## Charge Deficiency, Charge Transport and Comparison of Dimensions (1994)

Venue: | COMMUNICATIONS IN MATHEMATICAL PHYSICS |

Citations: | 30 - 0 self |

### BibTeX

@MISC{Avron94chargedeficiency,,

author = {Joseph E. Avron and Ruedi Seiler and Barry Simon},

title = {Charge Deficiency, Charge Transport and Comparison of Dimensions},

year = {1994}

}

### OpenURL

### Abstract

We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.

### Citations

1770 |
Perturbation Theory of Linear Operators
- Kato
- 1966
(Show Context)
Citation Context ...he result follows. (b) Let B⃗a ≡ e i⃗a·⃗x , a ∈ C, be a complex boost. Then: Ba H(A, V )B−a = H(A, V ) +⃗a ·⃗a +⃗a · (−i ⃗ ∇ − ⃗ A). (A.2) This gives an analytic family of type B in the sense of Kato =-=[21]-=- if the form domain is independent of ⃗a. In particular, this is the case if V is form bounded relative to the kinetic energy. By the diamagnetic inequality it is enough to check that V is bounded rel... |

264 |
Trace ideals and their applications
- Simon
- 2005
(Show Context)
Citation Context ...e last term. Since Eǫ and Fǫ converge strongly to zero as ǫ goes to 0, EǫK and FǫK go to zero in trace norm (as can be seen by writing K as a finite rank plus small trace norm), and since a result in =-=[36]-=- says that Tr (GǫKGǫ) is the integral over Gǫ of K(x, x) the result follows by taking the limit using the fact that K(x, x) is L 1 . This proves proposition (3.3). □ Proposition (3.3) could be replace... |

234 |
Field theories of condensed matter systems (Addison-Wesley
- Fradkin
- 1994
(Show Context)
Citation Context ...in the absence of dissipation, often lends itself to geometric interpretation. A good part, but not all, of this research has been motivated by, and applied to, the integer and fractional Hall effect =-=[2,8,11,17,20,26,32,35,34,38,44]-=-. The framework that will concern us here is that of (non-relativistic) quantum mechanics. Within this framework common models of the integer Hall effect are Schrödinger operators associated with non ... |

177 |
Fractional Statistics and Anyon Superconductivity , World Scientific Singapore
- Wilczek
- 1990
(Show Context)
Citation Context ...in the absence of dissipation, often lends itself to geometric interpretation. A good part, but not all, of this research has been motivated by, and applied to, the integer and fractional Hall effect =-=[2,8,11,17,20,26,32,35,34,38,44]-=-. The framework that will concern us here is that of (non-relativistic) quantum mechanics. Within this framework common models of the integer Hall effect are Schrödinger operators associated with non ... |

171 | Non-commutative differential geometry - Connes - 1985 |

168 | Schrödinger Operators - Cycon, Froese, et al. - 1987 |

158 |
Girvin (Eds.), The Quantum Hall effect
- Prange, M
- 1987
(Show Context)
Citation Context ...in the absence of dissipation, often lends itself to geometric interpretation. A good part, but not all, of this research has been motivated by, and applied to, the integer and fractional Hall effect =-=[2,8,11,17,20,26,32,35,34,38,44]-=-. The framework that will concern us here is that of (non-relativistic) quantum mechanics. Within this framework common models of the integer Hall effect are Schrödinger operators associated with non ... |

148 |
Non-commutative differential geometry, Publ
- Connes
- 1986
(Show Context)
Citation Context |

57 | Schrödinger semigroups - Simon - 1982 |

53 |
Group Theory
- Wigner
- 1959
(Show Context)
Citation Context ...d, time reversal invariance says that 3.7 is odd under conjugation., so the index must vanish. To see this, recall that time reversal says that (in the spinless case) the integral kernel of P is real =-=[43]-=-. It follows that the first triple product in 3.7, p(x, y)p(y, z)p(z, x), is even under conjugation. The second triple product of 3.7, ( 1 − u(x) u(y) )( 1 − u(y) u(z) )( 1 − u(z) u(x) It follows that... |

