"Short range correlations in nuclear matter"

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Short range correlations

in nuclear matter

Vittorio Somà

Henryk Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences,

Kraków, Poland

Thesis submitted for the Doctor degree prepared under the supervision of Dr. Piotr Bożek

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Abstract

In this thesis we present self-consistent spectral calculations of nuclear matter prop-erties at zero and finite temperature. Results are reported for different nucleon-nucleon potentials and with the inclusion of Urbana three-body forces for both isospin sym-metric and pure neutron matter. The calculations are performed within the Green’s functions approach in the in-medium T-matrix approximation with the full off-shell propagation of nucleons taken into account.

A new method of computing the grand canonical potential from a diagrammatic ex-pansion is presented. An important consequence is that all thermodynamic quantities are computed directly from diagram series without further approximations, and ther-modynamic consistency is automatically fulﬁlled.

Three-body forces are included via an eﬀective two-body potential derived from the Urbana interaction after averaging over the spatial, spin and isospin degrees of freedom of the third nucleon.

Results for the energy per particle, thermodynamic observables and single particle properties, without and with three-body forces, are presented for a range of densities and temperatures, for symmetric and neutron matter. Modifications of nuclear matter properties induced by three-body forces are studied in detail. Saturation properties of symmetric matter are significantly improved with respect to the two-body case. The critical temperature of the liquid-gas phase transition is reduced by about 5 MeV. The effects on single particle properties are found to be opposite in symmetric and pure neutron matter. In the first case spectral function are broadened, in the second an enhancement of the quasiparticle peak is observed. Three-body forces modifications become larger as the density increases, are almost independent of the temperature and are suppressed for nucleon momenta above the Fermi surface.

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Streszczenie

Tematem poniższej rozprawy są samouzgodnione obliczenia spektralne własności materii jądrowej w zerowej i skończonej temperaturze. Przedstawione są wyniki dla szeregu oddziaływań nukleon-nukleon, w tym również z uwzględnieniem oddziaływań trójciałowych typu Urbana, dla materii jądrowej symetrycznej i czysto neutronowej.

Przedstawiona jest nowa metoda obliczania potencjału termodynamicznego poprzez wysumowanie rozwinięcia na diagramy. W ramach przyjętego rozwinięcia, wszys-tkie wielkości termodynamiczne są wyliczone na podstawie tego samego (niepertur-bacyjnego) rozwinięcia. Dzięki temu, spójność termodynamiczna jest automatycznie zachowana.

Oddziaływania trójciałowe są włączone poprzez efektywne, zależne od gęstości, siły dwuciałowe, otrzymane z oddziaływania Urbana uśredniając po przestrzennych, spinowych i izospinowych stopniach swobody jednego z trzech nukleonów.

Wyliczono energię na cząstkę, wielkości termodynamiczne i jednocząstkowe w sze-rokim zakresie temperatur i gęstości, dla materii symetrycznej i neutronowej, zarówno dla przypadku z, jak i bez sił trójciałowych. Szczegółowo przebadano wpływ uwzględ-nienia sił trójciałowych na własności materii jądrowej. Przede wszystkim, wprowadze-nie oddziaływań trójciałowych umożliwia odtworzewprowadze-nie w rachunkach punktu saturacji materii jądrowej, a temperatura krytyczna dla przejścia ciecz-gaz ulega obniżeniu o 5 MeV. Wpływ sił trójciałowych na własności jednocząstkowe jest przeciwny dla materii symetrycznej i materii neutronowej. Dla tej pierwszej funkcje spektralne ulegają posz-erzeniu, a dla drugiej pik kwazicząstkowy jest wzmocniony. Wpływ sił trójciałowych rośnie z gęstością, słabo zależy od temperatury i jest niewielki dla pędów powyżej pędu Fermiego.

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Introduction

The microscopic determination of the equation of state of nuclear matter remains one of the most important open issues in modern physics, both as a theoretical challenge and as a crucial input in diﬀerent frontier research ﬁelds, from heavy-ion reactions to astrophysics. Presently, experimental estimates exist only around nuclear saturation density and for zero or small isospin asymmetries. In order to extrapolate the equa-tion of state to higher densities or neutron rich matter reliable theoretical models are necessary.

The starting point of the investigation of nuclear matter properties is the bare interaction between its constituents, described as hadronic degrees of freedom. These interactions yet cannot be derived directly from quantum chromodynamics and are modeled, for example in meson exchange ﬁeld theories, on the basis of all the infor-mation coming from nucleon-nucleon scattering experiments. In the last years con-siderable improvements have been made, and presently there exist few models of the nucleon-nucleon potential which are able to reproduce the huge amount of scattering data with excellent precision.

As larger computational power has become available, a big progress has been made also in the calculations addressing the many-body nuclear problem. Estimates of nu-clear matter properties have been made by using several microscopic approaches such as the Brückner-Hartree-Fock method [1], variational calculations [2], Monte-Carlo tech-niques [3, 4, 5], propagator methods [6] and the relativistic Dirac-Brückner-Hartree-Fock approach [7, 8].

This renewed interest is motivated also by new important projects involving heavy-ion experiments and the strong development of astro-nuclear physics, in particular researches related to compact stars, for which the nuclear equation of state constitutes an essential ingredient.

The upcoming FAIR facility at GSI has the goal of studying baryonic matter com-pressed at high densities (CBM experiment [9]) as well as the implementation of rare isotope beams, including large asymmetric nuclei and hypernuclei (NuSTAR, HypHI [10, 11]). A new generation of exotic beam accelerators, in which the behaviour of nuclear matter up to extreme ratios of neutron to proton numbers can be studied, is also being developed in France (Spiral), USA (RIA) and Japan (RIKEN).

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asym-metric low-temperature baryonic matter over a large range of densities, constitute an ideal laboratory for testing the nuclear equation of state. In the past years there has been an increasing amount of observational data and future satellite (Chandra, XMM-Newton, Integral) and earth-based experiments (LIGO, VIRGO) are hopefully going to provide a wide spectrum of new observations. A research network, CompStar, has been recently created in order to gather together and promote the collaboration be-tween experts from the various fields and to achieve a unified picture of all the different processes taking place in these stellar objects [12].

One of the main challenges of nuclear theory is to provide a consistent description of the fundamental properties of nuclear matter. If we start from the nucleon-nucleon interaction in vacuum, in order to study the properties of the dense system we must take into account the modification of the propagation of nucleons in the medium and, consequently, how these modifications affect the interaction itself. To address this problem in a consistent way one has to go necessarily beyond mean field approxima-tions. In particular the strong repulsive core of nuclear interactions induces short-range correlations in the medium, which have to be taken properly into account: in this sense the dense nuclear system is far from the picture of a gas of free quasiparticles.

A new approach which addresses this issue and aims at advances in this field is the object of this thesis. Our method is based on the many-body field theoretical techniques provided by finite temperature real-time Green’s functions, which allow to study consistently the macroscopic and the microscopic properties of uniform nuclear matter, accounting for the particle propagation in the presence of strong short-range correlations. Green’s functions can be considered first of all as particle propagators, thus they address microscopically the dynamics of the system. At the same time, as nuclear matter is viewed as a grand canonical statistical ensemble, they are defined as averages in the ensemble, thus they have access to all the statistical mechanics information of the system. In particular, from the knowledge of the single-particle propagators it is possible to derive the expectation values of all one-body operators which correspond to the observables in the ensemble.

In nuclear matter, in order to tackle the short-range repulsive part of the potential, one usually introduces the so-called ladder or T-matrix approximation, which resums subsequent scatterings between two particles. Even if the application of the Green’s functions theory to nuclear matter and the T-matrix approximation date back to the early 60’s [13, 14], only recently modern computers have enabled numerical calcula-tions which take into account the full off-shell in-medium propagation of nucleons. Since the late 90’s several groups have proposed self-consistent T-matrix approaches with reliable quantitative results [15, 16, 17, 18, 19, 20, 21, 22, 23] In this direction goes the first achievement presented in this thesis, the first self-consistent calculations of thermodynamic quantities obtained with the in-medium T-matrix with dressed nu-cleons, at zero and finite temperature. In particular we introduce a new method, fully diagrammatic, of computing the grand canonical potential, from which all ther-modynamic observables can be derived without further approximations. Moreover,

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thermodynamic consistency is automatically fulﬁlled.

