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Title Theoretical Study of Quasiparticle Excitations in Bose-Einstein condensates
Author(s) 筒井, 和政
Citation 北海道大学. 博士(理学) 甲第12365号
Issue Date 2016-06-30
DOI 10.14943/doctoral.k12365
Doc URL http://hdl.handle.net/2115/62455
Type theses (doctoral)
File Information Kazumasa_Tsutsui.pdf
Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP
DOCTORAL THESIS
Theoretical Study of Quasiparticle Excitationsin Bose-Einstein condensates
(ボーズ・アインシュタイン凝縮相における準粒子励起の研究)
Kazumasa Tsutsui
Department of Physics, Graduate school of science, Hokkaido University
June, 2016
Contents
I Preface 4
II Results of Conserving-Gapless Theory 7
1 Perturbation expansion for BEC 71.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Expressions of first and second self-energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Self-consistent equation in momentum space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Analysis including 1PR diagrams 132.1 Structure of 1PR diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 First-order analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Discussion for second order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Construction of third-order equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Equations for FLEX approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Numerical results at T = 0 193.1 Energy per particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Lifetime of one-particle excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
III Monte Carlo Study of Quasiparticle Excitations 25
4 Construction of trial wave function 254.1 Hamiltonian and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Variational optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Expression of quasiparticle excitations by moments 275.1 Density correlation function and Dynamic structure factor . . . . . . . . . . . . . . . . . . . . . . 285.2 Expressions of moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6 Numerical results 296.1 Density matrices and some functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2 lifetime estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
IV Summary and Conclusion 31
7 Summary and Conclusion 33
V Appendix 34
A Conserving-Gapless Theory 34A.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
A.1.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34A.1.2 Matsubara Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
A.2 Relations for BEC system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34A.2.1 Time evolution operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35A.2.2 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35A.2.3 Dyson-Beliaev equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.2.4 Hugenholtz-Pines relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1
A.3 Construction of Luttinger-Ward functional Φ for BEC . . . . . . . . . . . . . . . . . . . . . . . . 38A.3.1 Interaction Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38A.3.2 Luttinger-Ward functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.3.3 De Dominicis-Martin theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40A.3.4 Exact relations with Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40A.3.5 Constructing Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
A.4 Two-particle Geen’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42A.4.1 Extra perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.4.2 Linear response of G and Ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.4.3 Bethe-Salpeter equation for BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
B Quantum Monte Carlo Methods 47B.1 Green’s Function Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
B.1.1 Outline of DQMC method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47B.1.2 GFMC method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
B.2 Variational Quantum Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51B.2.1 Algorithm of VQMC method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51B.2.2 Trial wave function for bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
B.3 Finite size effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52B.4 Expressions of Physical Quantities for Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
B.4.1 Density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52B.4.2 The moment method for two-particle excitation . . . . . . . . . . . . . . . . . . . . . . . . 53B.4.3 Extensiton for one-particle excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
C Two-particle scattering theory 63C.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63C.2 Appications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2
Acknowledgements
I would like to express my gratitude to Professor Takafumi Kita for his cooperation and persistent helpthroughout my present study. I also would like to acknowledge Professor Koji Nemoto, Professor Migaku Odaand Professor Kenji Kondo for many helpful suggestions and a critical reading of my thesis. I also thankProfessor Yusuke Kato for providing me fruitful discussion regarding the moment method.
Finally, I thank all members of Condensed Matter Theory II, my parents and all my friends for theirencouragement.
3
Part I
PrefaceBose-Einstein condensation (BEC) [1, 2, 3] is typical of the spontaneous symmetry breaking which is one offundamental concepts in modern physics. BEC has been studied actively in both experimental and theoreticalapproaches. In the present thesis, we will confirm validities of results by previous studies. We first summarizethe historical backgrounds of BEC and point out some issues.
In an assembly of N identical bosons, a macroscopic number of particles occupy the lowest single-particlestate below a finite temperature Tc. If we assume an ideal Bose gas, all particles of the system lies in thelowest-energy state at T = 0. Such a phase transition is called Bose-Einstein condensation, which is one oftypical examples of U(1) symmetry breaking [4]. BEC was first pointed out in 1924. Bose reproduced conciselythe Planck’s formula by introducing a new occupation rule for the phase space of phonon [5]. The next year,Einstein predicted the existence of BEC by extending the theory to an ideal gas with mass [6, 7]. Afterward,superfluid 4He was discovered by Kapitza in 1938 [8]. London pointed out that the superfluidity may be causedby BEC [9]. Superfluid 4He also exhibits unique phenomena such as film flow and fountain effect. Tisza andLandau gave a phenomenological description called two fluid model, which regarded the condensate and excitedphases as perfect and normal fluid respectively[10, 11, 12]. However, since 4He particles interact strongly witheach other, superfluid 4He cannot be described qualitatively by the theory of Einstein. Thus, a realization ofBEC in a weakly interacting system was desired. On the other hand, in 1947, Bogoliubov started a study ofBEC with quantum field theory [13]. The perturbation theory of BEC in a dilute Bose system by using Green’sfunction was constructed in 1958 by Beliaev [14]. This method is used even now. In experiments, the lasercooling method for atomic gases was invented by Hansch and Shallow in 1975 [15]. Finally in 1995, Cornelland Wieman et.al. realized the BEC in dilute 87Rb gas [16]. The atomic gases are weakly interacting systemsso that a good agreement between experimental and theoretical results are expected. The realization of atomicBEC has caused a revival of theoretical interest in BEC.
We next outline the previous results of theoretical studies on BEC. We here consider a weakly interactingBose system, which was first considered by Bogoliubov as a microscopic model for superfluid 4He. It considersidentical bosons with mass m and spin 0 interacting via a contact potential V (r − r′) ≡ gδ(r − r′). Thecorresponding Hamiltonian H in the second quantized form is given by
H =
∫dr ψ†(r)Kψ(r) +
g
2
∫dr ψ†(r)ψ†(r)ψ(r)ψ(r),
where K ≡ p2
2m−µ is the kinetic energy operator measured from the chemical potential µ. Bogoliubov consideredthe uniform system at T ≈ 0. We express the field operator as a sum of the condensate wave function Ψ(r) ≡⟨ψ(r)⟩ and the quasiparticle field ϕ(r) as
ψ(r) = Ψ(r) + ϕ(r),
where ⟨· · · ⟩ denotes the grand canonical average, and ⟨ϕ⟩ = 0 by definition. We also introduce the condensatedensity n0 ≡ N0/V with N0 the condensate particle number. In uniform systems, we can set Ψ(r) → n0. Wethen substitute the decomposed expression into the Hamiltonian and approximate it in bilinear forms in termsof ϕ. Diagonalizing the resulting Hamiltonian for uniform systems yields the famous Bogoliubov spectrum
E(B)k ≡
√εk(εk + 2gn0),
where εk ≡ ~2k2/2m. Note that the Bogoliubov mode has linear dispersion at k→ 0.
In 1958, Beliaev constructed a perturbation theory for the quasiparticle field ϕ. Using this theory, thefollowing theorems for excitation of BEC were “proved”:
(i): One-particle excitation, which corresponds to the pole of one-particle Green’s function, must be a gaplessexcitation. (Hugenholtz-Pines theorem) [17]
(ii): Two-particle Green’s function share common poles with one-particle Green’s function. Therefore theone-particle excitation is same as the two-particle excitation. (Gavoret, Nozieres) [18]
4
The theorem (i) corresponds to the Goldstone’s theorem of quantum field theory [19, 20]. Also, the converseof the theorem (ii) doesn’t necessarily hold. However, the “proof” of the theorem (ii) has an ambiguity in thedefinition of Green’s functions, raising a fundamental question of whether their proof is really correct or not.This ambiguity originates from how to definite the skeleton diagrams under the existence of the condensatefields.
Next, we consider theories of BEC at finite temperatures. Girardeau and Arnowitt extended the Bogoliubovtheory in 1959 so as to incorpolate interaction between excitations approximately by a variational treatment(Girardeau-Arnowitt theory) [21]. It amount to retaining the pair correlation term ⟨ϕϕ⟩, which had been ignoredin the Bogoliubov theory, and diagonalizing the quadratic Hamiltonian. The Girardeau-Arnowitt theory wasrenamed “Hartree-Fock-Bogoliubov theory” (HFB theory) by Griffin after the realization of the atomic BEC [22].However, it gives rise to afinite energy gap in the single-particle excitation in contradiction to the Hugenholtz-Pines theorem. On the other hand, Shono proposed a theory which yields a gapless excitation for one-particleexcitation [23, 24]. It omits the pair correlation term in the HFB theory by hand so as to satisfy the Hugenholtz-Pines theorem. However the pair correlation terms must be considered at finite temperatures, so it may lackthe consistency. Also, this approximation cannot be used to describe a dynamical situation, because it breaksconservation laws such as particle number, momentum, energy, and so on. We also note that Griffin called thisapproximation “Popov” approximation [22], whose name becomes widely accepted today. However, there isn’tsuch an approximation in the literature by Popov [25] which was referenced by Griffin, so this approximationcshould be called more apparently as “Shono” approximation.
In 1965, on the other hand, Hohenberg and Martin [26] pointed out another important property:
(iii): Conservation laws should be satisfied.
Hohenberg and Martin classified the existing theories into “conserving” and “gapless” approximations. Theconserving approximation, which is also called Φ-derivable approximation, was first formulated by Baym for anormal state in 1962 [27, 28]. The advantage is to satisfy conservation laws automatically when applying it toa time-dependent problem. It can be also used for the thermal equilibrium state by using the Matsubara form.The HFB theory belongs to this approximation scheme together with the Hartree-Fock theory and BCS theory[29]. Another advantage is to be able to construct the Dyson equation and Berthe-Salpeter equation from thesame functional Φ. On the other hand, the gapless approximation such as the Bogoliubov theory and the Shonoapproximation may describe low-energy properties qualitatively, while it can’t be applied to time-dependentproblems. Hohenberg and Martin named this situation “conserving v.s. gapless” dilemma, which was alsoemphasized by Griffin [22]. Thus, no theory that satisfies (i) and (ii) had been known until recently.
In 2009, Kita constructed a new perturbation theory for BEC which satisfies the Hugenholtz-Pines theoremand conservation laws simultaneously (conserving-gapless theory) [30]. This theory is the extension of the Φ-derivative approximation, which is summarized in the following two steps. First, we construct the functional Φby including diagrams characteristic of BEC, whose numerical wights are still undetermined. Next, we determinethese weights by using the Hugenholtz-Pines theorem. By considering the resulting Green’s function, it turnsout that poles of the one- and the two-particle Green’s function are not shared. Therefore, the collective modeis not same as the one-particle excitation. The new one-particle excitation is called “bubbling mode” [31, 32].Thus in BEC system, there still remain fundamental problems concerning the quasiparticle excitations. We areespecially interested which of the conserving-gapless theory or that by Gavoret and Nozieres (the theorem (ii))is correct.
In this thesis, we will investigate the quasi-particle excitation by the conserving-gapless theory and theexact computational simulation. We here shortly summarize the results by our study. First, we will study thebubbling mode for weakly interacting Bose system at T = 0. We will find that the bubbling mode has a finitelifetime proportional to the s-wave scattering length a [33]. Also, the ground state energy proposed by Lee,Huang, Yang [34] should be modified as
EN
= 4πan
[1 +
16
5
(8
3√π+ cip
)√na3],
where cip is the new additional term of O(1) [35]. The result by Lee et.al. is reproduced by setting cip → 0.We will find that these new results are given by a new class of Feynman diagrams which has been overlookedso far. We will see that they bring one-particle reducible diagrams for the self-energy.
5
Next, we will perform an exact calculation for a Bose system at T = 0 by using diffusion Monte Carlomethod [36, 37, 38, 39, 40]. We will also apply the moment method [41, 42] to the one- and the two-particleexcitations. This approximation is originally used for the analysis of collective modes such as those superfluid4He and magnon system, which corresponds to taking a first-order moment of the frequency ω of the dynamicstructure factor S(q, ω). In this thesis, we will extend it to the higher-order moments and apply it also to thecase of the one-particle spectral function A(k, ε). We can thereby estimate the spectrum of the one- and thetwo-particle excitation and their lifetimes. We will obtain the result that these quasi-particle excitations havequalitatively different behavior in agreement with the conserving-gapless theory but against the theorem (ii).
This thesis is organized as follows: In chapter II, we summarized the conserving gapless theory and use itto estimate ground state energy and the lifetime of the bubbling mode. Next, in chapter III, we present how tosimulate a Bose system by using the diffusion quantum Monte Carlo method, and give the expressions of themoments. Afterward, we show the results by these methods. Finally, we conclude and summarize those resultsin chapter IV.
6
Part II
Results of Conserving-Gapless TheoryRecently, a new self-consistent perturbation expansion (conserving-gapless theory [30]) has been formulatedfor condensed Bose systems which satisfies the Hugenholtz-Pines theorem[17] and conservation laws [27, 26]simultaneously. We will introduce results of the conserving-gapless theory, which produces several differentresults from the previous studies. Notations and equations in the following section are defined and derived inAppendix A. We also choose the following units:
~ = 1, m =1
2, kB = 1, T 0
c = 1,
where m, kB, and T0c are the particle mass, the Boltzmann constant, the transition temperature of ideal Bose-
Einstein condensation respectively.
1 Perturbation expansion for BEC
1.1 Notations
We consider N identical bosons with mass m and spin 0 and interacting with a two-body potential V (r − r′).The corresponding Hamiltonian H is given by
H ≡ H0 + Hint
=
∫d3r ψ†(r)K1(r)ψ(r) +
1
2
∫d3r
∫d3r′ ψ†(r)ψ†(r)′V (r− r′)ψ(r′)ψ(r), (1)
where (ψ(r), ψ†(r)) are annihilation and creation operators satisfying the bosonic commutation relations, andK1(r) ≡ p2/2m + U(r) − µ with U(r) the external field and µ the chemical potential. Field operators in theHeisenberg representation are defined as
ψ1(1) ≡ e−τ1H ψ(r1)eτ1H , ψ2(1) ≡ e−τ1H ψ†(r1)e
τ1H , (2)
respectively. We next divide these field operators into two parts as
ψi(1) = Ψi(1) + ϕi(1), (i = 1, 2), (3)
where Ψi(1) ≡ ⟨ψ1(1)⟩ and ϕi(1) are the condensate wave function and the quasiparticle operator respectively,and ⟨· · ·⟩ is the grand-canonical average.
Next, we define the Matsubara Green’s function in the Nambu space as
Gij = −⟨Tτϕi(1)ϕ3−j(2)⟩(−1)j−1, (4)
where Tτ is the time-ordering operator. Gij also satisfies the following relations:
Gij(1, 2) = (−1)i+j−1G3−j,3−i(2, 1) = (−1)i+jG∗ji(r2τ1, r1, τ2). (5)
We can also express Gij in a matrix form as
G(1, 2) ≡[G11(1, 2) G12(1, 2)G21(1, 2) G22(1, 2)
]=
[G(1, 2) F (1, 2)−F (1, 2) −G(1, 2)
].
(6)
We now give the expression of the thermodynamic potential in a Luttinger-Ward form [28] for BEC as,
Ω = − 1
β
∫d1
∫d2 Ψ2(1)G
−10 (1, 2)Ψ1(2) +
1
2βTr[ln(−G−1
0 + Σ) + ΣG] + Φ, (7)
7
where G−10 ≡ (−σ0∂/∂τ1 − σ3K1)δ(1, 2), and σ0, σ3 are 2× 2 unit matrix and third Pauli matrix respectively,
the self-energy Σ ≡ Σij can be introduced as
Σij(1, 2) ≡ −2βδΦ
δGji(2, 1), (8)
also
Σ(1, 2) ≡[Σ(1, 2) ∆(1, 2)−∆(1, 2) −Σ(1, 2)
]. (9)
And the trace TrΣG is defined by
Tr ΣG ≡∫d1
∫d2 Tr
[Σ(1, 2) ∆(1, 2)−∆(1, 2) −Σ(1, 2)
] [G(2, 1+) F (2, 1)−F (2, 1) −G(2, 1−)
],
(10)
where the subscript of ± denote presence of additional infinitesimal positive (negative) constant. The grandcanonical average of Hint is given by
⟨Hint⟩ =1
4β
∫d1
∫d22Σ(1, 2)[Ψ(2)Ψ(1)−G(2, 1+)]
− ∆(1, 2)[Ψ(2)Ψ(1)− F (2, 1)]−∆(1, 2)[Ψ(2)Ψ(1)− F (2, 1)], (11)
which can be connected with the functional Φ order by order as
Φ = −T∞∑
n=1
1
2nTr Σ(n)G. (12)
By comparing Eqs. (11) and (12), we thus find the following relation:
Φ(n) =1
n⟨Hint⟩(n). (13)
G and Ψ ≡ Ψi satisfy the following equations:∫d3 [G−1
0 (1, 2)− Σ(1, 3)]G(3, 2) = σ0δ(1, 2), (14a)
∫d2 [G−1
0 (1, 2)− Σ(1, 2)]Ψ(2) =
[00
],
(14b)
which are the Dyson-Beliaev equation [14] and the Hugenholtz-Pines theorem [17] respectively.For a later convenience, we define the symmetrized vertex as
Γ(0)(11′, 22′) ≡ V (r1 − r2)δ(τ1 − τ2)[δ(1, 1′)δ(2, 2′) + δ(1, 2′)δ(2, 1′)], (15)
which also satisfies the following relations:
Γ(0)(11′, 22′) = Γ(0)(22′, 11′) = Γ(0)(1′1, 2′2) = Γ(0)(2′1, 21′). (16)
8
By using Eq. (15), the expression of Φ can be written explicitly order by oder in terms of the interaction as
Φ(1) =T
4
∫d1
∫d1′∫d2
∫d2′ Γ(0)(11′, 22′)
× 2G(1.1′)G(2, 2′) + c(1)2b F (1.2)F (1
′, 2′) + c(1)1a G(1, 1
′)Ψ(2)Ψ(2′)
+ c(1)1b [F (1, 2)Ψ(1′)Ψ(2′) + F (1′, 2′)Ψ(1)Ψ(2)] + Ψ(1′)Ψ(2′)Ψ(2)Ψ(1) (17a)
Φ(2) =− T
8
∫d1 · · ·
∫d4′ Γ(0)(11′, 22′) Γ(0)(33′, 44′)
× G(2, 3′)G(3, 2′)G(1, 4′)G(4′, 1) + c(2)4b F (2
′, 3′)F (2, 3)G(1, 4′)G(4, 1′)
+ c(2)4c F (2
′, 3′)F (2, 3)F (1′, 4′)F (1, 4) + c(2)3a G(2, 3
′)G(3, 2′)G(1, 4′)Ψ(4)Ψ(1′)
+ c(2)3b [F (2
′, 3′)Ψ(2)Ψ(3) + F (2, 3)Ψ(2′)Ψ(3′)]G(1, 4′)G(4, 1′)
+ c(2)3c F (2
′, 3′)F (2, 3)G(1, 4′)Ψ(4)ψ(1′)
+ c(2)3d [F (2
′, 3′)Ψ(2)Ψ(3) + F (2, 3)Ψ(2′)Ψ(3′)]F (1′, 4′)F (1, 4)
+ c(2)2a G(2, 3
′)G(3, 2′)Ψ(1)Ψ(4)Ψ(4′)Ψ(1′) + c(2)2b G(2, 3
′)Ψ(3)Ψ(2′)G(1, 4′)Ψ(4)Ψ(1′)
+ c(2)2c [F (2
′, 3′)Ψ(2)Ψ(3) + F (2, 3)Ψ(2′)Ψ(3′)]G(1, 4′)Ψ(4)Ψ(1′)
+ c(2)2d F (2
′, 3′)F (2, 3)Ψ(1)Ψ(4′)Ψ(4)Ψ(1′)
+ c(2)2e [F (2, 3)Ψ(3′)Ψ(2′)F (1, 4)Ψ(4′)Ψ(1′) + F (2′, 3′)Ψ(3)Ψ(2)F (1′, 4′)Ψ(4)Ψ(1)], (17b)
where
c(1)2b = −1, c(1)1a = −4, c(1)1b = 1, (18a)
c(2)4b = −2, c
(2)4c = 1,
c(2)3a = −4, c
(2)3b = 2, c
(2)3c = 4, c
(2)3d = −2,
c(2)2a = c
(2)2b = 2, c
(2)2c = −4 c
(2)2d = 2, c
(2)2e = 1. (18b)
1.2 Expressions of first and second self-energies
We introduce a matrix G ≡ Gij [32] as
Gij(1, 2) ≡ Gij − (−1)i−1Ψi(1)Ψ3−j(2). (19)
Using this function, Eq. (11) can be rewritten as
⟨Hint⟩ = −1
4βΣij(1, 2)Gji(2, 1), (20)
where summation over barred arguments such as 1 and i are implied.We can construct Φ as a functional of G, which can be confirmed by considering the following conditions:
Σii′(1, 1′) = −2β δΦ
δGjj′(2, 2′)δGjj′(2, 2′)δGii′(1′, 1)
= −2β δΦ
δGii′(1, 1′), (21a)
9
βδΦ
δΨ3−i(1)= β
δΦ
δGj′j(2′, 2)δGj′j(2′, 2)δΨ3−i(1)
= −1
2Σjj′(2
′, 2)δGj′j(2′, 2)δΨ3−i(1)
=1
2[Σij′(1, 2
′)(−1)j′−1Ψj′(2′) + Σj,3−i(2, 1)(−1)iΨ3−j(2)]
=1
2[Σij′(1, 2
′)(−1)j′−1Ψj′(2′) + Σi,3−j(1, 2)(−1)jΨ3−j(2)]
= Σij(1, 2)(−1)j−1Ψj(2) (21b)
Let us introduce
G(1, 2) ≡ G11(1, 2)− G22(2, 1)2
, F(1, 2) ≡ G12(1, 2)− G12(2, 1)2
, F(1, 2) ≡ G21(1, 2)− G21(2, 1)2
. (22)
Using these notations, we can transform Eq. (12) as
Φ(n) =1
2n
δΦ(n)
δGij(2, 1)Gij(2, 1)
=1
2n
[δΦ(n)
δG(2, 1)G(2, 1) + δΦ(n)
δF(2, 1)F(2, 1) + δΦ(n)
δF(2, 1)F(2, 1)
]. (23)
We can rewrite Eq. (17a) and (17b) as
Φ(1) ≡ 1
4βΓ(0)(1′1, 2′2)[2G(1, 1′)G(2, 2′)−F(1, 2)F(1′, 2′)], (24a)
Φ(2) = − 1
8βΓ(0)(1′1, 2′2)[G(2, 3′)G(3, 2′)−F(2′, 3′)F(2, 3)]
×Γ(0)(3′3, 4′4)[G(1, 4′)G(4, 1′)−F(1′, 4′)F(1, 4)] (24b)
respectively.Using Eq. (21a), self-energies are also calculated as
Σ(1)(1, 1′) ≡ −T−1 δΦ(1)
δG(1′, 1)= −Γ(0)(11′, 2′2)G(2, 2′), (25a)
∆(1)(1, 1′) ≡ 2T−1 δΦ(1)
δF(1′, 1)= −1
2Γ(0)(12′, 12)F(2′, 2), (25b)
Σ(2)(1, 1′) ≡− T−1 δΦ(2)
δG(1′, 1)
=1
2Γ(0)(14, 2′2)Γ(0)(3′3, 4′1′)[G(2, 3′)G(3, 2′)− F(2′, 3′)F(2, 3)]
=1
2Γ(0)(14, 2′2)Γ(0)(3′3, 4′1′)
× G(2, 3′)G(3, 2′)G(4, 4′)− F (2′, 3′)F (2, 3)G(4, 4′)− 2Ψ(2)Ψ(3′)G(3, 2′)G(4, 4′)−Ψ(3)Ψ(2′)G(2, 3′)G(4, 4′)
+ [F (2′, 3′)Ψ(2)Ψ(3) + F (2′, 3′)Ψ(2′)Ψ(3′)]G(4, 4′)
+ F (2′, 3′)F (2, 3)Ψ(4)Ψ(4′)
+G(4, 4′)Ψ(2)Ψ(3′)Ψ(3)Ψ(2′) +G(3, 2′)Ψ(2)Ψ(3′)Ψ(4)Ψ(4′)
− [F (2′, 3′)Ψ(2)Ψ(3) + F (2, 3)Ψ(2′)Ψ(3′)]Ψ(4)Ψ(4′), (26a)
10
∆(2)(1, 1′) ≡2T−1 δΦ(2)
δF(1′, 1)
=1
2Γ(0)(3′3, 1′2)Γ(0)(12′, 4′4)[G(2, 3′)G(3, 2′)− F(2′, 3′)F(2, 3)]
=1
2Γ(0)(3′3, 1′2)Γ(0)(12′, 4′4)
× G(3, 4′)G(4, 3′)F (2, 2′)− F (3′, 4′)F (3, 4)F (2, 2′)− [G(3, 4′)Ψ(4)Ψ(3′) +G(4, 3′)Ψ(3)Ψ(4′)]F (2, 2′)
−G(3, 4′)G(4, 3′)Ψ(2)Ψ(2′)
+ 2F (3, 4′)F (4, 3′)Ψ(2)Ψ(2′) + F (3, 4)Ψ(3′)Ψ(4′)F (2, 2′)
− [G(3, 4′)Ψ(4)Ψ(3′) +G(4, 3′)Ψ(3)Ψ(4′)]Ψ(2)Ψ(2′)
−Ψ(3)Ψ(4′)Ψ(4)Ψ(3′)F (2, 2′)− F (3′, 4′)Ψ(4)Ψ(3)Ψ(2′)Ψ(2), (26b)
respectively.
