arXiv:1405.5291 · Journal of the Physical Society of Japan

Effective Action for Bose–Einstein Condensates

Takafumi Kita

Published May 21, 2014• 8 citations

Abstract

We clarify basic properties of an effective action (i.e., self-consistent perturbation expansion) for interacting Bose-Einstein condensates, where field $ψ$ itself acquires a finite thermodynamic average $\langle ψ\rangle$ besides two-point Green's function $\hat G$ to form an off-diagonal long-range order. It is shown that the action can be expressed concisely order by order in terms of the interaction vertex and a special combination of $\langleψ\rangle$ and $\hat G$ so as to satisfy both Noether's theorem and Goldstone's theorem (I) corresponding to the first proof. The self-energy is predicted to have a one-particle-reducible structure due to $\langle ψ\rangle\neq 0$ to transform the Bogoliubov mode into a bubbling mode with a substantial decay rate.

DOI: 10.7566/jpsj.83.064005

Related papers

A Renormalization-Group Study of Interacting Bose–Einstein Condensates: Absence of the Bogoliubov Mode below Four ( T > 0) and Three ( T = 0) Dimensions

Takafumi Kita

arXiv:1903.05230Journal of the Physical Society of JapanApr 16, 20196 citations

Self-consistent perturbation expansion for Bose-Einstein condensates satisfying Goldstone’s theorem and conservation laws

Takafumi Kita

arXiv:0911.3681Physical Review BDec 3, 200927 citations

Quantum Field Theory of Correlated Bose–Einstein Condensates: I. Basic Formalism

Takafumi Kita

arXiv:2101.02389Journal of the Physical Society of JapanJan 7, 20214 citations

Nonequilibrium dynamics of condensates in a lattice with the two-particle-irreducible effective action in the 1∕N expansion

Kristan Temme, Thomas Gasenzer

arXiv:cond-mat/0607116Physical Review ANov 2, 200616 citations

Green's-function formalism for a condensed Bose gas consistent with infrared-divergent longitudinal susceptibility and Nepomnyashchii-Nepomnyashchii identity

Shohei Watabe, Yoji Ohashi

arXiv:1403.0005Physical Review AJul 3, 20148 citations

Perturbation theory of spin-1 Bose-Einstein condensates

Shingo Ohtsuka, Susumu Kurihara

Physical Review AJul 7, 20036 citations

arXiv:1405.5291 · Journal of the Physical Society of Japan

Effective Action for Bose–Einstein Condensates

Takafumi Kita

Published May 21, 2014• 8 citations

arXiv HTML PDF

Abstract

We clarify basic properties of an effective action (i.e., self-consistent perturbation expansion) for interacting Bose-Einstein condensates, where field

ψ

itself acquires a finite thermodynamic average

\langle ψ\rangle

besides two-point Green's function

\hat G

to form an off-diagonal long-range order. It is shown that the action can be expressed concisely order by order in terms of the interaction vertex and a special combination of

\langleψ\rangle

and

\hat G

so as to satisfy both Noether's theorem and Goldstone's theorem (I) corresponding to the first proof. The self-energy is predicted to have a one-particle-reducible structure due to

\langle ψ\rangle\neq 0

to transform the Bogoliubov mode into a bubbling mode with a substantial decay rate.

DOI: 10.7566/jpsj.83.064005

Related papers

A Renormalization-Group Study of Interacting Bose–Einstein Condensates: Absence of the Bogoliubov Mode below Four ( T > 0) and Three ( T = 0) Dimensions

Takafumi Kita

We derive exact renormalization-group equations for the $n$ -point vertices ( $n=0,1,2,\cdots$ ) of interacting single-component Bose-Einstein condensates based on the vertex expansion of the effective action. They have a notable feature of automatically satisfying Goldstone's theorem (I), which yields the Hugenholtz-Pines relation $Σ(0)-μ=Δ(0)$ as the lowest-order identity. Using them, it is found that the anomalous self-energy $Δ(0)$ vanishes below $d_{\rm c}=4$ ( $d_{\rm c}=3$ ) dimensions at finite temperatures (zero temperature), contrary to the Bogoliubov theory predicting a finite "sound-wave" velocity $v_{\rm s}\propto [Δ(0)]^{1/2}>0$ . It is also argued that the one-particle density matrix $ρ({\bf r})\equiv\langle\hatψ^\dagger({\bf r}_1)\hatψ({\bf r}_1+{\bf r})\rangle$ for $d0$ . The anomalous dimension $η$ at finite temperatures is predicted to behave for $d=4-ε$ dimensions ($0We derive exact renormalization-group equations for the $n$-point vertices ($n=0,1,2,\cdots$) of interacting single-component Bose-Einstein condensates based on the vertex expansion of the effective action. They have a notable feature of automatically satisfying Goldstone's theorem (I), which yields the Hugenholtz-Pines relation $Σ(0)-μ=Δ(0)$ as the lowest-order identity. Using them, it is found that the anomalous self-energy $Δ(0)$ vanishes below $d_{\rm c}=4$ ($d_{\rm c}=3$) dimensions at finite temperatures (zero temperature), contrary to the Bogoliubov theory predicting a finite "sound-wave" velocity $v_{\rm s}\propto [Δ(0)]^{1/2}>0$. It is also argued that the one-particle density matrix $ρ({\bf r})\equiv\langle\hatψ^\dagger({\bf r}_1)\hatψ({\bf r}_1+{\bf r})\rangle$ for $d0$. The anomalous dimension $η$ at finite temperatures is predicted to behave for $d=4-ε$ dimensions ($0

