Ground state properties of a two dimensional Fermi superfluid with an anisotropic spin-orbit coupling
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l Ground state properties of a two dimensional Fermisuperfluid with an anisotropic spin-orbit coupling
Kezhao Zhou ∗ , Zhidong Zhang ∗ Shenyang National Laboratory for Materials Science, Institute of Metal Research, ChineseAcademy of Sciences, 72 Wenhua Road, Shenyang 110016, China
Abstract
We performed a theoretical investigation on the ground state properties of a twodimensional ultra-cold Fermi superfluid with an anisotropic spin-orbit coupling(SOC). In the absence of Zeeman field, the system evolves from weak couplingBCS regime to strongly interacting BEC regime (BCS-BEC crossover) withincreasing either the two-particle interaction strength or SOC parameters. Wefocused on the behaviors of pairing parameter and density of states (DOS) whenincreasing the anisotropic parameter of the SOC. Surprisingly, we discoveredthat the gap parameter decreases with increasing the anisotropic parameters,but the DOS at the Fermi surface shows non-monotonic behavior as a functionof the anisotropic parameter. In the presence of the Zeeman field, we discussed aparticular type of topological phase transition by obtaining the analytical resultof the topological invariant and directly related this quantum phase transitionwith a sudden change of the ground state wave-function. Effects of higher partialwave pairing terms on this topological phase transition were briefly discussed.
Keywords: ultra-cold atoms, BCS-BEC crossover, spin-orbit coupling
PACS: ∗ Corresponding author.
Email addresses: [email protected] (Kezhao Zhou), [email protected] (ZhidongZhang)
Preprint submitted to Journal of Physics and Chemistry of Solids November 7, 2018 . Introduction.
In recent years, spin-orbit coupling effects (SOC) in condensed matter sys-tems have received lots of interest [1]. Firstly, SOC is a key ingredient in realizingnontrivial topological phases [2, 3, 4, 5, 6]. For example, combined effects of SOCand an external Zeeman field in superconductor systems can generate a non-Abelian quantum order [6]. Secondly, SOC can induce a nontrivial spin-tripletpairing field which significantly changes the properties of non-central-symmetricsuperconductors [7]. Furthermore, effects of SOC on the unconventional super-conductivity also attract lots of attention recently [8].In order to observe these novel phenomena, much effort has been investedto synthesize solid state materials with sizable SOC. Another promising plat-form is the artificial materials, especially ultra-cold atoms system where SOC,Zeeman field can be readily generated and superfluidity has been observed withcurrent experimental technique [9, 10, 11]. In ultra-cold atom community, con-struction of model systems on the Hamiltonian level is now available [12, 13].Due to its highly controllability, ultra-cold atom system has been proven to bea ideal platform for the investigation of many fundamental problem in solidstate chemistry and physics, such as the creation and manipulation of variouscrystalline structure using optical lattice trap and characterization of its energyband structure and other physical properties [14].There are mainly two types of SOC, namely Rashba [15] and Dresselhaus [16]SOC. In ultra-cold atoms systems, current experimental set-up can produce SOCwith arbitrary combination of these two types of SOC [17, 18]. Many theoreticalinvestigations have been performed to study effects of SOC on various superfluidproperties [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,37, 38, 39, 40, 41]. In the absence of Zeeman field, SOC can produce a novelbound-state called Rashbons and therefore induce a crossover from weakly cor-related BCS to strongly interacting BEC regime (BCS-BEC) even for very weakparticle-particle interaction [31]. Effects of anisotropic SOC on the ground stateproperties have been discussed in [ ? ]. It was found that Rashba SOC is the opti-2al one for superconductivity/superfluidity. Anisotropy of SOC suppresses pair-ing and condensation. Furthermore, combined effect of SOC and Zeeman fieldcan host a non-trivial topological order [6, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51].In two dimensional (2D) superfluid system with Rashba SOC, transition fromtrivial superfluid state to non-trivial topological state can be characterized bya topological invariant which has been obtained analytically in [47]. The infor-mation contained in the topological invariant and its physical consequence hasalso been discussed in [47]. However extension of this discussion to anisotropicSOC is still remained undone which is our main focus.In this paper, we investigate the effect of the anisotropic SOC on the groundstate properties of a 2D superfluid system using mean-field theory. In the ab-sence of Zeeman field, we calculate the density of states (DOS) at the Fermisurface with the self-consistent solutions of the mean-field number and gap equa-tions. Surprisingly, we find that the DOS at the fermi surface as a function ofthe anisotropic parameter is not monotonic and has local maximum for certainparameter space. But the gap parameter decreases with increasing degree ofanisotropy of SOC. This means that gap parameter is not sensitive to the den-sity of state at the Fermi surface and increasing DOS at the Fermi surface doesnot necessarily enhance pairing and the transition temperature. Furthermore,the maximum of the DOS as a function of the anisotropic parameter increaseswith increasing total strength of the SOC. In the presence of an external Zeemanfield, we also study the topological phase transition characterized by a topolog-ical invariant. We obtain the analytical result of the topological invariant forarbitrary SOC and found that the anisotropic SOC does not change nature ofthe topological phase transition.
