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Dynamic Generation of Spin-Orbit Coupling

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Dynamic Generation of Spin-Orbit Coupling PHYSICAL REVIEW LETTERS week ending VOLUME 93, NUMBER 3 16 JULY 2004 Dynamic Generation of Spin-Orbit Coupling Congjun Wu and Shou-Cheng Zhang Department of Physics, McCullough Building, Stanford University, Stanford California 94305-4045, USA (Received 24 December 2003; published 15 July 2004) Spin-orbit coupling plays an important role in determining the properties of solids, and is crucial for spintronics device applications. Conventional spin-orbit coupling arises microscopically from relativ- istic effects described by the Dirac equation, and is described as a single particle band effect. In this work, we propose a new mechanism in which spin-orbit coupling can be generated dynamically in strongly correlated, nonrelativistic systems as the result of Fermi surface instabilities in higher angular momentum channels. Various spin-orbit couplings can emerge in these new phases, and their magni- tudes can be continuously tuned by temperature or other quantum parameters. DOI: 10.1103/PhysRevLett.93.036403 PACS numbers: 71.10.Ay, 71.10.Ca, 71.10.Hf Most microscopic interactions in condensed matter model do not break rotational symmetry, and some of physics can be accurately described by nonrelativistic them preserve time-reversal and parity symmetries as physics. However, spin-orbit (SO) coupling is a notable well. Most correlated phases in condensed matter physics exception, which arises from the relativistic Dirac equa- are characterized by their broken symmetries [16]. Solids tion of the electrons [1]. The emerging science of break translational symmetry, liquid crystals break rota- spintronics makes crucial use of the SO coupling to tional symmetry, superfluids and superconductors break manipulate electron spins by purely electric means. The gauge symmetry, and ferromagnets break time-reversal proposed Datta-Das device [2] modulates the current flow symmetry and rotational symmetry. As far as we are through the spin procession caused by the SO coupling. aware, the new phase reported in this Letter is the only More recently, Murakami, Nagaosa, and Zhang [3,4] one besides the Fermi liquid which does not break any of proposed a method of generating the dissipationless the above symmetries. It is distinguished from the Fermi spin current by applying an electric field in the p-doped liquid by only breaking the ‘‘relative spin-orbit symme- semiconductors. This effect and the similar proposal for try,’’ a concept first introduced in the context of the 3He the n-doped semiconductors [5] both make crucial use liquid [17]. of the SO coupling. In contrast to the generation of the We first discuss the dynamic generation of SO coupling spin current by coupling to the ferromagnetic moment, from the LP instability within the Landau-Fermi liquid a purely electric manipulation has an intrinsic advantage. theory triggered by the negative Landau parameter F1 , However, unlike the ferromagnetic moment, which can be and then present its exact definition. This instability lies spontaneously generated through the strong correlation of in particle-hole channel with total spin one and relative spins, the conventional wisdom states that the SO cou- orbital angular momentum one. Operators in matrix a y ^ a pling is a noninteracting one-body effect, whose micro- forms are defined as Q r r ÿir r, scopic magnitude is fixed by the underlying relativistic where Greek indices denote the direction in the spin physics. space, Latin indices denote the direction in the orbital On the other hand, recent interest has been revived in space, and the operation of r^ a on the plane wave is the Landau-Pomeranchuk (LP) [6] Fermi surface insta- defined as r^ aeik~r~ ra=jrjeik~r~ k^aeik~r~. Qa r is bilities, largely in connection with electronic liquid crys- essentially the spin-current operator up to a constant tal states with spontaneously broken rotational symmetry factor. We use a Hamiltonian similar to that of Ref. [8], [7–12], and in connection with hidden orders in heavy a but in the F1 channel: Fermion systems [13–15]. Varma’s recent work showed Z 3 y ~ a that the LP instability could lead to the opening of an H d r~ r~ rr ÿ r~haQ r~ anisotropic gap at the Fermi surface [13]. In this Letter, Z 1 3 3 0 a 0 a a 0 we show that the SO coupling can be generated dynami- d rrd~ r~ f1 r~ ÿ r~ Q r~Q r~ ; (1) cally in a nonrelativistic system through strong correla- 2 tion effects as the LP instability in the spin channel with where is