Sine-Gordon model coupled with a free scalar field emergent in the low-energy phase dynamics of a mixture of pseudospin-1/2 Bose gases with interspecies spin exchange
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Sine-Gordon model coupled with a free scalar field emergent inthe low-energy phase dynamics of a mixture of pseudospin- Bosegases with interspecies spin exchange
Li Ge and Yu Shi ∗ Department of Physics and State Key Laboratory of Surface Physics,Fudan University, Shanghai, 200433, China
Abstract
Using the approach of low-energy effective field theory, the phase diagram is studied for a mixtureof two species of pseudospin- Bose atoms with interspecies spin-exchange. There are four mean-field regimes on the parameter plane of g e and g z , where g e is the interspecies spin-exchangeinteraction strength, while g z is the difference between the interaction strength of interspeciesscattering without spin-exchange of equal spins and that of unequal spins. Two regimes, with | g z | > | g e | , correspond to ground states with the total spins of the two species parallel or antiparallelalong z direction, and the low energy excitations are equivalent to those of two-component spinlessBosons. The other two regimes, with | g e | > | g z | , correspond to ground states with the totalspins of the two species parallel or antiparallel on xy plane, and the low energy excitations aredescribed by a sine-Gordon model coupled with a free scalar field, where the effective fields arecombinations of the phases of the original four Boson fields. In (1+1)-dimension, they are describedby Kosterlitz-Thouless renormalization group (RG) equations, and there are three sectors in thephase plane of a scaling dimension and a dimensionless parameter proportional to the strength ofthe cosine interaction, both depending on the densities. The gaps of these elementary excitationsare experimental probes of the underlying many-body ground states. PACS numbers: 03.75.Mn, 11.10.Hi, 11.10.Kk ∗ Electronic address: [email protected] . INTRODUCTION Sine-Gordon model is important in field theory and statistical mechanics, and its renor-malization group equations define the Kosterlitz-Thouless universality class for a class of(1+1)-dimensional quantum systems and two-dimensional classical systems [1, 2]. In thispaper, we show that a sine-Gordon model coupled with a free scalar field emerges in thephase dynamics of a mixture of two different species of pseudospin- Bose gases with in-terspecies spin-exchange interaction. Interestingly, here the scalar field described by thesine-Gordon model and the free scalar field are two different combinations of the phases ofthe four bosonic fields in this system.A mixture of two different species of pseudospin- Bose gases with interspecies spin-exchange interaction exhibits novel features beyond a single species of spinor Bose gas aswell as a mixture of two species without interspecies spin exchange [3–10]. In this model,each atom has an internal degree of freedom represented as a pseudospin with basis states | ↑i and | ↓i , while there are two species of atoms, with the atom number of each speciesconserved. It can be described by the following Hamiltonian density, H = X ασ ψ † ασ ( − m α ∇ + V ) ψ ασ + 12 X ασσ ′ g ( αα ) σσ ′ | ψ ασ | | ψ ασ ′ | + X σσ ′ g ( ab ) σσ ′ | ψ aσ | | ψ bσ ′ | + g e ( ψ † a ↑ ψ a ↓ ψ † b ↓ ψ b ↑ + ψ † a ↓ ψ a ↑ ψ † b ↑ ψ b ↓ ) , (1)where α = a, b represents the two species and σ = ↑ , ↓ . V = V ( x ) is the external poten-tial, g ( αα ) σσ ′ , g ( ab ) σσ ′ and g e are the interaction strengths for intraspecies scattering, interspeciesscattering without spin exchange, and interspecies spin-exchange scattering respectively,proportional to the corresponding scattering