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Stationary Entropy Principle and Renormalization in Normal and Superfluid Systems. I. Algebraic Formulation
Abstract
This paper shows how the dynamical and thermodynamical properties of an interacting quantum mechanical system with many degrees of freedom may be expressed and calculated solely in terms of renormalized propagators and renormalized vertices or interactions. The formulation employed is sufficiently general to encompass systems which have several components, with Fermi or Bose statistics, whether or not they exhibit superfluidity or superconductivity. The process of renormalization is the functional generalization of the thermodynamic transformation from the chemical potential and temperature to the energy and matter densities. With each set of variables (here, functions) is associated a natural thermodynamic function (here a functional). The natural functional for the unrenormalized potentials which occur in the Hamiltonian is the logarithm of the grand partition function; the natural functional for the fully renormalized variables, the distribution functions, is the entropy. In particular, a stationarity principle for a functional F(2) of distribution functions subject to constraints is shown to provide a fully renormalized description of the system. The numerical value of this functional, at the stationarity point at which the distribution functions take their actual value, is the entropy of the system. The equations of stationarity are expressions for the unrenormalized ν‐body potentials vν in terms of the ν′‐body distribution functions Gν′. The functionals F(2) and vν (of the distribution functions Gν′) are expressed as the solutions of closed functional differential equations which may be used to generate their power‐series expansions. For a superfluid Bose system, as for the electromagnetic field interacting with matter, it is necessary to consider expectation values of odd, as well as even, numbers of field operators. In particular it is necessary to employ the expectation values Gν for 2ν = 1, 2, 3, 4 field operators. For a fermion system, even if it is superconducting, only the functions Gν for 2ν = 2, 4 are required. In contrast to other thermodynamical functionals, the entropy functional F(2) makes no reference to equilibrium parameters such as temperature and chemical potential.
... The Baym-Kadanoff scheme is, however, self-consistent only for the one-particle functions; no two-particle vertices explicitly enter the generating Lutinger-Ward functional. A two-particle self-consistency was added to the Baym-Kadanoff scheme by Dominicis and Martin by including two-particle vertices [10][11][12]. ...
... To determine the susceptibility, we also need to evaluate the triplet irreducible vertex δ σ /δG σ . It compensates the second term on the right-hand side of Eq. (11) in the susceptibility of the spin-symmetric solution. ...
... That is why the dynamical vertex defined via a controlled sum of explicitly selected Feynman diagrams has been used to evaluate both the self-energy and the response functions in the self-consistent approximations [18,23,[33][34][35][36][37][38][39][40]. The situation does not improve even if we extend the Luttinger-Ward functional by two-particle vertex functions via the De Dominicis-Martin scheme, leading to a parquet construction [11,12,20,21,41,42]. ...
We disclose a serious deficiency of the Baym-Kadanoff construction of thermodynamically consistent conserving approximations. There are two vertices in this scheme: dynamical and conserving. The divergence of each indicates a phase instability. We show that each leads to incomplete and qualitatively different behavior at different critical points. The diagrammatically controlled dynamical vertex from the Schwinger-Dyson equation does not obey the Ward identity and cannot be continued beyond its singularity. The standardly used dynamical vertex alone cannot, hence, conclusively decide about the stability of the high-temperature phase. On the other hand, the divergence in the conserving vertex, obeying the conservation laws, does not invoke critical behavior of the spectral function and the specific heat. Moreover, the critical behavior of the conserving vertex may become spurious in low-dimensional systems. Consequently, the description of the critical behavior of correlated electrons becomes consistent and reliable only if the fluctuations of the order parameter in the conserving vertex lead to a divergence coinciding with that of the dynamical one.
... The classification of diagrams in terms of their two-particle (ir)reducibility started with the seminal work by Salpeter and Bethe in 1951 [1], published just two years after Feynman's [2,3] invention of the diagrammatic technique and Dyson's [4,5] related concepts of one-particle (ir)reducibility. A decade later, in 1964, De Dominicis and Martin [6,7] classified the twoparticle diagrams further into irreducible and three distinct types of reducible diagrams: the famous parquet equation. Here, De Dominicis and Martin used combinatorial arguments and also a Legendre transform of the free energy in order to analyze irreducibility properties. ...
... where we have introduced G n as the n-point expectation value or equivalently the n-point (connected and) disconnected Green's function [30]. On the other hand, by repeatedly performing derivatives of W with respect to V 1 , one obtains the connected n-point (n/2-particle) Green's functions G n [30,31] δ n W (δV 1 ) n =: G n (6) where δ n (δV 1 ) n denotes the n th functional derivative with respect to V 1 . The reason these Green's functions are connected is that each functional derivative δ δV 1 removes one external one-point vertex in the diagrams for W (see Fig. 1) leaving the initial connectedness of W intact. ...
... In this work we have shown how the parquet equation [6,7] can be derived from the free-energy functional employing methods of functional analysis. This has been done for the case of Φ 4 theory but can readily be extended to fermionic field theories with a two-particle scattering term by promoting the scalar field to spinors [7,29]. ...
The parquet equation is an exact field-theoretic equation known since the 60s that underlies numerous approximations to solve strongly correlated Fermion systems. Its derivation previously relied on combinatorial arguments classifying all diagrams of the two-particle Green's function in terms of their (ir)reducibility properties. In this work we provide a derivation of the parquet equation solely employing techniques of functional analysis namely functional Legendre transformations and functional derivatives. The advantage of a derivation in terms of a straightforward calculation is twofold: (i) the quantities appearing in the calculation have a clear mathematical definition and interpretation as derivatives of the Luttinger--Ward functional; (ii) analogous calculations to the ones that lead to the parquet equation may be performed for higher-order Green's functions potentially leading to a classification of these in terms of their (ir)reducible components.
... An exact and compact representation of the generating functional Z[J] can be achieved through the n-particle-irreducible (nPI) EA [107][108][109]. While the diagrammatic representation of Z[J] consists of vacuum diagrams involving the bare propagator and vertices of the theory, 2 Throughout this thesis manuscript, fluctuating fields, i.e. fields displaying quantum fluctuations treated within a PI, are denoted with an upper tilde, like ϕ α . ...
... As will be discussed throughout chapter 4 in particular, the 2PI EA framework 7 is a direct reformulation of the Green's function formalism based on Dyson equation and the Luttinger-Ward functional [116,118]. It was pioneered by the work of Lee, Yang, De Dominicis and others in statistical physics [108,[118][119][120][121][122], and subsequently extended by Cornwall, Jackiw and Tomboulis [109] to the framework of field theory discussed here, which is why the 2PI EA approach is also coined as CJT formalism. ...
... On étudie donc ensuite le cas de l'EA 2PI dans le cadre de ces deux dernières représentations. Il convient de rappeler que des degrés de liberté collectifs sont automatiquement introduits lorsque l'on manipule des EAs nP(P)I avec n ≥ 2. Les EAs 2PI [108,109,[118][119][120][121][122] et 2PPI [297,298] peuvent être considérées respectivement comme des fonctionnelles de fonctions de Green et des fonctionnelles de la densité, d'où le lien avec l'approche EDF nucléaire. Dans le cadre du développement par rapport à et par rapport à λ, les EAs 2PI originale et mixte sont toutes les deux poussées jusqu'à leur troisième ordre non-trivial, ce qui est une première pour le cas mixte [153, 168, 291-293, 295, 296] à notre connaissance (la détermination des diagrammes contribuant à l'EA 2PI mixte est aussi détaillée dans l'annexe C). ...
The core of this thesis is the path-integral formulation of quantum field theory and its ability to describe strongly-coupled many-body systems of finite size. Collective behaviors can be efficiently described in such systems through the implementation of spontaneous symmetry breaking (SSB) in mean field approaches. However, as the thermodynamic limit does not make sense in finite-size systems, the latter can not exhibit any SSB and the symmetries which are broken down at the mean field level must therefore be restored. The efficiency of theoretical approaches in the treatment of finite-size quantum systems can therefore be studied via their ability to restore spontaneously broken symmetries. In this thesis, a zero-dimensional O(N) model is taken as a theoretical laboratory to perform such an investigation with many state-of-the-art path-integral techniques: perturbation theory combined with various resummation methods (Padé-Borel, Meijer-G, conformal mapping), enhanced versions of perturbation theory (transseries derived via Lefschetz thimbles, optimized perturbation theory), self-consistent perturbation theory based on effective actions (auxiliary field loop expansion (LOAF), Cornwall-Jackiw-Tomboulis (CJT) formalism, 4PPI effective action,...), functional renormalization group (FRG) techniques (FRG based on the Wetterich equation, DFT-FRG, 2PI-FRG). Connections between these different techniques are also emphasized. In addition, the path-integral formalism provides us with the possibility to introduce collective degrees of freedom in an exact fashion via Hubbard-Stratonovich transformations: the effect of such transformations on all the aforementioned methods is also examined in detail.
... An exact and compact representation of the generating functional Z[J] can be achieved through the n-particle-irreducible (nPI) EA [107][108][109]. While the diagrammatic representation of Z[J] consists of vacuum diagrams involving the bare propagator and vertices of the theory, 2 Throughout this thesis manuscript, fluctuating fields, i.e. fields displaying quantum fluctuations treated within a PI, are denoted with an upper tilde, like ϕ α . ...
