arXiv:2101.02396 · Journal of the Physical Society of Japan

Quantum Field Theory of Correlated Bose–Einstein Condensates: II. Ward–Takahashi Identities and Correlation Functions

Takafumi Kita

Published Jan 7, 2021• 3 citations

Abstract

We derive Ward-Takahashi identities for correlated Bose-Einstein condensates based on the expressions of the first-order variations $(δΨ,δG)$ due to perturbations obtained in the preceding paper [T. Kita, J. Phys. Soc. Jpn. $\bf 90$, 024001 (2021)] for the condensate wave function $Ψ$ and Green's function $G$. They enable us to obtain several exact results on the density and current correlation functions $K_{νν'}^{}$, and also express $K_{νν'}^{}$ in terms of low-energy Green's functions and vertices. The latter expressions open up the possibility of constructing theory of superfluid Bose liquids in the same way as that for fermions at low temperatures. The vertices are found to have different limits depending on which of frequency $ω$ and wavenumber $q$ is set equal to zero first.

DOI: 10.7566/jpsj.90.024002

Related papers

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

Conserving Gapless Mean-Field Theory for Weakly Interacting Bose Gases

Takafumi Kita

arXiv:cond-mat/0604241Journal of the Physical Society of JapanApr 13, 200627 citations

Perturbation theory of spin-1 Bose-Einstein condensates

Shingo Ohtsuka, Susumu Kurihara

Physical Review AJul 7, 20036 citations

Phase Invariant Perturbation Scheme in Bose-Einstein Condensed States

H. Matsumoto, S. Sakamoto

Progress of Theoretical PhysicsApr 1, 20021 citation

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

Extended self-energy functional approach for strongly correlated lattice bosons in the superfluid phase

Enrico Arrigoni, Michael Knap, Wolfgang von der Linden

arXiv:1103.3664Physical Review BJul 29, 201117 citations

arXiv:2101.02396 · Journal of the Physical Society of Japan

Quantum Field Theory of Correlated Bose–Einstein Condensates: II. Ward–Takahashi Identities and Correlation Functions

Takafumi Kita

Published Jan 7, 2021• 3 citations

arXiv HTML PDF

Abstract

We derive Ward-Takahashi identities for correlated Bose-Einstein condensates based on the expressions of the first-order variations

(δΨ,δG)

due to perturbations obtained in the preceding paper [T. Kita, J. Phys. Soc. Jpn.

\bf 90

, 024001 (2021)] for the condensate wave function

Ψ

and Green's function

G

. They enable us to obtain several exact results on the density and current correlation functions

K_{νν'}^{}

, and also express

K_{νν'}^{}

in terms of low-energy Green's functions and vertices. The latter expressions open up the possibility of constructing theory of superfluid Bose liquids in the same way as that for fermions at low temperatures. The vertices are found to have different limits depending on which of frequency

ω

and wavenumber

q

is set equal to zero first.

