# Berry phase and topological effects of phonons

Berry phase and topological effects of phonons Phonons as collective excitations of lattice vibrations are the main heat carriers in solids. Tremendous effort has been devoted to the investigation of phonons and related properties, giving rise to an intriguing field of phononics, which is of great importance to many practical applications, including heat dissipation, thermal barrier coating, thermoelectrics and thermal control devices [1]. Meanwhile, research into topology-related physics, awarded the 2016 Nobel Prize in Physics, has led to discoveries of various exotic quantum states of matter, including the quantum (anomalous/spin) Hall [Q(A/S)H] effects, topological insulators/semimetals and topological superconductors [2,3]. An emerging research field is to bring topological concepts for a new paradigm phononics—‘topological phononics’ [4–6]. In this Perspective, we will briefly introduce this emerging field and discuss the use of novel quantum degrees of freedom like the Berry phase and topology for manipulating phonons in unprecedented new ways. Berry phase and topology are well established concepts for electrons and have shown important consequences for electronic properties [2,3,7]. When varying the wavevector $$\mathbf {k}$$ along a closed loop within the Brillouin zone (BZ), the Bloch wavefunction $$\left| \psi _\mathbf {k} \right\rangle$$ gains a geometric phase, called the Berry phase, $$\gamma = \oint \mathrm{d}\mathbf {k} \cdot \mathbf {A}_\mathbf {k}$$, where $$\mathbf {A}_\mathbf {k} = \left\langle \psi _\mathbf {k} \right| -i\nabla _\mathbf {k} \left| \psi _\mathbf {k} \right\rangle$$ is the Berry connection. The Berry curvature $$\mathbf {B}_\mathbf {k} = \nabla _\mathbf {k} \times \mathbf {A}_\mathbf {k}$$ plays the role of magnetic field in the momentum space, which generates anomalous/spin Hall effects [7]. The integration of $$\mathbf {B}_\mathbf {k}$$ over the whole BZ gives a topological invariant, the Chern number $$\mathcal {C}$$, which characterizes the Q(A)H states [2,3]. One may generalize topological concepts from electrons to phonons straightforwardly by writing the phonon equation in a Schrödinger-like form, $$H_\mathbf {k} \psi _\mathbf {k} = \omega _\mathbf {k} \psi _\mathbf {k}$$ [6], where \begin{eqnarray} H_{\mathbf {k}} &=& \left( \begin{array}{ccc}0 &\quad i D_{\mathbf {k}}^{1/2} \\ -i D_{\mathbf {k}}^{1/2} &\quad -2i{\eta }_{\mathbf {k}} \end{array} \right),\nonumber\\ {\mathbf {\psi }_{\mathbf {k}}} &=& \left( \begin{array}{c}D_{\mathbf {k}}^{1/2} {\mathbf {u}_{\mathbf {k}}} \\ {\dot{\mathbf {u}}_{\mathbf {k}}} \end{array}\right). \end{eqnarray} (1)Dk is the dynamic matrix. ηk is the time-reversal symmetry (TRS) breaking term. $$\mathbf {u}_{\mathbf {k}}$$ and $$\dot{\mathbf {u}}_{\mathbf {k}}$$ are the displacement and its time derivative. $$\omega _\mathbf {k}$$ is the phonon dispersion. $$\psi _\mathbf {k}$$ is the phonon wavefunction expressed in an extended coordinate-velocity space. Similar equations with different wavefunction definitions have been proposed [4,8]. Based on the Schrödinger-like equation, the Berry connection, Berry curvature and other topological quantities are defined for phonons as they are for electrons. However, the problem of phonons differs significantly from that of electrons in the following aspects [6]. Firstly, phonons as bosons are not limited by the Pauli exclusion principle, suggesting that the whole phonon spectrum is physically probed. Secondly, spin-related effects (like QSH) cannot be directly applied to phonons that typically are spinless, but can be simulated by utilizing pseudo-spin and pseudo-spin–orbit coupling [9]. Last not least, while phonon Hamiltonians can be mapped to tight-binding electron Hamiltonians, they are required to satisfy the acoustic sum rule by the rigid translational symmetry. Moreover, their basis functions are always px, y, z, independent of their atomic nature, implying distinct orbital physics between phonons and electrons. A honeycomb lattice is an ideal platform to explore the Berry phase and the topological effects of phonons [4,6]. The longitudinal optical and acoustic modes form linearly crossed bands and Dirac points at K/K΄ (Fig. 1(a)), as protected by inversion symmetry (IS) and TRS. Phonons around the Dirac points are described by an effective Hamiltonian (referenced to the Dirac frequency) [6]: \begin{equation*} H_0 = v_D(k_y\tau _z\sigma _x - k_x\sigma _y), \end{equation*} where $$v$$D is the group velocity, σ and τ are the Pauli matrices, σz = ±1 and τz = ±1 refer to A/B sublattices and K/K΄ valleys, respectively. This Hamiltonian resembles that for spinless electrons of graphene. In the presence of IS and TRS, the Berry curvature is zero except at the Dirac points that are characterized by Berry phases of ±π, leading to suppression of backscattering. Figure 1. View largeDownload slide (a) A honeycomb lattice composed of A/B sublattices, and the Dirac cones of phonons near the K/K΄ valleys. (b), (c) Berry curvature distribution (upper), schematic bulk bands together with topological boundary states (lower) of phonons with broken (b) inversion symmetry P and (c) time-reversal