Directional transport and propagation of quantum particle and current, such as electron, photon, spin, and phonon, are known to occur in the materials system with broken inversion symmetry, as exemplified by the diode in semiconductor p-n junction and the natural optical activity in chiral materials. Such a nonreciprocal response in the quantum materials of noncentrosymmetry occurs ubiquitously when the time-reversal symmetry is further broken by applying a magnetic field or with spontaneous magnetization, such as the magnetochiral effect and the nonreciprocal magnon transport or spin current in chiral magnets. In the nonlinear regime responding to the square of current and electric field, even a more variety of nonreciprocal phenomena can show up, including the photocurrent of topological origin and the unidirectional magnetoresistance in polar/chiral semiconductors. Microscopically, these nonreciprocal responses in the quantum materials are frequently encoded by the quantum Berry phase, the toroidal moment, and the magnetoelectric monopole, thus cultivating the fertile ground of the functional topological materials. Here, we review the basic mechanisms and emergent phenomena and functions of the nonreciprocal responses in the noncentrosymmetric quantum materials.

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Nonreciprocal responses from non-centrosymmetric quantum materials.

We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization approach that removes local entanglement and produces a coarse-grained lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixedpoint tensors Tinv plus the symmetry group Gsym of the tensors i.e., the symmetry group of the Lagrangian characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by two-dimensional 2D statistical Ising model, 2D statistical loop-gas model, and 1+1D quantum spin-1/2 and spin-1 models. In particular, using such a Gsym,Tinv characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry-protected topological phase. The Gsym,Tinv characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.

Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order

The law of reciprocity in acoustics and elastodynamics codifies a relation of symmetry between action and reaction in fluids and solids. In its simplest form, it states that the frequency-response functions between any two material points remain the same after swapping source and receiver, regardless of the presence of inhomogeneities and losses. As such, reciprocity has enabled numerous applications that make use of acoustic and elastic wave propagation. A recent change in paradigm has prompted us to see reciprocity under a new light: as an obstruction to the realization of wave-bearing media in which the source and receiver are not interchangeable. Such materials may enable the creation of devices such as acoustic one-way mirrors, isolators and topological insulators. Here, we review how reciprocity breaks down in materials with momentum bias, structured space-dependent and time-dependent constitutive properties, and constitutive nonlinearity, and report on recent advances in the modelling and fabrication of these materials, as well as on experiments demonstrating nonreciprocal acoustic and elastic wave propagation therein. The success of these efforts holds promise to enable robust, unidirectional acoustic and elastic wave-steering capabilities that exceed what is currently possible in conventional materials, metamaterials or phononic crystals. Nonreciprocal acoustic and elastic wave propagation may enable the creation of devices such as acoustic one-way mirrors, isolators and topological insulators. This Review presents advances in the creation of materials that break reciprocity and realize robust, unidirectional acoustic and elastic wave steering.

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Nonreciprocity in acoustic and elastic materials

Experimental perspective on three-dimensional topological semimetals

Topological states of matter are particularly robust, since they exploit global features insensitive to local perturbations. In this work, we describe how to create a Chern insulator of phonons in the solid state. The proposed implementation is based on a simple setting, a dielectric slab with a suitable pattern of holes. Its topological properties can be wholly tuned in-situ by adjusting the amplitude and frequency of a driving laser that controls the optomechanical interaction between light and sound. The resulting chiral, topologically protected phonon transport along the edges can be probed completely optically. Moreover, we identify a regime of strong mixing between photon and phonon excitations, which gives rise to a large set of different topological phases. This would be an example of a Chern insulator produced from the interaction between two physically very different particle species, photons and phonons.

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Topological Phases of Sound and Light

Guiding waves through a stable physical channel is essential for reliable information transport. However, energy transport in high-frequency mechanical systems, such as in signal-processing applications, is particularly sensitive to defects and sharp turns because of back-scattering and losses. Topological phenomena in condensed matter systems have shown immunity to defects and unidirectional energy propagation. Topological mechanical metamaterials translate these properties into classical systems for efficient phononic energy transport. Acoustic and mechanical topological metamaterials have so far been realized only in large-scale systems, such as arrays of pendulums, gyroscopic lattices, structured plates and arrays of rods, cans and other structures acting as acoustic scatterers9. To fulfil their potential in device applications, mechanical topological systems need to be scaled to the on-chip level for high-frequency transport. Here we report the experimental realization of topological nanoelectromechanical metamaterials, consisting of two-dimensional arrays of free-standing silicon nitride nanomembranes that operate at high frequencies (10–20 megahertz). We experimentally demonstrate the presence of edge states, and characterize their localization and Dirac-cone-like frequency dispersion. Our topological waveguides are also robust to waveguide distortions and pseudospin-dependent transport. The on-chip integrated acoustic components realized here could be used in unidirectional waveguides and compact delay lines for high-frequency signal-processing applications.

