No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Topological Field Theory of Time-Reversal Invariant Insulators
Abstract
We show that the fundamental time reversal invariant (TRI) insulator exists in 4+1 dimensions, where the effective field theory is described by the 4+1 dimensional Chern-Simons theory and the topological properties of the electronic structure is classified by the second Chern number. These topological properties are the natural generalizations of the time reversal breaking (TRB) quantum Hall insulator in 2+1 dimensions. The TRI quantum spin Hall insulator in 2+1 dimensions and the topological insulator in 3+1 dimension can be obtained as descendants from the fundamental TRI insulator in 4+1 dimensions through a dimensional reduction procedure. The effective topological field theory, and the $Z_2$ topological classification for the TRI insulators in 2+1 and 3+1 dimensions are naturally obtained from this procedure. All physically measurable topological response functions of the TRI insulators are completely described by the effective topological field theory. Our effective topological field theory predicts a number of novel and measurable phenomena, the most striking of which is the topological magneto-electric effect, where an electric field generates a magnetic field in the same direction, with an universal constant of proportionality quantized in odd multiples of the fine structure constant $\alpha=e^2/\hbar c$. Finally, we present a general classification of all topological insulators in various dimensions, and describe them in terms of a unified topological Chern-Simons field theory in phase space. Comment: 47 pages, 21 figures. Submitted to PRB. For high resolution figures please see final published version
... It has been known that the band inversion in accompany of SOC can support the topological phase transition from the trivial to non-trivial gapped phases. Specifically in 2D case, this is known to be captured by the quantized Chern number as the topological invariant [79], ...
... At µ eff = 0, by increasing h z , C processes the changes from 9, 4, 1, to 0. It reveals that the transitions occur within three topological phases with nonzero C. When µ eff ̸ = 0, the topological phases are separated by the gapless phase. We remark that the phenomenon associated with the emergent large C is the feature of the multi-spin models that is distinct from the spin-1/2 one [79], because C is limited by its maximum N s . Generally, the topological phase transitions are in accompany with the closing and reopening of the band gaps. ...
... We use the cylindrical geometry by setting the open boundary condition in the x direction, while periodic boundary condition in the y direction. According to the bulk-edge correspondence, the number of the existing chiral edge modes is closely tied to C [79]. We calculate C n for each band, and obtain C n = {1, 3, 5, −5, −3, −1} from the bottom to top bands for panel (b), C n = {1, 3, 0, 0, −3, −1} for panel (c), C n = {1, 0, 0, 0, 0, −1} for panel (d), and C n = 0 for panel (e). ...
The essential role of synthetic spin-orbit coupling in discovering new topological matter phases with cold atoms is widely acknowledged. However, the engineering of spin-orbit coupling remains unclear for arbitrary-spin models due to the complexity of spin matrices. In this work, we develop a more general but relatively straightforward method to achieve spin-orbit coupling for multi-spin models. Our approach hinges on controlling the coupling between distinct pseudo-spins through two intermediary states, resulting in tunneling with spin flips that have direction-dependent strength. The engineered spin-orbit coupling can facilitate topological phase transitions with Chern numbers over 1, a unique characteristic of multi-spin models compared to spin-1/2 models. By utilizing existing cold atom techniques, our proposed method provides an ideal platform for investigating topological properties related to large Chern numbers.
... The topological magnetoelectric effect (TME) presents an even more stark puzzle. In three-dimensional (3D) crystalline band insulators, TRS leads to a Z 2 topological classification of the electronic ground state and in Z 2 -odd phases to a magnetoelectric linear-response coefficient that is argued [10,11] to equal (n + 1/2)e 2 /hc for n ∈ Z. On the other hand, in FIG. 1. Relationship between α CS in bulk insulators that exhibit time-reversal symmetry (TRS) and α me in thin films thereof. ...
... any finite-sized system with TRS, the usual symmetry arguments dictate that the magnetoelectric coefficient must vanish. Indeed, the requirement of magnetic surface dopants for realization of the TME has been noted previously [10][11][12][13][14]. There is no physical bulk response directly related to the 3D Z 2 invariant, but a nontrivial Z 2 invariant implies the existence of surface states that can then be gapped to activate the TME. ...
... where e > 0 is the elementary charge, is the finite volume of the sample, E F is the Fermi energy, and E (r) are the electronic energy eigenfunctions of the unperturbed Hamiltonian. 10 Unfortunately the PZW approach cannot be directly applied to crystalline solids since, even in the absence of electromagnetic fields, the electronic charge and currentdensity expectation values are expressed in terms of Bloch energy eigenfunctions, the support of which is all of space. 11 Indeed, this is related to the property that the usual position operator is not well-defined in the Hilbert space of Bloch functions [22]. ...
A delicate tension complicates the relationship between the topological magnetoelectric effect (TME) in three-dimensional (3D) Z 2 topological insulators (TIs) and time-reversal symmetry (TRS). TRS underlies a particular Z 2 topological classification of the electronic ground state of crystalline band insulators and the associated quantization of the magnetoelectric response coefficient calculated using bulk linear-response theory but, according to standard symmetry arguments, simultaneously forbids a nonzero magnetoelectric coefficient in any physical finite-size system. This contrast between theories of magnetoelectric response in formal bulk models and in real finite-sized materials originates from the distinct approaches required to introduce notions of (electronic) polarization and orbital magnetization in these fundamentally different environments. In this work we argue for a modified interpretation of the bulk linear-response calculations in nonmagnetic Z 2 TIs that is more plainly consistent with TRS and use this interpretation to discuss the effect's observation—still absent over a decade after its prediction. Our analysis is reinforced by microscopic bulk and thin-film calculations carried out using a simplified but still realistic effective model for the well established V 2 VI 3 [ V = (Sb, Bi) and VI = (Se, Te) ] family of nonmagnetic Z 2 TIs. When a uniform dc magnetic field is included in this model, the anomalous n = 0 Landau levels (LLs) play the central role, both in thin films and in bulk. In the former case, only the n = 0 LL eigenfunctions can support a dipole moment, which vanishes if there are no magnetic surface dopants and is quantized in the thick-film limit if magnetic dopants at the top and bottom surfaces have opposite orientation. In the latter case, the Hamiltonian projected into the n = 0 LL subspace is a one-dimensional Su-Schrieffer-Heeger model with ground-state polarization that is quantized in accordance with the bulk linear-response coefficient calculated for (a lattice regularization of) the starting 3D model. Motivated by analytical results, we conjecture a type of microscopic bulk-boundary correspondence: a bulk insulator with (generalized) TRS supports a magnetoelectric coefficient that is purely itinerant (which is generically related to the geometry of the ground state) if and only if magnetic surface dopants are required for the TME to manifest in finite samples thereof. We conclude that in nonmagnetic Z 2 TIs the TME is activated by magnetic surface dopants, that the charge-density response to a uniform dc magnetic field is localized at the surface and specified by the configuration of those dopants, and that the TME is qualitatively less robust against disorder than the integer quantum Hall effect.
Published by the American Physical Society 2024
... Introduction-The (electronic) ground state of 3dimensional (3D) time-reversal invariant band insulators can be classified by a Z 2 index [1][2][3][4][5]. In thin films of Z 2 -odd materials, the nontrivial bulk topology implies the existence of an odd number of surface state Dirac cones that are commonly revealed by angle-resolved photoemission spectroscopy [6][7][8]. ...
... In thin films of Z 2 -odd materials, the nontrivial bulk topology implies the existence of an odd number of surface state Dirac cones that are commonly revealed by angle-resolved photoemission spectroscopy [6][7][8]. One key property [1,9] of bulk Z 2 topological insulators (TIs) is the quantization of their magnetoelectric response. The formal bulk response manifests as an observable topological magnetoelectric effect (TME) only when the surface states are gapped by locally breaking time-reversal so that they support a surface Hall conductivity [1,[9][10][11][12][13][14][15][16][17][18][19][20]. ...
... One key property [1,9] of bulk Z 2 topological insulators (TIs) is the quantization of their magnetoelectric response. The formal bulk response manifests as an observable topological magnetoelectric effect (TME) only when the surface states are gapped by locally breaking time-reversal so that they support a surface Hall conductivity [1,[9][10][11][12][13][14][15][16][17][18][19][20]. Although first proposed more than 15 years ago, the TME has not been yet directly measured. ...
The topological magnetoelectric effect (TME) is a defining property of 3-dimensional $\mathbb{Z}_{2}$ topological insulators that was predicted on theoretical grounds more than a decade ago, but has still not been directly measured. In this Letter we propose a strategy for direct measurement of the TME, and discuss the precision of the effect in real devices with charge and spin disorder.
... Similarly to other topological invariants, the presence of a non-trivial Hopf invariant in the bulk of a system has consequences for its response functions. In particular, in the presence of a static electromagnetic field, the vacuum of a three-dimensional Hopf insulator may support a topological magnetoelectric effect [42,47]. In general, this phenomenon is described by the effective action for axion electrodynamics [47][48][49][50][51] ...
... In particular, in the presence of a static electromagnetic field, the vacuum of a three-dimensional Hopf insulator may support a topological magnetoelectric effect [42,47]. In general, this phenomenon is described by the effective action for axion electrodynamics [47][48][49][50][51] ...
... The model described here may be easily extended to represent a phase with indices (H; χ) = (p; 0, 0, 2p ′ ) for any integers p, p ′ , by modifying k z → pk z and k x → p ′ k x . Let us now briefly elaborate upon the procedure which we have used to construct these three-dimensional Hopf-Euler phases, namely dimensional extension [47] from a two dimensional Euler model. This is in direct analogue to the correspondence between Chern and Hopf-Chern insulators described in [40]. ...
We discuss a class of three-band non-Abelian topological insulators in three dimensions which carry a single bulk Hopf index protected by spatiotemporal ($\mathcal{PT}$) inversion symmetry. These phases may also host subdimensional topological invariants given by the Euler characteristic class, resulting in real Hopf-Euler insulators. Such systems naturally realize helical nodal structures in the 3D Brillouin zone, providing a physical manifestation of the linking number described by the Hopf invariant. We show that, by opening a gap between the valence bands of these systems, one finds a fully-gapped `flag' phase, which displays a three-band multi-gap Pontryagin invariant. Unlike the previously reported $\mathcal{PT}$-symmetric four-band real Hopf insulator, which hosts a $\mathbb{Z} \oplus \mathbb{Z}$ invariant, these phases are not unitarily equivalent to two copies of a complex two-band Hopf insulator. We show that these uncharted phases can be obtained through dimensional extension of two-dimensional Euler insulators, and that they support (1) an optical bulk integrated circular shift effect quantized by the Hopf invariant, (2) quantum-geometric breathing in the real space Wannier functions, and (3) surface Euler topology on boundaries. Consequently, our findings pave a way for novel experimental realizations of real-space quantum-geometry, as these systems may be directly simulated by utilizing synthethic dimensions in metamaterials or ultracold atoms.
