Emergent Weyl excitations in systems of polar particles
Weyl fermions are massless chiral particles first predicted in 1929 and once thought to describe neutrinos. Although never observed as elementary particles, quasiparticles with Weyl dispersion have recently been experimentally discovered in solid-state systems causing a furore in the research community. Systems with Weyl excitations can display a plethora of fascinating phenomena and offer great potential for improved quantum technologies. Here we show that Weyl excitations generically exist in three-dimensional systems of dipolar particles with weakly broken time-reversal symmetry (for example, by a magnetic field). They emerge as a result of dipolar-interaction-induced transfer of angular momentum between the and internal particle levels. We also discuss momentum-resolved Ramsey spectroscopy methods for observing Weyl quasiparticles in cold alkaline-earth-atom systems. Our results provide a pathway for a feasible experimental realisation of Weyl quasiparticles and related phenomena in clean and controllable atomic systems.
Recent predictionsWan et al. (2011); Burkov and Balents (2011) and experimental observationsXu et al. (2015); Lv et al. (2015); Lu et al. (2015) of Weyl semimetals in solid state systems have instigated intensive studies of their properties, such as non-local electrodynamics and chiral anomaly Parameswaran et al. (2014), topologically protected Fermi arcs on the surfaces Wan et al. (2011); Xu et al. (2015); Lv et al. (2015), non-Anderson disorder-driven transitionsFradkin (1986); Goswami and Chakravarty (2011); Syzranov et al. (2015a, b); Sbierski et al. (2014); Moon and Kim (2014); Kobayashi et al. (2014) and unusual dependencies of transport properties on doping and temperatureSarma et al. (2015); Skinner (2014); Rodionov and Syzranov (2015). In parallel, enormous research efforts are now directed at finding Weyl excitations in new systems. A promising platform for exploring Weyl physics is tunable and fully controllable ultracold atomic gasesLan et al. (2011); Jiang (2012); Anderson et al. (2012); Dubček et al. (2015); Ganeshan and Sarma (2015); Liu et al. (2015). However, henceforth proposed cold-atom realisations of Weyl quasiparticles have focussed mostly on non-interacting systems, and all have required implementations of externally imposed spin-orbit coupling through laser assisted tunneling schemesLan et al. (2011); Jiang (2012); Anderson et al. (2012); Dubček et al. (2015); Ganeshan and Sarma (2015), other optical means Xia-Ji et al. (2015); Gong et al. (2011); Seo et al. (2012), or external rotating fieldsLiu et al. (2015).
In this paper we demonstrate that excitations with Weyl dispersion generically emerge in three-dimensional (3D) arrays of dipolar particles in the presence of a weak magnetic field, as a result of the dipole-interactions-induced transitions between their internal angular-momentum and states. These excitations exhibit the same single-particle physics as Weyl fermions Weyl (1929) but, similarly to other non-fermionic Weyl excitations Lu et al. (2015), their many-particle properties are expected to be different, opening alternative research directions, new functionalities and applications beyond those accessible with solid-state systems Xu et al. (2015); Lv et al. (2015).
We also show that experimentally such excitations can be observed, for instance, in trapped alkaline-earth atoms (AEAs) in a 3D optical lattice with lattice spacings smaller than the wavelength of the electronic transition. The simple and unique internal structure of these atoms has already lead to record levels of precision and accuracy in atomic clocksBloom et al. (2014). Taking advantage of the well developed spectroscopic tools to interrogate and manipulate AEAs, we propose to probe the Weyl quasiparticle dispersion and non-trivial chirality by means of momentum-resolved Ramsey spectroscopy. Our proposal opens a path for a feasible experimental realisation of Weyl quasiparticles in clean and controllable atomic systems. Moreover, it lays the groundwork for the yet unexplored regime of topologically protected sensing, owing to the topological robustness of Weyl quasiparticles that could be used to push the stability and accuracy of optical-lattice AEA-based clocks.
Phenomenological argument. We assume that the system has long-lived excitations (quasiparticles) with (integer) angular momentum . Due to the translational invariance, the (quasi)momentum is a good quantum number. In the long-wave limit the effective quasiparticle Hamiltonian is insensitive to the details of the potential of the periodic lattice that the particles may be placed in. To preserve rotation and inversion symmetries in the absence of magnetic field the Hamiltonian has to be an even function of and a function of and . In the presence of a sufficiently weak uniform magnetic field, , that creates a perturbation independent of in the limit , the most generic form of the quasiparticle Hamiltonian is given by
where is an arbitrary function of three arguments.
The small quasimomentum can be measured from any high-symmetry point in the Brillouin zone characterised by an isotropic dispersion of non-interacting particles in the limit .
