Spin-Orbit Ordering, Momentum Space Coexistence, and Cuprate Superconductivity
Abstract
Motivated by the energetic advantage of achieving coherent enhancement of effective spin-dependent interactions through approximate nesting, we propose specific forms of spin ordering, whose form varies over the Fermi surface, for the cuprate superconductors. Competing “spin-orbit” orderings involving order parameters in spatial and waves at commensurate and incommensurate wavevectors, phase separated in momentum space, support behavior suggestive of observed phenomena. The spin-orbit fluctuation induces an effective interaction that favors -wave pairing, as required for the observed superconductivity. Anisotropic spin susceptibility is a crucial prediction of our mechanism.
High superconductivity in cuprates has been an outstanding problem in condensed matter physics since it was first discovered in 1986. In the meantime, though no consensus theoretical understanding has developed, several striking features of the phenomenology have emerged clearly. These include the strongly 2-dimensional character of the essentially new physics; its proximity to antiferromagnetic spin ordering; the anomalous character of the normal state, especially in the underdoped region, suggestive of emergent energy gaps; the acute sensitivity of the low-temperature state to doping and impurities; the d-wave character of the superconductivity; and the apparent uniqueness or near-uniqueness of cuprate layers in supporting this overall phenomenological profile. On the face of it, several of these features suggest that the origin of these phenomena will involve forms of ordering that involve spin and depend sensitively on the precise form of the lattice and the Fermi surface.
There is a simple heuristic that appears to be broadly consistent with these indications. As is familiar from the BCS theory of superconductivity, the effect of weak attractive interactions can be amplified, and can lead to drastic qualitative effects, if there are many low-energy pairs sharing the same quantum numbers. Correlations among these pairs can then be arranged so that their interactions contribute coherently to lowering the energy. The BCS mechanism of superconductivity involves particle-particle and hole-hole correlations. In this context the existence of low-energy pairs with total momentum zero, arising from time-reversed states with momenta both near the Fermi surface, is generic. By contrast spin-density-wave ordering, at the level of electron creation and destruction operators, involves particle-hole correlations. In this context one finds that many low-energy pairs sharing a common (lattice) momentum only for specially shaped (“nested”) Fermi surfaces. Since the shape of the Fermi surface changes with doping, one might anticipate that at best a nesting condition would be approximately fulfilled at a specific doping level. Our point of departure is to consider that the Fermi surface is not an end in itself, but a step toward constructing the ground state. If changing the pattern of occupied levels — effectively, engineering the Fermi surface — can encourage favorable coherence factors, it might be favorable to make nesting persist. Realizations of this possibility and the properties of the emergent states will, on the face of it, depend sensitively on details of the interactions, lattice structure, and doping level. Two dimensional antiferromagnets on a square (or nearly square) lattice near half filling, as in cuprate layers, provide an especially favorable area for these ideas.
Proposed Ordering
At half filling antiferromagnetic (AF) spin ordering is observed. Electrons in the real materials may well be best described as strongly coupled and the spins as localized, but we shall construct our states heuristically by extrapolation from weak or intermediate coupling, anticipating that universal properties, specifically symmetry breaking patterns, might be successfully inferred. (Also, photoemission experiments ARP can be interpreted as revealing a Fermi surface, even at quite small doping.) In that spirit AF ordering can be regarded as follows. On-site Coulomb repulsion induces, in the crossed channel, an attractive interaction between electrons and holes of opposite spin at momentum transfer (modulo reciprocal lattice vectors). This makes it favorable to deform the effective Fermi surface into a diamond shape, which for half filling nests at (see Fig. 1a), and allow the electrons to form spin-triplet particle-hole pairs, according to . Here, are Pauli matrices, and and are electron annihilation and creation operators, with two spin components united into a column: .
At finite doping no ansatz seems so uniquely compelling, but the possibility illustrated in Fig. 1b is suggestive. The free Fermi surface has been deformed in two distinct ways, one operating near to and the other far from the zone diagonals. This geometry supports nesting for orderings with wavevector in the off-diagonal region and with wavevectors slightly off , namely, and , along the diagonals. This proposal embodies a new phenomenon, phase separation in momentum space, that might find wider application.
Order parameters with near-uniform -wave structure do not easily accommodate such phase separation. It is more natural when zeroes of one condensate correspond to maxima of the other. This leads us to propose spin-orbit (SO) orders of the form: rem
( off-diagonal) | ||||
(; diagonal) |
Here are constant, real vectors in spin space, and are orbital wave basis functions of lattice group , defined by and . (The lattice constant is set to a unit.) The order is purely imaginary, as required by hemiticity. Roughly speaking, it describes a state with microscopic spin currents flowing around each plaquette in real space.