The second development of the self-consistent Green’s functions approach shown here, and central part of this thesis, concerns the inclusion of three-body forces into the T-matrix formalism. A defect common to all non relativistic microscopic approaches based on realistic bare nucleon-nucleon potentials is the incapacity of reproducing cor-rectly the saturation properties of symmetric nuclear matter, well known from the study of finite nuclei. This deficiency is due to the absence of three-body forces, which play a non negligible role in the dense medium. Three-body forces has been imple-mented into variational and Brückner-Hartree-Fock methods with significant improve-ments in the description of nuclear saturation. We include three-nucleon interactions into the T-matrix approach and analyse their effects on the equation of state, the macroscopic observables as well as on the single-particle properties, addressing the dependence on the density, the temperature, the nucleon momenta, the choice of the nucleon-nucleon potential and the isospin asymmetry.

Part of the results presented in this thesis have been published in the articles I) V. Somà, P. Bożek

“Diagrammatic calculation of thermodynamical quantities in nuclear matter” Phys. Rev. C 74 045809 (2006)

(Ref. [24])

II) V. Somà, P. Bożek

“Nuclear matter with three-body forces from self-consistent spectral calculations” Acta Phys. Pol. B 39 1405 (2008)

(Ref. [25])

III) V. Somà, P. Bożek

“In medium T-matrix for nuclear matter with three-body forces: Binding energy and single-particle properties”

Phys. Rev. C 78 054003 (2008) (Ref. [26])

and in the conference proceedings IV) V. Somà, P. Bożek

“Thermodynamic properties of nuclear matter at finite temperature” Acta Phys. Pol. B 37 3399 (2006)

(Ref. [27]) V) V. Somà

“Realistic and effective interactions in the study of nuclear matter” Phys. Part. Nucl. 39 1052 (2008)

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VI) V. Somà, P. Bożek

“Modifications of single-particle properties in nuclear matter induced by three-body forces”

e-print arXiv:0811.4088 [nucl-th]

(Ref. [29])

The work is organized as follows. In Chapter 1 we introduce the concept of nuclear matter and brieﬂy outline the problems related to the study of its properties and the possible applications and constraints of the nuclear equation of state. We then review the Green’s functions methods for fermionic many-body systems at zero and ﬁnite temperature, with particular emphasis on the T-matrix theory.

In the second chapter we report the results of the calculations of the thermodynamic properties when two-body realistic interactions are used. We compute the energy per particle, the pressure and the entropy of symmetric nuclear matter and check the thermodynamic consistency for diﬀerent potentials. Part of this calculations has been published in Ref. [24].

Chapter 3 is devoted to the implementation of three-body forces into the Green’s function formalism. After a theoretical discussion, an eﬀective two-body mean-ﬁeld-like potential is obtained and added to the nucleon-nucleon potential in the T-matrix [26]. The steps of the derivation are outlined, with most of the technical details reported in the Appendix.

Finally, in the fourth chapter calculations with the inclusion of three-body forces are shown. The new equation of state is discussed and compared to other existing microscopic estimations. Thermodynamic quantities are computed without and with three-body forces, and the instability region related to the liquid-gas phase transition in nuclear matter is identified. The effects of three-body forces on the single particle properties are studied both in symmetric and pure neutron matter, in particular the modifications of the nucleon spectral functions, self-energies and effective masses are considered. Most of these calculations have appeared in Ref. [26], except for the study of the liquid-gas transition which is unpublished. An analysis devoted to single particle properties can be found in Ref. [29]. Finally, in the last chapter some conclusive remarks and a to-do list for the future can be found.

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Chapter

1 The nuclear many-body problem

1.1 The concept of nuclear matter

By nuclear matter we mean a quantum system composed of nucleons interacting via nuclear forces. We address only the case of infinite nuclear matter, i.e. imagining N particles in a box of volume V, we consider the thermodynamic limit N, V → ∞. This allows us to neglect all finite size effects and ensures that all macroscopic quantities are well defined.

The neutron and the proton are regarded as the same particle with vacuum mass1

m = 939 MeV, spin s = 1₂ and isospin t = 1₂_{, corresponding to two isospin z−projections}

tn_z = 1₂ and tpz = −1₂. We assume the system to be homogeneous in space and

spin-unpolarized. The ﬁrst property allows us to deﬁne a constant particle density2

ρ ≡ N/V. We then study only two situations: the case with an equal number of pro-tons and neutrons, symmetric nuclear matter, or a system composed of neutrons only, neutron matter.

The nucleons interact via strong forces, addressed in detail in the next section. Weak interactions enter only indirectly in some applications (such as neutron star matter) and are not present when pure nucleonic matter is considered. The eﬀects of Coulomb interactions between protons are assumed to be negligible and thus electro-magnetic forces are switched oﬀ from the beginning.

We keep ourselves in a non-relativistic framework. This assumption is supported by the fact that the nucleon mass is large enough; we shall discuss in the following (Section 3.1) the possible inclusion of relativistic corrections and their connection with (non-relativistic) three-body forces.

In this work only the case of nuclear matter in thermodynamic equilibrium will be considered. In the framework of statistical mechanics such a system may be regarded as a grand-canonical ensemble and we can investigate its thermodynamic properties, evaluating macroscopic observables which can be compared to experiments. For this

1

We use natural units, i.e. c = ~ = kB = 1 . 2

The number of particles per unit volume is sometimes denoted by n; however in the literature, when speaking of nuclear matter, one finds more often the notation ρ.

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purpose one has to develop a many-body technique capable of treating the hard core present in nuclear interactions and suitable for taking into account the modiﬁcations induced by the presence of a dense and possibly hot medium.

Which physical systems, and in which conditions, can be described with such an idealized picture? If we look at the famous Weizsäcker semi-empirical mass formula

(Np and Nn are respectively the proton and neutron densities, with Np+ Nn= N )

E(Np, Nn) = EBN + EsurfN2/3+ ECoulNp2N−1/3+ EPauli(Nn− Np)2/N (1.1)

we see that we can relate the energy per particle of nuclear matter, computed at the

density found in atomic nuclei (nuclear saturation density, ρ0 ≈ 0.16 fm−3), to the

ﬁrst coeﬃcient EB(all other terms go to zero), the binding energy of a single nucleon.

Estimated experimentally, EB ≈ −16 MeV represents the ﬁrst quantity that every

model of nuclear matter should reproduce. In fact, presently the few other constraints on the nuclear matter equation of state (EOS) come from the properties of nuclei, hence at ρ = ρ0, as we shall see in the following.

In terrestrial experiments it is extremely diﬃcult to create a state of nuclear matter at higher densities with the purpose of studying its thermodynamic properties. A dense and rather hot nuclear matter is formed in intermediate-energy heavy ion experiments, where a large initial energy density is deposited into a small volume. However, one has always to be careful when applying a model which assumes thermodynamic equilibrium to such a short-lived and rapidly expanding environment.

In the outer space, on the other hand, the core of neutron stars (representing 99 % of their total mass) can be imagined in a ﬁrst approximation to be formed of zero-temperature asymmetric nuclear matter spanning a large range of densities. Neutron stars and their formation (protoneutron stars) and cooling process constitute from this point of view an unique laboratory for testing the nuclear EOS beyond the nuclear saturation density.