2. Formalism.
We consider an anisotropic SOC which can be written as an arbitrary com-bination of Rashba and Dresselhaus type SOC. In momentum space, it can bedescribed by: 3 soc = λ R ( σ x p y − σ y p x ) + λ D ( σ x p y + σ y p x ) (1)where λ R and λ D denote the Rashba and Dresselhaus SOC parameters respec-tively and σ i = x,y,z are the Pauli matrices. The system under consideration canbe described by the Hamiltonian: H = Z d r ψ † ( r ) [ ε ˆp − hσ z + H soc ] ψ ( r ) − g Z d r ϕ †↑ ( r ) ϕ †↓ ( r ) ϕ ↓ ( r ) ϕ ↑ ( r ) (2)where g > ϕ σ (= ↑ , ↓ ) ( r ) and ϕ † σ ( r ) arethe annihilation and creation field operators, respectively, ψ ( r ) = [ ϕ ↑ ( r ) , ϕ ↓ ( r )] T and kinetic energy ε ˆp = ˆp / m − µ with m , µ and h being the mass of the Fermiatoms, the chemical potential and the effective Zeeman field, respectively. Forsimplicity we set ¯ h = 1 throughout this paper. As can be seen from Eq. (1),the system is isotropic when λ D = 0 or λ R = 0 and anisotropic when λ D = λ R .For convenience, we define an anisotropic parameter as η = λ D λ R (3)When η increases from 0 to 1, the system evolves from isotropic Rashba case toanisotropic case with equal Rashba and Dresselhaus SOC.Within mean-field theory, the interacting part can be approximated by − R d r (cid:16) ∆ ( r ) ϕ †↑ ( r ) ϕ †↓ ( r ) + h.c. (cid:17) + R d r | ∆ ( r ) | /g with ∆ ( r ) being the pair-ing field. For our purpose, we only consider translational invariant solutionswhere the paring field becomes a constant ∆ ( r ) = ∆. Therefore, the Hamil-tonian can be represented in momentum space and its matrix form reads: H = P p > Φ † p H BdG ( p ) Φ p + P p ε p + V ∆ /g where V denotes the size of thesystem, Φ p = h a p , ↑ , a p , ↓ , a †− p , ↑ , a †− p , ↓ i T and the BdG Hamiltonian H BdG ( p ) is H BdG ( p ) = ε p − h Γ p − ∆Γ ∗ p ε p + h ∆ 00 ∆ − ε p + h Γ ∗ p − ∆ 0 Γ p − ε p − h (4)with Γ p = λ R ( p y + ip x ) + λ D ( p y − ip x ).4sing the standard diagonalization procedure, we obtain the ground-statefree energy E g = P p ,s = ± ( ε p − E p ,s ) / V ∆ /g where the excitation spec-trum E p ,s = q E p ,s + ∆ p , with E p ,s = E p − s q h + | Γ p | , E p = q ε p + ∆ p , ,∆ p , = ∆ | cos θ p | , ∆ p , = ∆ sin θ p and θ p = π − arctan ( | Γ p | /h ). From vari-ation of ground state energy with respect to the gap parameter and chemicalpotential, we obtain the gap and number equations1 g = 1 V X p ,s s cos θ p hE p E p ,s , (5) N = 12 X p ,s (cid:18) − E p ,s E p ,s ε p E p (cid:19) . (6)Divergence of the integral over momenta in Eq. (5) is removed by replac-ing contact interaction parameter g by binding energy E b through V /g = P p / (2 ǫ p + E b ).Furthermore, the ground-state wave-function can be obtained as: | G i = Y p > ,s (cid:16) u p ,s + e isϕ p v p ,s β † p ,s β †− p ,s (cid:17) | g i (7)where β p ,s = u p c p ,s − v p c †− p , − s with u p = p (1 + ε p /E p ) / u p + v p = 1, c p ,s = sin ( θ p / a p ,s − s cos ( θ p / e isϕ p a p , − s with ϕ p = arctan [( λ R − λ D ) p x / ( λ R + λ D ) p y ]and u p ,s v p ,s = s (cid:18) ± E p ,s E p ,s (cid:19) . (8)