symmetry T. Phonon states mainly located at A (B) sublattices are colored red (blue). (d) The two valleys that are nondegenerate in frequency caused by breaking P and T simultaneously. Figure 1. View largeDownload slide (a) A honeycomb lattice composed of A/B sublattices, and the Dirac cones of phonons near the K/K΄ valleys. (b), (c) Berry curvature distribution (upper), schematic bulk bands together with topological boundary states (lower) of phonons with broken (b) inversion symmetry P and (c) time-reversal symmetry T. Phonon states mainly located at A (B) sublattices are colored red (blue). (d) The two valleys that are nondegenerate in frequency caused by breaking P and T simultaneously. One may introduce mass terms into the Hamiltonian by breaking IS (mI) or TRS (mT): \begin{equation*} H^\prime = m_\mathrm{I}\sigma _z + m_\mathrm{T}\sigma _z\tau _z, \end{equation*} which opens Dirac band gaps. The first term (the Semenoff term) can be realized by using different atomic masses for A/B sublattices. The second term (the Haldane term) can be realized extrinsically by magnetic/Coriolis fields or intrinsically by spin–lattice interactions in magnetic lattices. Importantly, the effective Hamiltonian is based on px, y orbitals rather than pz, which is essential to get a nonzero Haldane term [6]. When slightly breaking TRS or IS, $$\mathbf {B}_\mathbf {k}$$ keeps vanishing except near K/K΄ (Fig. 1(b), (c)). Then the Berry flux around each valley is ±π, which gives the valley Chern number $$\mathcal {C}_{K(K^\prime )} = \pm 1/2$$. By definition, $$\mathcal {C} = \mathcal {C}_{K}+\mathcal {C}_{K^\prime }$$. If only TRS is broken, $$\mathbf {B}_\mathbf {k} = \mathbf {B}_{-\mathbf {k}}$$ and $$\mathcal {C} = \pm 1$$, corresponding to the Q(A)H-like phase, which is topologically protected to have one-way edge phonon modes within the Dirac gap (Fig. 1(b)). These one-way edge modes are perfect transport channels, enabling conducting phonons without scattering. If only IS is broken, $$\mathbf {B}_\mathbf {k} = -\mathbf {B}_{-\mathbf {k}}$$ and $$\mathcal {C} = 0$$, corresponding to a topologically trivial phase. Nevertheless, for interfaces having opposite mI on the two sides, there exist topologically protected interfacial phonon modes that are valley-momentum locked (Fig. 1(c)). When both TRS and IS are broken, the frequency degeneracy between the two valleys is removed, and thus band gaps at K/K΄ can be tuned independently. One could realize zero band gap at only one valley in extreme cases (Fig. 1(d)), which could be used for valley filters. The Berry phase and topological effects of phonons can find promising applications. One possible direction is based on valley phononics that utilize the valley degree of freedom to control phonons. Valley phonons can be selectively excited by Raman processes using polarized photons [10]. Moreover, valley-polarized currents could be generated by the valley-polarized conduction channels (Fig. 1(c)) or by valley filters (Fig. 1(d)) [6], which can be further detected by the valley phonon Hall effect [10]. Another possible direction is based on the one-way edge states or, more generally, the topologically protected boundary states. These states are able to conduct phonons with little or no scattering, useful for designing phononic circuits. Moreover, an ideal phonon diode with 100% efficiency is possible in a multi-terminal transport system with one-way edge states [6]. As an outlook, many interesting research topics are waiting to be explored in this field: (i) finding efficient ways to break the TRS of phonons in solids; (ii) searching for topological phonon states protected by symmetries like space group symmetries, with no need to break the TRS; (iii) exploring Dirac or Weyl phonons in 3D lattices; and (iv) realizing the Berry phase and topological effects of phonons in realistic materials. Funding Y.X. acknowledges support from the National Thousand-Young-Talents Program and Tsinghua University Initiative Scientific Research Program. Y.L. and W.D. acknowledge support from the Ministry of Science and Technology of China (2016YFA0301001) and the National Natural Science Foundation of China (11674188 and 11334006). REFERENCES 1. Li N , Ren J , Wang L et al. Rev Mod Phys 2012 ; 84 : 1045 – 66 . CrossRef Search ADS 2. Hasan MZ , Kane CL . Rev Mod Phys 2010 ; 82 : 3045 – 67 . CrossRef Search ADS 3. Qi XL , Zhang SC . Rev Mod Phys 2011 ; 83 : 1057 – 110 . CrossRef Search ADS 4. Zhang L , Ren J , Wang JS et al. Phys Rev Lett 2010 ; 105 : 225901 . CrossRef Search ADS PubMed 5. Kane C , Lubensky T . Nat Phys 2014 ; 10 : 39 – 45 . CrossRef Search ADS 6. Liu Y , Xu Y , Zhang SC et al. Phys Rev B 2017 ; 96 : 064106 . CrossRef Search ADS 7. Xiao D , Chang MC , Niu Q . Rev Mod Phys 2010 ; 82 : 1959 – 2007 . CrossRef Search ADS 8. Süsstrunk R , Huber SD . Proc Natl Acad Sci USA 2016 ; 113 : E4767 – 75 . CrossRef Search ADS PubMed 9. Süsstrunk R , Huber SD . Science 2015 ; 349 : 47 – 50 . CrossRef Search ADS PubMed 10. Zhang L , Niu Q . Phys Rev Lett 2015 ; 115 : 115502 . CrossRef Search ADS PubMed © The Author(s) 2017. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png National Science Review Oxford University Press