... Further gaping out the surface states by an out-of-plane magnetism [46] gives rise to various topologically distinct phases. Within the scheme of magnetic topological insulators, two such phases have been discovered as the Chern insulator [47][48][49], aka quantum anomalous Hall effect (QAHE) that is characterized by Chern invariant and quantized Hall plateau, and the axion insulator [50,51], signatured by zero Hall plateau and non-vanishing longitudinal conductance. A semi-magnetic topological insulator, on the other hand, bears with the half-quantized quantum anomalous Hall effect (half QAHE) [31,52,53] with a half quantized Hall conductance and unusual bulk-boundary correspondence, signed by the absence of edge state but the appearance of the pow-law decaying current from boundary to bulk. ...
... With confined geometry, the topological insulator film is predicted [48,50,101] to host the quantum anomalous Hall effect (QAHE) with proper magnetism, either by magnetic doping approach [49,[102][103][104][105][106] or establishing intrinsic magnetic order [107][108][109]. In this sense three typical cases realizing the Chern insulating phase is presented in Fig. 12, with uniform Zeeman field (to make consistence with discussion here, the Zeeman strength here is still chosen to be weak, while the uniformly strong strength case is left to be discussed in the higher Chern number case later on), symmetric top and bottom surface Zeeman fields configuration and an asymmetric configuration which does not break the holistic polarization, by which we mean that the symmetric ingredient in the configuration overwhelms the asymmetric one. ...
... Top & Bottom Anti-Symmetric Along with the special 3 + 1 space-time dimension, the Maxwell electrodynamics is allowed to be decorated with an extra θ term, which generates axion electrodynamics [115,116] to the space-time dependent θ axion field that couples with the ordinary electromagnetic field. On a practical level, based on the picture of surface Hall effect [64,117] and analogical mathematical structure between Hall current and magnetization current, people generalize and propose the topological field theory [50], where a θ term is introduced to describe the magnetoelectric effect [110][111][112][118][119][120][121][122][123] in a topological insulator medium, where the axion field is forced to gain a magnitude of π [124] by symmetry and topological requirement. ...
We develop a Dirac fermion theory for topological phases in magnetic topological insulator films. The theory is based on exact solutions of the energies and the wave functions for an effective model of the three-dimensional topological insulator (TI) film. It is found that the TI film consists of a pair of massless or massive Dirac fermions for the surface states, and a series of massive Dirac fermions for the bulk states. The massive Dirac fermion always carries zero or integer quantum Hall conductance when the valence band is fully occupied while the massless Dirac fermion carries a one-half quantum Hall conductance when the chemical potential is located around the Dirac point for a finite range. The magnetic exchange interaction in the magnetic layers in the film can be used to manipulate either the masses or chirality of the Dirac fermions and gives rise to distinct topological phases, which cover the known topological insulating phases, such as quantum anomalous Hall effect, quantum spin Hall effect and axion effect, and also the novel topological metallic phases, such as half quantized Hall effect, half quantum mirror Hall effect, and metallic quantum anomalous Hall effect.
... Conventionally, the 4D Euclidean QHI is defined by a Dirac tight-binding model on the hypercubic lattice [44,45]: ...
... with d μ k ¼ ðsin k 1 ; sin k 2 ; sin k 3 ; sin k 4 ; m k Þ and m k ¼ m þ P j cos k j . Evidently, these expressions are equivalent to the Fourier transform of the Hamiltonian in Eq. (1), such that U(1)-HBT reproduces results from the 4D Euclidean case: at half-filling, the model undergoes topological phase transitions at jmj ¼ 0, 2, 4, but is gapped otherwise with a nonvanishing C 2 [45], as depicted in Fig. 1(b). The corresponding gapped DOS for m ¼ 3 is plotted with the red curve in Fig. 3(a). ...
... On the experimental front, to probe the transport associated with C 2 [44], one needs to reconcile a dimensionality mismatch: the hyperbolic plane is two-dimensional, whereas four orthogonal directions enter (via the four-component Levi-Civita symbol) the nonlinear response to applied fields [45,64]. Experimental realizations would also benefit from a generalization of the hyperbolic non-Abelian semimetal to fp; qg lattices with a smaller curvature per site than the {8,8} lattice assumed here. ...
We extend the notion of topologically protected semi-metallic band crossings to hyperbolic lattices in a negatively curved plane. Because of their distinct translation group structure, such lattices are associated with a high-dimensional reciprocal space. In addition, they support non-Abelian Bloch states which, unlike conventional Bloch states, acquire a matrix-valued Bloch factor under lattice translations. Combining diverse numerical and analytical approaches, we uncover an unconventional scaling in the density of states at low energies, and illuminate a nodal manifold of codimension five in the reciprocal space. The nodal manifold is topologically protected by a nonzero second Chern number, reminiscent of the characterization of Weyl nodes by the first Chern number.
... The lattice wave vectors k x,y,z are defined in the first Brillouin zone with k = k 2 x + k 2 y + k 2 z . The first four terms in Eq. 1 describe a topological insulator preserving both parity P = σ z and time reversal T = iτ y K (K is the complex conjugation operator) symmetry, whose axion field is quantized to θ = 0 when B = 0 while θ = π otherwise 10 . Whereas the fifth term describes the exchange interaction between the topological electron and the dynamic, antiparallel magnetic moments, which breaks these two symmetries explicitly and thus introduces an additional dynamic part to the static axion field, giving rise to the DAI. ...
... Quantitative expression for the dynamic axion field in the DAI To establish a quantitative expression for this dynamic axion field driven by the linearly polarized AFMR, we employ the gauge invariant expression for the axion field in terms of the Chern-Simons 3-form 10 . While the timescale of the AFMR hence θ(t) is about 7 orders of magnitude larger than the typical electron response time 37 , the electrons inside the DAI can adjust adiabatically to the instantaneous configuration of Néel vector, viewing the AFMR as a static magnetic vector almost frozen in time. ...
Dynamic axion insulators feature a time-dependent axion field that can be induced by antiferromagnetic resonance. Here, we show that a Josephson junction incorporating this dynamic axion insulator between two superconductors exhibits a striking doubled Shapiro steps wherein all odd steps are completely suppressed in the jointly presence of a DC bias and a static magnetic field. The resistively shunted junction simulation confirms that these doubled Shapiro steps originate from the distinctive axion electrodynamics driven by the antiferromagnetic resonance, which thus not only furnishes a hallmark to identify the dynamic axion insulator but also provides a method to evaluate its mass term. Furthermore, the experimentally feasible differential conductance is also determined. Our work holds significant importance in condensed matter physics and materials science for understanding the dynamic axion insulator, paving the way for its further exploration and applications.
... Introduction -Integrating magnetism into topological insulators (TIs) can break time reversal symmetry and lead to the emergence of magnetic topological phases, e.g. the quantum anomalous Hall (QAH) insulators [3][4][5][6][7][8][9][10] and axion insulators (AIs) [11][12][13][14]. Magnetism has been successfully achieved in TIs by either doping magnetic impurities, e.g. ...
... Cr and/or V doped (Bi,Sb) 2 Te 3 , or growing stoichiometric antiferromagnetic topological compound, MnBi 2 Te 4 . When ferromagnetism is achieved in TI films, both surface states are gapped, leading to the QAH effect [3,10,15,16]. The quantized Hall response for QAH states have been unambiguously observed in several systems, including magnetically doped TIs [5][6][7][8], MnBi 2 Te 4 films [9] and twisted graphene and transition metal dichalcogenides materials [17][18][19][20][21][22][23][24][25][26]. ...
Nonlinear Hall effect (NHE) can originate from the quantum metric mechanism in antiferromagnetic topological materials with PT symmetry, which has been experimentally observed in MnBi2Te4. In this work, we propose that breaking PT symmetry via external electric fields can lead to a dramatic enhancement of NHE, thus allowing for an electric control of NHE. Microscopically, this is because breaking PT symmetry can lift spin degeneracy of a Kramers' pair, giving rise to additional contributions within one Kramers' pair of bands. We demonstrate this enhancement through a model Hamiltonian that describes an antiferromagnetic topological insulator sandwich structure.
... Subsequently, the phenomenon was generalized to three-dimensional systems with theoretical predictions in Refs. [22][23][24][25][26][27], followed by an experimental confirmation in Ref. [28]. ...
... It is important to bear in mind that the effective equations emerging from the addition of the term L can describe diverse physical phenomena according to the different choices of (t, x). For instance: real and arbitrary yields the electromagnetic response of general magnetoelectrics, 0, π describes TIs [25], ∈ C gives the electrodynamics of metamaterials [39,40] and (x, t) 2b · x − 2b 0 t provides the response of Weyl semimetals [39,41,42]. It is interesting to observe that in high energy physics the term L describes the interaction with the electromagnetic field of an hypothetical dynamical axionic field (t, x) [27,43], which remains good a candidate for the particle that constitute dark matter [44]. ...
Motivated by the recent interest aroused by non-dynamical axionic electrodynamics in the context of topological insulators and Weyl semimetals, we discuss a simple model of the magnetoelectric effect in terms of a $$\theta$$ θ -scalar field that interacts through a delta-like potential located at a planar interface. Thus, in the bulk regions the field is constructed by standard free waves with the absence of evanescent components. These waves have to be combined into linear superposition to account for the boundary conditions at the interface in order to yield the corresponding normal modes. Our aim is twofold: first we quantize the $$\theta$$ θ -scalar field using the normal modes in the canonical approach and then we look for applications emphasizing the effect of momentum non-conservation due to the presence of the interface. To this end, we calculate the decay of a standard scalar particle into two $$\theta$$ θ -scalar particles showing the opening of new decay channels. As a second application, we deal with the two-body scattering of standard charged scalar particles mediated by a $$\theta$$ θ -scalar particle, focusing on the momentum non-conserving contribution of the scattering amplitude $${{\mathcal {M}}}^{NC}$$ M NC . We define a generalization of the usual cross section in order to quantify the emergence of these events. We also study the allowed kinematical region for momentum non-conservation as well as the position of the poles of the amplitude $${{\mathcal {M}}}^{NC}$$ M NC . Finally, the ratio of the magnitudes between $${{\mathcal {M}}}^{NC}$$ M NC and the momentum conserving amplitude is discussed in the appropriate region of momentum space.