1.3 Self-consistent equation in momentum space
We now construct the self-energy up to the second order in the momentum space. From here we considerparticles interacting with a contact potential V (r−r′) ≡ gδ(r−r′) in a homogeneous system. The correspondingHamiltonian is given by
H ≡∫dr ψ†(r)
p2
2mψ(r) +
∫dr
∫dr′ ψ†(r)ψ†(r′)gδ(r− r′)ψ(r′)ψ(r)
=∑p
εp +g
2V
∑pp′q
a†p+qa†p′−qap′ ap, (27)
where (ap, a†p) are bosonic annihilation and creation operators respectively. By introducing the cut-off momen-
tum pc, we can express g (see Appendix.C) as
g =( m
4πa− pc
4π2
)−1
.(28)
If we set pc as 0≪ pc ≪ πm/a, which can remove the cut-off dependence as
g =4πa
m .(29)
We next introduce the Fourier transformation of G as
G(1, 2) =∑p
G(p)eip·r, (30)
where r ≡ (r, iτ), p ≡ (p, zℓ), and zℓ ≡ 2iπTℓ is the Matsubara frequency, and the summation over p is givenby
∑p
≡ T∞∑
ℓ=−∞
∫d3p
(2π)3. (31)
Function (5) has the following symmetry relations:
Gij(p) = (−1)i+j−1G3−j,3−i(−p) = (−1)i+jG∗ji(p
∗). (32)
We define matrix Green’s function and self-energy by
Gp ≡[G11(p) G12(p)G21(p) G22(p)
]=
[Gp Fp
−Fp −Gp
], (33)
11
Σ(p) ≡[Σ11(p) Σ12(p)Σ21(p) Σ22(p)
]=
[Σp ∆p
−∆p −Σp
], (34)
respectively, where lower elements with bars are connected with upper ones generally by
fp = f∗−p∗ . (35)
We also give the Fourier transform of the vertex Γ(0)(11′, 22′) by
Γ(0)(p, p ′, q) = 2g. (36)
In these notations, the Dyson-Beliaev equation (14a) can be expressed as
G(p) =
[zℓ − ξp − Σp −∆p
∆p zℓ + ξp + Σp
]−1
=1
Dp
[zℓ + ξp + Σp ∆p
−∆p zℓ − ξp − Σp
], (37)
where ξp ≡ p2/2m− µ, and Dp is given by
Dp ≡ (zℓ − ξp − Σp)(zℓ + ξp + Σp) + ∆p∆p
= z2ℓ − (Σp − Σp)zℓ −(ξp +
Σp + Σp
2
)2
+
(Σp − Σp
2
)2
+∆p∆p
= (zℓ − Ep)(zℓ + Ep), (38)
with
Ep ≡Σp − Σp
2+
[(ξp +
Σp + Σp
2
)2
−∆p∆p
]1/2, (39a)
Ep ≡Σp − Σp
2+
[(ξp +
Σp + Σp
2
)2
−∆p∆p
]1/2.
(39b)
The Hugenholtz-Pines theorem (14b) reduces to
µ = Σ(0)−∆(0). (40)
The particle density n is given by
n = n0 −∑p
Gp, (41)
where n0 is the condensate density Ψ(1)→ n0.Expressions of Σp and ∆p up to second order are written as
Σp =2gn+ 2[ΞGGG(p)− ΞGFF (p)]
+ 2gn0[−2ΠGG(p)−ΠGG(p) + 2ΠGF (p) + ΠFF (p)]
+ 2(gn0)2(Gp +G−p − 2Fp) (42a)
∆p =g
(n0 −
∑p
Fp
)+ 2[ΞFGG(p)− ΞFFF (p)]
+ 2gn0[−ΠFG(p)−ΠFG(−p)−ΠGG(p) + 3ΠFF (p)]
+ 2(gn0)2(Gp +G−p − 2Fp), (42b)
12
where we have introduced
ΠGG(q) ≡ g∑p
Gp+qGp = ΠGG(−q), (43a)
ΠGG(q) ≡ g∑p
Gp+qG−p = ΠGG(−q), (43b)
ΠFF (q) ≡ g∑p
Fp+qFp = ΠFF (−q), (43c)
ΠGF (q) ≡ g∑p
Gp+qFp = ΠFG(−q), (43d)
ΞGGG(p) ≡ g∑q
Gp+qΠGG(q), (43e)
ΞGFF (p) ≡ g∑q
Gp+qΠFF (q), (43f)
ΞFGG(p) ≡ g∑q
Fp+qΠGG(q), (43g)
ΞFFF (p) ≡ g∑q
Fp+qΠFF (q). (43h)
2 Analysis including 1PR diagrams
A notable prediction of the conserving-gapless theory is that there is a new class of Feynman diagrams [32]for the self-energy, i.e., those that may be classified as one-particle reducible (1PR) following the conventionalterminology [4, 20] and should certainly be excluded from its definition in the normal state.
In this section, we will estimate physical quantities by numerical calculation including those 1PR diagrams atT = 0. We present the numerical results for energy, condensate density, and lifetime of one-particle excitation.
2.1 Structure of 1PR diagrams
In previous studies, perturbation analysis has been performed by incorporating only irreducible (1PI) struc-tures for the self-energy Σ. Such diagrams are represented in Fig.1. However, as we can see in the previoussection and Appendix.A, 1PR diagrams are essential to construct the Luttinger-Ward functional Φ satisfyingthe Hugenholtz-Pines theorem and conservation laws simultaneously. We note that these 1PR diagrams appearfrom second order as shown in Fig.2. Those of the second order are proportional to (Gp +G−p − 2Fp). In thecase of T = 0, we only need to consider a specific class of 1PR diagrams, whose structures are proportional to(Gp+G−p− 2Fp)
m−1 every mth-order self-energy, which are given by polygonal structures of Φ(ip) as shown inFig.3. The other structures of Φ contribute only gmnm−1
0 (n−n0)1, which is negligible near T = 0 (n−n0 ≪ 1).On the other hand, our 1PR diagrams are proportional to (gn0)
m (see also Eq. (42)). Therefore, we canapproximate the functional Φ as
Φ[G, Ψ] ≈ Φ(1)[G, Ψ] + Φ(ip)[G, Ψ]. (44)
We estimate physical quantities containing the first-order self-energy and 1PR structures in the higher-orderself-energies up to second and third-order and FLEX approximation. Such functionals are given by
Φ(ip) ≈ −V∑p
1
2(gn0fp)
2, (45a)
Φ(ip) ≈ −V∑p
[1
2(gn0fp)
2+
5
3(gn0fp)
3
], (45b)
Φ(ip) ≈ −V∑p
[1
2(gn0fp)
2+
∞∑n=3
1 + 2n−1
n(gn0fp)
n
], (45c)
13
respectively [35, 45], where fp ≡ (Gp +G−p − 2Fp). Let us estimate the internal energy E and the condensatedensity n0 in this approximation and compare the results to those by Lee, Huang, Yang [34, 35].
Fig 1: 1PI structures in Σ.
Fig 2: 1PR structures in Σ. These diagrams are characterized by straight line structures. Such structure hadnever been considered in previous studies.
Fig 3: Polygonal structures in Φ. These diagrams have the dominant contribution to physical properties inBEC system for T ≈ 0.
2.2 First-order analysis
To start with, we calculate the first-order self-energy, and confirm that the results by Bogoliubov [13] andLee, Huang, Yang [34] can be reproduced appropriately. The first-order Green’s functions G(1), F (1) and theself-energies Σ(1) and ∆(1) are given by
G(1)p ≡ zℓ + p2 +Σ(1) − µ(1)
z2ℓ − p2(p2 + 2∆(1)),
(46a)
F(1)p ≡ ∆(1)
z2ℓ − p2(p2 + 2∆(1)),
(46b)
Σ(1) ≡ 2gn, (47a)
∆(1) ≡ g(n0 −
∑p
F(1)p
),
(47b)
respectively, where the chemical potential µ(1) is given by
µ(1) ≡ Σ(1) −∆(1). (48)
14
Using Eq. (37), we can transform G(1)p , F
(1)p as
G(1)p =
zℓ + p2 +∆(1)
(zℓ − p√p2 + 2∆(1))(zℓ + p
√p2 + 2∆(1))
=1
2E(1)p
(p2 +∆(1) + E
(1)p
zℓ − E(1)p
− p2 +∆(1) − E(1)p
zℓ + E(1)p
)(49a)
F(1)p =
∆(1)
(zℓ − p√p2 + 2∆(1))(zℓ + p
√p2 + 2∆(1))
=∆(1)
2E(1)p
(1
zℓ − E(1)p
− 1
zℓ + E(1)p
), (49b)
where E(1)p corresponds to the Bogoliubov mode as
E(1)p ≡ p
√p2 + 2∆(1). (50)
Summing over p for Eq. (49), we obtain
∑p
G(1)p =
∫d3p
(2π)3
∫C
dz
2πi
n(z)
2E(1)p
(p2 +∆(1) + E
(1)p
z − E(1)p
− p2 +∆(1) − E(1)p
z + E(1)p
)
= −∫
d3p
(2π)3
[p2 +∆(1) + E
(1)p
2E(1)p
n(E(1)p ) +
p2 +∆(1) − E(1)p
2E(1)p
(1 + n(E(1)p ))
]
= − 1
2π2
∫ ∞
0
dp p2[p2 +∆(1)
E(1)p
n(E(1)p ) +
p2 +∆(1) − E(1)p
E(1)p
]= − 1
4π2
∫ ∞
0
dϵ ϵ1/2[
ϵ+∆(1)√ϵ(ϵ+ 2∆(1))
n(E(1)p ) +
ϵ+∆(1) −√ϵ(ϵ+ 2∆(1))
2√ϵ(ϵ+ 2∆(1))
],
(51a)
∑p
F(1)p =
∫d3p
(2π)3
∫C
dz
2πi
n(z)∆(1)
2E(1)p
(1
z − E(1)p
− 1
z + E(1)p
)
= −∫
d3p
(2π)3n(z)∆(1)
2E(1)p
[1 + 2n(E(1)p )]
= − 1
2π2
∫ pc
0
dp p2∆(1)
2E(1)p
[1 + 2n(E(1)p )]
= − 1
4π2
∫ p2c
0
dϵ ϵ1/2∆(1)√
ϵ(ϵ+ 2∆(1))
[1
2+ n(E(1)
p )
].
(51b)
where n(z) ≡ (eβz − 1)−1 and the integral path C has been taken as Fig.4 [43, 44].
15
C
C
Re z
Im z
Fig 4: integration path C
Taking the limit T → 0 for Eq. (51) gives the following:
∑p
G(1)p −−−→
T→0− 1
4π2
∫ ∞
0
dϵ ϵ1/2ϵ+∆(1) −
√ϵ(ϵ+ 2∆(1))
2√ϵ(ϵ+ 2∆(1))
=− (2∆(1))3/2
8π2
∫ ∞
0
dx
(x+ 1√x+ 2
− x1/2)
=− (2∆(1))3/2
8π2
∫ ∞
0
dx
[(x+ 2)1/2 − 1
(x+ 2)1/2− x1/2
]=− (2∆(1))3/2
8π2
[2
3(x+ 2)3/2 − 2(x+ 2)1/2 − 2
3x3/2
]=− (2∆(1))3/2
24π2, (52a)
∑p
F(1)p −−−→
T→0− ∆(1)
8π2
∫ p2c
0
dϵ ϵ1/21√
ϵ+ 2∆(1)≈ −∆(1)
4π2
(pc −
√2∆(1)
)= −∆(1)
4π2pc +
(∆(1))3/2
8π2. (52b)
By using these, Eq. (47) can be transformed as
n0n
= 1− (2∆(1))3/2
24π2n,∆(1) = g
[n0 +
∆(1)
4π2pc −
(2∆(1))3/2
8π2
]= gn
[n0n
+∆(1)
4π2npc −
(2∆(1))3/2
8π2n
].
(53)
Considering Eq. (28), we obtain
Σ(1) ≈ 16πan
(1 +
1
πpca
), (54a)
∆(1) ≈ 8πan
(1 +
2
πpca
)[1− 2∆(1)
24π2n+
∆(1)
4π2npc −
(2∆(1))3/2
8π2n
]≈ 8πan
(1 +
2
πpca
)[1 +
8πna
4π2n− (16πan)3/2
6π2n
]≈ 8πan
[1 +
4
πpca−
32
3√πa3/2n1/2
].
(54b)
Substituting them into Eqs. (53) and (48), we obtain
n0n≈ 1− (16πan)3/2
24π2n= 1− 8
3√π
√na3 ≈ 1− 1.504
√na3, (55)
16
µ(1) = 8πan
(1 +
32
3π1/2
√na3)
(56)
respectively. By using µ = ∂E/∂N , we thereby obtain
E(1)
N≈ 4πan
(1 +
128
15√π
√na3)≈ 4πan
(1 + 4.815
√na3). (57)
We conclude that the first-order analysis reproduces the results by Lee, Huang, Yang [34].
2.3 Discussion for second order
We now construct Green’s function containing 1PR diagrams up to second oder. Using Eq. (47) and (45a), we
obtain Σ(2)p and ∆
(2)p
Σ(2)p = Σ(1) +∆
(2c)p , (58a)
∆(2)p = ∆(1) +∆
(2c)p , (58b)
respectively, where ∆(2c)p is defined by Eq. (42) as
∆(2c)p ≡ 2(gn0)
2(Gp +G−p − 2Fp) = ∆(2c)−p . (59)
The chemical potential µ(2) = Σ(1) −∆(1) is equal to that of first-order Eq. (48).
The second-order Green’s functions G(2)p , F
(2)p are expressible as
G(2)p =
zℓ + p2 +∆(2)p
z2ℓ − p2(p2 + 2∆(2)p )
, (60a)
F(2)p =
zℓ + p2 +∆(2)p
z2ℓ − p2(p2 + 2∆(2)p )
, (60b)
respectively.
Introducing f(2)p ≡ (G
(2)p +G
(2)−p − 2F
(2)p ) and using Eq. (60), we obtain
f(2)p =
2p2
z2ℓ − p2(p2 + 2∆(2)p )
=1
ηp −∆(c2)p ,
(61)
where ηp is defined by
ηp ≡z2ℓ − p2(p2 + 2∆(1))
2p2 .
(62)
Thus we can construct a self-consistent equation for ∆(2c)p by using Eq. (59) as
∆(2c)p =
2(gn0)2
ηp −∆(2c)p ,
(63)
which reduces to the following quadratic equation:
(∆(2c)p )2 − ηp∆(2c)
p + 2(gn0)2 = 0. (64)
Noting this equation contain only real coefficients and ∆(1) ≈ gn0, we can’t find a solution which satisfies thesymmetry relation (59). Therefore, we need to include higher-order contribution to self-energies.
17
2.4 Construction of third-order equations
We here include third-order 1PR diagrams whose numerical weights were given by [30]. We also choose series
of the straight line diagrams. Such self-energies Σ(3)p , ∆
(3)p and Green’s functions G
(3)p , F
(3)p can be written by
Σ(3)p = Σ(1) +∆
(2c)p +∆
(3c)p ≡ Σ(1) + ∆
(3c)p , (65a)
∆(3)p = ∆(1) +∆
(2c)p +∆
(3c)p ≡ ∆(1) + ∆
(3c)p , (65b)
and
G(3)p =
zℓ + p2 +∆(3)p
z2ℓ − p2(p2 + 2∆(3)p )
(66a)
F(3)p =
zℓ + p2 +∆(3)p
z2ℓ − p2(p2 + 2∆(3)p )
(66b)
respectively, where ∆(3c)p can be represented by using Eq. (45b) as
∆(3c)p ≡ 2(gn0)
2f(3)p +
5
4(2gn0)
3(f(3)p )2, (67)
with f(3)p ≡ (G
(3)p +G
(3)−p − 2F
(3)p ).
By using ηp, the equation for ∆(3c)p is given by
(∆(3c)p )3 − 2ηp(∆
(3c)p )2 + [η2p + 2(gn0)
2]∆(3c)p − 2(gn0)
2(ηp + 5gn0) = 0, (68)
which is a cubic equation. So, there is a real solution with ∆(3c)p at least.
Here we perform the following transformation:
xp ≡ ∆(3c)p /gn0, (69)
ap ≡ ηp/gn0. (70)
Equation (68) can be transformed into
x3p − 2apx2p + (a2p + 2)xp − 2(ap + 5) = 0. (71)
We can solve the equation by using the Cardano formula as
xp =
2
3ap +
(− b0 +
√b20 + b31
)1/3
+
(− b0 −
√b20 + b31
)1/3
: |ap| ≥√6
2
3ap +
(− b0 +
√b20 + b31
)1/3
−(b0 +
√b20 + b31
)1/3
: |ap| <√6,
(72)
where
b1 ≡3(a2p + 2)− (−2ap)2
9=−a2p + 6
9, (73)
b0 ≡27(−2)(ap + 5)− 9(−2ap)(a2p + 2) + 2(−2ap)3
54=a3p − 9ap − 135
27, (74)
respectively.
18
2.5 Equations for FLEX approximation
We next consider the case of the FLEX approximation. Self-energies Σ(Fc)p , ∆
(Fc)p and Green’s functions are
given by
Σ(Fc)p = Σ(1) +∆
(Fc)p , (75a)
∆(Fc)p = ∆(1) +∆
(Fc)p , (75b)
G(Fc)p =
zℓ + p2 +∆(Fc)p
z2ℓ − p2(p2 + 2∆(Fc)p )
, (76a)
F(Fc)p =
zℓ + p2 +∆(Fc)p
z2ℓ − p2(p2 + 2∆(Fc)p )
.
