Two-dimensional Schrödinger symmetry and three-dimensional breathers and Kelvin-ripple complexes as quasi-massive-Nambu-Goldstone modes
Daisuke A. Takahashi, Keisuke Ohashi, Toshiaki Fujimori, Muneto Nitta
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug APS / Two-dimensional Schr¨odinger symmetry and three-dimensional breathers and Kelvin-ripplecomplexes as quasi-massive-Nambu-Goldstone modes
Daisuke A. Takahashi, ∗ Keisuke Ohashi, Toshiaki Fujimori, and Muneto Nitta
1, 2 Research and Education Center for Natural Sciences, Keio University,Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan Department of Physics, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan (Dated: November 6, 2018)Bose-Einstein condensates (BECs) confined in a two-dimensional (2D) harmonic trap are known to possessa hidden 2D Schr¨odinger symmetry, that is, the Schr¨odinger symmetry modified by a trapping potential. Spon-taneous breaking of this symmetry gives rise to a breathing motion of the BEC, whose oscillation frequencyis robustly determined by the strength of the harmonic trap. In this paper, we demonstrate that the concept ofthe 2D Schr¨odinger symmetry can be applied to predict the nature of three dimensional (3D) collective modespropagating along a condensate confined in an elongated trap. We find three kinds of collective modes whose ex-istence is robustly ensured by the Schr¨odinger symmetry, which are physically interpreted as one breather modeand two Kelvin-ripple complex modes, i.e., composite modes in which the vortex core and the condensate sur-face oscillate interactively. We provide analytical expressions for the dispersion relations (energy-momentumrelation) of these modes using the Bogoliubov theory [D. A. Takahashi and M. Nitta, Ann. Phys. , 101(2015)]. Furthermore, we point out that these modes can be interpreted as “quasi-massive-Nambu-Goldstone(NG) modes,” that is, they have the properties of both quasi-NG and massive NG modes: quasi-NG modes ap-pear when a symmetry of a part of a Lagrangian, which is not a symmetry of a full Lagrangian, is spontaneouslybroken, while massive NG modes appear when a modified symmetry is spontaneously broken.
I. INTRODUCTION
Conformal symmetry plays crucial roles in many branchesof modern physics to extract nontrivial consequences whichare far from intuition. The non-relativistic d -dimensionalSchr¨odinger equation, i ∂ t ψ = −∇ ψ with ∇ = ( ∂ , . . . , ∂ d ), hasa spacetime scaling symmetry, whose transformation groupis called the Schr¨odinger group (or non-relativistic conformalgroup) denoted by Sch( d ) (e.g., Refs. [1–3]). In particular, thegenerators of the special Schr¨odinger transformation, the di-latation, and the time-translation form the sl (2 , R ) algebra.The Schr¨odinger symmetry survives under a specificnonlinear generalization. Let us consider the nonlinearSchr¨odinger (NLS) equation,i ∂ t ψ = −∇ ψ + g | ψ | α ψ, (1.1)where g is a nonzero constant. Then, this equation preservesthe Sch( d ) covariance if the power is given by α = , , and for d = , , and 3, respectively. The cases ( d , α ) = (1 ,
4) and(2 ,
2) are important in the physics of ultracold atomic gases,since they represent the dynamics of the Tonks-Girardeau gasin one dimension [4, 5] and the Gross-Pitaevskii (GP) equa-tion describing the dynamics of Bose-Einstein condensates(BECs) in two-dimensional (2D) systems.The above-mentioned Sch( d ) symmetries can be applied, infact, even in the presence of the harmonic trap,i ∂ t ψ = −∇ ψ + g | ψ | α ψ + ω r ψ, (1.2)which can be experimentally realized by standard techniquesin ultracold atomic gases. Due to the existence of the har-monic trap, the original Schr¨odinger symmetry is explicitly ∗ [email protected] broken. However, the harmonic trap term can be eliminatedby a certain transformation, so that a modified symmetry ex-ists. Thus, nontrivial time-dependent solutions can be gen-erated from a stationary solution [5–8]. The breathing mo-tion of the 2D BEC in a harmonic trap originating from the sl (2 , R ) is theoretically predicted and experimentally observedin Refs. [6, 9]. The Schr¨odinger symmetry in a harmonic trapwas discussed in the context of non-relativistic conformal fieldtheory in Ref. [10, 11]. Similar investigations in supersym-metric gauge theories with a harmonic trap ( Ω -background)are found in Ref. [12].The aim of this paper is a di ff erent application of 2DSchr¨odinger symmetry — we apply it to predict the collectivemodes in three-dimensional (3D) BECs trapped in an elon-gated trap. Our setup is a harmonic trapping potential in the xy -directions and translationally invariance in the z -direction,representing a trap elongating in the z -direction. We empha-size that the 3D GP equation itself is not covariant under theSch(3) operations. Nevertheless, we find three kinds of col-lective modes whose existence is robustly guaranteed by the2D Schr¨odinger symmetry. The physical interpretation of thethree modes is: (i) a breather mode, which can be regardedas a generalization of the breathing mode in 2D BEC [6, 8],now propagating along the elongated z direction, and (ii) twoKelvin-ripple (KR) complex modes, i.e., the composite modesconsisting of the helical oscillation of the vortex core andthe condensate surface. See Fig. 1 for their physical picture.Moreover, we derive an analytical expression for the disper-sion relations (energy-momentum relations) of these modes,using the Bogoliubov theory approach [13–15].The above-mentioned collective modes are one of the vari-ant concepts of the Nambu-Goldstone (NG) modes emerg-ing in the systems with spontaneous symmetry breaking(SSB). Recently, a general framework of NG modes in non-relativistic systems has been explored [16–18]. Our modeshave properties of two variants of NG modes: massive NGmodes and quasi-NG modes.When a generalized chemical potential term generated bya symmetry generator is added to the original Hamiltonian orLagrangian, NG modes are gapped and such modes are calledmassive NG modes [19–21]. They are gapful, but their exis-tence is still robustly ensured and the value of the energy gapis determined only by the Lie algebra of the symmetry group.Although the authors in Refs. [19–21] interpreted that they be-come massive because of explicit symmetry breaking by thechemical potential term, it is not the case; adding chemicalpotential does not explicitly break the original symmetry butjust modifies it. This fact was found for the Schr¨odinger sym-metry [8] following the cases of the O (2 ,
