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PHYSICAL REVIEW LETTERS week ending VOLUME 93, NUMBER 3 16 JULY 2004

Dynamic Generation of -Orbit Coupling

Congjun Wu and Shou-Cheng Zhang Department of , 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 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 [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 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~. Qar 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 Š 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- vfk‰1 ‡ bk=kf Š, 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 . The symmetry of correlation functions of operators Qa acquire a long the Hamiltonian (1) is a direct product SO3L SO3S 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 cos1 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 ÿŠ 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 SO2L SO2S with the Goldstone mani- Z fold S2 S2. Two Goldstone modes are the of d3k~ L S na jfaj h ykk^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 yk‰kÿ ÿ nn~  k^Š 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 FnA tr‰n nŠ‡B tr‰n nŠ2 ‡ B tr‰n n2Š: (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. Explicitly, s T a a 2 2 2 T 2 tr‰n nŠn n 1 ‡ 2 ‡ 3,andtr‰n n Š k2 a b a b 4 4 4 2 n n n n 1 ‡ 2 ‡ 3, where 1;2;3 are eigenval- δ k1 T kf2 ues of n n.ForB2 < 0 or B2 > 0, Eq. (4) favors the the s 2 2 2 s structures of 1;2;3 to be proportional to 1; 0; 0 or 1; 1; 1, respectively. We name them as or phases whose general order parameter matrix structures are α− phase β−phase given by a  FIG. 1. The LP instability in the F1 channel, with dashed nd^ e^ phase; for B < 0; na  a 2 (5) lines marking the Fermi surface before symmetry breaking. nnD a phase; for B2 > 0; The Fermi surface distortion is anisotropic in the phase, while it is isotropic in the phase with dynamic generation of where d^ and e^ are two arbitrary unit vectors in the spin spin-orbit coupling. 036403-2 036403-2 PHYSICAL REVIEW LETTERS week ending VOLUME 93, NUMBER 3 16 JULY 2004

2 2 kf1;2=kf x1 ÿ bx ÿx and the chemical poten- nally studied under the name of ‘‘spin-split state’’ by 2 tial shift =vfkfÿx . Similar to the superfluid Hirsch [22] to explain the phase transition at TN 3He-B phase, the phase is essentially isotropic. The 311 K in the chromium system. orbital L~ and the spin S~ angular momenta are no longer By reducing the space dimension to 2, the mean-field separately conserved, but the total angular momentum Hamiltonian for the phase reduce to the familiar J~ L~ ‡ S~ 0 remains conserved. The Goldstone mani- Rashba [23] and Dressselhaus Hamiltonians [24] in the fold is ‰SO3L SO3SŠ=SO3L‡S SO3 with three 2D semiconductor heterostructures. The order parameter branches of Goldstone modes. For the general case na is a 3  2 matrix with  x; y; z and a x; y.Its of na nnD a, it is equivalent to a redefinition of third row of  z can be transformed to zero by per- 0 spin operators as S SDaa, thus Fermi surface forming suitable SO(3) rotation on the index , thus we ~0 ~ ~0 distortions remain isotropic and J L ‡ S is con- take na as a 2  2 matrix. The GL free energy is also the served. A similar symmetry breaking pattern also same as in Eq. (4), but with the new coefficients appears in the where the 1 1 N N two-flavor chiral symmetry SU2  SU2 is broken A ÿ f ;B f ; L R 2 jfaj 2 1 32v2k2 into the diagonal SU2L‡R [19]. In that case, both 1 f f SU2 are internal symmetries, and thus there is no bN L;R B f ; (10) flavor-orbit coupling. 