... These effects arise from an additional term in the Lagrangian, namely the Chern-Simons term, which modifies the action of the electromagnetic field at low frequencies [5][6][7][8]. In 3D strong topological insulators, a modest magnetic field that induces a gap in the surface states leads to Faraday and Kerr effects similar to those observed in 2D Chern insulators [4,9,10]. Here, these effects stem from an axion term, coupling the electric and magnetic fields [9,11,12]. ...
... In 3D strong topological insulators, a modest magnetic field that induces a gap in the surface states leads to Faraday and Kerr effects similar to those observed in 2D Chern insulators [4,9,10]. Here, these effects stem from an axion term, coupling the electric and magnetic fields [9,11,12]. While this modification has little impact on the bulk of 3D topological insulators at low energies, it produces observable consequences on their surfaces. ...
Extensive research has explored the optical properties of topological insulating materials, driven by their inherent stability and potential applications. In this study, we unveil a novel functionality of three-dimensional integer quantum Hall (3D IQH) states as broad-band filters for circularly polarized light, particularly effective in the terahertz (THz) frequency range under realistic system parameters. We also investigate the impact of practical imperfections, demonstrating the resilience of this filtering effect. Our findings reveal that this phenomenon is independent of the microscopic origin of the 3D IQH state, prompting discussions on its feasibility across diverse candidate materials. These results contribute to our understanding of fundamental optical properties and hold promise for practical applications in optical technologies.
... where σ 0,1,2,3 (τ 0,1,2,3 ) are the 2 × 2 identity matrix and three Pauli matrices respectively, operating on the spin (orbital) indices. This is the celebrated Bernevig-Hughes-Zhang model [43,44] Again, details of the code for defining equation (7) as a PyBinding instance are available in the tutorial on the accompanying webpage [27]. The results in figure 2 demonstrate the resulting fully connected WCC spectra indicating a non-trivial Z 2 index. ...
... Accounting for defects is critical to the investigation of any realistic experimental setup but computing topological invariants in the presence of defects such as vacancies can be exceedingly challenging. In this example, we consider the BHZ model of a two-dimensional quantum spin-Hall insulator with a non-trivial Z 2 index [43,44]. We will create a supercell of 21 × 21 unit cells. ...
Multiple software packages currently exist for the computation of bulk topological invariants in both idealized tight-binding models and realistic Wannier tight-binding models derived from density functional theory. Currently, only one package is capable of computing nested Wilson loops and spin-resolved Wilson loops. These state-of-the-art techniques are vital for accurate analysis of band topology. In this paper we introduce BerryEasy, a python package harnessing the speed of graphical processing units to allow for efficient topological analysis of supercells in the presence of disorder and impurities. Moreover, the BerryEasy package has built-in functionality to accommodate use of realistic many-band tight-binding models derived from first-principles.
... Although the internal magnetic structure does not manifest as a global magnetization, it can profoundly affect many other macroscopic properties, leading to novel physics: In strongly correlated systems, the anti-parallel spin structure promotes virtual hopping, making AFM a favorable ground state in un-doped Mott insulators [1]. In topological physics, the internal AFM spin structures can lead to novel topological phases such as the AFM topological insulator [2], AFM Dirac semimetals [3], and the Axion insulator [4,5]. The absence of net magnetization and the fast dynamics make them the ideal candidates for next-generation spintronic platforms, which motivates the fast-developing AFM spintronics [6]. ...
In a PN junction, the separation between positive and negative charges leads to diode transport. In the past few years, the intrinsic diode transport in noncentrosymmetric polar conductors has attracted great interest, because it suggests novel nonlinear applications and provides a symmetry-sensitive probe of Fermi surface. Recently, such studies have been extended to noncentrosymmetric superconductors, realizing the superconducting diode effect. Here, we show that, even in a centrosymmetric crystal without directional charge separation, the spins of an antiferromagnet (AFM) can generate a spatial directionality, leading to an AFM diode effect. We observe large second-harmonic transport in a nonlinear electronic device enabled by the compensated AFM state of even-layered MnBi2Te4. We also report a novel electrical sum-frequency generation (SFG), which has been rarely explored in contrast to the well-known optical SFG in wide-gap insulators. We demonstrate that the AFM enables an in-plane field-effect transistor and harvesting of wireless electromagnetic energy. The electrical SFG establishes a powerful method to study nonlinear electronics built by quantum materials. The AFM diode effect paves the way for potential device concepts including AFM logic circuits, self-powered AFM spintronics, and other applications that potentially bridge nonlinear electronics with AFM spintronics.
... TIs are characterized by the boundary state showing linearly dispersive energy bands with a band crossing at the Dirac point [1,2]. Symmetries that guarantee such a Diracband crossing play a key role in realizing various topological quantum phenomena [3][4][5]. ...
Antiferromagnetic topological insulators (AF TIs) are predicted to exhibit exotic physical properties such as gigantic optical and topological magnetoelectric responses. While a key to achieving such phenomena relies on how to break the symmetry protecting the Dirac-cone surface state (SS) and acquire the mass of Dirac fermions, the mechanism has yet to be clarified. To address this issue, we carried out micro-focused angle-resolved photoemission spectroscopy for GdBi hosting the type-II AF order, and uncovered the stripe-type 2$\times$1 reconstruction of the Fermi surface associated with the AF band folding. Intriguingly, in contrast to NdBi with the type-I AF order displaying the surface-selective Dirac-fermion mass, GdBi shows massless behavior irrespective of AF domains due to the robust topological protection. These results strongly suggest a crucial role of the ThetaTD (time-reversal and translational) symmetry to create the Dirac-fermion mass in AF TIs.
... Consequently, QAHI is distinguished by a quantized Hall effect, even in the absence of external magnetic fields. The existence of QAHI states has been theoretically predicted in a range of materials [5][6][7] and experimentally verified in Cr-doped topological insulator films for the first time [8]. Subsequently, QAHI has been experimentally demonstrated in various materials, including Cr-doped (Bi, Sb) 2 Te 3 films [9][10][11][12][13][14][15][16][17], V-doped (Bi, Sb) 2 Te 3 films [10,[18][19][20][21][22], MnBi 2 Te 4 [23][24][25], and recently, moiré superlattice systems [26][27][28]. ...
The intriguing interplay between topology and superconductivity has attracted significant attention, given its potential for realizing topological superconductivity. In the quantum anomalous Hall insulators (QAHIs)-based junction, the supercurrents are carried by the chiral edge states, characterized by a 2 Φ 0 magnetic flux periodicity ( Φ 0 = h / 2 e is the flux quantum, h the Planck constant, and e the electron charge). However, experimental observations indicate the presence of bulk carriers in QAHI samples due to magnetic dopants. In this study, we reveal a systematic transition from edge-state to bulk-state dominant supercurrents as the chemical potential varies from the bulk gap to the conduction band. This results in an evolution from a 2 Φ 0 -periodic oscillation pattern to an asymmetric Fraunhofer pattern. Furthermore, a Fraunhoher-like pattern emerges due to the coexistence of chiral edge states and bulk states caused by magnetic domains, even when the chemical potential resides within the gap. These findings not only advance the theoretical understanding but also pave the way for the experimental discovery of the chiral Josephson effect based on QAHI doped with magnetic impurities.
Published by the American Physical Society 2024
... Therefore, Tellegen materials offer the promise of nonreciprocal wave propagation, with practical applications such as electromagnetic isolators and nonreciprocal twist polarizers 14,15 . In addition, the Tellegen response is closely associated with axion electrodynamics, which has drawn great attention due to its connections with topological insulators, high-energy physics, and dark matters [16][17][18][19] . The electrodynamic equations of a media with a pure Tellegen response have the same form as those of the axion media [20][21][22] , making it an exciting opportunity to explore axion-related physics with Tellegen materials. ...
Tellegen medium has long been a topic of debate, with its existence being contested over several decades. It was first proposed by Tellegen in 1948 and is characterized by a real-valued cross coupling between electric and magnetic responses, distinguishing it from the well-known chiral medium that has imaginary coupling coefficients. Significantly, Tellegen responses are closely linked to axion dynamics, an extensively studied subject in condensed matter physics. Here, we report the realization of Tellegen metamaterials in the microwave region through a judicious combination of subwavelength metallic resonators, gyromagnetic materials, and permanent magnet discs. We observe the key signature of the Tellegen response, i.e. a Kerr rotation for reflected wave, while the polarization remains the same in the transmission direction. The retrieved effective Tellegen parameter is several orders of magnitude greater than that of natural materials. Our work opens door to a variety of nonreciprocal photonic devices and may provide a platform for studying axion physics.
... The numerical results presented in the paper were obtained using a standard theoretical model of a Chern insulator introduced in Ref. 13 . We consider a singleparticle Hamiltonian on a square lattice with dimensions N x , N y , in presence of a parabolic confining potential and on-site disorder. ...
The discovery of the quantum Hall effect founded the field of topological condensed matter physics. Its amazingly accurate quantisation of the Hall conductance, now enshrined in quantum metrology, is topologically protected: it is stable against any reasonable perturbation. Conversely, topological protection thus implies a form of censorship, as it completely hides any local information from the observer. The spatial distribution of the current in the sample is such a piece of information, which however has now become accessible thanks to spectacular experimental advances. It is an old question whether an original, and intuitively compelling, picture of the current flowing in a narrow channel along the sample edge is the physically correct one. Motivated by recent experiments $\textit{locally}$ imaging the quantized current flow in a Chern insulating (Bi, Sb)$_2$Te$_3$ heterostructure, [Rosen et al., PRL 129, 246602 (2022); Ferguson et.al, Nat. Mater. 22, 1100-1105 (2023)], we theoretically demonstrate the possibility of a broad `edge state' meandering away from the sample boundary deep into the sample bulk. Further, we show that varying experimental parameters permits continuously tuning between narrow edge states and meandering channels all the way to incompressible bulk transport. This accounts for various features observed in, and differing between, experiments. Overall, this underscores the robustness of topological condensed matter physics, but it also unveils a phenomenological richness hidden by topological censorship -- much of which we believe remains to be discovered.
... Introduction. -Topological Insulators (TIs) are materials that behave as gapped insulators in bulk whereas also hosting metallic (gapless) topological helical states localized at their edges in 2D TIs [1][2][3][4][5][6][7][8][9][10] or surfaces in 3D TIs [11][12][13][14][15][16]. For that reason, attention to topological materials has been mainly focused on edge-and surfacelike phenomena. ...