The picture of a homogeneous matter composed of nucleons only has of course

limits of applicability. Surely, one expects it to be valid around ρ0. When the system

becomes more and more dilute, at some point clusters of nucleons become energetically

favored and begin to be formed, starting from α-particles. At densities below 0.1 ρ0 the

matter is not uniform anymore but can be pictured as a gas of nuclides: free nucleons, deuterons, α-particles and heavier nuclear clusters. When we increase the density, on the other hand, we cross the threshold for the hyperon production and heavier baryons start to populate the system. Nucleons become more and more packed close to each other and a transition to uniform quark matter eventually occurs. Evidently in these situations the picture of pure nucleonic matter looses its eﬃciency, so that a safe range of validity can be set ρ ∈13ρ0, 3 ρ0

.

At these densities, however, the uniformity of nuclear matter is conditioned by two known critical behaviours: the ﬁrst-order liquid-gas phase transition and the second-order superﬂuid onset. A liquid-gas transition, similar to the one observed for a van-der-Waals gas, is believed to take place in nuclear matter. Below a critical temperature

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1.2 The nucleon-nucleon interaction 7

characterized by negative derivates, with respect to the density, of the pressure and chemical potential ∂ P ∂ ρ T < 0 ; ∂ µ ∂ ρ T < 0 . (1.2)

In this region the system tends to separate into a liquid (more dense) and a gas (more dilute) phase with equal pressure and chemical potential. We address this instability in detail in Sec. 4.3.

The formation of Cooper pairs between nucleons is possible because of the

attrac-tive nuclear forces. A critical temperature T_c∆ signals in nuclei or nuclear matter the

onset of superfluidity, which affects microscopic (correlations) as well as macroscopic (viscosity) properties. Several theoretical studies have attempted to determine the fea-tures of superfluid phase, which remain however rather uncertain from the quantitative point of view. In this thesis we do not take pairing between nucleons into account. Its effect on the energy per particle, from BCS theory [30], is of the order of

δEpairing≈ ∆2

µ ≈ O(0.1 MeV) , (1.3)

where ∆ is the pairing gap, hence smaller than the accuracy of the calculations. We discuss the possible appearance of instabilities related to nucleon pairing at low tem-peratures when discussing the numerical methods in Sec. 2.1.

1.2 The nucleon-nucleon interaction

In principle, forces between nucleons should be derived from Quantum Chromody-namics (QCD), in which hadrons are seen as bound states of quarks, the fundamental constituents of strongly interacting matter. Since such a reliable theory is not yet avail-able, we need to consider a Hamiltonian with hadronic degrees of freedom together with some model of the basic nucleon-nucleon (NN) interaction.

As a ﬁrst step it is possible to obtain information about the main features of the interaction from low energy NN scattering experiments: it has a relatively short range of about 1 fm, it is attractive for intermediate distances with a maximum amplitude of about 40 MeV ending with an attractive tail, it is strongly repulsive at short distances (< 0.5 fm); moreover it depends on the spin and isospin of the two nucleons. One can then try to construct a general form of the potential by requesting the invariance under the general symmetries of QCD. For a pair ij of interacting nucleons, the various dependences come in through invariant operators acting in coordinate (the relative po-sition rij), momentum (the relative momentum pij), spin (σi, σj) and isospin (τi, τj)

spaces. It turns out that, in order to explain the experimental data, the potential must contain couplings with the spin space via spin-orbit ( ˆL_{· ˆS, where ˆ}L = rij × pij

is the total orbital angular momentum and ˆS= 1₂(σi+ σj) the total spin) and tensor

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into account all the constraints one can write the general expression Vij = 18 X a=1 Va_(|rij|) ˆOija , (1.4)

where ˆO_ija=1,···18 are combinations of all the fore mentioned operators. The ﬁrst 14

terms are charge-independent, i.e. invariant with respect to rotations in the isospin space; the last four are small and break the charge independence, implying that V_npT =1 _{≈ V}nn= Vpp, T being the total isospin of the pair.

At the level of field theory, the most successful models have been built describing NN interactions with meson exchanges. The first pioneering work dates back to 1935 when Yukawa proposed that nucleons interact via the exchange of a virtual pion [31]. Modern potentials include scalar, pseudoscalar and vector meson exchanges, whose contributions account for the rich structure of (1.4). In particular three of these pa-rameterizations are able to fit all the NN phase shifts and most deuteron properties

with a χ2_{/datum ≈ 1, the CD-Bonn [32], the Nijmegen [33], and the Argonne V18}

(AV18) [34] interactions. For their capacity of precisely reproducing a large amount of experimental scattering data they are referred to as bare or realistic NN potentials. When employing these interactions in calculations of the properties of large nuclear systems or nuclear matter, one expects that the scattering behavior is modiﬁed by the presence of other nucleons. In particular the strong repulsion at short distances in-duces short-range correlations which have to be consistently taken into account. While ordinary perturbation theory is helpless, yielding a diverging series at every order in powers of the potential, one has instead to perform a partial sum over the most im-portant classes of interaction diagrams, as discussed in Sections 1.4 and 1.5, in order to obtain a ﬁnite result.

In the last years so-called low-momentum interactions have been proposed [35]. The

idea is to incorporate in the potential, denoted with V_{low k}, some of the short-range

correlations by "integrating out" the momenta larger than a certain cutoﬀ Λ (hence distances smaller then 1/Λ), for example applying a renormalization group equation. It

has been shown that the diﬀerent NN interactions converge to same V_{low k} if the cutoﬀ

is suﬃciently small (Λ ≤ 2 fm−1). This in some way also overcomes the uncertainty in

parameterizing a region which is instead not accessible by the experimental data. All ab-initio microscopic calculations based on these bare or low-momentum two-body potentials, however, are able to reproduce only qualitatively, and not quantita-tively, the properties of few-nucleon systems and the saturation behavior of nuclear matter. It is commonly established that this deﬁciency derives from neglecting three-and four-body forces between nucleons in the dense media. We shall discuss in detail in Chapter 3 the nature of many-body forces and how phenomenological three-body forces can and should be included in realistic calculations.

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1.3 Real time Green’s functions 9

1.3 Real time Green’s functions

Green’s functions constitute a powerful theoretical method to to tackle the problem of nuclear matter. They are particularly suitable to investigate consistently the modifica-tions of the single-particle propagation, with respect to the free case, influenced by the strong interactions with the surrounding medium, and of the interaction itself, which is affected by the changes of the particle properties. This connection is formalized with the concept of self-energy, a functional of the potential and the propagator accounting for medium-induced alterations of the one-body properties. The central objects of the theory, the particle propagators, have direct interpretation and are connected with all physical observables. The formulation in terms of Feynman diagrams, in addition, represents a possible help in the calculation of the different quantities.

In the following we introduce the theory of Green’s functions at ﬁnite tempera-ture3

(sometimes called thermodynamic Green’s functions) for a many-body system at equilibrium and apply it to the picture of nuclear matter. From this point of view the propagators have the double role of describing microscopically the particle dynamics and at the same time containing information about the statistical mechanical proper-ties of the system, related to the macroscopic physical observables.

Let us ﬁrst consider, in the Heisenberg representation, the second-quantization an-nihilation operator4

ψ(r, t), which destroys a particle at the point r, t, and creation

operator ψ†_{(r, t), which creates a particle at the point r, t. They fulﬁll the equal-time}

anticommutation relations

{ψ(r, t), ψ†(r′_{, t)} = δ(r − r}′) , (1.5a)

{ψ(r, t), ψ(r′, t)} = {ψ†(r, t), ψ†(r′_{, t)} = 0 ,} (1.5b)

which in second quantization guarantee the correct antisymmetrization of the fermionic wave functions. All the macroscopic operators of physical interest can be expressed in

terms of ψ(r, t) and ψ†(r, t). For example

n(r, t) = ψ†(r, t) ψ(r, t) (1.6)

represents the density of particles at point r at time t. The total number of particles is then

N (t) = Z

dr ψ†(r, t) ψ(r, t) . (1.7)

3

In this thesis we work with the real time approach of finite temperature Green’s functions, alter-native but completely equivalent to the the Matsubara or imaginary time formulation.

