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Universal features of canonical phonon angular momentum without time-reversal symmetry
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Abstract
It is known that phonons have angular momentum, and when the time-reversal symmetry (TRS) is broken, the total phonon angular momentum in the whole system becomes nonzero. In this paper, we propose that as an angular momentum of phonons for a crystal without TRS, we need to consider the canonical angular momentum, as opposed to the kinetic angular momentum in previous works. Next, we show that the angular momentum of phonons without TRS exhibits universal behaviors near the $\Gamma$ point. We focus on in-plane oscillations in two-dimensional crystals as an example. By breaking the TRS, one of the acoustic phonon branches at the $\Gamma$ point acquires a gap. We show that the angular momentum of its acoustic phonon with a gap has a peak with the height $\pm \hbar$ regardless of the details of the system. From this, we find that this peak height changes discontinuously by changing the sign of the TRS-breaking parameter.
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In condensed matter systems it is necessary to distinguish between the momentum of the constituents of the system and the pseudomomentum of quasiparticles. The same distinction is also valid for angular momentum and pseudoangular momentum. Based on Noether's theorem, we demonstrate that the recently discussed orbital angular momenta of phonons and magnons are pseudoangular momenta. This conceptual difference is important for a proper understanding of the transfer of angular momentum in condensed matter systems, especially in spintronics applications.
We theoretically investigate pumping of phonons by the dynamics of a magnetic film into a nonmagnetic contact. The enhanced damping due to the loss of energy and angular momentum shows interference patterns as a function of the resonance frequency and magnetic film thickness that cannot be described by viscous (“Gilbert”) damping. The phonon pumping depends on the magnetization direction as well as geometrical and material parameters and is observable, e.g., in thin films of yttrium iron garnet on a thick dielectric substrate.
Recent advances in the emerging field of magnon spintronics have stimulated renewed interest in phenomena involving the interaction between spin waves, the collective excitations of spins in magnetic materials that quantize as magnons, and the elastic waves that arise from excitations in the crystal lattice, which quantize as phonons. In magnetic insulators, owing to the magnetostrictive properties of materials, spin waves can become strongly coupled to elastic waves, forming magnetoelastic waves—a hybridized magnon–phonon excitation. While several aspects of this interaction have been subject to recent scrutiny, it remains unclear whether or not phonons can carry spin. Here we report experiments on a film of the ferrimagnetic insulator yttrium iron garnet under a non-uniform magnetic field demonstrating the conversion of coherent magnons generated by a microwave field into phonons that have spin. While it is well established that photons in circularly polarized light carry a spin, the spin of phonons has had little attention in the literature. By means of wavevector-resolved Brillouin light-scattering measurements, we show that the magnon–phonon conversion occurs with constant energy and varying linear momentum, and that the light scattered by the phonons is circularly polarized, thus demonstrating that the phonons have spin.
We show that topological order and vibrational edge modes can exist in a classical mechanical system consisting of a two-dimensional honeycomb lattice of masses and springs. The band structure shows the existence of Dirac cones and unconventional edge states that are similar to the vibrational modes in graphene. Interestingly, as the system is placed on a constantly rotational coordinate system, the Coriolis force resulting from the non-inertial reference frame introduces time-reversal symmetry breaking and leads to topologically nontrivial band gaps. The nontrivial topological orders are further verified by the calculation of Chern numbers for corresponding bands.
constantly rotational coordinate system, the Coriolis force resulted from the
non-inertial reference frame provides a possibility to break the time-reversal
symmetry. Thus, caused from topologically non-trivial band gaps, phononic edge
states are present between bands, which are verified by the calculation of
Chern numbers for corresponding bands.