53 | The index of a pair of projections - Avron, Seiler, et al. - 1994 |

52 | The Weyl calculus of pseudo-differential operators - Hörmander - 1979 |

32 |
Adiabatic theorems and applications to the quantum Hall effect
- Avron, Seiler, et al.
- 1987
(Show Context)
Citation Context ... quantum transport has been extended to a large class of quantum mechanical systems, including models of the integer Hall effect [17,24,25, 29,30,31,41], to models with electron-electron interactions =-=[3,23,30]-=- and to other systems that bear only little resemblance to the Integer Hall effect [8,14,29,35,38]. The Index approach has not been as popular, and has not Part of this work was written while the auth... |

28 |
The Quantum Hall Effect, World Scientific
- Stone
- 1992
(Show Context)
Citation Context |

27 |
Asymptotic behavior of eigenfunctions for multiparticle Schrödinger operators
- Combes, Thomas
- 1973
(Show Context)
Citation Context ...e i a·x −i a·y p(x, y)e (A.3) is real analytic in ⃗a uniformly in x and y. In particular, (A.2) says that p(x, y) is exponentially decaying in |x − y|. This is a version of the Combes–Thomas argument =-=[10]-=-. □ Remarks. 1. For potentials V which are perturbations of Landau Hamiltonian, an adaptation of the above method gives decay which is faster than any exponential. 2. It is easy to construct families ... |

20 |
The quantum Hall effect for electrons in a random potential
- Kunz
- 1987
(Show Context)
Citation Context ...2,44] . We shall not address these issues. The Chern number approach to quantum transport has been extended to a large class of quantum mechanical systems, including models of the integer Hall effect =-=[17,24,25, 29,30,31,41]-=-, to models with electron-electron interactions [3,23,30] and to other systems that bear only little resemblance to the Integer Hall effect [8,14,29,35,38]. The Index approach has not been as popular,... |

17 |
Ordinary quantum Hall effect and noncommutative cohomology
- Bellissard
- 1986
(Show Context)
Citation Context ...teracting electrons in the plane, with (constant) magnetic field perpendicular to the plane and random (or periodic) potential. The Hall conductance has been related to a Fredholm Index by Bellissard =-=[5]-=-, and to a Chern number by Thouless, Kohmoto, Nightingale and den-Nijs [40]. The Fractional Hall effect is associated with electron-electron interaction and this goes beyond what we do here. Quantum f... |

15 |
Adiabatic quantum transport in multiply connected systems
- Avron
- 1988
(Show Context)
Citation Context |

15 |
Elementary theory : the incompressible quantum fluid, in The quantum Hall effect
- Laughlin
- 1987
(Show Context)
Citation Context |

12 |
Gvishiani A. Theorems and Problems in Functional Analysis
- Kirillov
- 1982
(Show Context)
Citation Context ...t, QU1QU2Q − QU1U2Q = Q[U1, Q]U −1 1 U1U2Q. This follows from the compactness of [U1, Q]U −1 1 and the fact that all the remaining terms are bounded. By a basic result of stability theory for indices =-=[22]-=- the index is invariant under perturbations by compacts. Furthermore by the product formula for Fredholm indices one gets This proves the proposition. □ Related questions are addressed in [9,12,15]. I... |

11 |
Power-Law Corrections to the Kubo Formula Vanish in Quantum Hall Systems
- Klein, Seiler
- 1990
(Show Context)
Citation Context ... quantum transport has been extended to a large class of quantum mechanical systems, including models of the integer Hall effect [17,24,25, 29,30,31,41], to models with electron-electron interactions =-=[3,23,30]-=- and to other systems that bear only little resemblance to the Integer Hall effect [8,14,29,35,38]. The Index approach has not been as popular, and has not Part of this work was written while the auth... |

