Ferromagnetic and Nematic Non-Fermi Liquids in Spin-Orbit Coupled Two-Dimensional Fermi Gases
FFerromagnetic and Nematic Non-Fermi Liquids in Spin-Orbit CoupledTwo-Dimensional Fermi Gases
Jonathan Ruhman and Erez Berg
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
We study the fate of a two-dimensional system of interacting fermions with Rashba spin-orbitcoupling in the dilute limit. The interactions are strongly renormalized at low densities, and give riseto various fermionic liquid crystalline phases, including a spin-density wave, an in-plane ferromagnet,and a non-magnetic nematic phase, even in the weak coupling limit. The nature of the ground statein the low-density limit depends on the range of the interactions: for short range interactions itis the ferromagnet, while for dipolar interactions the nematic phase is favored. Interestingly, theferromagnetic and nematic phases exhibit strong deviations from Fermi liquid theory, due to thescattering of the Fermionic quasi-particles off long-wavelength collective modes. Thus, we arguethat a system of interacting fermions with Rashba dispersion is generically a non-Fermi liquid atlow densities.
I. INTRODUCTION
The realization of strongly spin-orbit coupled fermionsystems in low dimensions, either in solid state or coldatomic setups, calls for an understanding of the in-terplay between many-body interactions and spin-orbitcoupling. One of the effects of spin-orbit coupling insolids is to modify the dispersion relation of electrons;as a result, inter-particle interactions can be effectivelyenhanced. For example, in the case of Rashba-type spinorbit coupling (which occurs in two-dimensional electrongases in quantum wells without inversion symmetry), thedispersion minimum occurs on a nearly-degenerate ring in momentum space, instead of a single minimum at k = 0 . This leads to quenching of the kinetic energyat low densities, and hence many-body interactions be-come increasingly important. It has been argued that inthe low-density limit and in the presence of short-rangerepulsive interactions, a host of “electronic liquid crystal”phases can be stabilized, including nematic, ferromag-netic nematic, and anisotropic Wigner crystal phases. In bosonic systems, Rashba spin-orbit coupling can leadto unusual phases, as well.
Here, we analyze the fate of an interacting two-dimensional system of fermions with strong Rashba-typespin orbit coupling in the low-density limit. In thislimit, the two-particle effective low-energy interaction isstrongly renormalized, and obtains a universal form. ? We analyze the phase diagram, and find a competitionbetween several symmetry-broken liquid states, includinga spin-density wave, nematic, and an in-plane ferromag-netic nematic phase (Fig. 1); in the case of short-rangeinteractions, the ground state in the extreme low-densitylimit is the ferromagnetic nematic, whereas with inter-actions that decay as /r , where r is the inter-particledistance (which is the case, e.g., for Coulomb interactionsscreened by a nearby metallic gate, or for dipolar parti-cles with dipole moments pointing perpendicular to theplane), the ground state is a non-polarized nematic.Finally, we argue that the ferromagnetic and nematicphases are expected to be non-Fermi liquids, due to the FL FM N (a)(b) 𝑈 𝑛 FL FM SDW N FIG. 1. (a) The Fermi surface in the Fermi liquid (FL), fer-romagnetic (FM), and nematic (N) phases. (b) The phasediagram of the dilute Rashba gas with short range interac-tions, as a function of the dimensionless density ˜ n = n/k and dimensionless bare interaction strength ˜ U = ( k / ε ) U . strong scattering of quasi-particles near the Fermi surfaceoff the Goldstone modes of the ordered state. In theferromagnetic phase the strong coupling to the magneticGoldstone modes is generated by spin-orbit coupling .Rashba spin-orbit coupling thus offers a natural routeto realizing a non-Fermi liquid phase. This phase hasbeen studied extensively in the literature ; althoughits nature is still not completely understood, it is believedto be characterized by anomalous power law temperaturedependence of physical quantities, such as the specificheat and the resistivity.Our results are particularly relevant for cold atom ex-periments. We present an alternative method to studythe strongly interacting regime of cold Fermi gases with- a r X i v : . [ c ond - m a t . s t r- e l ] O c t out tuning too close to the Feshbach resonance. In aspin-orbit coupled gas, the interactions are effectivelyenhanced due to the large density of states in the low-density limit. This is crucially different from tuning tothe Feshbach resonance from the repulsive side, wherethe decay time to the bound state becomes very short. Formation of molecules limits the range of accessible in-teraction strength and has prevented the observation ofthe ferromagnetic instability thus far. This paper is organized as follows. Sec. II describesthe model Hamiltonian. In Sec. III we calculate the exacttwo-particle T-matrix for the case of short ranged inter-actions. The T-matrix is then used to approximate theeffective interactions in the low-density limit. In Sec. IVwe numerically compute the phase diagram presented inFig. 1.b. Sec. V analyzes the case of dipolar interac-tions. We then turn to discuss the validity of our resultsfor systems that do not possess perfect rotational sym-metry in Sec. VI. Finally, we analyze the effects of thecollective mode fluctuations including the stability of theordered phases to quantum fluctuations and their effecton the lifetime of quasi-particles near the Fermi surfacein Sec. VII.
