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C. and , where neetoantcfield, electromagnetic an n pn0cagdbsn fmass of bosons charged spin-0 ing ∂ urn est a aumadmdu contributions, medium and vacuum has density current F tem- at critical vanishes the energy at gap roton- whose disappears plasmon a perature, that exhibits longitudinal minimum which local gas, the Bose type of relativistic the relation of mode dispersion the in work [12]. seminal Landau the of to according major re- superfluidity, ultimately a for were sponsible was excitations phase collective superfluid Such discovery. (condensed) the phonon-roton in observa- collective of modes the [11] scattering There, neutron via tion scalars. charged self-interacting by h xrmmcniinlast awl’ equation Maxwell’s to leads ∂ condition extremum The o tutr ntecnesdpaeo h a,inspired liquid gas, as such the superfluids of of phase physics the condensed by the in structure for nEcienfntoa nerloe clr satisfying scalars of over logarithm integral the φ minus functional i.e., ob- Euclidean energy, be an free may the gas the from of tained response electromagnetic the rates ietmeaueadcag est s[13] is density charge and temperature nite the density into it charge drive phase. the and normal system, in the rotons disorder seeing that oscillations are we that suggests ¯ µ µ (0 µν S h ffcieato o classical a for action effective The ehv on tutrssmlrt uefli rotons superfluid to similar structures found have We h cinfrsaa unu lcrdnmc tfi- at electrodynamics quantum scalar for action The F ≡ ~x, µν = = ( = ) iu htdsper ttetransition the at disappears that nimum ∂ Z ffcieidxo refraction of index effective e k 4 ∂ 0 = crmgei rpgto oe that modes propagation ectromagnetic B β µ − S cfil narltvsi da a of gas ideal relativistic a in field ic dτ φ A etmeaueadcag density. charge and temperature te eff ( s,w hwta h longitudinal the that show we ase, J ,∂ ξ, Z = ,~xβ, ν ν = d ( − x i 3 ℏ ) ) = ) x 4 1 ∂ F ν ( = A 4 1 µν F µ F UFRJ, - o and , µν c δ µν rln( Tr F ,and 1, = µν rln Tr + φ L, 1 D + sacmlxsaa describ- scalar complex a is , − ¯ 3 zil µ D D ¯ ¯ φ 2 µ φ ( = + − n ∗ D eff β D ¯ m ¯ ∂ ¯ µ 2 m µ = A 2 φ T 1 = + ) µ − c /δA + − hsstrongly This . n charge and 4 m 1 htincorpo- that e described He, ieA m /T 2 ν 2 ( φ . µ x . ∗ ) .The ). φ φ ) with , A , µ (1) (2) e is , 2
(v) (m) Jν = Jν + Jν . The latter defines Pµν , through in the ground state, and another due to the particles in ′ (m) excited states. The longitudinal responses, ǫL = ǫ + ǫ iqµPµν (q) = −Jν (q), whose components are the po- ′ j and νL = ν + ν , become larization P~ , P4j = iP , and the magnetization M,~ Pij = k 2 2 2 −ǫijkM . ∗ q − 4m q ∗ ǫ (ω, |~q|)=1+ C − ω2 − B , Expanding for weak classical fields Aµ up to quadratic L e q4 − 4m2ω2 |~q|2 T (m) terms yields a linear response approximation Jµ = 2 2 2ωe ∗ ∗ ω ∗ Πµν (q)Aν (q), where Πµν is the field theory polarization νL(ω, |~q|)=1 − 2 +2C +2AT − 2 2 BT . (5) 2 q |~q| tensor defined below. One may show [2] that q Pµν = ΠµσFνσ − ΠνσFµσ , and define Hµν ≡ Fµν + Pµν , whose 2 2 (c) ωe = e η /m is the longitudinal electric plasmon fre- ~ j ∗ ∗ ∗ ∗ components are the electric displacement D, H4j = iD , quency; AT and BT are the functions A and B with k ′ and the magnetic induction H~ , Hij = −ǫijkH . the modification n(p) → n (p). ′ The field theory polarization tensor Πµν contributes c) For T = 0, all the charge condenses, so nc(p) → 0, ∗ ∗ two terms to the induced current, the tadpole and the and the scalar functions AT and BT vanish. Neglecting ∗ thermal bubble Feynman graphs the vacuum contribution (C << 1), one obtains