1) subgroup [5, 6]and the Galilean subgroup [7]. Here, a harmonic trappingpotential can be introduced as a generalized chemical poten-tial for the special Schr¨odinger symmetry. All possible gen-eralized chemical potentials including the rotation generatorwere introduced in Ref. [8]. When such modified symmetry isspontaneously broken, there appear massive NG modes. Thus,the breathing, harmonic oscillation and cyclotron motions ofthe 2D trapped BEC can be identified as massive NG modescorresponding to spontaneously broken modified Schr¨odingersymmetry [8].On the other hand, the equation of motion or a part of aLagrangian sometimes has a symmetry larger than the origi-nal Lagrangian. When such an enhanced symmetry is sponta-neously broken, there appear quasi-NG modes [22]. In thiscase, the ground-state order-parameter space (OPS) is en-larged from the original OPS of usual NG modes. Quasi-NGmodes become gapped when quantum corrections are takeninto account. Such situation occurs in pions for chiral sym-metry breaking [23], SSB in supersymmetric field theories[14, 24–30], and condensed matter systems such as super-fluid helium 3 [31], the spin-2 spinor BEC [32], and neu-tron P superfluids [33]. Quasi-NG modes which are quiteclose to the current study appear in a Skyrmion line in mag-nets [34], which has two gapless modes propagating along theline, that is, the dilaton-magnon mode and Kelvin mode. Here,the former corresponds to a spontaneous breaking of the dila-tional symmetry, which is a symmetry in the 2D section of theSkyrmion line but not a symmetry of full 3D.As mentioned above, we will point out that the collectivemodes found in this paper satisfy both quasi- and massive con-ditions — namely, they are the “quasi-massive-NG modes.” Rather surprisingly, a more fundamental low-energy exci-tation, i.e., the Kelvin mode [35], cannot be exactly treatedby the concept of symmetry in the presence of a trappingpotential, and we can find it only numerically. In the caseof the Kelvin mode, the vortex core solely oscillates and isnot coupled with the condensate-surface oscillation [Fig. 1(A)], in contrast to the KR modes mentioned above. Thismode shows the Landau instability (negative dispersion) inthe finite-size systems, which is consistent with the case studyof the cylindrical trap [15, 36]. Studies of the Kelvin modesin ultracold atomic gases in a trapping potential can be foundin Refs. [37–39]. In the trapless limit, the Kelvin mode be-comes an NG mode originating from an SSB of spatial trans- (A) (C)(B)
FIG. 1. The excited states of an elongated 3D BEC considered in thispaper. (A) The pure Kelvin mode, a helical motion of a vortex. (B)The breather mode, an oscillation of the outer condensate. (C) TheKR complex mode. Using the concept of the 2D Schr¨odinger sym-metry, we can rigorously find and determine the physical character-istics of excitations (B) and (C). There are one breather correspond-ing to (B) and two independent KR complex modes corresponding to(C). The pure Kelvin mode (A) cannot be discussed in the frameworkof the 2D Schr¨odinger symmetry, though we can find it numerically. lation, and showing the noninteger dispersion ǫ ∼ − k ln k [15, 40, 41].The organization of this paper is as follows. In Sec. II, weintroduce the 3D GP and Bogoliubov equations, and clarifythe problem discussed in this paper. In Sec. III, we summa-rize the result obtained from the 2D Schr¨odinger symmetry.In particular, we derive the zero-mode solutions of the Bo-goliubov equation, which is essential in the Bogoliubov the-ory approach. Section IV describes the main results of thispaper. We derive the collective excitations originating fromthe 2D Schr¨odinger symmetry and derive their dispersion re-lations. We also elucidate their physical picture. In Sec. V,we verify our main result by numerical calculations. We alsomention the existence of the Kelvin mode [Fig. 1, (A)], whichwe cannot treat by the concept of the Schr¨odinger symmetry.In Sec. VI, we prove that the KR complex and breather modes[Fig. 1, (B),(C)] can be identified as the quasi-massive-NGmodes. Section VII is devoted to a summary and outlook. II. DEFINITION OF THE PROBLEM
In this section, we clarify the problem which we want tosolve in this paper. We start from the 3D GP equation with aharmonic trap in the x - and y -directions:i ∂ t ψ = −∇ ψ + g | ψ | ψ + ω ( x + y )4 ψ, (2.1)where ∇ = ∂ x + ∂ y + ∂ z is the 3D Laplacian, g is a cou-pling constant of two-body interaction, and ω > g > g .We emphasize that the 3D GP equation (2.1) itself does not have the 3D Schr¨odinger symmetry, i.e., the equation is nevercovariant under the Sch(3) operations. The 3D Schr¨odingersymmetry is retained when the nonlinear term is modified to | ψ | ψ → | ψ | / ψ and the trap is isotropic: ω ( x + y + z )4 , but wedo not consider such a case in this paper.Let us introduce the Bogoliubov equation for the Bogoli-ubov quasiparticles as a linearized small oscillation around agiven solution of Eq. (2.1) [42–45]. Substituting ψ = ψ + δψ in Eq. (2.1) and linearizing the equation w.r.t. δψ , and writing( u , v ) = ( δψ, δψ ∗ ), we obtaini ∂ t uv ! = ˆ H + g | ψ | g ψ − g ψ ∗ − ˆ H − g | ψ | ! uv ! (2.2)with ˆ H = −∇ + ω ( x + y )4 . In condensed matter theory, theBogoliubov equation is frequently used to investigate the na-ture of low-energy excitations and linear stability for a givenstationary state (e.g., Refs. [43–45]).We are interested in a solution of the form ψ ( t , x , y , z ) = e − i µ t + i n θ f ( r ) , (2.3)where µ is a chemical potential and the cylindrical coordinatesare defined by ( x , y , z ) = ( r cos θ, r sin θ, z ). We assume that f ( r ) is a non-negative real function and n ∈ Z represents avortex charge. The di ff erential equation for f ( r ) is given by − f ′′ − f ′ r + n r − µ + ω r ! f + g f = . (2.4)The particle number per unit length along the z -axis, N = R | ψ | dxdy = π R r f dr , is a monotonically increasing func-tion of µ/ω which vanishes at µ = ω ( | n | + gN the profile of f ( r ) for a vortexless state ( n = f ( r ) ≃ f TF ( r ) : = s µ − ω r g θ (cid:16) √ µω − r (cid:17) , (2.5)which indicates that the position of the condensate surface isestimated as r ≃ r TF : = √ µω . (2.6)We can numerically check that this surface position r TF is alsovalid for vortex states with small n ’s. Therefore, we can use r TF as an e ff ective system size.The stationary Bogoliubov equation for the eigenenergy ǫ is obtained by setting u ( t , x , y , z ) v ( t , x , y , z ) ! = e i( − µ t + n θ ) σ e i( − ǫ t + m θ + kz ) u ( r ) v ( r ) ! , (2.7)and the resultant equation is ǫ uv ! = ( ˆ L m + σ k ) uv ! , (2.8) with ˆ L m = ˆ H m + n − µ + g f g f − g f − ˆ H m − n + µ − g f ! , (2.9) σ = − ! , (2.10)ˆ H m ± n = − ∂ r − r ∂ r + ( m ± n ) r + ω r . (2.11) ǫ is an eigenenergy of the Bogoliubov quasiparticles, and m ∈ Z , k ∈ R are quantum numbers labeling eigenstates, in-dicating the quantized angular momentums and wavenumbersin the z -direction, respectively.The stationary Bogoliubov equation (2.8) always providesa pair of eigenstates w = ( u , v ) T with numbers ( ǫ, k , m ) and σ w ∗ = ( v ∗ , u ∗ ) T with ( − ǫ ∗ , − k , − m ).Most eigenstates are determined only numerically. How-ever, as we will see later, several important low-energy ex-citations can be identified only by symmetry considerations.Besides, we can calculate their dispersion relations (energy-momentum relation) ǫ = ǫ ( k ) using the exact eigenfunctionsfound from the symmetry. The aim of this paper is thereforephrased as follows: Find all collective modes whose existenceis robustly ensured by the symmetry, and determine their en-ergy gaps and dispersion relations.