II. MODEL HAMILTONIAN
We consider a system of fermions in two dimensionswith a Rashba-type spin-orbit coupling. The single-particle Hamiltonian is H = (cid:88) k c † k ˆ H ( k ) c k (1)where ˆ H ( k ) = k m − µ + ε − α ( k × σ ) · ˆ z , α is the strength ofthe spin-orbit coupling, µ is the chemical potential, and ε = mα / is the spin-orbit energy scale. c † = ( c †↑ , c †↓ ) is a two component spinor and σ is the vector of Paulimatrices in the same basis. Since we are interested in thelow density limit, µ (cid:28) ε , we will consider only the lowenergy band, whose dispersion is ε k = ε ( k − k ) /k , (2)where k = mα is the radius of the circular disper-sion minimum. The annihilation operator for a par-ticle in this band is ψ k = (cid:0) c ↑ + i e iφ k c ↓ (cid:1) / √ , where φ k ≡ arctan ( k y /k x ) is the angle of the vector k (whichis perpendicular to the spin direction). For µ < ε theFermi sea has the topology of an annulus, with two con-centric Fermi surfaces at k = k ± k F , where k F = √ mµ .The single particle density of states is ρ ( µ ) = ρ k k F , where ρ ≡ m/π .The fermions interact via a two-particle repulsion. Wewill focus on two physically relevant cases: short range 𝑘 𝐹 𝑖 𝑘 𝑥 𝑘 𝑦 𝑓 𝐹𝑥 𝐹 (b)(a) 𝐹𝑦 FIG. 2. (a) The Fermi sea in the Fermi liquid state definedby the radii k = k ± k F and the two high energy shells at k − Λ i < k < k − Λ f and k + Λ f < k < k + Λ i . (b) Fermisea in the FM phase. The Fermi surface is highly anisotropiccharacterized by the Fermi wavelength in the radial direction k xF and in the azimuthal direction k yF ≈ k θ F . (contact) interactions, which are natural in the contextof cold atomic gases, and dipolar interactions that de-cay as /r , occurring in two-dimensional electron gaseswith a nearby screening metallic gate. For simplicity,we consider contact interactions first. The interactionHamiltonian projected to the lower band is written as H I = 14Ω (cid:88) kk (cid:48) Q Γ ( k , k (cid:48) ; Q ) ψ † k + Q ψ † k (cid:48) − Q ψ k (cid:48) ψ k , (3)where Γ ( k , k (cid:48) ; Q ) = U e iφ k , k + Q and U is the strength ofthe interaction, Ω is the volume, φ a , b ≡ φ a − φ b , and thefactor of e iφ k , k + Q arises from the projection to the lowerspin-orbit band. More extended dipolar interactions willbe considered later. III. RENORMALIZATION OF THETWO-PARTICLE VERTEX
We now turn to discuss the renormalization of the two-particle interactions in the case of a circular dispersionminimum. The derivation of the renormalized interactionproceeds along similar lines to that of Ref. ? .We are interested in the corrections to the bare inter-action vertex (3) generated by integrating out high en-ergy virtual states, which lie in the two momentum shells k − Λ i < q < k − Λ f and k + Λ f < q < k + Λ i (seeFig. 2.a). Here Λ i is the high momentum cutoff (whichis initially taken to be of order k ) and Λ f is the lowmomentum cutoff (which is of order k F ). This procedureresembles the momentum shell renormalization group ap-proach for fermions, except for the fact that here we areintegrating out empty states at energies greater than theFermi energy µ . In this case only the Bardeen-Cooper-Schrieffer diagram contributes (see Fig. 3.a). Summingall ladder diagrams we obtain the two particle T-matrix Γ( ω, k , k (cid:48) ; Q ) = Γ ( k , k (cid:48) ; Q )1 + B ( ω, P ) U , (4)where P = k + k (cid:48) , ω is the sum of the frequencies of theincoming particles, Q is the momentum transfer in thescattering, and B ( ω, P ) = (cid:82) d Λ d q (2 π ) − e iφ qP − q − iω + ξ q + ξ P − q . Here, d Λ denotes integration over the regions where both q and P − q belong to the shells that are integrated out, and ξ k ≡ ε k − µ .In the dilute limit, k, k (cid:48) (cid:39) k such that P (cid:39) k | cos φ k , k (cid:48) | . The renormalized forward scattering in-teraction ( Q = 0 and Q = k (cid:48) − k , ω = 0 ) assumes theform V ( φ k , k (cid:48) ) = 2 sin φ k , k (cid:48) U + B (0 , k | cos φ k , k (cid:48) | ) (5)The angular dependance of the interaction (5) for dif-ferent values of Λ f is presented in Fig. 3.b. We iden-tify two important features. First, for forward scattering( φ k , k (cid:48) ∼ ) the interaction vanishes as V ∼ U φ k , k (cid:48) / for all values of Λ f . This is because of the Pauli exclu-sion between spin states with equal orientation. Second,a strong renormalization occurs when the two incomingmomenta have opposite directions ( φ k , k (cid:48) ∼ π ), where thebare interaction is maximal. The strong renormalizationresults from the large phase space for scattering into thehigh energy shells when P (cid:39) .One can write analytic expressions for the renor-mailzed interactions near the points φ = 0 , π ? . As men-tioned above, in the case of φ k , k (cid:48) (cid:28) , V ( φ k , k (cid:48) (cid:39) ≈ U (cid:0) φ k , k (cid:48) (cid:1) . (6)On the other hand, for ( φ k , k (cid:48) (cid:39) π ) and Λ f (cid:28) k , theeffective interaction assumes a universal form V ( φ k , k (cid:48) (cid:39) π ) ≈ Λ f ρ k K (cid:16) − P f (cid:17) , (7)where K ( x ) = (cid:82) π/ dx (1 − x cos x ) − / is the com-plete elliptic integral of the first kind, which decays as K ( − x ) ≈ log x √ x for x → ∞ . IV. MEAN-FIELD PHASE-DIAGRAM