2 2 2 8vfkf The coefficients of the GL free energy Eq. (4) can be a microscopically derived from the MF theory as and the LP instability occurs at F1 < ÿ2.The -and -phase structures are similar as before in Eq. (5). 1 1 N N b A ÿ f ;B f 1 ‡ ; However, there are two options in the phase with na 2 jfaj 3 1 20v2k2 3 1 f f nnD a, where Da is a O2 matrix. If detD 1, then Jz N ÿ1 L ‡ S is conserved. With the MF ansatz n nn , B f ‡ b ; (8) z z a a 2 30v2k2 3 we arrive at the Rashba-like Hamiltonian f f Z 2 y ~  ^ a where b describes the cubic part of the dispersion k,as HR d r~ frr ÿ nn za ÿir g : (11) a explained earlier. With the definition of  1=jF1 jÿ 1=3, the LP instability takes place at <0, i.e., If detD ÿ1, Jz is not conserved while the energy a F1 < ÿ3.Forb<1=3, i.e., B2 < 0,the phase appears spectrum and Fermi surface distortions are still the 2 with jnj ‰jAj=2B1 ‡ B2Š.Forb>1=3,i.e.,B2 > 0, same as the case of detD 1. With the MF ansatz na 2 the phase is stabilized at with jnj ‰jAj=23B1 ‡ ndiagf1; ÿ1g, we arrive at the 2D Dresselhaus-like B2Š. The largest Fermi surface distortion kf;max=kf in Hamiltonian as the phase is larger than the uniform one kf=kf in the Z 2 y ~ ^ x ^ y phase, thus a large positive b is helpful to the phase. HD d r~ frrÿn‰xÿir ÿyÿir Šg : (12) However, we emphasize that this is only one of the options to change the sign of B2. If we generalize the mechanism of dynamical genera- To apply the and phases in the lattice system, we tion of the spin-orbit coupling to the spin 3=2 Fermionic only need replace the SO3L symmetry with the lattice system, an interesting phase can be obtained which pre- point group. For example, for the simple cubic lattice, we serves all familiar symmetries including the parity sym- a y  define Q ifc x~ cx~ ‡ e^aÿH:c:g with e^a the metry, breaking only the relative spin-orbit symmetry. It base vector in the a direction. The unbroken symmetry has been recently shown that any generic model of spin is OL SO3S where OL is the orbit lattice octahedral 3=2 with local interactions has an exact SO5 symmetry group. The mean-field Hamiltonian for the phase reads in the spin space [25]. The four spin components form the X spinor representation of the SO5 group. Using the Dirac H y ÿ  ÿ nn k  ; MF k k  sin a a k (9) matrix defined there, the spin 3=2 Landau interaction k functions are classified into the SO5’s scalar, vector, and with lattice momentum k~ restricted in the first Brillouin tensor channels [25]: ~ zone. The helicity structure for each k is aligned along the 0 s 0 v 0 a a f ; pp;~ p~ f pp;~ p~ ‡f pp;~ p~ =2 =2  direction of sinkx; sinky; sinkz, which breaks the sym- t 0 ab ab metry down to the combined octahedral rotation in the ‡ f pp;~ p~ =2 =2 : orbit and spin space OL‡S. As a real space analogy, the (13) hexagonal noncolinear antiferromagnet YMnO3 [20,21] has the spin order pattern inside the unit cell which is also We further pick out its L 2 part of the orbital angular momentum in the spin two vector channel de- invariant under the combined spin-orbit point group ro- v a tations. The difference is that the spin order in the phase noted as the F2 channel. We define operators Q r~ yr~ d^arr~ r~1  ; a  5, where d^arr~ lies in the momentum space and no spatial spin order p  p p p ^ ^ ^ ^ ^ ^ 1 ^ 2 ^ 2 3 ^ 2 exists at each lattice site. The lattice phase was origi- ‰ 3rxry; ÿ 3rxrz; 3ryrz; ÿ 2 3rz ÿ r ; 2 rx ÿ 036403-3 036403-3 PHYSICAL REVIEW LETTERS week ending VOLUME 93, NUMBER 3 16 JULY 2004

