A microscopic theory of the quantum Hall effect

Ishikawa, Kenzo; Matsuyama, Toyoki

doi:10.1016/0550-3213(87)90160-x

Cited by 159 publications

(125 citation statements)

References 26 publications

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“…This topological invariant N 2 was originally introduced in 2+1 dimensional relativistic field theory which is responsible for intrinsic quantum Hall conductivity [258,259,260]. The topological invariant is also applied to superfluid 3 He-A film [261,262,263].…”

Section: Topology Of Surface Majorana Fermionsmentioning

confidence: 99%

Symmetry protected topological superfluid³He-B

Mizushima

Tsutsumi

Sato

et al. 2015

J. Phys.: Condens. Matter

156

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Abstract.Owing to the richness of symmetry and well-established knowledge on the bulk superfluidity, the superfluid 3 He has offered a prototypical system to study intertwining of topology and symmetry. This article reviews recent progress in understanding the topological superfluidity of 3 He in a multifaceted manner, including symmetry consideration, the Jackiw-Rebbi's index theorem, and the quasiclassical theory. Special focus is placed on the symmetry protected topological superfuidity of the 3 He-B confined in a slab geometry. The 3 He-B under a magnetic field is separated to two different sub-phases: The symmetry protected topological phase and non-topological phase. The former phase is characterized by the existence of symmetry protected Majorana fermions. The topological phase transition between them is triggered off by the spontaneous breaking of a hidden discrete symmetry. The critical field is quantitatively determined from the microscopic calculation that takes account of magnetic dipole interaction of 3 He nucleus. It is also demonstrated that odd-frequency evenparity Cooper pair amplitudes are emergent in low-lying quasiparticles. The key ingredients, symmetry protected Majorana fermions and odd-frequency pairing, bring an important consequence that the coupling of the surface states to an applied field is prohibited by the hidden discrete symmetry, while the topological phase transition with the spontaneous symmetry breaking is accompanied by anomalous enhancement and anisotropic quantum criticality of surface spin susceptibility. We also illustrate common topological features between topological crystalline superconductors and symmetry protected topological superfluids, taking UPt 3 and Rashba superconductors as examples.

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Section: Topology Of Surface Majorana Fermionsmentioning

confidence: 99%

Symmetry protected topological superfluid³He-B

Mizushima

Tsutsumi

Sato

et al. 2015

J. Phys.: Condens. Matter

156

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“…Note that even in the interacting case, this quantity is proportional to the spin-Hall conductivity. [52][53][54] In Fig. 2(b), the SChN is plotted as a function of interaction strength.…”

Section: Chern Number (Schn)mentioning

confidence: 99%

Topological antiferromagnetic phase in a correlated Bernevig-Hughes-Zhang model

et al. 2013

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Topological properties of antiferromagnetic phases are studied for a correlated topological band insulator by applying the dynamical mean field theory to an extended Bernevig-Hughes-Zhang model including the Hubbard interaction. The calculation of the magnetic moment and the spin Chern number confirms the existence of a non-trivial antiferromagnetic (AF) phase beyond the Hartree-Fock theory. In particular, we uncover the intriguing fact that the topologically non-trivial AF phase is essentially stabilized by correlation effects but not by the Hartree shifts alone. This counterintuitive effect is demonstrated, through a comparison with the Hartree-Fock results, and should apply for generic topological insulators with strong correlations.

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“…In systems with broken T symmetry a finite charge/mass Hall current generally appear. In two spatial dimension D = 2, this situation is often described in terms of ChernSimons gauge field, accounting for the quantized Hall conductance, [35] as well as fractional charge and statistics. [36] Chern-Simons terms also appear in electrically neutral systems.…”

Section: F Kµkν Induces Hall Type Currents: Ahe and Spin Hall Effectmentioning

confidence: 99%

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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A microscopic theory of the quantum Hall effect

Cited by 159 publications

References 26 publications

Symmetry protected topological superfluid³He-B

Symmetry protected topological superfluid³He-B

Topological antiferromagnetic phase in a correlated Bernevig-Hughes-Zhang model

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

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A microscopic theory of the quantum Hall effect

Cited by 159 publications

References 26 publications

Symmetry protected topological superfluid3He-B

Symmetry protected topological superfluid3He-B

Topological antiferromagnetic phase in a correlated Bernevig-Hughes-Zhang model

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

Contact Info

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Symmetry protected topological superfluid³He-B

Symmetry protected topological superfluid³He-B