Abstract
Gauge fields, typified by the electromagnetic field, often appear as emergent phenomena due to geometrical properties of a curved Hilbert subspace, and provide a key mechanism for understanding such exotic phenomena as the anomalous and topological Hall effects. Nonabelian gauge potentials serve as a source of nonsingular magnetic monopoles. Here we show that unlike conventional solid materials, the nonabelianness of emergent gauge potentials in spinor BoseFermi atomic mixtures can be continuously varied by changing the relative particlenumber densities of bosons and fermions. The nonabelian feature is captured by an explicit dependence of the measurable spin current density of fermions in the mixture on the variable coupling constant. Spinor mixtures also provide us with a method to coherently and spontaneously generate a pure spin current without relying on the spin Hall effect. Such a spin current is expected to have potential applications in the new generation of atomtronic devices.
Introduction
Gauge fields are the fundamental ingredient in the Standard Model of elementaryparticle physics^{1}. In condensed matter physics, a large number of phenomena including optical and transport properties of solids appear as the responses of the system to an applied electromagnetic field. Remarkably, gauge fields often emerge due to a nontrivial topology of the Hilbert subspace to which the particle’s wavefunction is restricted^{2}. It is analogous to the parallel transport of a vector on the surface of the twodimensional unit sphere S^{2} as it picks up the surface’s curvature to change its direction while going around a closed loop. Here the vector and the S^{2} surface corresponds to the wavefunction and the restricted Hilbert subspace, respectively. If a particle with spin degrees of freedom moves in a spatially varying magnetic field and its spin adiabatically follows the field’s direction, the particle accumulates a quantummechanical phase known as the Berry phase^{3}. This phase arises from geometric properties of the Hilbert subspace composed of the spin and motional degrees of freedom, and it serves as an emergent gauge potential in the Hamiltonian description^{4}. Since the spin degrees of freedom of the particle are frozen by the adiabatic orientation of its magnetization to the local direction of the external magnetic field, the emergent gauge potential is abelian just like the vector potential of a conventional electromagnetic field. The potential generates forces from its spatial and temporal variations^{5}, and these forces act on a spincarrying particle in the same way as the Lorentz forces act on a charged particle. Examples include the anomalous Hall effect in magnetic materials due to noncoplanar spin configurations^{6,7} and the topological Hall effect, which has been observed in both chiral magnets^{8,9,10} and atomic Bose–Einstein condensates (BECs)^{11} with skyrmion spin textures.
On the other hand, in twodimensional material systems, almost every possible linear combination of the Rashba^{12} and Dresselhaus^{13} spin–orbit interactions can be represented by a spatially uniform nonabelian vector potential A=∑_{α=x,y,z} A^{α}σ_{α}, where σ_{α}’s are the Pauli matrices^{14}. Due to the noncommutativity [σ_{α}, σ_{β}]≠0, the nonabelian gauge potential generates an effective magnetic field proportional to σ_{z}. This magnetic field points in the opposite directions for spinup and spindown particles, thereby giving rise to the spin Hall effect that underlies a large number of spintronics devices^{15,16,17,18,19,20,21}. The spin Hall effect has also been extensively studied and recently observed in systems of ultracold atoms with lightinduced gauge potentials^{22,23,24}. The idea of a nonabelian gauge potential was first proposed by Wilczek and Zee who generalize the adiabatic theorem to the case in which there are a group of eigenstates that remain degenerate and well isolated from other levels in the course of time evolution^{25}. This analysis has led to applications in many areas such as molecular and condensed matter physics^{26,27}. In particular, the possibility of generating nonabelian magnetic monopoles was proposed and demonstrated in the rotational dynamics of diatomic molecules^{28,29,30,31}, nuclear quadrupole resonance^{32} and spinor BECs^{33}. The studies of nonabelian dynamics of ultracold atoms in laser fields have been conducted extensively^{34,35,36,37}.