DOI: 10.7566/jpsj.90.024002

Related papers

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

Conserving Gapless Mean-Field Theory for Weakly Interacting Bose Gases

Takafumi Kita

This paper presents a conserving gapless mean-field theory for weakly interacting Bose gases. We first construct a mean-field Luttinger-Ward thermodynamic functional in terms of the condensate wave function $Ψ$ and the Nambu Green's function $\hat{G}$ for the quasiparticle field. Imposing its stationarity respect to $Ψ$ and $\hat{G}$ yields a set of equations to determine the equilibrium for general non-uniform systems. They have a plausible property of satisfying the Hugenholtz-Pines theorem to provide a gapless excitation spectrum. Also, the corresponding dynamical equations of motion obey various conservation laws. Thus, the present mean-field theory shares two important properties with the exact theory: ``conserving'' and ``gapless.'' The theory is then applied to a homogeneous weakly interacting Bose gas with s-wave scattering length $a$ and particle mass $m$ to clarify its basic thermodynamic properties under two complementary conditions of constant density $n$ and constant pressure $p$ . The superfluid transition is predicted to be first-order because of the non-analytic nature of the order-parameter expansion near $T_{c}$ inherent in Bose systems, i.e., the Landau-Ginzburg expansion is not possible here. The transition temperature $T_{c}$ shows quite a different interaction dependence between the $n$ -fixed and $p$ -fixed cases. In the former case $T_{c}$ increases from the ideal gas value $T_{0}$ as $T_{c}/T_{0}= 1+ 2.33 an^{1/3}$ , whereas it decreases in the latter as $T_{c}/T_{0}= 1- 3.84a(mp/2π\hbar^{2})^{1/5}$ . Temperature dependences of basic thermodynamic quantities are clarified explicitly.This paper presents a conserving gapless mean-field theory for weakly interacting Bose gases. We first construct a mean-field Luttinger-Ward thermodynamic functional in terms of the condensate wave function $Ψ$ and the Nambu Green's function $\hat{G}$ for the quasiparticle field. Imposing its stationarity respect to $Ψ$ and $\hat{G}$ yields a set of equations to determine the equilibrium for general non-uniform systems. They have a plausible property of satisfying the Hugenholtz-Pines theorem to provide a gapless excitation spectrum. Also, the corresponding dynamical equations of motion obey various conservation laws. Thus, the present mean-field theory shares two important properties with the exact theory: ``conserving'' and ``gapless.'' The theory is then applied to a homogeneous weakly interacting Bose gas with s-wave scattering length $a$ and particle mass $m$ to clarify its basic thermodynamic properties under two complementary conditions of constant density $n$ and constant pressure $p$. The superfluid transition is predicted to be first-order because of the non-analytic nature of the order-parameter expansion near $T_{c}$ inherent in Bose systems, i.e., the Landau-Ginzburg expansion is not possible here. The transition temperature $T_{c}$ shows quite a different interaction dependence between the $n$-fixed and $p$-fixed cases. In the former case $T_{c}$ increases from the ideal gas value $T_{0}$ as $T_{c}/T_{0}= 1+ 2.33 an^{1/3}$, whereas it decreases in the latter as $T_{c}/T_{0}= 1- 3.84a(mp/2π\hbar^{2})^{1/5}$. Temperature dependences of basic thermodynamic quantities are clarified explicitly.

arXiv:cond-mat/0604241Journal of the Physical Society of JapanApr 13, 200627 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

Phase Invariant Perturbation Scheme in Bose-Einstein Condensed States

H. Matsumoto, S. Sakamoto

In the Bose-Einstein condensation of a dilute alkaline atomic gas trapped in a magnetic potential, one has to deal with a strongly space- (and time-) dependent order parameter, which violates the phase symmetry. In this case, certain relations are required among propagators and vertices, known as Ward-Takahashi identities. In this paper, we show that a systematic self-consistent perturbation scheme can be developed even with a space-time order parameter in such a way as to satisfy the requirement of phase symmetry. The key procedure is that the information concerning the full order parameter, which is self-consistently determined, is included in the unperturbed Hamiltonian.In the Bose-Einstein condensation of a dilute alkaline atomic gas trapped in a magnetic potential, one has to deal with a strongly space- (and time-) dependent order parameter, which violates the phase symmetry. In this case, certain relations are required among propagators and vertices, known as Ward-Takahashi identities. In this paper, we show that a systematic self-consistent perturbation scheme can be developed even with a space-time order parameter in such a way as to satisfy the requirement of phase symmetry. The key procedure is that the information concerning the full order parameter, which is self-consistently determined, is included in the unperturbed Hamiltonian.

Progress of Theoretical PhysicsApr 1, 20021 citation

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

Extended self-energy functional approach for strongly correlated lattice bosons in the superfluid phase