(76b)
The nth-order term of ∆(Fc)p can be written by using Eqs. (67) and (45c) as
∆(nc)p =
(1 +
1
2n−1
)(2gn0)
nfn−1p (n ≥ 3), (77)
where f(Fc)p is defined by f
(Fc)p ≡ (G
(Fc)p +G
(Fc)−p − 2F
(Fc)p ). Using Eqs. (59) and (77), we can transform it as
∆(Fc)p ≡ ∆
(2c)p +
∞∑n=3
∆(nc)p = 2(gn0)
2f(Fc)p +
∞∑n=3
[(2gn0)n + 2(gn0)
n](f(Fc)p )n−1
=(2gn0)
2f(Fc)p
1− 2gn0f(Fc)p
+2(gn0)
2f(Fc)p
1− gn0fp− 4(gn0)
2f(Fc)p
=(2gn0)
2
ηp − 2gn0 −∆(Fc)p
+2(gn0)
2
ηp − gn0 −∆(Fc)p
− 4(gn0)2
ηp −∆(Fc)p
, (78)
where we have used f−1p = ηp−∆(Fc)
p . We then find that the equation for ∆(Fc)p forms the fourth-order equation:
0 =−∆(Fc)p (ηp − 2gn0 −∆
(Fc)p )(ηp − gn0 −∆
(Fc)p )(ηp −∆
(Fc)p )
+ 2(gn0)2[2(ηp − gn0 −∆
(Fc)p )(ηp −∆
(Fc)p ) + (ηp − 2gn0 −∆
(Fc)p )(ηp −∆
(Fc)p )
− 2(ηp − 2gn0 −∆(Fc)p )(ηp − gn0 −∆
(Fc)p )]
=(∆(Fc)p )4 − 3(ηp − gn0)(∆(Fc)
p )3 + [3η2p − 6gn0ηp + 4(gn0)2](∆
(Fc)p )2
− [η3p − 3gn0η2p + 6(gn0)
2ηp + 4(gn0)3]∆
(Fc)p + 2(gn0)
2[η2p + 2gn0ηp − 4(gn0)2]. (79)
By introducing xp ≡ ∆(Fc)p /gn0 and using ap defined by Eq. (70), we can write down as
x4p − 3(ap − 1)x3p + (3a2p − 6ap + 4)x2p − (a3p − 3a2p + 6ap + 4)xp + 2(a2p + 2ap − 4) = 0. (80)
Considering ap → −1 for |p| → 0, a function h(x) ≡ x4 − 6x3 + 13x2 + 6x− 10 satisfies h(0) = −10 < 0 for thelimit h(|x| → ∞)→∞. So, we can find there is a real solution in Eq. (80) for ap.
3 Numerical results at T = 0
We now present the results of including 1PR diagrams for T = 0. We perform the calculation for the third-orderand the FLEX approximations which are introduced in the previous section.
At first, we will discuss the energy per particle E/N . 1PR diagrams add an extra term cip to the resultby Lee, Huang, Yang [34]. Next, we estimate a lifetime of the one-particle excitation by using the one-particlespectral function Ap(ω). We will see the one-particle excitation for BEC has a finite lifetime proportional tos-wave scattering length a.
19
3.1 Energy per particle
In this subsection, we first explain how to calculate the energy per particle in the third-order and the FLEXapproximations. So, we introduce the self-energies Σp and ∆p by using first-order self-energies in Eq. (47) and
its correction term ∆(c)p as
Σp = Σ(1) +∆(c)p , (81a)
∆p = ∆(1) +∆(c)p . (81b)
We also express the corresponding Matsubara Green’s functions by Gp, and Fp. Using first-order Green’sfunctions in Eq. (46), we define difference functions for Gp, and Fp as
∆Gp ≡ Gp −G(1)p , (82a)
∆Fp ≡ Fp − F (1)p , (82b)
respectively. We hence obtain
Gp = G(1)p +∆Gp, (83a)
Fp = F(1)p +∆Fp. (83b)
We can express the correction terms for non condensate density ∆n(nc) and the summation of F(1)p over p,
∆Fsum as
n(nc) =∑p
Gp =∑p
G(1)p +∆n(nc)
≡ n(1) +∆n(nc) ≈ 8
3√πa3/2n3/2 +∆n(nc), (84a)∑
p
Fp =∑p
F(1)p +∆Fsum, (84b)
where n(1) is the non condensate density of the first order as seen in Eq. (55). We note that the summationsof ∆Gp, ∆Fp are free from unphysical ultraviolet divergences for p→ 0. So, we can replace ∆(1) in ∆n(nc) and
∆Fsum in Eq. (84) as gn(1)0 ≡ 8πan
(1)0 , and can ignore the pc dependence in the summation over p (taking the
limit pc →∞). Therefore, we can rewrite the summation over p for T → 0 as∑p
−−−→T→0
∫ ∞
−∞
dω
2π
∫d3p
(2π)3=
1
(2π)3
∫ ∞
−∞dω
∫ ∞
0
de√e,
where the Matsubara frequency iεℓ and the one-particle kinetic energy p2 are replaced by iω and e respectively.Next we take the following transformation for ω and e:
ω = ω/gn(1)0 , e = e/gn
(1)0 .
Introducing
xp ≡ xe,iω = ∆(c)p /gn
(1)0 , (85)
we can transform ∆n(nc) and ∆Fsum as
∆n(nc) = −∑p
(zℓ + p2 +∆p
z2ℓ − p2(p2 + 2∆p)− zℓ + p2 +∆(1)
z2ℓ − p2(p2 + 2∆(1))
)
= − (gn(1)0 )3/2
(2π)3
∫ ∞
−∞dω
∫ ∞
0
de√e
(iω + e+ 1 + xe,iω
−ω2 − ee+ 2(1 + xe,iω)− iω + e+ 1
−ω2 − e(e+ 2)
)= (8πan
(1)0 )3/2 × IG ≈ (8πan)3/2 × IG, (86a)
20
∆Fsum =∑p
(∆p
z2ℓ − p2(p2 + 2∆p)− ∆(1)
z2ℓ − p2(p2 + 2∆(1))
)
=(gn
(1)0 )3/2
(2π)3
∫ ∞
−∞dω
∫ ∞
0
de√e
(1 + xe,iω
−ω2 − ee+ 2(1 + xe,iω)− 1
−ω2 − e(e+ 2)
)= (8πan
(1)0 )3/2 × IF ≈ (8πan)3/2 × IF (86b)
respectively, where we have used Eq. (55) and n0 ≈ n in the last equality, and also introduced
IG ≡ −1
(2π)3
∫ ∞
−∞dω
∫ ∞
0
de√e
(iω + e+ 1 + xe,iω
−ω2 − ee+ 2(1 + xe,iω)− iω + e+ 1
−ω2 − e(e+ 2)
),
(87a)
IF ≡1
(2π)3
∫ ∞
−∞dω
∫ ∞
0
de√e
(1 + xe,iω
−ω2 − ee+ 2(1 + xe,iω)− 1
−ω2 − e(e+ 2)
).
(87b)
Therefore, we can express the condensate density n0 as
n0n
=(n− n(1) −∆n(nc))
n
= 1−(
8
3√π+ (8π)3/2 × (IG + IF )
)√na3
≡ 1−(
8
3√π+ cip
)√na3, (88)
where cip is defined by
cip ≡4√2
π3/2
∫ ∞
0
dω
∫ ∞
0
de(ω2 − e2)xe,iω
[ω2 + e(e+ 2xe,iω)][ω2 + e(e+ 2xe,iω)] .(89)
The chemical potential µ is also given by
µ = Σ(1) −∆(1) = gn
1− 1
n
∑p
(G
(1)p − F
(1)p
)− 1
n
∑p
(∆Gp −∆Fp
)
= µ(1) − g∑p
(∆Gp −∆Fp
)≈ µ(1) + g(gn)3/2(IG + IF )
≈ 8πan
[1 +
(32
3√π+ cip
)n1/2a3/2
],
(90)
where we have used Eq. (56) and
∆(1) = g
(n0 −
∑p
Fp
)
= gn
1 +
1
n
∑p
(Gp − Fp
)
= gn
1 +
1
n
∑p
(G
(1)p − F
(1)p
)+
1
n
∑p
(∆Gp −∆Fp
).
(91)
We hence obtain the energy per particle E/N as
EN
= 4πan
[1 +
16
5
(8
3√π+ cip
)√na3]. (92)
21
We here present numerical results for cip in each approximation, which can be obtained as
c(3)ip = 0.412, c
(F )ip = 0.563, (93)
where superscripts (3) and (F ) denote the third-order and FLEX approximations respectively [35].Thus cip for the two approximations are both of order 1. The exact value of cip can only be obtained by
collecting all the leading-order 1PR terms, however, meaning that further studies are required for a quantitativeestimate of cip. On the other hand, a diffusion Monte Carlo study [47] was performed on Eq. (57); the datain Table I for na3 ≤ 10−2 yield the value 4.3(1), rather than 128/15
√π = 4.815 from Eq. (57), thus showing
a clear deviation from the Lee-Huang-Yang expression beyond the numerical uncertainly. However, the exactestimation of the energy by the diffusion Monte Carlo simulation needs certain approximations for removeingfinite size effects [37] (see also Appendix.B.3). On the other hand, an experiment on Eq. (57) in a trap potential[48] reported a value 4.5(7), which still requires improvement.
3.2 Lifetime of one-particle excitation
We here estimate lifetime of the one-particle excitation in BEC system by using the conserving-gapless theory.The one-particle spectral function is obtained by performing the analytic continuation for Gp from imaginaryaxis to real axis:
Ap(ω) = −2 Im Gp |zℓ→ω+iη . (94)
The resultant values on the imaginary axis are used subsequently to calculate Eq. (94) based on Thiele’sreciprocal difference algorithm for Pade approximation [49]. Function Ap(ω) thereby calculated can be checkednumerically with a couple of exact relations as∫ ∞
−∞
dω
2πAp(ω) = 1, (95a)
Gp =
∫ ∞
−∞
dω
2π
Ap(ω)
zℓ − ω .
(95b)
We have confirmed that sum rule Eq. (95a) is satisfied beyond 99.7% and Green’s function on the imaginaryaxis are reproduced with an error of less than 0.02%.
To begin with, let us review the spectral function of the Bogoliubov theory, which is obtained by insertingEq. (46) into Eq. (94). It yields
A(1)p (ω) = π[(αp + 1)δ(ω − E(B)
p )− (αp − 1)δ(ω + E(B)p )], (96)
where E(B)p ≡
√εp(εp + 2∆(1)) is the Bogoliubov mode with a linear dependence for εp/∆
(1) ≪ 1, and
αp ≡ (εp +∆(1))/E(B)p . Thus, A
(1)p (ω) has a couple of sharp δ-function peaks at ω = ±E(B)
p corresponding towell-defined quasiparticles with the infinite lifetime.
However, the 1PR self-energy brings about a qualitative change in the spectral function. To see this explicitly,let us consider the second-order approximation of Eq. (63), which can be solved analytically as
∆(2c)p =
D(1)p
4εp−
(D(1)p
4εp
)2
− 2(∆(1))2
1/2
(97)
with
D(1)p ≡ (zℓ − E(B)
p )(zℓ + E(B)p ). (98)
Using this ∆(2c)p in Eq. (60a) and substituting resultant Gp into Eq. (94), we can obtain the spectral function
in the second-order approximation as
A(2)p (ω) = θ(ω2 − E2
p−)θ(E2p+ − ω2)[θ(ω)− θ(ω)] (ω + εp)
2
(4∆(1))2ε3p
√(E2
p+ − ω2)(ω2 − E2p−), (99)
22
where Ep± ≡√εp[εp + (2± 4
√2)∆(1)]. Figure.5 exhibits A
(2)p (ω) for εp/∆
(1) = 4. as seen clearly, the quasi-
particle excitation peaks are broadened due to ∆(2c)p . The half-width δω
(2)p is proportional to the inverse of its
lifetime τ−1. The main peak of A(2)p (ω) for ω > 0 is clearly of the order of a and approaches 57.3an as εp →∞
as seen in Fig.6. However, Eq. (99) is valid only for εp/∆(1) ≥ −2+ 4
√2; the second-order approximation fails
to describe the low-momentum region adequately.This unphysical behavior is removed in the third-order and the FLEX approximation. First, Fig.7 plots
the spectral function A(3)p (ω) in the third-order approximation for εp/∆
(1) = 4 around its peak for ω > 0.
As seen clearly, this peak also has a finite lifetime τ ∝ a−1. Figure.8 shows εp dependence of δω(3)p ∝ τ−1.
It apparently develops from zero as εp is increased, has the maximum around εp/∆(1) ≈ 300, and starts to
decrease thereafter towards 0. A qualitatively similar behavior is obtained by the FLEX approximation as
shown in Fig.9, 10. However, both the magnitude of δω(F)p and its peak location are quantitatively different
from δω(3)p . To resolve this point requires a better treatment of the infinite series of Fig.3.
-10 -5 5 10
0.5
1.0
1.5
Fig 5: Plot of the spectral function A(2)p (ω) in the
second-order approximation for εp/∆(1) = 4. The half-
width δω(2)p is proportional to a.
0.0100 200 300 400 500
0.5
1.0
1.5
2.0
Fig 6: Plot of the half-width δω(2)/p of A(2)p (ω). It
increasing with εp/∆(1), and approaches 57.3an for
εp →∞.
4.980 4.982 4.984 4.986 4.988
1.0
2.0
3.0
4.0
5.0
6.0
0.0
Fig 7: Plot of the spectral function A(3)p (ω) in the third-
order approximation for εp/∆(1) = 4. The half-width
δω(3)p is proportional to a.
0.0 1.0 2.0 3.0 4.0 5.0
0.001
0.002
0.003
0.004
Fig 8: Plot of the half-width δω(3)/p of A(3)p (ω). It
increasing with εp/∆(1), and approaches 0 for εp →
∞.
23
4.80 4.85 4.90 4.95 5.00
100
200
300
400
Fig 9: Plot of the spectral function A(F)p (ω) in the
second-order approximation for εp/∆(1) = 4. The half-
width δω(F)p is proportional to a.
0 100 200 300 400 5000.00
0.01
0.02
0.03
0.04
Fig 10: Plot of the half-width δω(F)/p of A(F)p (ω). It
increasing with εp/∆(1), and approaches 0 for εp →
∞.
24
Part III
Monte Carlo Study of QuasiparticleExcitationsIn the previous part, we discussed the one-particle excitation of BEC by using the conserving-gapless theory.We concluded that the one-particle excitation has the finite lifetime proportional to s-wave scattering length awithin perturbation theory (the bubbling mode). In this chapter, we will estimate the one- and the two-particleexcitations by combining the Green’s function Monte Carlo (GFMC) method [36, 37, 38, 39, 40] and the momentmethod [41, 42], which are introduced in Appendix.B.
For performing the Monte Carlo (MC) simulation, we need the trial wave function. So, we will first constructthe Bijl-Jastraw type trial wave function [53] by solving two particle scattering problem (see also Appendix.C).Afterward, we will present results and conditions of MC simulations.
4 Construction of trial wave function
4.1 Hamiltonian and Units
We consider a system of N identical bosons with mass m and spin 0 interacting with a two-body potential U(r)in a box of volume V. The Hamiltonian is given by
H ≡∫dr ψ†(r)
p2
2mψ(r) +
∫dr
∫dr′ ψ†(r)ψ†(r′)U(r− r′)ψ(r′)ψ(r)
=∑p
εp +1
2V
∑pp′q
Uq c†p+qc
†p′−qcp′ cp, (100)
where the potential can be expanded as
U(r) =1
V∑q
Uqeiq·r ←→ Uq =
∫dr U(r)e−iq·r. (101)
Here we use the following potential:
U(r) =U0
8πr30e−r/r0 , (102)
for which Uq is given by Uq = U0/(1 + q2r20)2. We can construct Bijl-Jastraw type trial wave function for H by
solving a two-particle scattering problem. We present the procedures as follows.
4.2 Variational optimization
We start from the following shrodinger equation:[− ~2
2mred
1
r
d2
dr2r + U(r)
]ψ(r) = Eψ(r), (103)
where mred ≡ m/2. By introducing u(r) ≡ rψ(r), we obtain[− ~2
2mred
d2
dr2+ U(r)
]u(r) = Eu(r). (104)
We also introduce dimensionless length ξ ≡ r/a in therms of the s-wave scattering length a. Equation (104)can be rewritten as
d2u(ξ)
dξ2=
1
2D[U(ξ)− E]u(ξ), (105)
25
0
0.5
1
1.5
2
0 2 4 6 8 10
"psi1.dat""pot1.dat"
Fig 11: Plots of f(ξ) (red line) and its potentilal U(ξ) (green line).
where D ≡ ~2/2ma2 and U(ξ) ≡ (U0/8πr30)e
−(a/r0)ξ. We can solve this equation numerically by using Runge-Kutta method. We also set ~ = 1, m = 1/2 respectively.
We can also estimate the s-wave scattering legth a by solving the following equation:
a =m
4πU (eff)(0, 0), (106)
with
U (eff)(p, p′) = U0(p, p′)−
∫d3p1(2π)3
U0(p, p1)1
2εp1
U (eff)(p1, p′)
= U0(p, p′)− 1
4π2
∫ ∞
0
dp1 U0(p, p1)1
2εp1
U (eff)(p1, p′), (107)
where U0(p, p′) can be given by
U0(p, p′) =
U0
(1 + r20p2 + r20p
′2)2 − 4r40p2p′2
, (108)
for our potential. Using this equations, we can set s-wave scattering length by fixing r0 and U0. We set a = 1and r0 = 0.5, then U0 can be calculated U0 ≈ 57.7465 numerically.
We first construct a function as f0(ξ)→ 1 for large ξ, so we try to connect ψ(ξ) to a constant ψ0 at ξ = ξmax
which satisfies ψ′(ξmax) = 0. By normalizing it, we obtain f0(ξ) as
f0(ξ) =
ψ(ξ)/ψ0 (ξ ≤ ξmax)
1 (ξ > ξmax),(109)
Next, we connect f0(ξ)(ξs ≤ ξmax) to another function at ξ = ξs. We choose the function f1(ξ) as
f1(ξ) ≡ 1−A exp
(− ξα
)(ξ ≥ ξs), (110)
where the variational parameters A,α and ξs can be obtained by variational optimization. Then we match f0(ξ)and f1(ξ) at ξ = ξs up to the second derivative, which can be represented as
f0(ξs) = f1(ξs), f ′0(ξs) = f ′1(ξs), f ′′0 (ξs) = f ′′1 (ξs).
26
These conditions can be transformed into
α =1− f0(ξs)f ′0(ξs) ,
(111a)
A = (1− f0(ξs)) exp(ξsα
),
(111b)
f0(ξs)
f ′′0 (ξs)+ α2
(1
Aexp
(ξsα
)− 1
)= 0, (111c)
with which we seek the solution in 1 > A,α > 0 and ξmax > ξs. If we fix the value of ξmax, then other variationalparameters yield automatically. ξmax can be optimized by using VQMC method, which can be estimated asξmax ≈ 14.8867. Other parameters are also determined as ξs = 6.0512, E = 2.06124× 10−3, A = 0.702866, α =2.76514, respectively. The function f(ξ) is plotted in Fig.11. We also present the results of VQMC calculation,which are shown in Fig.12. Our conditions are the following: Walker number is 400, and prestep count andobservation count are 105 and 2.0×105 respectively. We adopt the rigid boundary condition for a cubic box andparticle number N = 50, and particle density na3 = 1.0 × 10−3. We obtained the value of energy per particleE(VQMC)/N ≈ 8.751(1)× 10−3 at ξmax = 14.8867.
Thus we can construct N -body trial wave function Ψ(R) in the Bijl-Jastraw form as
Ψ(R) ≡∏i<j
f(rij), (112)
with
f(ξ) =
ψ(ξ)/ψ0 (ξ ≤ ξs)
1−A exp
(− ξα
)(ξ > ξs),
(113)
where R denotes positions of particles R = (r1, · · · , rN ), and rij = |rij |, rij ≡ ri − rj .Finally, we show the energy estimated by GFMC method. We chose the unit of time as τ ≡ ~(E(VQMC)/N)−1
and set ∆τ = 1.0×10−3τ . We also set walker number 400, prestep number 30000 and main step number 100000respectively. Then, we obtain the energy per particle E/N ≈ 8.3455(2)× 10−3.
0.0087
0.00875
0.0088
0.00885
0.0089
0.00895
0.009
0.00905
0.0091
11 12 13 14 15 16 17 18 19
Fig 12: Plots of the energy per particle E/N to variational parameter ξmax.
5 Expression of quasiparticle excitations by moments
We next present how to estimate quasiparticle excitation in N identical bosons. We introduce the momentmethod, which was used for investigating two-particle excitation spectrum for superfluid 4He and magnon
27
systems. Previously, it has been used only for estimating the peak position of the dynamic structure factorS(q, ω) with frequency ω based on the first moment. However, by extending the method to higher moments, wecan estimate lifetime of the two-particle excitation. And we can also obtain details of the one-particle excitationby using it for the one-particle spectral function A(k, ε). Here we will outline the method. These details aregiven in Appendix B.