The nature of these spin-orbit orderings may be more transparent in real space:
(3) | |||||
where and Note that these vanish for .
At mean field level such spin-orbit orderings with nonzero and are favored by the nearest and next nearest neighbor Coulomb repulsions, respectively. Indeed the nearest-neighbor Coulomb interaction can be transformed to
(4) |
plus an unimportant term proportional to the density operator. In this form the anticipated electron-hole attraction is manifest. (At this level , with , also favors charge-orbit (CO) ordering, known as orbital-antiferromagnetism, staggered flux phase Aff , or DDW Chakravarty et al. (2001); a non-static version was proposed in Wen and Lee (1996). We shall not discuss it further here.) Similarly, the next-nearest-neighbor Coulomb interaction favors both SO and CO orders.
Phase diagram
We shall focus on the following competing orders, that we believe play major roles: AF, -SO and -SO, and -wave superconducting state (dSC). The two SO orders compete for particle-hole pairing states, plausibly with different outcomes in different domains, whose size depends on overall doping level. This can be understood by reference to Fig. 1b. With increasing density of doped holes, the total area of the four triangles along the zone diagonals increases at first while the size of off-diagonal regions shrinks. For sufficient large doping, e.g., , the triangles are no long sustainable. This explains the trends of phase transition lines of (for -SO) and (for -SO) as functions of doping.
For low doping up to , there are several competing spin orders (including AF) that are connected by first-order phase transitions. On general grounds, one expects that phase separation (in real space) may occur, plausibly in the form of stripes Emery et al. (1999); Zaanen and Nussinov ).
Effective Theory
We now propose an effective Lagrangian for the unconventional normal state, based on the hypothesis of -SO ordering:
(5) |
where is a composite operator, is assumed, and collects the hopping terms. The interaction term contains part of the original next-nearest neighbor Coulomb interaction.
A -SO ordered state is characterized by a non-vanishing order parameter field , corresponding to the imaginary part of , pointing to (say) the -direction in spin space,
(6) |
This correlation spontaneously breaks SU(2) spin symmetry down to a U(1) corresponding to spin rotation about the -axis. Two Nambu-Goldstone bosons appear as gapless collective spin excitations, corresponding to transverse fluctuations of the order parameter field . We will call them orbital magnons.
We can describe the low energy interactions of these collective modes using an effective Lagrangian. Following the standard technique Weinberg (1996), we isolate the Nambu-Goldstone part (two transverse spin fluctuations) in the electron field. Expressing the electron field as a local SU(2) spin rotation acting on a new fermion :
(7) |
This defines a new fermion that carries the full charge and the spin quantum numbers but does not carry transverse spin. parameterize the slowly-varying orbital magnons. They are given by the -component via: and with , where represents the (gapped) longitudinal spin-orbit fluctuation. With the above transformations, we now have a prescription of deriving an effective Lagrangian for the orbital magnons and new fermions: from the Lagrangian (5).
The free part of the effective theory for the -fermion is
(8) |
where with . Due to unit cell doubling, the fermion energy spectrum is split into two bands and
(9) |
each being spin up and down degenerate. For weak -SO order, the Fermi surface of the band is near the diagonals of the Brillouin zone while the Fermi surface of the band is near the off-diagonal region (see Fig. 3). Note that the fermion field , appropriate to describing the low-energy excitations, does not carry the full spin quantum numbers of the electron. This is a form of partial spin-charge separation.
The effective theory for -SO can be constructed in the same manner from a model Lagrangian similar to (5), with and (-SO order parameter). The ordering wavevector is changed from to .
Origin of -wave Superconductivity
In the effective theory of -SO, fermions are coupled to the longitudinal spin-orbit fluctuation ,
(10) |
Broken spin symmetry implies that the propagator has the form with having minima at and the damping rate as . Given the coupling (10), we have calculated the effective interaction between fermions induced by the longitudinal SO fluctuation (Fig. 4).
In the low energy (static) limit, the spin-singlet component of the Cooper channel interaction is described by in the Hamiltonian
(11) |
where and the spin indices . Here is a positive normalization constant of the SO field. The interaction is repulsive in both -wave and -wave pairing. Note that , as function of , peaks at the incommensurate wavevectors (Fig. 4). By examining a typical superconducting gap equation, , one finds nontrivial solution if and have opposite sign. By the arguments similar to Ref. Monthoux et al. (1992), we conclude that the favored superconducting pairing is -wave. Fermions near and will pair as: The -fermion pair condensate is equivalent to an electron pair condensate
(12) |
for uniform . Thus an exotic normal state can be associated with conventional electron pairing.
