simple nearest neighbor coupling and h is the external magnetic field.
The interaction in Ising model is a completely classical since the spin operators szi
can be replaced simply by their diagonal values. The SU(2) group structure ordinary
reduces to its center Z(2). Therefore, in the ground state maximum two states are
expected. In other words, each spin can equally have one of the two directions up
and down.

11
 The entropy per spin at finite temperature for h = 0 reads
  
S(T )  −2J/T
 J J
= ln 1 + e + 1 − tanh (23)
N T T
We see these results in Fig. 3a, where we have purposely taken the energy scale for J
to compare with Sq,2 in Fig. 1a. It is clear that the entropy is vanishing for vanishing
temperatures but for high temperature, S → ln 2. This means that the ground state
of Ising model does not reflect the structure of SU(2) symmetry group. In SU(2)
there are two states even at zero temperature, which leads to one spin entropy equals
to ln 2. Therefore, the canonical results in Fig. 3a indicates the fact that the ground
state structure in the Ising model is not included in the grand canonical partition
function. It is clear that the deviation from the characterized SU(2) structure is
strongly depending upon the exchange coupling, J. For all temperatures but only
if J = 0, the entropy per spin always equals to ln 2.
In doing a comparison with our models given in section 3.1, we are left with
an ad hoc inclusion of an additional part of the entropy reflecting the ground state
structure in the Ising model. Therefore, we recall the structure of the Heisenberg
model, for which we can write down a wavefaction for the strong correlated spin
matrices σ z at T = 0. Thus the density matrix for the two possible states |0 >, |1 >
which are corresponding to the two spin directions | ↑>, | ↓>, respectively, reads
X 1
ρ= |Ψ > PΨ < Ψ| = √ (|0 > |0∗ > +|1 > |1∗ >)
2
1
√ (< 0| < 0∗ |+ < 1| < 1∗ |) , (24)
2
where PΨ is the probability of each eigenstate Ψ. From the traceable E. 24 we get
the reduced density matrix by projecting out the conjugated components.
1
ρr = Trb ρ = (|0 >< 0| + |1 >< 1|) (25)
2
Obviously, the probability of each spin direction and correspondingly the two eigen-
values from the reduced density matrix, Eq. 25, are equal (λ = 1/2). Plugging into
Eq. 3, results in the entropy for single spin of 1D Ising model at zero temperature,
S = ln 2. In getting this value, we obviously assumed that J = 0 and considered
only the spin eigenfluctuation in the ground state. i.e, quantum states.
As we have done in section 3.1 (Eq. 10), we subtract from this value the T -
depending entropy part resulting in the temperature dependence of the ground state
(quantum) entropy
S(T ) ≡ ln 2 − SIsing (T ) (26)
These results are given in Fig. 3b. At T = 0, the entropy starts from the value
ln 2. With increasing temperature the ground state entropy of the 1D Ising model
monotonically decays. It reaches its asymptotic zero value for high T . Qualitatively,
we got the same results for our models given in section 3.1 and graphically illustrated
in Fig. 1 and 2. Obviously, we left arbitrary the units of the energy density in the
Boltzmann term in Eg. 10 and of the exchange coupling in Eq. 23.

12
4.2 Quantum XY spin chain

0.7 0.7
(a) (b)
0.6 0.6

0.5 0.5

0.4 0.4
SXY

SXY
0.3 h=1.0, γ=1.00 0.3
h=1.0, γ=0.50 h=0.5, γ=1.00
h=1.0, γ=0.10 h=0.5, γ=0.50
0.2 h=1.0, γ=0.05 0.2
h=0.5, γ=0.10
h=0.5, γ=0.05
0.1 0.1

0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
T T

0.7
(c)
0.6

0.5

0.4
SXY

0.3
h=0.005, γ=1.00
0.2 h=0.005, γ=0.50
h=0.005, γ=0.10
h=0.005, γ=0.05
0.1

0
0 0.5 1 1.5 2
T

Figure 4: Entropy per spin site calculated for different temperatures for XY -model. (a) gives the
results for constant h = 1.0 but different γ values (see text). We note that with increasing T the
entropy S decreases exponentially and the T -range within which S → 0 becomes wider with smaller γ
values. Also deceasing γ increases the finite entropy and the system favors going toward the ground
state value, ln 2. Almost the same behavior can be seen in the plots (b) for h = 0.5 and (c) for
h = 0.005.