For the particular case of , the Hamiltonian (1) has nodes at momenta , such that , corresponding to two intersecting branches with angular momentum projections and or on the direction of magnetic field. We note that such nodes always exist for weak magnetic fields and Hamiltonians that are regular as a function of .
The excitation Hamiltonian near a node is obtained by expanding the function in small momentum . For a 3D system, it has Weyl dispersion of the form (see Methods):
with Pauli matrices acting in the space of the respective two angular-momentum projections.
Model. In what follows we confirm the above phenomenological argument by microscopic calculations for a 3D system of dipolar particles described by the Hamiltonian
where is the dipole moment operator of the -th particle, and
is the single-particle Hamiltonian that includes the particle kinetic energy (hereinafter ), the periodic potential of the lattice that the system may be placed in, the energy of internal levels with being the angular momentum of the -th particle, and the interaction with magnetic field (measured in units of the gyromagnetic ratio) that splits the levels.
The most generic form of the dipole-dipole interaction, which accounts for retardation effects, is given by Gross and Haroche (1982)(see also Methods)
where ; and for , with and being the -th-order spherical Bessel functions of the second and first kind respectively and – the wavevector of the to transition. The terms proportional to describe elastic interactions between dipoles a distance apart, while the terms with account for the inelastic collective photon emission (radiation). is the natural linewidth of the transition and is its dipole moment. If the dipoles are much closer to one another than the wavelength of the dipole transition, , retardation effects can be ignored, and one recovers the more familiar form of the dipolar interactions, , , common for NMR solid-state systemsChildress (2015), polar moleculesYan et al. (2013) and Rydberg atomsWeber et al. (2015); Robicheaux et al. (2004).
We note that the above phenomenological derivation of the dispersion of Weyl-type quasiparticles carries over straightforwardly to other dimensions. For example, a 2D system of dipolar particles with an in-plane magnetic field hosts 2D Dirac excitations with the dispersion of monolayer graphene Novoselov et al. (2004). We emphasise that such 2D excitations are distinct from the 2D “chiron” excitationsSyzranov et al. (2014) that exist in a perpendicular magnetic field and resemble electrons in bilayer graphene.
Atoms in a deep lattice. While the above phenomenological argument demonstrates the existence of Weyl quasiparticles in a generic 3D system of dipolar particles in magnetic field, below we focus on the experimentally important case of particles pinned in a deep unit-filled cubic lattice (Fig. 1a) with small lattice spacing ; .
We assume that all particles are initially prepared in the state and that the energy of internal levels significantly exceeds the interaction strength (usually in dipolar gasesYan et al. (2013); Olmos et al. (2013) ), leading to the conservation of the number of sites excited to the state to a good accuracy (cf. Methods).
If an excitation with the angular momentum is created on site , the dipole-dipole interaction can transfer it to another site , possibly changing the projection of the angular momentum on the direction of the magnetic field; . The quasiparticles in the system are thus hard-core bosons corresponding to the angular momentum-degrees of freedom that hop from site to site as described by the effective Hamiltonian (see Methods for a detailed derivation)
Due to the translational invariance, the single-excitation Hamiltonian can be diagonalised in the basis of momentum states , with the results shown in Figs. 2a,c. In accordance with the above general phenomenological argument, for the dispersion has Weyl nodes (six in the first Brillouin zone, Figs. 2a,c).
To demonstrate the chiral nature of Weyl quasiparticles we show in Figs. 2d,e,f the pseudospins (with the Pauli matrices acting in the space of the and angular-momentum projections) for the eigenstates with momenta in the horizontal (), tilted , and vertical () planes (Fig. 2a) that contain a Weyl node. Excitations in these planes are equivalent to quasiparticles in graphene, the 2D counterpart of a Weyl semimetal, and are characterised by the non-trivial Berry phase . Figs. 2d,e,f demonstrate that the pseudospins of these states are linked to their momenta , measured from the Weyl node, in agreement with the effective Hamiltonian (2).
Effects of quenched disorder and dissipation. In general, quasiparticles in interacting systems have finite lifetimes due to elastic and inelastic scattering processes. Indeed, deep optical lattices under consideration are ususlly not completely filled by particles and thus inherently disordered due to the randomness of the particle disptribution. Also, spontaneous and dipolar collective emission from the internal levels to the ground state can lead to the decay of the excitations.