4

For convenience we are not going to write explicitly, in the following, the spin and isospin in-dices. The operators should be introduced as ψAα(r, t), where A is the spin and α the isospin of the

annihilated particle, and accordingly all other quantities. We will write spin and isospin indices ex-plicitly when needed; in general one should be able to write down without difficulties the spin-isospin coordinates at every step of the discussion.

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The dynamics of the system is determined by the Hamilton operator H(t). Assuming

that the particles interact via a two-body potential local in time V (r, t; r′_{, t}′_{) =}

V (r, r′_{) δ(t − t}′), it can be written as H(t) = 1 2m Z dr ∇ψ†(r, t) ∇ψ(r, t) + 1 2 Z dr dr′ψ†(r, t) ψ†(r′, t) V (r, r′) ψ(r′, t) ψ(r, t) . (1.8) The physical observables are then obtained by computing expectation values of such operators. When we consider systems at zero temperature, these expectation values

are taken with respect to the ground state |Ψ0i. For ﬁnite temperature systems this

is not the case, since they are in general distributed over a certain number of excited (non-stationary) states. To describe the time evolution of the macroscopic observables we have then to introduce statistical mechanics and, by means of the density matrix

operator ˆρ, we compute the expectation value of any operator ˆO as

h ˆOi = trhρ ˆˆOi. (1.9)

The fundamental object of the theory, the single-particle Green’s function or propa-gator G, is deﬁned as the expectation value (in the sense of (1.9)) of the time-ordered product of an annihilation and a creation operator:

i G(r, t; r′, t′) =DT ψ(r, t) ψ†(r′, t′)E . (1.10)

This is also known as the causal Green’s function because of the presence of the time-ordering operator T. The knowledge of the single-particle propagator is suﬃcient to compute the expectation values of all one-body operators.

There exist various techniques to evaluate G. In the zero temperature theory one can express the perturbative expansion of the propagators by means of the Feynman diagrams. This is the standard procedure in the interaction picture which involves the adiabatic theorem and the Wick decomposition. The same scheme however cannot be applied to the finite temperature case because of the different structure of the time evolution operator, which reflects the impossibility of identifying the states at t = ∞ with any of the states at t = −∞. The formal analogy with the zero temperature case can be restored by substituting the time axis with a closed time contour (Schwinger [36] and Keldysh [37]) over which the time variable of any function is defined. The time-ordering operator in (1.10) is replaced by T, which orders times on the contour, and the single-particle propagator is defined as

i G(r, t; r′, t′) =DT_{ψ(r, t) ψ}†_(r′_{, t}′₎E _(1.11)

After performing the Wick decomposition one can obtain Feynman rules for the ﬁnite temperature case which result similar to the ground-state ones [38].

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1.3 Real time Green’s functions 11

We may introduce also the two-particle propagator on the contour i2G2(r1, t1, r2, t2; r′1, t′1, r′2, t′2) = D T_ψ(r₁_{, t}₁_{) ψ(r}₂_{, t}₂_{) ψ}†_(r′ 2, t′2) ψ†(r′1, t′1) E . (1.12)

The equations of motion for G involve the two-particle Green’s function G2, which

in its turn is determined through the three-particle propagator G3, and so on, giving

rise to a system of coupled integro-diﬀerential equations. As we will see in the next paragraph, in order to solve the system this hierarchy must be interrupted by means

of some approximation, usually introduced at the level of G2.

As this approach is very powerful in describing the behavior of many-body systems also out of the equilibrium, we will consider in this work, however, only systems in thermodynamic equilibrium, for which the appropriate statistical mechanical descrip-tion is given by the grand canonical ensemble. As a consequence, the expectadescrip-tion value (1.9) reads

h ˆOi =

trhe−β(H−µN )Oˆi trhe−β(H−µN )i ,

(1.13)

where the parameters µ, the chemical potential, and β = 1/T , the inverse temperature, characterize the thermodynamic state of the system. The quantity in the

denomina-tor is identiﬁed with the grand-canonical partition function ZGC, which can be also

expressed in terms of the thermodynamic potential Ω (T, µ)

ZGC = tr

h

e−β(H−µN )i= e−β Ω. (1.14)

For many applications it is often more convenient to work in momentum space. In order to Fourier-transform the propagators we need to work with times which lie on the real axis and not on a contour. Thus we can introduce a matrix structure for the contour Green’s function (we use now the notation 1 ≡ (r1, t1) and d1 ≡ dr1dt1)

i G(1, 1′) = i Gc(1, 1′) G<(1, 1′) G>(1, 1′) Ga(1, 1′) = T ψ(1) ψ†₍₁′₎ ₋_ψ†₍₁′_{) ψ(1)} ψ(1) ψ†(1′) Taψ(1) ψ†(1′) , (1.15)

where Ta is the anti-chronological time operator. The four Green’s functions (which

have now real time arguments) correspond to the four possible placements of the two times on the two branches of the contour. These functions are not independent of each other but fulﬁll the relation

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Instead of working with Gc _{and G}a_{, it is useful, because of their direct physical}

interpretation, to introduce other two quantities, the retarded and advanced Green’s

functions GR and GA GR(1, 1′) = θ(t1− t′1) [G>(1, 1′) − G<(1, 1′)] , (1.17a) GA(1, 1′) = θ(t′₁_{− t}1) [G<(1, 1′) − G>(1, 1′)] . (1.17b) It follows that GR(1, 1′_{) − G}A(1, 1′) = G>(1, 1′_{) − G}<(1, 1′) , (1.18) and since [GA(1, 1′)]∗ = GR(1, 1′) 2i Im GR(1, 1′) = G>(1, 1′_{) − G}<(1, 1′) . (1.19)

Some useful rules to derive equations for the diﬀerent components of the matrix G are the Langreth-Wilkins formulae [39]. Valid for any quantity deﬁned on the contour, they state that when we have a convolution of path-ordered functions

C(1, 1′) = Z C d2 A(1, 2) B(2, 1′) (1.20) if follows C≷(1, 1′) = Z d2hAR(1, 2) B≷(2, 1′) + A≷(1, 2) BA(2, 1′)i, (1.21a) CR(A)(1, 1′) = Z d2 AR(A)(1, 2) BR(A)(2, 1′) . (1.21b)

A feature of systems at the equilibrium is that the Green’s functions depend only on the diﬀerences of their arguments, i.e. G(r, t; r′, t′_{) = G(r − r}′_{, t − t}′). Then we can deﬁne the Fourier transforms as follows

G(r − r′, t − t′) = Z _dp (2π)3 dω 2π e i p (r−r′₎ e−i ω(t−t′)G(p, ω) . (1.22)

The smaller and larger Green’s functions G< and G> are particularly important

be-cause of they can be interpreted as densities of particles and holes in the medium. Starting from their deﬁnition as grand-canonical averages, in momentum space one can derive a useful relation, known as the Kubo-Martin-Schwinger condition [40, 41]:

−i G<(p, ω) = e−β (ω−µ)i G>(p, ω) . (1.23)

If now we introduce a new quantity, the spectral function A(p, ω), it is possible to re-express G> and G< such that they automatically fulﬁll (1.23)

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1.4 Approximations for the two-particle propagator 13

−i G<(p, ω) = f (ω) A(p, ω) , (1.24b)

where

f (ω) = 1

eβ (ω−µ)+ 1 (1.25)

is the Fermi-Dirac distribution. This implies that in equilibrium it is suﬃcient to determine only one quantity to have complete knowledge of all the diﬀerent one-body propagators. The physical interpretation of the spectral function is the following. We

can identify −i G< _{with the average density of particle with momentum p and energy}

ω

−i G<(p, ω) = hn(p, ω)i . (1.26)

While the distribution function tells us about the occupation of the various modes, characterized by an energy ω, A(p, ω) determines the spectrum of the possible energies for a particle with momentum p. The total weight must be equal to unity

Z dω

2πA(p, ω) = 1 . (1.27)

For free particles, since no scattering can spread the energy spectrum, A(p, ω) is proportional to a delta function

A0(p, ω) = 2π δ(ω − p2/2m) . (1.28)

When an interaction is present, this is no longer the case: the spectral function has a non trivial structure and is non-zero over the entire energy range. For some momenta

(close to the Fermi momentum pF), A(p, ω) still shows a peak which resembles the

one of free particles. One can associate this behavior with the concept of quasiparti-cle, and assume that the system is composed of these weakly-interacting long-living excited states. In nuclear matter, however, the strong correlations between particles in the medium cannot be neglected and the quasiparticle picture becomes a very crude approximation when moving away from the Fermi surface.

