Chiral Symmetry Breaking inFour-fermion Interaction Model with
Thermal and Finite-size Effects

Theoretical Particle and Hadron Physics Group,
Graduate School of Science, Hiroshima University
Hiromu Shimoji2022

Abstract
Symmetry is a basic concept in physics. It is expected that a higher symmetric phase isrealized in the early universe and the symmetry spontaneously breaks down to the one of thestandard model in particle physics through the spacetime evolution. Chiral symmetry plays
an important role in the mechanism of dynamical mass generation for matter. It is significant to reveal the phase structure in terms of the chiral symmetry. The phase structure isordinarily illustrated on a chemical potential-temperature (µ-T ) plane in high-energy physics
and it is well known that the structure varies with the physical conditions of the surrounding
environment. The study of the effects of various environments can lead to a better understanding of phase structures. Here, we focus on finite-size effects on chiral symmetry breaking
in strongly coupled systems of fermions.
Four-fermion interaction models are often used to describe spontaneous breaking of thechiral symmetry. At first, we briefly review the four-fermion interaction models at zero temperature and a finite temperature. In this thesis, the Gross–Neveu model is basically employed
as the simplest of these models of the chiral symmetry breaking. Evaluating the effective potential in the leading order of the 1/N expansion, we investigate the phase structure with thefinite-size effect.
We discuss the phase structure with the finite-size effect at zero temperature. In theprevious works, the divergent part of the effective potential are simply removed along with afinite part that depends on parameters of the system. We improve the procedure and evaluate
the effective potential, the dynamically generated fermion mass and the string tension. In
addition to the phase transition, we discuss a stable size of the system. The phase structure
are independent of the improvement, but the string tension is modified. The search for the
stable size of the system under the string tension revealed that the stable size is not realized.
Next, we consider the thermal effect in addition to the finite size effect. Caused by thefinite-size effect, more complex phase structures are realized at a finite temperature and afinite chemical potential. In order to investigate the precise phase structure, we evaluate
not only the minimum but also for the other extrema of the effective potential. Through theanalysis, we show the phase diagrams on parameter planes. Complex structures of boundaries
is found at a certain size of the system for a low temperature and a high chemical potential.
We also evaluate the behavior of the thermodynamic quantities and find the stable size of thesystem.

i

Contents
1 Introduction

1

2 Four-Fermion Interaction Model
6
2.1 Nambu–Jona-Lasinio Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Gross–Neveu Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Massive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Finite Temperature and Chemical Potential . . . . . . . . . . . . . . . . . . . . 19
3 Finite-size Effect
3.1 Effective Potential Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Generated Fermion Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Casimir Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23
23
26
29

4 Thermal and Finite-size Effects
4.1 Effective Potential Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Phase Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Thermodynamic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34
34
36
43

5 Summary and Discussions

53

A Notation

55

Bibliography

62

ii

Chapter 1

Introduction
It is well-known that the following four interactions are fundamental in physics: strong,electromagnetic and weak interactions and gravity. The standard model in particle physics
describing the three interactions except for the gravity has great successes, for example, theprediction of the Higgs boson and its discovery. These interactions in the standard modelare described quantum field theories based on a gauge symmetry, SU(3)c × SU(2)L × U(1)Y .
Quarks and leptons that construct matters are obtained as fundamental representations of
the group. Although the standard model has brilliantly succeeded, there are still unsolved
problems: mass gap, quantization of gravity, grand unified theories, neutrino masses, dark
matter, etc. To reveal physics beyond the standard model, many researchers have addressed
these problems.
Symmetry is essential in a gauge theory and several types of symmetry play importantroles in modern physics to construct theories. This can not be overemphasized. It is consideredthat a theory with a higher symmetry was realized in the early universe and the symmetry of
the theory spontaneously breaks down to the theory indicated by SU(3)c ×SU(2)L ×U(1)Y andthe other types of symmetry through the time evolution of the universe. In other words, it isexpected that, just as a state of water changes between gas, liquid and solid, phase transitions
associated with spontaneous symmetry breaking occurred in the universe during the evolutionof the spacetime. Spontaneous symmetry breaking is also a key concept in modern physics.
Inspired by the Bardeen–Cooper–Schrieffer theory [1] on superconductivity, Y. Nambu and
G. Jona-Lasinio introduced the concept to particle physics with chiral symmetry in 1961 [2,3].
In this thesis, we focus on a phase transition with chiral symmetry.
The chiral symmetry is related to the strong interactions. As is known that the strong
interactions between quarks and gluons are described by quantum chromodynamics (QCD),
a non-Abelian gauge theory in quantum field theories. Asymptotic freedom is a property of
QCD [4, 5]. At a high energy scale, perturbative approaches that work in quantum electrodynamics are valid even in QCD. However, at a low energy scale, perturbative approaches do notwork and a non-trivial vacuum structure, called the QCD vacuum, appears. In the non-trivial
vacuum, color confinement and chiral symmetry breaking occur: the former is a phenomenon

1CHAPTER 1. INTRODUCTION

that quarks and gluons are not observed in isolation and the latter a mechanism that hadrons
dynamically obtain their mass through the condensates of quarks and anti-quarks. To investigate the vacuum structure, several approaches has been attempted: numerical calculations
by using lattice QCD (LQCD) that is a part of lattice gauge theories and effective models
that are induced by approximations and phenomenological considerations. Because LQCD isconstructed on a discretized Euclidean spacetime, a path integral becomes finite-dimensionaland hence can run on computers from the first principles. The effective models are not thegenuine QCD, but they can be discussed on a continuous spacetime and calculated exactly.
Moreover, the effective models can also have theoretically rich structures, such as integrability.
Phase transitions are phenomena that appear at several scales, as in particle physics andcondensed matter physics. Revealing a phase structure of a target is often a research goalin several fields in physics, of course, QCD. For example, the schematic QCD phase diagram
is shown in Fig. 1.1: the vertical axis denotes temperature of the system and the horizontal
axis denotes quark chemical potential which is a conjugate variable of quark number density.
The chiral symmetry is broken in the hadron phase at a low temperature and a low chemical
potential, and restored in the other phase. It is an approximate symmetry, but this is a clue
to investigate the phase structure.
As is mentioned above, LQCD is a powerful non-perturbative tool to analyze the low
energy region, but there is the notorious problem called the sign problem at a finite quarkchemical potential [6,7]. The problem is caused by failure to interpret an integrand at a finite
chemical potential as a probability in the Monte Carlo method. Integrating degree of freedomof quarks with a finite chemical potential, µ, one obtains a path integral measure, M (µ) thatsatisfies (M (µ))∗ = M (−µ∗ ). This problem is common in particle physics and condensed
matter physics as long as the Monte Carlo method is employed. To resolve the problem, someapproaches are proposed: the Taylor expansion [8, 9], the analytic continuation from imaginary chemical potential [10–14], the complex Langevin method [15–19], the Lifschetz thimble
method [20, 21], and the tensor renormalization group [22–25]. The tensor renormalizationgroup is particularly a method that has been developed in condensed matter physics. Since
the area that can be analyzed by these approaches are still restricted, the bulk of the QCD
phase diagram has been illustrated by using the effective models. As the other approach close
to the effective model, two-color QCD is proposed. In this theory, there is no sign problem.
The two-color QCD is different from the QCD in terms of symmetry, but it is reported thatthe theory describes quantitatively three-color QCD [26]. Furthermore, the effective models
play important roles at present.

2

CHAPTER 1. INTRODUCTIONT
LQCD

Quark Gluon Plasma
Experiments

HadronColor Superconductor

μFig. 1.1: Schematic QCD phase diagram.
There are various effective models to investigate the phase structure. The Nambu–JonaLasinio (NJL) model [2,3] is a simple model that describes dynamical chiral symmetry breaking
via quark and anti-quark interactions, and is often used for the purpose in terms of chiralsymmetry [27, 28]. Some types of four-fermion interaction models exist: the Gross–Neveu
model [29] that has a discretized symmetry through a quark and anti-quark interaction, theextended NJL models with diquarks [30, 31], and the Polyakov-loop extended NJL model [32]that includes the effects of the color confinement. We adopt a four-fermion interaction model,
mainly the Gross–Neveu model, to analyze and discuss the phase structure with the chiralsymmetry, in this thesis.
States of matter depend on the physical conditions of surrounding systems. In Fig. 1.1,
the state of matter is designated by temperature and quark chemical potential. There are,of course, various parameters to specify the condition of the system: temperature, entropy,pressure, volume of the system, particle number, chemical potential, curvature of the spacetime, magnetic field, rotational velocity, boundary conditions, topology, dimensions of thespacetime, and so on. A phase is basically realized by a state with the lowest thermodynamic
potential under the parameters considered in the system. Thus, studying with parametersin addition to temperature and chemical potential leads to a better understanding of phasestructures. By using four-fermion interaction models, phase transitions in various conditions
are researched [27, 28, 33–35].
Related to high-energy experiments and compact stars (e.g. neutron stars), effects from themagnetic field [36] and the rotational velocity [37, 38] to the phase structure are interesting3

CHAPTER 1. INTRODUCTION

to many researchers. The characteristic points of the two conditions are respectively thedimensional reduction [39, 40] that is a phenomenon in which an effective dimension becomes
lower than the original dimension, D, such as D → D − 2, and finite-volume effects with
topology of boundaries [41]. Moreover, although LQCD contains artifacts, finite-volume effects
and topological contributions from boundaries are not negligible in condensed matter physics
and LQCD. Looking back at the early universe, all the matter might be confined inside
an extremely small size space with a non-trivial topology. In string theory, the existence ofcompactified extra dimensions is predicted. Thus, we believe that it is significant to reconsider
the finite-size effects, the contributions from the boundaries, and low-dimensional physics. In
this thesis, we discuss not only the thermal effects but also the finite-size effects and thecontributions from the boundaries on the phase structure for the chiral symmetry by using
the four-fermion interaction model.
There are some works studying the finite-size effect on the chiral symmetry in four-fermion
interaction models. Periodic and anti-periodic boundary conditions are often imposed [42–46](Refs. [47, 48] are in LQCD). It is found that the chiral symmetry breaking tends to beenhanced because of the finite-size effect for the periodic boundary condition. On the contrary,
it is observed that the anti-periodic boundary condition tends to restore the chiral symmetry.
In this context, the finite temperature and chemical potential in the Matsubara formalism
can be regarded as the size and boundary condition for the imaginary time direction. The
boundary conditions can be more generalized to a U(1)-valued boundary condition for acompactified space, such as the one-dimensional sphere S1 . Some works on the U(1)-valued
boundary condition have previously done [49–54]. In general, such a boundary condition is
introduced from the Aharonov–Bohm effect [55] in an Abelian gauge theory. For instance, thefinite-size effect with the U(1)-valued boundary condition can be realized by the presence ofmagnetic flux in a ring or cylinder. In condensed matter physics, inhomogeneous condensates
are discussed in a superconducting ring with magnetic flux [54]. As is discussed in [56] forhigh energy physics, it is also pointed out that inhomogeneous phases can be favored in a
finite-size system.
To clarify the physical picture, it is important to evaluate thermodynamic quantities in
addition to the dynamically generated fermion mass; we calculate particle number densityand pressure. A well-known finite-size effect in quantum systems is the Casimir effect [57–70]
which is a phenomenon of attraction between two uncharged conductive plates placed parallel.
This is understood to be a phenomenon caused by a non-trivial vacuum. The Casimir force
was firstly introduced as an attractive force, but later it was pointed out that the force
could also be repulsive [71–73]. There is a study to connect the attractive force with therepulsive force for perfect electromagnetic conductor plates [74]. These discussions are basedon electromagnetic fields, in other words, bosonic and vector fields. The Casimir effect has
been studied in a four-fermion interaction model [75–77]; particularly, the sign-flip phenomena
is also found in Ref. [76]. In the context of lattice simulations, the Casimir effect has been4CHAPTER 1. INTRODUCTION

also studied [47, 48, 78–80].
In this thesis, we study the finite-size effect on chiral symmetry breaking in a strongly coupled system of fermions by using a four-fermion interaction model. To clarify the discussion,
we consider a simple condition: one-dimensional sphere, S1 , as a part of the background. Our
purpose is to show the finite-size effect on phase diagrams and clarify the phenomenologicalconsequences by evaluating the thermodynamic quantities.
This thesis is organized as follows: In Chapter 2, we give brief introductions about chiralsymmetry breaking in four-fermion interaction models without finite-size effects. In this thesis,
we evaluate the effective potential in the leading order of the 1/N expansion with supposing ahomogeneous chiral condensate. Next, the model at a finite temperature and a finite chemical
potential is introduced with a phase diagram in terms of the chiral symmetry. In Chapter 3,
we show the finite-size effects on chiral symmetry breaking in the Gross–Neveu model at zerotemperature based on [81]. We also improve the effective potential analysis from the previous
work [50]. Behavior of the dynamically generated fermion mass affected by the finite-size
effects is shown. We discuss the stable size with the Casimir effect. In Chapter 4, chiralsymmetry breaking at a finite temperature and a finite chemical potential is discussed withthe finite-size effects based on [82]. To investigate the precise phase structure, we introduce
new symbols and show the structure on bifurcation diagrams. We discuss the phase structure
and the stability of the system with thermodynamic quantities. Finally, Chapter 5 is devoted
to the summary and discussions.

5Chapter 2

Four-Fermion Interaction ModelIn this chapter, we briefly review four-fermion interaction models: the Nambu–Jona-Lasinio(NJL) model [2,3] and the Gross–Neveu (GN) model [29]. Both the models are well-known aseffective models of the quantum chromodynamics (QCD) in terms of the chiral symmetry andthe asymptotic freedom. They are simple models that describe spontaneous chiral symmetry
breaking. These two models are distinguished by the interaction terms and enjoy slightly
different symmetries. Although this difference does not affect on our analysis, we will mention
a note in two dimensions later.

2.1

Nambu–Jona-Lasinio Model

Let us consider an action of the NJL model with Nf -flavor fermions on a D-dimensional
Minkowski spacetime, MD , (2 ≤ D < 4):


Z

λ0
(2.1)
(ψ̄(x)ψ(x))2 + (ψ̄(x)iγ5 τ a ψ(x))2 ,SNJL = dD x ψ̄(x)iγ µ ∂µ ψ(x) +2Nc
where λ0 denotes a coupling constant of the four-fermion interaction, Nc the number of thecolor, and τ a a generator of SU(Nf ) (a = 1, 2, . . . , Nf2 − 1). λ0 has the mass dimension−(D − 2). The indices of the flavors and colors are omitted below. Such a model is introduced
by Y. Nambu and G. Jona-Lasinio as first in the papers [2, 3]. The model is evaluated in four
dimensions (D = 4), and hence the coupling has the mass dimension −2. This action (2.1) isinvariant under a global transformation of U(Nf )L × U(Nf )R :a a

ψχ → eiθχ τ +iθχ ψχ ,(θχa , θχ ∈ R)

(2.2)

where ψχ (χ = L, R) denote left- and right-handed fermions, respectively. In particular, we
are interested in U(1)A , a part of the group. A transformation of the group is called a chiraltransformation:
ψ(x) → eiγ5 θ ψ(x),(θ ∈ R).
(2.3)
The invariance under the chiral transformation is due to no mass term.

6

CHAPTER 2. FOUR-FERMION INTERACTION MODEL

A generating functional of the NJL model is given byZ
ZNJL = Dψ Dψ̄ eiSNJL .