5.1 Density correlation function and Dynamic structure factor
We first introduce the one-particle spectral function A(k, ε) and the dynamic structure factor S(q, ω) for T = 0as
A(k, ε) =∑ν
|⟨Ψν |a†k|ΨG⟩|2δ(ε− Eν + EG), (114a)
S(q, ω) ≡∑ν
|⟨Ψν |ρ†q|ΨG⟩|2δ(ω − Eν + EG). (114b)
respectively, where |Ψν⟩ is the eigenket of our system, and Eν is its eigenenergy. We also express the groundstate ket and its eigen energy |ΨG⟩, EG respectively. The operator ρq is given by
ρq ≡∑k
a†q+kak. (115)
We also introduce the moments of A(k, ε) and S(q, ω) as
Jn(k) ≡∫ ∞
−∞dε εnA(k, ε) =
∑ν
|⟨Ψν |a†k|ΨG⟩|2(Eν − EG)n, (116a)
In(q) ≡∫ ∞
−∞dω ωnS(q, ω) =
∑ν
|⟨Ψν |ρ†q|ΨG⟩|2(EG − Eν)n. (116b)
respectively.We first review how to estimate the two-particle excitation. The dynamic structure factor S(q, ω) includes
the information of two-particle excitation. The pole of S(q, ω) and its half-width correspond to the spectrumand its lifetime at the frequency ω respectively. So, we can find that the integration of S(q, ω) over ω isthe distribution function for mode q. Thus the first-order moment I1(q) is proportional to the two-particledispersion ωq. However, I1(q) is not normalized, so we here introduce the following functions:
Sn(q, ω) ≡S(q, ω)ωn
I0(q) ,
In(q) ≡In(q)
I0(q) .(117)
Noting that the function Sn(q, ω) is normalized, we realize that I1(q) signifies the peak position of the two-particle excitation ωq. In addition, the second moment includes information about the width of the modeS(q, ω). To be more specific, the two-particle excitation spectrum ωq and its half-width of S(q, ω) can beexpressed as
ωq = I1(q), (118a)
δωq =
√I2(q)− I1(q)2. (118b)
This consideration can be extended for the one-particle excitation by replacing S(q, ω) by A(k, ε). Definingthe normalized moments of A(k, ε) as
An(k, ε) ≡A(k, ε)εn
J0(k) ,
Jn(k) ≡Jn(k)
J0(k) .(119)
We also obtain the one-particle excitation spectrum εk and its lifetime δεk as
εk = J1(k), (120a)
δεk =
√J2(k)− J2
1 (k), (120b)
respectively. So, we have to calculate the moments up to the second order for both the one- and two- particlechannels.
28
5.2 Expressions of moments
We can construct moments of A(k, ε) as
J0(k) = 1 +
∫dr g1(r)e
−ik·r, (121a)
J1(k) =~2k2
2m
[1 +
∫dr g1(r)e
−ik·r]+ nUq=0 +
∫dr U(r)g1(r)e
−ik·r +N(N − 1)
∫dr SA(r)e
−ik·r, (121b)
J2(k) =
(~2k2
2m
)2 [1 +
∫dr eik·rg1(r)
]+
~2k2
2m
[nUq=0 +
∫dr U(r)g1(r)e
ik·r +N(N − 1)
∫dr SA(r)e
ik·r]
+~2k2
2m
[nUq=0 +
∫dr U(r)g1(r)e
ik·r +N(N − 1)
∫dr SA(r)e
ik·r]∗
+ n
∫dr [U(r)]2 +
∫dr [U(r)]2g1(r)e
ik·r +
∫dr
∫dr′ U(r− r′)U(r)g2(r
′)
+N(N − 1)
[∫dr U(r)SA(r)e
ik·r +
(∫dr U(r)SA(r)e
ik·r)∗]
+N(N − 1)
∫dr SB(r)e
ik·r +N(N − 1)(N − 2)
∫dr SC(r)e
ik·r, (121c)
where g1(r) and g2(r) are the one-particle and the two-particle density matrices respectively, and the functionsSA, SB , SC are given by
SA(r) ≡1
V
∫dr1 · · ·
∫drN U(r1 + r− r2)Ψ
∗G(r1, r2, · · · , rN )ΨG(r1 + r, r2, · · · , rN ), (122a)
SB(r) ≡1
V
∫dr1 · · ·
∫drN U(r1 − r2)U(r1 − r2 + r)Ψ∗
G(r1, r2, · · · , rN )ΨG(r1 + r, r2, · · · , rN ) (122b)
SC(r) ≡1
V
∫dr1 · · ·
∫drN U(r1 − r2)U(r1 − r3 + r)Ψ∗
G(r1, r2, · · · , rN )ΨG(r1 + r, r2, · · · , rN ), (122c)
respectively, where ΨG(R) is the ground state wave function.Next, the moments of S(q, ω) can be represented as
I0(q) = N
1
n
∫dr[g2(r)− 1]eiq·r +
Vnδq0 + 1
, (123a)
I1(q) = N~2q2
2m, (123b)
I2(q) = N
(~2q2
2m
)2
+~2q2
3m
∑k
k2∫dr g1(r)e
−ik·r. (123c)
We can see that these moments (121) and (123) have apparently different forms. We estimate these functionsby using GFMC and VQMC methods.
Finally, we introduce the unit of moments (k, q) ≡ (ak, aq).
6 Numerical results
We will here present our numerical results obtained by using the Bijl-Jastraw type wave function constructed inSec. 4.2. We need to calculate the one- and the two- particle density matrices and several functions to obtainthe quasi-particle excitation.
6.1 Density matrices and some functions
To estimate lifetimes, we need to calculate the density matrices and functions in Eq. (122). So, we here presentplots of these functions estimated by VQMC and GFMC methods. The conditions of the simulations are as
29
follows: Particle number N , density na3, some variational parameters for the trial wave function, and units areused the same as in Sec. 4.2. For VQMC simulation, we usually set the number of walker n(v) = 400, prestepcount 105 and main step count 3×104. But for two-particle density matrix, we set the main step count 9×104,because it has the worst convergence property. For GFMC calculations, we set the same walker number and stepfor main sampling as the VQMC calculation. We give the plots of the one- and two- particle density matricesand other basic functions in Figs.13-17. We obtain the values of functions by GFMC and VQMC samplings.We apply the “extrapolated” estimation for each of these functions. Finally we calculate the moments (121),(123) numerically.
0
0.005
0.01
0.015
0.02
0 2 4 6 8 10 12 14 16 18 20
"rho1_DMC_50_full.dat""rho1_VQMC_50_full.dat"
Fig 13: Plots of the one-particle density matrices per thedensity n with their errorbars. The red and the greenlines are estimated by GFMC and VQMC methods re-spectively.
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
"g2_DMC_50_full.dat""g2_VQMC_50_full.dat"
Fig 14: Plots of the the two-particle density matrices persquare of the density n2 with their errorbars. The redand the green lines are estimated by GFMC and VQMCmethods respectively. These lines have good agreementfor large ξ.
0.02
0.025
0.03
0.035
0 2 4 6 8 10 12 14 16 18 20
"SA_DMC_50_full.dat""SA_VQMC_50_full.dat"
Fig 15: Plots of SA(ξ) with their errorbars. The redand the green lines are estimated by GFMC and VQMCmethods respectively.
0
0.005
0.01
0.015
0.02
0 2 4 6 8 10 12 14 16 18 20
"SB_DMC_50_full.dat""SB_VQMC_50_full.dat"
Fig 16: Plots of SB(ξ) with their errorbars. The redand the green lines are estimated by GFMC and VQMCmethods respectively. SB(ξ) goes to rapidly 0 for largeξ.
30
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0 2 4 6 8 10 12 14 16 18 20
"SC_DMC_50_full.dat""SC_VQMC_50_full.dat"
Fig 17: Plots of SC(ξ) with their errorbars. The red and the green lines are estimated by GFMC and VQMCmethods respectively.
6.2 lifetime estimation
Before we present the final results, we comment on the first moment of the one-particle spectral function (120a).We note that the first-order moment J1(k) is apparently nonzero at k = 0 as seen in Eq. (121b). However, thisis not inconsistent with the Hugenholtz-Pines theorem. To confirm it, we start from the following expression ofJ1(k):
J1(0) =⟨ΨG|a0[H, a†0]|ΨG⟩⟨ΨG|a0a†0|ΨG⟩
=⟨ΨG|a0Ha†0|ΨG⟩ − ⟨ΨG|Ha0a†0|ΨG⟩
⟨ΨG|a0a†0|ΨG⟩. (124)
As seen in the above calculation, J1(0) consists of two different expectations with dimension of energy. If weconsider a system of the grand canonical ensemble and take the Lehman representation for the one-particleGreen’s function [44, 51], a similar term remains at k = 0, which corresponds to the chemical potential µ. Inthe present case where we fix the particle number (the canonical ensemble), the term obtained in Eq. (124) isnot exactly the chemical potential in this system. However, we are interested in the lifetime of the one-particleexcitation. This term only specify the position of the one-particle spectral function. So, when we consider thesecond-order moment for the one-particle excitation, this therm is removed. We show the plots of Jn(k) in Fig.18, 19, 20.
Next, we show the results of the half-widths of the one- and the two-particle excitation in Fig.23. For lowmomentum region the two-particle excitation cannot be calculated. It turns out impossible to obtain a reliableresult for the second moment in the low-momentum region due to the lack of enough accuracy in the largeξ region in the coordinate space. However, I0(q) approaches to N at the high momentum regime as seen inFig.22. So we can obtain the half-width in the high momentum regime. That is derived from the limit forξ →∞ At this regime, using the Eq. (123), the half-width δωq can be approximated by
δωq =
√I2(q)
I0(q)−(I1(q)
I0(q)
)2
≈
√na3
2Dq2
3
∫dk
(2π)3k2∫dξ g1(ξ)e−ik·ξ ∝ q. (125)
Actually, we can see the linear behavior of δωq in Fig 23. On the other hand, the half-width of the one-particleexcitation δεk has an apparently different behavior, which decreases monotonously as a function of k.
Therefore, we conclude that these half-widths are qualitatively different, which contradicts the statementGavoret and Nozieres [18] but agrees with that of the conserving-gapless theory.
31
1
1.002
1.004
1.006
1.008
1.01
0 0.5 1 1.5 2
"J0.dat"
Fig 18: Plot of J0(k).
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2
"J1.dat"
Fig 19: Plot of J1(k).
0
2
4
6
8
10
12
14
16
18
0 0.5 1 1.5 2
"J2.dat"
Fig 20: Plots of J2(k).
0.8
0.85
0.9
0.95
1
0 0.5 1 1.5 2 2.5
"I0.dat"
Fig 21: Plots of I0(q) per particle number N . It con-verges to 1 for large momentum.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2
"spec1_50.dat""spec2_50.dat"
Fig 22: Plots of εk and ωq. The red and the greenlines are results for the one- and the two-particle exci-tations. εk starts from a finite value corresponding tothe chemical potential. They have similar behavior forlarge momentum region.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
"spec1_life_50.dat""spec2_life_50.dat"
f2(x)
Fig 23: Plots of the half-widths of the quasi-particleexcitations. The red, the green and the dashed linesare results the one- and the two-particle excitations andthe auxiliary line f(ξ) = 0.0577q for δωq. The low mo-mentum region of the two-particle excitation cannot becalculated because we couldn’t estimate the two-particledensity matrix for large ξ.
32
Part IV
Summary and Conclusion
7 Summary and Conclusion
In this thesis, we have investigated the details of the quasiparticle excitation of BEC. Our motivations of thestudy are summarized concisely as follows. Bogoliubov theory predicts that the one-particle excitation hasan infinite lifetime and a linear dispersion at k → 0 [13]. Also, Gavoret and Nozieres claim that the one-particle Green’s function for BEC share its pole with that of the two-particle Green’s function [18] . Thus,they conclude that the Bogoliubov mode is a collective mode. However, the proof by Gavoret and Nozieres hasan ambiguity in the definition of the Green’s functions. Moreover, the Bogoliubov theory has some difficulties.One is relevant to its extension for the finite temperature system (HFB theory) [21, 22]. It gives a gap theone-particle excitation, which is inconsistent with the Hugenholtz-Pines (Goldstone) theorem [17, 19]. There isthe approximation of setting the gap zero by hand (Shono approximation [23, 24]). However, such a procedurebreaks the conservation laws when used in dynamical problems. Such a situation is called “conserving v.s.gapless dilemma” by Hohenberg and Martin. Recently, conserving-gapless theory was constructed by Kita [30].It corresponds to the extension of the Luttinger-Ward [28] theory for BEC. It also predicts that the one- andthe two-particle excitation are different [31]. Especially, the one-particle excitation is not the Bogoliubov modebut the bubbling mode, which has the finite lifetime.
We have studied the one-particle excitation by using the conserving-gapless theory for weak interactingsystem at T = 0. We have obtained the result that the lifetime of the one-particle excitation is proportional tothe s-wave scattering length a [33]. We have found that the physical quantities proposed by Lee, Huang, Yang[34] are also modified [35]. These properties originate from the 1PR diagrams which are overlooked so far in theprevious studies. Next, we have studied the one- and the two-particle excitation of BEC system at T = 0 byusing the Monte Carlo method [37, 54]. We have used the moment method [41, 42] extended to higher orders,which is a powerful method to estimate them quantitatively. We have obtained the results that the one- andtwo-particle excitation have qualitatively different behavior, which implies the proof by Gavoret and Nozieresare incorrect and justifies the statement of the conserving-gapless theory. The Bogoliubov theory and statementby Gavoret and Nozieres have some flaws. On the other hand, the conserving-gapless theory describes morerealistic situation given by the Monte Carlo simulation. We are confident that the conserving-gapless theory isone of powerful tools to understand BEC theoretically.
33
Part V
Appendix
A Conserving-Gapless Theory
A.1 Notations
A.1.1 Hamiltonian
We introduce some notations to calculate thermodynamical properties of BEC. Here we consider N identicalbosons with mass m and spin 0 and interacting with a two-body potential V (r − r′). The correspondingHamiltonian H is given by [50]
H ≡∫d3r ψ†(r)K1(r)ψ(r) +
1
2
∫d3r
∫d3r′ ψ†(r)ψ†(r)′V (r− r′)ψ(r′)ψ(r), (A.1)
where (ψ(r), ψ†(r)) are annihilation and creation operators satisfying the bosonic commutation relations, andK1(r) ≡ p2/2m+U(r)− µ with U(r) the external field and µ the chemical potential. Then, we introduce fieldoperators of Heisenberg representation as
ψ1(1) ≡ e−τ1H ψ(r1)eτ1H , ψ2(1) ≡ e−τ1H ψ†(r1)e
τ1H , (A.2)
where the arabic number 1 is defined by 1 ≡ (r1, τ1), and τ is the imaginary time (0 ≤ τ ≤ β) with β ≡ (kBT )−1
and temperature T . In BEC system, we must divide these field operators into two parts as
ψi(1) = Ψi(1) + ϕi(1), (i = 1, 2), (A.3)
where Ψi(1) ≡ ⟨ψi(1)⟩ and ϕi(1) are the condensate wave function and quasi-particle operator respectively,and ⟨· · ·⟩ is the grand-canonical average. They have the following properties: (i) Ψ1(1) = Ψ∗(2) ≡ Ψ(r1), (ii)⟨ϕi(1)⟩ = 0.
A.1.2 Matsubara Green’s function
By using these operators, we define the Matsubara Green’s function in the Nambu space as
Gij = −⟨Tτϕi(1)ϕ3−j(2)⟩(−1)j−1, (A.4)
with the time-ordering operator Tτ . The off-diagonal elements Gij (i = 1 , j = 2) and (i = 2, j = 1) representexpectations of pair annihilation and pair creation respectively, which are characteristic terms of BEC system.We can also write the particle number N as
N =
∫d3r1 [Ψ2(1)Ψ1(1)−G11(1, 1+)], (A.5)
with 1± ≡ (r1, τ1 ± η), where η is the infinitesimal positive constant. Then, we find symmetry relations for Gij
as
Gij(1, 2) = (−1)i+j−1G3−j,3−i(2, 1) = (−1)i+jG∗ji(r2τ1, r1, τ2). (A.6)
Using these notations, we will give a representation of the thermodynamic potential Ω as a functional of theMatsubara Green’s function and formulate a self-consistent perturbation expansion.
A.2 Relations for BEC system
Here we derive basic relations for BEC to carry out the perturbation expansion for the thermodynamic potentialΩ [30].
34
A.2.1 Time evolution operator
First, we introduce the extra perturbation to H as
Hext(τ1) ≡∫d3r1[ψ
†(r1)ηext(1) + ψ(r1)η∗ext(1)], (A.7)
where ηext(1) is the periodic function in τ1 with period β. The total Hamiltonian in this section is defined by
H(τ1) ≡ H + Hext(τ1). (A.8)
By using this extra perturbation, we will derive exact relations for BEC, and finally ηext will be set equal to 0.To begin with, we introduce the time evolution operator U for H as
U(τ, τ0) ≡ 1 +∞∑
n=1
(−1)n∫ τ
τ0
dτn · · ·∫ τ2
τ0
dτ1H(τn) · · · H(τ1)
=
Tτ exp
[−∫ τ
τ0
dτ1H(τ1)]
(τ ≥ τ0)
T aτ exp
[−∫ τ
τ0
dτ1H(τ1)]
(τ < τ0),
(A.9)
where Tτ and T aτ are time-ordering and anti-time ordering operators respectively. We also confirm that U
satisfies U(τ0, τ0) = 1, U−1(τ, τ0) = U(τ0, τ), and
U(τ, τ1)U(τ1, τ0) = U(τ, τ0). (A.10)
Moreover, it also satisfies following equations:
dU(τ, τ0)dτ
= −H(τ)U(τ, τ0), (A.11a)
dU(τ, τ0)dτ0
= U(τ, τ0)H(τ0). (A.11b)
We note that U reduces to the normal time evolution operator for ηext → 0.
A.2.2 Equation of motion
We introduce the Heisenberg representation for H by using U asψ(1) ≡ U−1(τ1)ψ(r1)U (τ1)ψ(1) ≡ U−1(τ1)ψ
†(r1)U (τ1),(A.12)
where 0 ≤ τ1 ≤ β. Equations of motion for these operators are given by(− ∂
∂τ1−K1
)ψ(1) = ηext(1) +
∫d1′ V (1, 1′)ψ(1′)ψ(1′)ψ(1), (A.13a)(
∂
∂τ1−K1
)ψ(1) = η∗ext(1) +
∫d1′ V (1, 1′)ψ(1)ψ(1′)ψ(1′), (A.13b)
with V (1, 2) ≡ δ(τ1 − τ2)V (r1 − r2).The expectation value of O(1) ≡ U−1(τ1)O(r1)U(τ1) is also defined by
⟨O(1)⟩ ≡ TrTτ U(β)O(1)TrU(β)
, (A.14)
35
where ⟨O(1)⟩ reduces to the normal form of the grand canonical average if we put ηext → 0. By using thisnotation, we take thermodynamic average of Eqs. (A.13) to obtain(
− ∂
∂τ1−K1
)Ψ(1) = ηext(1) + η(1), (A.15a)(
∂
∂τ1−K1
)Ψ(1) = η∗ext(1) + η(1), (A.15b)
respectively, where we also define expectation values for ψ(1), ψ(1) as
Ψ(1) ≡ ⟨ψ(1)⟩, Ψ(1) ≡ ⟨ψ(1)⟩, (A.16)
respectively, and we used the following notations:
η(1) ≡∫d1′ V (1, 1′)⟨ψ(1′)ψ(1′)ψ(1)⟩, (A.17a)
η(1) ≡∫d1′ V (1, 1′)⟨ψ(1)ψ(1′)ψ(1′)⟩. (A.17b)
We note that the condensate wave function in this section is given as the expectation value of the field operatordifferent from that introduced in the previous section.
A.2.3 Dyson-Beliaev equation
In this section, we will derive the Dyson-Beliaev equation [14] by using equations in the previous sections. First,we differentiate Ψ(1) in Eq. (A.16) with ηext(2). By using the definition Eq. (A.14), we then obtain
δΨ(1)
δηext(2)= −⟨Tτψ(1)ψ(2)⟩+Ψ(1)Ψ(2) ≡ G(1, 2). (A.18)
where we can see that this quantity is equal to ⟨Tτϕ1(1)ϕ2(2)⟩ by using the definition of Eq. (A.3). Similarly,we introduce
δΨ(1)
δηext(2)≡ G(1, 2), δΨ(1)
δη∗ext(2)≡ −F (1, 2) (A.19a)
δΨ(1)
δηext(2)≡ −F (1, 2), δΨ(1)
δη∗ext(2)≡ G(1, 2). (A.19b)
Next, the self energies are defined by
Σ(1, 2) ≡ δη(1)
δΨ(2), ∆(1, 2) ≡ δη(1)
δΨ(2), (A.20a)
∆(1, 2) ≡ δη(1)
δΨ(2), Σ(1, 2) ≡ δη(1)
δΨ(2). (A.20b)
By using these functions, the functional differentiation of η(1) with respect to ηext(2) is transformed as
δη(1)
δηext(2)=
∫d3
[δη(1)
δΨ(3)
δΨ(3)
δηext(2)+δη(1)
δΨ(3)
δΨ(3)
δηext(2)
]=
∫d3 [Σ(1, 3)G(3, 2)−∆(1, 3)F (3, 2)]. (A.21a)
And, similarly,
δη(1)
δη∗ext(2)=
∫d3 [−Σ(1, 3)F (3, 2) + ∆(1, 3)G(3, 2)], (A.21b)
δη(1)
δηext(2)=
∫d3 [∆(1, 3)G(3, 2)− Σ(1, 3)F (3, 2)], (A.21c)
δη(1)
δη∗ext(2)=
∫d3[−∆(1, 3)F (3, 2) + Σ(1, 3)G(3, 2)]. (A.21d)
36
The functional differentiation of Eq. (A.15a) with respect to ηext(2) is given by(− ∂
∂τ1−K1
)δΨ(1)
δηext(2)= δ(1, 2) +
δη(1)
δηext(2). (A.22)
Substituting this equation to Eq. (A.21a) and using the notation of Eq. (A.18), we obtain(− ∂
∂τ1−K1
)G(1, 2) = δ(1, 2) +
∫d3 [Σ(1, 3)G(3, 2)−∆(1, 3)F (3, 2)], (A.23a)
where δ(1, 2) ≡ δ(τ1 − τ2)δ(r1 − r2). And, with similar procedures, we also obtain
−(− ∂
∂τ1−K1
)F (1, 2) =
∫d3 [−Σ(1, 3)F (3, 2) + ∆(1, 3)G(3, 2)], (A.23b)
−(− ∂
∂τ1−K1
)F (1, 2) =
∫d3 [−∆(1, 3)G(3, 2)− Σ(1, 3)F (3, 2)], (A.23c)
−(− ∂
∂τ1−K1
)G(1, 2) = δ(1, 2) +
∫d3 [∆(1, 3)F (3, 2) + Σ(1, 3)G(3, 2)]. (A.23d)
By combining these equations, we finally arrive at the Dyson-Beliaev equation in a matrix representation as∫d3 [G−1
0 (1, 2)− Σ(1, 3)]G(3, 2) = σ0δ(1, 2), (A.24)
where σ0 is a 2× 2 unit matrix, and we define
G−10 (1, 2) ≡
(− σ0
∂
∂τ1− σ3K1
)δ(1, 2), (A.25a)
Σ(1, 2) ≡[Σ(1, 2) ∆(1, 2)−∆(1, 2) −Σ(1, 2)
], (A.25b)
G(1, 2) ≡[G(1, 2) F (1, 2)−F (1, 2) −G(1, 2)
].