the chemical potential and the small ha higher orbital angular momentum. It emerges collectively is dubbed as the ‘‘spin-orbit field,’’ which plays a after a phase transition, which is continuously tunable role similar to the external magnetic field. For later either by temperature or by a quantum parameter at zero convenience [8], we keep both the linear and the cubic temperature. Unlike the ferromagnet, our ordered phase terms in the expansion of the single particle dispersion ~ keeps time-reversal symmetry. Also in contrast to the LP relation around the Fermi wave vector kf, k 2 instabilities considered by the majority of previous theo- vfk1 b k=kf , with k k ÿ kf. We assume a ries, most translationally invariant liquid phases in our that the Fourier components of f1 r~ take the form 036403-1 0031-9007=04=93(3)=036403(4)$22.50 2004 The American Physical Society 036403-1 PHYSICAL REVIEW LETTERS week ending VOLUME 93, NUMBER 3 16 JULY 2004 R a iq~ r~ a a a 2 f1 q drre~ f1 rf1 = 1 jf1 jq and define the and orbital space, respectively, Da is any SO(3) rotation a a dimensionless Landau parameter F1 Nff1 , where Nf matrix, and n is a real number. In other words, the is the density of states at Fermi energy. The symmetry of correlation functions of operators Qa acquire a long the Hamiltonian (1) is a direct product SO 3L SO 3S range part in the ordered states in the orbital and spin channels. n2 ^ We define the spin-orbit susceptibility as a;b a b 0 de^a phase; hQ r~Q r~ i ! ab a 2 (6) hQai=hb in the limit hb ! 0, which is diagonal, i.e., jf1 j Da phase; , in the normal Fermi liquid phase. The a;b a b 0 Fermi liquid correction to is given by as jr~ ÿ r~ j!1. This correlation function gives the rig- orous definition for the new phases, independent of the m 1 approximate Fermi liquid theory used here. FL 0 a ; (2) m 1 F1 =3 The phase is a straightforward generalization of the nematic Fermi liquid [8] to the triplet channel as shown in with m =m the ratio between the effective and bare Fig. 1, where the spin and orbital degrees of freedom masses. The spin-orbital susceptibility is enhanced for a a remain decoupled, and the rotational symmetry is broken. F < 0 and divergent as the critical point F ÿ3 is a 1 1 Taking a special case n nn zaz, the dispersion re- reached. A lations for spin up and down branches are k1;2 In the mean-field (MF) analysis, the p-h channelR triplet kÿ n a a cos , respectively, where is the angle order parameter is defined as n r~ÿ drrf~ 1 r ÿ k z 0 a 0 between and axis. The Fermi surfaces for the two r hQ r i, and the external spin-orbit field ha is a a spin components are distorted in an opposite way as set to zero. Using the uniform ansatzR n rn , k =k x cos 1 ÿ bx2cos2ÿ1=3x2, with 3 y ~ f1;2 f Eq. (1) is decoupled into HMF d r~ r~ rr ÿ the dimensionless parameter x nn= v k . The chemi- a ^ a a a a f f n ÿir ÿ r~Vn n = 2jf1 j, with V the cal potential is shifted to ensure the particle number space volume. The self-consistent equation for the order conservation as = v k ÿx2=3. The remaining parameters reads f f symmetry is SO 2L SO 2S with the Goldstone mani- Z fold S2 S2. Two Goldstone modes are the oscillations of d3k~ L S na jfaj h y kk^a ki; (3) the distorted Fermi surfaces, and the other two are the 1 23 p oscillations of the spin directions. a In the phase, the rotational symmetry is preserved which is valid when the interaction range r0 jf1 j is much larger than the distance between particles 1=k ; i.e., with the dynamic generation of spin-orbit coupling as p f a a shown in Fig. 1. For example, with the ansatz n the dimensionless parameter kf jf1 j 1. The phase structures can be determined from the nn a, the MF Hamiltonian is reduced to X Ginzburg-Landau (GL) free energy, which is similar to H y k kÿ ÿ nn~ k^ k: (7) the triplet pairing order parameter in the 3He system MF k [17,18]. Under the independent SO(3) rotations in the a orbital and spin spaces RL and RS, n transforms as The single particle states can be classified according a ;b ÿ1 a n ! RL;n RS;ba. Furthermore, n is even under to the eigenvalues 1 of the helicity operator ~ k^, B the time-reversal but odd under the parity transformation. with dispersion relations k1;2 kÿ n.The With these symmetry requirements, the GL free energy Fermi surface distortions of two helicity bands are can be constructed up to the quartic order as T T T δk δ F nA trn nB trn n2 B trn n2: (4) f1 kf1 1 2 s Compared with the complex order parameter in the super- δkf2 3 fluid He case, the reality of the na restricts the free energy to contain only two quartic terms.
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