The Anomalous Hall Effect and Magnetic Monopoles in Momentum Space

Fang, Zhong; Nagaosa, Naoto; Takahashi, Kei; Asamitsu, A.; Mathieu, Roland; Ogasawara, T.; Yamada, Hiroyuki; Kawasaki, Masashi; Tokura, Y.; Terakura, Kiyoyuki

doi:10.1126/science.1089408

Cited by 998 publications

(958 citation statements)

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“…(25,28) and the MM in momentum space has been of much theoretical interest. [3,4,10,31]. From a more general point of view, physics in noncommutative space-time coordinates has been of great theoretical interest, rather in high-energy physics community, in particular, in the context of string and M theories.…”

Section: Origin Of the Gauge Fieldmentioning

confidence: 99%

“…4 In the following section, we will derive the EOM forx(t) andk(t), following the quantum mechanical motion of the wave packet we have just prepared. In order to make the set of EOM self-contained, however, we need also to take care of the motion of internal pseudospin degrees of freedom spanned by N bands.…”

Section: Construction Of a Wave Packetmentioning

confidence: 99%

“…[22] Recently, the Berry phase in AHE has attracted a renewed attention, revealing its rich topological structures. [3,4,6,7,23,24,25,26] The Berry phase has also been generalized to a non-Abelian case. [27] The equations of motion (EOM) for a wave packet of Bloch functions 1 is instrumental in all the analyses done in the paper.…”

Section: Introductionmentioning

confidence: 99%

“…[1,2] Recently a group of condensed-matter physicists [3,4] embodied the idea of detecting a monopole in the band structure. [5] In crystal momentum space, monopoles appear as a source or a sink of the reciprocal magnetic field [6,7] associated with the geometric phase of Bloch electrons.…”

Section: Introductionmentioning

confidence: 99%

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Noncommutative geometry and non-Abelian Berry phase in the wave-packet dynamics of Bloch electrons

2005

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Motivated by a recent proposal on the possibility of observing a monopole in the band structure, and by an increasing interest on the role of Berry phase in spintronics, we studied the adiabatic motion of a wave packet of Bloch functions, under a perturbation varying slowly and incommensurately to the lattice structure. We show, using only the fundamental principles of quantum mechanics, that the effective wave-packet dynamics is conveniently described by a set of equations of motion (EOM) for a semiclassical particle coupled to a nonabelian gauge field associated with a geometric Berry phase.Our EOM can be viewed as a generalization of the standard Ehrenfest's theorem, and their derivation was asymptotically exact in the framework of linear response theory. Our analysis is entirely based on the concept of local Bloch bands, a good starting point for describing the adiabatic motion of a wave packet. One of the advantages of our approach is that the various types of gauge fields were classified into two categories by their different physical origin: (i) projection onto specific bands, (ii) timedependent local Bloch basis. Using those gauge fields, we write our EOM in a covariant form, whereas the gauge-invariant field strength stems from the noncommutativity of covariant derivatives along different axes of the reciprocal parameter space. On the other hand, the degeneracy of Bloch bands makes the gauge fields nonabelian.For the purpose of applying our wave-packet dynamics to the analyses on transport phenomena in the context of Berry phase engineering, we focused on the Hall-type and polarization currents. Our formulation turned out to be useful for investigating and classifying various types of topological current on the same footing. We highlighted their symmetries, in particular, their behavior under time reversal (T ) and space inversion (I).

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Section: Origin Of the Gauge Fieldmentioning

confidence: 99%

Section: Construction Of a Wave Packetmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Noncommutative geometry and non-Abelian Berry phase in the wave-packet dynamics of Bloch electrons

2005

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“…5a the band crossing occurs at k = k 0 , which gives a diverging emergent magnetic field and acts as a magnetic monopole 68 . Physical consequences of Weyl fermions have been discussed in the context of the anomalous Hall effect 68 and magnon dynamics 69 in the itinerant ferromagnet SrRuO 3 , as shown in Fig. 5b.…”

Section: Berry Phase Engineeringmentioning

confidence: 99%

Emergent functions of quantum materials

2017

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Materials can harbour quantum many-body systems, most typically in the form of strongly correlated electrons in solids, that lead to novel and remarkable functions thanks to emergence-collective behaviours that arise from strong interactions among the elements. These include the Mott transition, high-temperature superconductivity, topological superconductivity, colossal magnetoresistance, giant magnetoelectric e ect, and topological insulators. These phenomena will probably be crucial for developing the next-generation quantum technologies that will meet the urgent technological demands for achieving a sustainable and safe society. Dissipationless electronics using topological currents and quantum spins, energy harvesting such as photovoltaics and thermoelectrics, and secure quantum computing and communication are the three major fields of applications working towards this goal. Here, we review the basic principles and the current status of the emergent phenomena and functions in materials from the viewpoint of strong correlation and topology.E mergence is a concept developed for many-body systems, indicating the properties, phenomena, and functions that never appear in the individual elements but are realized only when a huge number of elements get together 1 . Inside materials, emergent phenomena are often seen due to the interaction of electronic states; with recent developments on electronic states in solids schematically summarized in Fig. 1a. Electrons in solids have several degrees of freedom: charge, spin and orbital, and are characterized by the topological nature determined by the atomic potential on the crystal lattice structure, as schematically shown in Fig. 1b. These five attributes are coupled together and determine the overall responses to stimuli, which appear as materials' electrical, magnetic, optical, thermal, and mechanical properties.These collective electrons demonstrate various macroscopic quantum phenomena, the representative example of which is superconductivity. One of the big challenges in condensed matter physics is to increase the temperature (T c ) at which superconductivity can be observed to above room temperature. However, there are many other macroscopic quantum phenomena that are already observable above room temperature. One is magnetism (by the Bohr-van Leeuwen theorem), and the other is ferroelectricity 2,3 . Remarkably, there are common features of these three macroscopic quantum phenomena: the order parameter, which is of quantum origin, behaves as a classical quantity, and the quantum topology plays a crucial role.The topological nature of the electronic states is the key concept in the most recent developments in understanding quantum materials. The quantum Hall effect, topological insulators, and topological superconductors are all characterized by nontrivial topologies in Hilbert space. The Berry phase, which describes the connection and curvature of the subspace of Hilbert space, plays the central role in the unified principle to describe this topological nature 4 . ...

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The B erry Phase in Magnetism and the Anomalous Hall Effect

Bruno

2007

Handbook of Magnetism and Advanced Magnetic Materials

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The concept of geometrical phase, introduced by Berry in 1984, has proved to be an extremely fruitful unifying concept in quantum mechanics. In this Chapter, an elementary introduction to the Berry phase is presented, with special emphasis to its role in modern magnetism. In particular, the problem of the anomalous Hall effect, which benefited consederably from new light shed by the Berry phase concept, is discussed in detail.

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The Anomalous Hall Effect and Magnetic Monopoles in Momentum Space

Cited by 998 publications

References 20 publications

Noncommutative geometry and non-Abelian Berry phase in the wave-packet dynamics of Bloch electrons

Noncommutative geometry and non-Abelian Berry phase in the wave-packet dynamics of Bloch electrons

Emergent functions of quantum materials

The B erry Phase in Magnetism and the Anomalous Hall Effect

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