To obtain the zero temperature phase diagram(Fig. 1.b) we use a mean-field approximation with therenormalized interactions (4). First, we compare theenergy of two uniformly ordered (translationally invari-ant) trial states: the ferromagnet (FM) and nematic (N)phase. We then check the stability of these phases to-wards a spin-density wave state (SDW).Let us start from the uniform phases. The Fermi sur-faces of the FM and N phases presented in Fig. 1.a arenaturally favored by the angular dependance of the renor-malized interaction (5). This is because the interactionis minimal at φ kk (cid:48) = 0 and φ kk (cid:48) = π and therefore quasi-particles pairs have the lowest interaction energy when π /2 π π /2 ϕ kk ' =0 U U Λ f = k Λ f = k Λ f = k Λ f = k Λ f = k bare (a)(b) Γ = + …+ + FIG. 3. (a) A diagrammatic representation of the ladder seriesfor the T-matrix. (b) The angular dependance of the effectiveinteraction (5) for different values of the lower momentumcutoff Λ f . their momenta are collinear. The FM state is obtainedby confining the particles to a finite segment of the ringcentered around a specific direction in momentum space,for example ˆ k = ˆ x (as in Fig. 2.b). The spin, which islocked perpendicular to the momentum direction, has anon-zero average value. As a result the FM phase breakstime-reversal and rotational symmetry. The N state isobtained similarly by confining the fermions to two suchFermi surfaces residing on two opposite sides of the de-generacy ring. In this case the spin-density is zero onaverage, and therefore this state breaks only rotationalsymmetry.To compare the ground state energy of the FM and Nstates we expand the interaction (5) in Fourier compo-nents V ( φ kk (cid:48) ) = (cid:88) l V l e il ( φ k − φ k (cid:48) ) . (8)The trial wave functions are generated by the mean-fieldHamiltonian H NMF = H − (cid:88) k l h l cos( lφ k ) n k , (9)where we will restrict ourselves only to l = 1 , so-lutions (which are ferromagnetic and nematic, respec-tively). Minimizing the expectation value of the fullHamiltonian with respect to µ , h and h yields the equa-tions (see Appendix A) n = 1Ω (cid:88) k (cid:104) n k (cid:105) , (10) h = − V Ω (cid:88) k cos φ k (cid:104) n k (cid:105) , (11) h = − V Ω (cid:88) k cos 2 φ k (cid:104) n k (cid:105) . (12)The FM state is characterized by h (cid:54) = 0 , while the Nphase corresponds to h (cid:54) = 0 and h = 0 . At sufficientlylow density, the ground state is always the FM state;upon increasing the density, there is a first-order transi-tion to a N state, followed by another first-order transi-tion to a rotationally invariant FL state. The fact thatthe transitions are of first order is associated with thepresence of a nearby van Hove singularity in the densityof states. We now turn to discuss the stability of the uniformlyordered states towards textured phases (either spin orcharge density waves). First we note that in the low-density limit the Fermi surface contains nearly nestedsegments which are separated by q = 2 k F , where k F is the Fermi wavelength along the radial direction. Asa result, the charge and spin susceptibility χ ρ,σ ( q ) aresharply peaked at q = 2 k F ? (see Appendix B). For suffi-ciently short-range interactions, the FM phase is alwaysstable to SDW and CDW formation in the low densitylimit. This is because the system is nearly spin polar-ized, and the interaction between fermions on the Fermisurface is small.The FL and N phases become unstable to SDW forma-tion when the Stoner criterion V χ ⊥ ( q ) = 1 is satisfied,where χ ⊥ ( q ) is the in-plane spin-susceptibility transverseto q (see Appendix B). The transition lines to the SDWphase are shown as dashed lines in Fig. 1.In the low density limit, the angular size of the Fermisurfaces in the FM and N phases becomes small. Wecan then utilize the asymptotic analytic expressions forthe effective interaction near φ k , k (cid:48) = 0 , π [Eqs. (6,7) andRef. ? ] to estimate the ground state energy. The shapeof the Fermi surfaces is highly anisotropic, k xF (cid:28) k yF ,where k xF ( k yF ) is the Fermi wavelength along the radial(azimuthal) direction (see Fig. 2.b).In the N phase, the Fermi surface consists of two suchanisotropic patches. In this case, the inter- and intra-patch interactions are given by (6) and (7), respectively.(The lower cutoff for the renormalization procedure ofthe interaction is taken to be Λ f = 2 k xF .) The totalmomentum P = | k + k (cid:48) | , which enters the inter-patchinteraction (7), is much greater than k xF , over most ofthe Fermi surface. We can therefore use the approximateform of (7) for P (cid:29) Λ f : V ( φ kk (cid:48) (cid:39) π ) ≈ | φ kk (cid:48) − π | ρ log k /k xF . (13)On the other hand, the intra-patch interaction (6) decaysquadratically at small angles.The total energy per particle in the N phase scales as ε N ∝ (cid:18) ε ρ k (cid:19) / n / (cid:16) log k n (cid:17) / . (14)while in the FM phase the energy per particle is ε F M ∝ √
U ε k n / . We conclude that for short-ranged interactions in the zerodensity limit, the ground state is FM, in agreement withRef. 7.