arXiv:1903.05230Journal of the Physical Society of JapanApr 16, 20196 citations

Self-consistent perturbation expansion for Bose-Einstein condensates satisfying Goldstone’s theorem and conservation laws

Takafumi Kita

Quantum-field-theoretic descriptions of interacting condensed bosons have suffered from the lack of self-consistent approximation schemes satisfying Goldstone's theorem and dynamical conservation laws simultaneously. We present a procedure to construct such approximations systematically by using either an exact relation for the interaction energy or the Hugenholtz-Pines relation to express the thermodynamic potential in a Luttinger-Ward form. Inspection of the self-consistent perturbation expansion up to the third order with respect to the interaction shows that the two relations yield a unique identical result at each order, reproducing the conserving-gapless mean-field theory [T. Kita, J. Phys. Soc. Jpn. 74, 1891 (2005)] as the lowest-order approximation. The uniqueness implies that the series becomes exact when infinite terms are retained. We also derive useful expressions for the entropy and superfluid density in terms of Green's function and a set of real-time dynamical equations to describe thermalization of the condensate.Quantum-field-theoretic descriptions of interacting condensed bosons have suffered from the lack of self-consistent approximation schemes satisfying Goldstone's theorem and dynamical conservation laws simultaneously. We present a procedure to construct such approximations systematically by using either an exact relation for the interaction energy or the Hugenholtz-Pines relation to express the thermodynamic potential in a Luttinger-Ward form. Inspection of the self-consistent perturbation expansion up to the third order with respect to the interaction shows that the two relations yield a unique identical result at each order, reproducing the conserving-gapless mean-field theory [T. Kita, J. Phys. Soc. Jpn. 74, 1891 (2005)] as the lowest-order approximation. The uniqueness implies that the series becomes exact when infinite terms are retained. We also derive useful expressions for the entropy and superfluid density in terms of Green's function and a set of real-time dynamical equations to describe thermalization of the condensate.

arXiv:0911.3681Physical Review BDec 3, 200927 citations

Quantum Field Theory of Correlated Bose–Einstein Condensates: I. Basic Formalism

Takafumi Kita

Quantum field theory of equilibrium and nonequilibrium Bose-Einstein condensates is formulated so as to satisfy three basic requirements: the Hugenholtz-Pines relation; conservation laws; identities among vertices originating from Goldstone's theorem I. The key inputs are irreducible four-point vertices, in terms of which we derive a closed system of equations for Green's functions, three- and four-point vertices, and two-particle Green's functions. It enables us to study correlated Bose-Einstein condensates with a gapless branch of single-particle excitations without encountering any infrared divergence. The single- and two-particle Green's functions are found to share poles, i.e., the structure of the two-particle Green's functions predicted by Gavoret and Nozières for a homogeneous condensate at $T=0$ is also shown to persist at finite temperatures, in the presence of inhomogeneity, and also in nonequilibrium situations.Quantum field theory of equilibrium and nonequilibrium Bose-Einstein condensates is formulated so as to satisfy three basic requirements: the Hugenholtz-Pines relation; conservation laws; identities among vertices originating from Goldstone's theorem I. The key inputs are irreducible four-point vertices, in terms of which we derive a closed system of equations for Green's functions, three- and four-point vertices, and two-particle Green's functions. It enables us to study correlated Bose-Einstein condensates with a gapless branch of single-particle excitations without encountering any infrared divergence. The single- and two-particle Green's functions are found to share poles, i.e., the structure of the two-particle Green's functions predicted by Gavoret and Nozières for a homogeneous condensate at $T=0$ is also shown to persist at finite temperatures, in the presence of inhomogeneity, and also in nonequilibrium situations.