∞ 2 2 2 + 3 2 q − 4m 2e d p δµν ǫL(ω, |~q|)=1 − ωe , Πµν (q)= − q4 − 4m2ω2 β (2π)3 p¯2 + m2 n=−∞ X Z 2ω2 +∞ ν (ω, |~q|)=1 − e , (6) e2 d3p (2¯p + q )(2¯p + q ) L 2 + µ µ ν ν . (3) q β (2π)3 (¯p2 + m2)[(¯p + q)2 + m2] n=−∞ Z for q2 < 4m2. For q2 > 4m2, imaginary parts appear X 2 2 2 3/2 ImǫL(ω, |~q|) = [e /48π][(1 − (4m )/(q )] . As |~q| → 0, The sum is over bosonic Matsubara frequencies p4 = (v) (m) the longitudinal responses at T = 0 are the same as those 2nπT . Writing Πµν = Πµν + Πµν , symmetry and obtained in [2] for the relativistic electron gas, ǫL =1 − (m) 2 (m) 2 2 2 2 gauge invariance [14, 15] lead to Π44 = −q B, Π4i = ωe /ω and νL =1 − 2ωe /ω . 2 (m) 2 2 −q qˆ4qˆiB, Πij = −q (δij − qˆiqˆj ) A + δij qˆ4 B , and We now turn our attention to the propagation of col- (v) lective modes in the gas, and evaluate how the medium Π = −(q2δ −q q )C(q2), whereq ˆ ≡ q /|~q|, and q2 = µν µν µ ν µ µ affects the photon propagator in the Bose gas. Just as ~q2 + q2. The scalar functions A(|~q|, q ), B(|~q|, q ), and 4 4 4 in the case of the relativistic electron gas, we obtain the C(q2) are given by integrals containing the Bose-Einstein photon propagator Γ−1 in the form occupation number for bosons and antibosons n(p) = µν − − −1 −1 n+(p)+ n (p)= eβ(ωp ξ) − 1 + eβ(ωp+ξ) − 1 L T −1 Pµν Pµν λ qµqν The tensors Hµν and Pµν provide the constitutive Γµν = 2 + 2 + 2 2 , (7) −q ǫL −q (νL + 1) q q equations, Dj = ǫjkEk + τjkBk, Hj = νjkBk + τjkEk, −1 ′ L with ν ≡ µ . We may write ǫjk = ǫδjk + ǫ qˆj qˆk, where we have introduced the projectors Pµν ≡ P + ′ µν νjk = νδjk + ν qˆj qˆk, and τjk = τǫjklqˆl. The eigenval- T 2 T T ′ ′ Pµν = δµν − qµqν /q , with Pij = δij − qˆiqˆj , and P00 = ues of ǫjk are ǫ + ǫ and ǫ. The eigenvector of ǫ + ǫ is T P0i = 0, and λ a gauge parameter. The poles of the alongq ˆ, thus longitudinal, whereas the two eigenvectors photon propagator correspond to collective excitations with eigenvalues ǫ are transverse toq ˆ. The same occurs ′ and yield their dispersion relations. for νjk, with eigenvalues ν + ν and ν. τjk is clearly In the longitudinal propagator, there is a pole transverse. whenever the longitudinal electric permittivity vanishes a) For T >Tc, the permittivities and inverse per- ǫL(ω, ~q) = 0, leading to a longitudinal plasmon mode. meabilities are determined by the three scalar functions ∗ ∗ ∗ For ǫL(ω, ~q) nonzero, Maxwell’s equations lead to trans- A , B , and C . The asterisk denotes continuation to ~ + verse fields ~q · E = 0 that vanish when contracted with Minkowski space q4 → iω − 0 of the Euclidean scalar L 2 2 2 Pµν , so that q = 0 is not realized in this case. The functions A(|~q|, q4), B(|~q|, q4) and C(~q + q4 ). From now −1 on, we shall use the Minkowski definition q2 = ω2 − |~q|2 transverse propagator has a pole whenever µL (ω, ~q) = 2 2 νL(ω, ~q) = −1, which corresponds to collective oscilla- (qE →−qM ). (c) tions of the current density; it has another pole whenever b) For T 2 2 2 ωT = ωe + |~q| . (8) The existence of a photonic mode ωγ = |~q| with the speed of light in vacuum is another indication that the 2 The expression for ωL has two branches: one beginning modulus of the effective index of refraction of the gas − at ωL = ωe, at |~q| = 0; the other, which leads to is equal to one. Since, both ǫeff < 0 and νeff < 0, for pair creation, beginning at ωL+ = 2m, at |~q| = 0. For ωγ(|~q|) ≤ ω<ωT (|~q|), for any temperature and chemical 2 2 ωe < 4m , the longitudinal mode will show a local min- potential, one may use Snell’s law to show that neff = −1 imum at low values of |~q|, known as negative dispersion in that region, a negative refraction typical of a left- [16]. This behavior will appear only in the relativistic handed material [6, 7]. The shadowed sections of Fig.2 regime, since in the non-relativistic limit our expression illustrate those regions for both T = 0 and T 6= 0. Sim- 2 2 4 2 reduces to ωL = ωe + |~q| /4m , the same result obtained ilar behavior was obtained in the analysis of the rela- in [17] for a charged Bose gas within the random phase tivistic electron gas, confirming our hypothesis that the approximation. Fig.1 shows the dispersion curves of the relativistic nature of the gas was the key ingredient to two possibilities for the longitudinal mode, and the trans- achieve left-handed behavior. verse modes. Fig.2(a) shows that, in the condensed phase, the re- gion where the gas exhibits left-handed behavior shrinks + ωL with increasing temperature, at least up to Tc. Some- where above Tc (see Fig.2.