V. DIPOLAR INTERACTIONS
We now turn to discuss the case of dipolar interactions,which decay as /r at large distances. In Fourier space,the interaction is given by U d ( q ) ≈ v − v q for small q .The corresponding bare interaction vertex assumes theform H dI = 1Ω (cid:88) p,p (cid:48) ,P Γ d ( p , p (cid:48) ; P ) ψ † p (cid:48) ψ † P − p (cid:48) ψ P − p ψ p , (15)where the vertex function is given by Γ d ( p , p (cid:48) ; P ) = U d ( | p − p (cid:48) | ) × (16) (cid:2) e iφ p , p (cid:48) + e iφ P − p , P − p (cid:48) + e iφ p , p (cid:48) + iφ P − p , P − p (cid:48) (cid:3) . The one-loop correction to the effective interaction thentakes the form B d ( ω, P ) = (cid:90) d Λ d q (2 π ) (cid:20) Γ d ( p , q ; P )Γ d ( q , p (cid:48) ; P ) − iω + ξ q + ξ P − q (17) − Γ d ( p , q ; P )Γ d ( P − q , p (cid:48) ; P ) − iω + ξ q + ξ P − q (cid:21) . The interaction vertex (16), which now depends non-trivially on the momentum transfer Q = p − p (cid:48) , be-comes particularly simple in the limits of interest: (i)the Cooper channel ( P (cid:28) k ) and (ii) forward scattering( P (cid:39) k ). For Cooper channel scattering, case (i), thebare vertex (16) becomes a function of a single angle Γ d ( φ pp (cid:48) ) = U d ( φ pp (cid:48) ) (cid:104) e iφ pp (cid:48) (cid:105) . (18)We expand (18) in Fourier components Γ d ( φ pp (cid:48) ) = (cid:80) m Γ d ,m e i m φ pp (cid:48) and insert it into (17) to obtain ? B d ( P (cid:39)
0) = − ρ k Λ f K (cid:32) − P f (cid:33) ∞ (cid:88) m =1 [1 − ( − m ](Γ d ,m ) (19)in the Λ f → limit. Note that m denotes the totalangular momentum (orbital plus spin), and that only theodd ones contribute. The different angular momentumchannels decouple in the ladder series (Fig. 3.a) due toconservation of angular momentum, just as in equation(4). In the low density limit they all assume a universalform Γ d m +1 ( φ kk (cid:48) (cid:39) π ) ≈ C d ( φ kk (cid:48) ) ≡ | φ kk (cid:48) − π | ρ log (cid:16) k Λ f (cid:17) . (20)where k = p , k (cid:48) = P − p . The total angle dependantinteraction then assumes the form V d ( φ kk (cid:48) (cid:39) π ) = C d ( φ kk (cid:48) ) × (21) lim N →∞ N N (cid:88) m =0 e i (2 m +1) φ pp (cid:48) = C d ( φ kk (cid:48) ) δ φ pp (cid:48) , Overall for Cooper channel scattering we obtain the sameresult as for short-ranged interactions (7).In the case of forward scattering, P (cid:39) k , the inte-grand of (17) diverges at q = p and q = P − p . Theintegral is dominated by the vicinity of these two points,whose most divergent contribution as Λ f → is B d ( φ kk (cid:48) ) ≈ (cid:104) ˜ U d ( φ kk (cid:48) ) (cid:105) | sin φ kk (cid:48) | ρ log (cid:18) k Λ f (cid:19) (22)where ˜ U d ( φ kk (cid:48) ) ≡ U d (0) − (1+cos φ kk (cid:48) ) U d (2 k | sin φ kk (cid:48) | ) which decays linearly near φ kk (cid:48) = 0 ˜ U d ( φ kk (cid:48) ) ≈ v k | φ k , k (cid:48) | . Summing up the ladder series we have V d ( φ kk (cid:48) (cid:39)
0) = ˜ U d ( φ kk (cid:48) )1 + B d ( φ kk (cid:48) ) ˜ U d ( φ kk (cid:48) ) . Therefore, just as in the case of short-ranged interactions,we recover the bare interactions for small angle scattering( φ kk (cid:48) (cid:39) ): V d ( φ k , k (cid:48) (cid:39) ≈ v k | φ k , k (cid:48) | . (23)Using the asymptotic form of the effective interaction,Eqs. (21) and (23), we can estimate the ground state inthe zero density limit, just as we have for short rangedinteractions at the end of section IV. The crucial differ-ence is that now the forward scattering term (23) decayslinearly to zero near φ kk (cid:48) (cid:39) and not quadratically as itdid for short-ranged interactions (6). As a result, the en-ergy of the FM phase scales as ε F M ∝ n / , whereas thescaling of the energy of the N phase is unmodified com-pared to contact interactions [Eq. (14)]. Therefore, weconclude that for dipolar interactions, the ground statein the zero density limit is the nematic state, due to thelogarithm in Eq. (14). This connects to the results ofRef. 7, which predicted that for interactions that decaylike /r a the value a = 3 is critical, separating betweenthe anisotropic Wigner crystal (AWC) and the FM. TheN phase can be viewed as a melted version of the AWCphase. VI. EFFECTS OF BREAKING OF THEROTATIONAL SYMMETRY