So far the abelian and nonabelian gauge potentials have been investigated in different systems, and we have no way to control the nonabelianness of the gauge potentials in a single system. Moreover, there has been no observable indicator of the nonabelianness for partially nonabelian gauge potentials. In this work, we show that a spinor BoseFermi mixture can be used to serve this purpose. Such a mixture offers a platform for the study of the abelian–nonabelian crossover, where a change in the singularity of a topological defect such as Dirac’s magnetic monopole should play an important role. The mixture we consider consists of an optically trapped ultracold Fermi gas that overlaps with a spinor BEC, which forms a helical spin texture (see Fig. 1). A noncollinear spin texture gives rise to an emergent gauge potential that acts on fermions in the mixture due to their spindependent interaction with bosons. We can switch from the abelian to nonabelian regimes of the emergent potential by gradually liberating the spin degrees of freedom of fermions, or equivalently by weakening the spin adiabaticity in the sense of decreasing the ratio of the magnitude of the fermion–boson interaction to the Fermi energy. This can be implemented by varying the relative particlenumber densities of bosons and fermions in the mixture. The nonabelian gauge potential then generates a generalized nonabelian electric field, which causes a pure spin current of fermionic atoms to flow in the mixture (see discussions below equation (6)). The nonabelianness of the emergent gauge potential is characterized by an explicit dependence of the spin current density in the Fermi gas on the variable coupling constant. Unlike conventional solid materials, a pure spin current density can be measured in ultracold atomic gases by using the timeofflight absorption imaging in conjunction with the Stern–Gerlach experiment^{38}. The flows of particles of opposite spins in opposite directions would result in two spatially separated expanding clouds of upspin and downspin atoms as the trap is removed. Alternatively, the spin accumulations at the edges of the system as a result of the generated spin current can be observed by an in situ spintextureresolved measurement^{39}. Using the nonequilibrium Green’s function method, we calculate the spin current density in both strongcoupling and weakcoupling limits and obtain its analytic expression in terms of system’s fundamental parameters. Our result is valid over a large parameter region, and thus allows a quantitative comparison between theoretical predictions and experimental measurements. Spinor mixtures also provide us with a new tool to coherently and spontaneously generate a pure spin current, which is not a simple task in solid materials, without relying on the spin Hall effect. Such a spin current is expected to have potential applications in the new generation of ultracold atombased ‘atomtronic’ devices^{40,41,42,43,44}.
Results
System
A spinor BoseFermi mixture considered in this work is illustrated in Fig. 1. A helical spin texture is formed in a spinor BEC with a ferromagnetic interaction, such as a spin1 ^{87}Rb condensate, through application of a π/2 radiofrequency (RF) pulse to the zaxis polarized BEC to rotate the atomic magnetization to the xy plane before applying a transient magnetic field gradient dB_{z}/dz for a period of τ_{B} (ref. 45). The Larmor precession of atomic spins with a spacedependent frequency results in a helical spin texture F_{+}(r)≡F_{x}(r)+iF_{y}(r)=e^{iκ˙r} and F_{z}(r)=0, where F(r) represents the unit vector showing the direction of the magnetization of the condensate^{46}. The wavevector of the helical spin structure is given by κ=(g_{F}μ_{B}τ_{B}e_{z}/ℏ) (dB_{z}/dz), where g_{F} is the Lande gfactor for the hyperfine spinF manifold, μ_{B} is the Bohr magneton and e_{z} denotes the unit vector in the z direction.
Since the mixture of atomic gases is a dilute system at temperature much lower than the energy scale of the shortrange interactions between atoms, the boson–fermion interactions can be well approximated by contact interactions whose strengths are determined by the swave scattering lengths: , where is the effective mass for the relative motion of a boson with mass M_{b} and a fermion with mass M_{f}, and a_{F} and are the scattering length and the projection operator onto the total hyperfine spinF scattering channel, respectively. For the scattering of a spinS boson (S is an integer) and a spin1/2 fermion, F can take one of the two values . On the other hand, the projection operators can be linearly expressed in terms of the identity operator and the spin product operator as and ; thereby the interaction can be rewritten as
where and are the spinindependent and spindependent interaction strengths, respectively. In the secondquantization representation, the spindependent interaction between bosons and fermions is given by
where is the annihilation operator of the fermions (T denotes the transpose), σ is the vector of Pauli matrices and J=gSn_{b}/2 is the coupling constant with n_{b} being the density of condensate particles. Here we used the fact that the BEC has a macroscopic occupation of atoms in a singleparticle state so that their field operators can be approximated by a classical field with the magnetization vector given by F(r). In the following calculations, the spin texture is assumed to be time independent. The validity of this assumption and the roles of other interactions will be discussed in the Discussion section.