3. Balanced case.
In the absence of Zeeman field, h = 0, the ground state properties have beeninvestigated in [ ? ]. The self-consistent solution of the gap and number equa-tions show that the pairing parameter ∆ decreases with increasing anisotropicparameter η . In this paper, we focus on the dependence of pairing parameterson the DOS at the Fermi surface. For h = 0, Hamiltonian in the helicity basis c p ,s becomes 5 = ∆ g + X p ,s = ± E p ,s c † ks c ks − ∆2 X k (cid:16) e iϕ k c † k, + c †− k, + + e − iϕ k c † k, − c †− k, − (cid:17) (9)As can been seen from the above equation, pairing happens only betweenthe same helicity basis. And DOS of the helicity basis is defined as ρ F = X p ,s = ± δ ( −E p ,s ) (10)Performing the momentum integral, we obtain ρ F = Θ ( µ )2 π + Θ ( − µ )(2 π ) Z π dx Ψ ( x ) Θ (cid:2) Ψ ( x ) + µ (cid:3)p Ψ ( x ) + µ (11)with Ψ ( x ) = p λ R + λ D − λ R λ D cos (2 x ). As already known that, in the ab-sence of SOC, ∆ depends on ρ F explicitly in the weak interacting limit becausepairing happens only around the Fermi surface. Meanwhile, for an isotropicRashba SOC, ∆ and ρ F both increase when increasing the SOC strength [40].Therefore, it is believed that ∆ depends on ρ F in a monotonic manner. Andincreasing density of states at the Fermi surface is considered as an efficientway of increasing the pairing strength and transition temperature. However, wefind that the anisotropic nature of the SOC significantly changes this picture.The numerical results of DOS and gap parameter are presented in Fig. 1 andFig. 2. Without loss of generality, in the numerical calculations, we have set E b = 0 . E F with E F = k F / m and k F = √ πn . Fig. 1 represents DOS and gapparameter as functions of anisotropic parameter and different lines correspondto different λ = p λ R + λ D , ρ and ∆ are the DOS at the Fermi surface andgap parameter for λ = 0. Fig. 2 shows DOS and gap parameter as functions ofdimensionless parameter ˜ λ = mλ/k F and different lines correspond to different η . From Fig. 1(a), we find that the DOS at the Fermi surface as a function ofthe anisotropic parameter is not a monotonic function. However, as seen fromFig. 1 (b), the gap parameter ∆ decreases as η increases and ∆ reduces to∆ for equal Rashba and Dresselhaus SOC. Therefore, pairing does not increase6 Η Ρ F (cid:144) Ρ Η D (cid:144) D H a L H b L Figure 1: (Color online) DOS at the Fermi surface ρ F (a) and pairing parameter ∆ (b) asfunctions of the anisotropic parameter η . Different lines correspond to different values of ˜ λ .The brown solid, green solid, red dashed, long blue dashed and black dotted lines correspondto ˜ λ = 0 .
2, 0 .
8, 1 .
0, 1 . . monotonically with DOS at the Fermi surface. Furthermore, for large enough λ ,it has a maximum value and for small λ , the chemical potential remains positiveand ρ F = ρ as can be seen from Eq. (11 ) and the brown solid line in Fig. 1(a). Last but not least, the maximum value of DOS increases with increasing λ .More interestingly, as can be seen from Fig. 2, both the DOS at the Fermisurface and gap are non-monotonic functions of λ for for 0 < η <
1. When η = 0,the system is isotropic and ∆ increases with increasing λ [35, 40, 41]. However,for η = 1, the SOC terms reduces to equal Rashba and Dresselhaus case. Inthis case and without Zeeman field, the SOC does not affect the thermodynamicproperties and therefore ∆ does not change with increasing λ . Furthermore, for0 < η <
1, the gap parameter as a function of lambda has a local minimum ascan be seen from Fig. 2 (b).