(A.25c)
A.2.4 Hugenholtz-Pines relation
Here we will derive the Hugenholtz-Pines relation. This relation states that the quasi-particle excitation in BECsystem has gapless (massless) mode, which was proved by Hugenholtz and Pines in 1959 [17]. It is regarded asGoldstone’s theorem for the U(1) symmetry breaking.
We first consider the following gauge transformation:
ηext(1)→ eiχηext(1), ψ(r1)→ eiχψ(r1), (A.26)
where χ is a constant. By using Eq. (A.15) and (A.21), we obtain
0 =
(− ∂
∂τ1−K1
)δΨ(1)− δηext(1)− δη(1)
= iχ
(− ∂
∂τ1−K1
)Ψ(1)− ηext(1)−
∫d2 [Σ(1, 2)Ψ(2)−∆(1, 2)Ψ(2)]
, (A.27)
0 = iχ
(− ∂
∂τ1−K1
)Ψ(1)− η∗ext(1)−
∫d2 [∆(1, 2)Ψ(2)− Σ(1, 2)Ψ(2)]
, (A.28)
where we used the following variational relations: δηext(1) = iχηext(1), δη∗ext(1) = −iχη∗ext(1), δΨ(1) = iχΨ(1),
δΨ(1) = −iχΨ(1). By considering an arbitrariness of χ and taking the limit ηext → 0, we can represent aboveequations in a combined form as ∫
d2 [G−10 (1, 2)− Σ(1, 2)]Ψ(2) =
[00
],
(A.29)
37
where we also define the vector Ψ(1) by
Ψ(1) ≡[Ψ(1)−Ψ(1)
].
(A.30)
This equation is the inhomogeneous extension of the Hugenholtz-Pines relation. By imposing the uniformcondition U(r)→ 0, Ψ(1)→ n0 (n0: condensate density), we can express it as
µ =
∫d2 [Σ(1, 2)−∆(1, 2)]. (A.31)
Then by performing the Fourier transform to this relation, we arrive at the well-known form [44, 51] :
µ = Σp=0 −∆p=0, (A.32)
where p = (p, iϵn ≡ 2πinβ−1) and n is an integer.
A.3 Construction of Luttinger-Ward functional Φ for BEC
In this section, we derive the expression of the Luttinger-Ward functional for BEC. Our goal is to construct theLuttinger-Ward functional which satisfies the Dyson-Beliaev equation (A.24) and the Hugenholtz-Pines relation(A.29) simultaneously.
A.3.1 Interaction Energy
We first give an expression of a thermodynamic average of interaction term Hint. Let us introduce Hint by
Hint ≡1
2
∫dr
∫dr′ ψ(r)ψ†(r′)V (r− r′)ψ(r′)ψ(r′). (A.33)
By taking the limit ηext → 0 for Eqs. (A.13a,b), then multiplying ϕ2(1′), ϕ2(1
′) from the right sides respectively,and operating Tτ and finally taking the thermodynamic average, we obtain the following two equations
−(− ∂
∂τ1−K1
)G(1, 1′) + δ(1, 1′) =
∫d2 V (1, 2)⟨Tτ ψ(2)ψ(1)ψ(2)ϕ2(1′)⟩, (A.34a)
−(− ∂
∂τ1−K1
)G(1, 1′) + δ(1, 1′) =
∫d2 V (1, 2)⟨Tτ ψ(2)ψ(1)ψ(2)ϕ1(1′)⟩, (A.34b)
respectively, where we used the definition of quasi particle operator ⟨ϕi⟩ = 0 and the relation
−⟨Tτ∂ϕ1(1)
∂τ1ϕ2(1
′)⟩ = ∂
∂τ1⟨Tτϕ1(1)ϕ2(1′)⟩+ δ(1, 1′).
We also take the same limit for Eqs. (A.17a,b), and multiply Ψ(1), Ψ(1) from the left sides respectively. Weadd these equations and Eqs. (A.34a,b) to obtain
Ψ(1)
(− ∂
∂τ1−K1
)Ψ(1) + Ψ(1)
(∂
∂τ1−K1
)Ψ(1)
+
[−(− ∂
∂τ1−K1
)G(1, 1′) + δ(1, 1′)
]1′=1+
+
[−(∂
∂τ1−K1
)G(1, 1′) + δ(1, 1′)
]1′=1−
=
∫d2 V (1, 2)
[⟨Tτ ψ(2)ψ(2)ψ(1)ϕ2(1+)⟩+ ⟨Tτ ψ(1)ψ(2)ψ(2)ϕ1(1−)⟩
+Ψ(1)⟨Tτ ψ(2)ψ(2)ψ(1)⟩+ Ψ(1)⟨Tτ ψ(1)ψ(2)ψ(2)⟩], (A.35)
where we take 1′ → 1+, 1′ → 1+.
38
Next we integrate these equations over 1. Then we can find the results multiplied by 1/4β is equal to thethermodynamic average of Eq. (A.33). Also considering V (1, 2) = V (r2 − r1)δ(τ2 − τ1) and
∫dτ1δ(τ1) = β, we
obtain
⟨Hint⟩ =1
4β
∫d1
Ψ(1)
(− ∂
∂τ1−K1
)Ψ(1) + Ψ(1)
(∂
∂τ1−K1
)Ψ(1)
+
[−(− ∂
∂τ1−K1
)G(1, 1′) + δ(1, 1′)
]1′=1+
+
[−(∂
∂τ1−K1
)G(1, 1′) + δ(1, 1′)
]1′=1−
. (A.36)
Subsequently, we use the Dyson-Beliaev equation (A.24) and the Hugenholtz-Pines relation (A.29). Hence wecan transform Eq. (A.36) into
⟨Hint⟩ =1
4β
∫d1
∫d22Σ(1, 2)[Ψ(2)Ψ(1)−G(2, 1+)]
− ∆(1, 2)[Ψ(2)Ψ(1)− F (2, 1)]−∆(1, 2)[Ψ(2)Ψ(1)− F (2, 1)], (A.37)
where we use Eq. (A.6), and Σ(2, 1) = Σ(1, 2), G(1−, 2) = G(1, 2+).
A.3.2 Luttinger-Ward functional
Luttinger and Ward [28] gave an expression of the thermodynamic potential Ω ≡ −β lnTr[exp(−βH)] for anormal Fermi system as a functional of the self energy Σ. Their formulation can be easily extended for thenormal Bose system. For convenience, we regard our thermodynamic potential in normal state Ωnom as afunctional of G, which can be written as
Ωnom ≡ −β−1Tr[ln(−G−10 +Σ) + ΣG] + Φnom (A.38)
where the trace is defined by TrAB ≡∫d1∫d2 A(1, 2)B(2, 1+) and
G−10 (1, 2) ≡
(− ∂
∂τ1− K1
)δ(1, 2). (A.39)
The functional Φnom denotes all contribution of skelton diagrams in the perturbation expansion for Ωnom withthe replacement G0 → G. The functional differentiation of Φnom with respect to G give the self energy Σ as
Σ(1, 2) = −β−1 ∂Φnom
∂G(2, 1) .(A.40)
The stationary condition δΩ/δG(2, 1) = 0 yields the Dyson equation G = (G−10 − Σ)−1. We can also express
the functional Φnom order by order as
Φnom = −β−1∞∑
n=1
1
2nTrΣnG (A.41)
where we note Σ(n) has 2n− 1 Green’s function lines. Comparing it with Eq. (A.37) for the normal state limitF,Ψ→ 0, we obtain the following relation for ⟨Hint⟩ order by order:
⟨Hint⟩(n) =1
nΦ(n), (A.42)
where the factor 1/n is due to the extra Hint present in the evaluation of ⟨Hint⟩(n) as compared with that ofΦ(n).
39
A.3.3 De Dominicis-Martin theorem
Now we construct the Luttinger-Ward functional for Ψ = 0. Dominicis and Martin [52] prove that the thermo-
dynamic potential in thermal equilibrium is expressible as a functional such as Ω[T, µ, G, Ψ]. Like the Luttinger-Ward functional in the normal state, the exact Ω for BEC system is stationary with respect to variations for Gand Ψ:
δΩ
δG(2, 1)=
δΩ
δG(2, 1)=
δΩ
δF (2, 1)=
δΩ
δF (2, 1)= 0,
δΩ
δΨ(1)=
δΩ
δΨ(1)= 0. (A.43)
Note that our desirable thermodynamic potential Ω must satisfy both Dyson-Beliaev equation (A.24) and
Hugenholtz-Pines relation (A.29), and reduce to the normal-state functional Ωnom in the limit of Ψ→ 0. Sucha functional is given by
Ω = − 1
β
∫d1
∫d2 Ψ2(1)G
−10 (1, 2)Ψ1(2) +
1
2βTr[ln(−G−1
0 + Σ) + ΣG] + Φ, (A.44)
where G−10 ≡ (−σ0∂/∂τ1 − σ3K1)δ(1, 2), σ0, σ3 are 2× 2 unit matrix and third Pauli matrix respectively, and
the trace TrΣG is defined by
Tr ΣG ≡∫d1
∫d2 Tr
[Σ(1, 2) ∆(1, 2)−∆(1, 2) −Σ(1, 2)
] [G(2, 1+) F (2, 1)−F (2, 1) −G(2, 1−)
].
(A.45)
We also give an expression of the self energy Σ ≡ (Σij) similarly as in the case of the normal state (A.40) as
Σij(1, 2) ≡ −2βδΦ
δGji(2, 1), (A.46a)
where we used notations G ≡ (Gij), Ψ ≡ (Ψi), and Φ = Φ[G, Ψ] in Eq. (A.44). By using this definition, we canobtain the following relation:
βδΦ
δΨ3−i(1)= Σij(1, 2)(−1)j−1Ψj(2), (A.46b)
where summation and integration over barred indices j, 2 are implied. We haven’t found how Φ can beconstructed as the functional of G and Ψ yet. However, diffentiating Eq. (A.44) with respect to G and Ψ, thesereduce to the Dyson-Beliaev equation (A.24) and Hugenholtz-Pines relation (A.29) respectively.
A.3.4 Exact relations with Φ
We derive exact relations for BEC. Substituting the nth-order contribution of Eq. (A.37) into Eq. (A.42) andusing Eq. (A.46a), we obtain
2nΦ(n) +
∫d1
∫d2
δΦ(n)
δG(2, 1)[Ψ(2)Ψ(1)−G(2, 1)]
+δΦ(n)
δF (2, 1)[Ψ(2)Ψ(1)− F (2, 1)] + δΦ(n)
δF (2, 1)[Ψ(2)Ψ(1)− F (2, 1)]
= 0. (A.47)
We also substitute Eq. (A.46a) into Eq. (A.46b) to obtain
δΦ
δΨ(1)+
∫d2
[δΦ
δG(2, 1)Ψ(2) + 2
δΦ
δF (2, 1)Ψ(2)
]= 0. (A.48)
By using these equations, we can determine the specific structure of Φ order by order. For a later convenience,we define the symmetrized vertex as
Γ(0)(11′, 22′) ≡ V (r1 − r2)δ(τ1 − τ2)[δ(1, 1′)δ(2, 2′) + δ(1, 2′)δ(2, 1′)], (A.49)
which also satisfies the following relations:
Γ(0)(11′, 22′) = Γ(0)(22′, 11′) = Γ(0)(1′1, 2′2) = Γ(0)(2′1, 21′). (A.50)
40
A.3.5 Constructing Φ
We here present how to construct Φ which satisfies the Dyson-Beliaev equation (A.24) and the Hugenholtz-Pinesrelation (A.29) . Feynman diagrams of G,F, F are expressible in the same way as those of superconductivity[51] and Γ(0) is denoted by a filled circle. And we suppress drawing the condensate wave function Ψ, Ψ followingPopov [25], which can be easily recovered by considering the relation between the number of Green’s functionlines and its perturbative order. We summarize the steps for constructing nth-order functional Φ(n) as follows:
(i) : Draw all the normal-state diagrams contributing to Φ(n), whose numerical weights are equal to those ofthe perturbation analysis for the normal state.
(ii) : Draw all the distinct diagrams obtained from those of (i) by successively replacing G by F, F , whosenumerical weights are still undetermined.
(iii) : Draw all the distinct diagrams constructed by the steps (i) and (ii) by successively removing a Green’sfunction line so as to satisfy the condition that a further removal of any line from each diagram would notbreak itself into two unconnected parts.
(iv) : Determine numerical weights of these diagrams by using Eqs. (A.47), (A.48).
By following these procedures, we can construct the Luttinger-Ward functional order by order. We draw thediagrams Φ(1) and Φ(2) as Fig.24 and 25 respectively. Finally, we note that the diagram with only vertex inΦ(1) has the weight T/4.
(a) 2 (b) c(1)2b (c) c
(1)1a (c) c
(1)1b (c) 1
Fig 24: Diagramatic expression of Φ(1)
(a) 1 (b) c(2)4b (b) c
(2)4c
(c) c(2)3a (c) c
(2)3b (c) c
(2)3c (c) c
(2)3d
(c) c(2)2a (c) c
(2)2b (c) c
(2)2c
(c) c(2)2d (c) c
(2)2e
Fig 25: Diagramatic expression of Φ(1)
41
Based on these diagramatic expressions, we can express Φ(1) and Φ(2) as
Φ(1) =T
4
∫d1
∫d1′∫d2
∫d2′ Γ(0)(11′, 22′)
× 2G(1.1′)G(2, 2′) + c(1)2b F (1.2)F (1
′, 2′) + c(1)1a G(1, 1
′)Ψ(2)Ψ(2′)
+ c(1)1b [F (1, 2)Ψ(1′)Ψ(2′) + F (1′, 2′)Ψ(1)Ψ(2)] + Ψ(1′)Ψ(2′)Ψ(2)Ψ(1) (A.51)
Φ(2) =− T
8
∫d1 · · ·
∫d4′ Γ(0)(11′, 22′) Γ(0)(33′, 44′)
× G(2, 3′)G(3, 2′)G(1, 4′)G(4′, 1) + c(2)4b F (2
′, 3′)F (2, 3)G(1, 4′)G(4, 1′)
+ c(2)4c F (2
′, 3′)F (2, 3)F (1′, 4′)F (1, 4) + c(2)3a G(2, 3
′)G(3, 2′)G(1, 4′)Ψ(4)Ψ(1′)
+ c(2)3b [F (2
′, 3′)Ψ(2)Ψ(3) + F (2, 3)Ψ(2′)Ψ(3′)]G(1, 4′)G(4, 1′)
+ c(2)3c F (2
′, 3′)F (2, 3)G(1, 4′)Ψ(4)ψ(1′)
+ c(2)3d [F (2
′, 3′)Ψ(2)Ψ(3) + F (2, 3)Ψ(2′)Ψ(3′)]F (1′, 4′)F (1, 4)
+ c(2)2a G(2, 3
′)G(3, 2′)Ψ(1)Ψ(4)Ψ(4′)Ψ(1′) + c(2)2b G(2, 3
′)Ψ(3)Ψ(2′)G(1, 4′)Ψ(4)Ψ(1′)
+ c(2)2c [F (2
′, 3′)Ψ(2)Ψ(3) + F (2, 3)Ψ(2′)Ψ(3′)]G(1, 4′)Ψ(4)Ψ(1′)
+ c(2)2d F (2
′, 3′)F (2, 3)Ψ(1)Ψ(4′)Ψ(4)Ψ(1′)
+ c(2)2e [F (2, 3)Ψ(3′)Ψ(2′)F (1, 4)Ψ(4′)Ψ(1′) + F (2′, 3′)Ψ(3)Ψ(2)F (1′, 4′)Ψ(4)Ψ(1)], (A.52)
respectively. By imposing Eq. (A.47) for Eq. (A.51), we obtain
4 + c(1)1a = 0, c
(1)2b + c
(1)1b = 0, c
(1)1a + 2c
(1)1b + 2 = 0,
which can be solved easily as
c(1)2b = −1, c(1)1a = −4, c(1)1b = 1 (A.53)
Similarly, we also obtain the equations for Eq. (A.52) as
0 = c(2)3a + 4 = c
(2)3b + c
(2)4b = c
(2)3c + 2c
(2)4b = c
(2)3d + 2c
(2)4c ,
0 = c(2)2a + c
(2)3a + c
(2)3b = 2c
(2)2b + c
(2)3a = 2c
(2)2c + c
(2)3c + 2c
(2)3b
= 2c(2)2b + c
(2)3c + 4c
(2)3d = 2c
(2)2e + c
(2)3d ,
0 = c(2)2a + c
(2)2b + c
(2)2c = c
(2)2c + c
(2)2d + 2c
(2)2e .
We solve these equations as
c(2)4b = −2, c
(2)4c = 1
c(2)3a = −4, c
(2)3b = 2, c
(2)3c = 4, c
(2)3d = −2
c(2)2a = c
(2)2b = 2, c
(2)2c = −4 c
(2)2d = 2, c
(2)2e = 1. (A.54)
Thus we can determine numerical weights of the Feynman diagrams uniquely by using exact relations for BEC(A.47), (A.48). We note that we can obtain the same numerical weights by using either of them [30].
A.4 Two-particle Geen’s function
Here, we discuss two-particle Green’s function [31]. First, we introduce an extra perturbation U given in anS-matrix form. And, we differentiate it with respect to U , then take limit of U → 0. The procedure yields theBethe-Salpeter equation for BEC.
42
A.4.1 Extra perturbation
First, we define two-particle Green’s function by using Eq. (A.3) as
Kij,kl(12, 34) ≡⟨Tτψi(1)ψk(3)ψ3−l(4)ψ3−j(2)⟩(−1)j+l
− ⟨Tτψi(1)ψ3−j(2)⟩⟨Tτψk(3)ψ3−l(4)⟩(−1)j+l, (A.55)
with the nonlocal potential U ≡ Uij . We note that collective modes correspond to the two-particle Green’function.
Second, we introduce an extra perturbation described by the S-matrix as
S(β) ≡ Tτ exp[−1
2ψi(1)ψ3−j(2)(−1)j−1Uij(2, 1)
]. (A.56)
The full Matsubara Green’s function is also defined by
Gij(1, 2) ≡ −⟨TτS(β)ψi(1)ψ3−j(2)⟩
S(β)(−1)j−1
= −⟨TτS(β)ψi(1)ψ3−j(2)⟩c(−1)j−1, (A.57a)
where the subscript c represents contribution of Feynman diagrams which are connected with ψi(1) and (or)ψ3−j(2), and Ψi(1) is rewritten as Ψi(1) ≡ ⟨TτS(β)ψi(1)⟩c. Equation (A.57a) is also expressible as
Gij = Gij(1, 2)−Ψi(1)Ψ3−j(2)(−1)j−1, (A.57b)
where Gij(12) ≡ −⟨TτS(β)ϕi(1)ϕ3−j(2)⟩c(−1)j−1. Note this quantity reduces to Eq. (A.4) for U → 0. FromEq. (A.57a), it can be seen that two-particle Green’s function (A.55) is given by
Kij,kl(12, 34) = 2δGij(1, 2)δUlk(4, 3) ,
(A.58)
where taking the limit U → 0 after differentiation is implied. We will use this procedure implicitly below. Next,substituting Eq. (A.57b) into Eq. (A.58), we obtain
Kij,kl(12, 34) = 2δGij(1, 2)
δUlk(4, 3)− 2
[Ψi(1)
δΨ3−j(2)
δUlk(4, 3)+
δΨi(1)
δUlk(4, 3)Ψ3−j(2)
](−1)j−1, (A.59)
where we find that we only need to know linear responses of G and Ψ to U for writing K down explicitly.