To analyse the effects of dissipation in a unit-filled lattice we compute numerically the quasiparticle dispersion for retarded dipolar interactions, Eq. (5), with parameters of the to transition of the electronic levels of bosonic atoms trapped in a magic optical lattice with nm considered in Ref. Olmos et al., 2013. The wavelength and the dipole moment for this transition are m and D, leading to the linewidth s and the dissipation parameter . Albeit quasiparticle damping in this regime is rather strong, it is significantly suppressed (by more than three orders of magnitude) near the Weyl nodes, as our simulations show, Fig. 3a,b. Our results indicate that the quasiparticle scattering in such a system would be dominated by quenched disorder rather than by collective radiative decay or spontaneous emission.
To account for the effects of disorder we evaluate numerically the quasiparticle dispersion for a lattice filling of . This filling fraction could be achieved by preparing a cold bosonic Mott insulator using moderate atom numbers that allow one to suppress doubly occupied states at the trap centre. Mott insulators have already been realised with bosonic AEAs in the ground state Sugawa et al. (2013); Stellmer et al. (2014). These atoms can be excited to the desired state by laser pulsesAkatsuka et al. (2008).
As our simulations demonstrate, the characteristic energy scales of Weyl excitations significantly exceed the elastic scattering rate, demonstrating that the excitations could be conveniently observed in current experiments.
Experimental observation. For probing the Weyl character of the excitations we propose a Ramsey protocol illustrated in Fig. 4a. After preparing a Mott insulator of particles in the state, a pulse of interfering Raman beams is used to create excitations in the angular-momentum state with translational momentum . Here we consider the case when is set to be close to the Weyl point with intersecting and branches. For the proposed electronic levels in Sr, two intermediate states could be used to create the Raman pulses, imparting a net momentum to the atoms proportional to (see Fig. 4a). A possible excitation level scheme consists in using and as the intermediate and levels respectively. After a waiting time , another pulse is applied to measure the fraction of particles in the angular-momentum state. Because of the interference of the two branches, this fraction oscillates with the frequency , where the energy splitting between the two branches is linear in and vanishes near the Weyl node.
Another signature of the Weyl node is the strong dependency of the amplitude of such oscillations on momentum near the node, as the amplitude is determined by the projection of the Bloch vector on (the magnetic field direction).
In Fig. 4(c-f) we show the fraction of particles in the state as a function of time at the end of the above described Ramsey protocol, for the six different quasi-momenta in the plane near the Weyl point indicated in Fig. 4(b). Panel (c) shows the dynamics for an ideal unit filled lattice in the dissipationless limit . Panel (d) shows the dynamics in the presence of dissipation for the experimentally relevant scenario discussed above. The population dynamics in disordered systems is shown in panels (e) and (f) for and filled lattices respectively. Quasiparticles scattering on empty sites in a disordered system leads to the decay of the oscillations.
We demonstrated that Weyl quasiparticles generically emerge in 3D systems of polar particles in magnetic field. This opens intriguing prospects of observing chiral anomaly, non-local electrodynamics, non-Anderson disorder-driven transitions, and other fascinating phenomena in the realm of fully controllable atomic systems. We showed that observing Weyl excitations is currently possible in arrays of AEA in 3D lattices, in particular, using the levels of bosonic atoms. Other experimentally convenient schemes, that deserve further exploration, include using metastable levels of or atoms that can be trapped in magic lattices with spacings smaller than the wavelengthSafronova et al. (2015) or arrays of polar molecules with the rotational levels dressed to avoid the splitting of levels in the presence of hyperfine interactionsYan et al. (2013). The long lifetimes and the topological character of Weyl excitations in interacting dipolar systems also open new possibilities for implementing optical-lattice clocks with sensing capabilities beyond those of non-interacting systems.
Dispersion near Weyl nodes. In this work, we define the quasiparticle dispersion as the poles of the retarded Green’s function averaged with respect to quenched disorder.
While long-wave quasiparticles () are insensitive to the details of the lattice potential, their effective Hamiltonian preserves rotation and inversion symmetries, and in the absence of magnetic field– time-reversal symmetry, with the generic form of the Hamiltonian given by Eq. (1) and with the vector parallel to the magnetic field.
We assume the existence of excitations with momentum and focus on the respective manifold of states in what follows. The dispersion of such excitations has three branches for each momentum .
For momenta parallel to the respective excitations have momentum projections and on the direction . The branch with intersects the branch with at momenta such that
where we used that for the states under consideration.
The quasiparticle dispersion near the nodes can be found by expanding the Hamiltonian in small momenta . Momentum deviation from a node along the axis leads to the splitting between the intersecting branches. Using that , with being the Pauli matrices in the space of momentum projections and , we obtain the quasiparticle Hamiltonian (2) with
where the upper (lower) sign in Eq. (10) applies to the intersection of the branch with (), and and are the derivatives of the function with respect to the first and the second argument.