... As will be discussed throughout chapter 4 in particular, the 2PI EA framework 7 is a direct reformulation of the Green's function formalism based on Dyson equation and the Luttinger-Ward functional [116,118]. It was pioneered by the work of Lee, Yang, De Dominicis and others in statistical physics [108,[118][119][120][121][122], and subsequently extended by Cornwall, Jackiw and Tomboulis [109] to the framework of field theory discussed here, which is why the 2PI EA approach is also coined as CJT formalism. ...
... On étudie donc ensuite le cas de l'EA 2PI dans le cadre de ces deux dernières représentations. Il convient de rappeler que des degrés de liberté collectifs sont automatiquement introduits lorsque l'on manipule des EAs nP(P)I avec n ≥ 2. Les EAs 2PI [108,109,[118][119][120][121][122] et 2PPI [297,298] peuvent être considérées respectivement comme des fonctionnelles de fonctions de Green et des fonctionnelles de la densité, d'où le lien avec l'approche EDF nucléaire. Dans le cadre du développement par rapport à et par rapport à λ, les EAs 2PI originale et mixte sont toutes les deux poussées jusqu'à leur troisième ordre non-trivial, ce qui est une première pour le cas mixte [153, 168, 291-293, 295, 296] à notre connaissance (la détermination des diagrammes contribuant à l'EA 2PI mixte est aussi détaillée dans l'annexe C). ...
The core of this thesis is the path-integral formulation of quantum field theory and its ability to describe strongly-coupled quantum many-body systems of finite size. Collective behaviors can be efficiently described in such systems through the implementation of spontaneous symmetry breaking (SSB) in mean-field approaches. However, as the thermodynamic limit does not make sense in finite-size systems, the latter can not exhibit any SSB and the symmetries which are broken down at the mean-field level must therefore be restored. The efficiency of theoretical approaches in the treatment of finite-size quantum systems can therefore be studied via their ability to restore spontaneously broken symmetries. In this thesis, a zero-dimensional $O(N)$ model is taken as a theoretical laboratory to perform such an investigation with many state-of-the-art path-integral techniques: perturbation theory combined with various resummation methods (Pad\'e-Borel, Borel-hypergeometric, conformal mapping), enhanced versions of perturbation theory (transseries derived via Lefschetz thimbles, optimized perturbation theory), self-consistent perturbation theory based on effective actions (auxiliary field loop expansion (LOAF), Cornwall-Jackiw-Tomboulis (CJT) formalism, 4PPI effective action, ...), functional renormalization group (FRG) techniques (FRG based on the Wetterich equation, DFT-FRG, 2PI-FRG). Connections between these different techniques are also emphasized. In addition, the path-integral formalism provides us with the possibility to introduce collective degrees of freedom in an exact fashion via Hubbard-Stratonovich transformations: the effect of such transformations on all the aforementioned methods is also examined in detail.
... The 2PI effective action furthermore allows for the convenient investigation of the system dynamics starting from arbitrary Gaussian initial states [12]. Note that general n-particle irreducible (nPI) effective actions up to n = 4 are discussed at length in references [17,18]. This paper is organized as follows. ...
... Note that it is also possible to interpret the functional derivative with respect to K symmetrically, i.e. δ/δK ab + δ/δK ba , which removes the factor of 1/2 in equation (16) and the ensuing equations below. In close connection to the 1PI effective action in equation (10), the 2PI effective action [9,12] can now be shown to read (17) where ...
... From this perspective, applying the 2PI effective action to non-linear stochastic processes is a first step toward the possible implementation of higher-order effective actions (e.g. four-particle irreducible [17,18,32]), which would enable the computation of fourth cumulants that should otherwise require a numerical integrator of high weak order or unfeasibly many integration steps. ...
By combining the two-particle-irreducible (2PI) effective action common in non-equilibrium quantum field theory with the classical Martin-Siggia-Rose formalism, self-consistent equations of motion for the first and second cumulants of non-linear classical stochastic processes are constructed. Such dynamical equations for correlation and response functions are important in describing non-equilibrium systems, where equilibrium fluctuation-dissipation relations are unavailable. The method allows to evolve stochastic systems from arbitrary Gaussian initial conditions. In the non-linear case, it is found that the resulting integro-differential equations can be solved with considerably reduced computational effort compared to state-of-the-art stochastic Runge-Kutta methods. The details of the method are illustrated by several physical examples.
... An exact and compact representation of the generating functional Z[J] can be achieved through the n-particle-irreducible (nPI) EA [107][108][109]. While the diagrammatic representation of Z[J] consists of vacuum diagrams involving the bare propagator and vertices of the theory, 2 Throughout this thesis manuscript, fluctuating fields, i.e. fields displaying quantum fluctuations treated within a PI, are denoted with an upper tilde, like ϕ α . ...
... As will be discussed throughout chapter 4 in particular, the 2PI EA framework 7 is a direct reformulation of the Green's function formalism based on Dyson equation and the Luttinger-Ward functional [116,118]. It was pioneered by the work of Lee, Yang, De Dominicis and others in statistical physics [108,[118][119][120][121][122], and subsequently extended by Cornwall, Jackiw and Tomboulis [109] to the framework of field theory discussed here, which is why the 2PI EA approach is also coined as CJT formalism. ...
... On étudie donc ensuite le cas de l'EA 2PI dans le cadre de ces deux dernières représentations. Il convient de rappeler que des degrés de liberté collectifs sont automatiquement introduits lorsque l'on manipule des EAs nP(P)I avec n ≥ 2. Les EAs 2PI [108,109,[118][119][120][121][122] et 2PPI [297,298] peuvent être considérées respectivement comme des fonctionnelles de fonctions de Green et des fonctionnelles de la densité, d'où le lien avec l'approche EDF nucléaire. Dans le cadre du développement par rapport à et par rapport à λ, les EAs 2PI originale et mixte sont toutes les deux poussées jusqu'à leur troisième ordre non-trivial, ce qui est une première pour le cas mixte [153, 168, 291-293, 295, 296] à notre connaissance (la détermination des diagrammes contribuant à l'EA 2PI mixte est aussi détaillée dans l'annexe C). ...
The core of this thesis is the path-integral formulation of quantum field theory and its ability to describe strongly-coupled many-body systems of finite size. Collective behaviors can be efficiently described in such systems through the implementation of spontaneous symmetry breaking (SSB) in mean field approaches. However, as the thermodynamic limit does not make sense in finite-size systems, the latter can not exhibit any SSB and the symmetries which are broken down at the mean field level must therefore be restored. The efficiency of theoretical approaches in the treatment of finite-size quantum systems can therefore be studied via their ability to restore spontaneously broken symmetries. In this thesis, a zero-dimensional O(N) model is taken as a theoretical laboratory to perform such an investigation with many state-of-the-art path-integral techniques: perturbation theory combined with various resummation methods (Padé-Borel, Meijer-G, conformal mapping), enhanced versions of perturbation theory (transseries derived via Lefschetz thimbles, optimized perturbation theory), self-consistent perturbation theory based on effective actions (auxiliary field loop expansion (LOAF), Cornwall-Jackiw-Tomboulis (CJT) formalism, 4PPI effective action,...), functional renormalization group (FRG) techniques (FRG based on the Wetterich equation, DFT-FRG, 2PI-FRG). Connections between these different techniques are also emphasized. In addition, the path-integral formalism provides us with the possibility to introduce collective degrees of freedom in an exact fashion via Hubbard-Stratonovich transformations: the effect of such transformations on all the aforementioned methods is also examined in detail.
... Note that general n-particle irreducible (nPI) effective actions up to n = 4 are discussed at length in Refs. [17,18]. This paper is organized as follows. ...
... From this perspective, applying the 2PI effective action to non-linear stochastic processes is a first step towards the possible implementation of higher-order effective actions (e.g. fourparticle irreducible [17,18,32]), which would enable the computation of fourth cumulants that should otherwise require a numerical integrator of high weak order or unfeasibly many integration steps. ...
... SDEs with polynomial non-linearities of higher degree than investigated here. Furthermore, an extension to non-Gaussian initial states is possible in principle via nPI effective actions (where n > 2 Legendre transforms are performed with respect to higher moments) [17,18,48]. Evaluating the performance of nPI effective actions, if numerically feasible, on stochastic processes with strong non-linearities is another interesting perspective. ...
By combining the two-particle-irreducible (2PI) effective action common in non-equilibrium quantum field theory with the classical Martin-Siggia-Rose formalism, self-consistent equations of motion for the first and second cumulants of non-linear classical stochastic processes are constructed. Such dynamical equations for correlation and response functions are important in describing non-equilibrium systems, where equilibrium fluctuation-dissipation relations are unavailable. The method allows to evolve stochastic systems from arbitrary Gaussian initial conditions. In the non-linear case, it is found that the resulting integro-differential equations can be solved with considerably reduced computational effort compared to state-of-the-art stochastic Runge-Kutta methods. The details of the method are illustrated by several physical examples.
... We refer the reader to Chap. 15 of Ref. [12] for more details on the oriented diagrammatics in the formalism of Nambu. Among subsequent developments in diagrammatic approaches dealing explicitly with symmetry-breaking, we highlight here the pioneering work by De Dominicis and Martin on superfluidity [13,14]. In their work, perturbative contributions were expressed in terms of diagrams with un-oriented lines and totally antisymmetric vertices so that any further development is simplified. ...