1.4 Approximations for the two-particle propagator

Starting from the equations of motion for ψ(r, t) or ψ†(r, t) it is possible to derive an

equation of motion for the single-particle Green’s function on a contour i ∂ ∂t1′ +∇ 2 1′ 2m G(1, 1′) = δ(1, 1′_{) − i} Z dr2V (r1− r2) G2(1, r2, t1; 1′, r2, t+₁) (1.29)

where t+₁ represents a time inﬁnitesimally larger than t1 on the contour. Similarly one

can obtain an equation for G2 which involves the three-particle propagator G3 and

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of coupled integro-diﬀerential equations, known as the Martin-Schwinger hierarchy [41], is usually solved by making some approximation at the level of the two-particle

propagator, i.e. expressing G2 in terms of combinations of G’s and the potential

V . A convenient way of introducing such approximations is to deﬁne a new quantity, the self-energy Σ, constructed of V and G only and connected with the two-particle Green’s function via

Z

C

d3 Σ(1, 3) G(3, 1′_{) = −i} Z

dr2V (r1− r2) G2(1, r2, t1; 1′, r2, t+1) . (1.30)

Any approximate G2 corresponds to an approximation of Σ, with the advantage of

working at the one-particle level. In fact, by substituting (1.30) in the right hand side of (1.29) one derives the Dyson equation

G(1, 1′)−1= G0(1, 1′)−1− Σ(1, 1′) (1.31)

for the single-particle propagator. Here the power −1 has to be understood also as the inverse of the matrix (1.15). The inverse functions of the various matrix elements are well deﬁned in momentum space because of the homogeneity of the system. According to the Langreth-Wilkins rules, eq. (1.21) assumes a simple form for the retarded and advanced propagators.

It is often convenient to express the self-energy by means of diagrams. Because of the hard core of nuclear interactions, every term of the perturbation expansion is divergent and ordinary perturbation theory is not applicable. One has instead to select a class of diagrams which yield the most relevant contributions and perform sums of these objects up to inﬁnite order in powers of the potential. In nuclear matter this

par-tial summation usually includes all the so-called ladder diagrams, which incorporate

subsequent scatterings between two particles. First proposed by Brückner [42], these repeated interactions assure that the short distance region, in which the nuclear force is strongly repulsive, is properly treated. The selection of the ladder contributions leads to the T-matrix approximation, which is treated in detail in the next section. First we brieﬂy introduce some simpler approximation which may be useful as an introduction and will be used during the work for particular purposes.

The simplest case one can think of is the two-particle Green’s function being con-stituted by two uncorrelated single-particle propagators

G2(12; 1′2′) = G(1, 1′) G(2, 2′) . (1.32)

We can imagine (1.32) expressed diagrammatically in Fig. 1.1, where the lines repre-sent the propagation of a nucleon in the medium (i.e. a dressed nucleon). This is the Hartree approximation, and, in momentum space, results only in a shift of the pole of the one-particle propagator with respect to the free case

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1.4 Approximations for the two-particle propagator 15

G2 =

Figure 1.1: Hartree approximation for the two-particle Green’s function. with the Hartree self-energy (ρ is the number density of particles)

ΣH = ρ V (0) . (1.33b)

If we take into account the possibility of an exchange between the two (indistinguish-able) particles, we come to the Hartree-Fock approximation (Fig. 1.2)

G2(12; 1′2′) = G(1, 1′) G(2, 2′) − G(1, 2′) G(2, 1′) . (1.34)

The minus sign implies that G2(1, 2; 1′2′) = −G2(1, 2; 2′1′). The spectral function

G2 = −

Figure 1.2: Hartree-Fock approximation for the two-particle Green’s function. has still a delta-function structure, but the self-energy is now momentum dependent

AHF(p, ω) = 2π δ(ω − p2/2m + ΣHF(p)) , (1.35a) ΣHF(p) = ρ V (0) − Z dp′ (2π)3V (p − p ′_{) n(p}′_{) ,} _(1.35b)

where n(p) is the momentum distribution n(p) =

Z _dω

2π A(p, ω) f (ω) . (1.36)

The Hartree and Hartree-Fock approximations do not take into account any corre-lations between particles in the medium. The particles propagate independently in a mean ﬁeld which reﬂects of the presence of the other nucleons, but cannot jump out of their stable single-particle states.

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If we allow scatterings between the two nucleons they can instead be excited to other states, with a consequent spread of the possible energies. In Fig. 1.3 we show the second order Born approximation, in which the two particles either propagate freely or scatter once (including exchange processes). The two-body Green’s function is written

G2 = +

− −

Figure 1.3: Second order Born approximation for the two-particle Green’s function. as follows G2(12; 1′2′) = G(1, 1′) G(2, 2′) − G(1, 2′) G(2, 1′) + i Z C d3 dr4V (r3− r4) [G(1, 3) G(3, 1′) G(2, 4) G(4, 2′) − G(1, 3) G(3, 2′) G(2, 4) G(4, 1′)] t4=t3 . (1.37)

The self-energy can be divided into two parts

ΣBorn= ΣHF + Σd, (1.38)

where the new contribution Σd is now fully deﬁned on the contour and accounts for

the collisions between particles. We can write down its expression in momentum space as follows Σd(p, ω) = Z _dp′ (2π)3 dp′′ (2π)3 dω′ 2π dω′′ 2π 1 2 V (p − p′) − V (p′− p′′)2 ×G(p′, ω′) G(p′′, ω′′) G(p + p′_{− p}′′, ω+, ω′_{−, ω}′′) . (1.39)

The real and imaginary parts of Σd are related via the dispersion formula

Re ΣR (A)_d (p, ω) = P Z dω′ π Im ΣR (A)_d (p, ω′₎ ω′_{− ω} , (1.40)

where P denotes the principal value of the integral.

1.5 T-matrix

In the diagrammatic expansion of the nucleon self-energy, the partial summation of the

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1.5 T-matrix 17

matter the inclusion of such diagrams is necessary to handle the strong repulsive core of nuclear interactions, which induce short-range correlations between particles in the medium.

In the T-matrix approximation [14] the two-particle propagator is then expressed as an inﬁnite sum of terms in which two nucleons, propagating in the medium, interact zero, one, two, .. n times as in Fig. 1.4. It follows that the integral equation for the

G2 = + + + ...

− − − + ...

Figure 1.4: Ladder approximation for the two-particle Green’s function. two-particle Green’s function is

G2(12; 1′2′) = G(1, 1′) G(2, 2′) − G(1, 2′) G(2, 1′)

+ i Z

C

d3 d4 G(1, 3) G(2, 4) V (3, 4) G2(34; 1′2′) . (1.41)

This equation can be written in another form after introducing the T-matrix

(dia-T = + + + ... Figure 1.5: T-matrix. grammatically in Fig. 1.5) T as T(12; 34) = V (1, 2) δ(1, 3) δ(2, 4) + Z C d5 d6 V (1, 2) G(1, 5) G(2, 6) T(56; 34) . (1.42) Then (1.41) is equivalent to G2(12; 1′2′) = G(1, 1′) G(2, 2′) − G(1, 2′) G(2, 1′) + i Z Cd3 d4 d5 d6 [G(1, 3) G(2, 4) − G(1, 4) G(2, 3)] × T(34; 56) G(5, 1′) G(6, 2′) (1.43)

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G2 = + T

− − T

Figure 1.6: T-matrix approximation for the two-particle Green’s function. (diagrammatically in Fig. 1.6). Alternatively, deﬁnes the self-energy as

Σ(1, 2) = i Z

C

d3 d4 G(4, 3) T(12; 34) . (1.44)

In this way we obtain a closed set of equations, namely (1.42) and (1.44), together with the Dyson equation (1.31), which have to be solved self-consistently.