lengths. For pseudospin- atoms, intraspeciesscattering strengths with and without spin-exchange are the same [11]. For simplicity, weassume g ( αα ) σσ ′ = g α for any σ and σ ′ , g ( ab ) ↑↑ = g ( ab ) ↓↓ = g s and g ( ab ) ↑↓ = g ( ab ) ↓↑ = g d . We define S αi ( x ) = Ψ † α s i Ψ α , where Ψ α ( x ) ≡ ( ψ ↑ ( x ) , ψ ↓ ( x )) T , s i = τ i / τ i being the Pauli matrix,( i = x, y, z ). Then H can be rewritten as H = X α Ψ † α ( − m α ∇ + V )Ψ α + g a | Ψ a | + g b | Ψ b | + g ab | Ψ a | | Ψ b | +2 g e ( S ax S bx + S ay S by ) + 2 g z S az S bz , (2)where g ab ≡ g s + g d , g z ≡ g s − g d . It can be seen that g a , g b , and g ab characterize the usualdensity-density interactions, while g z and g e characterize the spin coupling between the twospecies. 2e make the presumption that g a > g b > g a g b > g ab , which is needed for thestability of the system and can be naturally satisfied in reality [10]. We study the phasediagram in the space of the parameters g e and g z , by using the approach of low energyeffective field theory. The regime of g e > g z > g z > | g e | , the ground state is withthe total spins of the two species antiparallel along z direction. In the regime of g z < −| g e | ,the ground state is with the total spins of the two species parallel along z direction. In theregime of g e > | g z | , the ground state is with the total spins of the two species antiparallelon xy direction. In the regime of g e < −| g z | , the ground state is with the total spins ofthe two species parallel on xy direction. Then we focus on the case of | g e | > | g z | in (1+1)-dimension. Without approximating the cosine interaction term, the low energy excitationscan be described by a sine-Gordon model coupled with a free scalar field, both fields beingcombinations of the phases of the original four boson field. It turns out that for given g e and g z with | g e | > | g z | , there are three phases according to a scaling dimension and adimensionless parameter proportional to | g e | . Both these two parameters depend on thedensities of the two species. II. PHASE DIAGRAM ON g e − g z PARAMETER PLANE
There is a symmetry between parameter points ( g e , g z ) and ( − g e , g z ). Consider the trans-formation ψ ′ a ↑ ≡ − ψ a ↑ , ψ ′ a ↓ ≡ ψ a ↓ , ψ ′ b ↑ ≡ ψ b ↑ , ψ ′ b ↓ ≡ ψ b ↓ . The Hamiltonian density in terms ofthe primmed operators in the parameter point ( g e , g z ) has the same form as the Hamiltoniandensity in terms of the unprimed ones in the parameter point ( − g e , g z ).Now consider the case of g z > | g e | . Then from the Hamiltonian density (2), it is easy tosee that in the ground state, S a and S b must align oppositely in the z direction. We choosethe mean field values in the ground state to be with ψ a ↑ = √ n a , ψ a ↓ = 0, ψ b ↑ = 0, ψ b ↓ = √ n b so that S a = n a ˆ z and S b = − n b ˆ z , where ˆ z is the unit vector in the z direction, n α is the totaldensity of species α , ( α = a, b ). The low energy dynamics is dominated by the fluctuationsof ψ a ↑ and ψ b ↓ , as the fluctuations of ψ a ↓ and ψ b ↑ , whose mean field values are zero, must beof the amplitudes rather than the phases, and thus increase the energy.3imilarly, in the case of g z < −| g e | , the ground state is that with S a and S b parallel in the z direction. We can choose the mean field values in the ground state to be with ψ a ↑ = √ n a , ψ a ↓ = 0, ψ b ↑ = √ n b , ψ b ↓ = 0, so that S a = n a ˆ z and S b = n b ˆ z . The low energy dynamicsis dominated by the fluctuations of ψ a ↑ and ψ b ↑ , as the fluctuations of ψ a ↓ and ψ b ↓ , whosemean field values are zero, must be of the amplitudes and thus increase the energy.In these two cases, which can be represented in a unified form as | g z | > | g e | , the