... Later, a similar formalism was reintroduced by Kleinert, in his work on collective excitations [15,16], as well as by Haussmann, in his work on the BCS-BEC crossover [17,18]. In this context, the present work can be seen as rooting the specific formalisms developed in [13][14][15][16][17][18] into the formalism of Nambu tensors, thus extending them to the case of a general Hamiltonian expressed in a general extended basis. ...
... To tackle this recurrent problem, we introduce a reformulation of the quantum many-body problem in the present paper. The present work can be seen as the natural continuation of the pioneering work of De Dominicis and Martin [13,14], Kleinert [15,16] and Haussmann [17,18]. These developments provided a solid starting point but also introduced restrictive hypotheses, either on the choice of the working basis or the type of interaction, to simplify some formal steps. ...
Symmetry-breaking considerations play an important role in allowing reliable and accurate predictions of complex systems in quantum many-body simulations. The general theory of perturbations in symmetry-breaking phases is nonetheless intrinsically more involved than in the unbroken phase due to non-vanishing anomalous Green's functions or anomalous quasiparticle interactions. In the present paper, we develop a formulation of many-body theory at non-zero temperature which is explicitly covariant with respect to a group containing Bogoliubov transformations. Based on the concept of Nambu tensors, we derive a factorisation of standard Feynman diagrams that is valid for a general Hamiltonian. The resulting factorised amplitudes are indexed over the set of un-oriented Feynman diagrams with fully antisymmetric vertices. We argue that, within this framework, the design of symmetry-breaking many-body approximations is simplified.
... The classification of diagrams in terms of their two-particle (ir)reducibility started with the seminal work by Salpeter and Bethe in 1951 [1], published just two years after Feynman's [2,3] invention of the diagrammatic technique and Dyson's [4,5] related concepts of one-particle (ir)reducibility. A decade later, in 1964, De Dominicis and Martin [6,7] classified the twoparticle diagrams further into irreducible and three distinct types of reducible diagrams: the famous parquet equation. Here, De Dominicis and Martin used combinatorial arguments and also a Legendre transform of the free energy in order to analyze irreducibility properties. ...
... In this work we have shown how the parquet equation [6,7] can be derived from the freeenergy functional employing methods of functional analysis. This has been done for the case of Φ 4 theory but can readily be extended to fermionic field theories with a two-particle scattering term by promoting the scalar field to spinors [7,29]. ...
The parquet equation is an exact field-theoretic equation known since the 60s that underlies numerous approximations to solve strongly correlated Fermion systems. Its derivation previously relied on combinatorial arguments classifying all diagrams of the two-particle Green’s function in terms of their (ir)reducibility properties. In this work we provide a derivation of the parquet equation solely employing techniques of functional analysis namely functional Legendre transformations and functional derivatives. The advantage of a derivation in terms of a straightforward calculation is twofold: (i) the quantities appearing in the calculation have a clear mathematical definition and interpretation as derivatives of the Luttinger-Ward functional; (ii) analogous calculations to the ones that lead to the parquet equation may be performed for higher-order Green’s functions potentially leading to a classification of these in terms of their (ir)reducible components.
... To construct equations that are satisfied by the Green's functions that characterize composite operators, it is convenient to use the formalism of functional Legendre transforms. In statistical physics it was used firstly in [8][9][10] and then began to be applied to the problems of quantum field theory as well. ...
... Knowing it, one can obtain equations for the functions (6) characterizing composite operators. To do this, denote S (ϕ, ρ) the action functional of the form S (ϕ, ρ) = S(ϕ) + ρV(ϕ), where S(ϕ) is the action of the model, in which the propagator and the mean field are solutions of the Equations (8) and (9). For quadratic with respect to the field ϕ functional V(ϕ) = 1 2 ϕV 2 ϕ and given function V 2 , the equations ...
The technique of functional Legendre transforms is used to develop an effective method for calculating the characteristics of critical phenomena in quantum field theory models in the Euclidean space of dimension d. Based on the diagrammatic representation of the second Legendre transform in the theory with a cubic interaction potential, the construction of self-consistent equations is carried out, the solution of which makes it possible to find the dimensions not only of the main fields, but also of the quadratic on the composite operators within the 1/n-expansion. Application of the proposed methods in the model F has given the opportunity to calculate in the main approximation by 1/n the anomalous dimensions of both scalar and tensor composite operators quadratic on the fields ϕ. For them, as functions of the spatial dimension d, we obtained explicit analytical expressions in the form of relations of two polynomials with integer coefficients.
... The single-boson exchange (SBE) decomposition, recently introduced [15] to rationalize the treatment of parquet-type diagrams [13,[16][17][18][19][20][21][22][23], leads to a significant reduction of the computational effort [24,25]. Due to the qualitative similarity of the diagrammatic structure in parquet-and fRG-based approximations, it seems quite natural to exploit similar ideas also in an fRG context, recasting the fRG flow equations within the SBE formalism. ...
... By inserting the limits (11) and (16) into Eq. (12), we hence obtain the flow equations for the screened interactions, Yukawa couplings, and rest functions: ...
We present a reformulation of the functional renormalization group (fRG) for many-electron systems, which relies on the recently introduced single-boson exchange (SBE) representation of the parquet equations [F. Krien, A. Valli, and M. Capone, Phys. Rev. B 100, 155149 (2019)]. The latter exploits a diagrammatic decomposition, which classifies the contributions to the full scattering amplitude in terms of their reducibility with respect to cutting one interaction line, naturally distinguishing the processes mediated by the exchange of a single boson in the different channels. We apply this idea to the fRG by splitting the one-loop fRG flow equations for the vertex function into SBE contributions and a residual four-point fermionic vertex. Similarly as in the case of parquet solvers, recasting the fRG algorithm in the SBE representation offers both computational and interpretative advantages: The SBE decomposition not only significantly reduces the numerical effort of treating the high-frequency asymptotics of the flowing vertices, but it also allows for a clear physical identification of the collective degrees of freedom at play. We illustrate the advantages of an SBE formulation of fRG-based schemes, by computing through the merger of dynamical mean-field theory and fRG the susceptibilities and the Yukawa couplings of the two-dimensional Hubbard model from weak to strong coupling, for which we also present an intuitive physical explanation of the results. The SBE formulation of the one-loop flow equations paves a promising route for future multiboson and multiloop extensions of fRG-based algorithms.
... The single boson exchange (SBE) decomposition, recently introduced [15] to rationalize the treatment of * d.vilardi@fkf.mpg.de parquet-type diagrams [13,[16][17][18][19][20][21][22][23], leads to a significant reduction of the computational effort [24,25]. Due to the qualitative similarity of the diagrammatic structure in parquet-and fRG-based approximations, it seems quite natural to exploit similar ideas also in an fRG context, recasting the fRG flow equations within the SBE formalism. ...
... By inserting the limits (11) and (16) into Eq. (12), we hence obtain the flow equations for the screened interactions, Yukawa couplings and rest functions: ...
We present a reformulation of the functional renormalization group (fRG) for many-electron systems, which relies on the recently introduced single boson exchange (SBE) representation of the parquet equations [Phys. Rev. B 100, 155149 (2019)]. The latter exploits a diagrammatic decomposition, which classifies the contributions to the full scattering amplitude in terms of their reducibility with respect to cutting one interaction line, naturally distinguishing the processes mediated by the exchange of a single boson in the different channels. We apply this idea to the fRG by splitting the one-loop fRG flow equations for the vertex function into SBE contributions and a residual four-point fermionic vertex. Similarly as in the case of parquet solvers, recasting the fRG algorithm in the SBE representation offers both computational and interpretative advantages: the SBE decomposition not only significantly reduces the numerical effort of treating the high-frequency asymptotics of the flowing vertices, but it also allows for a clear physical identification of the collective degrees of freedom at play. We illustrate the advantages of an SBE formulation of fRG-based schemes, by computing through the merger of dynamical mean-field theory and fRG the susceptibilities and the Yukawa couplings of the two-dimensional Hubbard model from weak to strong coupling, for which we also present an intuitive physical explanation of the results. The SBE formulation of the one-loop flow equations paves a promising route for future multiboson and multiloop extensions of fRG-based algorithms.
... Notice the difference between the quark-gluon vertex in the isospin basis (27) and the Nambu-Gorkov basis (17); in the latter case, the lower component of Γ μ a is in the antifundamental representation −T ⊤ a . The quark part of the two-particle-irreducible (2PI) action Γ can be written as the following functional of the full quark propagator S [62][63][64][65]: ...
In this paper QCD at large isospin density is studied, which is known to be in the superfluid state with Cooper pairs carrying the same quantum number as pions. The gap equation derived from the perturbation theory up to the next-to-leading-order corrections is solved. The pairing gap at large isospin chemical potential is found to be enhanced compared to the color-superconducting gap at large baryon chemical potential due to the 2 difference in the exponent arising from the stronger attraction in one-gluon exchange in the singlet channel. Then, using the gap function, the contribution of the condensation energy of the superfluid state to the QCD equation of state is evaluated. At isospin chemical potential of a few GeV, where the lattice QCD and the perturbative QCD can be both applied, the effect of the condensation energy becomes dominant even compared to the next-to-leading order corrections to the pressure in the perturbation theory. It resolves the discrepancy between the recent lattice QCD results and the perturbative QCD result.