Most physical realizations of the dispersion (2) will in-clude additional terms which break the rotational sym-metry. In solid state systems such terms arise from theunderlying lattice, while in cold atom systems they are due to the Raman lasers. To study the effects of theseterms on our results, we add the symmetry breaking term H (cid:15) = − α ( (cid:15) − k y σ y . (24)to the Hamiltonian (1). Here (cid:15) ≤ is a parame-ter that describes the degree of a two-fold anisotropy( (cid:15) = 1 corresponds to the isotropic case). In thiscase, the low-energy helical quasi-particles are given by ψ k = (cid:16) c k ↑ + i e i ˜ φ k c k ↓ (cid:17) , with tan ˜ φ k = (cid:15)k y /k x . We cal-culate the effect of the symmetry breaking term on thesolution of the self-consistency equations (10-12) for thecase of a FM transition.Before discussing the results, we note that the sym-metry breaking term (24) does not modify the renormal-ization of interactions presented in section III as long asthe Fermi energy is much greater than the energy scaleassociated with the anisotropy, ∆ (cid:15) = ε (1 − (cid:15) ) . In thislimit the Fermi sea of the non-interacting gas still hasthe form of an annulus with a density of states whichincreases with decreasing density (see inset of Fig. 4). Inthe opposite limit, µ (cid:28) ∆ (cid:15) , the Fermi surface is brokeninto two elliptic surfaces (similar to the Fermi surfacesof the N phase in Fig. 1.a) and the density of states de-creases with decreasing density and chemical potential.In this limit we expect that the renormalization of in-teractions will be closer to that of a Fermi gas withoutspin-orbit coupling. In the regime µ (cid:29) ∆ (cid:15) , the renor-malized interactions are approximated by V ( φ ) ≈ sin φ kk (cid:48) U + B (0 , P ) , where P = k + k (cid:48) and k , k (cid:48) lie on the elliptic dispersionminimum. Therefore, we can decouple the interactionin the same way we did in (9) and solve using the sameself-consistency equations (10-12) with ˜ φ k instead of φ k .Fig. 4 presents the critical density for the transitioninto the FM phase normalized by the critical densityat (cid:15) = 1 vs. the anisotropy energy ∆ (cid:15) divided bythe chemical potential at the transition. We find thatthe symmetry breaking term H (cid:15) has a negligible effectwhen the transition occurs at µ (cid:29) ∆ (cid:15) . However, when ∆ (cid:15) approaches µ at the transition, the critical densitydrops rapidly, and the FM order is obstructed by theanisotropy. VII. COLLECTIVE EXCITATIONS ANDNON-FERMI LIQUID BEHAVIOR
We now turn to discuss the effects of fluctuations of theorder parameter in the FM and N phases. These phasesbreak the (continuous) rotational symmetry of the sys-tem. The resulting gapless Goldstone mode associatedwith this symmetry breaking is strongly coupled to thequasi-particles at the Fermi energy.
This couplinggives rise to two important effects: first, the Goldstone Δ ϵ / μ n n c ρ / ρ ε / ε Δ ϵ FIG. 4. The critical density, n c , of the ferromagnetic transi-tion for ˜ U = 1 . in the presence of anisotropy in the single-particle dispersion (Eq. 24) normalized by the critical densitywithout the anisotropy term ( (cid:15) = 1 ), n , vs. the anisotropyenergy scale ∆ (cid:15) ≡ ε (1 − (cid:15) ) normalized by the chemical po-tential at the transition. The inset shows the density of statesas a function of energy for the anisotropic Rashba dispersion. modes become Landau damped by the particle-hole ex-citations near the Fermi surface. Second, the Landauquasi-particles are strongly scattered by the Goldstone mode, leading to the break down of the Fermi liquid be-havior. Below, we use Hertz-Millis type arguments todemonstrate that such a strongly coupled state indeedarises in the N and FM phases in our setup. Hertz-Millistheory is known to ultimately fail in d = 2 ; never-theless, following Ref. 16, we argue its application showsthat a Fermi liquid ground state is inconsistent.We consider, for example, the FM phase withshort-ranged interactions. We employ the Hubbard-Stratonovich transformation to decouple the imaginarytime action using the magnetization field M q = M yq − iM xq : S = (cid:88) k ψ † k ( − iω + ξ k ) ψ k − V (cid:88) k q (cid:16) e iφ k + q M q ψ † k + q ψ k − q + c.c. (cid:17) + V (cid:88) q | M q | where q = ( ω, q ) and k = ( ν, k ) denote componentvectors in frequency and momentum space. We expandthe action around the broken symmetry state δM q = M q − M δ q, : S = (cid:88) k ψ † k (cid:0) − iω + ξ F M k (cid:1) ψ k − V (cid:88) k q (cid:16) e iφ k + q δM q ψ † k + q ψ k − q + c.c. (cid:17) + V (cid:88) q (cid:0) | δM yq − M δ q, | + | δM xq | (cid:1) (25)where (cid:104) M q (cid:105) = M δ q, is taken to be real, and M = h /V is given by the solution of the self-consistent equation (11).The dispersion of the fermions is given by ξ F M k = ε k − h cos φ k − µ . The effective Ginzburg-Landau theory for δM is obtained by integrating out the Fermionic degrees of freedom. To second order in δM we get S (2) = − V (cid:88) q ω [ δ ij − V Π ij ( q )] δM iq δM j − q , (26)where the Lindhard function Π ij ( q ) is given by Π( q , iω ) ≈ (cid:90) d k (2 π ) n F (cid:16) ξ F M k + q (cid:17) − n F (cid:16) ξ F M k − q (cid:17) iω − v ( k ) · q (cid:18) sin φ k + q sin φ k − q sin φ k + q cos φ k − q cos φ k + q sin φ k − q cos φ k + q cos φ k − q (cid:19) . (27)Here n F is the Fermi function, v i ( k ) = 2 ε k (cid:16) kk − (cid:17) ˆ k i − h k ε ij ˆ k y ˆ k j