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

arXiv:cond-mat/9410004 · Springer eBooks

Renormalization group approach to zero temperature bose condensation

G. Benfatto

Published Apr 6, 2008• 11 citations

arXiv HTML PDF

DOI: 10.1007/3-540-59190-7_31

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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

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