In order to work in momentum space (always under the assumption of a system at equilibrium), we apply the Langreth-Wilkins rules to the previous formulae. We ﬁrst notice that, from (1.42) and from the fact that the potential interaction is local in time, i.e. V (1, 2) = V (r1− r2) δ(t1, t2), the time arguments of the T-matrix pairwise

coincide

T(12, 34) = T(r1, r2; r3, r4; t1, t3) δ(t1, t2) δ(t3, t4) . (1.45)

Because of the translational invariance in the system T will depend in fact only on

the time diﬀerence t1− t3, which means that its Fourier transform is a function of

one frequency or energy, denoted by Ω. Of the four momenta that we obtain, because of the total momentum conservation at every scattering only three are independent. We then use for the two-particle functions the notation (see Appendix A.1 for details)

hk|T (P, Ω)|k′_{i, where k is the relative momentum of one of the incoming particles}

(with respect to the total momentum of the pair P), and k′ the relative momentum

of the same particle after the interaction. Then k − k′ is the momentum exchanged

during the scattering.

We indicate with non-correlated Green’s function the simplest approximation for

G2 already introduced as the Hartree propagator (1.32). Its smaller and larger

com-ponents (the retarded and advanced terms in general can be derived through (1.18) and (1.19)) read hk′_|Gnc <(>) 2 (P, Ω)|ki = i (2π)3_{δ(k − k}′) Z dω′ 2π G <(>) (P/2 + k, Ω − ω′) G<(>)_{(P/2 − k, ω}′) . (1.46)

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1.5 T-matrix 19

In the T-matrix approximation the two-particle Green’s function in momentum space is then obtained from (1.43) [38]

×hp|TR(A)(P, Ω)|qihq|Gnc R(A)2 (P, Ω)|k′i . (1.47)

From (1.44) and using (1.45) the larger and smaller components of the self-energy can be obtained: i Σ≷(p, ω) = Z dp′ (2π)3 dω′ 2π G ≶_(p′_{, ω}′₎ p_{− p}′ 2 T≷(p + p′, ω + ω′) p− p ′ 2 − p_{− p}′ 2 T≷(p + p′, ω + ω′) p ′_{− p} 2 . (1.50)

The imaginary part of ΣR _{is then computed through (1.19) and reads}

Im ΣR(p, ω) = Z _dk (2π)3 dω′ 2π p_{− k} 2 Im TR_{(p + k, ω + ω}′₎p− k 2 − p_{− k} 2 Im TR_{(p + k, ω + ω}′₎k− p 2 × b(ω + ω′) + f (ω′)A(k, ω′) , (1.51) where b(ω) = 1 eβ (ω−µ)_{− 1} (1.52)

is the Bose-Einstein distribution. The real part of the self-energy is ﬁnally calculated through the dispersion relation (cf. (1.40))

Re ΣR(p, ω) = ΣHF(p, ω) + P Z dω′ π Im ΣR(p, ω′) ω′_{− ω} (1.53)

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Eqs. (1.51) and (1.53) allow us to obtain the self-energy once the retarded T-matrix is known. The self-energy is then used to calculate the spectral function (which deﬁnes the single-particle propagator) via the retarded Dyson equation

GR (A)(p, ω)−1 _{= ω −} p

2

2m − Σ

R (A)_{(p, ω) .} _(1.54)

The spectral function can be inserted again in (1.48) through Gnc R

2 : (1.48), (1.51),

(1.53) and (1.54) thus constitute, in momentum space, the set of equations which has to be solved iteratively until convergence is achieved.

One can derive an additional equation which relates the spectral function to the real and imaginary parts of the self-energy, thus alternative to (1.54), valid in general

in the presence of a dispersive contribution Σd:

A(p, ω) = −2 Im Σ

R_{(p, ω)}

ω − p2/2m − Re ΣR(p, ω)2+Im ΣR(p, ω)2 . (1.55)

1.6 Energy of the interacting system

We have seen that the from the single-particle propagator one can compute the ex-pectation values of all one-body operators. A basic quantity which characterizes a many-body system is the total internal energy, i.e. the expectation value of the total

Hamiltonian: E = hHtoti. Since we are dealing with an inﬁnite system it is convenient

to consider the total energy per particle E N = 1 ρ hHtoti V = 1 ρ hHkini V + hHpoti V . (1.56)

Here N represents the total number of particles, V the volume of the system and ρ = N/V the particle number density (ρ remains ﬁnite when N, V → ∞). The Hamiltonian can be split in two parts: a single-particle operator accounting for the

kinetic energies hHkini and the contribution from the interaction energy hHpoti. While

the ﬁrst contribution can be easily evaluated hHkini = V Z dp (2π)3 dω 2π p2 2mA(p, ω)f (ω) , (1.57)

all the diﬃculties appear in the second term hHpoti = V 2 Z dP (2π)3 dΩ 2π dk (2π)3 dk′ (2π)3V (k, k ′_)hk′_|G< 2(P, Ω)|ki , (1.58)

which involves the calculation of the two-particle propagator. When inserting the

chosen approximation for G2 into (1.58) one obtains a corresponding diagrammatic

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1.7 Thermodynamic properties and consistency 21

Under the assumption that the potential contains two-body forces only one can

express hHpoti in terms of single-particle operators and come to a simple formula for

the total energy known as the Galitskii-Koltun sum rule [43, 41, 44]

EGK N = 1 2 ρ Z dp (2π)3 dω 2π p2 2m + ω A(p, ω) f (ω) . (1.59)

This expression yields a convenient way of calculating together the kinetic and the interaction energies. When the exact solutions for the spectral function are consid-ered, as well as for conserving approximations, it is equivalent to the direct calculation (1.56). The Galitskii-Koltun formula, however, cannot be always applied. In particu-lar when three-body forces are present it looses its validity, and the interaction energy has to be evaluated directly through the diagrammatic expansion (1.58).

After some algebra (it is convenient here to work in the imaginary time formalism) one obtains a more useful expression involving the retarded components in which appears the Bose-Einstein distribution b(ω)

The series of diagrams which contribute to the interaction energy is shown in Fig. 1.7. It is expressed as a sum over the number of interaction lines n of ladder diagrams, with the external legs closed by a non-correlated two-particle propagator (direct and exchange contributions). This inﬁnite sum can be more conveniently written and more easily computed in terms of the T-matrix.

Once the solution of the set of the T-matrix equations is obtained, the last step

thus is the calculation of hHpoti and the energy per particle. As we will see in the

following, the diagrammatic method presents some advantages in terms of consistency and convergence. The whole expansion constitutes the full T-matrix approximation to the interaction energy. Moreover one can cut the series at the ﬁrst or second order in n and obtain respectively the Hartree-Fock or second Born approximations.

1.7 Thermodynamic properties and consistency

The ﬁnite-temperature Green’s functions, deﬁned as averages in the grand-canonical ensemble, contain information about the statistical mechanics properties of the

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many-hHpoti = X n 1 4 " ... ₋ ... # = 1 2 " T − T #

Figure 1.7: Interaction energy; n represents the number of interaction lines in each diagram.

body system. They are connected to the grand-canonical partition function, which can be constructed in terms of diagrams that contain the single-particle propagator and from which one can obtain all the thermodynamic observables at the equilibrium. There are various methods of deriving the partition function from G. They are all equivalent in the exact theory, but they might lead to diﬀerent results when an approximation for the two-body Green’s function is used. This inconsistency is then reﬂected into the thermodynamic observables when they are calculated as derivatives of the partition function with respect to the thermodynamic parameters.