The state of -SO is different. A similar coupling involving the longitudinal SO fluctuation in the leads to an effective interaction that disfavors the -SC pairing.
-SO order does not forbid the existence of Fermi nodal points in the dSC state. It splits the quasiparticle energy band around the four triangles along the zone diagonals (Fig. 1), with Fermi surfaces being carved out from the lowest possible band.
Incommensurate spin excitations
The translational symmetry broken by the SO orderings leads to multi-bands of orbital magnons in a reduced zone scheme, each having the degeneracy of two transverse spin modes. The orbital magnons associated with the -SO are gapless at the four incommensurate momenta and . With coexisting and SO orders, there will be another band that is gapless at . Dynamical spin-spin correlation functions are affected by particle-hole pair mixing with two orbital magnon channels. Combining magnons from and can yield two branches of spin excitation resonance at finite energies, one dispersing upward and another downward from momentum . The two branches cross at , with zero gap energy. This structure seems qualitatively consistent with the inelastic neutron scattering experiments in YBCO (e.g, as in D. Reznik, et al. D. Reznik, P. Bourges et al. and references therein; see also Dai et al. (2001)). A full investigation is in progress.
Anisotropic spin susceptibility
The state of SO ordering has , so the static correlation of local spins does not exhibit conventional long range order. A straightforward exercise confirms . as
We have calculated the uniform, static spin susceptibility for the -SO state, using the effective Lagrangian (8). We find that the susceptibility is anisotropic,
(13) |
where , , and the ordering direction is assumed arbitrary (cf. Eq. (8)). Fig. 5 shows the anisotropy of susceptibility predicted by the mean field theory (without corrections due to orbital magnon scatterings). Measurement of and could therefore provide important tests of our proposals. The result is quite different from the AF state, for which .
The authors are grateful for comments and advices by C. Honerkamp, P. A. Lee, T. Senthil, S. Weinberg and X.-G. Wen. This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818.
References
- (1) For recent reviews, see, A. Damascelli, Z. X. Shen, and Z. Hussain, in Rev. Mod. Phys. 75, 473 (2003); J. C. Campuzano, M. R. Norman, and M. Randeria, cond-mat/0209476.
- Schulz (1990) H. J. Schulz, Phys. Rev. Lett. 64, 1445 (1990).
- Littlewood et al. (1993) P. B. Littlewood, J. Zaanen, G. Aeppli, and H. Monien, Phys. Rev. B 48, 487 (1993), and references therein.
- (4) Under different names, the -SO (but not the ) appeared before as one of possible candidate orders in F. Bouis et al., cond-mat/9906369; C. Nayak, Phys. Rev. B62, 4880 (2000); A. P. Kampf and A. A. Katanin, Phys. Rev. B67, 125104 (2003); and C. Wu, W. V. Liu, and E. Fradkin, Phys. Rev. B68, 115104 (2003). To the best of our knowledge, the state of this order, however, was never fully studied. In a recent paper [K. Maki, B. Dóra and A. Virosztek, cond-mat/0306567], USDW (unconventional spin-density-wave) was proposed to interpreted experiments of pseudogap for high cuprates. Their USDW order features local spin component in the - plane.
- (5) I. Affleck and J. B. Marston, Phys. Rev. B 37, 3774 (1988); T. Hsu, J. B. Marston and I. Affleck, ibid. 43, 2866 (1991).
- Chakravarty et al. (2001) S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Phys. Rev. B 63, 094503 (2001).
- Wen and Lee (1996) X.-G. Wen and P. A. Lee, Phys. Rev. Lett. 76, 503 (1996).
- Emery et al. (1999) V. J. Emery, S. A. Kivelson, and J. M. Tranquada, Proc. Natl. Acad. Sci. 96, 8814 (1999).
- (9) J. Zaanen and Z. Nussinov, cond-mat/0006193.
- Weinberg (1996) S. Weinberg, The Quantum Theory of Fields II: Modern Applications (Cambridge University Press, Cambridge, England, 1996), chap. 19.
- Monthoux et al. (1992) P. Monthoux, A. V. Balatsky, and D. Pines, Phys. Rev. B 46, 14803 (1992).
- (12) D. Reznik, P. Bourges et al., cond-mat/0307591.
- Dai et al. (2001) P. Dai, H. A. Mook, R. D. Hunt, and F. Dogan, Phys. Rev. B 63, 054525 (2001).