(2.4)

Inserting a constant that is expressed as a path integral of auxiliary fields, σ̂(x) and π̂ a (x),


Z
Z

Nc
D
2
a
2
a
,
(2.5)
d x σ̂(x) + π̂ (x)Dσ̂ Dπ̂ exp −i
2λ0into the generating functional (2.4), and using the invariance of a path integral measure undershifts, we obtainZ
ZNJL = Dψ Dψ̄ Dσ̂ Dπ̂ a eiSaNJL ,
(2.6)


Z

Nc
SaNJL = dD x ψ̄(x) (iγ µ ∂µ − σ̂(x) − iγ5 τ a π̂ a (x)) ψ(x) −σ̂(x)2 + π̂ a (x)2 . (2.7)
2λ0
The Euler–Lagrange equation for σ̂(x) and π̂ a (x) gives us constraints, σ̂(x) = −λ0 ψ̄(x)ψ(x)/Nc
and π̂ a (x) = −λ0 ψ̄(x)iγ5 τ a ψ(x)/Nc , respectively. Substituting these into the action (2.7), theoriginal one (2.1) is reproduced. At the classical level, the auxiliary fields are proportional tocomposite operators of a fermion and an anti-fermion. Thus, the finite vacuum expectation
value of the auxiliary fields indicates spontaneous symmetry breaking of the chiral symmetry.
The action (2.7) is written as quadratic forms of the fermions and auxiliary fields. Integrating out the fermion and anti-fermion, the generating functional and the action are rewrittenas
Z
a
ZNJL = Dσ̂ Dπ̂ a eiNc S[σ̂,π̂ ] ,
(2.8)
Z

1
a
S[σ̂, π̂ ] = −dD x σ̂(x)2 + π̂ a (x)2 − i ln Det [iγ µ ∂µ − σ̂(x) − iγ5 τ a π̂ a (x)] ,
(2.9)
2λ0
with Det denoting the determinant over the flavor indices, the spinor indices, and the continuous spacetime coordinates. This generating functional (2.8) (or this action (2.9)) is represented
only by the degrees of freedom of the auxiliary fields. The above manner that makes a fourfermion interaction term into a quadratic form by introducing a (pseudo)scalar field, is called
the Hubbard–Stratonovich transformation [83, 84], or the auxiliary field method [85].
Although there is no degree of freedom of the fermion in the generating functional (2.8),
it is still difficult to evaluate analytically the action (2.9). To resolve this problem, we apply
the 1/Nc expansion and assume a homogeneous chiral condensate, in other words the vacuum
expectation values of σ̂(x) and π̂(x) are independent of the temporal and spatial coordinates.
We separate the auxiliary fields into the vacuum expectation values and fluctuations,π̂ a (x) = π a (x) + π̃ a (x).

σ̂(x) = σ(x) + σ̃(x),(2.10)

σ(x) denotes an order parameter of chiral symmetry. Under the 1/Nc expansion, we obtain
the effective action,
Γ[σ, π a ] = −i ln ZNJL = Nc (S[σ, π a ] + O (1/Nc )) .

7

(2.11)

CHAPTER 2. FOUR-FERMION INTERACTION MODEL

Assuming the homogeneous chiral condensate, σ(x) = σ and π a (x) = π a , we obtain theeffective potential in the leading order of the 1/Nc expansion,S[σ, π a ]σ 2 + (π a )2 Nf tr I−
V (σ, π a ) = − R D =
2λ0
2
d x

Z


dD k
2
2
a 2
ln
−k
+
σ
+
(π
)
,
i(2π)D(2.12)

where the trace, tr, denotes the diagonal sum with respect to the spinor indices. In our
numerical calculations, we set tr I = 2D/2 . A vacuum state (equivalently, a ground state) isdetermined by the minimum of the effective potential in the parameter space of σ and π a .
The value of the effective potential does not change as long as σ 2 + (π a )2 has the same value.
Therefore, if σ 2 + (π a )2 ̸= 0, the vacuum states distinguished by the values of σ and π a areinfinitely degenerated. Because these states are equivalent, we can choose the vacuum statesatisfying π a = 0 without loss of generality (V (σ) = V (σ, π a = 0)). In addition, we normalize
the effective potential at σ = 0 to V (0) = 0.
The effective potential (2.12) contains an UV divergence. It is known that results producedby the NJL model depend on regularization schemes because the model is not renormalizable[86, 87] as indicated by the mass dimension of the coupling, −2, in four dimensions. Here,
we show some regularization schemes: the sharp cutoff regularizations, and the dimensional
regularization.

Three-momentum cutoff regularization
First, we introduce a three-momentum cutoff parameter Λ3 in D = 4 as
Z
Z
Z
d3 k
dk 0
d4 k
→
,
i(2π)4i2π k2 ≤Λ23 (2π)3(2.13)

then the effective potential is given byNf
σ2VΛ3 (σ) =−2λ0 8π 2Λ3 2Λ23 + σ 2

q 2
σ4
Λ3 + σ 2 +
ln
2

σ2
p
(Λ3 + Λ23 + σ 2 )2!!
+

Nf Λ43.
4π 2(2.14)

Four-momentum cutoff regularization
In a similar way, we can also introduce a four-momentum cutoff parameter Λ4 in D = 4 after
the Wick rotation as
Z
Z
d4 k
d4 k
2
f
(−k
)
→
f (k2 ),
(2.15)
42i(2π)4(2π)
2k ≤Λ4where f is an arbitrary analytic function. The effective potential is give by






Nf
σ2
σ2
Λ24
4
4
2 2
VΛ4 (σ) =−Λ4 ln 1 + 2 − σ ln 1 + 2 + σ Λ4 .
2λ0 16π 2σΛ48(2.16)

CHAPTER 2. FOUR-FERMION INTERACTION MODEL

Proper-time regularization
Another regularization scheme that uses a sharp cutoff is the proper-time method [88, 89]. It
is useful to keep in mind the formal identity for a operator O, related to the zeta function
regularization [90, 91],Z ∞d 1
dτ τ s−1 e−τ O .
(2.17)
tr ln O = − limtrs→0 ds Γ (s)0
To treat the divergent term, we introduce a regularization parameter Λpt as
Z
Z
Z ∞

d4 k −τ (k2 +σ2 )
d4 k
−1
2
2
dτ
τ
ln
−k
+
σ
→
−
e
.
i(2π)4(2π)41/Λ2pt
The effective potential in the proper-time method reads
Z ∞Nf Λ4ptNf
σ2−3 −τ σ 2+dτ τ e−.
Vpt (σ) =2λ0 8π 2 1/Λ2pt
16π 2(2.18)(2.19)In this method, the effective potential is written as an integral over a proper-time space witha lower limit, not as an integral over a momentum space with a upper limit.

Dimensional regularization
Finally, we introduce the dimensional regularization that is one of regularization schemes byusing analytic continuation. Considering a complex number D ̸= 4 (in particular, considering
2 ≤ D < 4), we can perform the integral over a momentum space,

Z
D
2 D/2

Γ
1
−
1
dD k
2
2
2 (σ )
ln
−k
+
σ
=
.
(2.20)
2i(2π)DD(4π)D/2
Thus, we obtain the effective potential,
VDR (σ) =σ2(σ 2 )D/2− Nf CD,
2λ0
D(2.21)

with a constant depending on the dimension D,
CD =



tr I
D
Γ
1
−
.
2
(4π)D/2(2.22)

This term diverges in two and four dimensions. There is no scale parameter in this expression(2.21). We renormalize the coupling λ0 with a renormalization scale, µr . The renormalization
condition is given by∂ 2 VDR (σ)
µD−2
r
=
,
(2.23)∂σ 2λr
σ=µrwhere λr indicates a renormalized coupling that has the mass dimension zero. The renormalized effective potential with the dimensional regularization is
1 1(σ 2 )D/2
VDR (σ) =+ (D − 1)Nf CD σ 2 µD−2
− Nf CD.
(2.24)r
2 λr
D

9CHAPTER 2. FOUR-FERMION INTERACTION MODEL

This expression is finite at 2 ≤ D < 4.
The NJL model at D = 2 is also called the chiral Gross–Neveu model. In particular, thismodel is renormalizable (The mass dimension of the coupling becomes zero in two dimensions.) and shares some features with the four-dimensional theory because these dimensionsof a background spacetime is even. Hence, the results produced by the model in two dimensions does not depend on regularization schemes. Because of the well-known no-go theorem
stated by N. Mermin, H. Wagner, P. Hohenberg and S. Coleman [92–94], spontaneous chiralsymmetry breaking is prohibited in two dimensions, but the theorem becomes invalid in theleading order of the 1/Nc expansion. This is because fluctuations that destroy the spontaneous symmetry breaking, are suppressed in the next to leading order of the expansion (seeEq. (2.11)).
We comment on the dimensional regularization in the NJL model. Targets which is analyzed by using this model is usually in four dimensions, but one can not take the fourdimensional limit in this regularization because of the non-renormalizability. As is discussed
in Refs. [95–98], the results from the dimensional regularization in 2 < D < 2.5 is similar to
the results from the sharp cutoff; some properties of the nonet meson are reproduced.
As examples, we show the behavior of the effective potential for comparison of the different
cutoff regularizations in Figs. 2.1 and 2.2 (Nf = 1). It can be seen that there is only one
minimum of the effective potential at the origin, σ = 0, for the coupling constant smaller
than a certain value, but when the coupling constant becomes larger than the value, theminimum is located at a point that is not the origin, σ ̸= 0. In other words, spontaneouschiral symmetry occurs for the coupling constant larger than a certain value, and then thefermion mass is dynamically generated. Such a value of the coupling constant at which a
second-order phase transition between a chiral symmetric phase and a chiral broken phaseoccurs, is called a critical coupling constant.

100.010

0.010

0.008

0.008

0.006

0.006

V/Λ4

V/Λ4

CHAPTER 2. FOUR-FERMION INTERACTION MODEL

0.004

0.004

0.002

0.002

0.000

0.000

- 0.002
- 1.0

- 0.5

0.0

0.5

1.0

- 0.002
- 1.0

σ/Λ

- 0.5

0.0

0.5

1.0

σ/Λ

Fig. 2.1: The behavior of the effective potential: the three-momentum cutoff regularization. The coupling constant smaller
than the critical one corresponds to theblue dashed curve (λ0 Λ23 = π 2 ), and larger
than the critical one the orange solid curve(λ0 Λ23 = 3π 2 ).

Fig. 2.2: The behavior of the effective potential: the four-momentum cutoff regularization/the proper-time regularization. In
the both cases, the coupling constant smaller
than the critical one corresponds the blue
dotted/green dot-dashed curve (λ0 Λ24/pt =3π 2 ), and larger than the critical one the orange solid/red dashed curve (λ0 Λ24/pt = 8π 2 ).

The minimum of the effective potential corresponds to a vacuum state that has the lowest
energy. Therefore, to discuss spontaneous chiral symmetry breaking in the model, we need tofind out the minimum. This is a solution of the gap equation,∂V (σ)= 0,∂σ σ=m(2.25)

with m denoting a dynamically generated fermion mass. The behavior of the generated
fermion mass is shown in Fig. 2.3, and we can observe the second-order phase transitions.
The value of the generated fermion mass monotonically increases as the coupling constantincreases. Rewriting the gap equation as0=∂V (σ)= mf (m; λ0 ),∂σ σ=m(2.26)

where f (m; λ0 ) is a function of zero or more orders of m, and taking the chiral limit, m → 0, we
obtain a critical coupling λc from f (m; λc ) = 0. From Eq. (2.13), (2.15), and (2.19), the critical coupling constant is λ3,c Λ23 = 2π 2 /Nf in the three-momentum cutoff regularization, andλ4,c Λ24 = λpt,c Λ2pt = 4π 2 /Nf in the four-momentum cutoff and proper-time regularizations.

11CHAPTER 2. FOUR-FERMION INTERACTION MODEL

1.0

0.8

m/Λ

0.6

0.4

0.2

0.0
0

2

4
2

λΛ /π

6

8

2

Fig. 2.3: The dynamically generated fermion mass as a function of the coupling constant,
λ0 Λ23/4/pt /π 2 (Nf = 1). The blue dot-dashed curve corresponds to the three-momentum cutoff
regularization, the orange solid curve the four-momentum cutoff regularization, and the green
dashed curve the proper-time regularization.

2.2

Gross–Neveu Model

The GN model is a four-fermion interaction model similar to the NJL model. The action ona D-dimensional Minkowski spacetime, MD , (2 ≤ D < 4) reads


Z
λ0
2
D
µ
(ψ̄(x)ψ(x)) ,(2.27)
SGN = d x ψ̄(x)iγ ∂µ ψ(x) +2Nwith λ0 denoting a coupling constant of a four-fermion interaction that has the mass dimension−(D − 2), and N the number of copies of fermions. This action (2.27) is invariant under adiscrete chiral (Z2 ) transformation, ψ(x) → γ5 ψ(x). The action, of course, has the symmetry,ψ → −ψ. In the original paper [29], the model is constructed in two dimensions, and thus thecoupling has no mass dimension.
Applying the same manner with the NJL model, we can obtain the effective potential of
the GN model in the leading order of the 1/N expansion,(σ 2 )D/2
1 1+ (D − 1)CD σ 2 µD−2−C,(2.28)V (σ) =Dr
2 λr
D
where the coupling is renormalized according to the same procedure (2.23). We have usedthe dimensional regularization. Due to the discrete symmetry, the same expression as in
Eq. (2.24), except for Nf , is obtained without choosing a vacuum state. And also, becauseof the same reason, spontaneous discrete chiral symmetry breaking is not forbidden in two

12CHAPTER 2. FOUR-FERMION INTERACTION MODEL

dimensions. Therefore, we employ the GN model to discuss spontaneous chiral symmetry
breaking in two dimensions. The expression of the effective potential in two dimensions is 2 !
σ2σ2
σ
V (σ) =+
−3 + ln.
(2.29)
2λr
4πµr
As mentioned above, the GN model in two dimensions is independent of regularization schemes
due to its renormalizablity. Furthermore, the model in three dimensions, or in any dimension
2 ≤ D < 4, is renormalizable in the sense of the leading order of the 1/N expansion [99, 100].
In Fig. 2.4, the generated fermion mass in two and three dimensions are shown as a functionof the renormalized coupling, and the latter exhibits the second-order phase transition. The
chiral symmetry is always broken except for λr = 0 in two dimensions. The explicit expressionof the generated fermion mass is obtained as
 1

D−2
1
+D−1
.
(2.30)
m = µrλ r CDAnd also, the critical coupling, λc,D , is given by
λc,D = −1.
CD (D − 1)

(2.31)The behavior of the critical coupling is plotted in Fig. 2.5. For instance, λc,D=2 = 0 and√
λc,D=3 = π/ 2 ≃ 2.22. On the boundary, the second-order phase transition takes place.
2.0

2.5

2.0

1.5

Broken

D=2

λc

m/μr

1.5
1.0

D=3

0.5

0.0
0

Symmetric

1.0

0.5

2

4

6

8

0.0
2.0

2.5

3.0

3.5

4.0

D

λr

Fig. 2.4:
The dynamically generated
fermion mass as a function of the coupling
constant in two (the blue solid curve) andthree (the orange dashed curve) dimensions.

Fig. 2.5: The critical coupling as a functionof the dimension, D. The chiral symmetry
is restored in the area below the black curve,
and broken in the other area.

As is shown by no mass dimension of the coupling, the GN model is renormalizable intwo dimensions, and hence there is no regularization dependence. However, differences resulting from regularization schemes can also appear in two dimensions before one applies an13

CHAPTER 2. FOUR-FERMION INTERACTION MODEL

appropriate renormalization. We here compare the effective potentials derived from the cutoff
regularization and the dimensional regularization in two dimensions. We can easily obtain theexpression for a momentum cutoff regularization by replacing three with one in Eq. (2.13),
Z
dk p 2σ2VΛ1 (σ) =−k + σ2
2
2λ0
2π
2
k ≤Λ1


 2 
p(2.32)
2 + σ2
q
Λ
Λ
+
2
2
1
1
σ
1  2
σ


2
2
=
−
ln 
Λ1 Λ1 + σ + ,2λ0 2π2
σ2with Λ1 denoting the cutoff in two dimensions. Expanding this in powers of σ/Λ1 , we have
 2
 2 !
 3σσσVΛ1 (σ)σ21−1 − ln 4 + ln+O=+,
(2.33)
22
Λ1
Λ1
2λ0 Λ1 4π Λ1Λ31that is shifted as VΛ1 (σ = 0) = 0. The effective potentials (2.33) and (2.29) that is derived
from the dimensional regularization are obviously distinguished in terms of the higher orderterms of Λ1 , but also in terms of the coefficients in the second order.
Next, we briefly review the asymptotic freedom. The scale dependence of the renormalized
coupling, λr , is determined by the renormalization group equation,µrdV (σ)= 0,dµr(2.34)

in the leading order of the 1/N expansion. This equation based on the fact that physicalquantities are explicitly independent of the renormalization scale, µr , which is artificiallyintroduced. This equation is rewritten asµr∂V (σ; λr , µr )∂V (σ; λr , µr )+ β(λr )
= 0,
∂µr
∂λr

(2.35)

where β(λr ) is a beta function that stands for a dependence of the coupling on µr , and theparameter dependence of the effective potential is written explicitly. Note that there is norenormalization of σ in the leading order of the 1/N expansion because of our definition of
the auxiliary field, σ̂(x) ∼ λ0 ψ̄(x)ψ(x) (below Eq. (2.7) at the page 7). Solving the equation(2.35), we obtainD−2(λc,D − λr )λr ,(2.36)β(λr ) =λc,Din the leading order of the 1/N expansion. This result is consistent with Ref. [34].
The behavior of the beta function is plotted in Fig. 2.6. In two dimensions, the beta
function is always negative for λr > 0 (It means that the renormalized coupling is a monotonically decreasing function of µr .), and thus the theory is asymptotically free. On the other
hand, there is an UV fixed point at λr = λc,D=3 in the three-dimensional theory; with increasing µr , the value of the coupling monotonically decreases for λr > λc,D=3 , and increases
for λr < λc,D=3 .