where ε ij is the antisymmetric tensor, and ˆ k ≡ (cos φ k , sin φ k ) . To lowest order in ( ω, q ) (assuming that ω (cid:28) q ) the Lindhard function can be written as Π( q , ω ) ≈ (cid:32) x − η x ( φ q ) | ω | q − κ x q − ˜ κ x ( q x − q y ) γq x q y γq x q y y − η y ( φ q ) | ω | q − κ y q + ˜ κ y ( q x − q y ) (cid:33) . (28)Here, η x,y ( φ q ) are the (direction dependent) Landaudamping coefficients, κ x,y , ˜ κ x,y , and γ describe the stiff-ness of the order parameter to slow spatial modulations,and ∆ x,y determine the gaps of the collective modes. Theanisotropy in the parameters of Π( q , ω ) is due to the factthat we are working in an ordered phase that breaks ro- tational invariance. We have determined the parametersby numerically integrating Eq. (27) [ η x,y can also be ex-pressed analytically - see Eq. (30) below]. We find that ∆ x = V , such that transverse fluctuations of the orderparameter are gapless, as required from Goldstone’s the-orem (the order parameter is assumed to point along the y axis).From the effective action (26) we can compute the zeropoint fluctuations of magnetization field. Deep in the or-dered phase, these are dominated by the transverse fluc-tuations. The deviation of the angle of the order param-eter from the y direction is δϕ = δM x /M , and the meanfluctuations in δϕ are given by (cid:104) δϕ (cid:105) ≡ (cid:90) dωd q (2 π ) (cid:104) δϕ q δϕ − q (cid:105) (29) ≈ (cid:90) dωd q (2 π ) /h η x ( φ q ) | ω | q + κ x q + ˜ κ x ( q x − q y ) where we have kept only the most singular contribution inthe long-wavelength, low-frequency limit, and used h = V M . η x ( φ q ) can be expressed as η x ( φ q ) ≈ (cid:88) j k ,j (2 π ) v j sin ( φ k j ) . (30)The sum runs over the points k j on the Fermi surfacewhere the Fermi velocity v j is perpendicular to q . k ,j is the radius of curvature of the Fermi surface at thesepoints. As a crude approximation, we replace η x ( φ q ) and the order parameter stiffness in Eq. (29) by the av-erage values, η and κ , respectively. (For ˜ U = 0 . and n/k = 10 − − − we found that these anisotropies arenumerically small.) Eq. (29) then assumes the simpleform (cid:104) δϕ (cid:105) ≈ π ) Q h η log (cid:18) η Ω κQ (cid:19) , (31)where Q ∼ k xF and Ω are the momentum and frequencyultraviolet cutoffs, respectively. Interestingly in our nu-merically determined values for k xF , h and η x we findthat (cid:104) δϕ (cid:105) (cid:28) for a broad range of interaction strengthsand densities. Therefore, we conclude that the FM or-der is stable against quantum fluctuations in the range ofdensities n/k = 10 − − − , where we have computedthe phase diagram Fig. 1.b.It is interesting to note that in the N and FM phasesdiscussed here, the Goldstone modes are over-damped in any direction of propagation [i.e., η x ( φ q ) never vanishes].This is in contrast to the case of a distorted circular Fermisurface (analyzed in Ref. 16), where the Goldstone modesremain under-damped along a discrete set of angles closeto , ± π/ . This is because of the banana-like shape ofthe Fermi surfaces in our case (see Fig. 5). On such aFermi surface, there are points for which v ( k ) ⊥ q and sin φ k (cid:54) = 0 for any q . E.g., for q (cid:107) ˆ y , there are pointswhere the Fermi surface is parallel to ˆ y with k y (cid:54) = 0 (marked in red in Fig. 5). By Eq. (30), this implies thatthe Landau damping term η x is never zero.Finally, we turn to discuss the fate of the low-energy Fermionic quasi-particles in the FM and N phase.These quasi-particles are coupled to the Goldstone modethrough the following term [see Eq. (25)]: L ϕ,ψ = λ sin φ k + q δϕ q ψ † k + q ψ k − q + c.c , (32) 𝑘 𝑥 𝑘 𝑦 𝑘 𝑦 = 0 𝒒𝒗 𝑗 𝑘 ′ 𝒗 𝑗 ′ FIG. 5. The Fermi surface in the FM phase for three dif-ferent densities. The red lines denote regions in which v ( k ) points along the x direction and perpendicular to q whichis taken to be along y . These points give rise to the Landaudamping term in Eq. (27). The blue lines mark similar pointswhere v ( k ) is aligned along x but the form factor sin φ k inEq. (25) vanishes. The arrows denote the direction of the ve-locity v ( k ) , which is perpendicular to the vector q . The smalldashed circle denotes the radius of curvature k ,j (cid:48) of the point k j (cid:48) , which appears in the definition of the Landau dampingterm Eq. (30). where the coupling strength is λ = h . Following Ref. 16we consider the coupling term perturbatively, and showthat it necessarily leads to the breakdown of Fermi liquidbehavior on the entire Fermi surface, except for a discreteset of points. Using (26), we obtain the following leadingorder self-energy correction to the fermion propagator: Σ( ω, k ) = λ (2 π ) h (cid:90) sin φ k d q dν (cid:16) η | ν | q + κq (cid:17) ( i ( ν + ω ) + v ( k ) · q ) , (33)where k lies on the Fermi surface. As before, we havereplaced