The problem was addressed by Baym and Kadanoﬀ [45, 46] who proved that there exists a class of approximations which automatically fulﬁll all the consistency require-ments. These approximations are related to the existence of a closed functional of the single-particle propagator G and the potential V , indicated by Φ[G, V ], from which the self-energy Σ[G, V ] must be derived according to

Σ(1, 1′) = δ Φ[G, V ]

δ G(1, 1′₎ . (1.62)

The functional Φ is directly connected to the logarithm of the partition function, which

at the equilibrium can be identiﬁed with the grand-canonical potential Ω, as follows5

Ω = −tr{ln[G−1]} − tr{Σ G} + Φ . (1.63)

In Ref. [46] general prescriptions for constructing Φ are outlined. The Hartree, the Hartee-Fock and the T-matrix are all Φ-derivable approximation. In Fig. 1.8 we show the diagrammatic expansion of Φ in the T-matrix case. The series is similar to the one for the interaction energy (Fig. 1.7), diﬀering only by a factor 1/n in front of each diagram.

This is in principle all we need in order to compute all other thermodynamic properties

5

A standard alternative method of calculating the partition function is to integrate the expectation value of the potential energy (1.58) with respect to a varying coupling constant.

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1.7 Thermodynamic properties and consistency 23 Φ = X n 1 2n " ... ₋ ... #

Figure 1.8: Functional Φ; n represents the number of interaction lines in each diagram. of the system. Since Ω = −P V one has also the big advantage of being able to calcu-late the pressure P directly from the diagrammatic expansion, whereas the standard method of deriving P from the free energy

P = ρ µ −_NE + T S N (1.64) requires the prior knowledge of the entropy per particle S/N . If the quantities in (1.64) are not evaluated with suﬃcient precision, the propagation of numerical inaccuracies may lead to big uncertainties on the pressure calculations. From (1.64) we compute instead the entropy

S N = 1 T E N − µ + P ρ (1.65) which can be reliably estimated for suﬃciently large temperatures.

It is possible to derive the entropy directly as a derivative of the grand-canonical potential S = ∂Ω ∂T µ . (1.66)

Carneiro and Pethick [47] proved that the main contribution is given by the so-called dynamical quasi-particle entropy

SDQ N = 1 ρ Z dp (2π)3 dω 2πσ(ω) A(p, ω) 1 − ∂ ReΣ R_{(p, ω)} ∂ω +∂ ReG R_{(p, ω)} ∂ω Γ(p, ω) , (1.67) where σ(ω) = −f(ω) ln[f(ω)] − [1 − f(ω)] ln[1 − f(ω)] , (1.68)

and that all other terms can be usually neglected in the actual calculations.

In a conserving approximation all the diﬀerent ways of getting the thermodynamic observables should lead to the same result, constituting a tool for monitoring the thermodynamic consistency. At zero temperature this check is given by general re-quirements for many-body fermionic systems. In particular hold the Hugenholtz-Van Hove and Luttinger identities [48, 49], which are as well preserved by Φ-derivable approximations. The Hugenholtz-Van Hove theorem states that at saturation density

E

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This is related to two possible ways of calculating the pressure at T = 0

P = ρ2∂(E/N )

∂ρ , (1.70a)

P = ρ (EF − E/N) (1.70b)

(EF is the Fermi energy). At the saturation point, where P = 0, a useful property

follows:

EF = E/N . (1.71)

When applied to the equilibrium the condition (1.62) ensures a consistent thermo-dynamic description; moreover, being derived from the equations of motions for G, leads to a fully conserving transport theory out of equilibrium. It is then a general powerful tool to select certain classes of diagram in the inﬁnite order expansion of the self-energy and to judge the validity of a chosen approximation [46, 50].

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Chapter

2 Self-consistent calculation of

thermodynamic quantities

2.1 Iterative scheme and numerical methods

The knowledge of the nucleon spectral function is sufficient to compute the single-particle propagator, from which we can obtain all one-body properties of the system. In Sec. 1.5 we have seen that, within the T-matrix approximation, the calculation involves a set of coupled equations which have to be solved simultaneously, namely eqs. (1.48), (1.51), (1.53) and (1.55). The solution is achieved via an iterative scheme: first an ansatz for the spectral function is inserted in (1.48) and the T-matrix is computed. From the T-matrix, through (1.51) and (1.53), we derive the self-energy, from which a new spectral function can be obtained using (1.55). This procedure is then repeated until convergence is reached, i.e. until the spectral function has assumed a stable form. A simplification of the T-matrix equation (1.48) may be achieved after a decom-position into partial waves. In vacuum, the NN potential commutes with the total

angular momentum ˆJ = ˆL+ ˆS, with the total spin ˆS2 and isospin ˆT2, so that its

matrix elements between states with diﬀerent (JST ) are vanishing. Because of the tensor force, however, spin-triplet states with L = J ± 1 can mix: we can then write the potential as V_LL(JST )′ (k, k′), where the oﬀ-diagonal elements are non-zero only for

these coupled waves. Not all the (JST ) combinations are allowed since the Pauli prin-ciple (i.e. the antisymmetrization of the wave function) selects only the ones for which (−1)L+T +S = −1. In nuclear matter, however, this constitutes an approximation, since the diﬀerent partial waves do not separate exactly because of the presence of the Pauli operator in the T-matrix equations.

In order to apply the partial wave decomposition it is necessary to use an angle averaged two-particle Green’s function

< hk′|Gnc R2 (P, Ω)|ki >Θ= Z dΘ 4π hk ′_|Gnc R 2 (P, Ω)|ki (2.1)

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in which we are left with a one-dimensional momentum integration only. The imaginary self-energy is then computed as a sum over all partial waves

Im ΣR(p, ω) = 1 8π X (JST ) L (2T + 1) (2J + 1) × Z dk (2π)3 dω′ 2π |p − k| 2 Im TLLR (JST )(|p + k|, ω + ω′) |p − k|₂ − |p − k| 2 Im TLLR (JST )(|p + k|, ω + ω′) |k − p|₂ × b(ω + ω′) + f (ω′)A(k, ω′) . (2.3)

From (2.3) the real part of ΣR is derived using the dispersion relation (1.53), where

the Hartree-Fock contribution is

ΣHF(p) = 1 8π X (JST ) L (2T + 1) (2J + 1) ρ V_LL(JST )(0) − Z dk (2π)3 dω 2πV (JST ) LL (|p − k|/2, |p − k|/2) A(k, ω) f(ω) . (2.4)

The evaluation of the multi-dimensional integrals which appear in the set of equa-tions requires a huge computational eﬀort. We discretize the various quantities in the momentum and energy domains and use for the calculations a Fortran 77 code.

The energy integrations we encounter in the equations are often convolution inte-grals of the type

J(p, ω) = Z dω′ 2π Z q2dq 8π2 Z d cos Θ F1(k+, ω′− ω) F2(k−, ω′) , (2.5) where k±= |p ± q| = p

p2_{+ q}2_{± 2 p q cos Θ. Such formulae can be more conveniently}

calculated by means of the convolution theorem, which expresses a convolution integral as the inverse transform of the product of the two Fourier-transformed functions

Z

dω′F1(ω′− ω) F2(ω′) =F1T(t)F2T(t)

T

. (2.6)

In the program they are performed using fast Fourier transform (FFT) algorithms for numerical convolutions [51] as follows [52]: ﬁrst the two functions are Fourier

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2.1 Iterative scheme and numerical methods 27

transformed so that the energy integral becomes F F T Z _dω′ 2πF1(k+, ω ′_{− ω) F} 2(k−, ω′) = F₁T(k+, t) F2T(k−, t) . (2.7)

The 3-dimensional integral JT(p, t) = Z q2dq 8π2 Z d cos Θ F₁T(k+, t) F2T(k−, t) (2.8)

is then computed with standard methods and the ﬁnal result is obtained by an inverse Fourier transform

J(p, ω) = F F T−1JT(p, t) . (2.9)

This procedure signiﬁcantly fastens the code, but has the drawback of requiring a discretization on an energy grid with a ﬁxed spacing. This can cause problems at low temperatures, where near the Fermi surface the spectral functions rapidly change, presenting very narrow quasi-particle peaks. Following Ref. [20], we tackle these cases by treating separately the sharp peak, approximated by a δ-function, and the background, which instead can be discretized, i.e. we expressed the spectral function as

A(p, ω) = B(p, ω) + 2π Zpδ(ω − ωp) . (2.10)

The separation is controlled by a cut-oﬀ parameter η as we use for the background the deﬁnition B(p, ω) =      −2 Im ΣR(p, ω) (ω − ωp)2+ Im ΣR(p, ω)2 for (ω − ωp) 2₊_{Im Σ}R_{(p, ω)}2_{> η} −2 Im ΣR(p, ω)/η for (ω − ωp)2+Im ΣR(p, ω)2< η . (2.11)

This parameter is set to a value η ≈ 4∆ω2, where ∆ω is the energy spacing in the

discretization. The quantity which determines the position of the quasi-particle peak,

ωp, is obtained from the dispersion relation

ωp =

p2

2m + Re Σ

R_{(p, ω}

p) . (2.12)

Finally, the strenght of the quasi-particle component Zp is ﬁxed by requiring for each

momentum the conservation of the sum rule (1.27).