# Berry phase and topological effects of phonons

, Volume Advance Article (3) – Aug 2, 2017
3 pages

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Oxford University Press
Copyright
© The Author(s) 2017. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd.
ISSN
2095-5138
eISSN
2053-714X
D.O.I.
10.1093/nsr/nwx086
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### Abstract

Phonons as collective excitations of lattice vibrations are the main heat carriers in solids. Tremendous effort has been devoted to the investigation of phonons and related properties, giving rise to an intriguing field of phononics, which is of great importance to many practical applications, including heat dissipation, thermal barrier coating, thermoelectrics and thermal control devices [1]. Meanwhile, research into topology-related physics, awarded the 2016 Nobel Prize in Physics, has led to discoveries of various exotic quantum states of matter, including the quantum (anomalous/spin) Hall [Q(A/S)H] effects, topological insulators/semimetals and topological superconductors [2,3]. An emerging research field is to bring topological concepts for a new paradigm phononics—‘topological phononics’ [4–6]. In this Perspective, we will briefly introduce this emerging field and discuss the use of novel quantum degrees of freedom like the Berry phase and topology for manipulating phonons in unprecedented new ways. Berry phase and topology are well established concepts for electrons and have shown important consequences for electronic properties [2,3,7]. When varying the wavevector $$\mathbf {k}$$ along a closed loop within the Brillouin zone (BZ), the Bloch wavefunction $$\left| \psi _\mathbf {k} \right\rangle$$ gains a geometric phase, called the Berry phase, $$\gamma = \oint \mathrm{d}\mathbf {k} \cdot \mathbf {A}_\mathbf {k}$$, where $$\mathbf {A}_\mathbf {k} = \left\langle \psi _\mathbf {k} \right| -i\nabla _\mathbf {k} \left| \psi _\mathbf {k} \right\rangle$$ is the Berry connection. The Berry curvature $$\mathbf {B}_\mathbf {k} = \nabla _\mathbf {k} \times \mathbf {A}_\mathbf {k}$$ plays the role of magnetic field in the momentum space, which generates anomalous/spin Hall effects [7]. The integration of $$\mathbf {B}_\mathbf {k}$$ over the whole BZ gives a topological invariant, the Chern number $$\mathcal {C}$$, which characterizes the Q(A)H states [2,3]. One may generalize topological concepts from electrons to phonons straightforwardly by writing the phonon equation in a Schrödinger-like form, $$H_\mathbf {k} \psi _\mathbf {k} = \omega _\mathbf {k} \psi _\mathbf {k}$$ [6], where \begin{eqnarray} H_{\mathbf {k}} &=& \left( \begin{array}{ccc}0 &\quad i D_{\mathbf {k}}^{1/2} \\ -i D_{\mathbf {k}}^{1/2} &\quad -2i{\eta }_{\mathbf {k}} \end{array} \right),\nonumber\\ {\mathbf {\psi }_{\mathbf {k}}} &=& \left( \begin{array}{c}D_{\mathbf {k}}^{1/2} {\mathbf {u}_{\mathbf {k}}} \\ {\dot{\mathbf {u}}_{\mathbf {k}}} \end{array}\right). \end{eqnarray} (1)Dk is the dynamic matrix. ηk is the time-reversal symmetry (TRS) breaking term. $$\mathbf {u}_{\mathbf {k}}$$ and $$\dot{\mathbf {u}}_{\mathbf {k}}$$ are the displacement and its time derivative. $$\omega _\mathbf {k}$$ is the phonon dispersion. $$\psi _\mathbf {k}$$ is the phonon wavefunction expressed in an extended coordinate-velocity space. Similar equations with different wavefunction definitions have been proposed [4,8]. Based on the Schrödinger-like equation, the Berry connection, Berry curvature and other topological quantities are defined for phonons as they are for electrons. However, the problem of phonons differs