η and κ by their averages over φ q . Integratingover q and ω yields Σ( ω, k ) ≈ (cid:18) λh (cid:19) sin φ k π ηv ( k ) sign ω (cid:18) η | ω | κ (cid:19) / (34)For small ω , the self-energy becomes dominant over thebare iω term in the Fermionic Green’s function. This in-validates the perturbative approach, and signals a break-down of Fermi liquid behavior. E.g., treating (34) naivelyimplies that there is no discontinuity on the Fermionicoccupancy n k on the Fermi surface, except at the twopoints where φ k = 0 .From (34) we can extract a momentum-dependent en-ergy scale where the leading order self-energy correctionbecomes comparable to | ω | : E ( k ) = sin ( φ k )8 π ηv ( k ) κ . At thisenergy scale, a crossover from Fermi liquid to non-Fermiliquid behavior occurs. Using our numerically obtainedvalues of η , v , and κ at the tip of the banana-shapedFermi surface, we find that this energy scale is alwaysmuch greater than the Fermi energy. This implies thatnear the tips, there is no observable Fermi-liquid regime.Near the φ k = 0 “cold spots”, on the other hand, thenon-Fermi liquid scale vanishes rapidly as sin φ k .In the presence of a weak anisotropy in the disper-sion, as in Eq. (24), the Goldstone modes are ultimatelygapped at low energy. At this energy, another crossoveroccurs, and at asymptotically low energies Fermi liquidbehavior is recovered. VIII. DISCUSSION
In conclusion, we have analyzed the fate of a Rashbaspin-orbit coupled Fermi gas in the low density limit.The Fermi liquid state is unstable towards a varietyof competing liquid crystalline phases: ferromagnetic-nematic, nematic, and spin-density wave states. In thecase of short-ranged interactions, a cascade of phase tran-sitions occurs as the density decreases. The high densityisotropic Fermi liquid undergoes a transition to a spindensity wave, followed by a nematic state, and finally theground state becomes the ferromagnetic-nematic state atasymptotically low densities. In the case of dipolar in-teractions, the nematic state is the ground state all theway to the zero density limit.We have also analyzed the stability of theferromagnetic-nematic phase against terms that breakthe rotational symmetry, e.g., due to the underlyingcrystalline lattice. We found that the ferromagneticorder is stable as long as the chemical potential at thetransition is greater than the energy scale associatedwith the symmetry breaking term.Finally, we have discussed the effects of long-wavelength fluctuations of the FM and N order parame-ter on the low-energy quasi-particles. Scattering off thesefluctuations gives rise to the breakdown of Fermi liquidtheory, as generally expected for a phase that breaks acontinuous rotational symmetry . We therefore arguethat a system of interacting fermions with a Rashba-likedispersion offers a simple, generic route to realize a non-Fermi liquid phase.In contrast to the ferromagnetic and nematic phases,in the SDW phase the coupling between the fermionsand the Goldstone modes vanishes at long wavelengths (as it does for the case of CDW order ). It is likelythat it such coupling leads to qualitatively weaker effectscompared to the ferromagnetic or nematic cases.The nematic and spin-density wave phases may becomesuperconducting at sufficiently low temperature; such aninstability has been argued to be strongly enhanced inthe presence of gapless nematic fluctuations.
Theferromagnetic phase, however, does not posses a super-conducting instability, since it breaks the symmetries oftime reversal, inversion, and rotation by π around the z axis. Therefore, the non-Fermi liquid phase may berobust down to arbitrarily low temperatures.It is interesting to comment on the effect of disorderon the different symmetry breaking phase. Non-magneticdisorder couples linearly to the nematic order parame-ter. The system therefore maps onto a random field XYmodel; therefore, the nematic phase is expected to bedestroyed by disorder by the Imry-Ma argument. How-ever, since non-magnetic disorder does not couple linearlyto the magnetic component of the order parameter, thereis a possibility that the SDW and FM phases still posses afinite temperature transition in the presence of disorder.In the case of the ferromagnet, the system maps into therandom anisotropy XY model. It is an interesting openquestion whether an Ising finite-temperature transitioncan occur in this system at d = 2 for weak disorder. ACKNOWLEDGMENTS
We would like to thank Ehud Altman, Gareth Conduit,Nir Davidson, Sarang Gopalakrishnan, Anna Kesslman,Daniel Podolsky, Jonathan Schattner, and Senthil To-dadri for helpful discussions. E. B. was supported by theIsrael Science Foundation, by the Minerva foundation,and by an Alon fellowship. E. B. also thanks the AspenCenter for Physics, where part of this work was done. J.R. was supported by the ERC synergy grant UQUAM.