It is known that the T-matrix presents some instabilities at low temperature, re-lated to the onset of superfluidity [15]. In order to treat properly this low temperature regime, one should introduce anomalous propagators, which account for pairing, in ad-dition to the ordinary nucleon Green’s functions. This procedure is rather complex and requires serious theoretical and computational efforts. The first attempts have been made by Bożek with a quasiparticle approximation or a separable interaction [52, 53]. On the other hand one can try to study the effects of the dressing of nucleons on the gap equation, in general finding a substantial reduction of the gap [52, 54, 55, 56].

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In this thesis we do not include anomalous propagators. Physically, this approxima-tion does not aﬀect much the macroscopic observables because of the smallness of the superﬂuid gap. Thanks to the technical solutions outlined in this section, moreover,

our code is very stable also when going below T_c∆, yielding reliable results at zero

temperature. This is however one of the problems which should be addressed in the future developments of the Green’s functions approach to nuclear matter.

2.2 Parameters and cutoff dependence

In each calculation the temperature T and the density ρ are ﬁxed at the beginning and are inputs to the program. The chemical potential µ is adjusted at each iteration by requiring ρ = Z µ −∞ dω 2π Z p2dp 8π2 A(p, ω) , (2.13)

so that there no free parameters when two-body forces are used.

The ansatz for the spectral function to be used in the ﬁrst iteration can be, for

each momentum p, a Gaussian in the energy domain, centered at some ω0and fulﬁlling

the sum rule (1.27). Then it typically takes 10-12 iterations to get stable solutions, with the spectral function converging within an accuracy of 1%. If converged spectral functions (at similar T and ρ) are taken as input, this reduces to 4-6 iterations.

The use of a grid with fixed spacing implies that the momentum and energy variables are restricted to finite ranges. We consider single-particle momenta p < 1700 MeV, which means that the total momentum of the nucleon pair is limited to 3400 MeV. Also the energy range [−Λ, Λ] is controlled by a cut-off. We tested several values of Λ between 2 GeV and 10 GeV and found that some of the computed quantities do depend on the choice of this energy scale.

An example concerns the calculation of the energy per particle with the Galitskii-Koltun sum rule (1.59), which yields rather different estimations as we change the cutoff. This can be seen in Figs. 2.2 and 2.2, where E/N is displayed as a function of the inverse cutoff for symmetric nuclear matter, with the CD-Bonn potential, for

diﬀerent densities and temperatures. The energy computed directed from the

dia-grammatic expansion (1.56), on the other hand, turns out to be independent of the choice of the cutoﬀ scale at all densities and all temperatures, within a 0.5 MeV inaccu-racy at T = 0 (1 MeV at high temperatures) which can be attributed to the numerical discretization.

In order to estimate correctly the sum rule output we need to extrapolate its dependence to inﬁnite cutoﬀ. By means of a linear interpolation we can get a value

of EGK/N which can then compared with the stable results of Ediag/N . We notice

that in general the two methods are in agreement, diﬀering by at most 1 MeV over the whole temperature and density range. Supported by this solid behaviour of the

energy observable, in the following we assume the energy per particle to be Ediag/N

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2.3 Energy per particle for different potentials 29 0 1x10-4₂x10-4₃x10-4₄x10-4₅x10-4 -18 -16 -14 -12 -10 0 1x10-4 ₂x10-4 ₃x10-4 ₄x10-4 -22 -20 -18 -16 -14 -12 -10 1/Λ ρ = 2 ρ₀ 0 1x10-4 ₂x10-4 ₃x10-4 ₄x10-4 -18 -16 -14 -12 -10 -8 -6 -4 1/Λ ρ = 3 ρ₀ E/N [MeV] 1/Λ ρ = ρ₀

Figure 2.1: Cutoﬀ dependence of the energy per particle at T = 0 for three diﬀerent densities ρ = ρ0, 2 ρ0, 3 ρ0. Empty symbols represent calculations from the

Galitskii-Koltun sum rule (1.59), ﬁlled squares the energy per particle from the diagrammatic expansion (1.60). 1x10-4 2x10-4 3x10-4 4x10-4 5x10-4 -16 -14 -12 -10 -8 -6 0 ₁x10-42x10-43x10-44x10-45x10-4 -12 -10 -8 -6 -4 T = 14 MeV 1/Λ 0 ₁x10-42x10-43x10-44x10-45x10-4 -4 -2 0 2 T = 20 MeV T = 8 MeV E/N [MeV] 1/Λ 1/Λ

Figure 2.2: Cutoﬀ dependence of the energy per particle at ρ = ρ0 for three

dif-ferent temperatures T = 8, 14, 20 MeV. Empty symbols represent calculations from the Galitskii-Koltun sum rule (1.59), ﬁlled squares the energy per particle from the diagrammatic expansion (1.60).

dependent quantities, we shall use the extrapolation at Λ → ∞ as for the Galitskii-Koltun calculations.

2.3 Energy per particle for different potentials

The modern realistic NN potentials are the ones which reproduce all the available data from nucleon-nucleon scattering and deuteron properties with an accuracy of

χ2_{/datum ≈ 1. Models belonging to this class are the Argonne V18 [34], the}

CD-Bonn [32] and the Nijmegen [33] potentials. We implemented them in the T-matrix formalism and computed the energy per particle at zero temperature for diﬀerent

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0.5 1 1.5 2 2.5 3 -20 -15 -10 -5 0 5 10 T CD-Bonn T Nijmegen T A18 T Reid APR BHF E/N [MeV] ρ / ρ₀

Figure 2.3: Energy per particle in symmetric nuclear matter at zero temperature from T-matrix calculations with diﬀerent potentials. Variational [2] and BHF [57] calculations with the Argonne v18 potential are reported for reference.

densities in the range ρ ∈ [0.4 ρ0, 3 ρ0]. In addition, in a fourth calculation we tested

the Reid potential [58], more easily treatable because of its soft core behaviour. Results are displayed in Fig. 2.3, where we reported for reference also the energy per particle obtained in other two microscopic approaches, the variational method [2] and the Brückner-Hartree-Fock approach [57].

None of the calculations succeeds in reproducing correctly the saturation properties of symmetric nuclear matter. In general they yield more binding than the experimental

measurement EB = −16 ± 1 MeV and most importantly the saturation density is

much higher than the empirical value ρ0 = 0.16 ± 0.01 fm−3. The four potentials we

considered have a qualitative similar behaviour but do yield different estimations of the energy per particle and the saturation point. At low densities they differ by few MeV, while the disagreement becomes larger at high density, with the Reid and AV18 being much stiffer than CD-Bonn and Nijmegen.

We compute E/N also using the V_{low k} interaction [35] both within the full

T-matrix and the second Born approximation, i.e. diagrams up to the second order in the interaction. The results are shown in Fig. 2.3 and compared with the CD-Bonn