14CHAPTER 2. FOUR-FERMION INTERACTION MODEL

0.6

D=3

0.4

β

0.2
0.0
- 0.2

D=2

- 0.4
0.0

0.5

1.0

1.5

2.0

2.5

3.0

λr
Fig. 2.6: The beta function as a function of the coupling in two (the blue solid curve) andthree (the orange dashed curve) dimensions.
To make clear the behavior of the renormalized coupling, we solve a differential equation,
D−2dλr= β(λr ) =(λc,D − λr )λr ,dt
λc,D(2.37)

with a dimensionless parameter, t = ln(µr /µr,0 ), and a reference scale, µr,0 . Given λr (t =0) = λr,0 at t = 0 (µr = µr,0 ) as an initial condition, we obtain a running coupling constant,λr,0 − λc,D −(D−2)t −1
λr (t) = λc,D 1 −
e
.
λr,0

(2.38)The behavior of the running coupling is shown in Fig. 2.7. The coupling constant for a strong
coupling, λr,0 > λc,D , monotonically decreases with increasing the renormalization scale, but
for a weak coupling, λr,0 < λc,D , shows the opposite behavior.

15

6

6

5

5

4

4

λr

λr

CHAPTER 2. FOUR-FERMION INTERACTION MODELλr (0) = π > λc = 0

3

2

1

1

-2

0

2

4

6

8

10

2 > λc320

λr (0) = π0λr (0) = 1 < λc = π 
-2

0

2

4

t

t

D=2

D=3

2
6

8

10

Fig. 2.7: The running coupling constant as a function of the dimensionless parameter, t =ln(µr /µr,0 ) (left: two dimensions, and right: three dimensions ). The blue solid curves indicatethat the initial state is the strong coupling, λr,0 > λc,D . The orange dashed curve indicatesthat the initial state is the weak coupling, λr,0 < λc,D .

2.3

Massive Model

We have mentioned the no-go theorem for continuous symmetries in Chapter 2.1 in two
dimensions, and also mentioned that the theorem becomes invalid in the leading order of the
1/N expansion. Alternatively, we can also avoid the theorem by choosing a discrete symmetry
as the symmetry of the theory (Chapter 2.2).
In this section, we consider a massive model in four-fermion interaction models. For
simplicity, we use the massive Gross–Neveu model,


Z
λ0
2
D
µ
(ψ̄(x)ψ(x)) ,(2.39)
SmGN = d x ψ̄(x) (iγ ∂µ − mbare ) ψ(x) +2Nwith mbare denoting a bare fermion mass. The other parameters are the same ones in themassless model (2.27). Because of the mass term that is a explicitly breaking , mbare ψ̄(x)ψ(x),the model does not enjoys the discrete chiral symmetry; in that sense, the discrete chiralsymmetry becomes an approximate symmetry. The massive model is not only realistic, butalso can avoid the no-go theorem. Thus, it is worthwhile to consider the massive model.
After shifting the auxiliary field by mbare , we apply the same manner in the massless GN
model, and then obtain the effective potential in the leading order of the 1/N expansion,V (σ) =(σ 2 )D/2
σ 2 − 2mbare σ
− CD
,
2λ0
D

(2.40)

2
where the constant term, λ−10 mbare , is omitted. Following the renormalization procedure

16

CHAPTER 2. FOUR-FERMION INTERACTION MODEL(2.23), the effective potential is rewritten asV (σ) = 2
σ(σ 2 )D/2
1− CD (D − 1)
− mbare σ µD−2
− CD
.
r
λr
2
D

(2.41)Taking the two-dimensional limit, we can see that the effective potential contains a divergenceproportional to the mass term. Hence, in order to construct the finite theory in 2 ≤ D < 4,
we need to renormalize the divergence to the mass.
The physical mass is defined as a pole position of the propagator, and realized as theminimum of the effective potential [101, 102]. In that sense, mbare itself does not imply
physical mass unless it satisfies the minimum of the effective potential. In the weak coupling
limit, λr → 0, a mass parameter mbare is dominant in determining the physical mass. Whenthe mass parameter is zero, the effective potential is even and can have the minimum at σ = 0
for the weak coupling; intuitively, the mass parameter means the tilt of the effective potential.
Let us reconsider the massless model. The first-order derivative of the effective potential
with respect to σ always satisfies zero at the origin, σ = 0. This implies the fact that there
is always the trivial solution. Back to the massive theory again, the derivative at the origin
satisfies,1∂V (σ)=−+ CD (D − 1) mbare µD−2.
(2.42)
r∂σ σ=0λr
This indicates that the gap equation in the massive theory, mbare ̸= 0, does not always havethe trivial solution, and also the tilt of the effective potential at the origin becomes divergent
due to CD in the two-dimensional limit.
To resolve this problem, we apply a renormalization condition to mbare by using thederivative at the origin (2.42),
mr D−2∂V (σ)=−µ,∂σ σ=0λr r(2.43)

where mr is a renormalized mass parameter. Under this condition, the divergence in thetwo-dimensional limit is removed, and the trivial solution is excluded as long as mr ̸= 0. The
renormalized effective potential reads
1 1
mr D−2(σ 2 )D/2+ CD (D − 1) σ 2 µD−2
−
σµr
− CD
.
(2.44)V (σ) =r
2 λr
λr
DIn the weak coupling limit, λr → 0, the gap equation is approximately expressed as,0=∂V (σ)
m − mr D−2≃µr ,∂σ σ=mλr(2.45)

then we can confirm the solution, m = mr , being the minimum of the effective potential.
Therefore, the renormalization condition for the mass parameter (2.43) means that the renormalized mass parameter gives a physical mass in the weak coupling limit. In Fig. 2.8, thebehavior of the generated fermion mass is shown as a function of λr . We can observe that the17

CHAPTER 2. FOUR-FERMION INTERACTION MODEL

generated fermion mass monotonically and smoothly increases with increasing the coupling,
and the chiral symmetry is not restored in the weak coupling limit, λr → 0.
2.0

2.0

1.5

1.5

mr = 0.2

m/μr

m/μr

mr = 0.1
1.0

1.0

mr = 00.5

0.00mr = 0

0.5

2

4

6

8

0.0
0

2

4

λr

λr

D=2

D=3

6

8

Fig. 2.8: The dynamically generated fermion mass as a function of λr (left: two dimensions,and right: three dimensions). The blue solid curve corresponds to the massive theory, andthe orange dashed curve the massless theory.
Finally, we check the scale dependence of the mass parameter, mr , in this section. In themassive model, the renormalization group equation (2.34) is written asµr∂V (σ; λr , mr , µr )∂V (σ; λr , mr , µr )∂V (σ; λr , mr , µr )+ β(λr )+ mr γm (λr )
= 0,
∂µr
∂λr
∂mr

(2.46)

where γm (λr ) is a coefficient related to the renormalized mass parameter, mr . Solving thisequation, the solution of β(λr ) is Eq. (2.36). The solution of γm (λr ) is given by
γm (λr ) = −β(λr )D−2λr =− (D − 2).
λc,D
λc,D

(2.47)γm (λr ) is linear with respect to λr and monotonically decreases from γm (λr = 0) = 0 inarbitrary dimension.. Since we have solved β(λr ), we can also solve the differential equation,
λr,0 − λc,D −(D−2)t −1
dγm (t)= −(D − 2) 1 −
e
mr ,
dt
λr,0
then
mr (t) =λc,D mr,0,(D−2)te− 1 λr,0 + λc,D

(2.48)(2.49)

with mr (t = 0) = mr,0 at t = 0 as an initial condition. The behavior of the renormalized
mass parameter, mr (t), is shown in Fig. 2.9. Regardless of whether the coupling is strongly
or weakly coupled, the value of the mass parameter decreases as the renormalization scale
increases.

18

CHAPTER 2. FOUR-FERMION INTERACTION MODEL

3.0

3.0

2.5

2.5

2.0

2.0

2 > λcmr /mr,0

mr /mr,0

λr (0) = πλr (0) = π > λc = 0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

-2

0

2

4

6

8

10

0.0

-2

0

λr (0) = 1 < λc = π 

2

2

8

4

t

t

D=2

D=3

6

10

Fig. 2.9: The renormalized mass parameter as a function of t = ln(µr /µr,0 ) (left: twodimensions, and right: three dimensions). The blue solid curves indicate that the initial stateis the strong coupling, λr,0 > λc,D , and the orange dashed curve indicates that the initial stateis the weak coupling, λr,0 < λc,D .

2.4

Finite Temperature and Chemical Potential

In order to deal with more realistic situations in four-fermion interaction models, we extend
the models with a finite temperature and a finite chemical potential. Only thermal equilibrium
systems are treated. In this section and the following chapters, we use basically the GN model,
and the dimensional regularization.
The Matsubara formalism (or called the imaginary time formalism) [103–106] is well known
as an extension method to a finite temperature systems. In this formalism, a time direction isextended to a complex plane by using analytic continuation, and a time, x0 , is replaced with
an imaginary time, x0 → xD = −ix0 (xD ∈ R). The range of an integral over the imaginary
time is limited to a finite interval, [0, 1/T ], for a temperature T . In fermion systems, ananti-periodic boundary condition is imposed to the imaginary time direction,
ψ(x1 , x2 , . . . , xD + 1/T ) = −ψ(x1 , x2 , . . . , xD )(2.50)

Thus, we can regard the imaginary time direction as a one-dimensional sphere, S1 , with acircumference of 1/T : the background spacetime RD−1 × S1 . The generating functional (2.6)
is replaced with a partition function in thermodynamics,Z
ZGN,T = Dψ Dψ̄ e−SGN,T ,(2.51)


Z
Z 1/T
λ0
(ψ̄(x)ψ(x))2 ,(2.52)
SGN,T = dD−1 x
dxD ψ̄(x)γ̃ k ∂k ψ(x) −
2N
0

19

CHAPTER 2. FOUR-FERMION INTERACTION MODEL

where γ̃ k is a Euclidean gamma matrix, (γ̃ k )† = −γ̃ k . Because of the finite range of theintegral, the Fourier transformation of fermions is discretized to the Matsubara frequencies,
ωn = (2n + 1)πT (n ∈ Z).
Next, we introduce a chemical potential, µ. According to the Noether’s theorem, a particle
number density is obtained as a conserved quantity from a global transformation, ψ(x) →eiα ψ(x) (α ∈ R): ψ̄(x)γ 0 ψ(x). A chemical potential is a quantity that is conjugate to aparticle number in thermodynamics. Hence, we can introduce a chemical potential into thethermal theory by replacing the action as follows,Z
SGN,T → SGN,T + µ

D−1

d

Z 1/T
x

dxD ψ̄(x)γ D ψ(x).

(2.53)

0

Combining the discussions at a finite temperature and at a finite chemical potential, andapplying the procedures described in Chapter 2.1 and 2.2, we obtain the effective potential in
the leading order of the 1/N expansion,
∞ Z
X

tr I
σ2
dD−1 k
2
2
2
−
T
ln
k
+
ω̃
+
σ
,
V (σ; T, µ) =µ,n2λ02 n=−∞ (2π)D−1(2.54)

with ω̃µ,n = ωn − iµ. The integral over in the continuous momentum space is replaced withthe summation from −∞ to ∞.
We perform the Matsubara sum. By using the useful identity related to the zeta function
regularization (2.17), the second term in Eq. (2.54) can be written as
tr I
lim
s→0 2

Z

dD−1 k d 1(2π)D−1 ds Γ (s)

Z ∞
dτ τ
0

s−1 −τ (k2 +σ 2 )

e

T

∞
X

2

e−τ ω̃µ,n .

(2.55)n=−∞After some calculations, we can confirm this term separated into terms come from the vacuum
state (not thermal), and the Fermi–Dirac distribution in finite temperature systems,(σ 2 )D/2
T tr I
−
−CD

D−1
D
(4π) 2 Γ D−1
2

Z ∞
dq q

D−2





√
 
−
q 2 +σ 2 +µ /T
ln 1 + e+ (µ → −µ) ,0(2.56)
where (µ → −µ) represents the term where µ is replaced by −µ in the previous term. As canbe seen from this expression, the contribution from a finite temperature does not include thedivergence. The Matsubara sum has been performed and the integral over the momentum
space remains. Thus, the effective potential at a finite temperature and a finite chemical
potential reads(σ 2 )D/2
1 1+ CD (D − 1) σ 2 µD−2−C
V (σ; T, µ) =
D
r
2 λr
D
 

√
 
Z ∞
T tr I−q 2 +σ 2 +µ /T
D−2
−
dq q
ln 1 + e+ (µ → −µ) .
D−10(4π) 2 Γ D−1
2(2.57)

20

CHAPTER 2. FOUR-FERMION INTERACTION MODEL

We have applied the same renormalization procedure (2.23) as in the model at zero temperature. This is because there is no divergence in the finite temperature term. The contribution
from a finite temperature appears to push down the effective potential strongly near the origin
and weakly away from the origin. As a result, the chiral symmetry gets restored continuouslywith increasing a temperature, and the second-order phase transition occurs.
To clarify the contribution from a finite chemical potential, we takes the zero temperature
limit. At zero temperature, the expression is given by(σ 2 )D/2
1 1+ CD (D − 1) σ 2 µD−2
− CD
V (σ; T → 0, µ) =
r
2 λr
D
Z ∞


p
p
tr I
dq q D−2 |µ| − q 2 + σ 2 θ(|µ| − q 2 + σ 2 ),
−

D−1
0
(4π) 2 Γ D−1
2(2.58)
where θ(x) is a step function defined by θ(x) = 1 for x > 0, and θ(x) = 0 for x < 0. From thisexpression, the contribution from a chemical potential appears to push the effective potential
down in a range, |σ| < µ, without modifications of the shape of the effective potential outside
the range, |σ| > µ. The chiral symmetry can be restored discontinuously when a chemical
potential becomes larger than a certain value (Shown in Fig. 2.10).
0.10

T/μr=0.2, chem/μr=0.0
T/μr=0.0, chem/μr=0.8
T/μr=0.6, chem/μr=0.0

0.05

V/μr

0.00
-0.05
-0.10
-0.15
-0.20
-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

σ/μr

Fig. 2.10: Behavior of the effective potential at a finite temperature and chemical potential.
A phase diagram on a µ-T plane plotted by using Eq. (2.57) in two dimensions is shownin Fig. 2.11. In this calculation, the coupling is fixed so that the generated fermion mass
normalized by µr is unity at zero temperature and chemical potential. A second-order phasetransition boundary is drawn in high temperature and low chemical potential area, and the line
ends at lower temperature and higher chemical potential; this point is called the critical end-21

CHAPTER 2. FOUR-FERMION INTERACTION MODEL

point. When the model becomes massive, the critical line disappears and a crossover region
takes place. A first-order phase transition boundary exists in the massless and massive theory.
The critical temperature where the chiral condensate vanishes at zero chemical potential iseγ /π ≃ 0.57 (on the vertical axis) and the chemical potential where the condensate jump from√a finite value to zero at zero temperature is 1/ 2 ≃ 0.71 (on the horizontal axis) [107, 108].
Similarly in three dimensions, the former value is 1/ ln 4 ≃ 0.72 and the latter is 1 [33, 109].
0.6
0.5

T/μr

0.4
0.3
0.2
0.1
0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Chem/μr

Fig. 2.11: Phase diagram on the µ-T plane in two dimensions. The solid and dashed lines
show the first-order and second-order phase transition boundaries respectively. The chiralsymmetry is broken inside the boundary and the chiral condensate vanishes outside.