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Experimental realization of on-chip topological nanoelectromechanical metamaterials

Spin current plays a central role in spintronics. In particular, finding more efficient ways to generate spin current has been an important issue and has been studied actively. For example, representative methods of spin-current generation include spin-polarized current injections from ferromagnetic metals, the spin Hall effect, and the spin battery. Here, we theoretically propose a mechanism of spin-current generation based on nonlinear phenomena. By using Boltzmann transport theory, we show that a simple application of the electric field $\mathbit{E}$ induces spin current proportional to ${\mathbit{E}}^{\mathbit{2}}$ in noncentrosymmetric spin-orbit coupled systems. We demonstrate that the nonlinear spin current of the proposed mechanism is supported in the surface state of three-dimensional topological insulators and two-dimensional semiconductors with the Rashba and/or Dresselhaus interaction. In the latter case, the angular dependence of the nonlinear spin current can be manipulated by the direction of the electric field and by the ratio of the Rashba and Dresselhaus interactions. We find that the magnitude of the spin current largely exceeds those in the previous methods for a reasonable magnitude of the electric field. Furthermore, we show that application of ac electric fields (e.g., terahertz light) leads to the rectifying effect of the spin current, where dc spin current is generated. These findings will pave a route to manipulate the spin current in noncentrosymmetric crystals.

Nonlinear spin current generation in noncentrosymmetric spin-orbit coupled systems

Topological mechanical metamaterials translate condensed matter phenomena, like non-reciprocity and robustness to defects, into classical platforms. At small scales, topological nanoelectromechanical metamaterials (NEMM) can enable the realization of on-chip acoustic components, like unidirectional waveguides and compact delay-lines for mobile devices. Here, we report the experimental realization of NEMM phononic topological insulators, consisting of two-dimensional arrays of free-standing silicon nitride (SiN) nanomembranes that operate at high frequencies (10-20 MHz). We experimentally demonstrate the presence of edge states, by characterizing their localization and Dirac cone-like frequency dispersion. Our topological waveguides also exhibit robustness to waveguide distortions and pseudospin-dependent transport. The suggested devices open wide opportunities to develop functional acoustic systems for high-frequency signal processing applications.

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The generation of photocurrent in condensed matter is of main interest for photovoltaic and optoelectronic applications. Shift current, a nonlinear photoresponse, has attracted recent intensive attention as a dominant player of bulk photovoltaic effect in ferroelectric materials. In three-dimensional topological insulators ${\text{Bi}}_{2}{X}_{3}$ ($X=\text{Te}$, Se), we find that Dirac surface states with a hexagonal warping term support shift current by linearly polarized light. Moreover, we study ``shift spin current'' that arises in Dirac surface states by introducing time-reversal symmetry breaking perturbation. The estimate for the magnitudes of the shift charge and spin current densities are $0.13{I}_{0}$ and $0.40{I}_{0}$ (nA/m) for ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$ with the intensity of light ${I}_{0}$ measured in $(\text{W}/{\text{m}}^{2})$, respectively, which can offer a useful method to generate these currents efficiently.

Shift charge and spin photocurrents in Dirac surface states of topological insulator

We investigate the quantum phase transition of the Su-Schrieffer-Heeger (SSH) model by inspecting the two-site entanglements in the ground state. It is shown that the topological phase transition of the SSH model is signified by a nonanalyticity of local entanglement, which becomes discontinuous for finite even system sizes, and that this nonanalyticity has a topological origin. Such a peculiar singularity has a universal nature in one-dimensional topological phase transitions of noninteracting fermions. We make this clearer by pointing out that an analogous quantity in the Kitaev chain exhibiting the identical nonanalyticity is the local electron density. As a byproduct, we show that there exists a different type of phase transition, whereby the pattern of the two-site entanglements undergoes a sudden change. This transition is characterised solely by quantum information theory and does not accompany the closure of the spectral gap. We analyse the scaling behaviours of the entanglement in the vicinities of the transition points.

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Kun Woo Kim

Papers

Experimental realization of on-chip topological nanoelectromechanical metamaterials

Nonlinear spin current generation in noncentrosymmetric spin-orbit coupled systems

Experimental realization of on-chip topological nanoelectromechanical metamaterials

Shift charge and spin photocurrents in Dirac surface states of topological insulator

Quantum Phase Transition and Entanglement in Topological Quantum Wires.