^ 2 ryŠ. The model Hamiltonian is constructed as follows: conjecture that besides the familiar superfluid A and B Z phases, 3He may contain the new phases proposed in this 3 y ~ a 3 H d r~ r~‰rr ÿ Š r~ work. The Landau parameter F1 in He was determined Z to be negative from various experiments [27–30] such as 1 3 3 0 v 0 a a 0 ‡ d rrd~ r~ f2 r~ ÿ r~ Q r~Q r~ ; (14) the normal-state spin diffusion constant, spin-wave spec- 2 trum, and the temperature dependence of the specific with the symmetry of SO3L SO5S. TheR order pa- heat, etc. It varies from around ÿ0:5 to ÿ1:2 with in- a 3 v rameter is defined as before n rÿ dr~ f2 r~ ÿ creasing pressures to the melting point, reasonably close 0 a 0 v a r~ hQ r~ i and the LP instability occurs when F2 to the instability point F1 ÿ3. Even though we pre- v sented mean-field descriptions of the new phases with Nff2 q 0 < ÿ5. The ordered phases after the LP instability can also be dynamically generated spin-orbit coupling, the existence classified into two categories as before: the phase with of these phase can obviously be studied by exact micro- anisotropic Fermi surface distortions and phase with scopic calculations of the correlation function Eq. (6) for spin-orbit coupling. The detail phase structures are much realistic models. more complicated here. For example, the phase has two We thank J. P. Hu and S. Kivelson for helpful dis- nonequivalent configurations because the L 2 channel cussions. C.W. especially thanks E. Fradkin and Fermi surface distortions can be either uniaxial or biax- V. Oganesyan for their education on nematic Fermi ial. A comprehensive classification of all the possible liquids. This work is supported by the NSF under Grant phases is quite involved and is deferred to a future No. DMR-0342832, and the U.S. Department of Energy, work. We focus here on the phase with the order pa- Office of Basic Energy Sciences under Contract No. a DE-AC03-76SF00515. C.W. is supported by the rameter structure n nn a.Inthiscase,theMFR 3 y Stanford SGF. Hamiltonian is reduced into HMF d r~ r~fr ÿ  a ^  ÿ nn a d rrg r~. From the relation between the  matrices and the quadratic form of spin 3=2 matrices S~ [4], it can be easily recognized the Luttinger-like Hamiltonian [26] [1] C. Kittel, Quantum Theory of Solids (Wiley Press, New Z York, 1987). H d3r~ yr~fr ÿ  ÿ nÿirr^ S~2g r~; (15) [2] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990). L [3] S. Murakami, N. Nagaosa, and S. C. Zhang, Science 301, 1348 (2003). which is the standard model for the hole-doped III-V [4] S. Murakami et al., cond-mat/0310005 [Phys. Rev. B (to semiconductors. The original symmetry SO3L be published)]. SO5S in Eq. (14) is broken into SO3L‡S with ten [5] J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004). branches of Goldstone modes. Different from the spin [6] I. I. Pomeranchuk, Sov. Phys. JETP 8, 361 (1959). 1=2 case, the order parameter na here is even both under [7] C. J. Halboth et al. Phys. Rev. Lett. 85, 5162 (2000). the time-reversal and parity transformations. [8] V. Oganesyan et al., Phys. Rev. B 64, 195109 (2001). It is remarkable that most standard spin-orbit systems, [9] D. G. Barci et al. Phys. Rev. B 67, 205108 (2003). including the familiar Rashba, Dresselhaus, and [10] W. Metzner et al., Phys. Rev. Lett. 91, 66402 (2003). Luttinger-like Hamiltonians can all be generated dy- [11] H.Y. Kee, Phys. Rev. B 67, 73105 (2003). namically by our mechanism, which can compliment [12] H.Y. Kee et al., Phys. Rev. B 68, 245109 (2003). the conventional mechanism based on relativistic single [13] C. M. Varma, cond-mat/0311145 (unpublised). [14] T.T. M. Palstra et al., Phys. Rev. Lett. 55, 2727 (1985). . If a system has a small spin-orbit cou- [15] A. P. Ramirez et al., Phys. Rev. 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