22

Chapter 3

Finite-size EffectWe show finite-size effects on the dynamical chiral symmetry breaking by using a four-fermion
interaction model. In this chapter, we consider a situation at zero temperature in the GNmodel based on our paper [81] (The GN model is also used in the next chapter.). The background spacetime where we analyze the finite-size effects is a (D − 1)-dimensional Minkowski
spacetime with a one-dimensional sphere, MD−1 × S1 . To treat a theory with a finite-size
space, one usually needs to consider a boundary condition. We impose a U(1)-valued boundary
condition to fermions in the compactified direction,
ψ(x0 , x1 , . . . , xD−1 + L) = e−iπδ ψ(x0 , x1 , . . . , xD−1 ),(3.1)

with L denoting the size of S1 and δ a U(1) phase. In other words, going around the onedimensional sphere, the fermion gets the U(1) phase, δ: δ = 0 and 1 corresponding to periodic
and anti-periodic boundary conditions, respectively. The phase, δ, is a parameter that isdetermined by conditions outside of the system or a non-trivial topology of the early universe.
When we use the term finite-size effect, we include both contributions from a finite volume
and a boundary condition.

3.1

Effective Potential Analysis

To analyze the theory with the compactified space, we follow the manner already described in
Chapter 2.4 as treating a model at a finite temperature. This is because the imaginary time
direction at finite temperature is also compactified to S1 . The difference between the two liesin the boundary condition. To apply the following replacement in the effective potential at
finite temperature (2.54),

1


→
T
L

,
(3.2)
2π
δ


n+
ω̃µ,n → kδ,n =L
223CHAPTER 3. FINITE-SIZE EFFECT

we obtain the effective potential with the compactified space in the leading order of the 1/N
expansion,(σ 2 )D/2
1 1+ CD (D − 1) σ 2 µD−2
− CD
V (σ; L, δ) =
r
2 λr
D
Z ∞
(3.3)
√
h 
i
2 tr I
D−2−L q 2 +σ 2 +iπδ
−
dq
q
Re
ln
1
+
e
.

D−1
(4π) 2 Γ D−1
L 02
In the large size limit, L → ∞, this effective potential returns to Eq. (2.28) and reproduces thetheory without the compactified space. Because of the periodicity of the effective potential,
V (σ)|δ = V (σ)|δ+2nπ (n ∈ Z), it is enough to discuss the effective potential within the interval,
0 ≤ δ ≤ 1 or 0 ≤ δ ≤ 2.
In this massless model, it is convenient to use the solution of the gap equation at L → ∞instead of the renormalization scale, µr , to clarify finite-size effects. The expression of theeffective potential rewritten by using the generated fermion mass (2.30) reads"D/2 #V (σ; L, δ)
σ 2
1 σ2
(−1)α
−
=CD
mD
2
mα
D m2α
α


Z ∞
√
i
h 
dq
q D−2
2 tr I−L q 2 +σ 2 +iπδ
,
Re
ln
1
+
e
−

D−1
mα mα
0
(4π) 2 Γ D−1
Lm
α
2
(3.4)
where α is a parameter taking 0 or 1. m0 is the usual dynamical mass in the large size limit,
L → ∞, for the strong coupling, λr > λc,D , (shown in Fig. 2.4), and m1 is a mass parameter
defined by an absolute value of the solution of the gap equation (2.30) for the weak coupling,
λr < λc,D ; as a result, the coupling constant is removed form this expression. Because thecritical coupling is vanishing in two dimensions, λc,D=2 = 0, we can describe the dynamicalchiral symmetry breaking for D = 2 only by using m0 . Using this notation makes clear finitesize effects; in other words, if we observe chiral symmetry restoration (breaking) by using theeffective potential written in m0 (m1 ), we find that it is due to the finite-size effects. The
same applies to the situations at a finite temperature and a finite chemical potential (using
the next chapter).
In the large size limit, L → ∞ (Chapter 2), we have normalized the effective potential at
the origin to V (σ = 0) = 0. This is the usual way to remove the zero-point energy that isdivergent. However, when dealing with theories with a finite compactified space (as well as ata finite temperature), it is necessary to pay attention to how to remove the zero-point energy;
this is the improvement from [50]. By using some formulae [90, 91, 109, 110], the effective
potential with the finite size (3.4) at the origin, σ = 0, is reduced to


tr I
Γ (D) Re LiD eiπδ

,
(3.5)
V (σ = 0; L, δ) =
D−1
D+1
LD
(4π) 2 Γ 2P−s n
where Lia (x) is the polylogarithm function defined by Lis (z) = ∞n=1 n z . Thus, in thenormalization we have adopted, the value of the effective potential at the origin depends on

24

CHAPTER 3. FINITE-SIZE EFFECT

the size, L, and the boundary condition, δ. In the large size limit, L → ∞, or when the phase,δ, takes a certain value, the value of the effective potential at the origin becomes zero. Such aparameter dependence has a decisive contribution to the thermodynamic properties discussed
later.
The behavior of the effective potential is plotted in Fig. 3.1 for a fixed size, Lmα = 1.7.
Since the effective potential is axisymmetric, we show the region, σ ≥ 0. We can observethat the effective potential lifts up and the position of the minimum moves away from theorigin with varying the boundary condition form δ = 1 (anti-periodic) to δ = 0 (periodic) for
both the strong coupling and the weak coupling. When the value of the effective potential
at the origin is close to zero, this means that the finite-size effects is small. Therefore, thechiral symmetry tends to be restored near the anti-periodic boundary condition, δ = 1, andenhanced near the periodic one, δ = 0, for the fixed size, Lmα = 1.7. These results are
consistent with Ref. [34, 42, 43, 111].

25

CHAPTER 3. FINITE-SIZE EFFECT

0.4
0.3

δ=0

V/m0 D

0.2
0.1
0.0
- 0.1

δ = 0.4
δ = 0.6
δ =1

- 0.2
0.0

0.5

1.0

1.5

σ/m0

(a) D = 2, λr > λc,D (α = 0)
0.15

δ=0

δ=0

0.10

0.05

V/m1 D

V/m0 D

0.10

0.15

δ = 0.4
0.00

0.05
0.00

δ = 0.6- 0.05

δ = 0.4

δ = 0.6- 0.05
δ=1- 0.10
0.0

0.5

1.0

- 0.10
0.0

1.5

σ/m0

δ=1
0.5

1.0

1.5

σ/m1

(b) D = 3, λr > λc,D (α = 0)(c) D = 3, λr < λc,D (α = 1)

Fig. 3.1: The effective potential as a function of σ for Lmα = 1.7. The two figures on theleft are in the strong coupling, λr > λc,D (α = 0), and the figure on the right is in the weakcoupling, λr < λc,D (α = 1).

3.2

Generated Fermion Mass

The dynamically generated fermion mass, denoted by m, is given by the minimum of theeffective potential, V (σ; L, δ). As reviewed in Chapter 2.2, this is obtained as a non-trivial
solution of the gap equation,∂V (σ; L, δ)
= 0.
(3.6)
∂σ
σ=m
Since an exact solution can not be obtained, we evaluate the gap equation (3.6) numericallyand plot the behavior of the generated fermion mass as a function of the phase, δ, and thesize, L. In Fig. 3.2, the behavior of the generated fermion mass as a function of δ for the
fixed size, Lmα = 1.7. For the strong coupling, λr > λc,D , it is seen that the generated

26

CHAPTER 3. FINITE-SIZE EFFECT

fermion mass monotonically decreases as the phase approaches one, δ → 1; in particular, thechiral symmetry gets restored as the second-order phase transition in two dimensions. Near
the periodic boundary condition, the chiral symmetry breaking is enhanced (m/m0 > 1), andeven for the weak coupling, λr < λc,D , observed in the narrow regions.
0.30

1.2

0.25

1.0

m/m1

m/m0

D= 3

0.6

0.15
0.10

0.4
0.2
0.0
0.0

D= 3

0.20

0.8

0.05

D= 2
0.5

1.0

1.5

2.0

0.00
0.0

0.5

1.0

1.5

2.0

δ

δ

(a) λr > λc,D (α = 0), Lm0 = 1.7(b) λr < λc,D (α = 1), Lm1 = 1.7

Fig. 3.2: The dynamically generate fermion mass as a function of δ (left: strong coupling,
and right: weak coupling). The blue solid curves is in two dimensions, and the orange dashedcurve in three dimensions.
In Fig. 3.3, the behavior of the generated fermion mass is shown as a function of L. At theperiodic boundary condition, δ = 0, the dynamical mass increases with decreasing the size,
L, and the chiral symmetry breaking is enhanced; as a result, the dynamical mass diverges
in the small size limit, L → 0. In contrast, at the anti-periodic boundary condition, δ = 1,
the finite-size effect restores the broken chiral symmetry as the second-order phase transition.
At δ = 0.4 for the strong coupling, the behavior of the generated mass indicates features ofboth the periodic and anti-periodic boundary conditions. In this case, the generated fermion
mass converges to 1 from above in the large size limit, m/m0 → 1+ for L → ∞, but the chiralsymmetry is restored in the small size limit, m/m0 → 0 for L → 0.

27

CHAPTER 3. FINITE-SIZE EFFECT

2.0
δ=0

m/m0

1.5
δ = 0.4
1.0

δ=1

0.5

0.0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

Lm0

(a) D = 2, λr > λc,D (α = 0)
2.0

2.0

δ=0

1.5

1.5
δ=0

m/m1

m/m0

δ = 0.4
1.0

0.5

0.0
0.0

0.5

δ=1

0.5

1.0

1.5

1.0

2.0

2.5

3.0

Lm0

δ = 0.2

δ = 0.4
0.0
0.0
0.5

1.0

1.5

2.0

2.5

3.0

Lm1

(b) D = 3, λr > λc,D (α = 0)(c) D = 3, λr < λc,D (α = 1)

Fig. 3.3: The dynamically generated fermion mass as a function of L. The two figures on
the left are in the strong coupling, λr > λc,D (α = 0), and the figure on the right is in the
weak coupling, λr < λc,D (α = 1).
Since the order of the phase transition is second order, the phase boundary is obtainedin the chiral limit, m → 0, of the gap equation, in the same way as Eq. (2.26). The criticallength, denoted by Lcr , that divides the symmetric and broken phases is given by
 "
# 1


D−2

2(−1)α Γ 3−D

2


√

2π
ζ
(3
−
D)
(δ = 0, 2)
π Γ 1− D2,
Lcr mα =
1
"

 


# D−23−Dα(−1) Γ 2δδ(0 < δ < 2)
2π √π Γ 1 − D  ζ 3 − D, 2 + ζ 3 − D, 1 − 22(3.7)
where ζ (s) is the Riemann zeta function, and ζ (s, a) is the Hurwitz zeta function defined byP−sζ (s, a) = ∞n=0 (n + a) : ζ (s, 1) = ζ (s). At the anti-periodic boundary condition, δ = 1,28

CHAPTER 3. FINITE-SIZE EFFECT

this formula coincides with the one of the critical temperature that is derived in Ref. [109]:
Lcr m0 ≃ 1.76 for (D, δ) = (2, 1), and Lcr m0 = ln 4 ≃ 1.39 for (D, δ) = (3, 1).
By using the critical length (3.7), the phase diagram is plotted on a δ-L plane in Fig. 3.4.
For the strong coupling, λr > λc,D , the chiral symmetric phase can be observed around theanti-periodic boundary condition, δ = 1, while only the broken phase is observed at theperiodic boundary condition, δ = 0. For the weak coupling, λr < λc,D , the chiral symmetryis broken near the periodic boundary condition, and the critical length is divergent at δ = 0
and 2 in three dimensions. As long as the size is finite, L < ∞, the chiral symmetry is always
broken at the periodic boundary condition. It can be also observed that the finite-size effect ismore suppressed, and the critical length is smaller in three dimensions than two dimensions.
2.0

2.0
D= 2

Broken1.5

1.0

Lcr m1

Lcr m0

1.5

D= 3

0.5

1.0

Symmetric

D= 3

0.5
Symmetric

0.0
0.0

0.5

1.0

1.5

2.0

δ

0.0
0.0

0.5

1.0

1.5

2.0

δ

(a) λr > λc,D (α = 0)(b) λr < λc,D (α = 1)

Fig. 3.4: The phase diagram on a δ-L (left: strong coupling, and right: weak coupling).
The chiral symmetry is restored below the curves for λr > λc,D , and between the curves forλr < λc,D .

3.3

Casimir EffectTo consider a finite-size effect in the four-fermion interaction model on MD−1 ×S1 , we evaluate
a response for the size given by the first derivative of the effective potential with respect toL,∂V (σ; L, δ),(3.8)
T (L, δ) = −∂L
σ=mwhere m is the fermion mass that is given by the minimum of the effective potential, V (σ; L, δ).
T (L, δ) is a string tension. Because the effective potential is a thermodynamic potential
density, the string tension is also a thermodynamic quantity. By using this thermodynamic
quantity, we discuss the phase transition and the stability of the system.

29

CHAPTER 3. FINITE-SIZE EFFECT

The explicit expression of the string tension reads


Z ∞
dq
2 tr I
q D−2
T (L, δ)
=−

D−1
mD+1
α
(Lmα )2 0 mα mα(4π) 2 Γ D−1
2
"
#
p
√


2 + m2Lq
2
2√× Re ln 1 + e−L q +m +iπδ +.
22
1 + eL q +m −iπδ

(3.9)

In Fig. 3.5, the behavior of the string tension is shown as a function of the phase, δ, for a
fixed length, Lmα = 1.7. For the strong coupling, λr > λc,D , sharp bends appear on the blue
curve (in two dimensions) near δ = 1. These sharp bends are observed at the same boundary
conditions where the chiral phase transition occurs in Fig. 3.2. On the other hand, it is shownthat the string tension is repulsive at the periodic boundary condition, δ = 0 and 2, and thetension monotonically decreases as approaching the anti-periodic boundary condition, δ = 1.
During this decreasing, the sign of the tension, T (L, δ), changes near δ ∼ 0.5. An attractive
string tension is realized near the periodic boundary condition.
0.2
0.1

0.0

- 0.1

D= 3

- 0.2

D= 2

T/m1 D+1

T/m0 D+1

0.1

0.0

- 0.1

0.0

0.5

1.0

1.5

2.0

δ

D= 3

0.0

0.5

1.0

1.5

2.0

δ

(a) λr > λc,D (α = 0), Lm0 = 1.7(b) λr < λc,D (α = 1), Lm1 = 1.7

Fig. 3.5: The string tension as a function of δ for a fixed length Lmα = 1.7 (left: strong
coupling, and right: weak coupling).
Next, we plot the behavior of the string tension as a function of the size, L, in Fig. 3.6.
This string tension diverges and vanishes in the small and large size limit, L → 0 and L → ∞,respectively. At the periodic (anti-periodic) boundary condition, δ = 0 (δ = 1), the tension
remains repulsive (attractive) and monotonically decreases (increases) with increasing thesize, L. A sharp bend appears on the orange curve (δ = 0.4) in Fig. 3.6 (a). From comparison
with Fig. 3.3, this bend exhibits the phase transition. The sign flip seen in Fig. 3.5 is notobserved in Fig. 3.6.