Mechanical graphene, which is a spring-mass model with the honeycomb
structure, is investigated. The vibration spectrum is dramatically changed by
controlling only one parameter, spring tension at equilibrium. In the spectrum,
there always exist Dirac cones at K- and K'-points. As the tension is modified,
extra Dirac cones are created and annihilated in pairs. When the time reversal
symmetry is broken by uniform rotation of the system, creation and annihilation
of the Dirac cones result in a jump of the appropriately defined Chern number.
Then, a flip of the propagation direction of the chiral edge modes takes place,
which gives an experimental way to detect the topological transition. This is a
bulk-edge correspondence of the classical system. We also demonstrate the other
important concept, symmetry protection of the topological states, is at work in
the classical system. For the time reversal invariant case, the topological
edge modes exist for the fixed boundary condition but not for the free boundary
condition. This contrast originates from the symmetry breaking at the free
boundary.
In monolayer hexagonal lattices, two inequivalent valleys appear in the
Brillouin zone. With inversion symmetry breaking, we find chiral phonons with
valley contrasting circular polarization and ionic magnetic moment. At valley
centers, there is a three-fold rotational symmetry endowing phonons with a
quantized pseudo angular momentum, which includes spin and orbital parts. From
conservation of the pseudo angular momentum, crystal momentum and energy,
selection rules in intervalley scattering of electrons by phonons are obtained.
The chiral valley phonons are verified and the selection rules are predicted in
monolayer Molybdenum disulfide. Due to valley contrasting phonon Berry
curvature, one can also detect a valley phonon Hall effect. The
valley-contrasting chiral phonon, together with phonon circular polarization,
ionic magnetic moment, phonon pseudo angular momentum, valley phonon Hall
effect, will form the basis for valley-based electronics and phononics
applications in the future.
We study angular momentum of phonons in a magnetic crystal. In the presence
of a spinphonon interaction, we obtain a nonzero angular momentum of phonons,
which is an odd function of magnetization. At zero temperature, phonon has a
zero-point angular momentum besides a zero-point energy. With increasing
temperature, the total phonon angular momentum diminishes and approaches to
zero in the classical limit. The nonzero phonon angular momentum can have a
significant impact on the Einstein-de Haas effect. To obtain the change of
angular momentum of electrons, the change of phonon angular momentum needs to
be subtracted from the opposite change of lattice angular momentum.
Furthermore, the finding of phonon angular momentum gives a potential method to
study the spin-phonon interaction. Possible experiments on phonon angular
momentum are also discussed.
Using the effective Lagrangian approach, we clarify general issues about
Nambu-Goldstone bosons without Lorentz invariance. We show how to count their
number and study their dispersion relations. Their number is less than the
number of broken generators when some of them form canonically conjugate pairs.
The pairing occurs when the generators have a nonzero expectation value of
their commutator. For non-semi-simple algebras, central extensions are
possible. The underlying geometry of the coset space in general is partially
symplectic.
We establish the general phonon dynamics of magnetic solids by incorporating
the Mead-Truhlar correction in the Born-Oppenheimer approximation. The
effective magnetic-field acting on the phonons naturally emerges, giving rise
to the phonon Hall effect. A general formula of the intrinsic phonon Hall
conductivity is obtained by using the corrected Kubo formula with the energy
magnetization contribution incorporated properly. The resulting phonon Hall
conductivity is fully determined by the phonon Berry curvature and the
dispersions. Based on the formula, the topological phonon system could be
rigorously defined. In the low temperature regime, we predict that the phonon
Hall conductivity is proportional to $T^{3}$ for the ordinary phonon systems,
while that for the topological phonon systems has the linear $T$ dependence
with the quantized temperature coefficient.
We provide a topological understanding of the phonon Hall effect in dielectrics with Raman spin-phonon coupling. A general expression for phonon Hall conductivity is obtained in terms of the Berry curvature of band structures. We find a nonmonotonic behavior of phonon Hall conductivity as a function of the magnetic field. Moreover, we observe a phase transition in the phonon Hall effect, which corresponds to the sudden change of band topology, characterized by the altering of integer Chern numbers. This can be explained by touching and splitting of phonon bands.