Note added.–
A related paper, Ref. 20, has appearedin parallel to this work. Our results are consistent wherethey overlap.
Appendix A: Variational calculation
In this appendix we derive the self-consistency equa-tions (10-12) using the variational principle. We seek thebest candidate ground state for the Rashba Hamiltonian(1) with the interaction term (8) H I = 1Ω (cid:88) k , k (cid:48) ∞ (cid:88) l =0 V l e iφ kk (cid:48) n k n k (cid:48) (A1)The variational trial state, |{ h l }(cid:105) , is taken to be theground state of the mean-field Hamiltonian (9). Thevalue of the variational parameters h l and µ are de-termined by minimizing the energy functional E V ≡ 𝑛 FM N ℎ × 10 𝜖 ℎ × 10 𝜖 FIG. 6. The mean-field variational parameters h and h vs.the density as obtained by equations (10-12) for ˜ U = 0 . . (cid:104){ h l }|H + H I |{ h l }(cid:105) ∂E V ∂h l = 0 and n = 1Ω (cid:88) k (cid:104) n k (cid:105) To simplify the variational equations we use the identity ∂∂h l (cid:104)H NMF (cid:105) = − n l , where n l ≡ (cid:80) k cos lφ k (cid:104) n k (cid:105) ∂E V ∂h l = ∂∂h l (cid:34) (cid:104)H NMF (cid:105) + (cid:88) l (cid:48) (cid:0) h l (cid:48) n l (cid:48) + V l (cid:48) n l (cid:48) (cid:1)(cid:35) (A2) = (cid:88) l (cid:48) ( h l (cid:48) + 2 V l (cid:48) n l (cid:48) ) ∂n l (cid:48) ∂h l = 0 (A3)Thus, we find that the minimum solution for l = 0 , , isobtained by the equations (10-12) as long as the matrix Q ll (cid:48) = ∂n l (cid:48) ∂h l is not singular. The solution of the equationsfor ˜ U = 0 . is presented in Fig. 6.We note that it is straightforward to generalize thederivation of the self-consistency equations (10-12) to thecase of a textured order parameter (not translationallyinvariant). We simply substitute n l by n l ( q ) = (cid:88) k cos lφ k (cid:104) ψ † k + q ψ k − q (cid:105) (A4) in Eq. (A1) - Eq. (A2). An important outcome of thisgeneralization is that the coupling constant that couplesto the SDW order n ( q ) is V . This validates using V as the corresponding coupling constant in the Stoner cri-terion for the SDW instability in accord with the maintext.It is also useful to point out that the solution of theequations (10 - 12) can be simplified to some extent inthe case of a single order parameter. Performing theintegration over k in the case of h = 0 we find thatthese equations reduce to n = k k xF π E (cid:18)
21 + a ; φ (cid:19) (A5) h = 4 k k xF V π (cid:20) a E (cid:18)
21 + a ; φ (cid:19) + (1 − a ) K (cid:18)
21 + a ; φ (cid:19)(cid:21) (A6)where k xF = k (cid:113) h + µε , a ≡ µ/h , φ ≡ arccos( − a ) and E ( x ; φ ) = (cid:82) φ dφ (cid:112) − x sin φ and K ( x ; φ ) = (cid:82) φ dφ √ − x cos φ the partial elliptic integral of the sec-ond kind. Similarly, in the nematic case where h = 0 we have h = 2 k k xF V π (cid:20) a E (cid:18)
21 + a ; φ (cid:19) + (1 − a ) K (cid:18)
21 + a ; φ (cid:19)(cid:21) (A7)where a = µ/h and there are two Fermi surfaces. Appendix B: The spin susceptibility of the Rashbagas in the dilute limit
In this appendix we calculate the in-plane spin-susceptibility of the Rashba gas, which is given by χ ij ( iω, q ) = 1Ω (cid:88) k n F ( ξ k ) − n F ( ξ k + q ) − iω + ε k + q − ε k P ( φ k + φ k + q ) (B1)where P ( x ) = 12 (cid:18) − cos x sin x sin x x (cid:19) In the FL phase we can compute (B1) analytically inthe static limit. First, we linearize the denominator term ε k + q − ε k ≈ v qδkk cos φ k , q where v = ε /k , δk = k − k c and k c ≈ k (cid:16) − q k cos φ k , q (cid:17) . Integrating over δk and φ k yields χ ij ( iω, q ) = ρ ( µ )2 ( F − (˜ ω, ˜ q ) − cos 2 φ q [2 F (˜ ω, ˜ q ) − F − (˜ ω, ˜ q )] σ z ) (B2)0where ˜ ω ≡ mω k F q , ˜ q ≡ q k F and F n ( x, y ) = 18 π (cid:90) π dφ cos n φ log (cid:18) x + (1 + y cos φ ) cos φx + (1 − y cos φ ) cos φ (cid:19) y
21 0 1 2
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