arXiv:2101.02389Journal of the Physical Society of JapanJan 7, 20214 citations

Nonequilibrium dynamics of condensates in a lattice with the two-particle-irreducible effective action in the 1∕N expansion

Kristan Temme, Thomas Gasenzer

The dynamical evolution of a Bose-Einstein condensate trapped in a one-dimensional lattice potential is investigated theoretically in the framework of the Bose-Hubbard model. The emphasis is set on the far-from-equilibrium evolution in a case where the gas is strongly interacting. This is realized by an appropriate choice of the parameters in the Hamiltonian, and by starting with an initial state, where one lattice well contains a Bose-Einstein condensate while all other wells are empty. Oscillations of the condensate as well as noncondensate fractions of the gas between the different sites of the lattice are found to be damped as a consequence of the collisional interactions between the atoms. Functional integral techniques involving self-consistently determined mean fields as well as two-point correlation functions are used to derive the two-particle-irreducible (2PI) effective action. The action is expanded in inverse powers of the number of field components $\mathcal{N}$ , and the dynamic equations are derived from it to next-to-leading order in this expansion. This approach reaches considerably beyond the Hartree-Fock-Bogoliubov mean-field theory, and its results are compared to the exact quantum dynamics obtained by Rey et al. [Phys. Rev. A 69, 033610 (2004)] for small atom numbers.The dynamical evolution of a Bose-Einstein condensate trapped in a one-dimensional lattice potential is investigated theoretically in the framework of the Bose-Hubbard model. The emphasis is set on the far-from-equilibrium evolution in a case where the gas is strongly interacting. This is realized by an appropriate choice of the parameters in the Hamiltonian, and by starting with an initial state, where one lattice well contains a Bose-Einstein condensate while all other wells are empty. Oscillations of the condensate as well as noncondensate fractions of the gas between the different sites of the lattice are found to be damped as a consequence of the collisional interactions between the atoms. Functional integral techniques involving self-consistently determined mean fields as well as two-point correlation functions are used to derive the two-particle-irreducible (2PI) effective action. The action is expanded in inverse powers of the number of field components $\mathcal{N}$, and the dynamic equations are derived from it to next-to-leading order in this expansion. This approach reaches considerably beyond the Hartree-Fock-Bogoliubov mean-field theory, and its results are compared to the exact quantum dynamics obtained by Rey et al. [Phys. Rev. A 69, 033610 (2004)] for small atom numbers.

arXiv:cond-mat/0607116Physical Review ANov 2, 200616 citations

Green's-function formalism for a condensed Bose gas consistent with infrared-divergent longitudinal susceptibility and Nepomnyashchii-Nepomnyashchii identity

Shohei Watabe, Yoji Ohashi

We present a Green's-function formalism for an interacting Bose-Einstein condensate (BEC) satisfying the two required conditions: (i) the infrared-divergent longitudinal susceptibility with respect to the BEC order parameter, and (ii) the Nepomnyashchii-Nepomnyashchii identity stating the vanishing off-diagonal self-energy in the low-energy and low-momentum limit. These conditions cannot be described by the ordinary mean-field Bogoliubov theory, the many-body $T$ -matrix theory, or the random-phase approximation with the vertex correction. In this paper, we show that these required conditions can be satisfied, when we divide many-body corrections into singular and nonsingular parts, and separately treat them as different self-energy corrections. The resulting Green's function may be viewed as an extension of the Popov's hydrodynamic theory to the region at finite temperatures. Our results would be useful in constructing a consistent theory of BECs satisfying various required conditions, beyond the mean-field level.We present a Green's-function formalism for an interacting Bose-Einstein condensate (BEC) satisfying the two required conditions: (i) the infrared-divergent longitudinal susceptibility with respect to the BEC order parameter, and (ii) the Nepomnyashchii-Nepomnyashchii identity stating the vanishing off-diagonal self-energy in the low-energy and low-momentum limit. These conditions cannot be described by the ordinary mean-field Bogoliubov theory, the many-body $T$-matrix theory, or the random-phase approximation with the vertex correction. In this paper, we show that these required conditions can be satisfied, when we divide many-body corrections into singular and nonsingular parts, and separately treat them as different self-energy corrections. The resulting Green's function may be viewed as an extension of the Popov's hydrodynamic theory to the region at finite temperatures. Our results would be useful in constructing a consistent theory of BECs satisfying various required conditions, beyond the mean-field level.