(b)), the region expands and, 2m for the case of ultra-relativistic densities (η ≥ (mc/ℏ)3), ] ℏ / ωT 2 it begins to expand at the critical temperature T = Tc, mc [ as illustrated in Fig.2(c). Fig.3 shows the plasmon en- ω ωe ergy at |~q| = 0, normalized by its value at T = 0, as a - ωL function of T/Tc for various densities. In contrast to the non-relativistic case, the transverse ωγ plasmon frequency at |~q| = 0, known as plasmon gap 0.0 0.5 1.0 1.5 2.0 energy, is a function of temperature for relativistic densi- q[mc/ℏ] ties. In fact, the two regimes just described correspond to T boson density η = 10(mc/ℏ)3. the transverse plasmon gap energy begins to increase linearly with temperature. This temperature increases with decreasing density, and has the critical temperature as its lower bound in the ultra-relativistic regime. Nu- We stress that the responses discussed thus far are merical estimates for τ ≡ k T /mc2, η˜ ≡ η(~/mc2) give NOT the ones appearing in Maxwells equations. In or- t B t (τt, η˜) = (5.47, 10); (1.84, 1); (0.74, 0.1); (0.37, 0.01). We der to find the exact analog of the situation proposed by note that similar behavior is observed in the relativis- Veselago [7], we need the effective electric permittivity tic electron gas, where kBTt may be associated with the and effective magnetic permeability that appear in the Fermi energy. two equations with sources. Following [6], we may define For ultra-relativistic densities, one may perform a high the effective responses as temperature hard thermal loop expansion [13], and derive µ|~q| |~q| analytic solutions for the transverse plasmon frequency µ = ; ǫ = ǫ + τ. (9) 2 2 2 2 eff eff ωT (|~q|). At |~q| = 0, we obtain ωT (0) = ωe + e T /9, for |~q|− ωµτ ω 2 2 2 T ≤ Tc, and ωT (0) = e T /9, for T ≥ Tc. For T (a) (b) The existence of roton-like structures has been pre- 1.5 1.5 dicted for condensates of dipolar particles [18–22], of ] ] ¦ / / nonpolar atoms under the action of an intense laser light 2 1.0 2 1.0 [mc [mc [23], and for Rydberg-excited condensates [24]. Recently, ¥ T=0 T=13.9 0.5 0.5 these vortex-like quasi-particles have been observed for £ TTc T§ Tc → → the first time in a Bose-Einstein condensate of ultra-cold γ=|q| ¨γ=|q| 0.0 0.0 atoms [25]. However, in a charged superfluid, it has been 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 → → shown that the phonon mode of the neutral superfluid © q|| [mc/¤] q|| [mc/ ] is pushed to a finite plasmon frequency ωp, whereas the 1.4 roton mode is more or less unaffected [26, 27]. In the (c) ] Normal phase charged case, the spectrum of the superfluid field shows ¢ 1.2 / Condensed phase 2 a plasmon excitation that turns into a roton excitation, ] mc 1.0 with a gap energy ∆(|~qrot|) for higher |~q|. (0) T 0.8 0.05 ¡ (a) 0.6 0.04 T > Tc T = 0 ] ℏ 0.4 / 0.03 0.0 0.5 1.0 1.5 2.0 2.5 2 mc [ T/T L 0.02 0 < T < Tc c ω 0.01 Δ FIG. 2: The shadowed regions illustrate where the RBG is a LHM: gap (a) in the condensed phase, the LHM region decreases with the 0.00 temperature until we reach the critical temperature Tc; (b) above 0.00 0.05 0.10 0.15 0.20 0.25 0.30 the critical temperature Tc, the LHM region starts to increase with q[mc/ℏ] the temperature; (c) transverse plasmon ωT at |~q| = 0 as a function 0.04 of the relative temperature. Calculations were performed for η = 3 2 (b) 10(mc/ℏ) and Tc = 5.47mc /kB . 