Another exactly solvable model relating to the Heisenberg model is the

XY model [15, 16]. In one spatial dimension it is a highly correlated quantum me-
chanical system also in the ground state [17]. Furthermore, it is known [7, 17, 18]
that the XY -model in the ground state has a very complex structure containing
an oscillating region for small values of h and γ in addition to the expected ferro-
magnetic and paramagnetic phases for larger values of these parameters. Also, for
a class of time dependent magnetic fields the magnetization has been shown to be
nonergodic [17]. Recently, the correlations of the spins has been calculated using
the reduced density matrix [7] in order to find the ground state entropy for a block
of spins yielding the expected logarithmic behavior.
The Hamiltonian can be written in terms of the spin components in the x and y
spin-directions with an external field in the z-direction
1 X X
(1 + γ)sxi sxi+1 + (1 − γ)syi syi+1 − h szi

HXY = (27)
2 i i

13
where γ is the spin anisotropy parameter in the x and y spin-directions. The exact
solution of this model has been carried out long ago [15, 16]. The spins are allowed
to take an arbitrary angle, k ∈ [0, 2π] with cyclic boundary conditions. By using
the fermionic variables, a†k , ak the Hamiltonian3 may be written as
X  † 1

H = Λk ak ak − (28)
k
2
    
2 1 2 2

2 k 2

4 k
Λk = 2 (h − 1) + 4 h − 1 + γ sin − 4 γ − 1 sin (29)
γ 2 2

Then the entropy per spin site at finite temperature reads

"Z #
π/2 Z π/2 Λk /T
Sγ (T ) 1 1 e (µ − Λ k )
= dk ln(1 + e−(Λk −µ)/T ) + dk Λ /T (30)
N 2π 0 T 0 (e k + eµ/T )

The entropy per spin given in Eq. 30 represents the change of the mixing of the
finite temperature state form the ground state due to the thermal effects. Since the
quantum Ising model with γ = 1 has always the smallest value of SXY , the effects of
spin mixing disappear at the lower temperature. From these plots we see that SXY
yields results analogous to our models for SU(2) and SU(3) discussed in section 3,
depending upon the units of energy density, coupling, and the parameters, h and γ.
In Fig. 4 we plot the entropy per spin site versus the temperature T and for different
value of h and γ. Depending on these parameters the results of XY model given in
Fig. 4 can be compared with that of the Ising model, Eq. 3. The condition is that
γ = 1 and h → 0. We should notice the arbitrary units in both figures. For γ → 0
the value of the ground state entropy favours to stay constant for all temperarutes.
The same results are found in 1D Ising model for the limits J → 0. We can conclude
that the ground state entropy for the XY spin chain are qualitatively comparable
with our models for the SU(2) and SU(3).

5 Discussion
We have calculated the entropy for colored quark states in the hadron singlet and
octet structures and extended our considerations to finite temperatures. According
to the third law of thermodynamics the pure states, such as completely specified
enclosed hadronic systems, possess a vanishing entropy at very low temperatures.
Therefore, the hadron bags as macroscopic isolated objects are expected to have
zero entropy at a vanishing temperature. However, if we further consider the
microscopic hadron subsystems at low temperatures up to the quark mass, the
quantum entropy is seen to have a finite value. It is a curious fact that this
value is usually simply ignored or approximately taken to be zero especially for
classical systems, although it had been already recognized since the nineteenth
3
The positive sign of Hamiltonian refers to attractive nearest neighbor spin interactions. By
rotating the chain along the spin z direction to every second spin one can flip this sign [7].

14
century [3] and confirmed for the laws of quantization of statistical states for the
thermodynamics in last century [2, 4, 5]. Therefore, we think that there is no
compelling reason to assume that the value of the ground state entropy of quantum
or even classical subsystems is zero, just because one believes that the system to
which they are assigned is enclosed and thereby must have a vanishing entropy.
The subsystems have other degrees of mixing and thereby a finite entropy, even
if the whole closed system is isolated and consequently has a vanishing entropy.
Nevertheless, the ground state entropy is important and therefore should be taken
into consider at low temperatures. As we have seen the upper limit of temperature
of the validity or the importance of the ground state entropy is characterized by the
quark mass, i.e., T ∈ [0, m]. For light quarks and if we are interested on the QCD
phase transition or on the quark matter at very high temperatures, the ground
state entropy of colored quarks is no longer significant. On the other hand, in the
interior of stellar compact objects, cold dense quark matter is highly expected.
for which the ground state entropy would play an important role, especially, on
understanding the superconductivity on cold dense quark matter [19] and the phase
transition from neutron to quark matter in the hybrid stars and the stability and
structure of these compact stars [20, 21].

The quark and antiquark mathematically build up the Schmidt decompositions

of meson-state, where the Schmidt numbers simply represent the normalization of
their wavefunctions. Furthermore, from the quantum teleportation we know that
the colored quark and antiquark can be considered as mutual purifications for each
other. Each single state is equally weighted in the decomposition of meson states,
from which each state possesses an equal probability. On the other hand each single
quark state represents a certain degree of mixing and therefore has a finite entropy
although the meson state – as a pure isolated state – must have zero entropy. The
baryon states have the doubly reduced density matrix for each single quark state
appearing twice. The reduced density matrix give the spatial mixing of subsystems.
The resulting quantum entropy at zero temperature gives the maximal quantum
entropy for completely mixed states.
As we have seen the octet states are much more complex than the singlet states,
since the octets have many more and quite different states. When we looked at the
Pauli matrices we realized that the eighth Gell-Mann matrix, λ8 , counts also the
other states and generally no one of them can be considered as a purification of the
other states. At finite temperature the octet structure becomes much more complex.
The reason for this is that with increasing temperature the correlations between
the states become stronger, and at high temperature the correlated states reach the
asymptotic value. At high temperatures the states of the Pauli type become more
probable. Furthermore, the pure ground state becomes no longer possible. Thus
only the unoccupied states are the most available at high temperature.