30

CHAPTER 3. FINITE-SIZE EFFECT

1.0

0.5
δ=0

T/m0 D+1

δ = 0.4
0.0

δ=1

- 0.5

- 1.0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

Lm0

(a) D = 2, λr > λc,D (α = 0)
1.0

0.5

δ=0

δ = 0.4

T/m1 D+1

T/m0 D+1

0.5

1.0

0.0

- 0.5

- 1.0
0.0

1.0

1.5

δ=0

0.0

δ=1

- 0.5

δ=1

0.5

δ = 0.4

2.0

2.5

3.0

Lm0

- 1.0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

Lm1

(b) D = 3, λr > λc,D (α = 0)(c) D = 3, λr < λc,D (α = 1)

Fig. 3.6: The string tension as a function of L. The two figures on the left are in the strong
coupling, λr > λc,D (α = 0), and the figure on the right is in the weak coupling, λr < λc,D
(α = 1).
In order to clarify the sign-flip points of the string tension, we numerically solve T (L, δ) =0. The results of this numerical calculation are plotted in Fig. 3.7. Since the sign-flip boundaries appear in limited ranges of the phase, 0.4 < δ < 0.5 and 1.5 < δ < 1.6, we show only therange, 0.4 < δ < 0.5. The string tension is repulsive and attractive on the left and right sideof the curve, respectively. For the strong coupling, λr > λc,D , the sign-flip boundary asymptotically approaches δ = 0.5 as the size increases. Referring the phase diagram (Fig. 3.4), thesystem with the compactified space larger than Lm0 ≃ 0.5 is in the broken phase, and thisbehavior of the sign-flip boundary is caused by the generated fermion mass; in other words,
the generated fermion mass extends the domain where the repulsive tension is induced. For
Lm0 ≲ 0.5, the system is in the chiral symmetric phase, and then the boundary is represented
by a vertical line. In the symmetric phase, the expression of the string tension (3.9) can be

31

CHAPTER 3. FINITE-SIZE EFFECT

D

2

3

δ

0.423

0.462

Tab. 3.1: The phase δ where the sign of the string tension flips in the symmetric phase.
reduced to
T (L, δ)sym
tr I Γ (D + 1) Re LiD eiπδ.
=(4π)D−1 Γ D+1(Lmα )D+1mD+1
α2(3.10)

From this expression, it is clear that the solution of T (L, δ)sym = 0 is independent of L aslong as L is finite, and given byhi
Re LiD eiπδ = 0.
(3.11)The specific value of the sigh-flip phase is shown in Tab. 3.1.
For the weak coupling, λr < λc,D , the sign-flip boundary for arbitrary Lm1 (< ∞) is
determined by the equation (3.11) because, as can be seen from Fig. 3.4, the system is in thechiral symmetric phase.
Through the discussions of the string tension, it is found that the string tension is only
repulsive (attractive) near the periodic (anti-periodic) boundary condition. Therefore, nearboth the boundary conditions, the compactified space can not remains finite. While the signflip point exists at the intermediate boundary condition, 0.4 < δ < 0.5, as we have shown, thispoint does not means a stable size. The stable size we call is the point where the string tension
becomes negative with increasing the system size and positive with decreasing the system size.
This is because when the size of the compactified space becomes larger (smaller) than thispoint, the tension becomes repulsive (attractive). Here, let us go back to the discussion in
Chapter. 3.1. At the periodic (anti-periodic) boundary condition, the effective potential at a
finite size is larger than the one in the large (small) limit, L → ∞ (L → 0), which is consistent
with the string tension being repulsive (attractive).

32

3.0

3.0

2.5

2.5

2.0

2.0
Repulsive

1.5

Lm1

Lm0

CHAPTER 3. FINITE-SIZE EFFECT

1.0

Repulsive

1.5
1.0

D= 2

D= 3

0.5
0.0
0.40

0.42

0.44

0.46

D= 3
Attractive

0.48

0.5

0.50

δ

0.0
0.40

0.42

0.44

0.46

Attractive

0.48

0.50

δ

(a) λr > λc,D (α = 0), Lm0 = 1.7(b) λr < λc,D (α = 1), Lm1 = 1.7

Fig. 3.7: The sign-flip boundaries between the repulsive (left side of the curves) and attractive(right side of the curves) tension on a δ-L plane.

We have discussed dynamical chiral symmetry breaking in the GN model on MD−1 × S1
with the U(1)-valued boundary condition. Assuming the homogeneous chiral condensate andusing the zeta function regularization, we have obtained the explicit expression of the effective
potential for the vacuum expectation value of the fermion and anti-fermion composite field in
the leading order of the 1/N expansion.
In particular, we have focused on the topological effect stemming from the boundary
condition at zero temperature. We calculated the dynamically generated fermion mass as afunction of L and δ, and checked that only the second-order phase transition occurs. The
phase diagram has shown on a δ-L plane for the strong and weak couplings in Fig. 3.4.
To find a phenomenological consequence, we has discussed a string tension as the Casimir
effect in the model on MD−1 × S1 . The critical lengths for chiral symmetry breaking are
observed as slightly sharp bends of the string tension. Moreover, we have found the explicit
expression for the sign-flip points in the symmetric phase, and the boundary lines dividing
the repulsive tension and attractive tension on the L-δ plane, Fig. 3.7. The phase transition
is observed as sharp bends in Fig. 3.7 (a). However, following our analysis, there is no stablesize for the string tension in the four-fermion interaction model at zero temperature.

33

Chapter 4

Thermal and Finite-size Effects
In the preceding chapter, we have discussed finite-size effects in the GN model at zero temperature. We here consider finite-size effects in the GN model at a finite temperature and afinite chemical potential.

4.1

Effective Potential Analysis

Let us derive the effective potential in the leading order of the 1/N expansion on a Ddimensional background spacetime, RD−2 × T2 , a torus T2 ≃ S1 × S1 , with a finite-size spaceat a finite temperature. The formulation at a finite temperature and a finite chemical potential
is based on the discussion in Chapter 2.4.
The action of the model is


Z 1/T
Z
Z L
λ0
dxD ψ̄(x)γ̃ k (∂k + µδkD )ψ(x) −SGN,L,T = dD−2 x
dxD−1(ψ̄(x)ψ(x))2 ,
2N
0
0
(4.1)
with the U(1)-valued boundary condition in the compactified direction,
ψ(x1 , . . . , xD−1 + L, xD ) = e−iπδ ψ(x1 , . . . , xD−1 , xD ),(4.2)

the anti-periodic boundary condition in the imaginary time direction,
ψ(x1 , . . . , xD−1 , xD + 1/T ) = −ψ(x1 , . . . , xD−1 , xD ),(4.3)

and a chemical potential, µ. Applying the same procedures described in Chapter 2.1 and 2.2,
we obtain the effective potential in the leading order of the 1/N expansion,
σ2
tr I T−
V (σ; L, δ, T, µ) =
2λ0
2 L

∞
X
n,n′ =−∞

Z
dD−2 k2
22
ln k2 + kδ,n+ ω̃µ,n
.
′ + σ
D−2
(2π)

(4.4)

There are two sums: quantized momenta, kδ,n , on the compactified direction and the Matsubara frequencies, ω̃µ,n′ , on the imaginary-time direction.

34

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

By using the identity related to the zeta function regularization (2.17) again, the second
term in this expression (4.4) is separated into a non-thermal part and a thermal part by
performing the Matsubara sum. We takes the sum over the momentum space in the nonthermal term, while the sum in the thermal term are left. The expression of the effective
potential on RD−2 × T2 reads
V (σ; L, δ, T, µ)(σ 2 )D/2
1 1+ CD (D − 1) σ 2 µrD−2 − CD
=
2 λr
D
Z ∞
√
h

i
2 tr I22−
dq q D−2 Re ln 1 + e−L q +σ +iπδC̃D+1 L 0
q
 
 

∞ Z ∞
2 +σ 2 +µ /T
tr I T X−
q 2 +kδ,n
D−3
−
dq q
ln 1 + e+ (µ → −µ) ,C̃D L n=−∞ 0(4.5)

with a constant, C̃D = (4π)D/2−1 Γ (D/2 − 1). In this expression, the coupling of the fourfermion interaction is also renormalized by the same procedure (2.23) since these extensions
with L, δ, T , and µ have nothing to do with the UV divergence. The second term represents
the contribution from the vacuum, the third term the pure contribution from finite size, andthe last term the contribution from finite size, a finite temperature, and a finite chemical
potential. The first through third terms are the same as the effective potential with thecompactified space (3.3). We can confirm that this effective potential (4.5) reproduces theone at a finite temperature and a finite chemical potential (2.57) in the large size limit,
L → ∞, and the one with the compactified space (3.3) at zero temperature and zero chemical
potential, T = µ = 0.
We are interested in the situation in which the chiral symmetry is broken in the large
size limit, L → ∞. As is mentioned in Chapter 3.1, the normalized coupling, λr , and thenormalization scale, µr , can be removed from Eq. (4.5) by using the generated fermion mass,
m0 , on MD . The expression (4.5) is simplified toV (σ; L, δ, T, µ)
mD
" 0


D/2 #
1 σ 2
1 σ2
=CD
−
2 m0
D m20


Z ∞
√
h 
i
dq
q D−2
2 tr I22−
Re ln 1 + e−L q +σ +iπδC̃D+1 Lm0 0 m0 m0
 
q


 

∞ Z ∞
X
2 +σ 2 +µ /T
tr I T m−1
dq
q D−3
−
q 2 +kδ,n0−
ln 1 + e+ (µ → −µ) .
C̃D Lm0 n=−∞ 0 m0 m0(4.6)
In the following, we numerically evaluate this effective potential (4.6) in two and three dimensions in the framework of the dimensional regularization. In Fig. 4.1, the typical behaviorof the effective potential is shown as a function of σ/m0 for a fixed size, Lm0 = 1.5, at the35

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

periodic boundary condition, δ = 0; only positive are is shown due to the axisymmetric. Forµ/m0 = 0 (left in Fig. 4.1), the minimum is located at σ/m0 > 1 by the finite-size effect at
δ = 0, and smoothly approaches the origin, σ/m0 = 0, by the thermal effect with increasing
T /m0 . For µ/m0 = 0.7 and 0.9 (right in Fig. 4.1), the other minimum appears at the origin
for T /m0 = 0.1 and 0.2. As mentioned in Chapter 3.1, we do not normalized the effective
potential as V (σ = 0; L, δ, T, µ) = 0. Our normalization makes it easy to regard the minimumof the effective potential as a thermodynamic potential.
0.08

0.35

0.06

0.3

0.04

0.25

0.02

0.2

V/m02

V/m02

0.4

0.15

0

-0.02

0.1

-0.04

0.05

-0.06

0

-0.08

-0.05

0

0.2

0.4

0.6

0.8

σ/m0

1

1.2

1.4

-0.1

1.6

0

(a) D = 2, Lm0 = 1.5, µ/m0 = 0

0.2

0.4

0.6

0.8

σ/m0

1

1.2

1.4

1.6

(b) D = 2, Lm0 = 1.5, µ/m0 = 0.7

0.16

0.06
0.05

0.14

0.04
0.12

0.03
0.02

V/m03

V/m03

0.1
0.08
0.06

0.01
0

-0.01
-0.02

0.04

-0.03
0.02
0

-0.04
0

0.2

0.4

0.6

0.8

σ/m0

1

1.2

1.4

-0.05

1.6

(c) D = 3, Lm0 = 1.5, µ/m0 = 0

0

0.2

0.4

0.6

0.8

σ/m0

1

1.2

1.4

1.6

(d) D = 3, Lm0 = 1.5, µ/m0 = 0.9

Fig. 4.1: The behavior of the effective potential for each chemical potential (above: D = 2,
and below: D = 3) for Lm0 = 1.5 and δ = 0 (purple: T /m0 = 0.1, green: T /m0 = 0.2, andlight blue: T /m0 = 0.4).

4.2

Phase Structure

An order parameter of chiral symmetry breaking is given by the expectation value of thecomposite operator of the fermion and anti-fermion, ψ̄ψ . As long as we consider the mass-36

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

less theory, the quantity is proportional to the auxiliary field, σ, in the ground state. One
finds this state by evaluating the global minimum of the effective potential that satisfies
V (m; L, δ, T, µ) ≤ V (σ; L, δ, T, µ) for any σ; the value of the minimum, m, is equivalent to a
generated fermion mass. On the other hand, the local minima of the effective potential implies
the existence of metastable states. The metastable states as well as the ground state has somephenomenological consequences. We assume a spatially homogeneous ground state as sameas the previous sections and evaluate the number of extrema. In that sense, our results do
not directly indicate the presence of the inhomogeneous phase, but the ground state can beinhomogeneous in the existence of multiple minima in a certain region of a parameter space.

Phase diagrams
In order to find a precise phase structure of the GN model, we evaluate not only the global
minima but also the local minima of the effective potential, and hence we plot bifurcation
diagrams. Since we consider the massless theory, the effective potential is an even function ofσ. We classify the states by (i) the number of extrema and (ii) the position of the minimumof the effective potential for σ ≥ 0. In this chapter, the class of the state is described by two
symbols that we introduce: ab S and ab B. The former and the latter denote a symmetric phaseand a broken phase, respectively. The superscript a is the number of extrema for σ ≥ 0 andthe subscript b is the number of the extrema from the origin to the minimum, i.e. the bth
extremum is the minimum and b ≤ a. As an example, we show the phase diagram on the µ-Tplane divided into four areas for D = 3, Lm0 = 8.0 and δ = 1.0. The correspondences between
the symbol and behavior of the effective potential are indicated in Fig. 4.2. The symmetricphase 11 S corresponding to (s) as the behavior of the effective potential, is realized at a hightemperature and chemical potential, outside the outer boundary. On the outer boundary, only
a second-order phase transition takes place. The broken phase, inside the outer boundary,contains three areas identified by the by the number of the extrema and the position of theminimum. A jump of the generated fermion mass between finite values is observed on theboundary between (b1), 44 B, and (b2), 42 B.

37CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

(b0)

2B
2

(b2)

4B
2

(b1)

(b0’)

4B
4

2B
2

(b0’)
(s)

(b2)
(b0) (b1)

(s)

1S
1

Fig. 4.2: Phase diagram on the µ-T plane (D = 3.0, Lm0 = 8.0 and δ = 1.0) and behavior of
the effective potential along the dashed line (T /m0 = 0.03). The arrows indicate the positionof the minimum.
Although we can also treat intermediate value boundary conditions, we analyze the phasestructure at the periodic (δ = 0) and anti-periodic (δ = 1) boundary conditions for simplicity.
The phase diagrams on the µ-T plane for some fixed sizes and boundary conditions in Figs.
4.3 (D = 2) and 4.4 (D = 3); the bottom figures are plotted at a larger size than the top
figures. As is mentioned in Chapter 2.4, it is known that, on usual non-compactified spaces
R and R2 , the critical temperatures at zero chemical potential are eγ /π ≃ 0.57 (D = 2) and1/ ln 4 (D = 3), and the chemical potentials where the generated fermion mass jumps at zero√temperature are 1/ 2 ≃ 0.71 (D = 2) and 1 (D = 3) with the normalization by m0 . The
finite-size effects primarily appear at a low temperature with a high chemical potential, anda high temperature with a low chemical potential. The effects of the boundary conditions aresuppressed for a large size, but at a certain size prominently appear.
For instance, in two dimensions at Lm0 = 8.0 and δ = 0.0 (Fig. 4.3, bottom left), we canobserve the complex behavior of the boundaries. As the chemical potential increases at a lowtemperature, the domain changes 44 B → 55 B → 53 B → 51 S → 31 S → 11 S and mass jumps appear
twice on the boundaries 55 B → 53 B and 53 B → 51 S.

38

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS
0.6

0.6

0.5

0.5

T/m0

2B
2

0.3

0.4

1S
1

0.3

0.2

0.2

0.1

0.1

0

3B
3
0

0.2

0.4

3S
1
0.6

μ/m0

0.8

1

1.2

0

1.4

0.6

0.6

0.5

0.5

T/m0

2B
2

0.3

3B
3
5B
3
@
5B @
3S
5
1
c
5B
c
5 PPc

0.2
0.1
0

0

0.2

4B
4 XX
0.4

μ/m0

0.6

c
P
P

0.8

3B
3
0

0.4

1S
1

1S
1

3S
1
0.2

0.4

0.6

μ/m0

0.8

0.3

1

1.2

1.4

1S
1

2B
2

T/m0

0.4

2B
2

T/m0

0.4

0.2
0.1

5S
1

0

1

0

0.2

0.4

3B
3

3S
1

0.6

0.8

μ/m0

1

Fig. 4.3: Phase diagrams in 2 dimensions on a µ-T plane (above: Lm0 = 3.0, below:
Lm0 = 8.0) for δ = 0.0 (lefts) and δ = 1.0 (rights).
The basic properties are common between two and three dimensions. Their differences
come from the continuous momentum in the additional non-compactified space, R. At
Lm0 = 4.0 and δ = 0.0 in three dimensions (Fig. 4.4, bottom left), only the second-orderphase transition takes place, and no critical end-point appears on the boundary dividing thesymmetric and the broken phases. In the broken phase, the generated fermion mass discontinuously changes on the boundary between the areas, 44 B and 42 B.

39CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS
0.8

0.8

0.7

0.7

0.6

0.6

T/m0

2B
2

0.4

0.5

1S
1

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

3B
3
0

0.2

0.4

3S
1
0.6

0.8

μ/m0

1

1.2

1.4

1.6

0

1.8

0.8

0.8

0.7

0.7

0.6

0.6

T/m0

2B
2

0.4

0

0.2

0.4

0.6

0.8

μ/m0

1

1.2

1.4

1.6

1.8

1S
1

2B
2

0.4

0.3

0.3

0.2

0.2

4B
2

0.1
0

3B
3

0.5

1S
1

1S
1

3S
1

T/m0

0.5

2B
2

T/m0

0.5

4B
4
0

0.2

0.4

0.6

μ/m0

3

0.8

! 1S
!
1

3S
1

0.1
0

1.2

3B
3
0

0.2

0.4

0.6

μ/m0

0.8

1

1.2

Fig. 4.4: Phase diagrams in 3 dimensions on a µ-T plane (above: Lm0 = 2.0, below:
Lm0 = 4.0) for δ = 0.0 (lefts) and δ = 1.0 (rights).
By observing the phase diagrams on the µ-T plane, we can recognize the differences thatemerge via the size of the system and the boundary conditions. In order to determine thesize dependence of the chiral symmetry, we next plot the phase structure on a L-T plane.
The phase diagrams on the L-T plane are shown in Figs. 4.5 (D = 2) and 4.6 (D = 3);
the bottom figures are plotted at a larger size than the top figures. The finite size effectat T = µ = 0 enhances the chiral symmetry breaking for the periodic boundary condition,and suppresses the symmetry breaking for the anti-periodic one [81]. However, it is also
observed that these relations are reversed for certain intervals of size at a finite temperatureand chemical potential. For low temperatures, the broken and the symmetric phases alternatewith increasing size of the system, L.

40

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS
0.5

0.5

1S
1

0.4

0.3

0.3

2B
2

T/m0

T/m0

2B
2

0.2

0.2

0.1

0

1S
1

0.4

0.1

3B
3
0

2

4

6

Lm0

8

10

0

12

0.5

3B
3
0

2

4

6

Lm0

8

10

12

0.5

0.4 2

1S
1

2B

1S
1

0.4

T/m0

0.3

T/m0

0.3

3B
3

0.2

5B
5

0.1

0

3B
3
0

0.1

5S
1

3S
1
2

4

6

Lm0

8

10

3B
3

0.2

0

12

2B
2
0

2

3S
1

3B
3
4

6

Lm0

8

10

12

Fig. 4.5: Phase diagrams in 2 dimensions on a L-T plane (above: µ/m0 = 0.5, below:µ/m0 = 0.7) for δ = 0.0 (lefts) and δ = 1.0 (rights).
0.5

0.5

0.4 2

1S
1

2B

0.4 1

1S

0.3

0.3

2B
2

T/m0

T/m0

2B
2

0.2

0.2

4B
2
0.1

0

0.1

3B
3
0

3S
1
2

4B
4
4

6

Lm0

8

10

0

12

0.5

0

2

4

6

Lm0

8

10

12

0.5

1S
1

0.4 2

2B

0.4 1

1S

T/m0

0.3

T/m0

0.3

0.2

0.2

3B
3
3S
1

0.1

0

0

2

2B
2
3B
3
4

6

Lm0

8

3S
1
10

2B
2
4B
2

0

12

4B
2

2B
2

0.1

3B
3
0

2

4

Lm0

5S
1

3B
3

3S
1

2B

@ 2
3S
@
@
1
6

5B
3

8

10

12

Fig. 4.6: Phase diagrams in 3 dimensions on a L-T plane (above: µ/m0 = 0.5, below:µ/m0 = 0.7) for δ = 0.0 (lefts) and δ = 1.0 (rights).

41

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

Dynamically generated fermion mass
The dynamically generated fermion mass, m, satisfies the gap equation,∂V (σ; L, δ, T, µ)
= 0.
∂σ
σ=m

(4.7)

1.4

1.4

1.2

1.2

1

1

0.8

0.8

m/m0

m/m0

We show the generated fermion mass as a function of the length of S1 , L, and the U(1) phase,
δ, in Figs. 4.7 (D = 2) and 4.8 (D = 3). The dashed black line (m/m0 = 1) indicates thevalue in the trivial condition (L → ∞, T → 0 and µ = 0). For µ = 0 (purple lines) thegenerated fermion mass at δ = 0.0 is heavier than that at δ = 1.0. In the two graphs on
the top of Figs. 4.7 and 4.8, the generated fermion mass vanishes and the chiral symmetry is
restored around the periodic boundary condition (δ = 0.0) slightly below the critical chemical
potential.
This situation depends on the size, L. Observing the generated fermion mass as a functionof L (green and yellow lines), we see that the broken and symmetric phases alternate withincreasing length of S1 , L. It is consistent with the phase diagrams, Figs. 4.5 and 4.6. The
generated fermion mass changes more smoothly in three dimensions than in two dimensions,because of the continuous momentum.

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.5

1

δ

1.5

0

2

0

(a-1) Lm0 = 3.0

1

δ

1.5

2

15

20

(a-2) Lm0 = 5.0

1.4

1.4

1.2

1.2

1

1

m/m0

1.6

m/m0

1.6

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0.5

0

5

10

Lm0

15

0

20

(b-1) δ = 0.0

0

5

10

Lm0

(b-2) δ = 1.0

Fig. 4.7: Dynamically generated fermion mass as a function of δ (above) and L (below) in two
dimensions: (T /m0 , µ/m0 ; color) = (0.005, 0.0; purple), (0.005, 0.7; green), (0.1, 0.7; yellow).

42

1.4

1.4

1.2

1.2

1

1

0.8

0.8

m/m0

m/m0

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.5

1

δ

1.5

0

2

0

(a-1) Lm0 = 2.0

1

δ

1.5

2

15

20

(a-2) Lm0 = 4.0

1.4

1.4

1.2

1.2

1

1

m/m0

1.6

m/m0

1.6

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0.5

0

5

10

Lm0

15

0

20

(b-1) δ = 0.0

0

5

10

Lm0

(b-2) δ = 1.0

Fig. 4.8: Dynamically generated fermion mass as a function of δ (above) and L (below) in three dimensions: (T /m0 , µ/m0 ; color) = (0.005, 0.0; purple), (0.005, 1.0; green),(0.2, 1.0; yellow).

4.3

Thermodynamic Quantities

In Chapter 4.1 and 4.2, we have evaluated the effective potential and plotted the phase
diagrams on the µ-T and L-T plane and the dynamically generated fermion mass. Other
thermodynamic quantities are also derived from the effective potential. We here calculate
and discuss particle number density and pressure.

Grand potential
A minimum value of the effective potential can be naively regarded as the density of the grandpotential, denoted by Ω(L, δ, T, µ), as is mentioned at the last in Chapter 4.1. We set the valueof the grand potential to zero in the trivial conditions: an infinite volume, zero temperatureand zero chemical potential with a homogeneous and non-vanishing chiral condensate. The
definition of the grand potential that wee consider is given byΩ(L, δ, T, µ) = V (m; L, δ, T, µ)LV − V (m0 LV,

43

(4.8)

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

where V (σ) denotes the effective potential on the trivial spacetime (derived in Chapter 2.2)
and LV is (D − 1)-dimensional volume.
The grand potential is shown in Figs. 4.9 (D = 2) and 4.10 (D = 3) as a function of theboundary condition, δ, and the length of S1 , L. Bends of a line are observed at the same
parameters at which the fermion mass jumps apper in Figs. 4.7 and 4.8. Thus, the phaseboundary is found by observing the sharp bends if Figs. 4.9 and 4.10.
A stable size is a state for which the pressure is zero and becomes negative (positive) withincreasing (decreasing) a length. We find the existence of a stable size for a finite chemical
potential at a low temperature. For a finite chemical potential (green line), it is observed thatthe grand potential at δ = 0.0 is minimized at Lm0 ∼ 4.0. This size is realized in the chirally
symmetric phase. This stable state disappears at µ = 0, because the grand potential (purpleline) is a monotonic function of L and divergent at the small L limit.

0.02

0.02

0

0

-0.02

-0.02

Ω2/m0

0.04

Ω2/m0

0.04

-0.04

-0.04

-0.06

-0.06

-0.08

-0.08

-0.1

-0.1

-0.12

0

0.5

1

δ

1.5

-0.12

2

0

(a-1) Lm0 = 3.0

1

δ

1.5

2

15

20

(a-2) Lm0 = 5.0

0

0

Ω2/m0

0.05

Ω2/m0

0.05

-0.05

-0.05

-0.1

-0.15

0.5

-0.1

0

5

10

Lm0

15

-0.15

20

(b-1) δ = 0.0

0

5

10

Lm0

(b-2) δ = 1.0

Fig. 4.9: Grand potential as a function of δ (above) and L (below) in two dimensions:(T /m0 , µ/m0 ; color) = (0.005, 0.0; purple), (0.005, 0.7; green), (0.1, 0.7; yellow).

44

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

0.04

0.04

0.02

0.02

0

0

Ω3/Vm020.06

Ω3/Vm02

0.06

-0.02

-0.02

-0.04

-0.04

-0.06

-0.06

-0.08

0

0.5

1

1.5

δ

-0.08

2

0

(a-1) Lm0 = 2.0

1

δ

1.5

2

15

20

(a-2) Lm0 = 4.0

0

0

-0.02

-0.02

Ω3/Vm02

0.02

Ω3/Vm02

0.02

-0.04

-0.04

-0.06

-0.06

-0.08

-0.08

-0.1

0.5

0

5

10

15

Lm0

-0.1

20

0

5

(b-1) δ = 0.0

10

Lm0

(b-2) δ = 1.0

Fig. 4.10: Grand potential as a function of δ (above) and L (below) in three dimensions:(T /m0 , µ/m0 ; color) = (0.005, 0.0; purple), (0.005, 1.0; green), (0.2, 1.0; yellow).

Particle number density
To investigate the origin of the complex behavior of the phase diagrams, we study the contribution of the chemical potential. The particle number density is defined by the derivative of
the grand potential with respect to the chemical potential,ρ (L, δ, T, µ) = −1 ∂Ω (L, δ, T, µ)
.
VL
∂µ

(4.9)

The explicit expression normalized by m0 is given by=ρ (L, δ, T, µ)
m0D−1


∞ Z ∞
X
tr I 1
dq
q D−3
C̃D Lm0 n=−∞

0

m0

m0

sinh µ/Tq.
2 + m2 /T + cosh µ/Tcosh q 2 + kδ,nFor instance, the expression in two dimensions reads∞X
ρ2 (L, δ, T, µ)1sinh µ/Tq=.
m0
Lm0 n=−∞ cosh k 2 + m2 /T + cosh µ/Tδ,n45(4.10)

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

We show the numerical results in Fig. 4.11 (D = 2)as a function of the boundary conditions,δ, and the length of S1 , L. In the limit T → 0, the each term of the sum is modified asqsinh µ/T
2 + m2 .
q−→ sgn(µ)θ |µ| − kδ,n(4.11)
2 + m2 /T + cosh µ/Tcosh kδ,n2 , except for the periodic boundary condition. ThisA non-zero lower bound appears for kδ,nimplies that the particle number, ρ2 (L, δ, T, µ) L, takes an integer value at zero temperature.
Because of the degeneracy of the states, the possible values are restricted to 0, 1, 3, 5, . . . forδ = 0.0 and 0, 2, 4, . . . for δ = 1.0. It is found that the correspondences between Figs. 4.7 (thegenerated fermion mass) and 4.11. The particle number density vanishes in the chiral broken
phase for m > 0. In the symmetric phase, the particle number density vanishes because2
the non-zero lower bound for kδ=1.0,nextremely suppresses the summation in Eq. (4.10) for
Lm0 ≲ 1.7 at δ = 1.0 (cf. the phase diagram in Chapter 3.2). At T /m0 = 0.1, we observed
pre-transitional phenomena because of the finite temperature effect.

0.35

0.35

0.3

0.3

0.25

0.25

ρ2/m0

0.4

ρ2/m0

0.4

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

0.5

1

δ

1.5

0

2

0

(a-1) Lm0 = 3.0

1

δ

1.5

2

(a-2) Lm0 = 5.0

0.7

0.7

0.6

0.6

0.5

0.5

ρ2/m0

0.8

ρ2/m0

0.8

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0.5

0

5

10

Lm0

15

0

20

(b-1) δ = 0.0

0

5

10

Lm0

15

20

(b-2) δ = 1.0

Fig. 4.11: Particle number density fixed to µ = 0.7 as a function of δ (above) and L (below)
in two dimensions. The green and yellow curve denotes T /m0 = 0.005 and T /m0 = 0.1
respectively.
As is shown in Fig. 4.12 (D = 3), the continuous momentum for the additional dimension

46

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

enhances the pre-transitional phenomena at a finite temperature and induces a finite number
density near the phase boundary. Through the analysis of the particle number density andthe comparison with the generated fermion mass, the complex behavior in the phase diagrams
is caused by a balance between the particle production and the mass generation.

0.2

0.2

0.15

0.15

ρ3/m020.25

ρ3/m02

0.25

0.1

0.05

0

0.1

0.05

0

0.5

1

1.5

δ

0

2

0

0.5

(a-1) Lm0 = 2.0

1.5

2

15

20

(a-2) Lm0 = 4.0

0.35

0.35

0.3

0.3

0.25

0.25

ρ3/m020.4

ρ3/m02

0.4

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

1

δ

0

5

10

15

Lm0

0

20

0

(b-1) δ = 0.0

5

10

Lm0

(b-2) δ = 1.0

Fig. 4.12: Particle number density fixed to µ = 1.0 as a function of δ (above) and L (below)
in three dimensions. The green and yellow curve denotes T /m0 = 0.005 and T /m0 = 0.2
respectively.

Pressure
We have analyzed the particle number density and mentioned the correspondence between
this quantity and the generated fermion mass in the preceding section. In order to find astable size, we evaluated pressure. The pressure is defined by the derivative of the grandpotential with respect to L,
P (L, δ, T, µ) = −1 ∂Ω (L, δ, T, µ)
,
V
∂L

47

(4.12)

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

and explicitly expressed as
!#

 D
1
m 2· 2
−1 −
−1
D
m0
 p


D−2 p 2Z ∞2 + m2 − cos πδ
exp
−L
q
2
dq
q
q +m
tr I
p
−
CD 0 m0 m0m0cosh L q 2 + m2 − cos πδ


∞ Z ∞
X
tr I 1
dq
q D−3+
C̃D Lm0 n=−∞ 0 m0 m0 q

2 + q 2 + m2 /T + cosh µ/T
2
exp
−
k
kδ,n
δ,n
q
q
.
×
2 + q 2 + m2 cosh2 + q 2 + m2 /T + cosh µ/T
m0 kδ,nkδ,ntr I · Γ 1 − D
P (L, δ, T, µ)
2
=
−
D
mD
2
0
(4π)

"

1
2



m
m0

2

!

In particular, the expression in two dimensions reads
"

 !
 #
P (L, δ, T, µ)
m 2
m 2
1
1 − 1 − ln
=−
4π
m0
m0
m20
 p

p
Z ∞
2 + m2 − cos πδ
exp
−L
q
2
2
dq q + m
1p−
π 0 m0m0
cosh L q 2 + m2 − cos πδ q
2 + m2 /T + cosh µ/T
∞
2
exp
−
k
X
kδ,n
δ,n
1
q
q
.
+
Lm0 n=−∞ m k 2 + m2 cosh k 2 + m2 /T + cosh µ/T
0
δ,n
δ,n
Behavior of the pressure as a function of the boundary conditions, δ, and the length of S1 , L,
is shown in Figs. 4.13 (D = 2) and 4.14 (D = 3). It is observed that, for a finite chemical
potential, the jumps of the pressure are located at the same points where the generated fermion
mass and the particle number density jump. As seen in Fig. 4.14 (D = 3), one can finds
the critical length, L, and U(1) phase, δ, for a second-order phase transition by observing the
sharp bends of the pressure line.
With increasing the length, L, the finite-size effects are suppressed and the absolute valueof the pressure tends to be smaller. At the same time, in particular D = 3, it becomesapparent that the pressure is positively pushed up by thermal effects (the yellow lines on the
below).