Emergent gauge potential
To find the emergent gauge potential, we go to the adiabatic reference frame in which the quantization axis of the spins of fermions is aligned parallel to the local spin texture of the BEC. Mathematically, it is carried out by means of a 2 × 2 unitary matrix U(r) such that U^{†}(r)[F(r)·σ]U(r)=σ_{z}. If the unit vector F(r) is represented by the polar angles as F(r)=(sin θ cosφ, sinθ sinφ, cosθ)^{T}, U(r) can be expressed in terms of the Pauli matrices as U(r)=m(r)·σ with . Note that U^{†}=U and U^{2}=1, implying the unitarity. The field operator of fermions in the adiabatic frame is related to its counterpart in the laboratory frame by . The Hamiltonian of the Fermi gas , where μ is the temperaturedependent chemical potential for a given particlenumber density and is the interaction given by equation (2), is then rewritten in terms of as^{47}
where each component A_{j} (j=x, y, z) of the generalized nonabelian gauge potential is a 2 × 2 matrix given by . Here and henceforth, the superscripts in greek characters represent the spin polarization, while the subscripts in roman characters denote the spatial direction. In terms of the polar angles θ and φ of F(r), the three spin polarization components of the emergent gauge potential are written as:
For the helical spin texture with θ(r)=π/2 and φ(r)=κ·r, we have A^{x}(r)=−κ cos(κ·r)/2, A^{y}(r)=−κ sin(κ·r)/2, and A^{z}(r) =κ/2. The last term on the righthand side of equation (3), which arises from the interaction (equation (2)) after the unitary transformation, is equivalent to a magnetic Zeeman energy in the adiabatic frame. It is the nonabelian feature of the gauge potential in combination with the Zeeman interaction that generates a generalized nonabelian electric field, which in turn causes a pure spin current to flow in the Fermi gas. This can be seen from the Heisenberg equation of motion for a spin1/2 particle under the Hamiltonian (equation (3)). Using the algebra of operators, we obtain , where the subscript H indicates operators in the Heisenberg picture, the [,] denotes the commutator of two operators, and A(r)=A^{x}(r)σ_{x}+A^{y}(r)σ_{y}+A^{z}(r)σ_{z} (see Supplementary Note 1 for more details). Due to the nonabelianness in the gauge potential A(r), the righthand side of the equation of motion, which depends only on r and σ, is nonzero, and it can be interpreted as a force exerted on the particle by a generalized nonabelian electric field. It can also be seen from equation (3) that the emergent gauge potential A(r) is coupled with the spin current density of fermions by , where the spin current density in the adiabatic frame is given by . As will be shown in the next section, it is the generalized nonabelian electric field generated by the nonabelian gauge potential that causes a pure spin current of fermionic atoms in the mixture, and the nonabelianness can be characterized by the dependence of the measurable spin current density on the variable coupling constant.
Spin current density
Using the unitary transformation U(r) introduced in the previous section, the spin current density operator for spin1/2 fermions in the laboratory frame is expressed in terms of the field operators in the adiabatic frame as . With U(r)=m(r)·σ, we obtain the j (current’s direction) and α (spin polarization) component of the spin current density as
Sinceand is the particlenumber density of fermions, the first term on the righthand side of equation (7) reduces to
This is the abelian contribution of the emergent gauge potential to the spin current density since it is equal to the value of at the limit of J→∞, at which the spin direction of fermions is always aligned parallel to the local magnetization of the spin texture and thus the spin degrees of freedom are frozen. It is also evident from equation (8) that the different spin polarization components A^{α} (α=x, y, z) of the gauge potential do not mix with each other as opposed to the nonabelian contribution. For the helical spin texture, we have . This type of spin current induced by a helical spin texture is related to the spin current mechanism of the electric polarization underlying multiferroic materials^{48,49,50,51}. There is also a similarity between this abelian component and the spin current in a fermionic system with the onedimensional spin–orbit interaction, that is, with equal weights of the Rashba and Dresselhaus components. In both cases, the spinlocking effect, which arises from either the adiabatic alignment of magnetization to the helical spin texture or the spin–orbit interaction, plays the key role in generating a pure spin current.