In ionic materials, circularly polarized phonons carry orbital magnetic moments that arise from circular motions of the ions, and which interact with other magnetic moments or fields. Here, we calculate the orbital magnetic moments of phonons in 35 different materials using density functional theory, and we identify the factors that lead to, and materials that show, large responses. We compute the resulting macroscopic orbital magnetic moments that can be induced by the excitation of coherent phonons using mid-infrared laser pulses, and we evaluate the magnitudes of the phonon Zeeman effect in a strong magnetic field. Finally, we apply our formalism to chiral phonons, in which the motions of the ions are intrinsically circular. The zoology presented here may serve as a guide to identifying materials for phonon and spin-phonon driven phenomena.
We investigated the magnetoterahertz response of the Dirac semimetal Cd3As2 and observed a particularly low frequency optical phonon, as well as a very prominent and field sensitive cyclotron resonance. As the cyclotron frequency is tuned with field to pass through the phonon, the phonon become circularly polarized as shown by a notable splitting in their response to right- and left-hand polarized light. This splitting can be expressed as an effective phonon magnetic moment that is approximately 2.7 times the Bohr magneton, which is almost four orders of magnitude larger than ab initio calculations predict for phonon magnetic moments in nonmagnetic insulators. This exceedingly large value is due to the coupling of the phonons to the cyclotron motion and is controlled directly by the electron-phonon coupling constant. This field tunable circular-polarization selective coupling provides new functionality for nonlinear optics to create light-induced topological phases in Dirac semimetals.
In crystals with time-reversal symmetry but without inversion symmetry, the phonon angular momentum can be generated by the temperature gradient, and it is called the phonon thermal Edelstein effect. On the other hand, when both symmetries are broken and their product is conserved, the phonon angular momentum for any phonon modes at any wave vectors vanishes and the phonon thermal Edelstein effect does not occur. In this paper, we propose another mechanism of generation of the phonon angular momentum. We show that in such crystals the electric field generates the phonon angular momentum via the lattice distortion due to the electric field. This effect is in the same symmetry class with the magnetoelectric effect, and we call this effect the phonon rotoelectric effect. We discuss the temperature dependence of the phonon angular momentum generated by the temperature gradient and by the electric field in the high- and low-temperature limits.
A mechanism for the phonon Hall effect (PHE) in nonmagnetic insulators under an external magnetic field is theoretically studied. PHE is known in (para)magnetic compounds, where the magnetic moments and spin-orbit interaction play an essential role. In sharp contrast, we here discuss that the PHE also occurs in nonmagnetic band insulators subject to the magnetic field. We find that a correction to the Born-Oppenheimer approximation gives rise to a Raman-type interaction between the magnetic field and the phonons; this interaction gives rise to the Berry curvature of a phonon band. This Berry curvature results in the finite thermal Hall conductivity κH in nonmagnetic band insulators. The value of κH is calculated for square and honeycomb lattices. The order of the magnitude estimation for κH is given for Si at room temperature.
Phonon modes in crystals can have angular momenta in general. It nevertheless cancels in equilibrium when the time-reversal symmetry is preserved. In this Letter, we show that when a temperature gradient is applied and heat current flows in the crystal, the phonon distribution becomes off equilibrium, and a finite angular momentum is generated by the heat current. This mechanism is analogous to the Edelstein effect in electronic systems. This effect requires crystals with sufficiently low crystallographic symmetries, such as polar or chiral crystal structures. Because of the positive charges of the nuclei, this phonon angular momentum induces magnetization. In addition, when the crystal can freely rotate, this generated phonon angular momentum is converted to a rigid-body rotation of the crystal, due to the conservation of the total angular momentum. Furthermore, in metallic crystals, the phonon angular momentum will be partially converted into spin angular momentum of electrons.