We investigate the Shubnikov-de Haas (SdH) magneto-oscillations in the resistivity of two-dimensional topological insulators (TIs). Within the Bernevig-Hughes-Zhang (BHZ) model for TIs in the presence of a quantizing magnetic field, we obtain analytical expressions for the SdH oscillations by combining a semiclassical approach for the resistivity and a trace formula for the density of states. We show that when the non-trivial topology is produced by inverted bands with ''Mexican-hat'' shape, SdH oscillations show an anomalous beating pattern that is {\it solely} due to the non-trivial topology of the system. These beatings are robust against, and distinct from beatings originating from spin-orbit interactions. This provides a direct way to experimentally probe the non-trivial topology of 2D TIs entirely from a bulk measurement. Furthermore, the Fourier transform of the SdH oscillations as a function of the Fermi energy and quantum capacitance models allows for extracting both the topological gap and gap at zero momentum.
... An axion insulator features (1) a quantized, bulk magnetoelectric coupling coefficient of π (refs. [15][16][17] that is challenging to access experimentally and (2) a gapless, time-reversal symmetry protected topological surface state 12 that is readily accessible Article https://doi.org/10.1038/s41567-024-02469-1 few-millimetre-long needle shape with an aspect ratio of approximately 10, reflecting its one-dimensional nature (see Fig. 1c, inset and Methods for details on the crystal growth). ...
Charge density waves appear in numerous condensed matter platforms ranging from high-temperature superconductors to quantum Hall systems. Despite such ubiquity, there has been a lack of direct experimental study of boundary states that can uniquely stem from the charge order. Here we directly visualize the bulk and boundary phenomenology of the charge density wave in a topological material, Ta2Se8I, using scanning tunnelling microscopy. At a monolayer step edge, we demonstrate the presence of an in-gap boundary mode persisting up to the charge ordering temperature with modulations along the edge that match the charge density wave wavevector along the edge. Furthermore, these results manifesting the presence of an edge state challenge the existing axion insulator interpretation of the charge-ordered phase in this compound.
... The reason why these superfluid stiffness cannot be employed to reflect the topological charges of band touching points in higher dimensions is due to the same reason for which the strong topological properties of a three-dimensional system cannot be captured by a lower-dimensional topological invariant. In general, topological systems can be classified into hierarchies which are related by dimensional extension and reduction [89,90]. For instance, Chern insulators in two and four dimensions are classified by the first and second Chern numbers C 1 and C 2 , and belong to two distinct hierarchies. ...
We bring forward a unified framework for the study of the superfluid stiffness and the quantum capacitance of superconducting platforms exhibiting conventional spin-singlet pairing. We focus on systems which in their normal phase contain topological band touching points or crossings, while in their superconducting regime feature a fully gapped energy spectrum. Our unified description relies on viewing these two types of physical quantities as the charge current and density response coefficients obtained for slow spatiotemporal variations of the superconducting phase. Within our adiabatic formalism, the two coefficients are given in terms of Berry curvatures defined in synthetic spaces. Our paper lays the foundation for the systematic description of topological diagonal superfluid responses induced by singularities dictating the synthetic Berry curvatures. We exemplify our approach for concrete one- and two-dimensional models of superconducting topological (semi)metals. We discuss topological phenomena which arise in the superfluid stiffness of bulk systems and the quantum capacitance of Josephson junctions. We show that both coefficients become proportional to a topological invariant which counts the number of topological touchings or crossings of the normal phase band structure. These topological effects can be equivalently viewed as manifestations of chiral anomaly. Our predictions appear experimentally testable in topological semimetals with proximity-induced pairing, such as in graphene-superconductor hybrids at charge neutrality.
Published by the American Physical Society 2024
... From these spectra, we can accurately derive the Zak phase [33,34] of the SL under varying lattice parameters. Consequently, we can obtain Chern numbers by observing the topological winding of the Zak phase through dimension reduction techniques [35][36][37][38][39]. ...
Quantum simulation offers an analog approach for exploring exotic quantum phenomena using controllable platforms, typically necessitating ultracold temperatures to maintain the quantum coherence. Superradiance lattices (SLs) have been harnessed to simulate coherent topological physics at room temperature, but the thermal motion of atoms remains a notable challenge in accurately measuring the physical quantities. To overcome this obstacle, we invent and validate a velocity scanning tomography technique to discern the responses of atoms with different velocities, allowing cold-atom spectroscopic resolution within room-temperature SLs. By comparing absorption spectra with and without atoms moving at specific velocities, we can derive the Wannier-Stark ladders of the SL across various effective static electric fields, their strengths being proportional to the atomic velocities. We extract the Zak phase of the SL by monitoring the ladder frequency shift as a function of the atomic velocity, effectively demonstrating the topological winding of the energy bands. Our research signifies the feasibility of room-temperature quantum simulation and facilitates their applications in quantum information processing.
... In general, the phason space augments the physical space and supplies additional virtual dimensions [3], hence enabling physical phenomena beyond what can be ordinarily observed in our physical space. In particular, it can enable two and higher dimensional quantum Hall physics without the need of breaking the time-reversal and this has spurred the vigorous experimental progress mentioned above on the investigation of topological un-correlated phases from class A of classification table [51][52][53][54]. ...
Topological phases supported by quasi-periodic spin-chain models and their bulk-boundary principles are investigated by numerical and K-theoretic methods. We show that, for both the un-correlated and correlated phases, the operator algebras that generate the Hamiltonians are non-commutative tori, hence the quasi-periodic chains display physics akin to the quantum Hall effect in two and higher dimensions. The robust topological edge modes are found to be strongly shaped by the interaction and, generically, they have hybrid edge-localized and chain-delocalized structures. Our findings lay the foundations for topological spin pumping using the phason of a quasi-periodic pattern as an adiabatic parameter, where selectively chosen quantized bits of magnetization can be transferred from one edge of the chain to the other.
... Since the theoretical proposal and experimental observation of 2D [1][2][3][4][5][6] and 3D [7][8][9][10][11][12] topological insulators (TIs), these materials have been the subject of substantial research in condensed matter physics and related areas of science [11,[13][14][15][16][17]. Further investigations soon resulted in the discovery of another important class of topological materials named topological crystalline insulators (TCIs) [18][19][20][21][22]. ...
Dual topological insulators (DTIs) are simultaneously protected by time-reversal and crystal symmetries, representing advantageous alternatives to conventional topological insulators. By combining ab initio calculations and the $\mathbf{k}\cdot\mathbf{p}$ approach, here we investigate the electronic band structure of a Na$_2$CdSn tri-atomic layer and derive a low-energy $4\times 4$ effective model consistent with all the symmetries of this material class. We obtain the effective Hamiltonian using the L\"owdin perturbation theory, the folding down technique, and the theory of invariants, and determine its parameters by fitting our analytical dispersion relations to those of ab initio calculations. We then calculate the bulk topological invariants of the system and show that the Na$_2$CdSn tri-atomic layer is a giant-gap (hundreds of meV) quasi-2D DTI characterized by both spin and mirror Chern numbers $-2$. In agreement with the bulk-boundary correspondence theorem, we find that a finite-width strip of Na$_2$CdSn possesses two pairs of counter-propagating helical edge states per interface. We obtain analytical expressions for the edge states energy dispersions and wave functions, which are shown to agree with our numerical calculations. Our work opens a new avenue for further studies of Na$_2$CdSn as a potential DTI candidate with room-temperature applications in areas of technological interest, such as nanoelectronics and spintronics.
... For example, in the presence of time-reversal symmetry, three dimensional insulator typically falls into two categories: one is trivial insulator while the other is topological insulator 5,6 . These divergent categories can be well described within the framework of the Chern-Simons theory, where the Lagrangian incorporates an additional symmetry allowed term L θ = R dtdr 3 αθ=ð4π 2 ÞE Á B with E and B the conventional electric and magnetic fields, α the fine structure constant, and θ the gauge dependent axion term 7 . Because of the 2π periodicity under a gauge transformation 8 , the axion term here is well defined within the region ð 0, 2π ½ Þ. Besides, as the quantity E ⋅ B flips sign under timereversal (T ) operation, the axion field manifests only two distinct values, that is, θ = 0 for normal insulator and θ = π for topological insulator 7 . ...
Axion insulators possess a quantized axion field θ = π protected by combined lattice and time-reversal symmetry, holding great potential for device applications in layertronics and quantum computing. Here, we propose a high-spin axion insulator (HSAI) defined in large spin-s representation, which maintains the same inherent symmetry but possesses a notable axion field θ = (s + 1/2)²π. Such distinct axion field is confirmed independently by the direct calculation of the axion term using hybrid Wannier functions, layer-resolved Chern numbers, as well as the topological magneto-electric effect. We show that the guaranteed gapless quasi-particle excitation is absent at the boundary of the HSAI despite its integer surface Chern number, hinting an unusual quantum anomaly violating the conventional bulk-boundary correspondence. Furthermore, we ascertain that the axion field θ can be precisely tuned through an external magnetic field, enabling the manipulation of bonded transport properties. The HSAI proposed here can be experimentally verified in ultra-cold atoms by the quantized non-reciprocal conductance or topological magnetoelectric response. Our work enriches the understanding of axion insulators in condensed matter physics, paving the way for future device applications.
... The discovery of topological insulators and superconductors has enlarged the notion of topological phases that owe their properties to symmetries [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. These topological phases are dubbed Symmetry-Protected Topological phases (SPTs), and their key features are bulk energy gaps and edge modes that are robust to symmetrypreserving perturbations. ...
We adapt the fluid description of Fractional Quantum Hall (FQH) states, as seen in (arXiv:2203.06516), to model a system of interacting two-component bosons. This system represents the simplest physical realization of an interacting bosonic Symmetry-Protected Topological (SPT) phase, also known as the integer quantum Hall effect (IQHE) of bosons. In particular, we demonstrate how the fluid dynamical boundary conditions of no-penetration and no-stress at a hard wall naturally give rise to the two counter-propagating boundary modes expected in these SPT phases. Moreover, we identify energy-conserving hydro boundary conditions that can either create a gap in these edge modes or completely isolate the edge states from the bulk, as described in (Physical Review X 14, 011057 (2024)), where they are termed fragile surface states. These fragile surface states are typically absent in K-matrix edge theories and require bulk dynamics to manifest. By leveraging insights from hydrodynamical boundary dynamics, we can further elucidate the intricate surface properties of SPTs beyond the usual topological quantum field theory based approaches.
... We mix the notation i ∈ (1, 2, 3, 4) ↔ (x, y, z, w) in the subscript of Γ i and label G ab = σ a ⊗ σ b through the 2 × 2 identity σ 0 and Pauli matrices σ i hereafter. As one can see that H 0 is the well-known 4D quantum Hall insulator (QHI) characterized by the second Chern number (SCN) C 2 [32], harboring |C 2 | Weyl cones at the origin of the 3D boundary BZ [33]. Under OBCs along the x (w)-direction with a chain length L, introducing a non-zero b xw term in H 0 leads to the shifted Weyl cones along the k w (k x )-axis in the boundary BZ, resulting in the second-order chiral Fermi-arc hinge states when considering OBCs at x and y directions. ...