0.03 ] ℏ / 2 mc 0.02 1.4 [ gap Δ 1.2 0.01 T 1.0 0.00 Ω 0.2 0.4 0.6 0.8 1.0 )/ 0 0.8 T/Tc ( T η =10 ω 0.6 η =1 FIG. 4: (a) Dispersion curves for longitudinal modes for T > Tc Normal phase -1 0.4 η =10 (solid and dotted-dashed line), 0 FIG. 3: Normalized transverse plasmon frequency in the long- wavelength limit (|~q| = 0) as a function of the relative temperature In the present study of a charged relativistic Bose gas, for various densities [in units of (mc/ℏ)3] of the Bose gas in units the dispersion relation of the longitudinal plasmon mode of ΩT ≡ ωT (|~q| = 0, T = 0). In the shadowed region, below the shows an ordinary plasmon, near |~q| = 0, and a roton critical temperature Tc, the gas is in the Bose-Einstein condensed excitation, near the value of |~q| that corresponds to the phase (BEC). For each density η, the critical temperatures [in units 2 of (mc /kB )] are Tc = 5.47(η = 10), Tc = 1.72(η = 1), Tc = local minimum. As the temperature is increased, the −1 −2 0.53(η = 10 ), and Tc = 0.14(η = 10 ). gap energy of the local minimum decreases, and Fig.4(b) shows how it depends on the temperature. We may view the gap energy as an order parameter, which vanishes at T = Tc. For 0 It should be noted that those results agree remarkably We remark that the Lagrangian density in our ap- well with the non-relativistic results reported in the lit- proach does not include a self-interaction term λ(φ∗φ)2. erature, obtained via a different route [28]. The condensate is introduced by performing the substitu- Figures 4 and 5 suggest that thermal effects induce the tion (4), which yielded T = 0 results for the longitudinal rotons that will contribute to disorder the system. Ther- plasmon mode in agreement with the literature in both mally induced rotons were also present in the bosonic relativistic [30] and nonrelativistic limits [17]. We plan to spectrum of a two-dimensional dilute Bose gas [29], where include self-interactions, either directly or induced by in- it was argued that their emergence is a consequence of tegration over quantum electromagnetic fields, in a future the strong phase fluctuation in two dimensions. investigation. In the non-relativistic limit, that inclusion will be useful in the description of the superfluidity of liq- 1.8 7 (a) uid Helium at low temperatures [31]. In the relativistic 1.6 6 (b) -5 -3 limit, it could possibly help in the study of the relativis- Tc = 6.8 × 10 Tc = 7.1 × 10 ] 1.4 ] 5 ℏ -4 -8 ℏ tic Bose plasma found in astrophysical scenarios such as / η = 9.3 × 10 / η = 10 2 2 4 mc 1.2 neutron stars, where the creation of charged pion pairs mc 4 3 - - 3 10 10 and pion condensation may take place [30, 32]. [ 1.0 [ L L ω ω 2 -3 0.8 T=5.3×10-5 T=6.8×10 T=2.6×10-5 1 T=5.3×10-3 0.6 T=0 T=0 0 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10 12 2 2 q[10- mc/ℏ] q[10- mc/ℏ] 2.0 5 (c) (d) T = 0.52 Tc = 1.72 1.5 c 4 -1 ] η = 10 ] η = 1 ℏ ℏ / / 2 2 mc mc 1 1.0 1 3 - - 10 10 [ [ L L ω T=0.47 ω 0.5 2 T=1.43 T=0.31 T=1 T=0 T=0 0.0 1 0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 -1 q[10 mc/ℏ] q[10-1mc/ℏ] Acknowledgments FIG. 5: Dispersion curves for longitudinal modes for various tem- 2 peratures below the critical temperature Tc (in units of mc /kB), and various densities η [in units of (mc/ℏ)3]. In (a) and (b), calcu- lations were performed for non-relativistic densities and tempera- tures; in (c) and (d), for relativistic and ultra-relativistic. The authors would like to thank the Brazilian Agencies CNPq and CAPES for partial financial support. [1] M. Grether, M. de Lanno, and G. A. Baker, Physical Schultz, Physical Review Letters 84, 4184 (2000). Review Letters 99, 200406 (2007); A. Haber and H. Wel- [9] N. Engheta and R.W. 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