... This observation paves the way for an investigation into the screened version of the eh T-matrix, which has demonstrated successes in diverse systems, such as ferromagnetic periodic structures, as reported in prior studies. [61][62][63][64] Another avenue for further exploration involves the combination of these three correlation channels, akin to "fluctuation exchange" (FLEX), 85-88 the Baym-Kadanoff approximation, 11,12 parquet theory, 89,90 and other similar approaches. 19,64,91-94 Although challenging, this task holds significant promise and represents a potential avenue for our future investigations. ...
We derive the explicit expression of the three self-energies that one encounters in many-body perturbation theory: the well-known GW self-energy, as well as the particle–particle and electron–hole T-matrix self-energies. Each of these can be easily computed via the eigenvalues and eigenvectors of a different random-phase approximation linear eigenvalue problem that completely defines their corresponding response function. For illustrative and comparative purposes, we report the principal ionization potentials of a set of small molecules computed at each level of theory. The performance of these schemes on strongly correlated systems (B2 and C2) is also discussed.
... This observation paves the way for an investigation into the screened version of the eh T -matrix, which has demonstrated successes in diverse systems, such as ferromagnetic periodic structures, as reported in prior studies. [61][62][63][64] Another avenue for further exploration involves the combination of these three correlation channels, akin to "fluctuation exchange" (FLEX), 85-88 the Baym-Kadanoff approximation, 11,12 parquet theory, 89,90 and other similar approaches. 19,64,91-94 Although challenging, this task holds significant promise and represents a potential avenue for our future investigations. ...
We derive the explicit expression of the three self-energies that one encounters in many-body perturbation theory: the well-known $GW$ self-energy, as well as the particle-particle and electron-hole $T$-matrix self-energies. Each of these can be easily computed via the eigenvalues and eigenvectors of a different random-phase approximation (RPA) linear eigenvalue problem that completely defines their corresponding response function. For illustrative and comparative purposes, we report the principal ionization potentials of a set of small molecules computed at each level of theory.
... σ i represents the spin quantum number and k i = (k i , ν i ) includes both the momentum and Matsubara frequency. The coupling function can be further decomposed into the magnetic, density, and superconducting channel by the parquet decomposition [51][52][53][54][55][56][57][58][59] ...
Using the recently introduced multiloop extension of the functional renormalization group, we compute the magnetic, density, and superconducting susceptibilities of the two-dimensional Hubbard model at weak coupling and present a detailed analysis of their evolution with temperature, interaction strength, and loop order. By breaking down the susceptibilities into contributions from the bare susceptibility and the individual channels, we investigate their relative importance as well as the channel interplay. In particular, we trace the influence of antiferromagnetic fluctuations on the $d$-wave superconductivity and provide an analytical understanding for the observed behavior.
... Note that the third Legendre transform, even though it is usually referred to as the 3PI effective action, is not the sum of all 3PI diagrams. Indeed, in a theory with only a cubic interaction there are no 3PI diagrams, but one can nevertheless construct an effective action that generates a self-consistent vertex function [40,41]. ...
A bstract
We construct one- and two-particle irreducible (1PI and 2PI) effective actions for the stochastic fluid dynamics of a conserved density undergoing diffusive motion. We compute the 1PI action in one-loop order and the 2PI action in two-loop approximation. We derive a set of Schwinger-Dyson equations and regularize the resulting equations using Pauli-Villars fields. We numerically solve the Schwinger-Dyson equations for a non-critical fluid. We find that higher-loop effects summed by the Schwinger-Dyson renormalize the non-linear coupling. We also find indications of a diffuson-cascade, the appearance of n -loop correction with smaller and smaller exponential suppression.
... Going beyond these approximations has been shown to be rather challenging. 83,84,89,98,99,[102][103][104][105][106][107][108][109][110][111][112][113][114][115][116] Here, for the sake of simplicity, we consider only one-shot schemes where one does not self-consistently update the self-energy, 22,30,[117][118][119][120][121] but the same analysis can be performed in the case of (partially) self-consistent schemes. 26,46,[122][123][124][125][126][127][128][129][130] Another attractive point concerning Green's functionbased techniques is the Bethe-Salpeter equation (BSE) formalism 6,7,131,132 that allows access to the neutral (i.e., optical) excitations of a given system. ...
In recent years, Green's function methods have garnered considerable interest due to their ability to target both charged and neutral excitations. Among them, the well-established $GW$ approximation provides accurate ionization potentials and electron affinities and can be extended to neutral excitations using the Bethe-Salpeter equation (BSE) formalism. Here, we investigate the connections between various Green's function methods and evaluate their performance for charged and neutral excitations. Comparisons with other widely-known second-order wave function methods are also reported. Additionally, we calculate the singlet-triplet gap of cycl[3,3,3]azine, a model molecular emitter for thermally activated delayed fluorescence, which has the particularity of having an inverted gap thanks to a substantial contribution from the double excitations. We demonstrate that, within the $GW$ approximation, a second-order BSE kernel with dynamical correction is required to predict this distinctive characteristic.
... The physical theory of inhomogenous fluids goes essentially back to the 60s [19,20,58,72,100]. Functional methods and their applications to the theory of the structure of bulk fluids were described in [75,99]. ...
We provide upper and lower bounds on the lowest free energy of a classical system at given one-particle density ρ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho (x)$$\end{document}. We study both the canonical and grand-canonical cases, assuming the particles interact with a pair potential which decays fast enough at infinity.
... Unfortunately, defining a systematic way to go beyond GW via the inclusion of vertex corrections has been demonstrated to be a tricky task. 47,[74][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92] For example, Lewis and Berkelbach have shown that naive vertex corrections can even worsen the quasiparticle energies with respect to GW . 93 We refer the reader to the recent review by Golze and co-workers 4 for an extensive list of current challenges in Green's function methods. ...
The family of Green's function methods based on the $GW$ approximation has gained popularity in the electronic structure theory thanks to its accuracy in weakly correlated systems combined with its cost-effectiveness. Despite this, self-consistent versions still pose challenges in terms of convergence. A recent study \href{https://doi.org/10.1063/5.0089317}{[J. Chem. Phys. 156, 231101 (2022)]} has linked these convergence issues to the intruder-state problem. In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods. The SRG formalism enables us to derive, from first principles, the expression of a naturally static and Hermitian form of the self-energy that can be employed in quasiparticle self-consistent $GW$ (qs$GW$) calculations. The resulting SRG-based regularized self-energy significantly accelerates the convergence of qs$GW$ calculations, slightly improves the overall accuracy, and is straightforward to implement in existing code.
... In Fig. 18, ξ sp is shown for the half-filled 2D square lattice Hubbard model at constant interaction U = 2. Several methods are compared against each other, namely OG TPSC, TPSC+GG, DMFT+TPSC, DΓA, 63 DiagMC, 59,64 TRILEX 65,66 and the Parquet Approximation (PA). 67,68 The correlation length ξ sp is extracted from the Ornstein-Zernicke fit of the momentumdependent static spin susceptibility χ sp q−Q (iq n = 0) in the vicinity of the AFM scattering wave vector Q: ...
Nonlocal correlations play an essential role in correlated electron systems, especially in the vicinity of phase transitions and crossovers, where two-particle correlation functions display a distinct momentum dependence. In nonequilibrium settings, the effect of nonlocal correlations on dynamical phase transitions, prethermalization phenomena and trapping in metastable states is not well understood. In this paper, we introduce a dynamical mean field theory (DMFT) extension to the nonequilibrium Two-particle Self-Consistent (TPSC) approach, which allows to perform nonequilibrium simulations capturing short- and long-ranged nonlocal correlations in the weak-, intermediate- and strong-correlation regime. The method self-consistently computes local spin and charge vertices, from which a momentum-dependent self-energy is constructed. Replacing the local part of the self-energy by the DMFT result within this self-consistent scheme provides an improved description of local correlation effects. We explain the details of the formalism and the implementation, and demonstrate the versatility of DMFT+TPSC with lattice hopping quenches and dimensional crossovers in the Hubbard model.
... 1 A formalism of multilocal sources was elaborated in the framework of quantum statistics by De Dominisis and Martin [8] and elaborated in quantum field theory in works [9][10][11] (see also [12] for the modern applications). ...
As shown in the works [1-3], the asymptotic behavior of the propagator in the Euclidean region of momenta for the model of a complex scalar field φ and a real scalar field χ with the interaction drastically changes depending on the value of the coupling constant. For small values of the coupling, the propagator of the field φ behaves asymptotically as free, while in the strong-coupling region the propagator in the deep Euclidean region tends to be a constant. In this paper, the influence of the vacuum stability problem of this model on this critical behavior is investigated. It is shown that within the framework of the approximations used, the addition of a stabilizing term of type to the Lagrangian leads to a renormalization of the mass and does not change the main effect of changing the ultraviolet behavior of the propagator. PACS number: 11.10.Jj.
... Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 3 The relativistic 2PI effective action was described by Coleman et al [1], building on early applications of higher-order Legendre transforms in non-relativistic statistical mechanics, e.g. by De Dominicis and Martin [2,3]. 4 The importance of identifying the true convex-conjugate variables, in terms of the non-connected functions, was emphasised earlier in reference [6]. ...
By exploiting the convexity of the two-particle-irreducible effective action, we describe a procedure for extracting n -point vertex functions. This procedure is developed within the context of a zero-dimensional ‘quantum field theory’ and subsequently extended to higher dimensions. These results extend the practicability and utility of a recent, alternative approach to the functional renormalization group programme (see Alexander et al 2021 Phys. Rev. D 104 069906; Millington and Saffin 2021 J. Phys. A: Math. Theor. 54 465401), and clarify the relationship between the flow equations for coupling parameters and vertices.
... The same problem is also inherent in cluster extensions of the dynamical meanfield theory (DMFT) [44][45][46][47][48][49], such as, e.g., DCA [38][39][40]. Diagrammatic methods based on the parquet approximation [50][51][52][53][54][55] allow one to account for the interplay between charge and spin fluctuations [26] originating from the two-particle vertex functions in an unbiased and powerful fashion. These vertices are incorporated with full momentum-and frequency-dependence, and the approach is thus computationally very expensive, which severally limits its applicability. ...
Systems with strong electronic Coulomb correlations often display rich phase diagrams exhibiting different ordered phases involving spin, charge, or orbital degrees of freedom. The theoretical description of the interplay of the corresponding collective fluctuations giving rise to this phenomenology remains however a tremendous challenge. Here, we introduce a multi-channel extension of the recently developed fluctuating field approach to competing collective fluctuations in correlated electron systems. The method is based on a variational optimization of a trial action that explicitly contains the order parameters of the leading fluctuation channels. It gives direct access to the free energy of the system, facilitating the distinction between stable and meta-stable phases of the system. We apply our approach to the extended Hubbard model in the weak to intermediate coupling regime where we find it to capture the interplay of competing charge density wave and antiferromagnetic fluctuations with qualitative agreement with more computationally expensive methods. The multi-channel fluctuation field approach thus offers a promising new route for a numerically cheap treatment of the interplay between collective fluctuations in large systems.
... The physical theory of inhomogenous fluids goes essentially back to the 60s [MH61,DeD62,DDM64,SB62,LP63]. Functional methods and their applications to the theory of the structure of bulk fluids were described in [Per64,Ste64]. ...
We provide upper and lower bounds on the lowest free energy of a classical system at given one-particle density $\rho(x)$. We study both the canonical and grand-canonical cases, assuming the particles interact with a pair potential which decays fast enough at infinity.
... As a result, for both quantities we obtain a perturbation series in terms of Feynman diagrams. De Dominicis and Martin [43] have shown by a Legendre transformation that the entropy S = S[G, Γ] can be rewritten as a functional of the exact Green function G and the exact vertex function Γ. Furthermore, they have derived an explicit perturbation series in terms of irreducible Feynman diagrams where the lines are identified by G and the vertices by Γ [44]. ...
Starting from a general classical model of many interacting particles we present a well defined step by step procedure to derive the continuum-mechanics equations of nonlinear elasticity theory with fluctuations which describe the macroscopic phenomena of a solid crystal. As the relevant variables we specify the coarse-grained densities of the conserved quantities and a properly defined displacement field which describes the local translations, rotations, and deformations. In order to stay within the framework of the conventional density-functional theory we first and mainly consider the isothermal case and omit the effects of heat transport and warming by friction where later we extend our theory to the general case and include these effects. We proceed in two steps. First, we apply the concept of local thermodynamic equilibrium and minimize the free energy functional under the constraints that the macroscopic relevant variables are fixed. As results we obtain the local free energy density and we derive explicit formulas for the elastic constants which are exact within the framework of density-functional theory. Second, we apply the methods of nonequilibrium statistical mechanics with projection-operator techniques. We extend the projection operators in order to include the effects of coarse-graining and the displacement field. As a result we obtain the time-evolution equations for the relevant variables with three kinds of terms on the right-hand sides: reversible, dissipative, and fluctuating terms. We find explicit formulas for the transport coefficients which are exact in the limit of continuum mechanics if the projection operators are properly defined. By construction the theory allows the diffusion of particles in terms of point defects where, however, in a normal crystal this diffusion is suppressed.
... All known implementations of DFT break down in certain stronglycorrelated electron materials, illustrating the difficulty of developing sufficiently robust approximations for the exchange-correlation functional. Instead of creating functionals of the density, one can choose to create functionals of other observables, which might be more sensitive to capturing the physics of strong correlations, such as the single particle Green's functions [92,93,94]. In such approaches, there is an analogy to the exchange-correlation functional, which embodies the unknown portion of the energy as a functional of the Green's function, and the functional derivative of this quantity with respect to the Green's function is the well-known self-energy. ...
To efficiently capture the energy of the nuclear bond, advanced nuclear reactor concepts seek solid fuels that must withstand unprecedented temperature and radiation extremes. In these advanced fuels, thermal energy transport under irradiation is directly related to reactor performance as well as reactor safety. The science of thermal transport in nuclear fuel is a grand challenge due to both computational and experimental complexities. Here, we provide a comprehensive review of thermal transport research on two actinide oxides: one currently in use in commercial nuclear reactors, uranium dioxide (UO2), and one advanced fuel candidate material, thorium dioxide (ThO2). In both materials, heat is carried by lattice waves or phonons. Crystalline defects caused by fission events effectively scatter phonons and lead to a degradation in fuel performance over time. Bolstered by new computational and experimental tools, researchers are now developing the foundational work necessary to accurately model and ultimately control thermal transport in advanced nuclear fuel. We begin by reviewing research aimed at understanding thermal transport in perfect single crystals. The absence of defects enables studies that focus on the fundamental aspects of phonon transport. Next, we review research that targets defect generation and evolution. Here, the focus is on ion irradiation studies used as surrogates for damage caused by fission products. We end this review with a discussion of modeling and experimental efforts directed at predicting and validating mesoscale thermal transport in the presence of irradiation defects. While efforts into these research areas have been robust, challenging work remains in developing holistic tools to capture and predict thermal energy transport across widely varying environmental conditions.
... The formalism of the 2PI effective action was first introduced in [13,14] and an efficient computational procedure was developed in [15]. Here we recap the main points to establish our notation. ...
In this work we numerically explore the quantum behavior of a classically unstable relativistic Bose-Einstein condensate (BEC). The main goal is to study the phenomenon of so-called quantum break time which amounts to a significant departure from a semiclassical mean-field description. It has been suggested previously that the existence of Lyapunov instability is crucial for a fast quantum breaking, chaos and scrambling. In order to clarify the issue, we work within the 2-PI effective action formalism and introduce a simple and very widely applicable dynamical criterion for identifying the timescale of quantum breaking. We indeed observe that the fast quantum break time is controlled by the Lyapunov exponent of the unstable BEC.
... Restoring the conserving properties of the FLEX formalism amounts to solving a self-consistent parquet equation scheme [108,109]. The parquet equation scheme determines the full twoparticle vertex Γ in terms of the two-particle irreducible vertices. ...
This thesis studies the role of fluctuation dynamics as a precursor of a nonequilibrium phase transition. The performed numerical simulations of microscopic many-body models provide new insights into the questions how order is formed in a solid state system and how the fluctuations as a key quantity for the characterization of long-range order can be detected on ultrafast timescales. The first part of the thesis investigates the initial dynamics of superconducting fluctuations in the attractive three dimensional Hubbard model following a sudden quench from the unordered paramagnetic to the ordered phase of the system. The thermal relaxation dynamics in the vicinity of a phase transition are well understood in terms of time-dependent Ginzburg-Landau theory. However, significantly less is known about the initial nonthermal stages after a sudden excitation of the system, due to the inseparability of the fermionic electronic and bosonic fluctuation dynamics. This thesis investigates the nonthermal electronic dynamics in this regime and characterizes their consequent impact on the growth of the fluctuations utilizing the fluctuation exchange approximation (FLEX) in the Keldysh formalism. The Keldysh formalism contains the full complexity of the nonequilibrium many-body problem for the electronic degrees of freedom based on time-dependent Green’s functions, that are determined by the solution of the differo-integral Kadanoff-Baym equations. The fluctuation exchange approximation on the other hand allows a consistent construction of the bosonic fluctuation propagators in the same formalism. The obtained set of self-consistent equations is hence suitable for the description of competing instabilities in correlated solids and can in principle be propagated to arbitrary times. The numerical solution of the Kadanoff-Baym equations reveals a nonmonotonous growth of the fluctuations, which is beyond the scope of Ginzburg-Landau theory or any local approximation scheme for the self-energy. Furthermore, the growing fluctuations cause a redistribution of spectral weight and the opening of a pseudogap. This can be understood as the consequence of Andreev reflections of the electrons from the fluctuations. The further simulation of the impact of the growing fluctuations on the thermal electrons is limited by the memory demands for the solution of the equations in the three dimensional system. The study of the Kadananoff-Baym equations is hence complemented by test calculations for a systematic truncation of the temporal integrals within the Kadanoff-Baym evaluation scheme. Even though the truncated equations are not yielding physically relevant results, the analysis of the numerical error indicates that physically consistent results can be expected for many-body systems coupled to a heat reservoir. The final part of the thesis is devoted to the discussion of the spectral signatures of fluctua- tions dynamics. As the number of experimental techniques capable of resolving the ultrafast processes below the thermalization time of electrons in a solid is limited, we are proposing an experimental protocol for the detection of correlations in the shot-to-shot noise of time- and angular-resolved photoemission spectroscopy (tr-ARPES). The formula for the noise correlation measurement (NCM) is derived for a pump-probe experiment and the protocol is tested within the context of a vanishing superconducting phase. The NCM is hereby capable of identifying dephasing and melting dynamics, that are indistinguishable by the time-dependent spectra alone.