We report the discovery of several classes of novel topological insulators (TIs) with hybrid-order boundary states generated from the first-order TIs with additional crystalline symmetries. Unlike the current studies on hybrid-order TIs where different-order topology arises from merging different-order TIs in various energy, these novel TIs exhibit a remarkable coexsitence of first-order gapless modes and higher-order Fermi-arc states, behaving as a hybrid between the first-order TIs and higher-order topological semimetals within a single bulk gap. Our findings establish a profound connection between these novel d-dimensional (dD) TIs and (d − 1)D higher-order TIs (HOTIs), which can be understood as a result of stacking (d − 1)D HOTIs to dD with d = 3, 4, revealing unconventional topological phase transitions by closing the gap in certain first-order boundaries rather than the bulk. The bulk-boundary correspondence between these higher-order Fermi-arcs and bulk topological invariants acossiated with additional crystallline symmetries is also demonstrated. We then discuss the conventional topological phase transitions from these novel TIs to nodal-line/nodal-surface semimetal phases, where the gapless phases host new kinds of topological responses. Meawhile, we present the corresponding topological semimetal phases by stacking these unique TIs. Finally, we discuss potential ways to realize these novel phases in synthetic and real materials, with a particular focus on the feasible implementation in optical lattices using ultracold atoms. Introduction.-Topological insulators (TIs) and topo-logical semimetals (TSMs) have emerged as one of the most active fields of modern physics over the past two decades [1-3], attracting significant attention from researchers in condensed matter physics and artificial systems[4-10]. One significant property of topologi-cal phases is the presence of bulk-boundary correspondence , which guarantees the existence of gapless first-order boundary modes that are associated with certain bulk topological invariant. These first-order topological materials have been classifed within the framework provided by the real K-theory [11] and Altland-Zirnbauer (AZ) classes [12] for both gapped [13, 14] and gapless [15-17] systems, based on their fundamental symmetries including time-reversal T , charge-conjugation C, and chi-ral symmetry S.
... 1See also related recent studies [5][6][7][8][9][10][11][12][13][14][15][16] 2We find Refs. [17][18][19][20] on the microscopic approach in condensed matter physics. 3The curved domain-wall fermion is also studied in Refs. ...
... Topological insulators have aroused great interest in the scientific community in the past decade [1,2]. Band inversion caused by strong spin-orbit coupling (SOC) in the three-dimensional (3D) TIs induce gapless surface states (SSs) composed of spin helical Dirac fermions [3][4][5][6][7], which are predicted to host a wide range of topological quantum phenomenon [8,9]. The topological surface states (TSS) are constructed by the strong SOC effect and are immune to non-magnetic perturbation, because the time-reversal (TR) symmetry protects the topological order [10][11][12]. ...
... The dimension of a physical system is usually limited by its geometric dimension. The recent interest in exploring higher-dimensional physics [1][2][3][4][5][6] leads to the need for higherdimensional physical systems, which however are typically difficult for experimental implementation. To address this issue, researchers brought in synthetic dimensions that can extend the dimension of a physical system [7][8][9]. ...
Synthetic dimensions have drawn intense recent attention in investigating higher-dimensional topological physics and offering additional degrees of freedom for manipulating light. It has been demonstrated that synthetic dimensions can help to concentrate light with different frequencies at different locations. Here, we show that synthetic dimensions can also route light from different incident directions. Our system consists of an interface formed by two different photonic crystals. A synthetic dimension $\xi$ ξ is introduced by shifting the termination position of the photonic crystal on the right-hand side of the interface. We identify a correspondence between $\xi$ ξ and the interface state such that light incident from a specific direction can be collected. Thus, routing incident light from different directions is achieved by designing an interface with a proper distribution of $\xi$ ξ . Traditionally, this goal is achieved with a standard ${4f}$ 4 f optical system using a convex lens, and our approach offers the possibility for such a capability within a few lattice sites of photonic crystals. Such an approach reduces the size of the system, making it easier for integration. Our work provides, to our knowledge, a new direction for routing light with different momentums and possibly contributes to applications such as lidar.
Topological semimetals possess nodal or nodal-line phases where conduction and valence bands touch at points or lines in momentum space, respectively. Such band touching is symmetry protected and gives rise to exotic and interesting electronic properties. Coupling topological order with magnetism provides a platform for exploring time-reversal (TR) symmetry breaking topological physics, such as axion electrodynamics, inverse spin-galvanic effect, and the quantized anomalous Hall effect. The Weyl semimetal (WSM) requires breaking either TR symmetry or lattice inversion symmetry (I). By doping inversion-symmetry-broken WSM with magnetic dopants, one can expect to create a WSM with both symmetries breaking simultaneously. Here, structural, electrical, and magnetic properties of FexW1−xTe2 (x=0 and 0.011) are reported. It is revealed that, with a small Fe doping concentration (x=0.011), a ferromagnetism is induced at low temperature (<10 K). Scanning tunneling microscopy and spectroscopy measurements in Fe0.011W0.989Te2 further reveal only substitution and no intercalated dopants being observed. The probabilities of the Fe substitutions at the two nonequivalent W sites are quantified with equal probability. The dI/dV point spectra indicates that the Fe substitution in WTe2 manifests itself as electron doping regardless of doping sites. The results clearly reveal the possible coexistence of magnetism and Weyl points in the lightly Fe doped WTe2 at low temperature. This provides an ideal system for further study on the interplay between the topological Weyl points and the TR symmetry breaking.
We establish a connection between the electromagnetic Hall response and band topological invariants in hyperbolic Chern insulators by deriving a hyperbolic analog of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula. By generalizing the Kubo formula to hyperbolic lattices, we show that the Hall conductivity is quantized to $-e^2C_{ij}/h$, where $C_{ij}$ is the first Chern number. Through a flux-threading argument, we provide an interpretation of the Chern number as a topological invariant in hyperbolic band theory. We demonstrate that, although it receives contributions from both Abelian and non-Abelian Bloch states, the Chern number can be calculated solely from Abelian states, resulting in a tremendous simplification of the topological band theory. Finally, we verify our results numerically by computing various Chern numbers in the hyperbolic Haldane model.
In a recent study, we proposed an axion-electrodynamics model that consistently incorporates a lepton Dirac field into the gauge-invariant Lagrangian of a closed physical system. Our investigation delved toward potential applications of the model, with a focus on its implications in the realm of Dark Matter axions interacting with leptons in a nonlinear electrodynamics background. In the present work, we introduce an extended axion-electrodynamics model wherein the Bianchi identities are modified by the axion field. This leads to a modification of the energy conservation law for the fields: the Poynting theorem in a source-free region, in which the axion field is involved. By implementing a quantization scheme, our model can offer a novel approach for addressing the problem of axion production/conversion in the presence of electromagnetic and Dirac fields.
In the context of θ electrodynamics we find transverse electromagnetic wave solutions forbidden in Maxwell electrodynamics. Our results attest to different signatures of the topological magnetoelectric effect in topological insulators, resulting from a polarization rotation of an external electromagnetic field. Unlike Faraday and Kerr rotations, the effect does not rely on a longitudinal magnetic field, the reflected field, or birefringence. The rotation occurs due to transversal discontinuities of the topological magnetoelectric parameter in cylindrical geometries. The dispersion relation is linear, and birefringence is absent. Exact transversality allows electromagnetic waves to propagate in an optical fiber without successive total internal reflections, diminishing losses, and regardless of acute bends of the fiber. These results may open other possibilities in optics and photonics by utilizing topological insulators to manipulate light.
Материалы MnBi 2 Te 4 , Mn(Bi,Sb) 2 Te 4 и MnBi 2 Te 4 (Bi 2 Te 3 ) m (где m ≥ 1) относятся к классу магнитных топологических изоляторов. Для успешного применения данных материалов в устройствах наноэлектроники необходимо всестороннее изучение их электронной структуры и магнитных свойств в зависимости от соотношения атомов Bi/Sb и количества ( m ) блоков Bi 2 Te 3 . Изучались магнитные свойства поверхности соединений MnBi 2 Te 4 , MnBi 4 Te 7 и Mn(Bi 1– x Sb x ) 2 Te 4 (где x = 0.43, 0.32) при помощи магнитооптического эффекта Керра. Показано, что температуры магнитных переходов на поверхности и в объеме MnBi 4 Te 7 и Mn(Bi,Sb) 2 Te 4 существенно различаются.
In topological insulators, massive surface states resulting from local symmetry breaking were thought to exhibit a half-quantized Hall conductance, obtained from the low-energy effective model in an infinite Brillouin zone. In a lattice model, the surface band is composed of a combination of surface states and bulk states. The massive surface states alone may not be enough to support an exact one-half quantized surface Hall conductance in a finite Brillouin zone and the whole surface band always gives an integer quantized Hall conductance as enforced by the TKNN theorem. To explore this, we investigate the band structures of a lattice model describing the magnetic topological insulator film that supports the axion insulator, Chern insulator, and semi-magnetic topological insulator phases. We reveal that the gapped and gapless surface bands in the three phases are characterized by an integer-quantized Hall conductance and a half-quantized Hall conductance, respectively. We propose an effective model to describe the three phases and show that the low-energy dispersion of the surface bands inherits from the surface Dirac fermions. The gapped surface band manifests a nearly half-quantized Hall conductance at low energy near the center of Brillouin zone, but is compensated by another nearly half-quantized Hall conductance at high energy near the boundary of Brillouin zone because a single band can only have an integer-quantized Hall conductance. The gapless band hosts a zero Hall conductance at low energy but is compensated by another half-quantized Hall conductance at high energy, and thus the half-quantized Hall conductance can only originate from the gapless band. Moreover, we calculate the layer-resolved Hall conductance of the system. The conclusion suggests that the individual gapped surface band alone does not support the half-quantized surface Hall effect in a lattice model.
Non-Hermiticity leads to distinctive topological phenomena absent in Hermitian systems. However, connection between such intrinsic non-Hermitian topology and Hermitian topology has remained largely elusive. Here, considering the bulk and boundary as an environment and system, we demonstrate that anomalous boundary states in Hermitian topological insulators exhibit non-Hermitian topology. We study the self-energy capturing the particle exchange between the bulk and boundary, and demonstrate that it detects Hermitian topology in the bulk and induces non-Hermitian topology at the boundary. As an illustrative example, we show the non-Hermitian topology and concomitant skin effect inherently embedded within chiral edge states of Chern insulators. We also find the emergence of hinge states within effective non-Hermitian Hamiltonians at surfaces of three-dimensional topological insulators. Furthermore, we comprehensively classify our correspondence across all the tenfold symmetry classes of topological insulators and superconductors. Our work uncovers a hidden connection between Hermitian and non-Hermitian topology, and provides an approach to identifying non-Hermitian topology in quantum matter.