systembehaves like a mixture of two species of spinless Boson gases, with the effective Hamiltoniandensity H = X α Ψ † α ( − m α ∇ + V )Ψ α + g a | Ψ a | + g b | Ψ b | + g ab − | g z | | Ψ a | | Ψ b | , (3)whose excitation spectra are [12] ω = 12 ( ε a + ε b ) ± q ( ε a − ε b ) + 4 E a E b n a n b ( g ab − | g z | ) , (4)where we have introduced ε α ≡ E α (2 g α n α + E α ) , (5)with α = a, b , and E α = k m α being the free particle energy of species α . All spectra aregapless, as in the usual case of phonon-like Goldstone modes. That is, ω → k → g e > | g z | , which has been discussed previously [10], the ground state is thatwith S a and S b antiparallel on the xy plane. One can choose the ground state to be with ψ a ↑ = ψ a ↓ = q n a / ψ b ↑ = − ψ b ↓ = q n b / g e < −| g z | , the ground state is that with S a and S b parallel onthe xy plane. One can choose the ground state to be with ψ a ↑ = ψ a ↓ = q n a / ψ b ↑ = ψ b ↓ = q n b / | g e | > | g z | , theeffective Lagrangian describing the phase fluctuations is, with | g e | replacing g e in the resultfor g e > g z > L eff = 12 ( ∂ t Γ T ) A − ( ∂ t Γ) −
12 ( ∇ Γ T ) M − ( ∇ Γ) + | g e | n a n b cos (2 γ ) (6)where Γ = γ γ γ γ = √ √ √ √ −
12 12 − − −
12 12 Φ a ↑ Φ a ↓ Φ b ↑ Φ b ↓ , (7)4ith Φ ασ being the phase of ψ ασ , A ≡ g a g ab − | g e | g ab − | g e | g b | g e | η + + g z | g e | η − | g e | η − | g e | η + − g z , (8)with η ± ≡
12 ( n b n a ± n a n b ) ,M − ≡ n a m a n b m b ξ + ξ − ξ − ξ + , (9)with ξ ± ≡
12 ( n a m a ± n b m b ) . In (3+1)-dimension or (2+1)-dimension, the fluctuation of γ is largely suppressed andwe can make the approximation that cos(2 γ ) ≈ − γ , subsequently, the four excitationspectra can be obtained as [10], ω I,II = k (cid:20) g a n a m a + g b n b m b ∓ s ( g a n a m a − g b n b m b ) + ( g ab − | g e | ) n a n b m a m b (cid:21) , (10) ω III,IV = 12 (cid:20) Bk + ∆ ∓ √ Ck + Dk + ∆ (cid:21) , (11)where ∆ = | g e ( n b n a + n a n b ) − | g e | g z | n a n b , B ≡ | g e | ( n b m a + n a m b ), C ≡ g e ( n b m a − n a m b ) + g z n a n b m a m b , D ≡ | g e | n a n b [ g e ( n b m a − n a m b )( n b n a − n a n b ) − | g e | g z ( n b m a + n a m b ) + 2 g z ( n a m a + n b m b )]. Under the conditions g a >
0, 4 g a g b > g ab and | g e | > | g z | , all these excitations have real energies for any k ,guaranteeing the stability of the ground state.It can be seen that ω IV has a gap ∆ while the other three excitations are gapless. Thatis, as k → ω I,II,III →
0, but ω IV → ∆.Therefore in (3+1)-dimension, we obtain the mean-field phase diagram as shown in Fig. 1.5 IG. 1: Mean-field phase diagram on g e − g z phase plane in (3+1)-dimension. In regime A, g z > | g e | , the ground state is ψ a ↑ = √ n a , ψ a ↓ = 0, ψ b ↑ = 0, ψ b ↓ = √ n b . In regime B, g z < −| g e | ,the ground state is ψ a ↑ = √ n a , ψ a ↓ = 0, ψ b ↑ = √ n b , ψ b ↓ = 0. In regime C, g e > | g z | , the groundstate is ψ a ↑ = ψ a ↓ = p n a / ψ b ↑ = − ψ b ↓ = p n b /
2. In regime D, g e < −| g z | , the ground state is ψ a ↑ = ψ a ↓ = p n a / ψ b ↑ = ψ b ↓ = p n b / III. RENORMALIZATION GROUP ANALYSIS IN THE CASE OF | g e | > | g z | IN(1+1)-DIMENSION
Reconsider the case of | g e | > | g z | . In (1+1)-dimension, the fluctuation is important andwe must take into account the whole effect of cos(2 γ ) term, which is the only interactionterm in L eff , where γ and γ are both free fields and not coupled to γ . Hence we can focus6n the sine-Gordon field γ coupled to a free scalar field γ , L = 12( g e − g z ) (cid:18) ∂ t γ ∂ t γ (cid:19) | g e | η + − g z −| g e | η − −| g e | η − | g e | η + + g z ∂ t γ ∂ t γ − (cid:18) ∂ x γ ∂ x γ (cid:19) ξ + ξ − ξ − ξ + ∂ x γ ∂ x γ + | g e | n a n b cos (2 γ ) . (12)We now make a renormalization group (RG) analysis of the cosine interaction term dueto spin exchange in (1+1)-dimension, by following the approach in [1]. It turns out that itstill belongs to the Kosterlitz-Thouless universality class. But the novelty is that the scalarfields are now combinations of the phases of the original bosonic fields.Let us define x ≡ vt , with v ≡ r ( g e − g z ) ξ + | g e | η + + g z ) , and two dimensionless field variablescorresponding to γ and γ ϕ ≡ [ ξ + ( | g e | η + + g z )2( g e − g z ) ] γ ,χ ≡ [ ξ + ( | g e | η + + g z )2( g e − g z ) ] γ . Then the action can be written as S = S + S I , (13)where S ≡ R L d x , S I ≡ R L I d x , with L = 12 (cid:18) ∂ ϕ ∂ χ (cid:19) f f f ∂ ϕ∂ χ + 12 (cid:18) ∂ ϕ ∂ χ (cid:19) pp ∂ ϕ∂ χ , (14) L I = λa cos ( βχ ) , (15)where f ≡ | g e | η + − g z | g e | η + + g z ,f ≡ − | g e | η − | g e | η + + g z ,p ≡ ξ − ξ + ,β ≡ ξ + ( | g e | η + + g z )2( g e − g z ) ] − ,λ ≡ | g e | n a n b a v , is the short range cut-off, which is the coherence or healing length, and can be estimatedto be ¯ h [ m b ( g a n a + g b n b + ( g s + g d − g e ) n a n b )] − / , assuming m a ≥ m b . λ is dimensionless.From L the free propagator of χ is obtained as G χ ( k ) = f k + k k ( f k + k ) − ( f k + pk ) = 1 k f cos θ + sin θf cos θ + sin θ − ( f cos θ + p sin θ ) , (16)where k = k + k , tan θ = k k .Since the interaction term does not involve ϕ field, we only need to split χ into the fastand slow components, χ Λ ( x ) = χ Λ ′ ( x ) + h ( x ) , (17)where χ Λ ′ ( x ) ≡ X k< Λ ′ e ikx χ k , (18) h ( x ) ≡ X Λ ′
From the RG equations, one obtains d ( t ) − ακ (0) d ( λ ) = 0 , (33)which is similar to the equation in the pure SG model [1], except that α and κ are notconstant here. Moreover, dκ (0) dl and dαdl are both proportional to λ . Hence to the order of λ , one can replace ακ (0) d ( λ ) as d ( ακ (0) λ ). Thus we arrive at the following equation, t − y = µ , (34)where y ≡ √ ακ (0) λ, (35) µ represents a constant. For given g e and g z with | g e | > | g z | , Equation (34) determines thephase diagram of the model on the plane ( t, y ) in the regime where t and y are small, asschematically shown in Fig. 2.It can be seen that the ( t, y ) phase space is divided to three sectors, namely, weak coupling,strong coupling and crossover sectors. In the weak coupling sector, the effective theory scalesto a Gaussian model, y ( l ) → l → ∞ , and the spectrum is massless, while in the crossoverand strong coupling sectors, the coupling constants flow away from the Gaussian fixed lineand the spectrum has a mass gap.Note that both t and y depend not only on the interspecies spin-exchange coupling, butalso on the densities. Consequently the phase is dependent not only on the interactionstrengths, but also on the densities of the two species, which can be easily adjusted inexperiments. To illustrate this explicitly, let us set n a = n b = n so that f = | g e |− g z | g e | + g z , f = 0, p = m b − m a m b + m a and thus κ and α as well, are all independent of n , while β ∝ n − and λ ∝ n .10 IG. 2: Phase diagram of our emergent sine-Gordon model coupled with a free field in (1+1)-dimension, with y ≡ √ ακ (0) λ , t ≡ D − D is a scaling dimension. There are two separatrices t = ± y that divide the phase plane into three sectors: (1) t ≥ y , the weak coupling (WC) sector; (2) | t | < y , the crossover (C) sector; (3) t ≤ − y , the strong coupling (SC) sector. The RG flows aresimilar to the pure sine-Gordon model. Therefore D ∝ n − and y ∝ n . Then according to Fig. 2, for small enough n , the system isin the weak coupling phase. For large enough n , the system is in the strong coupling phase.Therefore, following the change of density, the system goes through phase transitions.The mass gap in the crossover