Enrico Arrigoni, Michael Knap, Wolfgang von der Linden

Among the various numerical techniques to study the physics of strongly correlated quantum many-body systems, the self-energy functional approach (SFA) has become increasingly important. In its previous form, however, SFA is not applicable to Bose-Einstein condensation or superfluidity. In this paper, we show how to overcome this shortcoming. To this end, we identify an appropriate quantity, which we term $D$ , that represents the correlation correction of the condensate order parameter, as it does the self-energy for Green's function. An appropriate functional is derived, which is stationary at the exact physical realization of $D$ and of the self-energy. Its derivation is based on a functional-integral representation of the grand potential followed by an appropriate sequence of Legendre transformations. The approach is not perturbative and, therefore, applicable to a wide range of models with local interactions. We show that the variational cluster approach based on the extended self-energy functional is equivalent to the ``pseudoparticle'' approach proposed in Phys. Rev. B 83, 134507 (2011). We present results for the superfluid density in the two-dimensional Bose-Hubbard model, which shows a remarkable agreement with those of quantum-Monte-Carlo calculations.Among the various numerical techniques to study the physics of strongly correlated quantum many-body systems, the self-energy functional approach (SFA) has become increasingly important. In its previous form, however, SFA is not applicable to Bose-Einstein condensation or superfluidity. In this paper, we show how to overcome this shortcoming. To this end, we identify an appropriate quantity, which we term $D$, that represents the correlation correction of the condensate order parameter, as it does the self-energy for Green's function. An appropriate functional is derived, which is stationary at the exact physical realization of $D$ and of the self-energy. Its derivation is based on a functional-integral representation of the grand potential followed by an appropriate sequence of Legendre transformations. The approach is not perturbative and, therefore, applicable to a wide range of models with local interactions. We show that the variational cluster approach based on the extended self-energy functional is equivalent to the ``pseudoparticle'' approach proposed in Phys. Rev. B 83, 134507 (2011). We present results for the superfluid density in the two-dimensional Bose-Hubbard model, which shows a remarkable agreement with those of quantum-Monte-Carlo calculations.

arXiv:1103.3664Physical Review BJul 29, 201117 citations

We derive Ward-Takahashi identities for correlated Bose-Einstein condensates based on the expressions of the first-order variations

(δΨ,δG)

due to perturbations obtained in the preceding paper [T. Kita, J. Phys. Soc. Jpn.

\bf 90

, 024001 (2021)] for the condensate wave function

Ψ

and Green's function

G

. They enable us to obtain several exact results on the density and current correlation functions

K_{νν'}^{}

, and also express

K_{νν'}^{}

ω

and wavenumber

q

is set equal to zero first.

T=0

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

Ψ

and the Nambu Green's function

\hat{G}

for the quasiparticle field. Imposing its stationarity respect to

Ψ

and

\hat{G}

yields a set of equations to determine the equilibrium for general non-uniform systems. They have a plausible property of satisfying the Hugenholtz-Pines theorem to provide a gapless excitation spectrum. Also, the corresponding dynamical equations of motion obey various conservation laws. Thus, the present mean-field theory shares two important properties with the exact theory: ``conserving'' and ``gapless.'' The theory is then applied to a homogeneous weakly interacting Bose gas with s-wave scattering length

a

and particle mass

m

to clarify its basic thermodynamic properties under two complementary conditions of constant density

n

and constant pressure

p

. The superfluid transition is predicted to be first-order because of the non-analytic nature of the order-parameter expansion near

T_{c}

inherent in Bose systems, i.e., the Landau-Ginzburg expansion is not possible here. The transition temperature

T_{c}

shows quite a different interaction dependence between the

n

-fixed and

p

-fixed cases. In the former case

T_{c}

increases from the ideal gas value

T_{0}

T_{c}/T_{0}= 1+ 2.33 an^{1/3}

, whereas it decreases in the latter as

T_{c}/T_{0}= 1- 3.84a(mp/2π\hbar^{2})^{1/5}

. Temperature dependences of basic thermodynamic quantities are clarified explicitly.

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.

D

, that represents the correlation correction of the condensate order parameter, as it does the self-energy for Green's function. An appropriate functional is derived, which is stationary at the exact physical realization of

D

and of the self-energy. Its derivation is based on a functional-integral representation of the grand potential followed by an appropriate sequence of Legendre transformations. The approach is not perturbative and, therefore, applicable to a wide range of models with local interactions. We show that the variational cluster approach based on the extended self-energy functional is equivalent to the ``pseudoparticle'' approach proposed in Phys. Rev. B 83, 134507 (2011). We present results for the superfluid density in the two-dimensional Bose-Hubbard model, which shows a remarkable agreement with those of quantum-Monte-Carlo calculations.