We solve the above-mentioned problem by the Bogoliubovtheory approach [13–15]. The method consists of two proce-dures: (i) First, we construct a one-parameter family of thesolutions to the GP equation, and by di ff erentiating it, we geta zero-mode solution to the Bogoliubov equation. (ii) Next,regarding the σ k term as a perturbation term, we solve thefinite-wavenumber problem by perturbation theory, and obtainthe expansion of the dispersion relation ǫ = ǫ + ǫ k + · · · . (For the linear dispersion relation, the perturbative expansionneeds a modification [13, 14]. See the example of the Bogoli-ubov sound wave in Subsec. IV B.)Equation (2.8) reduces to the 2D equation if k =
0, andhence the 2D result coming from the 2D Schr¨odinger symme-try is partially applicable. We see this in the next section.
III. CONSEQUENCES OF 2D SCHR ¨ODINGERSYMMETRY
In this section, we summarize the 2D Schr¨odinger symme-try of the harmonically-trapped 2D NLS systems, and derivethe SSB-originated zero-mode solutions for the Bogoliubovequation.Henceforth, the algebra of the 2D Schr¨odinger groupSch(2) is denoted by sch(2).
A. Schr¨odinger symmetry in the 2D NLS systems with aharmonic trap
Let us consider the 2D NLS (or GP) equation with and with-out a harmonic trap:i ψ t = − ψ xx − ψ yy + g | ψ | ψ, (3.1)i ψ t = − ψ xx − ψ yy + g | ψ | ψ + ω ( x + y )4 ψ. (3.2)Following Ref. [5], we consider the following function-to-function map T : T [ ψ ]( t , x , y ) : = p ˙ τ ( t ) exp − i¨ τ ( t )( x + y )8˙ τ ( t ) ! × ψ (cid:16) τ ( t ) , x p ˙ τ ( t ) , y p ˙ τ ( t ) (cid:17) , (3.3)where τ ( t ) is a non-decreasing function. Then we can provethe following: if ψ satisfies Eq. (3.1), T [ ψ ] satisfies Eq. (3.2)with the harmonic trap replaced by the time-dependent one ω ( t ) ( x + y )4 , where ω ( t ) = S τ ( t ), and S τ ( t ) : = ... τ ( t )˙ τ ( t ) − (cid:16) ¨ τ ( t )˙ τ ( t ) (cid:17) is the Schwarzian derivative.If τ ( t ) is chosen to be a fractional linear transformation τ ( t ) = at + bct + d , ad − bc =
1, then the Schwarzian deriva-tive vanishes and the map (3.3) produces a new solution ofEq. (3.1), and generators of such transformations forms the al-gebra sl (2 , R ), which is a subalgebra of sch(2). More generalfunctions τ ( t ) induce extended Schr¨odinger transformations(e.g., Ref. [3]), whose algebra is also called the Schr¨odinger-Virasoro algebra. The 2D NLS equation is not invariant un-der transformations with general τ ( t )’s and hence the equationchanges. The map (3.3) generally gives a time-dependent har-monic trap. The time-independent trap is obtained by [5] T trap [ ψ ]( t , x , y ) : = exp (cid:16) − i4 ( x + y ) ω tan ω t (cid:17) cos ω t × ψ (cid:18) tan ω t ω , x cos ω t , y cos ω t (cid:19) , (3.4)which is realized by the choice τ ( t ) = tan ω t ω . If ψ satisfies thefree 2D NLS equation (3.1), then T trap [ ψ ] satisfies the trapped2D NLS equation (3.2). We further define T release [ ψ ]( t , x , y ) : = exp (cid:18) i ω t ( x + y )4(1 + ω t ) (cid:19) √ + ω t ψ tan − ω t ω , x √ + ω t , y √ + ω t ! , (3.5)which is realized by τ ( t ) = tan − ω t ω . If ψ satisfies Eq. (3.2),then T release [ ψ ] satisfies Eq. (3.1). The latter is an inverse ofthe former, i.e., T release = T − , and the relation T trap ◦ T release [ ψ ] = T release ◦ T trap [ ψ ] = ψ (3.6)holds.Next, let us take ˆ Q ∈ sch(2), and let us define T Sch(2) ( α ) : = e i α ˆ Q . If ψ is a solution to the 2D NLS equation (3.1), then T Sch(2) ( α )[ ψ ] = e i α ˆ Q ψ is also a solution to the same equation.Finally, we define φ ( t , x , y , α ) = T trap ◦ T Sch(2) ( α ) ◦ T release [ ψ ]( t , x , y ) . (3.7)Following the above-mentioned properties, if ψ ( t , x , y ) satis-fies the trapped 2D NLS equation (3.2), then φ ( t , x , y , α ) alsosatisfies the same equation. Thus, we get a family of solutionsfor Eq. (3.2) parametrized by a parameter α . Taking infinites-imal α , a set of operators {T trap ◦ ˆ Q ◦ T release | ˆ Q ∈ sch(2) } turnsout to give the generators of the modified symmetry of themodel (2.1), which has been directly discussed in [8]. B. SSB-originated zero mode solutions for the Bogoliubovequation
Using the one-parameter family of solutions (3.7) forEq. (3.2), we can now apply the formulation of NG modesbased on the Bogoliubov theory [13–15]. The zero modesbased on the symmetry and parameter derivatives have beendiscussed in the earlier paper [46].Let us introduce the Bogoliubov equation as a linearizationof the GP equation (3.2):i ∂ t uv ! = ˆ H + g | ψ | g ψ − g ψ ∗ − ˆ H − g | ψ | ! uv ! , (3.8)with ˆ H = − ∂ x − ∂ y + ω ( x + y )4 . Now let us derive theSSB-originated zero-mode solutions [13]. If we di ff erentiateEq. (3.2) with ψ replaced by φ in Eq. (3.7) by α and set α = ff erentiation, we get a particular solution for the time-dependent Bogoliubov equation (3.8): w = [ ∂ α φ ] α = [ ∂ α φ ] ∗ α = ! = T trap ◦ (i ˆ Q ) ◦ T − [ ψ ] (cid:16) T trap ◦ (i ˆ Q ) ◦ T − [ ψ ] (cid:17) ∗ . (3.9)This expression provides a formula analogous to Eq.(G.8) inthe Appendix G of Ref. [13], where massive NG modes areformulated in terms of the Bogoliubov theory. Choosing ˆ Q from various elements of sch(2), we obtain several zero-modesolutions. (Note that in the case of massive NG modes, weget the finite-energy eigenstates of the Bogoliubov equation,but we still keep to use the term “zero-mode solutions” forbrevity.) Since sch(2) is nine-dimensional, we obtain nine so-lutions of the form (3.9), unless ψ preserves some symmetry.The list of the solutions is shown below. We also attach ex-pressions in the trap-free limit ω →