arXiv:1403.0005Physical Review AJul 3, 20148 citations

Perturbation theory of spin-1 Bose-Einstein condensates

Shingo Ohtsuka, Susumu Kurihara

We generalize the perturbation theory of weakly interacting bosons to Bose-Einstein condensates with hyperfine spin $F=1.$ Analytical expressions of second-order self-energies beyond the mean-field approximation are derived at zero temperature. We further extend the perturbation theory to the finite-temperature case, and derive the corresponding self-energies. We evaluate these self-energies near the poles of the first-order (mean-field) Green's functions within the on-shell approximation, and expand the results in powers of wave number in the long-wavelength limit. Excitation spectra for each mode are obtained analytically at low temperatures. We also derive the ground-state energy and thermodynamic quantities such as free energy of the system.We generalize the perturbation theory of weakly interacting bosons to Bose-Einstein condensates with hyperfine spin $F=1.$ Analytical expressions of second-order self-energies beyond the mean-field approximation are derived at zero temperature. We further extend the perturbation theory to the finite-temperature case, and derive the corresponding self-energies. We evaluate these self-energies near the poles of the first-order (mean-field) Green's functions within the on-shell approximation, and expand the results in powers of wave number in the long-wavelength limit. Excitation spectra for each mode are obtained analytically at low temperatures. We also derive the ground-state energy and thermodynamic quantities such as free energy of the system.

Physical Review AJul 7, 20036 citations

We clarify basic properties of an effective action (i.e., self-consistent perturbation expansion) for interacting Bose-Einstein condensates, where field

ψ

itself acquires a finite thermodynamic average

\langle ψ\rangle

besides two-point Green's function

\hat G

to form an off-diagonal long-range order. It is shown that the action can be expressed concisely order by order in terms of the interaction vertex and a special combination of

\langleψ\rangle

and

\hat G

so as to satisfy both Noether's theorem and Goldstone's theorem (I) corresponding to the first proof. The self-energy is predicted to have a one-particle-reducible structure due to

\langle ψ\rangle\neq 0

to transform the Bogoliubov mode into a bubbling mode with a substantial decay rate.

We derive exact renormalization-group equations for the

n

-point vertices (

n=0,1,2,\cdots

) of interacting single-component Bose-Einstein condensates based on the vertex expansion of the effective action. They have a notable feature of automatically satisfying Goldstone's theorem (I), which yields the Hugenholtz-Pines relation

Σ(0)-μ=Δ(0)

as the lowest-order identity. Using them, it is found that the anomalous self-energy

Δ(0)

vanishes below

d_{\rm c}=4

(

d_{\rm c}=3

) dimensions at finite temperatures (zero temperature), contrary to the Bogoliubov theory predicting a finite "sound-wave" velocity

v_{\rm s}\propto [Δ(0)]^{1/2}>0

. It is also argued that the one-particle density matrix

ρ({\bf r})\equiv\langle\hatψ^\dagger({\bf r}_1)\hatψ({\bf r}_1+{\bf r})\rangle

for

d0

. The anomalous dimension

η

at finite temperatures is predicted to behave for

d=4-ε

dimensions ($0

T=0

is also shown to persist at finite temperatures, in the presence of inhomogeneity, and also in nonequilibrium situations.

\mathcal{N}

, and the dynamic equations are derived from it to next-to-leading order in this expansion. This approach reaches considerably beyond the Hartree-Fock-Bogoliubov mean-field theory, and its results are compared to the exact quantum dynamics obtained by Rey et al. [Phys. Rev. A 69, 033610 (2004)] for small atom numbers.

T

-matrix theory, or the random-phase approximation with the vertex correction. In this paper, we show that these required conditions can be satisfied, when we divide many-body corrections into singular and nonsingular parts, and separately treat them as different self-energy corrections. The resulting Green's function may be viewed as an extension of the Popov's hydrodynamic theory to the region at finite temperatures. Our results would be useful in constructing a consistent theory of BECs satisfying various required conditions, beyond the mean-field level.

We generalize the perturbation theory of weakly interacting bosons to Bose-Einstein condensates with hyperfine spin

F=1.

Analytical expressions of second-order self-energies beyond the mean-field approximation are derived at zero temperature. We further extend the perturbation theory to the finite-temperature case, and derive the corresponding self-energies. We evaluate these self-energies near the poles of the first-order (mean-field) Green's functions within the on-shell approximation, and expand the results in powers of wave number in the long-wavelength limit. Excitation spectra for each mode are obtained analytically at low temperatures. We also derive the ground-state energy and thermodynamic quantities such as free energy of the system.