10 | Geometric invariants of the quantum Hall effect - Xia - 1988 |

10 | Ground states of the two-dimensional electron - Dubrovin, Novikov - 1980 |

9 |
Topological invariants and the quantization of the Hall conductance
- Kohmoto
- 1985
(Show Context)
Citation Context ...2,44] . We shall not address these issues. The Chern number approach to quantum transport has been extended to a large class of quantum mechanical systems, including models of the integer Hall effect =-=[17,24,25, 29,30,31,41]-=-, to models with electron-electron interactions [3,23,30] and to other systems that bear only little resemblance to the Integer Hall effect [8,14,29,35,38]. The Index approach has not been as popular,... |

9 | Magnetic translation group - Zak - 1964 |

8 | T.: Universality in quantum Hall systems - Fröhlich, Kerler - 1991 |

6 |
D.J.: Quantum Hall effect with realistic boundary conditions
- Niu, Thouless
- 1987
(Show Context)
Citation Context ...2,44] . We shall not address these issues. The Chern number approach to quantum transport has been extended to a large class of quantum mechanical systems, including models of the integer Hall effect =-=[17,24,25, 29,30,31,41]-=-, to models with electron-electron interactions [3,23,30] and to other systems that bear only little resemblance to the Integer Hall effect [8,14,29,35,38]. The Index approach has not been as popular,... |

5 | Noncommutative differential geometry", Pub - Connes - 1986 |

4 |
A proof of the Fredholm trace formula as an application of a simple embedding for kernels of integral operators of trace class
- Birman
- 1989
(Show Context)
Citation Context ...ing the fact that K(x, x) is L 1 . This proves proposition (3.3). □ Proposition (3.3) could be replaced by the following statement which is is a kind of a Lebesgue integral version of proposition 3.3 =-=[6]-=-. Its application to the concrete cases we have in mind requires however somewhat more care. 6Remark (3.4). Let K be trace class on L 2 (R n ). Then, its integral kernel K(x, y) may be chosen so that... |

4 |
Representations of quantized differential forms in noncommutative geometry
- Cuntz
(Show Context)
Citation Context ...indices [22] the index is invariant under perturbations by compacts. Furthermore by the product formula for Fredholm indices one gets This proves the proposition. □ Related questions are addressed in =-=[9,12,15]-=-. Index(QU1QU2Q) = Index(QU1Q) + Index(QU2Q), (2.20) 3. Gauge Transformations and Computations with Integral Kernels In this section we introduce additional structure into the general operator theoret... |

4 |
Vacuum degeneracy of chiral spin states in compactified space
- Wen
- 1989
(Show Context)
Citation Context ... Quantum field theory is another framework where transport properties and geometry are related. The focal point here has been the Fractional Hall effect and the associated Chern-Simons field theories =-=[7, 8,18,26,42,44]-=- . We shall not address these issues. The Chern number approach to quantum transport has been extended to a large class of quantum mechanical systems, including models of the integer Hall effect [17,2... |

4 |
den Nijs, M.: Quantum Hall conductance in a two dimensional periodic potential
- Thouless, Kohmoto, et al.
- 1982
(Show Context)
Citation Context ...ular to the plane and random (or periodic) potential. The Hall conductance has been related to a Fredholm Index by Bellissard [5], and to a Chern number by Thouless, Kohmoto, Nightingale and den-Nijs =-=[40]-=-. The Fractional Hall effect is associated with electron-electron interaction and this goes beyond what we do here. Quantum field theory is another framework where transport properties and geometry ar... |

4 | Magnetic translation group. Phys. Rev. A134 - Zak - 1964 |

3 |
Hamiltonians on symmetric space
- Avron, Pnueli, et al.
- 1992
(Show Context)
Citation Context ...ency for the pair of projectors P and Q = UPU −1 of the two Schrödinger operators H and UHU −1 let us introduce a canonical interpolation between the two: H(t) = (−i∇ − φ(t)(∇ arg z) − A0) 2 + V, t ∈ =-=[0, 1]-=- where φ(t) interpolats smoothly between zero and one. ∇ arg z denotes a vector field on the real two plane respectively the complex plane. H(t) has, by definition, a time independent domain of defini... |