We reexamine the relaxation process of a single spin embedded in an elastic medium, a problem studied recently by Garanin and Chudnovsky (GC) [Phys. Rev. B 92, 024421 (2015)] from the viewpoint of angular-momentum transfer. Using Noether's theorem, we identify two distinct angular momenta of the medium, one Newtonian discussed by GC and the other field-theoretical, both of which consist of an orbital part and a spin part. For both angular momenta, we found that the orbital part is as essential as the spin part in the relaxation process. In particular, the angular-momentum transfer from the (real) spin to the Newtonian orbital part may be considered as an incipient rotation that leads to the Einstein–de Haas effect.
A distinct thermal Hall signal is observed in a quantum spin liquid candidate Ba$_3$CuSb$_2$O$_9$. The transverse thermal conduction shows a power-law temperature dependence below 40 K where a spin gap opens. We suggest that, through the very low longitudinal thermal conductivity and the thermal Hall signals, a phonon Hall effect is induced by strong phonon scatterings by orphan Cu$^{2+}$ spins formed in random domains of Cu$^{2+}$-Sb$^{5+}$ dumbbells.
Generalizing the concept of topology from electrons to phonons could bring in an intriguing emerging field of "topological phononics". For this purpose we propose a Schr\"odinger-like equation of phonons where topology-related quantities, time reversal symmetry (TRS) and its breaking can be naturally introduced. A Haldane model of phonons for a two-dimensional honeycomb lattice is then developed to describe the interplay of symmetry and quantum (anomalous) Hall-like phonon states. The nontrivial topological phase supports one-way gapless edge states within the bulk gap, which can conduct phonons without dissipation. Moreover, breaking inversion symmetry and TRS simultaneously is suggested to open a route for valley phononics and phonon diode. The findings could help design unprecedented new phononic devices.
Significance
A mechanical metamaterial is an engineered material that is characterized by properties that go beyond the properties of its microscopic building blocks. Specifically, in topological metamaterials, one makes use of surface or boundary modes that are stable against imperfections or environmental influences and hence constitute reliable building blocks for various applications. In our work, we provide a classification scheme of possible topological metamaterials and an extensive number of examples illustrating this scheme. Our classification can serve as an important blueprint for many future applications that target such stable boundary modes for engineering purposes.
The ferrimagnetic insulator yttrium iron garnet (YIG) is an important material in the field of magnon spintronics, mainly because of its low magnetic losses. YIG also has very low acoustic losses, and for this reason the conversion of a state of magnetic excitation (magnons) into a state of lattice vibration (phonons), or vice versa, broadens its possible applications in spintronics. Since the magnetic parameters can be varied by some external action, the magnon-phonon interconversion can be tuned to perform a desired function. We present a quantum theory of the interaction between magnons and phonons in a ferromagnetic material subject to a dynamic variation of the applied magnetic field. It is shown that when the field gradient at the magnetoelastic crossover region is much smaller than a critical value, an initial elastic excitation can be completely converted into a magnetic excitation, or vice versa. This occurs with conservation of linear momentum and spin angular momentum, implying that phonons created by the conversion of magnons have spin angular momentum and carry spin current. It is shown further that if the system is initially in a quantum coherent state, its coherence properties are maintained regardless of the time dependence of the field.
Quantum theory of spin relaxation in the elastic environment is revised with
account of the concept of a phonon spin recently introduced by Zhang and Niu
(PRL 2014). Similar to the case of the electromagnetic field, the division of
the angular momentum associated with elastic deformations into the orbital part
and the part due to phonon spins proves to be useful for the analysis of the
balance of the angular momentum. Such analysis sheds important light on
microscopic processes leading to the Einstein - de Haas effect.