0, which obviously corre-sponds to T trap = id and hence given by w = (i ˆ Q ψ, − i ˆ Q ∗ ψ ∗ ) T .(i) the x -translation ˆ Q = − i ∂ x :e i α ˆ Q ψ ( t , x , y ) = ψ ( t , x + α, y ) . (3.10) w x -trans = i x ω sin ω t ψ − ψ ∗ ! + cos ω t ψ x ψ ∗ x ! (3.11) ω → −→ ψ x ψ ∗ x ! . (3.12)(ii) the y -translation ˆ Q = − i ∂ y :e i α ˆ Q ψ ( t , x , y ) = ψ ( t , x , y + α ) . (3.13) w y -trans = i y ω sin ω t ψ − ψ ∗ ! + cos ω t ψ y ψ ∗ y ! (3.14) ω → −→ ψ y ψ ∗ y ! . (3.15)(iii) the z -axis rotation ˆ Q = − i( x ∂ y − y ∂ x ):e i α ˆ Q ψ ( t , x , y ) = ψ ( t , x cos α − y sin α, x sin α + y cos α ) . (3.16) w z -rot = x ψ y − y ψ x x ψ ∗ y − y ψ ∗ x ! . ( ω -independent) (3.17)(iv) phase multiplication ˆ Q = i α ˆ Q ψ ( t , x , y ) = e i α ψ ( t , x , y ) . (3.18) w phase = i ψ − i ψ ∗ ! . ( ω -independent) (3.19)(v) the x -Galilei transformation ˆ Q = − x − t ∂ x :e i α ˆ Q ψ ( t , x , y ) = e − i( α x + α t ) ψ ( t , x + α t , y ) . (3.20) w x -Gal = − i x cos ω t ψ − ψ ∗ ! + ω t ω ψ x ψ ∗ x ! (3.21) ω → −→ − i x ψ − ψ ∗ ! + t ψ x ψ ∗ x ! . (3.22)(vi) the y -Galilei transformation ˆ Q = − y − t ∂ y :e i α ˆ Q ψ ( t , x , y ) = e − i( α y + α t ) ψ ( t , x , y + α t ) . (3.23) w y -Gal = − i y cos ω t ψ − ψ ∗ ! + ω t ω ψ y ψ ∗ y ! (3.24) ω → −→ − i y ψ − ψ ∗ ! + t ψ y ψ ∗ y ! . (3.25)(vii) the t -translation ˆ Q = − i ∂ t :e i α ˆ Q ψ ( t , x , y ) = ψ ( t + α, x , y ) . (3.26) w t -trans = cos ω t ψ t ψ ∗ t ! − ω sin 2 ω t ψ + x ψ x + y ψ y ψ ∗ + x ψ ∗ x + y ψ ∗ y ! + i( x + y ) ω cos 2 ω t ψ − ψ ∗ ! (3.27) ω → −→ ψ t ψ ∗ t ! . (3.28) (viii) the dilatation ˆ Q = − i(1 + x ∂ x + y ∂ y + t ∂ t ):e i α ˆ Q ψ ( t , x , y ) = e α ψ (e α t , e α x , e α y ) . (3.29) w dila = cos 2 ω t ψ + x ψ x + y ψ y ψ ∗ + x ψ ∗ x + y ψ ∗ y ! + sin 2 ω t ω ψ t ψ ∗ t ! + i( x + y ) ω sin 2 ω t ψ − ψ ∗ ! (3.30) ω → −→ ψ + x ψ x + y ψ y ψ ∗ + x ψ ∗ x + y ψ ∗ y ! + t ψ t ψ ∗ t ! . (3.31)(ix) the special Schr¨odinger transformationˆ Q = i t (1 + x ∂ x + y ∂ y + t ∂ t ) + x + y :e i α ˆ Q ψ ( t , x , y ) = exp( i α ( x + y + α t ) )1 + α t ψ ( t + α t , x + α t , y + α t ) . (3.32) w special = − sin ω t ω ψ t ψ ∗ t ! − sin 2 ω t ω ψ + x ψ x + y ψ y ψ ∗ + x ψ ∗ x + y ψ ∗ y ! + i( x + y ) cos 2 ω t ψ − ψ ∗ ! (3.33) ω → −→ − t ψ t ψ ∗ t ! − t ψ + x ψ x + y ψ y ψ ∗ + x ψ ∗ x + y ψ ∗ y ! + i( x + y )4 ψ − ψ ∗ ! . (3.34)Note that the ordinary t -translation zero mode is reproducedby w t -trans − ω w special = ( ψ t , ψ ∗ t ) T .We thus get nine solutions both for free ( ω =
0) and trapped( ω ,
0) systems. However, if we consider a one-vortex solu-tion in the trap-free ( ω =
0) system, where ψ tends to a con-stant value in the spatial infinity | ψ | → ρ ∞ , the zero-modesolutions created from the rotation (3.17), the Galilei transfor-mation (3.22) and (3.25), the dilatation (3.31), and the specialSchr¨odinger transformation (3.34) contain polynomial coef-ficients of x , y , and t . Hence, they are excluded from phys-ical consideration due to an unphysical divergence at spatialand / or temporal infinities. Thus, for the stationary solutionof the trap-free system, we only get three physical solutions: w x -trans , w y -trans , and w phase . This analysis provides a moresimplified solution for the redundancy problem of NG modes[47].On the other hand, in the trapped system ( ω , ψ decaysin a Gaussian fashion, and the polynomial coe ffi cient of x , y causes no problem in convergence. The t -dependence also ap-pears in the form of cos ω t or sin ω t , which remains finite forall time. Thus, all zero-mode solutions created from all ele-ments in sch(2) can contribute as independent physical modes.This situation is highly contrasted with the trap-free system. IV. COLLECTIVE EXCITATIONS PROPAGATING ALONGELONGATED BECS IN 3DA. Seed zero modes
Equation (2.8) reduces to the 2D equation if k =
0, and thusthe SSB-originated zero-mode solutions derived in Sec. III Bcan be applied. Though the solutions (3.11)-(3.33) are time-dependent, we can make a stationary eigenstate for the station-ary Bogoliubov equation (2.8) by taking their linear combina-tion. The eigenstates with their eigenenergies ǫ , quantized an-gular momentums m , and wavenumbers k [introduced in Eqs.(2.7)-(2.11)] are summarized as follows:(a) Seed of the Bogoliubov sound wave, ( ǫ, k , m ) = (0 , , w phonon = f − f ! . (4.1)This is made from w phase or w z -rot or w t -trans − ω w special .(b) Seed of the first KR complex, ( ǫ, k , m ) = ( ω, , − w KR1 = f ′ ! + (2 n − ω r ) f r − ! . (4.2)This is made by w x -trans − i w y -trans − i ω ( w x -Gal − i w y -Gal ).(c) Seed of the second KR complex, ( ǫ, k , m ) = ( ω, , w KR2 = f ′ ! − (2 n + ω r ) f r − ! . (4.3)This is made by w x -trans + i w y -trans − i ω ( w x -Gal + i w y -Gal ).