A.4.2 Linear response of G and Ψ
First we note that an extra term appear in G−1(1, 2) ≡ G−10 (1, 2)− Σ(1, 2) by adding the perturbation U as
G−1(1, 2) =
(−σ0
∂
∂τ1− σ3K1
)δ(1, 2)− Σ(1, 2)− U ′(1, 2), (A.60)
where the extra U ′ ≡ (U ′ij) is given by
U ′ij(1, 2) ≡
Uij(1, 2) + (−1)i+j+1U3−j,3−i(2, 1)
2 .(A.61)
By varying U → U + δU , then setting U = 0 for Dyson-Beliaev equation (A.24) and Hugenholtz-Pines relation(A.29), we obtain the following equations as
G−1(1, 3)δG(3, 2) = [δU(1, 3) + δΣ(1, 2)]G(3, 2), (A.62a)
G−1(1, 3)δΨ(3) = [δU(1, 3) + δΣ(1, 2)]σ3Ψ(2), (A.62b)
43
respectively. Let us introduce the following vertices for a later convenience:
Γ(4)ij,kl(12, 34) ≡ −
1
2
δΣij(1, 2)
δGlk(4, 3)=
1
T
δ2Φ
δGij(2, 1)δGlk(4, 3) ,(A.63a)
Γ(3)ij,k(12, 3) ≡
1
2(−1)k−1 δΣij(1, 2)
δΨk(3)
= 2(−1)k+lΓ(4)ij, 3− kl(12, 34)Ψl(4), (A.63b)
Γ(3)i,jk(1, 23) ≡ 2(−1)l−1Γ(4)(14, 23)Ψl(4) = (−1)iΓ(3)
jk,3−i(23, 1), (A.63c)
Γ(2)ij ≡ 2(−1)k−1Γ
(3)
ik,j(13, 2)Ψk(3), (A.63d)
where we used Eq. (A.46) for deriving the second expression of Γ(3),Γ(4). These are the irreducible vertices of ourcondensed Bose system, as seen below. In Fig.26, we give the diagramatic expressions of them. By considering
Eq. (A.6) and Φ = Φ∗, we can find symmetry relations of vertices such as Γ(4)ij,kl(12, 34) = Γ
(4)kl,ij(34, 12) =
(−1)i+j−1Γ(4)3−j,3−i,kl(21, 34) etc. Note that our Γ(3), Γ(4) correspond to I and J of Gavoret and Nozieres [18],
respectively. Our formulations may have an advantage over theirs because the vertices can be obtained from asingle functional Φ with clear relations among them.
By using Γ(4),Γ(3), we can express δΣ in Eq. (A.62) as
δΣij(1, 2) = −2Γ(4)
ij,lk(12, 43)δGkl(3, 4) + 2Γ
(3)
ij,k(12, 3)(−1)k−1δΨk(3). (A.64)
This relation enables us to obtain closed equations for δG and δΨ. To be specific, we multiply Eq. (A.62) byG from the left, substitute Eq. (A.64), and use Eqs. (A.63c,d). The procedure yields
δGij(1, 2) = Gil(1, 4)Gkj(3, 2)δU′lk(4, 3)− 2Gil(1, 4)Gkj(3, 2)Γ
(4)
lk,nm(43, 65)δGmn(5, 6)
+ 2Gil(1, 4)Gkj(3, 2)Γ(3)
lk,m(43, 5)(−1)m−1δΨm(5), (A.65a)
(−1)i−1δΨi(1) = Gik(1, 3)(−1)j−1Ψj(2)δU′kj(3, 2)−Gij(1, 3)Γ
(3)
i,jk(2, 43)δGkl(3, 4)
+Gij(1, 2)Γ(2)
jk(2, 3)(−1)k−1δΨk(3). (A.65b)
We also obtain from Eq. (A.61) the relation
Gil(1, 4)Gkj(3, 2)δU′lk(4, 3) =
1
2[Gil(1, 4)Gkj(3, 2) + (−1)k+l−1Gi,3−k(1, 3)G3−l,j(4, 2)]δUlk(4, 3). (A.66)
We can also transform
Gi,l(1, 4)Gkj(3, 2)Γ(4)(43, 56) = (−1)k+l−1Gi,3−k(1, 3)G3−l,j(4, 2)Γ
(4)
lkmn(43, 56), (A.67)
by using
Γ(4)ij,kl(12, 34) = (−1)i+j−1Γ
(4)3−j,3−i,kl(21, 34). (A.68)
Fig 26: Irreducible vertices
44
A.4.3 Bethe-Salpeter equation for BEC
We can express Eq. (A.65). We start to introduce the following notations for δG and δU as
⟨12ij |δG = δGij(1, 2), ⟨12ij |δU = δUij(1, 2). (A.69)
We also introduce the matrices K, Γ(4), χ(0), 1, Γ(3), Γ(3)
, Ψ(3), Ψ(3)
, and Γ(2) as
⟨12ij |K|43lk⟩ ≡ Kij,kl(12, 34), (A.70a)
⟨12ij |Γ(4)|43lk⟩ ≡ Γ(4)ij,kl(12, 34), (A.70b)
⟨12ij |χ(0)|43lk⟩ ≡ Gil(1, 4)Gkj(3, 2) + (−1)k+l−1Gi,3−k(1, 3)G3−l,j(4, 2), (A.70c)
⟨12ij |1|43lk⟩ ≡ δilδkjδ(1, 4)δ(3, 2), (A.70d)
⟨12ij |Γ(3)|3k⟩ ≡ Γ(3)ij,k(12, 3), (A.70e)
⟨1i|Γ(3)|32kj⟩ ≡ Γ
(3)i,jk(1, 23), (A.70f)
⟨12ij |Ψ(3)|3k⟩ ≡ (−1)j+k[Ψi(1)δk,3−jδ(3, 2) + δkiδ(3, 1)Ψ3−j(3)], (A.70g)
⟨1i|Ψ(3)|32kj⟩ ≡ (−1)j−1[δlkδ(1, 3)Ψj(2) + δi,3−jδ(1, 2)Ψ3−k(3)], (A.70h)
⟨1i|Γ(2)|2j⟩ ≡ Γ(2)ij (1, 2), (A.70i)
where the quantity χ(0) expresses independent propagation of two particles.
Using Eqs. (A.69) and (A.70), we can obtain the equations δG = 12χ
(0)δU −χ(0)Γ(4)δG+χ(0)Γ(3)σ3δΨ and
σ3δΨ = 12 GΨ
(3)δU − GΓ(3)
δG+ GΓ(2)σ3δΨ from Eq. (A.65). We finally obtain
δG =1
2χ(4)δU + χ(4)Γ(3)σ3δΨ, (A.71a)
σ3δΨ =1
2χ(2)Ψ
(3)δU − χ(2)Γ
(3)δG, (A.71b)
where
χ(4) ≡ (1 + χ(0)Γ(4))−1χ(0) = (χ(0)−1 + Γ(4))−1, (A.72a)
χ(2) ≡ (1− GΓ(2))−1G = (G−1 − Γ(2))−1. (A.72b)
It is convenient to define the quantities:
χ(q) ≡ (χ(4)−1 + Γ(3)χ(2)Γ(3)
)−1, (A.72c)
χ(c) ≡ (χ(2)−1 + Γ(3)χ(4)Γ(3))−1 = (G−1 − Γ(2) + Γ
(3)χ(4)Γ(3))−1, (A.72d)
where the superscripts q and c means “quasiparticle” and “condensate” respectively. These quantities arerepresented diagrammatically in Fig. 27. We thus obtain solution of Eq. (A.71) in terms of χ(q) and χ(c) as
δG =1
2χ(q)(1 + Γ(3)χ(2)Ψ
(3))δU , (A.73a)
δΨ =1
2σ3χ
(c)(Ψ(3) − Γ
(3)χ(4))δU . (A.73b)
By substituting Eq. (A.73) into Eq. (A.59) and using the notations of Eq. (A.70) and χ(q)Γ(3)χ(2) = χ(4)Γ(3)χ(c)
in Eq. (A.72), we obtain the Bethe-Salpeter equation:
K = χ(q) + χ(4)Γ(3)χ(c)Ψ(3)
+Ψ(3)χ(c)Γ(3)χ(4) −Ψ(3)χ(c)Ψ
(3). (A.74)
45
＝
＝
＝
＝
+
+
+
+
+
+
+
+
...
...
...
...
Fig 27: Diagrammatic structure of Eq. (A.72). Every long straight line in the second equation denotes G.
We now discuss poles of two-particle Green’s function K. Poles of χ(4) in Eq. (A.74) are cancelled by those
of χ(4) in the denominator of χ(c) in Eq. (A.72). It can be also seen that the poles of χ(c) are generally not
identical to those of the one particle Green’s function G due to the additional terms Γ(2) − Γ(3)χ(4)Γ(3), which
contradicts with the statement by Gavoret and Nozieres [18]. They erroneously identified χ(c) as the one particleGreen’s function’s line [31].
46
B Quantum Monte Carlo Methods
B.1 Green’s Function Monte Carlo Method
Here we will introduce the Green’s function Monte Carlo (GFMC) method [37, 54], which is one of the diffusionquantum Monte Carlo (DQMC) methods [55]. DQMC methods give exact results which have no dependenceson trial functions in principle. The GFMC method is a powerful tool for calculations in quantum many-bodysystems, especially for bosonic cases.
B.1.1 Outline of DQMC method
First, we consider a system of N identical bosons with mass m and spin 0 interacting via a two-body potentialV (r− r′). The corresponding Hamiltonian H is given by
H ≡N∑i=1
[~2
2m∆i + U(ri)
]+∑i<j
V (ri − rj), (B.1)
where ri ≡ (xi, yi, zi), ∆i ≡ ∇i · ∇i = (∂2xi+ ∂2yi
+ ∂2zi), and U(ri) is the external field. The exact N -body wavefunction Ψ(R, t) satisfies the time-dependent Schrodinger equation as
i~∂Ψ(R, t)
∂t= HΨ(R, τ), (B.2)
with R ≡ (r1, · · · , rN ). Next, we introduce the imaginary time τ ≡ −it/~ and a constant energy shift E. Thenwe can rewrite Eq. (B.2) as
−∂Ψ(R, τ)
∂τ= (H − E)Ψ(R, τ), (B.3)
whose formal solution can be written as Ψ(R, τ) = e−(H−E)τΨ(R, 0). By introducing eigenfunctions andeigenvalues as ϕn(R, τ) and En which are ordered by increasing order E0 < E1 < · · · respectively, Ψ(R, τ) canbe expanded as
Ψ(R, τ) =∞∑
n=0
cnϕn(R, 0)e−(En−E)τ , (B.4)
with cn the expansion prefactor. In the limit of τ →∞, excited states vanish and the ground state eigenfunctionwill have dominant contribution as,
Ψ(R, τ)→ c0ϕ0(R, 0)e−(E0−E)τ . (B.5)
If the energy shift E we choose is equal to the exact ground state energy E0, only the ground state eigenfunctionwill survive. Finally, we can determine the ground state eigenfunction ϕ0 and energy E0. We can also estimateother physical quantities.
B.1.2 GFMC method
In this subsection, we introduce the GFMC method which needs a trial wave function ψT (R), but final resultsare exact for physical quantities which commute with H. This will be explained later. We start from Eq. (B.3),which is expressible in terms of the distribution function f(R, τ) ≡ ⟨R|f(τ)⟩ ≡ ψT (R)Ψ(R, τ) as
−∂f(R, τ)∂τ
=N∑i=1
[−D∆if(R, τ) +D∇i · (Fif(R, τ))] + (Eloc(R)− E)f(R, τ)
≡ (A1 + A2 + A3)f(R, τ) ≡ Af(R, τ), (B.6)
47
where D ≡ ~2/2m, the drift force Fi and the local energy Eloc(R) are defined by Fi ≡ (2∇iψT (R))/ψT (R),Eloc(R) ≡ (HψT (R))/ψT (R) respectively [56], and we have used the following relation:
∆i
(f(R, τ)
ψT (R)
)= ∇i ·
[∇if(R, τ)
ψT (R)− 1
[ψT (R)]2(∇iψT (R))f(R, τ)
]=
1
ψT (R)
[∆if(R, τ)−∇i ·
(2∇iψT (R)
ψT (R)f(R, τ)
)+
∆iψT (R)
ψT (R)f(R, τ)
]. (B.7)
Its formal solution is given by
f(R′, τ +∆τ) =
∫dR G(R′,R;∆τ)f(R, τ), (B.8)
where the Green’s function can be written as
G(R′,R;∆τ) ≡ ⟨R′| exp(−A∆τ)|R⟩. (B.9)
Now we approximate exp(−A∆τ) by using Baker-Campbell-Hausdorff formula [56] as
exp(−A∆τ) ≈ exp(−A3∆
τ
2
)exp
(−A2∆
τ
2
)exp
(−A1∆τ
)exp
(−A2∆
τ
2
)exp
(−A3∆
τ
2
), (B.10)
which is an exact transformation up to (∆τ)2 order. So, Eq. (B.8) can be expressed by using Green’s functionsG1, G2 and G3 for A1, A2, A3 respectively as
f(R′, τ +∆τ) =
∫dRdR1 · · · dR4 G3
(R′,R1;
∆τ
2
)G2
(R1,R2;
∆τ
2
)G1 (R2,R3;∆τ)
×G2
(R3,R4;
∆τ
2
)G3
(R4,R;
∆τ
2
)f(R, τ), (B.11)
with
G1(R′,R; τ) ≡ 1
(4πDτ)3N/2exp
(− (R′ −R)2
4Dτ
), (B.12a)
G2(R′,R; τ) ≡ δ(R′ −R(τ)), where
R(0) = R
dR(τ)dτ = DF(R(τ)),
(B.12b)
G3(R′,R; τ) = exp (−(Eloc(R)− E)τ) δ(R−R′), (B.12c)
where F in Eq. (B.12b) is the vector of the drift forces defined as F ≡ (F1, · · · ,FN ). Especially, the validityof Eq. (B.12b) can be confirmed as follows: We introduce G2(k,R; τ) as the Fourier transform of G2(R
′,R; τ).It can be written as ∫
dk G2(k,R; τ)eik·R′=
∫dk eik·(R
′−R(τ)),
∴ G2(k,R; τ) = eik·R(τ). (B.13)
By differentiating both sides with respect to τ , we obtain the following equation
∂G2(k,R; τ)
∂τ= (−ik) · ∂R(τ)
∂τG2(k,R; τ),
∂G2(k,R; τ)
∂τeik·R
′= ∇R′ ·
(∂R(τ)
∂τG2(k,R; τ)e−ik·R′
). (B.14)
48
Next, we integrate this equation with respect to k and use the condition dR(τ)/dτ = DF(R(τ)). It can betransformed into
∂G2(R′,R; τ)
∂τ= D∇R′ · (F(R(τ))G2(R
′,R; τ)),
with ∇R′ ≡ (∇1′ , · · · ,∇N ′) for R′ = (r′1, · · · , r′N ). Noting the definition G2(R′,R; τ) = δ(R′ − R(τ)), we
obtain
∂G2(R′,R; τ)
∂τ= D∇R′ · (F(R′)G2(R
′,R; τ)). (B.15)
Thus, we have confirmed that G2 is the correct expression of the Green’s function of the operator A2.We will summarize the algorithm of GFMC method below. First, we prepare the initial state R which is
called walker, then we replicate a set of nw walkers. Now we label these nw walkers as Ri, (i = 1, · · · , nw), andthese walkers will move following Eq. (B.11) to τ → ∞ in steps of ∆τ . The corresponding GFMC simulationcan be performed as follows:
1. Choose a walker Ri, and calculate its local energy Eloc(R).
2. Move the walker under the drift force F(R) which can be obtained from Eq. (B.12b) as R′ → R +F(R)∆τ/2.
3. Take Gaussian steps for all components of Ri. Those steps are derivable from Eq. (B.12a) as R′′i →
R′i + ∆Ri for each 3N coordinates, where ∆Ri is generated by random numbers obeying a Gaussian
distribution function exp(−∆R2i /4D∆τ).
4. Repeat procedure 2 for R′′ (then we obtain R′′′), and calculate Eloc(R′′′).
5. Replicate the walker with ⟨n⟩i times. ⟨n⟩i can be calculated by the following relation,
⟨n⟩i = int
[exp
(−∆τ
(Eloc(R) + Eloc(R
′′′)
2− E
))+ ran()
],
(B.16)
where ran() is a uniform random number lying between 0 to 1.
6. By repeating above procedures for all walkers, we obtain a new set of walkers with the total numbern′w =
∑i⟨n⟩i. We then update the new value of Enew as
Enew = E +1
∆τ
(1− n′w
nw
),
(B.17)
where 1/∆τ doesn’t always have to be used as the prefactor of (1− n′w/nw) [55].
By repeating these procedures to large τ , walkers start to follow the distribution ψTϕ0 where ϕ0 is the exactground state wave function. We can estimate the ground state energy as an average of Eloc(R) and its staticerror σ(H). It can be expressed as
σ2(H) ≡ 1
NI − 1
1
NI
NI∑j=1
E2loc(Rj)−
1
NI
NI∑j=1
Eloc(Rj)
2
,
(B.18)
where NI is the number of observations and here j is the label of each obsarvations. In addition, we can alsoestimate other physical quantities. If we want to observe an expectation value of an operator A which commutewith H, we have only to take an average of Aloc(R) ≡ (AψT (R))/ψT (R). That’s because the estimation of thetotal energy ⟨H⟩ by using GFMC method (taking the average of Eloc(R)) is nothing but the calculation of thefollowing integration
⟨H⟩m ≡∫dR ϕ0(R)HψT (R)∫dR ϕ0(R)ψT (R)
=⟨ϕ0(R)|H|ψT (R)⟩⟨ϕ0(R)|ψT (R)⟩ ,
(B.19)
49
where such estimations are called “mixed” estimation. Therefore, we can simulate accurately expectation valuesof any operators which commute with H by mixed estimation (not biased by trial wave function ψT ).
For ageneral operator B ≡ B(R), estimations are more complicated. An average of B which does notcommute with H are biased in using mixed estimation. Then we use the “pure” estimation ⟨B⟩p which isdefined by [46]
⟨B⟩p ≡∫dR ϕ0(R)B(R)ϕ0(R)∫dR ϕ0(R)ϕ0(R)
=⟨ϕ0(R)|B(R)ϕ0(R)/ψT (R)|ψT (R)⟩
⟨ϕ0(R)|ϕ0/ψT (R)|ψT (R)⟩ .
(B.20)
In τ →∞ regime, ⟨B⟩p can be transformed as
⟨B⟩p =
∑iB(Ri)W (Ri)∑
iW (Ri) ,
(B.21)
where the weight W (R) is proportional to the number of future descendants at R as
W (R) ∝ n(R, τ →∞), (B.22)
as proved by Liu et.al. [58, 57]. How to perform the pure estimation is summarized as follows. First, we performGFMC simulation for the system from τ = 0 to large τ ′. Then, we continue GFMC steps with sampling B as,
Ri → R′i
Bi → B′i,in the same time interval ∆τ , and the number of walker N changes to N ′ in this step. We also introduce anauxiliary variable Pi which is evolved by the following law,
Pi → B′i + P ti, (B.23)
where P ti is the previous configuration of Pi. At the first step, P ti is set 0 for all i. If we have obtaineda set of Nf values Pi after additional M steps, the pure estimation of B can be written as
⟨B⟩p =
Nf∑i=1
PiMNf .
(B.24)
Finally in this subsection, we introduce an approximate estimation called “extrapolated” [54], which is givenby
⟨A⟩e ≡ 2⟨A⟩m − ⟨A⟩v, (B.25)
where ⟨A⟩v is variational estimation of A which is defined by
⟨A⟩v ≡∫dR ψT (R)HψT (R)∫dR ψT (R)ψT (R)
=⟨ψT (R)|H|ψT (R)⟩⟨ψT (R)|ψT (R)⟩ .
(B.26)
The next subsection explains how we can estimate ⟨A⟩v. On the other hand, Eq. (B.25) can be obtained bythe following procedure. Let us change the trial wave function ψT as ψT → ϕ0 + δψT . By using it, the pureestimation of A can be expressed as
⟨A⟩p =⟨ϕ0|A|ϕ0⟩⟨ϕ0|ϕ0⟩
=1
⟨ϕ0|ϕ0⟩
(⟨ψT |A|ψT ⟩+ 2⟨ψT |A|δψT ⟩+ ⟨δψT |A|δψT ⟩
).
(B.27)
If we can neglect the second-order term ⟨δψT |A|δψT ⟩, ⟨A⟩p can be approximated as
⟨A⟩p ≈ 2⟨ϕ0|A|ψT ⟩⟨ϕ0|ψT ⟩
− ⟨ψT |A|ψT ⟩⟨ψT |ψT ⟩
= ⟨A⟩e, (B.28)
where we also used ⟨ϕ0|A|δψT ⟩ = ⟨ψT |A|ϕ0⟩ − ⟨ψT |A|ψT ⟩. Advantages of this estimation are twofold. First, itis easier than the pure estimation; second, we can observe derivative operators such as p = −i~∇.
50
B.2 Variational Quantum Monte Carlo Method
Next we will introduce the variational Monte Carlo (VQMC) method [37, 39]. By using this method, we canestimate a many body integration, such as
Ev ≡ ⟨H⟩v =⟨ψT |H|ψT ⟩⟨ψT |ψT ⟩
=
∫dR Eloc(R)|ψT (R)|2∫
dR |ψT (R)|2.
(B.29)
If we try to perform this integration directly, we have to solve a numerical 3N -dimensional integration, which istoo difficult. Here |ψT (R)|2 is proportional to the stochastic density at R, so it can be used for the importancesampling. An appropriate choice of the trial wave function ψT is also very important to estimate the groundstate energy E0 because the value of Ev satisfies the following variational principle,
ET ≥ E0. (B.30)
Moreover, if we use the exact eigenstate wave function ϕk with eigenenergy Ek of H, the corresponding staticerror is equal to 0 exactly.