48

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS
0.08

0.01

0.06

0

0.04

P2/m02P2/m02

-0.01

0.02

-0.02

0
-0.03

-0.02

-0.04

0

0.2

0.4

0.6

0.8

1

δ

1.2

1.4

1.6

1.8

-0.04

2

0

(a-1) Lm0 = 3.0
0.04

0.02

0.02

P2/m020.04

P2/m02

0.06

0

-0.02

-0.04

-0.04

5

10

Lm0

1.5

2

15

20

0

-0.02

0

1

δ

(a-2) Lm0 = 5.0

0.06

-0.06

0.5

15

-0.06

20

(b-1) δ = 0.0

0

5

10

Lm0

(b-2) δ = 1.0

Fig. 4.13: Pressure as a function of δ (above) and L (below) in two dimensions.
(T /m0 , µ/m0 ; color) = (0.005, 0.0; purple), (0.005, 0.7; green), (0.1, 0.7; yellow).

49

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS
0.1

0.03
0.02

0.05

P3/m03P3/m03

0.01
0

0

-0.01
-0.05
-0.02

-0.1

0

0.5

1

δ

1.5

-0.03

2

0

(a-1) Lm0 = 2.0
0.03

0.02

0.02

0.01

0.01

P3/m030.03

P3/m03

0.04

0

-0.01

-0.02

-0.02
5

10

Lm0

1.5

2

15

20

0

-0.01

0

1

δ

(a-2) Lm0 = 4.0

0.04

-0.03

0.5

15

-0.03

20

(b-1) δ = 0.0

0

5

10

Lm0

(b-2) δ = 1.0

Fig. 4.14: Pressure as a function of δ (above) and L (below) in three dimensions.
(T /m0 , µ/m0 ; color) = (0.005, 0.0; purple), (0.005, 1.0; green), (0.2, 1.0; yellow).
We call a size a stable size where the sign of the pressure flips from negative to positive
as the size decreases. The sign-flip points of the pressure coincides with the local and global
minima of the grand potential as a function of the length of S1 , L. Figures 4.15 (D = 2)
and 4.16 (D = 3) show behavior of the sigh-flip boundaries on a δ-L plane. A value of thechemical potential is distinguished by line colors: the purple line corresponds µ/m0 = 0.0 andthe green lines correspond µ/m0 = 0.7 (D = 2) and µ/m0 = 1.0 (D = 3).
At a low temperature and zero chemical potential, denoted by the purple line on the left,
the pressure is repulsive (attractive) near the (anti-)periodic boundary condition. Moreover,the sigh-flip boundary approaches δ = 0.5, just the middle of the periodic and anti-periodic
boundary conditions, with increasing the size. The finite-size effects are suppressed for a large
size, L, and the thermal fluctuations induce a repulsive pressure. Thus, the repulsive pressure
becomes favored for a large length, L, at higher temperatures, T /m0 = 0.1 (D = 2; the right)
and T /m0 = 0.2 (D = 3; the right). The stable size at µ = 0 can be found only for Lm0 ≲ 1.0
and 0.4 < δ < 0.5 in both the dimensions.
For a finite chemical potential denoted by green lines, one observes the complex behaviorof the sign-flip boundaries on the δ-L plane. The stable size is found for a wider range of δ
than at zero chemical potential. The size, for instance on the left in Fig. 4.15, exists on the50

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

green lines at Lm0 ∼ 4.0 and 0 ≤ δ ≲ 0.5 and a metastable size at Lm0 ∼ 7.0 and 0.5 ≲ δ ≤ 1.
The second line of the effective potential (4.6) remains even at zero temperature and zero
chemical potential, and thus describes the pure finite-size effect. For a small L, a dominant
contribution to the potential energy comes from this term. For 1/2 < δ ≤ 1, this term equivalent to the finite-size effect is negative and decreases the energy. As a result, an attractive
pressure is induced. Whereas, the term is positive for 0 ≤ δ ≤ 1/3 and a repulsive pressure
is induced. Thus, the sign of the pressure flips
1/3
h betweeni As is read in the√ < δ < 1/2.
2 +σ 2 +iπδq−L
is understood as an
second line of the effective potential (4.6), Re ln 1 + eansatz of the Bose–Einstein (δ = 0) and Fermi–Dirac (δ = 1) distributions in finite temperature systems. An opposite contribution to the thermodynamic potential is caused by thisdifference of the distributions. The U(1) phase, δ, is also regarded as an imaginary chemical
potential [112, 113].
12

12

Rep./Rep.

10

10Att./Rep.

8

Lm0

Lm0

8

Att./Att.

6

Rep./Att.
4

Att./Att.

Att./Rep.

6

Rep./Att.
4

20

Rep./Rep.

Rep./Rep.
0

0.2

2

Att./Att.
0.4

δ

0.6

0.8

0

1

(a) T /m0 = 0.005

Rep./Rep.
0

0.2

Att./Att.
0.4

δ

0.6

0.8

1

(b) T /m0 = 0.1

Fig. 4.15: Sign-flip boundaries of the pressure on the δ-L plane in two dimensions. The
purple and green lines are µ/m0 = 0.0 and µ/m0 = 1.0, respectively. Xxx./Xxx. (Xxx. is
Rep. or Att.) denotes whether the pressure is repulsive or attractive in the area at µ/m0 = 0.0(left; purple) and µ/m0 = 0.7 (right; green).

51

CHAPTER 4. THERMAL AND FINITE-SIZE EFFECTS

12

12

10

Rep./Rep.

10

Att./Att.

8

6

Rep./Att.

Lm0

Lm0

8

Att./Rep.

4

6
4

2

0

Rep./Rep.

Rep./Rep.
0

0.2

δ

0.6

0.8

Att./Rep.

2

Att./Att.
0.4

Rep./Att.

01(a) T /m0 = 0.005

Att./Att.
0

0.2

0.4

δ

0.6

0.8

1

(b) T /m0 = 0.2

Fig. 4.16: Sign-flip boundaries of the pressure on the δ-L plane in three dimensions. The
purple and green lines are µ/m0 = 0.0 and µ/m0 = 1.0, respectively. Xxx./Xxx. (Xxx. is
Rep. or Att.) denotes whether the pressure is repulsive or attractive in the area at µ/m0 = 0.0(left; purple) and µ/m0 = 1.0 (right; green).

52

Chapter 5

Summary and Discussions
Symmetry is a central concept in particle physics and the chiral symmetry is particularly
important, albeit an approximate symmetry. Regardless of research fields, because the analysis
from ab initio is often difficult in principle, investigation of the phase structure by using
effective models is still significant, where the analysis with a wide range of parameters isnecessary. In this thesis, employing the four-fermion interaction model called the Gross–Neveu
model, we have evaluated the effective potential in the leading order of the 1/N expansion
in order to investigate the phase structure in terms of the chiral symmetry. The effective
potential is derived with supposing a homogeneous chiral condensate. To study the finitesize effect in addition to the thermal effect with chemical potential, we have compactified
a spacial direction to a one-dimensional sphere, S1 , and besides imposed the U(1)-valued
boundary condition to the fermion.
In Chapter 3, we have considered the finite-size effect at zero temperature. We havederived the improved effective potential compared with [50] paying attention to the finite partof the subtraction of the divergence. The subtraction before our improvement includes theparameters of the finite-size effect, L and δ, and hence prevents the correspondence between
the effective potential and the phase of the grand state. The chiral symmetry breaking tends
to be enhanced at the periodic boundary condition, δ = 0. On the contrary, the anti-periodic
boundary condition, δ = 1, shows opposite results for the chiral symmetry. This behavior
becomes clear with decreasing the size of S1 . The system with the anti-periodic boundary
condition can be regarded to mimic the finite temperature system filled with the fermions.
The phase transitions can be detected by the sharp bends on the lines of the dynamicallygenerated fermion mass and string tension. We have found sign-flip boundaries of the string
tension in the restricted region, 0.4 < δ < 0.5, for any finite size. At zero temperature, it
seems difficult to keep the system at a finite size.
In Chapter 4, we have extended the study on Chapter 3 and considered the finite-size effectwith a finite temperature and a finite chemical potential. In order to investigate the phasestructure more precisely, we have evaluated the effective potential by observing not only theminimum but also the extrema. We have found the boundaries at which the number of the53CHAPTER 5. SUMMARY AND DISCUSSIONS

extrema and the position of the minimum change on the µ-T and L-T planes. The finite-size
effect induces the complex structure at a low temperature and a high chemical potential on
the µ-T plane. Observing the L-T plane, the broken and the symmetric phases alternate withincreasing the size of S1 . We also have shown this behavior of the generated fermion mass asa function of L and δ. By using the grand potential defined as the zero-point is located at
L → ∞, T → 0 and µ = 0, we have found the stable size for a finite chemical potential near
the periodic boundary condition. Through the analysis of the particle number density, thetrade-off relationship between the generated fermion mass and the particle number density isshown. The jumps of the mass and density take place at the same L and δ. Based on theanalysis of the thermodynamic quantities, it is considered that the complex behavior of theboundaries on the phase diagrams is caused by the competition among the generated fermion
mass, the inverse of the size, 2π/L, and the chemical potential. Alternatively, the sign-flipboundaries gives us information about the stable and metastable sizes. It is found that thesizes additionally appear caused by the contribution of a finite chemical potential.
We have discussed the finite-size effect in the Gross–Neveu model and shown this effecton the phase structure along with thermodynamic quantities in terms of the chiral symmetry.
However, we have analyzed only the one-dimensional sphere S1 . There are many shapes ofsystems with various topologies. On the other hand, to consider more realistic situations,
it is necessary to introduce fermion mass, fermion flavors, and an electromagnetic field. Recently, the inhomogeneous chiral condensate are theoretically discussed. The works relatively
close to our study, as is already mentioned in Chapter 1 and Chapter 4.2, the Fulde–Ferrell–
Larkin–Ovchinnikov states are discussed in a superconducting ring with magnetic flux [54].
Furthermore, it is also pointed out that the possibility of the inhomogeneous chiral condensate can not be ignored in a finite-size system [56]. In Ref. [114], the inhomogeneous chiralcondensate are investigated by using lattice simulations.

54

Appendix A

NotationIn this thesis, we adopt Planck units,ℏ = c = kB = 1,

(A.1)

and diag(+, −, −, −) as a metric signature on a Minkowski spacetime, M4 . Extending themetric signature, we also use diag(+, −, . . . , −) on a D-dimensional Minkowski spacetime,
MD . The metric is denoted by ηµν and η µν . The sign of the zeroth component, the temporalcomponent, is positive and the others, the spatial components, are negative.
Greek indices, such as µ, ν, ρ, . . . , run from 0 to D −1, and Latin indices, such as i, j, k, . . . ,from 1 to D. When an index variable appears as upper and lower indices in a term, it
implies a summation over all a range of index variables: xµ pµ = x0 p0 + · · · + xD−1 pD−1 andk j kj = k 1 k1 + · · · + k D kD . We use a bold font to describe a quantity without a zeroth (ortemporal) component, such as k2 = k12 + · · · + k32 and p2 = p21 + · · · + p24 .
A set of gamma matrices, γ µ (µ = 0, 1, . . . , D − 1), is a basis of the Clifford algebra, aspace of spinors, that satisfies{γ µ , γ ν } = γ µ γ ν + γ ν γ µ = 2η µν I,(A.2)

with I denoting an identity matrix in the spinor space. γ 5 is a chiral operator that satisfies{γ 5 , γ µ } = 0.

55

(A.3)

Acknowledgements
I would like to thank Prof. Tomohiro Inagaki, as a supervisor and coauthor, for valuable
and helpful discussions and advise. Also I would like to thank Prof. Daiji Kimura, Dr.
Hiroki Sakamoto and Dr. Yamato Matsuo for fruitful discussions. I am grateful to staff andcolleagues in the laboratory. Finally, I give special thanks to my parents, Hiroaki and Akemi,
for their great supports.

56

Bibliography[1] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 106 (1957) 162,
doi:10.1103/PhysRev.106.162.
[2] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345–358, doi:10.1103/PhysRev.122.345.
[3] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124 (1961) 246–254, doi:10.1103/PhysRev.124.246.
[4] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343–1346,
doi:10.1103/PhysRevLett.30.1343.
[5] H. D. Politzer, Phys. Rev. Lett. 30 (1973) 1346–1349, doi:10.1103/PhysRevLett.30.1346.
[6] S. Muroya, A. Nakamura, C. Nonaka, and T. Takaishi, Prog. Theor. Phys. 110 (2003) 615–668,
doi:10.1143/PTP.110.615, arXiv:hep-lat/0306031.
[7] P. de Forcrand, PoS LAT2009 (2009) 010, doi:10.22323/1.091.0010, arXiv:1005.0539[hep-lat].
[8] C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann, C. Schmidt, and
L. Scorzato, Phys. Rev. D 66 (2002) 074507, doi:10.1103/PhysRevD.66.074507,
arXiv:hep-lat/0204010.
[9] WHOT-QCD Collaboration, S. Ejiri, Y. Maezawa, N. Ukita, S. Aoki, T. Hatsuda, N. Ishii,
K. Kanaya, and T. Umeda, Phys. Rev. D 82 (2010) 014508, doi:10.1103/PhysRevD.82.014508,
arXiv:0909.2121 [hep-lat].
[10] M. G. Alford, A. Kapustin, and F. Wilczek, Phys. Rev. D 59 (1999) 054502,
doi:10.1103/PhysRevD.59.054502, arXiv:hep-lat/9807039.
[11] P. de Forcrand and O. Philipsen, Nucl. Phys. B 642 (2002) 290–306,
doi:10.1016/S0550-3213(02)00626-0, arXiv:hep-lat/0205016.
[12] P. de Forcrand and O. Philipsen, Nucl. Phys. B 673 (2003) 170–186,
doi:10.1016/j.nuclphysb.2003.09.005, arXiv:hep-lat/0307020.
[13] P. de Forcrand and O. Philipsen, Phys. Rev. Lett. 105 (2010) 152001,
doi:10.1103/PhysRevLett.105.152001, arXiv:1004.3144 [hep-lat].
[14] K. Nagata and A. Nakamura, Phys. Rev. D 83 (2011) 114507,
doi:10.1103/PhysRevD.83.114507, arXiv:1104.2142 [hep-lat].
[15] G. Aarts, Phys. Rev. Lett. 102 (2009) 131601, doi:10.1103/PhysRevLett.102.131601,
arXiv:0810.2089 [hep-lat].