(d) Seed of the breather mode, ( ǫ, k , m ) = (2 ω, , w breather = ω h ′ ! + (2 µ − ω r ) h r − ! , h ( r ) : = r f ( r ) . (4.4)This is made by ω w dila + i( w t -trans + ω w special ).In addition to the above four solutions (4.1)-(4.4), bythe symmetry of the Bogoliubov equation, we have con-jugate solutions σ w KR1 , σ w KR2 , and σ w breather , whoseeigenenergies and quantum numbers are given by ( ǫ, k , m ) = ( − ω, , , ( − ω, , − , and ( − ω, , w phase , w z -rot , and w t -trans − ω w special are degenerate when ψ has the form (2.3), all possi-ble solutions obtainable from the nine solutions (3.11)-(3.33)are exhausted by these seven solutions.A physical picture of the above modes can be understoodby considering the density and phase oscillations as follows.If an eigenstate of the Bogoliubov equation α w = α ( u , v ) T isexcited, where α > ψ + δψ = ψ + α ( u + v ∗ ).In particular, when the solution is given by the form (2.3) and(2.7), we obtain ψ + δψ = e i( − µ t + n θ ) × (cid:20) f + α u + v kz − ǫ t + m θ ) + i α u − v kz − ǫ t + m θ ) (cid:21) . (4.5)From this, we can show that the density and phase fluctuationsare expressed by | ψ + δψ | = f ( r ) + αδρ ( r ) cos( kz − ǫ t + m θ ) , (4.6)arg( ψ + δψ ) = − µ t + n θ + αδϕ ( r ) sin( kz − ǫ t + m θ ) (4.7) with δρ ( r ) : = [ u ( r ) + v ( r )] f ( r ) , δϕ ( r ) : = u ( r ) − v ( r )2 f ( r ) . (4.8)Using the expressions (4.5)-(4.8), we can find the physical in-terpretation for a given eigenstate. For example, w phonon onlyhas a phase fluctuation, consistent with the fact that this modeoriginates from the SSB of the U (1) phase transformation.The details of other modes will be discussed in the next sub-section. B. Dispersion relation
We now determine the dispersion relations ǫ ( k ). We solveEq. (2.8) for finite k perturbatively, regarding ˆ L m and σ k as an unperturbed term and a perturbation term, respectively[13–15]. If we have an eigenstate w = ( u , v ) T with eigen-value ǫ , the leading-order correction of the eigenvalue isgiven by: ǫ ( k ) = ǫ + ( w , σ w ) σ ( w , w ) σ k + O ( k ) , (4.9)where the σ -inner product for w = ( u , v ) T and w = ( u , v ) T is defined by( w , w ) σ = Z π d θ Z ∞ r d r ( u ∗ u − v ∗ v ) . (4.10)This product is invariant under the Bogoliubov transforma-tions and used to formulate the perturbation theory [13]. w is said to have finite norm if ( w , w ) σ ,
0, and the perturba-tion theory for finite-norm eigenfunctions is almost the sameas that for the well-known hermitian operators. Among thezero-mode solutions (4.1)-(4.4), only w phonon has zero norm,while w KR1 , w KR2 , and w breather have finite and positive norm.Therefore, for the latter three modes, the coe ffi cient of k iscalculated by ( w , σ w ) σ ( w , w ) σ = R r d r ( | u | + | v | ) R r d r ( | u | − | v | ) . (4.11)In the following, we provide the dispersion relations for thesemodes.
1. w
KR1 and w
KR2 — the KR complex modes
We obtain ( w KR1 , w KR1 ) σ = ( w KR2 , w KR2 ) σ = πω I , ( w KR1 , σ w KR1 ) σ = π [( µ − n ω ) I − gI ],and ( w KR2 , σ w KR2 ) σ = π [( µ + n ω ) I − gI ] with I αβ = R ∞ r α f β d r . Therefore, the dispersion relations aregiven by ǫ KR1 ( k ) = ω + A − k + O ( k ) , (4.12) ǫ KR2 ( k ) = ω + A + k + O ( k ) , (4.13)where A ± = µ − g ηω ± n , η : = R ∞ r f d r R ∞ r f d r . (4.14)We emphasize that A ± is always positive for any n by defini-tion. Therefore, these modes show no Landau instability (i.e.,no negative dispersion relation for positive-norm eigenstates).Using Eqs. (4.5)-(4.8), the oscillation of the vortex core andthe condensate surface can be respectively estimated as (cid:0) r core ( t , z ) , θ core ( t , z ) (cid:1) = (cid:16) α | n | , m ( π − kz + ǫ t ) (cid:17) (if n , , (4.15) r surface ( t , z , θ ) = r TF − α cos( kz − ǫ t + m θ ) , (4.16)where (cid:0) ǫ, m (cid:1) = (cid:0) ǫ KR1 ( k ) , − (cid:1) and (cid:0) ǫ KR2 ( k ) , + (cid:1) should be usedfor KR1 and KR2, respectively. Here, ( r core , θ core ) is obtainedby solving | ψ + δψ | = w KR1,2 . Also notethat we define the condensate surface by the position suchthat the condensate density is equal to f ( r TF ) , where r TF isthe TF radius [Eq. (2.6)]. Since r surface ( t , z , θ + θ core ( t , z )) = r TF + α cos( θ ), the configuration looks static from an observerrotating with the vortex core.Since the KR1 and KR2 modes have the same density fluc-tuation δρ [Eq. (4.8)], their di ff erence appears in the phasefluctuation δϕ . It will be seen in Sec. V.
2. w breather — the breather mode
Denoting I αβ = R ∞ r α f β d r , the σ -inner products are( w breather , w breather ) σ = πω I , (4.17)( w breather , σ w breather ) σ = π (cid:16) [ µ + (1 − n ) ω ] I − g ω I (cid:17) . (4.18)Furthermore, we can prove2 µ I − ω I − gI = , (4.19)2(1 − n ) I + µ I − ω I − gI = , (4.20)by integrating r f ′ × [Eq. (2.4)] and r ( fr ) ′ × [Eq. (2.4)]. Suchidentities are also found by following Derrick’s argument ofthe scaling transformation for the energy functional [48].Thus, the dispersion relation is given by ǫ breather ( k ) = ω + Bk + O ( k ) , (4.21) B = µ (3 g η − µ ) + ω ˜ η ω ( µ − g η ) , ˜ η : = R ∞ r f d r R ∞ r f d r , (4.22)where η is defined by Eq. (4.14).Solving | ψ + δψ | = w breather with assum-ing f ∝ r | n | near the origin, we can check that this mode showsno vortex-core oscillation up to O ( α ). The surface oscillationis estimated as r surface ( t , z , θ ) ≃ r TF [1 − αω cos( kz − ǫ t )] , (4.23) with ǫ = ǫ breather ( k ). This expression is consistent with the fea-ture of Fig. 1 (B). Here, we have assumed r TF ≫ f ( r TF ) f ′ ( r TF ) .The node point of breathing motion, where the density os-cillation vanishes, can be found by solving δρ ( r ) =
0. Withinthe TF approximation (2.5), it is estimated as r node ≃ p µω = √ r TF . (4.24)The phase oscillation δϕ ( r ) also vanishes at the same point.The validity of Eq. (4.24) will be checked in Figs. 5 and 11.