... For a more general discussion on self-consistent vertices, we refer the reader to Refs. [68,69], where they are introduced by means of a Legendre transformations and related to a diagrammatic expansion. ...
This paper is devoted to the formulation of Self-Consistent Green's Function (SCGF) in an explicit Nambu-covariant fashion for applications to many-body systems at non-zero temperature in symmetry-broken phases. This is achieved by extending the Nambu-covariant formulation of perturbation theory, presented in the first part of this work, to non-perturbative schemes based on self-consistently dressed propagators and vertices. We work out in detail the self-consistent ladder approximation, motivated by a trade-off between numerical complexity and many-body phenomenology. Taking a complex general Hartree-Fock-Bogoliubov (HFB) propagator as the starting point, we also formulate and prove a sufficient condition on the stability of the HFB self-energy to ensure the convergence of the initial series of ladders at any energy. The self-consistent ladder approximation is written purely in terms of spectral functions and the resulting set of equations, when expressed in terms of Nambu tensors, are remarkably similar to those in the symmetry-conserving case. This puts the application of the self-consistent ladder approximation to symmetry-broken phases of infinite nuclear matter within reach.
... Beyond the HF, two popular approximations of this class are the GW approximation [16,21], involving G and W , and the fluctuation -exchange (FLEX) theory [22], involving G and χ's. More complicated schemes such as the parquet approximation [23,24] and covariant quartic approximation [25], in addition to the one -momentum functions, unfortunately have to consider multiple -momenta quantities, such as two -body vertex functions and high -order correlators. ...
A modified $GW$ approximation to many - body systems is developed. The approximation has the same computational complexity as the traditional $GW$ approach, but uses a different truncation scheme. This scheme neglects high order connected correlation functions. A covariant (preserving Ward identities due to charge conservation) scheme for two - body correlators is employed, which holds the relation between the charge correlator and charge susceptibility. The method is tested on the two - dimensional one - band Hubbard model. Results are compared with exact diagonalization, the fluctuation - exchange (FLEX) theory and determinantal quantum Monte Carlo (DQMC) approach. The comparison for the (one - body) Green's function demonstrates that it is more precise in strong - coupling regime (especially away from half - filling) than similar - complexity approximations $GW$ or FLEX. The charge correlator is in excellent agreement with the numerically exact result obtained from DQMC.
Electron dynamics in a two‐sites Hubbard model is studied using the nonequilibrium Green's function approach. The study is motivated by the empirical observation that a full solution of the integro‐differential Kadanoff–Baym equation (KBE) is more stable and often accompanied by artificial damping [Marc Puig von Friesen, C. Verdozzi, and C.‐O. Almbladh (2009)] than its time‐linear reformulations relying on the generalized Kadanoff–Baym ansatz (GKBA). Additionally, for conserving theories, numerical simulations suggest that KBE produces natural occupations bounded by one and zero in agreement with the Pauli exclusion principle, whereas, in some regimes, GKBA‐based theories violate this principle. As the first step for understanding these issues, the electron dynamics arising in the adiabatic switching scenario is studied. Many‐body approximations are classified according to the channel of the Bethe–Salpeter equation in which electronic correlations are explicitly treated. They give rise to the so‐called second Born, T ‐matrix, and GW approximations. In each of these cases, the model is reduced to a system of ordinary differential equations, which resemble equations of motion for a driven harmonic oscillator with time‐dependent frequencies. A more complete treatment of electronic correlations is achieved by combining different correlation channels, with parquet theory serving as a starting point.
The Matsubara Green’s function formalism stands as a powerful technique for computing the thermodynamic characteristics of interacting quantum many-particle systems at finite temperatures. In this manuscript, our focus centers on introducing MatsubaraFunctions.jl, a Julia library that implements data structures for generalized n-point Green’s functions on Matsubara frequency grids. The package’s architecture prioritizes user-friendliness without compromising the development of efficient solvers for quantum field theories in equilibrium. Following a comprehensive introduction of the fundamental types, we delve into a thorough examination of key facets of the interface. This encompasses avenues for accessing Green’s functions, techniques for extrapolation and interpolation, as well as the incorporation of symmetries and a variety of parallelization strategies. Examples of increasing complexity serve to demonstrate the practical utility of the library, supplemented by discussions on strategies for sidestepping impediments to optimal performance.
В безмассовых квантово-полевых теориях в логарифмических размерностях нарушается скейлинг. Рассматривается возможность интерпретации этого эффекта как спонтанного появления массы в рамках скелетных уравнений самосогласования с полным пропагатором в моделях с взаимодействиями $\varphi^3$, $\varphi^4$, $\varphi^6$ скалярного поля $\varphi$.
Systems with strong electronic Coulomb correlations often display rich phase diagrams exhibiting different ordered phases involving spin, charge, or orbital degrees of freedom. The theoretical description of the interplay of the corresponding collective fluctuations giving rise to this phenomenology, however, remains a tremendous challenge. Here, we introduce a multichannel extension of the recently developed fluctuating field approach to competing collective fluctuations in correlated electron systems. The method is based on a variational optimization of a trial action that explicitly contains the order parameters of the leading fluctuation channels. It gives direct access to the free energy of the system, facilitating the distinction between stable and metastable phases of the system. We apply our approach to the extended Hubbard model in the weak to intermediate coupling regime where we find it to capture the interplay of competing charge density wave and antiferromagnetic fluctuations with qualitative agreement with more computationally expensive methods. The multichannel fluctuating field approach thus offers a promising route for a numerically low-cost treatment of the interplay between collective fluctuations in small to large systems.
In recent years, Green’s function methods have garnered considerable interest due to their ability to target both charged and neutral excitations. Among them, the well-established GW approximation provides accurate ionization potentials and electron affinities and can be extended to neutral excitations using the Bethe–Salpeter equation (BSE) formalism. Here, we investigate the connections between various Green’s function methods and evaluate their performance for charged and neutral excitations. Comparisons with other widely known second-order wave function methods are also reported. Additionally, we calculate the singlet-triplet gap of cycl[3,3,3]azine, a model molecular emitter for thermally activated delayed fluorescence, which has the particularity of having an inverted gap thanks to a substantial contribution from the double excitations. We demonstrate that, within the GW approximation, a second-order BSE kernel with dynamical correction is required to predict this distinctive characteristic.
Nonlocal correlations play an essential role in correlated electron systems, especially in the vicinity of phase transitions and crossovers, where two-particle correlation functions display a distinct momentum dependence. In nonequilibrium settings, the effect of nonlocal correlations on dynamical phase transitions, prethermalization phenomena, and trapping in metastable states is not well understood. In this paper, we introduce a dynamical mean field theory (DMFT) extension to the nonequilibrium two-particle self-consistent (TPSC) approach, which allows to perform nonequilibrium simulations capturing short- and long-ranged nonlocal correlations in the weak-, intermediate-, and strong-correlation regimes. The method self-consistently computes local spin and charge vertices, from which a momentum-dependent self-energy is constructed. Replacing the local part of the self-energy by the DMFT result within this self-consistent scheme provides an improved description of local correlation effects. We explain the details of the formalism and the implementation, and demonstrate the versatility of DMFT+TPSC with interaction quenches and dimensional crossovers in the Hubbard model.
The family of Green's function methods based on the GW approximation has gained popularity in the electronic structure theory thanks to its accuracy in weakly correlated systems combined with its cost-effectiveness. Despite this, self-consistent versions still pose challenges in terms of convergence. A recent study [Monino and Loos J. Chem. Phys. 2022, 156, 231101.] has linked these convergence issues to the intruder-state problem. In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods. The SRG formalism enables us to derive, from first-principles, the expression of a naturally static and Hermitian form of the self-energy that can be employed in quasiparticle self-consistent GW (qsGW) calculations. The resulting SRG-based regularized self-energy significantly accelerates the convergence of qsGW calculations, slightly improves the overall accuracy, and is straightforward to implement in existing code.
The energy density functional (EDF) method is currently the only microscopic theoretical approach able to tackle the entire nuclear chart. Nevertheless, it suffers from limitations resulting from its empirical character and deteriorating its reliability. This paper is part of a larger program that aims at formulating the EDF approach as an effective field theory (EFT) in order to overcome these limitations. A relevant framework to achieve this is the path-integral formulation of quantum field theory. The latter indeed provides a wide variety of treatments of the many-body problem well suited to deal with non-perturbative interactions and to exploit a Lagrangian resulting from an EFT as a starting point. While developing the formalism in a general setting, we present below a comparative study of such techniques applied to a toy model, i.e. the (0 + 0)-D O(N)-symmetric \(\varphi ^{4}\)-theory. More specifically, our focus will be on the following diagrammatic techniques: loop expansion, optimized perturbation theory and self-consistent perturbation theory. With these methods, we notably address the spontaneous breakdown of the O(N) symmetry with care especially since spontaneous symmetry breakings play a paramount role in current implementations of the EDF approach.