57

BIBLIOGRAPHY[16] G. Aarts, L. Bongiovanni, E. Seiler, D. Sexty, and I.-O. Stamatescu, Eur. Phys. J. A 49 (2013)
89, doi:10.1140/epja/i2013-13089-4, arXiv:1303.6425 [hep-lat].
[17] D. Sexty, Phys. Lett. B 729 (2014) 108–111, doi:10.1016/j.physletb.2014.01.019,
arXiv:1307.7748 [hep-lat].
[18] J. Nishimura and S. Shimasaki, Phys. Rev. D 92 (2015) no. 1, 011501,
doi:10.1103/PhysRevD.92.011501, arXiv:1504.08359 [hep-lat].
[19] K. Nagata, J. Nishimura, and S. Shimasaki, Phys. Rev. D 94 (2016) no. 11, 114515,
doi:10.1103/PhysRevD.94.114515, arXiv:1606.07627 [hep-lat].
[20] AuroraScience Collaboration, M. Cristoforetti, F. Di Renzo, and L. Scorzato, Phys. Rev.
D 86 (2012) 074506, doi:10.1103/PhysRevD.86.074506, arXiv:1205.3996 [hep-lat].
[21] H. Fujii, D. Honda, M. Kato, Y. Kikukawa, S. Komatsu, and T. Sano, JHEP 10 (2013) 147,
doi:10.1007/JHEP10(2013)147, arXiv:1309.4371 [hep-lat].
[22] M. Levin and C. P. Nave, Phys. Rev. Lett. 99 (2007) no. 12, 120601,
doi:10.1103/PhysRevLett.99.120601, arXiv:cond-mat/0611687.
[23] Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang, Physical Review B 86
(Jul, 2012) no. 4, , doi:10.1103/physrevb.86.045139.
https://doi.org/10.1103%2Fphysrevb.86.045139.
[24] S. Akiyama, D. Kadoh, Y. Kuramashi, T. Yamashita, and Y. Yoshimura, JHEP 09 (2020) 177,
doi:10.1007/JHEP09(2020)177, arXiv:2005.04645 [hep-lat].
[25] S. Akiyama, Y. Kuramashi, T. Yamashita, and Y. Yoshimura, JHEP 01 (2021) 121,
doi:10.1007/JHEP01(2021)121, arXiv:2009.11583 [hep-lat].
[26] J. B. Kogut, M. A. Stephanov, D. Toublan, J. J. M. Verbaarschot, and A. Zhitnitsky, Nucl.
Phys. B 582 (2000) 477–513, doi:10.1016/S0550-3213(00)00242-X, arXiv:hep-ph/0001171.
[27] S. Klevansky, Rev. Mod. Phys. 64 (1992) 649–708, doi:10.1103/RevModPhys.64.649.
[28] T. Hatsuda and T. Kunihiro, Phys. Rept. 247 (1994) 221–367,
doi:10.1016/0370-1573(94)90022-1, arXiv:hep-ph/9401310.
[29] D. J. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235, doi:10.1103/PhysRevD.10.3235.
[30] D. Kahana and U. Vogl, Phys. Lett. B 244 (1990) 10–18, doi:10.1016/0370-2693(90)90260-D.
[31] D. Ebert, L. Kaschluhn, and G. Kastelewicz, Phys. Lett. B 264 (1991) 420–425,
doi:10.1016/0370-2693(91)90371-V.
[32] K. Fukushima, Phys. Lett. B 591 (2004) 277–284, doi:10.1016/j.physletb.2004.04.027,
arXiv:hep-ph/0310121.
[33] B. Rosenstein, B. Warr, and S. Park, Phys. Rept. 205 (1991) 59–108,
doi:10.1016/0370-1573(91)90129-A.
[34] T. Inagaki, T. Muta, and S. D. Odintsov, Prog. Theor. Phys. Suppl. 127 (1997) 93,
doi:10.1143/PTPS.127.93, arXiv:hep-th/9711084 [hep-th].
[35] V. A. Miransky and I. A. Shovkovy, Phys. Rept. 576 (2015) 1–209,
doi:10.1016/j.physrep.2015.02.003, arXiv:1503.00732 [hep-ph].

58

BIBLIOGRAPHY[36] K. Hattori and X.-G. Huang, Nucl. Sci. Tech. 28 (2017) no. 2, 26,
doi:10.1007/s41365-016-0178-3, arXiv:1609.00747 [nucl-th].
[37] Y. Jiang and J. Liao, Phys. Rev. Lett. 117 (2016) no. 19, 192302,
doi:10.1103/PhysRevLett.117.192302, arXiv:1606.03808 [hep-ph].
[38] K. Fukushima, Prog. Part. Nucl. Phys. 107 (2019) 167–199, doi:10.1016/j.ppnp.2019.04.001,
arXiv:1812.08886 [hep-ph].
[39] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Lett. B 349 (1995) 477–483,
doi:10.1016/0370-2693(95)00232-A, arXiv:hep-ph/9412257.
[40] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Nucl. Phys. B 462 (1996) 249–290,
doi:10.1016/0550-3213(96)00021-1, arXiv:hep-ph/9509320.
[41] M. N. Chernodub and S. Gongyo, Phys. Rev. D 95 (2017) no. 9, 096006,
doi:10.1103/PhysRevD.95.096006, arXiv:1702.08266 [hep-th].
[42] S. K. Kim, W. Namgung, K. S. Soh, and J. H. Yee, Phys. Rev. D36 (1987) 3172–3177,
doi:10.1103/PhysRevD.36.3172.
[43] K. Ishikawa, T. Inagaki, K. Yamamoto, and K. Fukazawa, Prog. Theor. Phys. 99 (1998)
237–255, doi:10.1143/PTP.99.237.
[44] A. V. Gamayun and E. V. Gorbar, Phys. Lett. B 610 (2005) 74–79,
doi:10.1016/j.physletb.2005.01.079, arXiv:hep-ph/0411157.
[45] D. Ebert and K. G. Klimenko, Phys. Rev. D 82 (2010) 025018,
doi:10.1103/PhysRevD.82.025018, arXiv:1005.0699 [hep-ph].
[46] L. M. Abreu and E. S. Nery, Phys. Rev. C 96 (2017) no. 5, 055204,
doi:10.1103/PhysRevC.96.055204, arXiv:1711.07934 [nucl-th].
[47] T. Ishikawa, K. Nakayama, and K. Suzuki, Phys. Lett. B 809 (2020) 135713,
doi:10.1016/j.physletb.2020.135713, arXiv:2005.10758 [hep-lat].
[48] T. Ishikawa, K. Nakayama, and K. Suzuki, Phys. Rev. Res. 3 (2021) no. 2, 023201,
doi:10.1103/PhysRevResearch.3.023201, arXiv:2012.11398 [hep-lat].
[49] A. Higuchi and L. Parker, Phys. Rev. D37 (1988) 2853–2871, doi:10.1103/PhysRevD.37.2853.
[50] S. Huang and B. Schreiber, Nucl. Phys. B 426 (1994) no. 3, 644–660,
doi:10.1016/0550-3213(94)90024-8, arXiv:cond-mat/9404080 [cond-mat].
[51] D. Y. Song, Phys. Rev. D48 (1993) 3925–3928, doi:10.1103/PhysRevD.48.3925.
[52] A. Flachi, Phys. Rev. D 88 (2013) no. 4, 041501, doi:10.1103/PhysRevD.88.041501,
arXiv:1305.5348 [hep-th].
[53] A. Flachi, Phys. Rev. D 88 (2013) no. 8, 085011, doi:10.1103/PhysRevD.88.085011,
arXiv:1304.6880 [hep-th].
[54] R. Yoshii, S. Takada, S. Tsuchiya, G. Marmorini, H. Hayakawa, and M. Nitta, Phys. Rev. B92(2015) no. 22, 224512, doi:10.1103/PhysRevB.92.224512, arXiv:1404.3519[cond-mat.supr-con].

59

BIBLIOGRAPHY[55] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485–491, doi:10.1103/PhysRev.115.485.
[56] A. Flachi, Phys. Rev. D 86 (2012) 104047, doi:10.1103/PhysRevD.86.104047,
arXiv:1209.4754 [hep-th].
[57] H. B. G. Casimir and D. Polder, Phys. Rev. 73 (1948) 360–372, doi:10.1103/PhysRev.73.360.
[58] H. Casimir, Proc. Kon. Ned. Akad. Wetensch. 51 (1948) 793–795.
[59] K. A. Milton, L. L. DeRaad, Jr., and J. S. Schwinger, Annals Phys. 115 (1978) 388,
doi:10.1016/0003-4916(78)90161-6.
[60] K. A. Milton, Annals Phys. 127 (1980) 49, doi:10.1016/0003-4916(80)90149-9.
[61] I. Brevik, Annals Phys. 138 (1982) no. 1, 36–52,
doi:https://doi.org/10.1016/0003-4916(82)90174-9.
[62] I. Brevik and H. Kolbenstvedt, Annals Phys. 143 (1982) no. 1, 179–190,
doi:https://doi.org/10.1016/0003-4916(82)90218-4.
[63] G. Plunien, B. Müller, and W. Greiner, Phys. Rept. 134 (1986) no. 2, 87–193,
doi:https://doi.org/10.1016/0370-1573(86)90020-7.
[64] M. Bordag, E. Elizalde, and K. Kirsten, J. Math. Phys. 37 (1996) 895–916,
doi:10.1063/1.531418, arXiv:hep-th/9503023.
[65] E. Elizalde, L. Vanzo, and S. Zerbini, Commun. Math. Phys. 194 (1998) 613–630,
doi:10.1007/s002200050371, arXiv:hep-th/9701060.
[66] M. Bordag, U. Mohideen, and V. M. Mostepanenko, Phys. Rept. 353 (2001) 1–205,
doi:10.1016/S0370-1573(01)00015-1, arXiv:quant-ph/0106045.
[67] K. A. Milton, “The Casimir Effect: Physical Manifestations of Zero-Point Energy”, World
Scientific, Hackensack, NJ, US, 2001.
[68] M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “Advances in the
Casimir Effect”, Oxford University Press, Oxford, UK, 2009.
[69] D. Dalvit, P. Milonni, D. Roberts, and F. Rosa, “Casimir Physics”, Springer, Berlin, Germany,
2011.
[70] W. M. R. Simpson and U. Leonhardt, “Forces fo the Quantum Vacuum: An Introduction to
Casimir Physics”, World Scientific, Hackensack, NJ, US, 2015.
[71] I. E. Dzyaloshinskii, E. M. Lifshitz, and L. P. Pitaevskii, Advances in Physics 10 (1961) no. 38,
165–209, doi:10.1080/00018736100101281.
[72] T. H. Boyer, Phys. Rev. 174 (1968) 1764–1774, doi:10.1103/PhysRev.174.1764.
[73] T. H. Boyer, Phys. Rev. A 9 (1974) 2078–2084, doi:10.1103/PhysRevA.9.2078.
[74] S. Rode, R. Bennett, and S. Y. Buhmann, New J. Phys. 20 (2018) no. 4, 043024,
doi:10.1088/1367-2630/aaaa44, arXiv:1710.01509 [quant-ph].
[75] A. Flachi, Phys. Rev. Lett. 110 (2013) no. 6, 060401, doi:10.1103/PhysRevLett.110.060401,
arXiv:1301.1193 [hep-th].

60

BIBLIOGRAPHY[76] A. Flachi, M. Nitta, S. Takada, and R. Yoshii, Phys. Rev. Lett. 119 (2017) no. 3, 031601,
doi:10.1103/PhysRevLett.119.031601, arXiv:1704.04918 [hep-th].
[77] M. Nitta and R. Yoshii, JHEP 12 (2017) 145, doi:10.1007/JHEP12(2017)145,
arXiv:1707.03207 [hep-th].
[78] M. N. Chernodub, V. A. Goy, and A. V. Molochkov, Phys. Rev. D 99 (2019) no. 7, 074021,
doi:10.1103/PhysRevD.99.074021, arXiv:1811.01550 [hep-lat].
[79] M. N. Chernodub, V. A. Goy, and A. V. Molochkov, PoS Confinement2018 (2019) 006,
doi:10.22323/1.336.0006, arXiv:1901.04754 [hep-th].
[80] T. Ishikawa, K. Nakayama, D. Suenaga, and K. Suzuki, Phys. Rev. D 100 (2019) no. 3, 034016,
doi:10.1103/PhysRevD.100.034016, arXiv:1905.11164 [hep-ph].
[81] T. Inagaki, Y. Matsuo, and H. Shimoji, Symmetry 11 (2019) no. 4, 451,
doi:10.3390/sym11040451, arXiv:1903.04244 [hep-th].
[82] T. Inagaki, Y. Matsuo, and H. Shimoji, PTEP 2022 (2022) no. 1, 013B09,
doi:10.1093/ptep/ptab160, arXiv:2108.03583 [hep-ph].
[83] R. L. Stratonovich, Soviet Physics Doklady 2 (1957) 416.
[84] J. Hubbard, Phys. Rev. Lett. 3 (1959) 77–78, doi:10.1103/PhysRevLett.3.77.
[85] T. Kashiwa and T. Sakaguchi, Phys. Rev. D 68 (2003) 065002,
doi:10.1103/PhysRevD.68.065002, arXiv:hep-th/0306008.
[86] H. Kohyama, D. Kimura, and T. Inagaki, Nucl. Phys. B 896 (2015) 682–715,
doi:10.1016/j.nuclphysb.2015.05.015, arXiv:1501.00449 [hep-ph].
[87] H. Kohyama, D. Kimura, and T. Inagaki, Nucl. Phys. B 906 (2016) 524–548,
doi:10.1016/j.nuclphysb.2016.03.015, arXiv:1601.02411 [hep-ph].
[88] J. S. Schwinger, Phys. Rev. 82 (1951) 664–679, doi:10.1103/PhysRev.82.664.
[89] C. Itzykson and J. Zuber, “Quantum Field Theory”, McGrow-Hill, New York, USA, 1980.
[90] E. Elizalde and S. D. Sergei, Odintsov, “Zeta regularization Techniques with Applications”,
World Scientific, Singapore, 1994.
[91] E. Elizalde, “Ten Physical Applications of Spectral Zeta Functions”, Springer, Berlin, Germany,
2012.
[92] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133–1136,
doi:10.1103/PhysRevLett.17.1133.
[93] P. C. Hohenberg, Phys. Rev. 158 (1967) 383–386, doi:10.1103/PhysRev.158.383.
[94] S. R. Coleman, Commun. Math. Phys. 31 (1973) 259–264, doi:10.1007/BF01646487.
[95] T. Fujihara, T. Inagaki, and D. Kimura, J. Phys. A 39 (2006) 6371–6376,
doi:10.1088/0305-4470/39/21/S30, arXiv:hep-ph/0511286.
[96] T. Inagaki, D. Kimura, and A. Kvinikhidze, Phys. Rev. D 77 (2008) 116004,
doi:10.1103/PhysRevD.77.116004, arXiv:0712.1336 [hep-ph].

61

BIBLIOGRAPHY[97] T. Fujihara, T. Inagaki, D. Kimura, and A. Kvinikhidze, Prog. Theor. Phys. Suppl. 174 (2008)
72–75, doi:10.1143/PTPS.174.72, arXiv:0806.1331 [hep-ph].
[98] T. Inagaki, D. Kimura, H. Kohyama, and A. Kvinikhidze, Phys. Rev. D 83 (2011) 034005,
doi:10.1103/PhysRevD.83.034005, arXiv:1010.0309 [hep-ph].
[99] T. Eguchi, Phys. Rev. D 17 (1978) 611, doi:10.1103/PhysRevD.17.611.
[100] K.-i. Shizuya, Phys. Rev. D 21 (1980) 2327, doi:10.1103/PhysRevD.21.2327.
[101] S. Aoki and K. Higashijima, Prog. Theor. Phys. 76 (1986) 521, doi:10.1143/PTP.76.521.
[102] D. Ebert and K. G. Klimenko, Phys. Rev. D 80 (2009) 125013,
doi:10.1103/PhysRevD.80.125013, arXiv:0911.1944 [hep-ph].
[103] T. Matsubara, Prog. Theor. Phys. 14 (1955) 351–378, doi:10.1143/PTP.14.351.
[104] H. Ezawa, Y. Tomozawa, and H. Umezawa, Nuovo Cim. 5 (1957) 810–841,
doi:10.1007/BF02903206.
[105] L. Dolan and R. Jackiw, Phys. Rev. D 9 (1974) 3320–3341, doi:10.1103/PhysRevD.9.3320.
[106] S. Weinberg, Phys. Rev. D 9 (1974) 3357–3378, doi:10.1103/PhysRevD.9.3357.
[107] L. Jacobs, Phys. Rev. D 10 (1974) 3956, doi:10.1103/PhysRevD.10.3956.
[108] U. Wolff, Phys. Lett. B 157 (1985) 303–308, doi:10.1016/0370-2693(85)90671-9.
[109] T. Inagaki, T. Kouno, and T. Muta, Int. J. Mod. Phys. A 10 (1995) 2241–2268,
doi:10.1142/S0217751X95001091, arXiv:hep-ph/9409413.
[110] T. M. Apostol, “Introduction to Analytic Number Theory”, Springer, New York, US, 1976,
pp. 257–259.
[111] D. K. Kim, Y. D. Han, and I. G. Koh, Phys. Rev. D 49 (1994) 6943–6946,
doi:10.1103/PhysRevD.49.6943.
[112] A. Roberge and N. Weiss, Nucl. Phys. B 275 (1986) 734–745,
doi:10.1016/0550-3213(86)90582-1.
[113] K. Kashiwa, Symmetry 11 (2019) no. 4, 562, doi:10.3390/sym11040562.
[114] J. Lenz, L. Pannullo, M. Wagner, B. Wellegehausen, and A. Wipf, Phys. Rev. D 101 (2020)
no. 9, 094512, doi:10.1103/PhysRevD.101.094512, arXiv:2004.00295 [hep-lat].

62