The remaining terms on the righthand side of equation (7) give the nonabelian contribution of the emergent gauge potential to the spin current density. Using the explicit expression of for the helical spin texture, we obtain the spin current density that is polarized in the z direction:
As shown below, and , which emerge purely from a nonabelian gauge potential, are nonvanishing for finite J, and the mixing of different spin polarization components due to the noncommutativity [σ_{x}, σ_{y}]=2iσ_{z} characterizes the nonabelian component.
To evaluate (α=x, y), we use the nonequilibrium Green’s function method in which the spin current density can be expressed in terms of the lesser Green’s function (see the Methods section). In the adiabatic limit, where either the spin texture varies smoothly in space or the coupling between fermions and the spin texture is strong, we can make a perturbative expansion of the Green’s function with respect to the emergent gauge potential A^{α}. This amounts to an expansion in powers of the dimensionless adiabatic parameter , where k_{F} and denote the Fermi wavevector and Fermi energy, respectively^{52}. Up to the linear order in A^{α}, the Fourier transform of the lesser Green’s function is given by
where the superscript < denotes the lesser component of the nonequilibrium Green’s function and the summation over α=x, y, z was taken. Here is the noninteracting Green’s function in the adiabatic frame which is diagonal in both the wavevector and the frequency, and with V being the volume of the system. Equation (10) is illustrated by the Feynman diagram in Fig. 2. A straightforward calculation of the nonabelian component of the spin current density yields (see Supplementary Note 2 for the derivation)
where is the difference in total kinetic energy density between fermions with spin up and those with spin down in the adiabatic frame due to the Zeeman energy (the last term in equation (3)). Here , and are the FermiDirac distributions with μ_{±}≡μ J. Therefore, the total spin current density is
In the strongcoupling limit and at low temperature , the difference in kinetic energy density between particles with opposite spins can be obtained analytically as . The spin current density then reduces to
In the opposite weakcoupling limit , where the perturbative expansion with respect to A^{α} does not converge if , we can make an alternative expansion of the spin current density in powers of in the laboratory frame. Using a similar nonequilibrium Green’s function approach, the total spin current density is given in terms of the noninteracting Green’s function g_{k,ω} up to the second order in J as
where F_{q}≡(1/V)∫d^{3}r e^{iq·r}F(r). The cross product of two magnetization vectors on the righthand side of equation (14) implies that the spin current emerges from a noncollinear spin texture. This factor arises from Tr{σ_{z}[F_{1}·σ][F_{2}·σ]}=2i[F_{1} × F_{2}]_{z}, where F·σ is the interaction between fermions and the spin texture (equation (2)). For the helical spin texture, we have . The lesser noninteracting Green’s function in the laboratory frame is given by , where . Substituting these in the righthand side of equation (14) and with a straightforward calculation, we obtain the zaxis spin polarization component at low temperature as (see Supplementary Note 3 for details)
The other spin polarization components j_{s}^{x} and j_{s}^{y} vanish. It should be noted, however, that the coupling constant J is also a function of temperature via the density of condensate particles , where is the critical temperature of Bose–Einstein condensation with n being the total particlenumber density of bosons. Here we use the model of free bosons, which is a good approximation in evaluating thermodynamic properties of a weakly interacting Bose gas^{53}. Therefore, we have
where . Using the same parameters as in ultracold atomic experiments (see the Discussion section below), the critical temperature is estimated to be . The temperature dependence of the spin current density at higher temperatures is evaluated numerically and the obtained result is shown in Fig. 3. The initial growth of with increasing temperature originates from the broadening of the FermiDirac distribution towards the states with high velocities due to thermal excitations. In contrast, at higher temperatures where the Fermi gas becomes nondegenerate, the thermal random motion of fermions and a decrease in the number of condensate particles suppress the directional flow of the spin current. The maximum value of the spin current density is attained at the temperature given by .