3. w phonon — Bogoliubov sound wave
This mode yields the famous Bogoliubov phonon (or soundwave) with the linear dispersion relation ǫ ∝ k [42]. Since w phonon is a zero-norm eigenstate, the perturbation theoryneeds a slight modification [13, 14], corresponding to the caseof the Jordan block of size 2. The perturbation theory for gen-eral larger Jordan blocks is given in Appendix F of Ref. [13]. f ( r ) can be regarded as a function of parameters ( µ, ω ). Dif-ferentiating Eq. (2.4) by µ , we findˆ L z phonon = w phonon , z phonon : = f µ f µ ! . (4.25) z phonon corresponds to the generalized eigenvector of the Jor-dan block. Using it, we solve the finite- k Bogoliubov equa-tion (2.8) by a perturbation expansion ( ˆ L + σ k )( w phonon + ǫ kz phonon + β k w + · · · ) = ( ǫ k + ǫ k + · · · )( w phonon + ǫ kz phonon + β k w + · · · ). See also Ref. [13], Sec. 6.1. Then we find ǫ = ( w phonon , σ w phonon ) σ ( w phonon , z phonon ) σ = R ∞ r f d r ∂ µ [ R ∞ r f d r ] . (4.26)We thus obtain the linear dispersion relation ǫ phonon ( k ) = ǫ k + O ( k ) . (4.27)In the trap-free limit ω →
0, the condensate behaves as f ( r → ∞ ) = µ g , and hence we get ǫ = p µ , which isconsistent with the phonon dispersion relation for a uniformcondensate.Here we remark on the density fluctuation of the Bogoli-ubov phonon. Recall Eq. (4.8) again. The eigenstate w phonon has only a phase fluctuation δϕ . However, it does not meanthat the Bogoliubov phonon induces no density fluctuation. Itappears from the first order in k (or ǫ ). Using the first-orderwavefunction w phonon + ǫ kz phonon + O ( k ), we find δϕ = δρ = ǫ k ∂ µ ( f ). Thus, the generalized eigenvector z phonon rep-resents the density fluctuation.Basically, the density fluctuation of phonon excitations ap-pears from the first order in ǫ . If it emerges from the zerothorder, it triggers the instability [49, 50]. ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ k - - - ϵ ω - ω ● phonon ■ breather ϵ = ± ϵ k ϵ = ±( ω + Bk ) FIG. 2. (Color online) The dispersion relation for the eigenstates ofthe Bogoliubov equation with ( n , m ) = (0 , g = , µ =
2, and ω = .
38. The red circle and green squarepoints represent the Bogoliubov phonon and the breather excitations,respectively. The curves are given by ǫ = ± ǫ k and ǫ = ± (2 ω + Bk ), as predicted in Eqs. (4.21), (4.22), (4.26), and (4.27). High-energy eigenstates with no SSB origin are shown by gray triangles.The coe ffi cients are evaluated as η = . , ˜ η = . × , ǫ = . , B = . ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ k - - - ϵω - ω ● KR1 ■ KR2 ϵ = ω + Ak ϵ = - ω - Ak FIG. 3. (Color online) The dispersion relation for the eigenstatesof the Bogoliubov equation with ( n , m ) = (0 , − A = A + = A − .The numerical integration shows A = .
98. The plot for ( n , m ) = (0 ,
1) is identical to this plot, but the KR1 and KR2 excitations areexchanged.
V. NUMERICAL CHECK
In this section, we numerically verify the analytical predic-tions derived in the last section. We plot the dispersion rela-tions, their zero-mode eigenfunctions and density and phasefluctuations (4.8). We also discuss the Kelvin mode appearingin the vortex state.
A. The vortexless state ( n = ) We first consider the collective excitations when the back-ground BEC is in the ground state, i.e., the state with no vor-
FIG. 4. (Color online) Plot of the KR1 complex zero mode [Eq. (4.2)] w KR1 = ( u KR1 , v KR1 ) T , δρ KR1 = ( u KR1 + v KR1 ) f , and δϕ KR1 = ( u KR1 − v KR1 ) / (2 f ) with a constant factor multiplied for visibil-ity. We also plot the condensate wavefunction f ( r ) and its TF ap-proximation f TF ( r ) [Eq. (2.5)] for reference. The parameters are g = , µ = , ω = .
38 (the same as Fig. 2), and the TF radiusis r TF = .
44. There is no vortex helical motion when n =
0, thoughit is named the “Kelvin-ripple complex.” Note that w KR2 is identicalto w KR1 .FIG. 5. (Color online) Plot of the breather zero mode [Eq. (4.4)] w breather = ( u breather , v breather ) T , δρ breather = ( u breather + v breather ) f , and δϕ breather = ( u breather − v breather ) / (2 f ) with a constant factor multipliedfor visibility. We also plot the condensate wavefunction f ( r ) and itsTF approximation f TF ( r ) [Eq. (2.5)] for reference. The same param-eters are used as Fig. 4. The node of the breathing motion (4.24) isestimated to be r node ≃ r TF √ = . tex. f ( r ) is a nodeless solution of Eq. (2.4) with n =
0, and theBogoliubov equation is given by Eqs. (2.8)-(2.11) with n = f ( r ) is well approximated by Eq. (2.5) exceptfor the vicinity of the condensate surface.Figures 2 and 3 show the energy spectra for the eigenstateswith the angular quantum numbers m = −
1, respec-tively. Note that if n =
0, the Bogoliubov equations for ± m arecompletely the same. In Fig. 2, we find two modes originatedfrom the SSB of the Sch(2) symmetry, i.e., the Bogoliubovphonon and the breather mode. Their dispersion relations arewell fitted by the theoretical expressions, given by Eqs. (4.21),(4.22), (4.26), and (4.27). In Fig. 3, we find the KR1 andKR2 modes. The theoretical curve is given by Eq. (4.12) with(4.14), showing a good agreement with the numerical points ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ k - - - ϵ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ k - - - ϵ ω - ω ● phonon ■ breather ϵ = ± ϵ k ϵ = ±( ω + Bk ) FIG. 6. (Color online) The dispersion relation for the eigenstatesof the Bogoliubov equation with ( n , m ) = (1 , g = , µ =
2, and ω = .