In the next subsection, we will summarize the algorithm of VQMC sampling with a trial ψT . Next, we willintroduce the trial wave function for many-body systems called “Bijl-Jastraw” type [53].
B.2.1 Algorithm of VQMC method
Here we summarize the VQMC calculation. First, we want to observe values of physical quantities such as Hand so on. Their expectations are denoted by ⟨H⟩v, which can be rewritten by using the probability densityp(R) ≡ |ψT (R)|2/
∫dR |ψT (R)|2 as
⟨H⟩v =
∫dR Eloc(R)p(R) ≈
NI∑i=1
Eloc(Ri). (B.31)
where NI is the total number of observations, and the final form can be calculated by importance sampling.The algorithm of VQMC calculation is summarized as follows:
1. Set up a Initial state R(0).
2. Take a movement R(1) → R(0) + (2× ran()− 1) ∆R, where ∆R is the maximum distance for each step.
3. If |ψT (R(1))|2/|ψT (R
(0))|2 > ran(), then R(0) = R(1). Unless, R(0) = R(0) (Metropolis step).
4. Sample physical quantities such as energy and its static error etc.
5. Go back to step 2.
By repeating these procedures, we can estimate various expectation values.
B.2.2 Trial wave function for bosons
Here we introduce one of many body trial wave functions for the ground sate, called Bijl-Jastraw type wavefunction ΨT (R). For bosonic systems, it can be given by
ΨT (R) ≡N∏i=1
u(ri)∏i<j
f(rij), (B.32)
where rij ≡ ri − rj , functions u(ri) and f(rij) are constructed so as to incorporate the external field and theinteraction potential respectively.
51
We can write down the expression of Eloc(R) for the Hamiltonian Eq. (B.1) as
Eloc(R) =N∑i=1
[− ~2
2m
∇iu(ri)
u(ri)+∑j =i
rijrij
f ′(rij)
f(rij)
2
+
(∆iu(ri)
u(ri)−(∇iu(ri)
u(ri)
)2)
+∑j =i
(2
rij
f ′(rij)
f(rij)+f
′′(rij)
f(rij)−(f ′(rij)
f(rij)
)2)
+ U(ri) +∑j>i
V (ri − rj)
],
(B.33)
where f ′(r) ≡ ∂f(r)/∂r and f ′′(r) ≡ ∂2f(r)/∂r2.
B.3 Finite size effect
We now discuss finite size effects for the Monte Carlo simulation [37, 54, 56].We consider the size effect of the volume V . If we want to estimate an energy for a homogeneous system,
we have to remove a “tail” effect. One of the methods is to adopt “uniform” approximation[37] as follows:We perform a simulation, we set particle number N , and fix the density ρ. The maximum length L = (ρ/N)1/3
can be obtained automatically. Here we assume that we can extract the information of the energy to the distanceL/2. The tail contributions of the potential energy (V/N)t(ρ) and the kinetic energy (K/N)t(ρ) from the regionlarger than L/2 can be included by the following integrals as(
V
N
)t
(ρ) = 2πρ
∫ ∞
L/2
dr r2V (r)g(r) = 2πρ
∫ ∞
L/2
dr r2V (r), (B.34a)(K
N
)t
(ρ) = −~2
m2πρ
∫ ∞
L/2
dr r2(2
r
f ′(r)
f(r)+f ′′(r)
f(r)
), (B.34b)
respectively, where we have assumed that the two-body radial distribution function [60] g(r) = 1 for r > L/2,and f(r) is the part of the Bijl-Jastraw type wave function in Eq. (B.32). There is also a dependence on particlenumber N . So, we have to consider the effect by performing several simulations changing the number N . Wecan adopt the lowest value of N where the energy is nearly constant.
B.4 Expressions of Physical Quantities for Sampling
We will introduce several expressions of basic physical quantities for importance sampling.
B.4.1 Density matrix
In the second quantized formulation, the one-particle density matrix [50] is given by
g1(r′, r) ≡ ⟨ψ†(r)ψ(r′)⟩, (B.35)
where ψ(r) is the annihilation operator at r, and ψ(r), ψ†(r) obey commutation relations [ψ(r), ψ†(r′)] =
δ(r− r′), [ψ(r), ψ(r′)] = 0. When a N -body normalized ground state |ΨG⟩ is considered, we can express g1 as
g1(r′, r) =⟨ΨG|ψ†(r)ψ(r′)|ΨG⟩
=
∫dR
∫dR′ Ψ∗
G(R)⟨R|ψ†(r)ψ(r′)|R′⟩ΨG(R′)
=N
∫dr2dr3 · · · drN
∫dr′2dr
′3 · · · dr′N Ψ∗
G(r, r2, r3, · · · , rN )
× ⟨r, r2, r3, · · · , rN |r′, r′2, r′3, · · · , r′N ⟩ΨG(r′, r′2, r
′3, · · · , r′N )
=N
∫dr2dr3 · · · drN Ψ∗
G(r, r2, r3, · · · , rN )ΨG(r′, r2, r3, · · · , rN ), (B.36)
52
where ΨG(R) ≡ ⟨R|ΨG⟩ and R ≡ (r1, r2, · · · , rN ). Then we transform Eq. (B.36) into a form suitable forGFMC and VMC methods. In a homogeneous system, we can express the one-particle density matrix for theground state by variational estimation as
gvar1 (r) ≡ N∫dr2dr3 · · · drN ψ∗
T (r1 + r, r2, · · · , rN )ψT (r1, r2, · · · , rN )∫dR |ψT (R)|2
.
(B.37a)
We here adopt the extrapolated method (B.25) for the GFMC estimation, which is given by
g(e)1 ≡ 2gmix
1 (r)− gvar1 (r), (B.37b)
where gmix1 (r) can be written as
gmix1 (r) ≡ N
∫dr2dr3 · · · drN ψ∗
T (r1 + r, r2, · · · , rN )ϕ0(r1, r2, · · · , rN )∫dR ψT (R)ϕ0(r) .
(B.37c)
These expressions can be approximated [54] as
gvar1 (r) = n
⟨ψ∗T (r1 + r, r2, · · · , rN )
ψT (R)
⟩v
, (B.38a)
gmix1 (r) ≈ n
∫dR ψ∗
T (r1 + r, r2, · · · , rN )(ψT (R))−1f(r1, r2, · · · , rN )∫dR f(R)
= n
⟨ψ∗T (r1 + r, r2, · · · , rN )
ψT (R)
⟩m
, (B.38b)
respectively, where n is the particle density n ≡ N/V.Next, we introduce two-particle density matrix g2(r, r
′) as
g2(r, r′) ≡ ⟨ΨG|ψ†(r)ψ†(r′)ψ(r′)ψ(r)|ΨG⟩
= N(N − 1)
∫dr3 · · · drN |ΨG(r, r
′, r3 · · · rN )|2. (B.39)
In homogeneous cases, we can express g2(r, r′)→ g2(r− r′), which can be transformed as
g2(r) =N(N − 1)
V
∫dr2 · · · drN |ΨG(r+ r2, r2, r3 · · · rN )|2
=2
V
∫dr1dr2 · · · drN
∑i<j
δ(rij − r)|ΨG(r1, r2, r3 · · · rN )|2. (B.40)
Introducing the step function θh(z), which is 1 if z < h and 0 otherwise, Eq. (B.40) can be written by [61]
g2(r) ≈2
V
1
4πr2h
∫dr1dr2 · · · drN
∑i<j
θh(|rij − r|)|ΨG(r1, r2, r3 · · · rN )|2. (B.41)
B.4.2 The moment method for two-particle excitation
We want to know details of quasiparticle excitation for a system of N particles. The moment mehod, developedmainly for the two-particle excitations, forms a tractable and reliable tool for this purpose [41, 42, 59, 60]. Inthis subsection we first summarize the moment method. Then we extend it up to second-order moment andapply to the one-particle excitation. We also give expressions of these excitation for sampling.
We start from the density correlation function S(q, t) for the ground state |ΨG⟩ given by
S(q, t) ≡∑ν
⟨ΨG|eiHt/~ρqe−iHt/~ρ†q|ΨG⟩, (B.42)
53
where the particle density operator ρq is defined by
ρ(r) = ψ†(r)ψ(r) =1
V∑q
ρqeiq·r, ρq ≡
∑k
a†q+kak, (B.43)
with
ψ(r) =1√V
∑k
akeik·r. (B.44)
Its Fourier transform with respect to t,
S(q, ω) ≡ 1
2π
∫ ∞
−∞dt S(q, t)eiωt, (B.45a)
defines the dynamic form factor. Let us insert the complete set∑
ν |Ψν⟩⟨Ψν | between eiHt/~ and ρ†q in Eq.
(B.42), transform ⟨Ψν |eiHt/~ = ⟨Ψν |eiEνt/~ and e−iHt/~|ΨG⟩ = e−iEGt/~|ΨG⟩, and substitute the resultingexpression of S(q, t) into Eq. (B.45a) to perform the time integration. We thereby obtain
S(q, ω) =∑ν
|⟨Ψν |ρ†q|ΨG⟩|2δ(ω − Eν + EG). (B.45b)
We next introduce the moments of S(q, ω) as
In(q) ≡∫ ∞
−∞dω ωnS(q, ω) =
∑ν
|⟨Ψν |ρ†q|ΨG⟩|2(EG − Eν)n. (B.46)
Substituting Eq. (B.45a) into the first expression and performing partial integrations with respect to t, we canexpress In(q) alternatively as
In(q) =
∫ ∞
−∞dω
∫ ∞
−∞
dt
2πS(q, t)
(−i ddt
)n
eiωt
=
∫ ∞
−∞
dt
2π
∫ ∞
−∞dω eiωt
(id
dt
)n
S(q, t)
=
(id
dt
)n
S(q, t)
∣∣∣∣t→0
(B.47)
By using Eq. (B.42), we can express the first few series of In(q) as
I0(q) = ⟨ΨG|ρqρ†q|ΨG⟩ (B.48a)
I1(q) = ⟨ΨG|[ρq, H]ρ†q|ΨG⟩ (B.48b)
I2(q) = ⟨ΨG|[[ρq, H], H]ρ†q|ΨG⟩. (B.48c)
I0(q) can be transformed by using Eq. (B.43) as
I0(q) = ⟨ΨG|ρqρ†q|ΨG⟩ =∫dr1
∫dr2 ⟨ΨG|ρ(r1)ρ(r2)|ΨG⟩e−iq·(r1−r2)
=
∫dr1
∫dr2 ⟨ΨG|ψ†(r1)ψ(r1)ψ
†(r2)ψ(r2)|ΨG⟩e−iq·(r1−r2)
=
∫dr1
∫dr2 [⟨ΨG|ψ†(r1)ψ
†(r2)ψ(r1)ψ(r2)|ΨG⟩+ δ(r1 − r2)⟨ΨG|ψ†(r1)ψ(r2)|ΨG⟩]e−iq·(r1−r2)
= V∫dr g2(r)e
−q·r +N
= N
1
n
∫dr[g2(r)− 1]eiq·r +
Vnδq0 + 1
.
(B.49)
54
We next give an alternative expression of I1. We consider the following quantity:
I1 ≡ ⟨ΨG|[[ρq, H], ρ†q]|ΨG⟩
=∑ν
[⟨ΨG|(ρqH − Hρq)|Ψν⟩⟨Ψν |ρ†q|ΨG⟩ − ⟨ΨG|ρ†q|Ψν⟩⟨Ψν |(ρqH − Hρq)|ΨG⟩
]=∑ν
(|⟨ΨG|ρ†q|ΨG⟩|2 + |⟨ΨG|ρq|ΨG⟩|2
)(Eν − EG)
= I1(q) + I1(−q). (B.50)
The time reversal symmetry implies
I1(q) = I1(−q). (B.51)
Hence, we can express Eq. (B.48b) as
I1(q) =1
2⟨ΨG|[[ρq, H], ρ†q]|ΨG⟩. (B.52)
Noting [ρq, Hint] = 0 in Eq. (B.46) , we can transform [ρq, H] as
[ρq, H] = [ρq, H0] =∑kk′
εk[a†k′ ak′+q, a
†kak] =
∑kk′
εk
(a†k′ [ak′+q, a
†k]ak + a†k[a
†k′ , ak]ak′+q
)=∑k
(εk+q − εk)a†kak+q =∑k
(εk+q/2 − εk−q/2)a†k−q/2ak+q/2
=~qm·∑k
~ka†k−q/2ak+q/2 ≡ Jq, (B.53)
where Jq is the current operator [62].We also transform the double commutator in Eq. (B.52) as
[[ρq, H], ρ†q] =∑kk′
(εk+q − εk)a†kak+q[a†kak+q, a
†k′+qak′ ]
=∑kk′
(εk+q − εk)(a†k[ak+q, a
†k′+q]ak′ + a†k′+q[a
†k, ak′ ]ak+q
)=∑k
(εk+q − εk)(a†kak + a†k+qak+q
)=∑k
[(εk+q − εk)− (εk − εk−q)]a†kak
=~2q2
m
∑k
a†kak = N~2q2
m .(B.54)
Substituting the result into Eq. (B.52), we obtain the f -sum rule
I1(q) = N~2q2
2m .(B.55)
By using
ak =1√V
∫dr ψ(r)eik·r, (B.56)
55
and substituting Eq. (B.53) into Eq. (B.48b), I1(q) can be transformed as
I1(q) =~qm·∑kk′
~k⟨ΨG|a†k−q/2ak+q/2a†k′+q/2ak′−q/2|ΨG⟩
=~qm·∫dr1
∫dr′1
∫dr2
∫dr′2
1
V2
∑kk′
~k⟨ΨG|ψ†(r′1)ψ(r1)ψ†(r′2)ψ(r2)|ΨG⟩
× e−i(k−q/2)·r′1−i(k+q/2)·r1+i(k′+q/2)·r′2−i(k′−q/2)·r′2
=~qm·∫dr1
∫dr′1
∫dr2
∫dr′2
1
V2
∑kk′
~k[⟨ΨG|ψ†(r′1)ψ
†(r′2)ψ(r1)ψ(r2)|ΨG⟩+ δ(r1 − r′2)⟨ΨG|ψ†(r′1)ψ(r2)|ΨG⟩]
× e−ik·(r1−r′1)−ik·(r2−r′2)−iq·(r1+r′1−r2−r′2)/2
=~qm·∫dr1
∫dr′1
∫dr2
∫dr′2
1
V2
∑k
~k[⟨ΨG|ψ†(r′1)ψ
†(r′2)ψ(r1)ψ(r2)|ΨG⟩+ δ(r1 − r′2)⟨ΨG|ψ†(r′1)ψ(r2)|ΨG⟩]
× e−ik·(r1−r′1)−iq·(r1+r′1−r2−r′2)/2δ(r2 − r′2)
=~qm·∫dr1
∫dr′1
∫dr2
1
V∑k
~k[⟨ΨG|ψ†(r′1)ψ
†(r2)ψ(r2)ψ(r1)|ΨG⟩+ δ(r1 − r2)⟨ΨG|ψ†(r′1)ψ(r2)|ΨG⟩]
× e−ik·(r1−r′1)−iq·(r1+r′1−2r2)/2. (B.57a)
The second term in the square bracket can be transformed as
~qm·∫dr1
∫dr′1
∫dr2
1
V∑k
~kδ(r1 − r2)⟨ΨG|ψ†(r′1)ψ(r2)|ΨG⟩e−ik·(r1−r′1)−iq·(r1+r′1−2r2)/2
=~qm·∑k
~k∫dr1
∫dr′1⟨ΨG|ψ†(r′1)ψ(r1)|ΨG⟩e−ik·(r1−r′1)−iq·(r′1+r1)/2
=~qm·∑k
~k∫dr1
∫dr′1g1(r1 − r′1)e
−i(k−q/2)·(r1−r′1)
=~qm·∑k
~(k+
q
2
)∫dr g1(r)e
−ik·r =~2q2
2m
∫dr g1(r)δ(r) = N
~2q2
2m(B.57b)
On the other hand, the first term in the square bracket of Eq. (B.57a) can be transformed as
~qm·∫dr1
∫dr′1
∫dr2
1
V∑k
~k⟨ΨG|ψ†(r′1)ψ†(r2)ψ(r2)ψ(r1)|ΨG⟩e−ik·(r1−r′1)−iq·(r1+r′1−2r2)/2
=~qm·∫dr1
∫dr′1
∫dr2
1
V∑k
⟨ΨG|ψ†(r′1)ψ†(r2)ψ(r2)ψ(r1)|ΨG⟩~
∇1′ −∇1
2ie−ik·(r1−r′1)−iq·(r1+r′1−2r2)/2
=~qm·∫dr1
∫dr′1
∫dr2
1
V∑k
e−ik·(r1−r′1)−iq·(r1+r′1−2r2)/2~∇1 −∇1′
2i⟨ΨG|ψ†(r′1)ψ
†(r2)ψ(r2)ψ(r1)|ΨG⟩
=~qm·∫dr1
∫dr′1
∫dr2 δ(r1 − r′1)e
−iq·(r1−r2)~∇1 −∇1′
2i⟨ΨG|ψ†(r′1)ψ
†(r2)ψ(r2)ψ(r1)|ΨG⟩. (B.57c)
Substituting Eqs. (B.57b) and (B.57c) into Eq. (B.57a), we obtain
I1(q) = N~2q2
2m+
~qm·∫dr1
∫dr2 e
−iq·(r1+r2)~∇1 −∇1′
2i⟨ΨG|ψ†(r′1)ψ
†(r2)ψ(r2)ψ(r1)|ΨG⟩∣∣∣r′1→r1
. (B.57d)
Comparing Eq. (B.55) with (B.57c), we obtain
~qm·∫dr1
∫dr2 e
−iq·(r1+r2)~∇1 −∇1′
2i⟨ΨG|ψ†(r′1)ψ
†(r2)ψ(r2)ψ(r1)|ΨG⟩∣∣∣r′1→r1
= 0. (B.58)
56
The second moment I2 can be also written as
I2(q) = ⟨ΨG|JqJ†q|ΨG⟩ = ⟨ΨG|[ρq, H][ρq, H]†|ΨG⟩. (B.59)
Let us substitute Eq. (B.53) into Eq. (B.59):
I2(q) =∑
µ,ν=x,y,z
~4qµqνm2
∑kk′
kµk′ν⟨ΨG|a†k−q/2ak+q/2a
†k′+q/2ak′−q/2|ΨG⟩
=∑
µ,ν=x,y,z
~4qµqνm2
∫dr1
∫dr′1
∫dr2
∫dr′2
1
V2
∑kk′
kµk′ν⟨ΨG|ψ†(r′1)ψ(r1)ψ
†(r′2)ψ(r2)|ΨG⟩
× e−i(k−q/2)·r′1−i(k+q/2)·r1+i(k′+q/2)·r′2−i(k′−q/2)·r′2
=∑
µ,ν=x,y,z
~4qµqνm2
∫dr1
∫dr′1
∫dr2
∫dr′2
1
V2
∑kk′
kµk′ν
[⟨ΨG|ψ†(r′1)ψ(r1)ψ
†(r′2)ψ(r2)|ΨG⟩
+ δ(r1 − r′2)⟨ΨG|ψ†(r′1)ψ(r2)|ΨG⟩]e−ik·(r1−r′1)−ik·(r2−r′2)−iq·(r1+r′1−r2−r′2)/2. (B.60a)
The second term in the square bracket can be transformed as∑µ,ν=x,y,z
~4qµqνm2
∫dr1
∫dr′1
∫dr2
∫dr′2
1
V2
∑kk′
kµk′νδ(r1 − r′2)⟨ΨG|ψ†(r′1)ψ(r2)|ΨG⟩
× e−ik·(r1−r′1)−ik·(r2−r′2)−iq·(r1+r′1−r2−r′2)/2
=∑
µ,ν=x,y,z
~4qµqνm2
∫dr1
∫dr′1
∫dr2
1
V2
∑kk′
kµk′ν⟨ΨG|ψ†(r′1)ψ(r2)|ΨG⟩ e−ik·(r1−r′1)−ik·(r2−r1)−iq·(r′1−r2)/2
=∑
µ,ν=x,y,z
~4qµqνm2
∫dr′1
∫dr2
∫dr′2
1
V2
∑kk′
kµk′ν⟨ΨG|ψ†(r′1)ψ(r2)|ΨG⟩eik·r
′1−ik′·r2−iq·(r′1−r2)/2
∫dr1 e
i(k−k′)·r1
=∑
µ,ν=x,y,z
~4qµqνm2
1
V∑k
kµkν
∫dr′1
∫dr2 ⟨ΨG|ψ†(r′1)ψ(r2)|ΨG⟩ei(k−q/2)·(r′1−r2)
=∑
µ,ν=x,y,z
~4qµqνm2
1
V∑k′
(kµ +
qµ2
)(kν +
qµ2
)∫dr′1
∫dr2 g1(r2 − r′1)e
−i(k′)·(r2−r′1)
=
(~2q2
2m
)2 ∫dr g1(r)
∑k′
e−ik′·r +∑
µ,ν=x,y,z
~4qµqνm2
∑k′
k′µk′ν
∫dr g1(r)e
−ik′·r
=N
(~2q2
2m
)2
+~2q2
3m2
∑k
k2∫dr g1(r)e
−ik·r. (B.60b)
57
On the other hand, the first term in the square bracket of Eq. (B.60a) can be written as∑µ,ν=x,y,z
~4qµqνm2
∫dr1
∫dr′1
∫dr2
∫dr′2
1
V2
∑kk′
kµk′ν⟨ΨG|ψ†(r′1)ψ(r1)ψ
†(r′2)ψ(r2)|ΨG⟩
× e−ik·(r1−r′1)−ik·(r2−r′2)−iq·(r1+r′1−r2−r′2)/2
=∑
µ,ν=x,y,z
~2qµqνm2
∫dr1
∫dr′1
∫dr2
∫dr′2
1
V2
∑kk′
⟨ΨG|ψ†(r′1)ψ(r1)ψ†(r′2)ψ(r2)|ΨG⟩
× ~∇′
1µ −∇1µ
2i~∇′
2ν −∇2ν
2ie−ik·(r1−r′1)−ik·(r2−r′2)−iq·(r1+r′1−r2−r′2)/2
=∑
µ,ν=x,y,z
~2qµqνm2
∫dr1
∫dr′1
∫dr2
∫dr′2
1
V2
∑kk′
e−ik·(r1−r′1)−ik·(r2−r′2)−iq·(r1+r′1−r2−r′2)/2
× ~∇1µ −∇′
1µ
2i~∇2ν −∇′
2ν
2i⟨ΨG|ψ†(r′1)ψ(r1)ψ
†(r′2)ψ(r2)|ΨG⟩
=∑
µ,ν=x,y,z
~2qµqνm2
∫dr1
∫dr′1
∫dr2
∫dr′2 δ(r1 − r′1)δ(r2 − r′2)e
−iq·(r1−r2)
× ~∇1µ −∇′
1µ
2i~∇2ν −∇′
2ν
2i⟨ΨG|ψ†(r′1)ψ(r1)ψ
†(r′2)ψ(r2)|ΨG⟩. (B.60c)
Substituting Eqs. (B.60b) and Eq. (B.60c) into Eq. (B.60a), we obtain
I2(q) = N
(~2q2
2m
)2
+~2q2
3m2
∑k
k2∫dr g1(r)e
−ik·r
+∑
µ,ν=x,y,z
~2qµqνm2
∫dr1
∫dr2 e
−iq·(r1−r2) ~∇1µ −∇′
1µ
2i~∇2ν −∇′
2ν
2i⟨ΨG|ψ†(r′1)ψ(r1)ψ
†(r′2)ψ(r2)|ΨG⟩∣∣∣∣r′1→r1,r′2→r2
.