The polarizatison conversion and the Goos-Hänchen (GH) shifts of the reflected electromagnetic wave for the multilayer structure made of topological insulator (TI) layers with finite surface energy gap are investigated. The transfer matrix formalism is adopted to analyze the reflection of electromagnetic wave through the multilayer structure, and the influences of surface energy gap, thickness and number of the TI layers are discussed. We find that maximum polarization conversion rate can be obtained with appropriate surface energy gap of TI, and within a certain range of finite energy gap, the polarization conversion effect is stronger than that for the case under the infinite surface energy gap limit. Greater polarization conversion rate for TI with small surface energy gap can be found than that for TI with larger energy gap in some range of layer numbers. At large incident angles the GH shifts vary considerably with the layer number for TI with relatively larger energy gap. Result of the combined influence of surface energy gap and layer number shows that, there exists both the positive and negative enhancement peaks of the GH shifts, and for smaller energy gap, fewer TI layers are required to obtain the transition between positive and negative GH shifts.
Artificial media provide a unique playground to test fundamental theories, allowing one to probe the laws of electromagnetism in the presence of hypothetical axions. While some materials are known to realize this physics, here we propose the nonlocal extension of axion electrodynamics. Compared to the usual axion case, the suggested metamaterial features similar optical properties including Kerr and Faraday rotation. However, the external sources in this structure do not induce dyon charges, eliminating the well-celebrated Witten effect. We put forward the design of such a nonreciprocal nonlocal metamaterial and discuss its potential applications.
We study the effective low-energy fermionic theory of the Kondo-Kitaev model to leading order in the Kondo coupling. Our main goal is to understand the nature of the superconducting instability induced in the proximate metal due to its coupling to spin fluctuations of the spin liquid. The special combination of the low-energy modes of a graphene-like metal and the form of the interaction induced by the Majorana excitations of the spin liquid furnish chiral superconducting order with px+ipy symmetry. Computing its response to a U(1) gauge field moreover shows that this superconducting state is topologically nontrivial, characterized by a first Chern number of ±2.
Quantum anomalous Hall (QAH) insulators exhibit chiral edge channels characterized by vanishing longitudinal conductance and quantized Hall conductance of Ce2/h, wherein the Chern number C is an integer equal to the number of the parallel chiral edge channels. These chiral edge channels conduct dissipationless transport in QAH insulators, making them pivotal for applications in low-consumption electronics and topological quantum computing. While the QAH effect with multiple chiral edge channels (i.e., C > 1) has been demonstrated in multilayers consisting of magnetic topological insulators and normal insulators, the channel number remains fixed for a given sample. Here, we unveil the tunability of the number of chiral edge channels within a single QAH insulator device. By tuning the magnetization of individual layers within the multilayer system, Chern insulating states with different Chern numbers are unveiled. The tunable Chern number was corroborated by our theoretical calculations. Furthermore, we conducted layer-dependent calculations to elucidate the contribution of the Chern number from different layers in the multilayer. Our findings demonstrate an extra degree of freedom in manipulating the chiral edge channels in QAH insulators. This tunability offers an extra dimension for the implementation of the QAH-based multichannel dissipationless transport.
In this article, we explore conditions of continuous emergent symmetries in gapless states, either as topological quantum critical points (TQCPs) or a stable phase with protecting symmetries and connections to smooth deformations of the gapped states around. For a wide class of gapless states that can be associated with fully isolated scale-invariant fixed points, we illustrate that there shall be emergent continuous symmetries that are directly related to smooth deformations of gapped states with symmetries lower than the protecting ones Gp. A short-distance invariance of gapped states under deformations can descend to be an emergent continuous symmetry when approaching the gapless limit. Around one TQCP with Gp=Z2T symmetry, we construct these deformations explicitly and show emergence of symmetries via fully gapped quaternion superconducting states that break the protecting symmetry Gp=Z2T. For a 3D TQCP in DIII classes with Gp=Z2T, UEM=U(1), and Nf=12 fermions but without charge U(1) symmetry, we further explicitly construct a corresponding boundary representation based on a 4D topological state with lattice symmetry H=Z2T⋉U(1) and Nf=1 fermions. The lattice model is shown to be dual to a conventional 4D topological insulator. Although emergent continuous symmetries appear to be robust at weakly interacting TQCPs, we further show the breakdown of such one-to-one correspondence between deformations of gapped states and emergent continuous symmetries when gapless states become strongly interacting. In a strongly interacting limit, gapless states can be represented by a smooth manifold of conformal-field-theory fixed points rather than a fully isolated one. A smooth manifold of strong coupling fixed points hinders emergence of a continuous emergent symmetry in the strongly interacting gapless limit, as deformations no longer leave a gapless state or a TQCP invariant, unlike in the more conventional weakly interacting case. This typically reduces continuous emergent symmetries to a discrete symmetry originating from duality transformations under the protection symmetry Gp.
Nontrivial bulk topological invariants of quantum materials can leave their signatures on charge, thermal, and spin transports. In two dimensions, their imprints can be experimentally measured from well-developed multiterminal Hall bar arrangements. Here, we numerically compute the low temperature (T) thermal (κxy) and zero temperature spin (σxysp) Hall conductivities, and longitudinal thermal conductance (Gxxth) of various prominent two-dimensional fully gapped topological superconductors, belonging to distinct Altland-Zirnbauer symmetry classes, namely p+ip (class D), d+id (class C), and p±ip (class DIII) paired states, in mesoscopic six-terminal Hall bar setups from the scattering matrix formalism using kwant. In both clean and weak disorder limits, the time-reversal symmetry breaking p+ip and d+id pairings show half-quantized and quantized κxy [in units of κ0=π2kB2T/(3h)], respectively, while the latter one in addition accommodates a quantized σxysp [in units of σ0sp=ℏ/(8π)]. By contrast, the time-reversal invariant p±ip pairing only displays a quantized Gxxth at low T up to a moderate strength of disorder. In the strong disorder regime, all these topological responses (κxy, σxysp, and Gxxth) vanish. Possible material platforms hosting such paired states and manifesting these robust topological thermal and spin responses are discussed.
Topological insulators are described by topological invariants that can be computed by integrals over momentum space but also as traces over local, real-space topological markers. These markers are useful to detect topological insulating phases in disordered crystals, quasicrystals, and amorphous systems. Among these markers, only the spectral localizer operator can be used to distinguish topological metals that show zero-modes of the localizer spectrum. However, it remains unclear whether trivial metals also display zero-modes and if their localizer spectrum is distinguishable from topological ones. Here we show that trivial metals generically display zero-modes of the localizer spectrum. The localizer zero-modes are determined by the zero-mode solutions of a Dirac equation with a varying mass parameter. We use this observation, valid in any dimension, to determine the difference between the localizer spectrum of trivial and topological metals and study the spectrum of the localizer for fractional quantum Hall edges. Because the localizer is a local, real-space operator, it may be used as a tool to differentiate between noncrystalline topological and trivial metals, and characterize strongly correlated systems, for which local topological markers are scarce.
Bi2−xTe2.4Se0.6 single crystals show gapless topological surface states, and doping (x) with vanadium allows to shift the chemical potential in the bulk bandgap. Accordingly, the resistivity, carrier density, and mobility are constant below 10 K, and the magnetoresistance shows weak antilocalization as expected for low-temperature transport properties dominated by gapless surface states of so-called three-dimensional topological insulators. However, the magnetoresistance also shows a hysteresis depending on the sweep rate and the magnetic field direction. Here, we provide evidence that such magnetoresistance hysteresis is enhanced if both three-dimensional bulk states and quasi-two-dimensional topological states contribute to the transport (x = 0 and 0.03), and it is mostly suppressed if the topological states govern transport (x = 0.015). It is proposed that the hysteresis in the magnetoresistance results from different spin-dependent scattering rates of the topological surface and bulk states. Generally, this observation is of relevance to the studies of topologically insulating materials in which both topological surface and bulk states exist.
We show that many-body fermionic non-Hermitian systems require two distinct sets of topological invariants to describe the topology of energy bands and quantum states, respectively, with the latter yet to be explored. We identify 10 symmetry classes—determined by particle-hole, linearized time-reversal, and linearized chiral symmetries. Each class has a topological invariant associated with each dimension, dictating the topology of quantum states. These findings pave the way for a deeper understanding of the topological phases of many-body non-Hermitian systems.
We investigate the topological properties of one-dimensional weakly interacting topological insulators using bosonization. To do that we study the topological edge states that emerge at the edges of a model realized by a strong impurity or at the boundary between topologically distinct phases. In the bosonic model, the edge states are manifested as degenerate bosonic kinks at the boundaries. We first illustrate this idea on the example of the interacting Su-Schrieffer-Heeger (SSH) chain. We compute the localization length of the edge states as the width of an edge soliton that occurs in the SSH model in the presence of a strong impurity. Next, we examine models of two capacitively coupled SSH chains that can be either identical or in distinct topological phases. We find that weak Hubbard interaction reduces the ground-state degeneracy in the topological phase of identical chains. We then prove that, similarly to the noninteracting model, the degeneracy of the edge states in the interacting case is protected by chiral symmetry. We then study topological insulators built from two SSH chains with interchain hopping that represent models of different chiral symmetric universality classes. We demonstrate in bosonic language that the topological index of a weakly coupled model is determined by the type of interchain coupling, invariant under one of two possible chiral symmetry operators. Finally, we show that a general one-dimensional model in a phase with topological index ν is equivalent at low energies to a theory of at least ν SSH chains. We illustrate this idea on the example of an SSH model with longer-range hopping.
We examine the impact of nonmagnetic disorder on the electronic states of a multilayer structure comprising layers of both topological and conventional band insulators. Employing the Burkov-Balents model with renormalized tunneling parameters, we generate phase diagrams correlating with disorder, demonstrating that nonmagnetic disorder can induce transitions between distinct topological phases. The subsequent section of our investigation focuses on the scenario where disorder is unevenly distributed across layers, resulting in fluctuations of the interlayer tunneling parameter, termed off-diagonal disorder. Furthermore, we determine the density of states employing the self-consistent single-site diagram technique, expanding the Green function in relation to the interlayer tunneling parameter (locator method). Our findings reveal that off-diagonal disorder engenders delocalized bulk states within the band gap. The emergence of these states may lead to the breakdown of the anomalous quantum Hall effect (AQHE) phase, a phenomenon that has garnered significant attention from researchers in the realm of topological heterostructures. Nonetheless, our results affirm the stability of the Weyl semimetal phase even under substantial off-diagonal disorder.