and strong coupling sectors can be qualitatively obtained.In the strong coupling sector, µ is real, while in the crossover sector, µ is purely imaginary.Following [1], it can be found that the mass gap in the strong coupling sector is M = Λ( y t ) /t , − t ≫ y , Λ exp( − /y ) , µ ≪ | y | , (36)where the subscript “0” means the bare values, that is, the values measured in experiments,while in the crossover sector, the mass gap is M = Λ exp( − π/ | µ | ) , (37)with | µ | ≫ | t | .The scaling behavior of the mass gap may be observed in experiments. Taking M =Λ exp( − /y ) as an example. If n a = n b = n , we have y = √ ακ (0) λ ∼ n , then the relationbetween M and n may be investigated.The last two of the RG equations (32) determine the RG flows of the couplings betweenthe fields ϕ and χ . Since α >
0, it is easy to find that f = 0, p = 0 is the only stablefixed point of the two equations, namely, whatever the initial values of f and p are, they11nevitably flow to 0. Moreover, the larger λ is, the more rapidly f and p flow to 0. It is likethat its strong self interaction “traps” the field χ and separate it from ϕ . If the bare valueof λ were 0, there would be no RG flows of f and p .Also note that if we diagonalize L in (14), then L I in (15) becomes a cosine term ofcosine of a linear combination of the two fields, of which the RG analysis is quite difficult.Hence we use the above approach instead.The elementary excitations studied here can be experimentally measured by using theBragg spectroscopy. The gap in a collective mode is a novel feature absent in the BECmixtures previously studied. The two key parameters g e and g z both originate from theinterspecies spin-dependent scattering, thus they are roughly of the same order of magnitude.We expect that the excitation gaps can be detected in experiments and are indications ofthe underlying many-body ground states. V. SUMMARY
We have developed a low energy effective theory for a mixture of two species of pseudospin- Bose gases and explore the phase transitions in the space of the parameters g e and g z ,where g e is the interspecies spin-exchange interaction strength, while g z is the differencebetween the strengths of equal-spin and unequal-spin interspecies interaction without spinexchange. The phase diagram on the plane of parameters g e and g z is shown in Fig 1. Inthe regime of | g z | > | g e | , the system is effectively described by a two component model, andthe excitation spectra are gapless. In the regime of | g e | > | g z | , the system is described by afour effective fields, which are combinations of the phases of the four original boson fields.There is a cosine interaction term of one of the effective field, which can be approximatedas a square in (3+1)-dimension. There are three gapless modes and one gapped mode.In (1+1)-dimension, the effective theory in the regime of | g e | > | g z | is a novel realiza-tion of a sine-Gordon model coupled with a free scalar field, on which we have made arenormalization analysis. Described by Kosterlitz-Thouless equations, the phase space isfurther divided into three sectors, as shown in Fig. 2, according to a scaling dimension t ≡ π [ g e − g z ) ξ + ( | g e | η + + g z ) ] / κ (0) − y = √ ακ (0) λ , where κ is a cor-relation function given in (24), α = R ∞ dr π r κ ( r ). Both t and y depend on the densities,through ξ + ≡ ( n a m a + n b m b ) and λ ≡ | g e | n a n b a v , respectively. Both the excitation gap in the12trong coupling regime and the density-dependent phase transition can be observed in ex-periments. On the theoretical side, it is interesting to make further studies of the model inthe framework of bosonization [2, 13]. Acknowledgments
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