Abelian–nonabelian crossover
From the result obtained in the previous section (equation (12)), the ratio between the nonabelian and abelian components of the spin current density is given by , which is equal to for and T=0. This implies that the abelian–nonabelian crossover should occur at . The investigation of this crossover should be of importance because it relates to the question of how the singularity of a topological defect such as the Dirac magnetic monopole changes as it transforms into a nonsingular nonabelian monopole. In contrast, if the coupling constant is so weak that , where typically , the fermions are essentially decoupled from the spin texture in the BEC. The classification of the emergent gauge potential into the abelian (), nonabelian (), and decoupled regimes () is schematically shown in Fig. 4, where these different regimes can be identified from the magnitude of the total spin current density relative to its abelian component based on equations (12) and (15).
Discussion
We now discuss possible experimental situations of a spinor BoseFermi mixture in which the nonabelianness of the emergent gauge potential can be controlled, and, consequently, the abelian–nonabelian crossover can be investigated. As shown in Fig. 4, we can move from the abelian to nonabelian regimes by changing the ratio of the magnitude of the coupling constant J to the Fermi energy . It is given by
Therefore, the nonabelianness of the emergent gauge potential can be continuously changed by varying the particlenumber density of condensed bosons n_{b} against fermions n_{f}. Since the abelian component of the spin current density is a constant if n_{f} is held fixed, the J dependence of the measurable total spin current density (with n_{b} being varied) will characterize the nonabelian feature of the gauge potential. Alternatively, if n_{f} is varied to linearly change the coupling constant J, the deviation of the measured from a linear function of n_{f} provides a telltale signal of the nonabelian gauge potential.
To give a concrete example, we consider a mixture of the spin1 ^{87}Rb BEC and the ^{6}Li Fermi gas with spin1/2 (refs 54, 55, 56, 57). Other choices of atomic species for the spinor BoseFermi mixture are also possible^{58}. To give an estimate of the difference a_{3/2}−a_{1/2} in the swave scattering length between the two possible total hyperfine spinF channels (F=1/2 and 3/2), we have used the numerically calculated atomic potentials^{59} and the corresponding values of the scattering lengths^{60} for the electronic spinsinglet and spintriplet scattering channels, and calculated the Clebsch–Gordan coefficients for the transformation between the hyperfine and electronicspin bases (see the Supplementary Note 4 for details). Here we take a_{3/2}−a_{1/2}=50 a_{B}, where a_{B} is the Bohr radius. We take the particlenumber density of condensed bosons n_{b}=10^{14} cm^{−3} and that of fermions n_{f}=10^{11} cm^{−3}. Substituting these parameters in equation (17), we obtain . The ratio between the nonabelian and abelian components of the spin current density is found to be . The pitch of the helical spin texture can be adjusted by controlling the duration of the applied magnetic field gradient, and here we take κ/k_{F}=0.1. The magnitude of the spin transport velocity for spin1/2 particles is found to be about 9.4 × 10^{−4} m s^{−1}, implying that a particle with a specific spin polarization propagates through a 100μmlong atomic cloud in ∼0.1 s. In other words, onehalf of particles in the Fermi gas with spin up moves with that velocity in one direction and the other half of particles with spin down moves with the same velocity but in the opposite direction. This kind of spin transport can be measured in ultracold atomic systems using timeofflight absorption imaging in conjunction with the Stern–Gerlach experiment.