We give an explanation of the CP conservation of strong interactions which includes the effects of pseudoparticles. We find it is a natural result for any theory where at least one flavor of fermion acquires its mass through a Yukawa coupling to a scalar field which has nonvanishing vacuum expectation value.
In this work we investigate properties of fermions in theSO(5) theory of highTcsuperconductivity. We show that the adiabatic time evolution of aSO(5) superspin vector leads to a non-AbelianSU(2) holonomy of theSO(5) spinor states. Physically, this non-trivial holonomy arises from the non-zero overlap between the SDW and BCS quasiparticle states. While the usual Berry's phase of aSO(3) spinor is described by a Dirac magnetic monopole at the degeneracy point, the non-Abelian holonomy of aSO(5) spinor is described by a Yang monopole at the degeneracy point and is deeply related to the existence of the second Hopf map fromS7toS4. We conclude this work by extending the bosonicSO(5) nonlinearσmodel to include the fermionic states around the gap nodes as 4 component Dirac fermions coupled toSU(2) gauge fields in 2+1 dimensions.
The excitation spectrum of a two-dimensional px+ipy fermionic superfluid, such as a thin film of 3He-A, includes a gapless Majorana-Weyl fermion which is confined to the boundary by Andreev reflection. There is also a persistent ground-state boundary current which provides a droplet containing N particles with angular momentum ħN/2. Both of these boundary effects are associated with bulk Chern-Simons effective actions. We show that the gapless edge mode is required for the gauge invariance of the total effective action, but the same is not true of the boundary current.
There are fundamental relations between three vast areas of physics: particle physics, cosmology, and condensed matter physics. The fundamental links between the first two areas - in other words, between micro- and macro-worlds - have been well established. There is a unified system of laws governing the scales from subatomic particles to the cosmos and this principle is widely exploited in the description of the physics of the early universe. This book aims to establish and define the connection of these two fields with condensed matter physics. According to the modern view, elementary particles (electrons, neutrinos, quarks, etc.) are excitations of a more fundamental medium called the quantum vacuum. This is the new 'aether' of the 21st century. Electromagnetism, gravity, and the fields transferring weak and strong interactions all represent different types of the collective motion of the quantum vacuum. Among the existing condensed matter systems, a quantum liquid called superfluid 3He-A most closely represents the quantum vacuum. Its quasiparticles are very similar to the elementary particles, while the collective modes are analogues of photons and gravitons. The fundamental laws of physics, such as the laws of relativity (Lorentz invariance) and gauge invariance, arise when the temperature of the quantum liquid decreases.
Solitons of a nonlinear field interacting with fermions often acquire a fermionic number or an electric charge if fermions carry a charge. We establish a correspondence between charge and statistics (or spin) of solitons showing how the same mechanism (chiral anomaly) gives solitons statistical and rotational properties of fermions. These properties are encoded in a geometrical phase, i.e., an imaginary part of a euclidian action for a nonlinear σ-model. In the most interesting cases the geometrical phase is non-perturbative and has a form of an integer-valued theta-term.
Strong coupling of the internal and external degrees of freedom of a cold atom to each other and to the spatially periodic
field of the standing light wave in a high-finesse cavity is responsible for the dynamic instability of the atomic center-of-mass
motion. Due to a weak interaction of the internal nonlinear resonances in the standard model of cavity QED, a stochastic layer
appears, whose width in the semiclassical approximation is estimated in terms of the main parameters of the system: atomic
recoil frequency, mean number of excitations, and detuning from the resonance. As a result, the atomic motion in the absolutely
regular potential has the fractal character, with long Lévy flights alternating with small chaotic oscillations in potential
wells.
I show that a lattice theory of massive interacting fermions in 2n+1 dimensions may be used to simulate the behavior of massless chiral fermions in 2n dimensions if the fermion mass has a step function shape in the extra dimension. The massless states arise as zero modes bound to the mass defect, and all doublers can be given large gauge invariant masses. The manner in which the anomalies are realized is transparent: apparent chiral anomalies in the 2n-dimensional subspace correspond to charge flow into the extra dimension.
We study the properties of two dimensional topological spin Hall insulators which arise through spontaneous breakdown of spin symmetry in systems that are spin rotation invariant. Such a phase breaks spin rotation but not time reversal symmetry and has a vector order parameter. Skyrmion configurations in this vector order parameter are shown to have an electric charge that is twice the electron charge. When the spin Hall order is destroyed by condensation of Skyrmions superconductivity results. This may happen either through doping or at fixed filling by tuning interactions to close the Skyrmion gap. In the latter case the superconductor-spin Hall insulator quantum phase transition can be second order even though the two phases break distinct symmetries.
We show that quasiparticle excitations with irrational charge and irrational exchange statistics exist in tight-binding systems described, in the continuum approximation, by the Dirac equation in (2+1)-dimensional space and time. These excitations can be deconfined at zero temperature, but when they are, the charge rerationalizes to the value 1/2 and the exchange statistics to that of "quartons" (half-semions).
Whenever the Fermi level lies in a gap (or mobility gap) the bulk Hall conductance can be expressed in a topologically invariant form showing the quantization explicitly. The new formulation generalizes the earlier result by Thouless, Kohmoto, Nightingale, and den Nijs to the situation where many-body interaction and substrate disorder are also present. When applying to the fractional quantized Hall effect, we draw the conclusion that there must be a symmetry breaking in the many-body ground state. The possibility of writing the fractionally quantized Hall conductance as a topological invariant is also discussed. Journal Article
We prove a general theorem on the relation between the bulk topological quantum number and the edge states in two-dimensional insulators. It is shown that whenever there is a topological order in bulk, characterized by a nonvanishing Chern number, even if it is defined for a nonconserved quantity such as spin in the case of the spin Hall effect, one can always infer the existence of gapless edge states under certain twisted boundary conditions that allow tunneling between edges. This relation is robust against disorder and interactions, and it provides a unified topological classification of both the quantum (charge) Hall effect and the quantum spin Hall effect. In addition, it reconciles the apparent conflict between the stability of bulk topological order and the instability of gapless edge states in systems with open boundaries (a known happening in the spin Hall case). The consequences of time reversal invariance for bulk topological order and edge state dynamics are further studied in the present framework. Journal Article
We propose models of two-dimensional paramagnetic semiconductors where the intrinsic spin Hall effect is exactly quantized in integer units of a topological charge. The model describes a topological insulator in the bulk and a “holographic metal” at the edge, where the number of extended edge states crossing the Fermi level is dictated by (exactly equal to) the bulk topological charge. We also demonstrate the spin Hall effect explicitly in terms of the spin accumulation caused by the adiabatic flux insertion. Journal Article
The quantum Hall effect is usually observed when a two-dimensional electron gas is subjected to an external magnetic field, so that their quantum states form Landau levels. In this work we predict that a new phenomenon, the quantum anomalous Hall effect, can be realized in Hg{1-y}Mn{y}Te quantum wells, without an external magnetic field and the associated Landau levels. This effect arises purely from the spin polarization of the Mn atoms, and the quantized Hall conductance is predicted for a range of quantum well thickness and the concentration of the Mn atoms. This effect enables dissipationless charge current in spintronics devices.
During past decades, concepts about the electrostatics of infinite systems have been a challenge for theoretical physicists. In particular, the question of whether the absolute macroscopic polarization or the difference between the polarizations of two states of an insulating crystal is a well-defined bulk property has remained a controversial one. Recently, King-Smith and Vanderbilt, and Resta have provided an approach in terms of the geometric Berrys phase of electronic orbitals in an independent-particle approximation. Here we extend the derivation of Niu and Thouless for quantized charge transport to the case where the quantum adiabatic evolution is noncyclic, and we show how this polarization difference can be written in terms of a Berrys phase for a system with many-body interactions. We also discuss the origin and magnitude of the ``quantum uncertainty'' that appears when a path-independent gauge is used to compute those geometric quantum phases. This geometric viewpoint not only helps us understand the issues raised above but provides a mathematical method to compute polarizations in a many-body framework.
If a three-dimensional semimetal or doped semiconductor is placed in a sufficiently strong magnetic field, then a change in its transport properties will occur. If the electron-impurity interaction is dominant, then the magnetic field will produce localization of the electron wavefunctions, sometimes described as magnetic freezeout. If the electron-electron interaction is more important, then some type of collective transition may occur. Spin-density waves, charge-density waves, valley-density waves, excitonic insulators, and Wigner crystallization have been proposed to occur under various circumstances. As a generalization to three-dimensions of the integral quantized Hall effect, we show that for electrons in periodic or quasiperiodic potential, when the Fermi level lies in an energy gap, the zero temperature conductivity tensor is given by σij=(e²/2πh)∑kεijkGk, where \vecG is a reciprocal-lattice-vector of the potential. We discuss the effect of impurities and dislocations on this result.
We show that a PbTe-type narrow-gap semiconductor with an antiphase boundary (or domain wall) has currents of abnormal parity and induced fractional charges. A model is introduced which reduces the problem to the physics of a Dirac equation with a soliton in background electric and magnetic fields. We show that this system is a physical realization of the parity anomaly.
We present a theoretical study of soliton formation in long-chain polyenes, including the energy of formation, length, mass, and activation energy for motion. The results provide an explanation of the mobile neutral defect observed in undoped (CH)x. Since the soliton formation energy is less than that needed to create band excitation, solitons play a fundamental role in the charge-transfer doping mechanism.
This paper gives a systematic review of a field theoretical approach to the fractional quantum Hall effect (FQHE) that has been developed in the past few years. We first illustrate some simple physical ideas to motivate such an approach and then present a systematic derivation of the Chern–Simons–Landau–Ginzburg (CSLG) action for the FQHE, starting from the microscopic Hamiltonian. It is demonstrated that all the phenomenological aspects of the FQHE can be derived from the mean field solution and the small fluctuations of the CSLG action. Although this formalism is logically independent of Laughlin's wave function approach, their physical consequences are equivalent. The CSLG theory demonstrates a deep connection between the phenomena of superfluidity and the FQHE, and can provide a simple and direct formalism to address many new macroscopic phenomena of the FQHE.
A quantized Hall plateau of ρxy=3h/e2, accompanied by a minimum in ρxx, was observed at T<5 K in magnetotransport of high-mobility, two-dimensional electrons, when the lowest-energy, spin-polarized Landau level is 1/3 filled. The formation of a Wigner solid or charge-density-wave state with triangular symmetry is suggested as a possible explanation.
A new quantum field-theoretical technique is developed and used to explore the relationship between even-space-time-dimensional axial anomalies and background-field-induced fermion numbers and Euler-Heisenberg effective actions in odd-dimensional space-times.