(B.61)
We note that if we choose the real wave function, the third term is equal to 0.We introduce the following functions:
Sn(q, ω) ≡S(q, ω)ωn
I0(q) ,
In(q) ≡In(q)
I0(q) .(B.62)
which satisfies the sum rule ∫dω S0(q, ω) = 1. (B.63)
S0(q, ω) can be regarded as the momentum distribution function for ω with mode q. We can see easily that I1and I2 correspond to expectation values for ωq and ω2
q respectively. Therefore, we can estimate the two-particleexcitation spectrum ωq and its lifetime δωq as
ωq = I1(q), (B.64a)
δωq =
√I2(q)− I1(q)2. (B.64b)
respectively.
B.4.3 Extensiton for one-particle excitation
We next consider the one-particle excitation. We start from the one-particle correlation function A(k, t) definedby
A(k, t) ≡ ⟨ΨG|eiHt/~ake−iHt/~a†k|ΨG⟩, (B.65)
58
whose Fourier transform defines the one-particle spectral function as
A(k, ε) =1
2π
∫dt A(k, t)eiεt. (B.66)
Using a similar procedure for deriving Eq. (B.45b), we obtain an expression for A(k, ε) as
A(k, ε) =∑ν
|⟨Ψν |a†k|ΨG⟩|2δ(ε− Eν + EG). (B.67)
The moment of A(k, ε) is defined by
Jn(k) =
∫dε A(k, ε)εn =
∑ν
|⟨Ψν |a†k|ΨG⟩|2(Eν − EG). (B.68)
It can be expressed in an alternative form similarly as in Eq. (B.47) as
Jn(k) =
(id
dt
)n
A(k, t)
∣∣∣∣t→0
. (B.69)
The first few series of Jn(k) are given by
J0(k) ≡ ⟨ΨG|aka†k|ΨG⟩, (B.70a)
J1(k) ≡ ⟨ΨG|ak[H, a†k]|ΨG⟩, (B.70b)
J2(k) ≡ ⟨ΨG|ak[H, [H, a†k]]|ΨG⟩. (B.70c)
So we can find the details of the one-particle excitation by these moments.We next transform them into forms suitable for importance sampling. To this end, we use the following
commutation relations for convenience:
[H0, ψ(r)] = −h(1)(r)ψ(r), (B.71a)
[Hint, ψ(r)] = −∫dr1V (r− r1)ψ
†(r1)ψ(r1)ψ(r), (B.71b)
[H0, ψ†(r)] = h(1)(r)ψ†(r), (B.71c)
[Hint, ψ†(r)] =
∫dr1V (r− r1)ψ
†(r)ψ†(r1)ψ(r1), (B.71d)
where h(1)(r) ≡ ~2∇2r/2m is the one-particle operator [50]. We now give the expression of J0(k), which can be
written by
J0(k) = 1 +1
V
∫dr
∫dr′ g1(r, r
′)e−ik·(r−r′)
= 1 +
∫dr g1(r)e
−ik·r. (B.72)
59
Next, we consider J1(k). Using Eq. (B.71), we can transform it as
J1(k) =1
V
∫dr1
∫dr′1 h
(1)(r1)⟨ΨG|ψ(r1)ψ†(r′1)|ΨG⟩e−ik·(r1−r′1)
+1
V
∫dr1
∫dr′1
∫dr2 V (r1 − r2)⟨ΨG|ψ†(r2)ψ(r1)ψ
†(r1)ψ†(r′1)|ΨG⟩e−ik·(r1−r′1)
=1
V
∫dr1
∫dr′1 h
(1)(r1)[δ(r1 − r′1) + g1(r1 − r′1)] e−ik·(r1−r′1)
+1
V
∫dr1
∫dr′1
∫dr2 V (r1 − r2)[δ(r1 − r′1)⟨ΨG|ψ†(r2)ψ(r2)|ΨG⟩
+ δ(r2 − r′1)⟨ΨG|ψ†(r2)ψ(r1)|ΨG⟩+ ⟨ΨG|ψ†(r′1)ψ†(r2)ψ(r2)ψ(r1)|ΨG⟩] e−ik·(r1−r′1)
=1
V
∫dr1
∫dr′1 [δ(r1 − r′1) + g1(r1 − r′1)]
~2k2
2me−ik·(r1−r′1)
+1
V
∫dr1
∫dr2 V (r1 − r2)⟨ΨG|ψ†(r2)ψ(r2)|ΨG⟩
+1
V
∫dr1
∫dr2 V (r1 − r2)⟨ΨG|ψ†(r2)ψ(r1)|ΨG⟩e−ik·(r1−r2)
+1
V
∫dr1
∫dr′1
∫dr2 V (r1 − r2)⟨ΨG|ψ†(r′1)ψ
†(r2)ψ(r2)ψ(r1)|ΨG⟩ e−ik·(r1−r′1). (B.73)
Noting ⟨ΨG|ψ†(r2)ψ(r2)|ΨG⟩ = N/V, we obtain J1(k) in terms of
SA(r1 − r′1) ≡1
N(N − 1)
∫dr1 V (r1 − r2)⟨ΨG|ψ†(r′1)ψ
†(r2)ψ(r2)ψ(r1)|ΨG⟩, (B.74)
as
J1(k) =~2k2
2m
[1 +
∫dr g1(r)e
−ik·r]+ nVq=0 +
∫dr V (r)g1(r)e
−ik·r
+N(N − 1)
V
∫dr1
∫dr′1 SA(r1 − r′1)e
−ik·(r1−r′1)
=~2k2
2m
[1 +
∫drg1(r)e
−ik·r]+ nVq=0 +
∫dr V (r)g1(r)e
−ik·r +N(N − 1)
∫dr SA(r)e
−ik·r.
(B.75)
Note that the second and the third terms are the Hartree and Fock energies, respectively.We next transform J2 as
J2(k) =1
V
∫dr1
∫dr′1 e
ik·(r1−r′1)h(1)(r1)h(1)(r′1)⟨ΨG|ψ(r1)ψ†(r′1)|ΨG⟩
+1
V
∫dr1
∫dr′1
∫dr2 e
ik·(r1−r′1)h(1)(r′1)V (r1 − r2)⟨ΨG|ψ†(r1)ψ(r′1)ψ(r2)ψ
†(r2)|ΨG⟩
+1
V
∫dr1
∫dr′1
∫dr2 e
ik·(r1−r′1)h(1)(r1)V (r′1 − r2)⟨ΨG|ψ(r1)ψ†(r′1)ψ†(r2)ψ(r2)|ΨG⟩
+1
V
∫dr1
∫dr′1
∫dr2
∫dr′2 V (r1 − r2)V (r′1 − r′2)
× eik·(r1−r′1)⟨ΨG|ψ†(r2)ψ(r2)ψ(r1)ψ†(r′1)ψ(r
′2)ψ(r2)|ΨG⟩
=
(~2k2
2m
)21
V
∫dr1
∫dr′1 e
ik·(r1−r′1)[δ(r1 − r′1) + g1(r1 − r′1)]
+~2
2m
1
V
∫dr1
∫dr′1
∫dr2 e
ik·(r1−r′1)V (r1 − r2)⟨ΨG|ψ†(r1)ψ(r′1)ψ(r2)ψ
†(r2)|ΨG⟩
+~2
2m
1
V
∫dr1
∫dr′1
∫dr2 e
ik·(r1−r′1)h(1)(r1)V (r′1 − r2)⟨ΨG|ψ(r1)ψ†(r′1)ψ†(r2)ψ(r2)|ΨG⟩
60
+1
V
∫dr1
∫dr′1
∫dr2
∫dr′2 V (r1 − r2)V (r′1 − r′2)[δ(r1 − r′1)δ(r2 − r′2)⟨ΨG|ψ†(r2)ψ(r2)|ΨG⟩
+ δ(r′1 − r2)δ(r1 − r′2)⟨ΨG|ψ†(r′1)ψ(r1)|ΨG⟩+ δ(r1 − r′1)⟨ΨG|ψ†(r2)ψ†(r′2)ψ(r
′2)ψ(r2)|ΨG⟩
+ δ(r1 − r′2)⟨ΨG|ψ†(r′1)ψ†(r2)ψ(r2)ψ(r1)|ΨG⟩+ δ(r2 − r′1)⟨ΨG|ψ†(r′1)ψ
†(r′2)ψ(r′2)ψ(r1)|ΨG⟩
+ δ(r2 − r′2)⟨ΨG|ψ†(r′1)ψ†(r2)ψ(r2)ψ(r1)|ΨG⟩+ ⟨ΨG|ψ†(r′1)ψ
†(r2)ψ†(r′2)ψ(r
′2)ψ(r2)ψ(r1)|ΨG⟩]
× eik·(r1−r′1)
=
(~2k2
2m
)2 [1 +
∫dr eik·rg1(r)
]+
~2k2
2m
[nVq=0 +
∫dr V (r)g1(r)e
ik·r +N(N − 1)
∫dr SA(r)e
ik·r]
+~2k2
2m
[nVq=0 +
∫dr V (r)g1(r)e
ik·r +N(N − 1)
∫dr SA(r)e
ik·r]∗
+1
V
∫dr1
∫dr′1
∫dr2
∫dr′2 V (r1 − r2)V (r′1 − r′2)[δ(r1 − r′1)δ(r2 − r′2)⟨ΨG|ψ†(r2)ψ(r2)|ΨG⟩
+ δ(r′1 − r2)δ(r1 − r′2)⟨ΨG|ψ†(r′1)ψ(r1)|ΨG⟩+ δ(r1 − r′1)⟨ΨG|ψ†(r2)ψ†(r′2)ψ(r
′2)ψ(r2)|ΨG⟩
+ δ(r1 − r′2)⟨ΨG|ψ†(r′1)ψ†(r2)ψ(r2)ψ(r1)|ΨG⟩+ δ(r2 − r′1)⟨ΨG|ψ†(r′1)ψ
†(r′2)ψ(r′2)ψ(r1)|ΨG⟩
+ δ(r2 − r′2)⟨ΨG|ψ†(r′1)ψ†(r2)ψ(r2)ψ(r1)|ΨG⟩+ ⟨ΨG|ψ†(r′1)ψ
†(r2)ψ†(r′2)ψ(r
′2)ψ(r2)ψ(r1)|ΨG⟩]
× eik·(r1−r′1) (B.76)
We now introduce the following functions:
SB(r1 − r′1) ≡1
N(N − 1)
∫dr2 V (r1 − r2)V (r′1 − r2)⟨ΨG|ψ†(r′1)ψ
†(r2)ψ(r2)ψ(r1)|ΨG⟩, (B.77)
SC(r1 − r′1) ≡1
N(N − 1)(N − 2)
∫dr2
∫dr′2 V (r1 − r2)V (r′1 − r′2)
× ⟨ΨG|ψ†(r′1)ψ†(r2)ψ
†(r′2)ψ(r′2)ψ(r2)ψ(r1)|ΨG⟩. (B.78)
Using them, we can express Eq. (B.70) concisely as
J2(k) =
(~2k2
2m
)2 [1 +
∫dr eik·rg1(r)
]+
~2k2
2m
[nVq=0 +
∫dr V (r)g1(r)e
ik·r +N(N − 1)
∫dr SA(r)e
ik·r]
+~2k2
2m
[nVq=0 +
∫dr V (r)g1(r)e
ik·r +N(N − 1)
∫dr SA(r)e
ik·r]∗
+ n
∫dr [V (r)]2 +
∫dr [V (r)]2g1(r)e
ik·r +
∫dr
∫dr′ V (r− r′)V (r)g2(r
′)
+N(N − 1)
[∫dr V (r)SA(r)e
ik·r +
(∫dr V (r)SA(r)e
ik·r)∗]
+N(N − 1)
∫dr SB(r)e
ik·r +N(N − 1)(N − 2)
∫dr SC(r)e
ik·r. (B.79)
We note that the functions SA(r), SB(r), SC(r) are expressible in forms suitable for the sampling as
SA(r1 − r′1) =
∫dr2 · · ·
∫drN V (r1 − r2)Ψ
∗G(r
′1, r2, · · · , rN )ΨG(r1, r2, · · · , rN )
SA(r) =1
V
∫dr1 · · ·
∫drN V (r1 + r− r2)Ψ
∗G(r1, r2, · · · , rN )ΨG(r1 + r, r2, · · · , rN )
=1
V2
N(N − 1)
∑i<j
⟨V (ri − rj + r)
ΨG(r1, r2, · · · , ri + r, · · · , rj , · · · rN )
ΨG(r1, r2, · · · , rN )
⟩, (B.80)
61
SB(r) =1
V
∫dr1 · · ·
∫drN V (r1 − r2)V (r1 − r2 + r)Ψ∗
G(r1, r2, · · · , rN )ΨG(r1 + r, r2, · · · , rN )
=1
V2
N(N − 1)
∑i<j
⟨V (ri − rj)V (ri − rj + r)
ΨG(r1, r2, · · · , ri + r, · · · , rj , · · · rN )
ΨG(r1, r2, · · · , rN )
⟩, (B.81)
SC(r1 − r′1) =
∫dr2
∫dr′2
∫dr4 · · ·
∫drN V (r1 − r2)V (r′1 − r′2)
×Ψ∗G(r1, r2, r
′2, r4, · · · , rN )ΨG(r1, r2, r
′2, r4, · · · , rN )
SC(r) =1
V
∫dr1 · · ·
∫drN V (r1 − r2)V (r1 − r3 + r)Ψ∗
G(r1, r2, · · · , rN )ΨG(r1 + r, r2, · · · , rN )
=1
V2
N(N − 1)(N − 2)
∑k =i,j
∑i<j
⟨V (ri − rj)V (ri − rk + r)
ΨG(r1, r2, · · · , ri + r, · · · , rj , · · · , rk, · · · rN )
ΨG(r1, r2, · · · , rN )
⟩(B.82)
respectively.Finally we introduce the following functions:
An(k, ε) ≡A(k, ε)εn
J0(k) ,
Jn(k) ≡Jn(k)
J0(k) .(B.83)
Hence we can express the one-particle excitation spectrum εk and its lifetime δεk as
εk = J1(k), (B.84a)
δεk =
√J2(k)− J2
1 (k), (B.84b)
respectively.
62
C Two-particle scattering theory
C.1 Formalism
We consider a two-particle scattering problem of identical particles with mass m described by reduced Hamil-tonian for the relative coordinate (~ = 1),
H ≡ − ∇2
2mred+ U(r), mred =
m
2, (C.1)
which in the momentum representation reads
⟨p′|H|p⟩ = εredp δp′p +Up′−p
(2π)3, εredp =
p2
2mred, Up ≡
∫d3r U(r)eip·r. (C.2)
The transition matrix T is given by Eq. (37.25) of Schiff, Quantum Mechanics [64]. It is expressed in themomentum representation as
⟨p′|T |p⟩≡⟨p′|U |p⟩+ ⟨p′|U(εredp − H − i0+)−1U |p⟩
=⟨p′|U |p⟩+ ⟨p′|U(εredp − H0 − i0+)−1U |p⟩
+ ⟨p′|U(εredp − H0 − i0+)−1U(εredp − H0 − i0+)−1U |p⟩+ · · ·
=1
(2π)3Up′−p +
1
(2π)6
∫d3p1 Up′−p1(ε
redp − εredp1
− i0+)−1Up1−p +1
(2π)9
∫d3p1
∫d3p2
× Up′−p1(εredp − εredp1
− i0+)−1Up1−p2(εredp − εredp2
− i0+)−1Up2−p + · · ·
=1
(2π)3
[Up′−p +
1
(2π)3
∫d3p1 Up′−p1(ε
redp − εredp1
− i0+)−1Up1−p +1
(2π)6
∫d3p1
∫d3p2
× Up′−p1(εredp − εredp1
− i0+)−1Up1−p2(εredp − εredp2
− i0+)−1Up2−p + · · ·].
(C.3)
We also introduce the scattering amplitude f(p′,p) by Eq. (37.21) of Schiff with C = (2π)−3/2 and µ = mred,i.e.,
− 2π
mredf(p′,p) ≡ (2π)3⟨p′|T |p⟩. (C.4)
We note that k′ = k in Eq. (C.3) and (C.4) for the relevant potential scattering by U(r). Next, we expandUp′−p and f(p′,p) in spherical harmonics as
Up′−p = 4π∞∑ℓ=0
Uℓ(k′, p)
ℓ∑m=−ℓ
Yℓm(p′)Y ∗ℓm(p), (C.5a)
f(p′,p) = 4π
∞∑ℓ=0
fℓ(p)
ℓ∑m=−ℓ
Yℓm(p′)Y ∗ℓm(p). (C.5b)
Substituting them into Eq. (C.4) and carrying out the angular integrations with∫dΩp Yℓm(p)Y ∗
ℓ′m′(p) = δℓ,ℓ′δm,m′ . (C.6)
Therefore we obtain
−2πmred
fℓ(p)
=Uℓ(p, p) +
∫d3p1(2π)3
Uℓ(p, p1)1
εredp − εredp1− i0+
Uℓ(p1, p) +
∫d3p1(2π)3
∫d3p2(2π)3
Uℓ(p, p1)
× 1
εredp − εredp1− i0+
Uℓ(p1, p2)1
εredp − εredp2− i0+
Uℓ(p2, p) + · · · . (C.7)
63
We now consider the low-energy limit p → 0, where fℓ=0(p) reduces to the s-wave scattering length a as[see, Landau and Lifschitz, Quantum Mechanics, Eq. (130.9) [63] ]
limp→0
f0(p) = −a, (C.8)
and ℓ > 0 channels can be neglected. Taking this limit in Eq. (C.7), we obtain
4π
ma = U0(0, 0)−
∫d3p1(2π)3
U0(0, p1)1
2εp1
U0(p1, 0)
+
∫d3p1(2π)3
∫d3p2(2π)3
U0(0, p1)1
2εp1
U0(p1, p2))1
2εp2
U0(p2, 0) + · · · , (C.9)
where we have used mred = m/2 and εredp = 2εp with
εp ≡p2
2m. (C.10)
Let us introduce U (eff)(p, p′) that satisfies the integral equation:
U (eff)(p, p′) = U0(p, p′)−
∫d3p1(2π)3
U0(p, p1)1
2εp1
U (eff)(p1, p′)
= U0(p, p′)− 1
4π2
∫ ∞
0
dp1 U0(p, p1)1
2εp1
U (eff)(p1, p′). (C.11)
Then, Eq. (C.9) can be expressed concisely as
a =m
4πU (eff)(0, 0). (C.12)
C.2 Appications
First, we consider the following potential:
U(r) = U0δ(r), (C.13)
whose Fourier transformation is given by Up = U0. It has no dependence on momentum p, so we also obtainU0(p, p
′) = U0. By introducing the cut-off momentum pc for the integration in Eq. (C.11), we can solve it as
U0 =( m
4πa− pc
4π2
)−1
.(C.14)
If we set pc as 0≪ pc ≪ πm/a, we can remove the cut-off dependence as
U0 =4πa
m .(C.15)
Next, we consider a finite-range potential (see reference [50], Chapter 6.):
U(r) ≡ U0
8πr30e−r/r0 . (C.16)
By performing Fourier transform for U(r), we obtain
Up =U0
(1 + r20p2)2
. (C.17a)
We then expand it in spherical harmonics and use Eq. (C.6) to obtain
U0(p, q) =U0
(1 + r20p2 + r20q
2)2 − 4r40p2q2
. (C.17b)
64
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