38. The red circle and greensquare points show the Bogoliubov phonon and breather excitations,respectively. High-energy eigenstates which have no SSB originare shown by gray triangles. The theoretical curves are given byEqs. (4.21), (4.22), (4.26), and (4.27). The coe ffi cients are evaluatedas η = . , ˜ η = . × , ǫ = . , B = . for small k .Figure 4 shows a plot of the KR1 zero mode w KR1 [Eq. (4.2)]. Here, we must admit that the naming “Kelvin-ripple complex” is a little inappropriate for the no-vortex state( n = w KR2 is the same as that for the KR1, up to the angularexponential factor e ± i θ .Figure 5 shows the zero mode of the breather mode w breather [Eq. (4.4)]. The breather mode has a node at r node = √ r TF . B. n = (the vortex state) We next consider the case n =
1, where there exists a vor-tex. f ( r ) vanishes at the origin r =
0, but the profile of theouter condensate is still well approximated by the TF wave-function (2.5). The dispersion relations for m = , ± n , m ) = (1 ,
0) shown inFig. 6 is almost the same as the no-vortex case ( n , m ) = (0 , ff erence appears in Fig 7. FromFig. 7, we first find that the dispersion curves of the KR1 andKR2 complex modes are not symmetric because A + , A − .Furthermore, we find a low-energy excitation shown by pur-ple diamond dots in Fig. 7, which cannot be predicted from theSch(2) symmetry. As shown in Fig. 12, the physical interpre-tation of this mode is just the Kelvin mode. This mode showsthe so-called Landau instability, i.e., a positive-norm eigen-state has a negative eigenvalue. Such a character is also foundfor the Kelvin modes confined in the cylindrical trap [15, 36].Since in the trap-free limit ( ω →
0) the Kelvin mode becomesa “genuine” NG mode associated with spontaneously brokentranslational symmetries in the presence of a vortex, the ex- ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ k - - - ϵω - ω ● KR1 ■ KR2 ◆ Kelvin ϵ = ω + A - k ϵ = -( ω + A + k ) FIG. 7. (Color online) The dispersion relation for the eigenstates ofthe Bogoliubov equation with ( n , m ) = (1 , − A − = . , A + = .
49. Anexpression for the dispersion coe ffi cient for the Kelvin mode is notknown, but we can find it by integrating the zero-mode eigenfunction[Fig. 12] and using the general formulas given by Eqs. (4.9) with(4.11). The purple dotted curve is plotted in this way. ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ k - - - ϵω - ω ● KR1 ■ KR2 ◆ Kelvin ϵ = -( ω + A - k ) ϵ = ω + A + k FIG. 8. (Color online) The dispersion relation for the eigenstates ofthe Bogoliubov equation with ( n , m ) = (1 , act zero-mode eigenfunction can be found in this limit [13].However, for finite ω , it does not have an exact expression, incontrast to the other modes whose zero-mode eigenfunctionsare found for finite ω using Sch(2) symmetry. Even if we donot know the exact dispersion relation, we can find a k coef-ficient by the general formula (4.9) with (4.11). This is shownby a dotted line in Fig 7. The eigenstates for ( n , m ) = (1 , ǫ, k , m ) and ( − ǫ ∗ , − k , − m ) always emerges in a pair inthe Bogoliubov equation.Figures 9 and 10 show plots of the zero-mode wavefunc-tions (4.2) and (4.3) for the KR1 and KR2 complex modes,respectively. In the vortex state n =
1, these two modes aredistinguished by the existence or absence of the node in thewavefunctions ( u , v ). The divergence of the phase fluctua-tions δϕ in the KR1 and KR2 modes indicates that the vor-0 FIG. 9. (Color online) Plot of w KR1 = ( u KR1 , v KR1 ) T [Eq. (4.2)], δρ KR1 = ( u KR1 + v KR1 ) f , and δϕ KR1 = ( u KR1 − v KR1 ) / (2 f ) with aconstant factor multiplied for visibility. We also plot the condensatewavefunction f ( r ) and its TF approximation f TF ( r ) [Eq. (2.5)] forreference. The parameters used are the same as Fig 6. The divergent δϕ around a vortex core suggests the existence of the core’s helicalmotion.FIG. 10. (Color online) Plot of w KR2 = ( u KR2 , v KR2 ) T [Eq. (4.3)], δρ KR2 = ( u KR2 + v KR2 ) f , and δϕ KR2 = ( u KR2 − v KR2 ) / (2 f ) with aconstant factor multiplied for visibility. We also plot the condensatewavefunction f ( r ) and its TF approximation f TF ( r ) [Eq. (2.5)] forreference. The parameters used are the same as Fig 6. The divergent δϕ around a vortex core suggests core’s helical motion. While theKR1 mode (Fig 9) has a node in the wavefunction ( u KR1 , v KR1 ), nonode is found for ( u KR2 , v KR2 ). tex core is oscillating in these modes, consistent with the ex-pression (4.15). Since the outer condensate is also oscillating[Eq. (4.16)], the picture in Fig. 1 (C) is indeed realized in thesemodes.Figure 11 shows a plot of the zero mode of the breathergiven in Eq. (4.4). We can see that the position of the node r node is well estimated by Eq. (4.24). In contrast to the KR1and KR2 modes, this mode has finite phase fluctuation δϕ at the vortex core r =
0, which indicates that this excita-tion shows no vortex-core motion. Therefore, this mode isregarded as a pure breather, not coupled to the Kelvin mode.Figure 12 is a zero-mode wavefunction of the Kelvin mode,i.e., the plot of the eigenfunction for the purple diamond dot ofthe k = FIG. 11. (Color online) Plot of w breather = ( u breather , v breather ) T [Eq. (4.4)], δρ breather = ( u breather + v breather ) f , and δϕ breather = ( u breather − v breather ) / (2 f ) with a constant factor multiplied for visibility. Wealso plot the condensate wavefunction f ( r ) and its TF approximation f TF ( r ) [Eq. (2.5)] for reference. The parameters used are the same asFig 6. The node position (4.24) is given by r node = .
26. The finite δϕ at r = k = f ( r ) and f TF ( r )are also plotted. This Kelvin-mode wavefunction does not have ananalytical expression, in contrast to other modes. is natural because both modes will reduce to the same zeromode w x -trans − i w y -trans in the trap-free limit ω →
0. However,this mode decays near the condensate surface, and thereforeit can be identified as a pure Kelvin mode, corresponding toFig. 1 (A).