The Hall conductance of a two-dimensional electron gas has been studied in a uniform magnetic field and a periodic substrate potential U. The Kubo formula is written in a form that makes apparent the quantization when the Fermi energy lies in a gap. Explicit expressions have been obtained for the Hall conductance for both large and small Uℏomegac.
Working within the framework of perturbation theory, we show that the axial-vector vertex in spinor electrodynamics has anomalous properties which disagree with those found by the formal manipulation of field equations. Specifically, because of the presence of closed-loop "triangle diagrams," the divergence of axial-vector current is not the usual expression calculated from the field equations, and the axial-vector current does not satisfy the usual Ward identity. One consequence is that, even after the external-line wave-function renormalizations are made, the axial-vector vertex is still divergent in fourth- (and higher-) order perturbation theory. A corollary is that the radiative corrections to νll elastic scattering in the local current-current theory diverge in fourth (and higher) order. A second consequence is that, in massless electrodynamics, despite the fact that the theory is invariant under γ5 tranformations, the axial-vector current is not conserved. In an Appendix we demonstrate the uniqueness of the triangle diagrams, and discuss a possible connection between our results and the π0→2γ and η→2γ decays. In particular, we argue that as a result of triangle diagrams, the equations expressing partial conservation of axial-vector current (PCAC) for the neutral members of the axial-vector-current octet must be modified in a well-defined manner, which completely alters the PCAC predictions for the π0 and the η two-photon decays.
This letter presents variational ground-state and excited-state wave functions which describe the condensation of a two-dimensional electron gas into a new state of matter.
A method is proposed to calculate quantum numbers on solitons in quantum field theory. The method is checked on previously known examples and, in a special model, by other methods. It is found, for example, that the fermion number on kinks in one dimension or on magnetic monopoles in three dimensions is, in general, a transcendental function of the coupling constant of the theories.
Measurements of the Hall voltage of a two-dimensional electron gas, realized with a silicon metal-oxide-semiconductor field-effect transistor, show that the Hall resistance at particular, experimentally well-defined surface carrier concentrations has fixed values which depend only on the fine-structure constant and speed of light, and is insensitive to the geometry of the device. Preliminary data are reported.
We propose a description of the vacuum in Yang-Mills theory and arrive at a physical interpretation of the pseudoparticle solution and the attendant violation of symmetries. The existence of topologically inequivalent classical gauge fields gives rise to a family of quantum mechanical vacua, parametrized by a CP-nonconserving angle. The requirement of vacuum stability against gauge transformations renders the vacua chirally noninvariant.
We study the structure of soliton-monopole systems when Fermi fields are present. We show that the existence of a nondegenerate, isolated, zero-energy, c-number solution of the Dirac equation implies that the soliton is a degenerate doublet with Fermi number +/- 1/2 . We find such solutions in the theory of Yang-Mills monopoles and dyons.
The topological current in the (2 + 1)-dimensional O(3) nonlinear sigma model is shown to be identical to the fermionic current induced by a Yukawa-type interaction with the classical field of the sigma model. Alternatively, the system can be described in terms of two species of fermions interacting with an abelian gauge field, and hence related to the recently discovered three-dimensional anomaly. The role of fermion zone modes is also discussed. Laboratoire propre du Centre National de la Recherche Scientifique.
The effective gauge field action due to fermions coupled to SU(N) gauge fields in three dimensions is found to change by ±π|n| under a homotopically nontrivial gauge transformation with winding number n. This gauge noninvariance can be eliminated by adding a parity-violating topological term to the action, or by regulating the theory in a way which produces this term automatically in the effective action. The Euler-Heisenberg effective action is calculated in the SU(2) theory and in QED.
It is shown that the quantization of the Hall conductivity of two-dimensional metals which has been observed recently by Klitzing, Dorda, and Pepper and by Tsui and Gossard is a consequence of gauge invariance and the existence of a mobility gap. Edge effects are shown to have no influence on the accuracy of quantization. An estimate of the error based on thermal activation of carriers to the mobility edge is suggested.
The integrated particle current produced by a slow periodic variation of the potential of a Schrödinger equation is evaluated. It is shown that in a finite torus the integral of the current over a period can vary continuously, but in an infinite periodic system with full bands it must have an integer value. This quantization of particle transport is used to classify the energy gaps in a one-dimensional system with competing or incommensurate periods. It is also used to rederive Prange's results for the fractional charge of a soliton.
We show that a non-Abelian gauge theory with Higgs fields exhibits
classical solutions which are both electrically and magnetically
charged. This represents a specific realization of the dyons discussed
some years ago by Schwinger. At the classical level the electric charge
of the dyon does not appear to be quantized. We present some remarks in
this connection.
The effective coupling constant for π0→γγ should vanish for zero pion mass in theories with PCAC and gauge invariance. It does not so vanish in an explicit perturbation
calculation in the σ-model. The resolution of the puzzle is effected by a modification of Pauli-Villars-Gupta regularization
which respects both PCAC and gauge invariance.
La costante di accoppiamento effettiva per π0→γγ si dovrebbe annullare per massa del pione nulla nelle teorie con PCAC e invarianza di gauge. Non si dovrebbe verificare
la stessa cosa in un esplicito calcolo perturbativo nel modello σ. Si risolve il rompicapo modificando la regolarizzazione
di Pauli. Villars e Gupta, che rispetta sia la PCAC che l’invarianza di gauge.
Зффективная константа связи для π0→γγ должна обращаться в нуль при нулевой массе пиона в теориях с РСАС и калибровочной инвариантностыо. Однако, она не обращается
в нуль при точном пертурбационном вычислении в σ модели. Сазрешение зтого затруднения осуществляется лутем видоизменения регуляризации
Паули-Вилларса-Гупта, которая учитывает и РСАС и калибровочную инвариантность.
We show that the mathematical relation between non-abelian anomalies in 2n dimensions, the parity anomaly in 2n+1 dimensions, and the Dirac index density in 2n+2 dimensions can be understood in terms of the physics of fermion zero modes on strings and domain walls. We show that the Dirac equation possesses chiral zero modes in the presence of strings in 2n+2 dimensions (such as occur in axion theories) or domain walls in 2n+1 dimensions. We show that the anomalies due to the chiral zero modes are exactly cancelled by anomalies due to the coupling of axion-like fields to the Dirac index density or by anomalies due to the induced topological mass term.
The finite action Euclidean solutions of gauge theories are shown to indicate the existence of tunneling between topologically distinct vacuum configurations. Diagonalization of the Hamiltonian then leads to a continuum of vacua. The construction and properties of these vacua are analyzed. In non-abelian theories of the strong interactions one finds spontaneous symmetry breaking of axial baryon number without the generation of a Goldstone boson, a mechanism for chiral SU(N) symmetry breaking and a possible source of T violation.
It is shown that in CP non-conserving theories, the electric charge of an 't Hooft-Polyakov magnetic monopole will not ordinarily be integral, or even rational in units of the fundamental charge e. If a non-zero vacuum angle θ is the only mechanism for CP violation, the electric charge of the monopole is exactly calculable and is −eθ/2π, plus an integer. If there are additional CP violating interactions, the monopole charge must be computed as a power series in the coupling constant. These results apply in realistic theories such as SU(5).
We derive an effective topological field theory model of the four dimensional quantum Hall liquid state recently constructed by Zhang and Hu. Using a generalization of the flux attachment transformation, the effective field theory can be formulated as a U(1) Chern–Simons theory over the total configuration space CP3, or as a SU(2) Chern–Simons theory over S4. The new quantum Hall liquid supports various types of topological excitations, including the 0-brane (particles), the 2-brane (membranes), and the 4-brane. There is a topological phase interaction among the membranes which generalizes the concept of fractional statistics.
The unusual electromagnetic properties of axionic domain walls are investigated particular, the origin of the extra unit of electric charge acquired by a magnetic monopole that traverses an axionic domain wall is elucidated.
We present a theoretical study of soliton formation in long-chain polyenes, including the energy of formation, length, mass, and activation energy for motion. The results provide an explanation of the mobile neutral defect observed in undoped ${(\mathrm{CH})}_{x}$. Since the soliton formation energy is less than that needed to create band excitation, solitons play a fundamental role in the charge-transfer doping mechanism.
The quantum spin Hall state is a topologically nontrivial insulator state protected by the time-reversal symmetry. We show that such a state always leads to spin-charge separation in the presence of a pi flux. Our result is generally valid for any interacting system. We present a proposal to experimentally observe the phenomenon of spin-charge separation in the recently discovered quantum spin Hall system.
In this Letter we construct a simple, controllable, two-dimensional model based on a topological band insulator. It has many attractive properties. (1) We obtain spin-charge separated solitons that are associated with dynamic pi fluxes. (2) These solitons obey Bose statistics and their condensation triggers a phase transition from a spin Hall insulator to an easy-plane ferromagnet. (3) It suggests an alternative way to classify the Z2 topological band insulator without resorting to the sample boundary.
The structure of axionic domain walls is investigated using the low-energy effective theory of axions and pions. We derive the spatial dependence of the phases of the Peccei-Quinn scalar field and the QCD quark-antiquark condensates inside an axionic domain wall. Thence an accurate estimate of the wall surface energy density is obtained. The equations of motion for axions, photons, leptons, and baryons in the neighborhood of axionic domain walls are written down and estimates are given for the wall reflection and transmission coefficients of these particles. Finally, we discuss the energy dissipation by axionic domain walls oscillating in the early universe due to the reflection of particles in the primordial soup.
Photons and electrons interacting with axion domain walls are considered in a simple model. A topological mass term for the photon and a parity-violating mass term for the electron arise naturally on axion domain walls. The quantization of topological mass and the interaction between domain walls and magnetic monopoles, along with other features, are discussed.
We consider the change in polarization $\Delta${}P which occurs upon making an adiabatic change in the Kohn-Sham Hamiltonian of the solid. A simple expression for $\Delta${}P is derived in terms of the valence-band wave functions of the initial and final Hamiltonians. We show that physically $\Delta${}P can be interpreted as a displacement of the center of charge of the Wannier functions. The formulation is successfully applied to compute the piezoelectric tensor of GaAs in a first-principles pseudopotential calculation.
The equations of axion electrodynamics are studied. Variations in the axion field can give rise to peculiar distributions of charge and current. These effects provide a simple understanding of the fractional electric charge on dyons and of some recently discovered oddities in the electrodynamics of antiphase boundaries in PbTe. Some speculations regarding the possible occurrence of related phenomena in other solids are presented.
A two-dimensional condensed-matter lattice model is presented which exhibits a nonzero quantization of the Hall conductance sigmaxy in the absence of an external magnetic field. Massless fermions without spectral doubling ccur at critical values of the model parameters, and exhibit the so-called ``parity anomaly'' of (2+1)-dimensional field theories.