Generic Hamiltonian of retarded dipole-dipole interactions. The dynamics of internal transitions in a system of particles is described by the Hamiltonian
where the operator excites the -th atom from the ground state to one of the Cartesian states of the manifold with energy ; is the dipole moment of such a transition; and are the creation and annihilation operators of a photon with momentum , frequency , and polarisation ; is the operator of electric field, and is the volume of the system.
where is the Hamiltonian of the internal states of the particles, and the effective interaction Hamiltonian is given by
with , and being the -th-order spherical Bessel function of the second kind, is the wavevector of the to transition, , , and .
The operator in Eq. (13) accounts for dissipation and is given by
where , , , with being the -th-order spherical Bessel function of the first kind, and is the so-called recycling operatorGross and Haroche (1982) that does not affect the dynamics of a single excitation and is thus omitted in the present paper. Combining the interaction and dissipation terms we obtain the effective (non-Hermitian) Hamiltonian (5) of the dipole-dipole interactions.
Excitation dispersion in a deep lattice. As particles cannot move from site to site in a deep optical lattice, the quasiparticles are represented by the angular-momentum degrees of freedom that propagate through the system. Assuming there is one particle per site and introducing bosonic operators and for creating and annihilating the particle state on site with angular momentum and projection and the operators and for creating and annihilating the momentum state on site , the system Hamiltonian can be rewritten as
The first term in the Hamiltonian (16) is responsible for moving angular-momentum excitations from site to site; the angular-momentum state can be transferred by the dipole-dipole interactions from site to another state on site that initially was in the state. The terms in the second sum in Eq. (16) change pairs of sites and from the to angular-momentum states or vice versa. The term is the Zeeman energy. The term accounts for the internal rotation (internal levels) of the particles. The terms in Eq. (16) enforce the hard-core constraints for the bosons created by the operators and , taking into account that there is one particle on each site.
In this paper we consider excitations on top of the ground state with all sites (particles) in the state. Exciting the internal degree of freedom of a particle on a site costs the rotation energy that significantly exceeds all the other energy scales, except , including the matrix elements of hopping of such angular-momentum degrees of freedom between sites (for instance, for dipolar molecules and alkaline-earth atoms ). As a result, the number of sites excited to the states is conserved to a good approximation, and the second sum in Eq. (16), that creates or annihilates pairs of excitations, can be neglected when considering the angular-momentum dynamics.
Details of disorder averaging. Realistic systems of particles pinned in deep optical lattices are inherently disordered due to the randomness of the spatial distribution of the particles. Each lattice site hosts either a particle with probability or a vacancy with probability .
For a small concentration of vacancies, excitations in the system are delocalised and their dispersion is close to that in the disorder-free system but acquires a small finite imaginary part due to the scattering on the vacancies.
In order to numerically obtain the quasiparticle spectra in such a disordered system we diagonalise the Hamiltonian , where is the excitation Hamiltonian in the clean case and the operator models vacancies as sites with infinite potential . We compute the retarded Green’s function
for multiple disorder realisations, where and are the eigenfunctions and eigenenergies for a particular disorder realisation, and label projections of the angular momentum , and is a small positive number introduced to ensure that the disorder-averaged Green’s function is a smooth function of its arguments for a given number of disorder realisations. At the same time, has to be chosen sufficiently small to not affect the results for the quasiparticle dispersion. The energy has to be chosen close to the energies of the quasiparticles of interest.
Disorder averaging restores translational invariance, yielding an averaged Green’s function that depends only on the coordinate difference . Computing the Fourier transform of the function with respect to and diagonalising it in the angular-momentum space gives , where labels the dispersion branch for a given , is the quasiparticle dispersion and – the scattering rate.
In this paper we perform averaging over disorder realisations on a cubic lattice with periodic boundary conditions for the filling fraction , close to that in the recent experimentsKöhl et al. (2005); Schneider et al. (2008); Greif et al. (2013). The results for the quasiparticle dispersion and scattering rates are shown in Fig. 3c.
The datasets generated in the current study are available from the corresponding author on reasonable request.
We thank M. Hermele and J. Ye for useful discussions and R. Nandkishore for feedback on the manuscript. This work was supported by the NSF (PIF-1211914 and PFC-1125844), AFOSR, AFOSR-MURI, NIST and ARO. MLW thanks the NRC postdoctoral fellowship programme for support. SVS has been also partially supported by the Alexander von Humboldt Foundation through the Feodor Lynen Research Fellowship.
All authors contributed to all aspects of this work.
Competing financial interests: The authors declare no competing financial interests.
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