significantly from that of electrons in the following aspects [6]. Firstly, phonons as bosons are not limited by the Pauli exclusion principle, suggesting that the whole phonon spectrum is physically probed. Secondly, spin-related effects (like QSH) cannot be directly applied to phonons that typically are spinless, but can be simulated by utilizing pseudo-spin and pseudo-spin–orbit coupling [9]. Last not least, while phonon Hamiltonians can be mapped to tight-binding electron Hamiltonians, they are required to satisfy the acoustic sum rule by the rigid translational symmetry. Moreover, their basis functions are always px, y, z, independent of their atomic nature, implying distinct orbital physics between phonons and electrons. A honeycomb lattice is an ideal platform to explore the Berry phase and the topological effects of phonons [4,6]. The longitudinal optical and acoustic modes form linearly crossed bands and Dirac points at K/K΄ (Fig. 1(a)), as protected by inversion symmetry (IS) and TRS. Phonons around the Dirac points are described by an effective Hamiltonian (referenced to the Dirac frequency) [6]: \begin{equation*} H_0 = v_D(k_y\tau _z\sigma _x - k_x\sigma _y), \end{equation*} where $$v$$D is the group velocity, σ and τ are the Pauli matrices, σz = ±1 and τz = ±1 refer to A/B sublattices and K/K΄ valleys, respectively. This Hamiltonian resembles that for spinless electrons of graphene. In the presence of IS and TRS, the Berry curvature is zero except at the Dirac points that are characterized by Berry phases of ±π, leading to suppression of backscattering. Figure 1. View largeDownload slide (a) A honeycomb lattice composed of A/B sublattices, and the Dirac cones of phonons near the K/K΄ valleys. (b), (c) Berry curvature distribution (upper), schematic bulk bands together with topological boundary states (lower) of phonons with broken (b) inversion symmetry P and (c) time-reversal symmetry T. Phonon states mainly located at A (B) sublattices are colored red (blue). (d) The two valleys that are nondegenerate in frequency caused by breaking P and T simultaneously. Figure 1. View largeDownload slide (a) A honeycomb lattice composed of A/B sublattices, and the Dirac cones of phonons near the K/K΄ valleys. (b), (c) Berry curvature distribution (upper), schematic bulk bands together with topological boundary states (lower) of phonons with broken (b) inversion symmetry P and (c) time-reversal symmetry T. Phonon states mainly located at A (B) sublattices are colored red (blue). (d) The two valleys that are nondegenerate in frequency caused by breaking P and T simultaneously. One may introduce mass terms into the Hamiltonian by breaking IS (mI) or TRS (mT): \begin{equation*} H^\prime = m_\mathrm{I}\sigma _z + m_\mathrm{T}\sigma _z\tau _z, \end{equation*} which opens Dirac band gaps. The first term (the Semenoff term) can be realized by using different atomic masses for A/B sublattices. The second term (the Haldane term) can be realized extrinsically by magnetic/Coriolis fields or intrinsically by spin–lattice interactions in magnetic lattices. Importantly, the effective Hamiltonian is based on px, y orbitals rather than pz, which is essential to get a nonzero Haldane term [6]. When slightly breaking TRS or IS, $$\mathbf {B}_\mathbf {k}$$ keeps vanishing except near K/K΄ (Fig. 1(b), (c)). Then the Berry flux around each valley is ±π, which gives the valley Chern number $$\mathcal {C}_{K(K^\prime )} = \pm 1/2$$. By definition, $$\mathcal {C} = \mathcal {C}_{K}+\mathcal {C}_{K^\prime }$$. If only TRS is broken, $$\mathbf {B}_\mathbf {k} = \mathbf {B}_{-\mathbf {k}}$$ and $$\mathcal {C} = \pm 1$$, corresponding to the Q(A)H-like phase, which is topologically protected to have one-way edge phonon modes within the Dirac gap (Fig. 1(b)). These one-way edge modes are perfect transport channels, enabling conducting phonons without scattering. If only IS is broken, $$\mathbf {B}_\mathbf {k} = -\mathbf {B}_{-\mathbf {k}}$$ and $$\mathcal {C} = 0$$, corresponding to a topologically trivial phase. Nevertheless, for interfaces having opposite mI on the two sides, there exist topologically protected interfacial phonon modes that are valley-momentum locked (Fig. 1(c)). When both TRS and IS are broken, the frequency degeneracy between the two valleys is removed, and thus band gaps at K/K΄ can be tuned independently. One could realize zero band gap at only one valley in extreme cases (Fig. 1(d)), which could be used for valley filters. The Berry phase and topological effects of phonons can find promising applications. One possible direction is based on valley phononics that utilize the valley degree of freedom to control phonons. Valley phonons can be selectively excited by Raman processes using polarized photons [10]. Moreover, valley-polarized currents could be generated by the valley-polarized conduction channels (Fig. 1(c)) or by valley filters (Fig. 1(d)) [6], which can be further detected by the valley phonon Hall effect [10]. Another possible direction is based on the one-way edge states or, more generally, the topologically protected boundary states. These states are able to conduct phonons with little or no scattering, useful for designing phononic circuits. Moreover, an ideal phonon diode with 100% efficiency is possible in a multi-terminal transport system with one-way edge states [6]. As an outlook, many interesting research topics are waiting to be explored in this field: (i) finding efficient ways to break the TRS of phonons in solids; (ii) searching for topological phonon states protected by symmetries like space group symmetries, with no need to break the TRS; (iii) exploring Dirac or Weyl phonons in 3D lattices; and (iv) realizing the Berry phase and topological effects of phonons in realistic materials. Funding Y.X. acknowledges support from the National Thousand-Young-Talents Program and Tsinghua University Initiative Scientific Research Program. Y.L. and W.D. acknowledge support from the Ministry of Science and Technology of China (2016YFA0301001) and the National Natural Science Foundation of China (11674188 and 11334006). REFERENCES 1. Li N , Ren J , Wang L et al. Rev Mod Phys 2012 ; 84 : 1045 – 66 . CrossRef Search ADS 2. Hasan MZ , Kane CL . Rev Mod Phys 2010 ; 82 : 3045 – 67 . CrossRef Search ADS 3. Qi XL , Zhang SC . Rev Mod Phys 2011 ; 83 : 1057 – 110 . CrossRef Search ADS 4. Zhang L , Ren J , Wang JS et al. Phys Rev Lett 2010 ; 105 : 225901 . CrossRef Search ADS PubMed 5. Kane C , Lubensky T . Nat Phys 2014 ; 10 : 39 – 45 . CrossRef Search ADS 6. Liu Y , Xu Y , Zhang SC et al. Phys Rev B 2017 ; 96 : 064106 . CrossRef Search ADS 7. Xiao D , Chang MC , Niu Q . Rev Mod Phys 2010 ; 82 : 1959 – 2007 . CrossRef Search ADS 8. Süsstrunk R , Huber SD . Proc Natl Acad Sci USA 2016 ; 113 : E4767 – 75 . CrossRef Search ADS PubMed 9. Süsstrunk R , Huber SD . Science 2015 ; 349 : 47 – 50 . CrossRef Search ADS PubMed 10. Zhang L , Niu Q . Phys Rev Lett 2015 ; 115 : 115502 . CrossRef Search ADS PubMed © The Author(s) 2017. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

### Journal

National Science ReviewOxford University Press

Published: Aug 2, 2017

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