We have postulated simple models for the thermal dependence of the ground
state entropy. We saw that the singlet state is a completely mixed state with the
maximum value of the entropy given by ln 3 at vanishing temperature. We have

15
used a Boltzmann-like factor for the thermal dependence of the entropy. With
increasing temperature the reduction of the ground state entropy is still continuing.
At high temperature the mixing and consequently the entropy vanish entirely.
The octet states show qualitatively the same results. The behavior of octet state
with the temperature reflects the complexity of their basic structure. We have
noted that by decreasing the quark masses the range of temperatures becomes
limited for reaching this asymptotic region. Since the asymptotic value at high
temperatures has remained more than three quarters of its ground state value of
ln 3, there are still considerable correlations between two of the three color states.
This observation is based on the complex structure of SU(3). Obviously, the root
structure forbids a complete cancellation of the real solutions which is needed
to maintain the trace condition on the reduced density matrix. Thus two states
are always matched against one, which forms the subsystem. Hence in the high
temperature limit one color vanishes while the other two remain mixed and thereby
correlated. We can conclude that ground state favors the color singlet state with
complete mixing, meanwhile SU(3) – in the high temperature limit – favors the
octet states involving mostly two colors. This situation for SU(3) can be contrasted
with SU(2) where the triplet and the singlet states have the same reduced density
matrices except for the pure triplet states.

In order to further model the ground state structure and the entropy of
SU(2), we utilized some known classical and quantum spin chains, which have
strong correlations at finite temperature. We investigate the behavior of their
Hamiltonians at finite temperature. The simplest and widely used classical spin
model is the one-dimensional Ising model. The other solvable model which is
related to Heisenberg model is XY spin chain. It is a highly correlated quantum
mechanical system in the ground state. One of the most valuable results we have
gotten from our investigation here is that the classical spin model is not able to
successfully describe the ground state. The entropy from the partition function
is simply zero at zero temperature. Thus we plot the entropy differences of the
finite temperature states to the ground state. If we additionally include thermal
terms similar to those in our models for SU(2) and SU(3) the entropy starts from
zero and monotonically increases up to the asymptotic value of ln 2. Also we
have found that this asymptotic limit does not depend on the spatial dimension.
Particularity with regard to the XY -model it has a nontrivial structure depending
on the parameters of spin asymmetry and external field, which has been recently
investigated through the correlation functions in the ground state [7].

6 Conclusions and Outlook

In this work we have compared the quantum definitions as contrasted to the
classical concepts of entropy in relation to the temperature. We have noticed that
in general the quantum definition is important in the low temperature limit, while
the classical concepts usually relate to higher temperatures. First we have discussed

16
the ground state entropy, which is strictly a quantum definition and does not
itself appear in classical physics. We have evaluated and contrasted the symmetry
structure for both the SU(2) and SU(3) color groups. For these symmetries
we have used simple thermodynamical models involving the color ground state
entropy to show how the quantum mixing entropy disappears for the entropy
differences with increasing temperature. From this result we have also discussed
how this disappearance can give rise to new pure states in the high temperature limit.

One motivation for this work has been to study the correlation between
the quarks and antiquarks. In recent QCD lattice simulations one can find
indications of ground state behavior at short distances and low temperatures [9–11].
These results could very well relate to the entropy we described above. As a next
step we want to look into the thermodynamical properties of these correlations in
relation to the present study [22]. There are many application of the finite entropy
of colored quarks at zero and very low temperatures much below the temperature of
QCD phase transition from hadron to quark-gluon plasma [23, 24]. This endeavor
could help demonstrate the usefulness of the quantum entropy in the description
of the thermal properties of the strong interaction at very low temperatures.
Furthermore, in two outcoming works we shall include the effects of the gluons and
the chiral symmetry in the future. We will also consider the effects of finite value
of entropy at low temperature on the pressure inside the hadron bag [25]. Also the
effects of the quantum entropy on the condensates of quark pairs with strong cor-
relations and at very low temperatures and very high quark chemical potentials [26].

Thus we have seen in several different models how the usual thermody-
namical entropy gotten from the evaluation of the partition function acts as a
means of disorganizing the ground state. It undoes the entanglement in some cases
completely and in others only partially. Therefore it has the effect of lowering the
correlations between the quarks and antiquarks seen in the ground state.

Acknowledgments
The authors would like to thank Frithjof Karsch, Krzysztof Redlich and
Helmut Satz for the very helpful discussions. D.E.M. is very grateful to the Penn-
sylvania State University Hazleton for the sabbatical leave of absence and to the
Fakultät für Physik der Universität Bielefeld.

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