In this work, we have concentrated on the spindependent interaction between bosons and fermions. The effects of the other interactions in the system can be neglected because the spinindependent interaction does not affect the spin transport under consideration. The interaction between fermions, which is typically small in an atomic gas compared with the Fermi energy, should give a negligible effect on the spin current density. However, the helical spin texture in the BEC tends to decay towards a modulated spin structure. There are two mechanisms for this decay. The first is due to the reflection of the supercurrent at the edges of the finitesize BEC as shown by a numerical simulation^{61}. The second is due to the backaction of the Fermi gas on the spin texture in the condensate. The backaction can be understood in terms of the RKKY interaction, which is an indirect interaction between two spins mediated by the Fermi sea. Similar to the case of ferromagnetic metals where two localized spins can interact with each other via conducting electrons, the RKKY interaction between two magnetizations F_{1} at r_{1} and F_{2} at r_{2} due to a threedimensional Fermi gas has the form of
where r=r_{1}−r_{2} and f(x)≡[sin(2x)−2x cos(2x)]/x^{4} (refs 62, 63, 64). As shown in Fig. 5, the RKKY interaction is ferromagnetic at small distances r≤r_{c}=2.25 k_{F}^{−1} but its sign will oscillate between positive and negative in addition to the rapid damping of its magnitude at large distances. Both of the two mechanisms, however, can be suppressed by loading bosons to an optical lattice with the lattice constant much larger than k_{F}^{−1} and the lattice depth adjusted by the laser intensity so that each of the subcondensates on the lattice sites has a dimension of the same order or smaller than r_{c}. The bosonic part of the mixture then consists of an array of almost independent and spinhomogeneous subcondensates whose magnetization’s direction varies from one subcondensate to another to form a helical spin texture. This would suppress the supercurrent in the system, while the RKKY interaction between atoms in a single subcondensate has a ferromagnetic nature that helps stabilize the spin configuration. It should be noted that the magnetic dipole–dipole interaction of ^{87}Rb is so weak that it gives a negligible effect on the spin structure in a lattice.
To conclude, we have demonstrated that spinor BoseFermi mixtures offer an excellent playground for the study of the abelian–nonabelian crossover of the emergent gauge potential as the nonabelianness can be continuously varied. The change in the singularity of a topological defect such as Dirac’s magnetic monopole at this crossover deserves further investigation. We can move from the abelian to nonabelian regimes by varying the relative particlenumber densities of condensed bosons and fermions in the mixture. The nonabelian feature of the emergent gauge potential is characterized by the dependence of the measurable spin current density on the variable coupling strength. The result of this study also suggests a method to coherently and spontaneously generate a pure spin current without relying on the spin Hall effect and the spin–orbit interaction unlike in conventional solid materials. It is expected that the nonabelian emergent gauge potential will have potential applications in the new generation of ultracold atombased spintronics, that is, ‘atomtronics’ devices.
Methods
Nonequilibrium Green’s function method
The spin current density (α=x, y, z) in the adiabatic frame can be expressed in terms of the lesser Green’s function (m, n=↑, ↓) as
where the trace is taken over the spin indices and ∇_{r} indicates the gradient with respect to r. On the other hand, the lesser Green’s function is related to the Keldysh nonequilibrium Green’s function by taking the two arguments on the forward (τ∈C_{←}) and backward (τ′∈C_{→}) parts of the Keldysh contour C^{65}. Here denotes the pathordering operator on C. Similar to the ordinary Green’s function with the timeordering operator defined on the realtime axis, the Keldysh Green’s function satisfies the Dyson’s equation. Making a perturbative expansion of the Dyson’s equation with respect to the emergent gauge potential, we obtain the Fourier transform of the lesser Green’s function up to the linear order in A^{α} as given by equation (10). Here for a timeindependent system under consideration, the Green’s function depends only on the time difference t−t′; thereby, its Fourier transform depends only on a single frequency ω. Substituting equation (10) in equation (9) and further in equation (9) and after a straightforward calculation, we obtain the nonabelian contribution of the emergent gauge potential to spin current density as given by equation (11). The details of the derivation are presented in Supplementary Note 2.
Additional information
How to cite this article: Phuc, N. T. et al. Controlling and probing nonabelian emergent gauge potentials in spinor BoseFermi mixtures. Nat. Commun. 6:8135 doi: 10.1038/ncomms9135 (2015).
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Acknowledgements
N.T.P. thanks S. Inoyue and P. Naidon for useful discussions. This work was supported by KAKENHI Grant No. 26287088 from the Japan Society for the Promotion of Science, and a GrantinAid for Scientific Research on Innovation Areas ‘ Topological Quantum Phenomena’ (KAKENHI Grant No. 22103005), and the Photon Frontier Network Program from MEXT of Japan. Y.K. acknowledges the financial support from KAKENHI Grant No.22740265 and Inoue Foundation.
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Phuc, N., Tatara, G., Kawaguchi, Y. et al. Controlling and probing nonabelian emergent gauge potentials in spinor BoseFermi mixtures. Nat Commun 6, 8135 (2015). https://doi.org/10.1038/ncomms9135
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DOI: https://doi.org/10.1038/ncomms9135
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