We illustrate the algorithmic advantages of the recently introduced single-boson exchange (SBE) formulation for the one-loop functional renormalization group (fRG), by applying it to the two-dimensional Hubbard model on a square lattice. We present a detailed analysis of the fermion-boson Yukawa couplings and of the corresponding physical susceptibilities by studying their evolution with temperature and interaction strength, both at half filling and finite doping. The comparison with the conventional fermionic fRG decomposition shows that the rest functions of the SBE algorithm, which describe correlation effects beyond the SBE processes, play a negligible role in the weak-coupling regime above the pseudo-critical temperature, in contrast to the rest functions of the conventional fRG. Remarkably, they remain finite also at the pseudo-critical transition, whereas the corresponding rest functions of the conventional fRG implementation diverge. As a result, the SBE formulation of the fRG flow allows for a substantial reduction of the numerical effort in the treatment of the two-particle vertex function, paving a promising route for future multiboson and multiloop extensions.
Graphical abstract
The degrees of freedom that confer to strongly correlated systems their many intriguing properties also render them fairly intractable through typical perturbative treatments. For this reason, the mechanisms responsible for these technologically promising properties remain mostly elusive. Computational approaches have played a major role in efforts to fill this void. In particular, dynamical mean field theory (DMFT) and its cluster extension, the dynamical cluster approximation (DCA) have allowed significant progress. However, despite all the insightful results of these embedding schemes, computational constraints, such as the minus sign problem in Quantum Monte Carlo (QMC), and the exponential growth of the Hilbert space in exact diagonalization (ED) methods, still limit the length scale within which correlations can be treated exactly in the formalism. A recent advance to overcome these difficulties is the development of multiscale many body approaches whereby this challenge is addressed by introducing an intermediate length scale between the short length scale where correlations are treated exactly using a cluster solver such QMC or ED, and the long length scale where correlations are treated in a mean field manner. At this intermediate length scale correlations can be treated perturbatively. This is the essence of multiscale many-body methods. We will review various implementations of these multiscale many-body approaches, the results they have produced, and the outstanding challenges that should be addressed for further advances.
To efficiently capture the energy of the nuclear bond, advanced nuclear reactor concepts seek solid fuels that must withstand unprecedented temperature and radiation extremes. In these advanced fuels, thermal energy transport under irradiation is directly related to reactor performance as well as reactor safety. The science of thermal transport in nuclear fuel is a grand challenge as a result of both computational and experimental complexities. Here we provide a comprehensive review of thermal transport research on two actinide oxides: one currently in use in commercial nuclear reactors, uranium dioxide (UO2), and one advanced fuel candidate material, thorium dioxide (ThO2). In both materials, heat is carried by lattice waves or phonons. Crystalline defects caused by fission events effectively scatter phonons and lead to a degradation in fuel performance over time. Bolstered by new computational and experimental tools, researchers are now developing the foundational work necessary to accurately model and ultimately control thermal transport in advanced nuclear fuels. We begin by reviewing research aimed at understanding thermal transport in perfect single crystals. The absence of defects enables studies that focus on the fundamental aspects of phonon transport. Next, we review research that targets defect generation and evolution. Here the focus is on ion irradiation studies used as surrogates for damage caused by fission products. We end this review with a discussion of modeling and experimental efforts directed at predicting and validating mesoscale thermal transport in the presence of irradiation defects. While efforts in these research areas have been robust, challenging work remains in developing holistic tools to capture and predict thermal energy transport across widely varying environmental conditions.
A modified GW approximation to many-body systems is developed. The approximation has the same computational complexity as the traditional GW approach, but uses a different truncation scheme. This scheme neglects the high-order connected correlation functions. A covariant (preserving the Ward identities due to the charge conservation) scheme for the two-body correlators is employed, which holds the relation between the charge correlator and the charge susceptibility. The method is tested on the two-dimensional one-band Hubbard model. The results are compared with exact diagonalization, the GW approximation, the fluctuation-exchange (FLEX) theory, and determinantal Monte Carlo approach. The comparison for the (one-body) Green's function demonstrates that it is more precise in the strong-coupling regime (especially away from half filling) than the GW and FLEX approximations, which have a similar complexity. More importantly, this method indicates a Mott-Hubbard gap as the Hubbard U increases, whereas the GW and FLEX methods fail. In addition, the charge correlator obtained from the covariant scheme not only holds the consistency of the static charge susceptibility, but also makes a significant improvement over the random phase approximation calculations.
We develop the general many-body perturbation theory for a superconducting quantum dot represented by a single-impurity Anderson model attached to superconducting leads. We build our approach on a thermodynamically consistent mean-field approximation with a two-particle self-consistency of the parquet type. The two-particle self-consistency leading to a screening of the bare interaction proves substantial for suppressing the spurious transitions of the Hartree-Fock solution. We demonstrate that the magnetic field plays a fundamental role in the extension of the perturbation theory beyond the weakly correlated 0 phase. It controls the critical behavior of the 0−π quantum transition and lifts the degeneracy in the π phase, where the limits to zero temperature and zero magnetic field do not commute. The response to the magnetic field is quite different in 0 and π phases. While the magnetic susceptibility vanishes in the 0 phase it becomes divergent in the π phase at zero temperature.
We combine two non-perturbative approaches, one based on the two-particle-irreducible (2PI) action, the other on the functional renormalization group (fRG), in an effort to develop new non-perturbative approximations for the field theoretical description of strongly coupled systems. In particular, we exploit the exact 2PI relations between the two-point and four-point functions in order to truncate the infinite hierarchy of equations of the functional renormalization group. The truncation is ”exact” in two ways. First, the solution of the resulting flow equation is independent of the choice of the regulator. Second, this solution coincides with that of the 2PI equations for the two-point and the four-point functions, for any selection of two-skeleton diagrams characterizing a so-called Φ-derivable approximation. The transformation of the equations of the 2PI formalism into flow equations offers new ways to solve these equations in practice, and provides new insight on certain aspects of their renormalization. It also opens the possibility to develop approximation schemes going beyond the strict Φ-derivable ones, as well as new truncation schemes for the fRG hierarchy.
A systematic generalization of the Mayer cluster integral theory has been developed to deal with the quantum statistics of interacting particles. The grand partition function appears in a natural way and the cluster integrals are integrals over propagators which are derived from the Green's function solution of the Bloch equation (which follows from the Schroedinger equation by replacing it/ℏ by β = 1/kT). Every cluster integral can be represented by a hybrid of a Mayer graph and a Feynman diagram in (β, r) space.
The generalization of classical ring cluster integrals has been analyzed. It is shown that in the case of the electron gas the classical limit of the contribution of these integrals to the grand partition function yields the Debye-Huckel theory while the low temperature limit leads to the Gell-Mann—Brueckner equation for the correlation energy of the ground state.
A prescription is given for the construction of the cluster integral associated with any given diagram.
A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain U functions defined for the same system with Boltzmann statistics. It is then shown that these U functions can be evaluated in successive approximations in terms of a binary kernel B which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.
Starting from Rules A and B of a previous paper (I), it is shown that the grand partition function can be evaluated in terms of the statistical averages of the occupation number in momentum space. The final formulation is in terms of a simple variational principle. The procedure represents a concise and complete separation of the effect of the Bose-Einstein or Fermi-Dirac statistical character of the particles from the dynamical problem. In the case of Bose statistics, this formulation makes possible a systematic computation of all thermodynamic functions near the Bose-Einstein transition point in the gaseous phase. Applications to a system of hard spheres are discussed.
The formal cell-cluster theory for a binary liquid solution is given. The configurational partition function is expanded in terms of cluster integrals which are small if the molecules are mainly confined to the lattice sites of a virtual lattice system. The evaluation of the partition function is performed in the quasi-chemical as well as in the Bragg-Williams approximation. The methods used are very similar to those employed by Guggenheim in his studies on polymer solutions. When only single cells are taken into account a generalization of the partition function used by Prigogine and his co-workers is obtained. The corrections due to the introduction of clusters of two cells are given and applied to a few simple cases.The same formalism can be used to construct the cell-cluster theory for a liquid with a fixed number of holes. The necessary modifications are indicated and the results given for the case that only clusters of one and two cells are considered. As an example a one dimensional ideal gas with holes is treated.
DOI:https://doi.org/10.1103/PhysRev.105.1119
This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.
Ideas and techniques known in quantum electrodynamics have been applied to the Bardeen-Cooper-Schrieffer theory of superconductivity. In an approximation which corresponds to a generalization of the Hartree-Fock fields, one can write down an integral equation defining the self-energy of an electron in an electron gas with phonon and Coulomb interaction. The form of the equation implies the existence of a particular solution which does not follow from perturbation theory, and which leads to the energy gap equation and the quasi-particle picture analogous to Bogoliubov's. The gauge invariance, to the first order in the external electromagnetic field, can be maintained in the quasi-particle picture by taking into account a certain class of corrections to the chargecurrent operator due to the phonon and Coulomb interaction. In fact, generalized forms of the Ward identity are obtained between certain vertex parts and the self-energy. The Meissner effect calculation is thus rendered strictly gauge invariant, but essentially keeping the BCS result unaltered for transverse fields. It is shown also that the integral equation for vertex parts allows homogeneous solutions which describe collective excitations of quasi-particle pairs, and the nature and effects of such collective states are discussed.