4. Imbalanced case.
In the presence of an external Zeeman field, the ground state of the systemunder consideration becomes far more complex. Many exotic phases may appear7 ΛŽ Ρ F (cid:144) Ρ ΛŽ D (cid:144) D H a L H b L Figure 2: (Color online) DOS at the Fermi surface ρ F (a) and pairing parameter ∆ (b) asfunctions of ˜ λ . Different lines correspond to different values of η . In (a), the brown solid,green solid, red dashed, long blue dashed and black dotted lines correspond to η = 0, 0 .
1, 0 . . . η = 0 . and the ground state phase diagram has been investigated extensively [42, 43,44, 45, 46, 47, 48, 49, 50, 51]. Most interestingly, there is a topological phasetransition driven by Zeeman field which is our main focus. The critical Zeemanfield reads h c = p µ + ∆ . For h < h c , the system is in a trivial gappedsuperfluid state. When h > h c , the ground state is topologically nontrivial andis characterized by a nonzero topological invariant N which is defined as [6, 2] N = 1 / π R + ∞−∞ d p B ( p ) with the Berry curvature being given by B ( p ) = − i X E α p < (cid:2) ∂ p x u † p ,α ∂ p y u p ,α − ∂ p y u † p ,α ∂ p x u p ,α (cid:3) (12)where u p ,α =1 , , , are the eigenvectors of Eq. ( 4) corresponding to the eigenval-ues − E p , + , E p , + , − E p , − , E p , − , respectively. Following the same procedure in[47], we obtain the eigen states as u p ,s = ± = (cid:2) e isϕ p F p ,s , F p ,s , F p ,s , e isϕ p F p ,s (cid:3) T with F p ,s = u p sin θ p v p ,s − v p cos θ p u p ,s (13) F p ,s = u p cos θ p v p ,s + v p sin θ p u p ,s (14)8 p ,s = u p sin θ p u p ,s + v p cos θ p v p ,s (15) F p ,s = u p cos θ p u p ,s − v p sin θ p v p ,s . (16)The only difference here is the anisotropic SOC characterized by the phasefactor ϕ p = arctan [ ϑp x /p y ] with ϑ = ( λ R − λ D ) / ( λ R + λ D ). Simple algebraicmanipulation leads to B ( p ) = ∂ p y ϕ p ∂ p x F p − ∂ p x ϕ p ∂ p y F p = − ϑ p p y + ( ϑp x ) · ∇ F p (17)and F p = P α =1 , ,s s (cid:0) F α p ,s (cid:1) . Clearly the Berry curvature B ( p ) is anisotropic.However, by proper scaling of the integral variables, the topological invariantdoes not depend on the anisotropic parameter and we obtain N = F = v , + = θ ( h − h c ) . (18)From this we can see that the topological phase transition corresponds to asudden change of the ground state wave function at zero momentum character-ized by v , + . Consequently, there is a sudden change of the ground-state wave-function associated with the component of triplet pairing of the quasi-particlesdenoted by β p , + at zero momentum. This is also reflected in the momentumdistribution as can be seen from Eq. (6) that E , + /E , + = sign ( h c − h ). Thisunique property provides a conclusive evidence that the topological phase tran-sition can be determined by measuring the momentum distribution in cold atomexperiments.Furthermore, in the presence of higher partial wave pairing terms, taking p and d wave pairing symmetry for example, since the topological phase transitiondepends only on the zero momentum parts of the ground state wave function,the p wave pairing does not affect the topological phase transition while d wavedoes [6].
5. Conclusion.
We investigated the ground-state properties of a 2D Fermi superfluid systemin the presence of a general anisotropic SOC and Zeeman coupling that sup-9orts non-trivial topological order. Particularly, we found that increasing theDOS at the Fermi surface is not a sufficient way of obtaining large △ and hightransition temperature. For the topological phase transition driven by an exter-nal Zeeman field, we found that the anisotropic nature of the system inducedby an anisotropic SOC does not change the topological phase transition. Andfrom the analytical result of the topological invariant, we discovered that thetopological phase transition can be determined by measuring the momentumdistribution in cold atomic experiments.
6. Acknowledgements.
This work has been supported by the National Natural Science Foundation ofChina under Grant 51331006, 51590883, 11204321, the National Basic ResearchProgram (No.2017YFA0206302) of China and the project of Chinese academyof Sciences with grant number KJZD-EW-M05-3.
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