3 |
The norm of the L p Fourier transform on unimodular groups
- Russo
- 1974
(Show Context)
Citation Context ... − Q) p , p > 2, is trace class it is enough to show that the function ∫ g(y) ≡ ( |p(x + y, y) 1 − 1 − u(x) u(y) ) u(x + y) | u(y) q dx ∈ L p−1 (R 2 ), 1/p + 1/q = 1, (3.8) because of Russo’s theorem =-=[33]-=-. To prove (3.8) notice that close to the diagonal x = 0 the second term of the integrand in (3.8) is small, off the diagonal it is the first one which is small. To put this 7in analytic form we( not... |

3 |
Universality in Quantum Hall systems, Nucl.Phys
- Fröhlich, Kerler
- 1991
(Show Context)
Citation Context ... Quantum field theory is another framework where transport properties and geometry are related. The focal point here has been the Fractional Hall effect and the associated Chern-Simons field theories =-=[7, 8,18,26,42,44]-=- . We shall not address these issues. The Chern number approach to quantum transport has been extended to a large class of quantum mechanical systems, including models of the integer Hall effect [17,2... |

3 |
On the quantum Hall effect. In: Recent developments in quantum mechanics
- Seiler
- 1991
(Show Context)
Citation Context |

3 | X.G.: Effective theories of the fractional quantum Hall effect at generic filling fractions. Phys. Rev. B42 - Block, Wen - 1990 |

3 | Quantisation of particle transport - Thouless - 1983 |

2 |
H.: The physics of interacting electrons in disordered systems
- Kanamura, Aoki
- 1989
(Show Context)
Citation Context |

2 | Theory of quantized Hall conductivity in two dimensions - Streda - 1982 |

2 |
Magnetic translation
- Zak
- 1964
(Show Context)
Citation Context ...uously differentiable multiplication operators i.e. nonsingular gauge transformations. This notion of covariance is motivated by the covariance for Schrödinger operators with constant magnetic fields =-=[46]-=-. It follows that the first triple product in the integrand in 3.7 is invariant under translation of all arguments x,y,z.: p(x, y)p(y, z)p(z, x) = p(x − t, y − t)p(y − t, z − t)p(z − t, x − t) t ∈ R 2... |

2 | A.: Landau Hamiltonians on symmetric spaces. In: Ideas and methods - Avron, Pnueli - 1992 |

1 | The index of a pair of projections", in preparation - Avron |

1 |
Effective theoreis of the Fractional quantum Hall effect at generic filling fractions”, Phys
- Block, Wen
- 1990
(Show Context)
Citation Context ... Quantum field theory is another framework where transport properties and geometry are related. The focal point here has been the Fractional Hall effect and the associated Chern-Simons field theories =-=[7, 8,18,26,42,44]-=- . We shall not address these issues. The Chern number approach to quantum transport has been extended to a large class of quantum mechanical systems, including models of the integer Hall effect [17,2... |

1 |
Some homogeneous spaces and representations of the Hilbert Lied group
- Carrey
- 1985
(Show Context)
Citation Context ...indices [22] the index is invariant under perturbations by compacts. Furthermore by the product formula for Fredholm indices one gets This proves the proposition. □ Related questions are addressed in =-=[9,12,15]-=-. Index(QU1QU2Q) = Index(QU1Q) + Index(QU2Q), (2.20) 3. Gauge Transformations and Computations with Integral Kernels In this section we introduce additional structure into the general operator theoret... |

1 |
Efros,"Why the circel is connected
- G
- 1989
(Show Context)
Citation Context ...indices [22] the index is invariant under perturbations by compacts. Furthermore by the product formula for Fredholm indices one gets This proves the proposition. □ Related questions are addressed in =-=[9,12,15]-=-. Index(QU1QU2Q) = Index(QU1Q) + Index(QU2Q), (2.20) 3. Gauge Transformations and Computations with Integral Kernels In this section we introduce additional structure into the general operator theoret... |

1 | Direct proof of th formula for the index of an elliptic system in Euclidean space", Funct - Fedosov - 1970 |

1 | Universality in Quantum Hall Systems", Nucl. Phys - Frohlich, Kerler - 1991 |

1 | The Weyl calcuclus of Pseudo-Differential operators - Hormander - 1965 |