A group-theoretic study is made of the degeneracies of the normal modes of vibration of a crystal and of the manner in which the polarization vectors describing these modes transform under the operations of the space group of the crystal. To describe the effects of the spatial symmetry operations a set of 3r-dimensional matrices is constructed, where r is the number of atoms in a primitive unit cell of the crystal, each of which commutes with the Fourier-transformed dynamical matrix for each value of the wave vector labeling the modes. These matrices are shown to provide a multiplier representation of the point group of the wave vector. The reduction of this representation yields the degeneracies (due to spatial symmetry) and transformation properties of the polarization vectors corresponding to a given wave vector, while the forms of the eigenvectors are obtained by projection operator techniques. For appropriate wave vectors, the consequences of time-reversal symmetry on the degeneracies and polarization vectors are investigated by introducing an anti-unitary matrix operator which commutes with the Fourier-transformed dynamical matrix. A criterion for the existence of extra degeneracies due to time-reversal symmetry is presented. The symmetries of lattice vibrations and selection rules for two-phonon absorption processes corresponding to several values of k in the first Brillouin zone of diamond are determined to illustrate the methods developed in this paper.
The phonon Hall effect in the paramagnetic dielectric garnet Tb3Ga5O12 has been investigated. It has been found that the coefficient of the phonon Hall effect is positive and is equal to (3.5
± 2) × 10−5T−1 in a magnetic field of 3 T at a temperature of 5.13 K. The results are experimental evidence of the phonon Hall effect in
the paramagnetic dielectric found by C. Strohm, G. L. J. A. Rikken, and P. Wyder, Phys. Rev. Lett. 95, 155901 (2005).
The Hamiltonian of an ionic crystal lattice in the harmonic approximation and in the presence of a static magnetic field is diagonalized in terms of creation and annihilation operators. Phonon energies and polarizations are determined in this theory by means of the solutions of an eigenvalue problem of the first kind with twice the dimension of the corresponding problem without magnetic field. The energy eigenvalues of the Hamiltonian are linear combinations of the phonon energies h{combining short stroke overlay}ωj(k) with integers as coefficients, just as in the case without a magnetic field. For crystals in which every ion is at a center of inversion symmetry the phonon spectrum will conserve its inversion symmetry in k-space. For crystals without inversion symmetry this will no longer be true. In every case the polarizations of the phonons will be elliptical.
Two exactly solvable models are presented to describe the dynamic coupling of magnetic transitions within Kramer's degenerate multiplets and lattice vibrations. The magnetic susceptibilities are calculated. It is shown that the spin-boson coupling can reduce the single ion Curic constant at low temperature. For special symmetries the Curie constant may vanish, yielding a temperature independent susceptibility at lowT.
A microscopic model is presented to show explicitly how dynamical charge and lattice fluctuations (at zero temperature the zero point motion) will-through spin orbit coupling-induce spin-flip terms off-electrons in intermediate valent and heavy fermion systems. A simple model Hamiltonian coupling charge, lattice, and spin degrees of freedom is derived and diagonalized exactly.
The theory of the anomalous Hall effect for the heat transfer in a parmagnetic dielectric, discovered by Strohm, Rikken, and Wyder [Phys. Rev. Lett. 95, 155901 (2005)]10.1103/PhysRevLett.95.155901, is developed. The appearance of the phonon heat flux normal to both the temperature gradient and the magnetic field is connected with the interaction of magnetic ions with the crystal field oscillations. In crystals with an arbitrary phonon spectrum this interaction creates the elliptical polarization of phonons. The kinetics related to phonon scattering induced by the spin-phonon interaction determines an origin of the off-diagonal phonon density matrix. The combination of both factors is decisive for the phenomenon under consideration.
In the electrical Hall effect, a magnetic field, applied perpendicular to an electrical current, induces through the Lorentz force a voltage perpendicular to the field and the current. It is generally assumed that an analogous effect cannot exist in the phonon thermal conductivity, as there is no charge transport associated with phonon propagation. In this Letter, we argue that such a magnetotransverse thermal effect should exist and experimentally demonstrate this "phonon Hall effect" in Tb3Ga5O12.
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