VI. QUASI-MASSIVE-NAMBU-GOLDSTONE MODES
The collective excitations w KR1 , w KR2 , and w breather in anelongated 3D BEC, which are found by using 2D Schr¨odingersymmetry, can be identified as both quasi- and massive NGmodes. Here we explain why this is so. (Note: w phonon is anordinary NG mode associated with the U (1) SSB.)First, let us recall the concept of the massive NG modes[19–21]. These modes emerge in the systems with an exter-1nal field term which lowers the symmetry of the Lagrangian,but could be eliminated by performing some time-dependenttransformation for physical variables. The Bogoliubov-theoretical treatment of these modes is available in Ref. [13],Appendix G. If the external field is a Noether charge − µ Q ,the elimination is achieved by the transformation ˜ ψ = e − i µ Qt ψ .Therefore, if the symmetry group for the system with no ex-ternal field is denoted by G , that with the external field canbe expressed as G ′ = e i µ Qt G e − i µ Qt . A well-known typical ex-ample is the magnetic field term − BS z which breaks the spin-rotation symmetry. The massive NG mode is not gapless, butits existence is still robustly ensured by symmetry, and thevalue of the gap is determined only from the Lie algebra ofthe symmetry group.Next, we explain the concept of quasi-NG modes based onRef. [14] with a slight generalization. Let us consider a sys-tem described by several classical fields, whose Lagrangian iswritten by two terms: L = L + L . We assume that each termhas a group symmetry denoted by G L and G L . The symme-try of the total Lagrangian is then given by G L = G L ∩ G L .As proved in Ref. [14], if we have a family of solutionsparametrized by several continuous parameters, and if all ele-ments in this family satisfies L =
0, then we can construct aparameter-derivative quasi-zero-mode solution for all genera-tors of G L . (Some of them may be a “genuine” zero modeoriginating from the true Lagrangian symmetry G L .) Fur-thermore, we can find a dispersion relation for these modesby perturbation theory. In Ref. [14], the theory has beenconstructed for the case of L = −V , L = L time − T ,where L time = R d x P i i( ψ ∗ i ˙ ψ i − ˙ ψ ∗ i ψ i )2 is a time-derivative termof the Lagrangian appearing for the Schr¨odinger-type equa-tions, and T and V are kinetic and potential terms, respec-tively. H = T + V can be regarded as a Hamiltonian. Inparticular, an example of the complex O ( N ) model, where G L = G V = O ( N , C ) and G L = G T = U (3), has beendiscussed. In the condensed-matter example of the spin-2 Bose condensate [51], we have L = L time − H and L = −H , with L time = R d x P m = − ψ ∗ m ˙ ψ m − ˙ ψ ∗ m ψ m )2 , H = R d x (cid:16)P m = − |∇ ψ m | + c ρ + c | Θ | (cid:17) , and H = R d x c M ,where ρ = P m = − | ψ m | is a particle number density, Θ = P m = − ( − ψ m ψ − m is a singlet pair amplitude, and M = ( M x , M y , M z ) is a magnetization vector. The symmetry groupfor each term is given by G L = U (1) × S O (5) and G L = U (1) × S O (3). While the symmetry of the Lagrangian is G L = U (1) × S O (3), by an appropriate choice of couplingconstants c , c , c , we obtain the nematic phase where all thestates with vanishing magnetization can be a ground state [51–53]. The large degeneracy of this phase cannot be resolvedunless the quantum many-body e ff ects are included [54–56],and thus we get quasi-NG modes originating from the SSB of G L , i.e., the S O (5) symmetry [32].On the basis of the above-mentioned concepts of quasi-and massive NG modes, we can now explain why the modes w KR1 , w KR2 , and w breather in the harmonic trap are quasi-massive-NG modes. In the present system, we can decom-pose the Lagrangian into the two terms L = L time − H and L = −H with L time = Z d x i( ψ ∗ ˙ ψ − ˙ ψ ∗ ψ )2 , (6.1) H = Z d x | ∂ x ψ | + | ∂ y ψ | + g | ψ | + ω ( x + y )4 | ψ | ! , (6.2) H = Z d x | ∂ z ψ | . (6.3)Then, as explained in Subsec. III A, L has a 2D Schr¨odingersymmetry modified by T trap , that is, G L = T trap Sch(2) T − .(Recall that T − = T release .) Moreover, if we consider so-lutions with the z -translational symmetry, we get H = G L , as derived in Secs. III B and IV. These can be re-garded as quasi-NG modes in the sense that they have anorigin in the SSB of the partial Lagrangian L , but not thetotal L . The quasi-NG modes with the same origin is alsofound for a Skyrmion line in Ref. [34]. Furthermore, in the L , the external harmonic-trap term ω ( x + y )4 , which breaksthe translational symmetry and hence lowers the symmetry ofthe total Lagrangian, can be eliminated by the transformation˜ ψ = T release [ ψ ], and the zero-mode solutions are expressed asEq. (3.9), which are analogous to Eq. (G.8) in the AppendixG of Ref. [13]. They have finite gaps determined by symme-try consideration, which are ǫ = ω for w KR1,2 and ǫ = ω for w breather . Therefore, they are also regarded as massive NGmodes. This is why we can refer to these modes as quasi-massive-Nambu-Goldstone modes . VII. SUMMARY AND FUTURE OUTLOOK
To summarize, we have provided exact characteristics ofthe 3D collective excitations in an elongated BEC confinedby a harmonic trap, using the concept of the 2D Schr¨odingersymmetry and the Bogoliubov theory. We found four kindsof low-energy excitations whose existence is robustly guaran-teed by the Schr¨odinger-group symmetry, that is, the two KRcomplex modes, the one breather mode, and the Bogoliubovsound wave. We have determined their dispersion relationsanalytically and clarified their physical picture (Secs. III, IV,and V). We also have pointed out that the most basic excita-tion, i.e., the Kelvin mode, cannot be treated in terms of theSSB of the 2D Schr¨odinger symmetry.Furthermore, we have pointed out in Sec. VI that the KRcomplex modes and the breather mode can be regarded as thequasi-massive-NG modes, extending the generalized conceptsof the NG modes.We have constructed the theory of the excitations propa-gating along the z -axis in this paper. If the system length isfinite in the z direction, these excitations will be observed as astanding wave, which will be a more natural setting in ultra-cold atomic experiments. Our formalism should be extendedto the case in which the dependence of the system in the z -direction is small.2The concept of the quasi-massive-NG modes proposed inthis paper should be further investigated from various view-points in closely related recent popular physical topics, e.g.,application to the quantum turbulence [57, 58], the Higgsmodes [59–62], and the Nambu sum rules [63, 64] in singleand / or multi- component bosonic / fermionic superfluids, andso on. The splitting instability of multiple vortices in an elon-gated trap which may undergo the quantum turbulence wasdiscussed in Ref. [65]. The analytical study of the Kelvinmode for the small g regime will be also worth investigatingas was done in 2D in Ref. [66]. In this paper, we have ignoredthe quantum many-body e ff ects. If the quantum correction isadded, the Schr¨odinger symmetry of the 2D GP system willdisppear due to the quantum anomaly. Therefore, studyingthese e ff ects through the dispersion relations of the breathingmodes will be an important future work. Another directionfor future work will be a singular perturbation analysis [67] for the extrapolation of the solutions to the trap-free system ω →
0, where we expect only two NG modes, i.e. the Bogoli-ubov sound wave with linear dispersion and the Kelvin modewith logarithmic dispersion.
ACKNOWLEDGMENTS
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