The formulation of the author's previous paper is extended so that it ; becomes applicable in an interacting system in the presence of a Bose-Einstein ; degeneracy. This extension is carried out by the introduction of an x-ensemble, ; which enables one to utilize an Ursell-type expansion even in the presence of a ; Bose-Einstein degeneracy. The variational principle of the previous paper is ; also extended. It is proved that in the presence of a Bose-Einstein degeneracy, ; the average occupation number of a single particle state with momentum p ; approaches infinity as p yields 0. The method is applied to a dilute system of ; Bose hard spheres. (auth);
In this paper properties of a boson gas at zero temperature are investigated by means of field-theoretic methods. Difficulties arising from the depletion of the ground state are resolved in a simple way by the elimination of the zero-momentum state. The result of this procedure when applied to the calculation of the Green's functions of the system is identical to that of Beliaev. It is then shown generally that for a repulsive interaction the energy E(k) of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap. The Green's function method is applied to the calculation of the properties of a low-density boson gas. The next order term beyond that calculated by Lee and Yang, and Beliaev for the ground-state energy is obtained and the general form of the series expansion is found to be (E0Omega)=12n2f0[1+a(nf03)12+b(nf03)lnnf03+c(nf03)+d(nf03)32ln(nf03)+...], where n is the density and f0 is the scattering length for the assumed two-body interaction between the bosons. The coefficients a and b are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient b is in agreement with the hard-sphere gas calculations of Wu and of Sawada. A discussion is given of the intermediate-density calculation of Brueckner and Sawada, and certain possible improvements in the method of summing a selected set of higher-order terms are proposed.
The perturbation series for the ground-state energy of a many-fermion ; system was investigated to arbitrary order for the "isotropic" case. This is the ; case of over-all spherical symmetry. both in the interaction and in the ; unperturbed single particle energies. It was shown that for spin one-half ; feninions the Brueckner-Goldstone perturbation series is valid to all orders in ; the perturbation. For spins greater than one-half it is in general incorrect ; even in the isotropic case. unless the interactions are spin independent. The ; discussion to arbitrary order in the interaction ts carried out by means of a ; Feynman-like propagator formalism, which is developed in detail. (auth);
Thermodynamical functions for classical and quantum systems are expressed in terms of the one‐particle density n1 and the two‐particle correlation matrix C12 (or quantities in direct relation to them). Use is made of topological relations valid for the diagram representations of the grand partition function expansions. The result considered as a functional of n1 and C12 is stationary under independent variations δn1 and δC12. In particular, the entropy functional of a classical system no longer contains any reference to the equilibrium parameters (or to the interactions) and the second functional derivative is a negative definite matrix. The entropy functional of a quantum system conserves traces of the equilibrium parameters in the Lee‐Yang formulation; the Green's function formulation does not, but in this case the second functional derivative is no longer a negative definite matrix.
It is shown that for a system composed of N identical molecules with mutual potential energy, the assumption that the total potential energy can be expressed as the sum of that between pairs of molecules allows the derivation of simple, accurate formal equations for the thermodynamic properties of the system. Under certain conditions, generally fulfilled at low temperatures, the equations predict a region where the pressure and Gibb's free energy are independent of volume, the characteristic of condensing systems. The equations permit calculation of the Gibb's free energy of the liquid in equilibrium with the vapor and all the properties of the saturated vapor, but not the volume or volume dependence of the condensed phase.
A classical system enclosed in a finite volume and exerted by external
forces is treated in quite a general way. The main results, which may be
applicable to solids as well as to fluids, are as follows. Exact
integral equations are found for the one- and two-particle distribution
functions. Some thermodynamic functions are expressed in terms of these
distribution functions. It is shown that there exist variational
principles saying that the grand partition function is to be maximum
with respect to the variations in the one- and two-particle distribution
functions. Variational principles are found also for the Helmholtz free
energy. It is suggested that Mayer's theory of condensation may in fact
give the end point of the metastable gaseous state. It is pointed out
that the hyper-netted chain approximation, proposed previously by one of
the present auhors, has a meaning in solids as well as in fluids.
A theory of superconductivity is presented, based on the fact that the interaction between electrons resulting from virtual exchange of phonons is attractive when the energy difference between the electrons states involved is less than the phonon energy, ℏω. It is favorable to form a superconducting phase when this attractive interaction dominates the repulsive screened Coulomb interaction. The normal phase is described by the Bloch individual-particle model. The ground state of a superconductor, formed from a linear combination of normal state configurations in which electrons are virtually excited in pairs of opposite spin and momentum, is lower in energy than the normal state by amount proportional to an average (ℏω)2, consistent with the isotope effect. A mutually orthogonal set of excited states in one-to-one correspondence with those of the normal phase is obtained by specifying occupation of certain Bloch states and by using the rest to form a linear combination of virtual pair configurations. The theory yields a second-order phase transition and a Meissner effect in the form suggested by Pippard. Calculated values of specific heats and penetration depths and their temperature variation are in good agreement with experiment. There is an energy gap for individual-particle excitations which decreases from about 3.5kTc at T=0°K to zero at Tc. Tables of matrix elements of single-particle operators between the excited-state superconducting wave functions, useful for perturbation expansions and calculations of transition probabilities, are given.
This paper investigates the criteria for maintenance of the macroscopic conservation laws of number, momentum, and energy by approximate two-particle correlation functions in many-body systems. The methods of generating such approximations are the same as in a previous paper. However, the derivations of the conservation laws given here clarify both why the approximation method works and the connection between the macroscopic conservation laws and those at the vertices. Conserving nonequilibrium approximations are based on self-consistent approximations to the one-particle Green's function. The same condition that ensures that the nonequilibrium theory be conserving also ensures that the equilibrium approximation has the following properties. The several common methods for determining the partition function from the one-particle Green's function all lead to the same result. When applied to a zero-temperature normal fermion system, the approximation procedure maintains the Hugenholtz-Van Hove theorem. Consequently, the self-consistent version of Brueckner's nuclear matter theory obeys this theorem.
Statistical mechanics is concerned primarily with what are known as “normal properties” of assemblies. The underlying idea is that of the generalised phase-space. The configuration of an assembly is determined (on classical mechanics) by a certain number of pairs of Hamiltonian canonical coordinates p, q, which are the coordinates of the phase-space referred to. Liouville's theorem leads us to take the element of volume dτ=Πdp dq as giving the correct element of a priori probability. Any isolated assembly is confined to a surface in the phase-space, for its energy at least is constant; when there are no other uniform integrals of the equations of motion, the actual probability of a given aggregate of states of the proper energy, i.e., of a given portion of the surface, varies as the volume, in the neighbourhood of points of this portion, included between two neighbouring surfaces of constant energies E, E + dE; it therefore varies as the integral of (∂E/∂n)−1 taken over the portion. If I be the measure of the total phase-space available, interpreted in this way, and i that of the portion in which some particular condition is satisfied, then i/I is the probability of that condition being satisfied.(Received February 01 1927)(Accepted January 31 1927)
A physical interpretation of the generalized free energy , introduced in part I of this work in connection with the variational principle, is obtained by relating it to the statistical distribution of the occupation numbers. The values of the occupation numbers derived from the variational principle are then interpreted as the most probable values in a grand canonical ensemble of identical systems. As a particular consequence of the theory, it is shown that in the case of an infinite system without Bose condensation, the probability distribution of any occupation number has the same form as in the absence of interactions. The classical limit of the theory is then considered. First, the freat similarity of our results with the classical expansions expressed in terms of the local density is exhibited by rederiving the well known classical expressions with the methods of part I. Finally the classical limit of the Gibbs potential is shown to be identical, term by term, with the classical virial expansion.
A Green's function method in quantum statistics is developed. It is shown that the equations obtained contain in a simple approximation various methods encountered in statistical physics and in the theory of many particles as well as their extensions to cases of non-vanishing temperature (e.g., the Debye-Hückel, Hartree-Fock, Thomas-Fermi, Gell-Mann-Brueckner methods). A transition to time-dependent Green's functions is considered and a method for the determination of the energy spectrum of the system is proposed.
The subject of discussion is the Hamiltonian for a system of bosons interacting by two body forces, as expressed in the formalism of second quantization. In this paper, we examine properties of the classical wave field governed by the Hamiltonian. For a general potential there is always an exact solution representing a uniform density. Exact solutions are exhibited, which represent disturbances of a definite velocity and of arbitrary amplitude. For small amplitudes the disturbances obey Bogolyubov's dispersion relation. Corresponding solutions are found for disturbances when the system moves as a whole. For suitably attractive potentials we find a class of exact solutions, degenerate in energy, with spatially periodic density. These solutions have a lower energy than the uniform type. Small amplitude excitations are investigated for the periodic case. They are phonons for long wavelengths, but show a band character at shorter wavelengths. A theory of the motion of foreign atoms in the boson fluid is formulated.