*Pier A. Mello and Narendra Kumar*

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.003.0007
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter treats the problem of finding the probability distribution of quantities related to quantum transport through a strictly one-dimensional (i.e., 1-channel) and through an N-channel ...
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This chapter treats the problem of finding the probability distribution of quantities related to quantum transport through a strictly one-dimensional (i.e., 1-channel) and through an N-channel quasi-one-dimensional disordered system. It uses the maximum entropy approach wherein the distribution for the random transfer matrix for an elementary building block is determined by maximizing the associated Shannon entropy, subject to the physically relevant constraints of flux conservation, time-reversal symmetry (when relevant), and the Ohmic small length-scale limit. The contents of this chapter include ensemble of transfer matrices; universality classes — the orthogonal and the unitary classes; invariant measure; the Fokker-Planck equation for a disordered one-dimensional conductor; the maximum-entropy ansatz for the building block; construction of the probability density for a system of finite length; the Fokker-Planck equation for a quasi-one-dimensional multi-channel disordered conductor; the diffusion equation for the orthogonal universality class, β = 1; the diffusion equation for the unitary universality class, β = 2; and universal conductance fluctuations in the good-metallic limit.Less

This chapter treats the problem of finding the probability distribution of quantities related to quantum transport through a strictly one-dimensional (i.e., 1-channel) and through an *N*-channel quasi-one-dimensional disordered system. It uses the maximum entropy approach wherein the distribution for the random transfer matrix for an elementary building block is determined by maximizing the associated Shannon entropy, subject to the physically relevant constraints of flux conservation, time-reversal symmetry (when relevant), and the Ohmic small length-scale limit. The contents of this chapter include ensemble of transfer matrices; universality classes — the orthogonal and the unitary classes; invariant measure; the Fokker-Planck equation for a disordered one-dimensional conductor; the maximum-entropy ansatz for the building block; construction of the probability density for a system of finite length; the Fokker-Planck equation for a quasi-one-dimensional multi-channel disordered conductor; the diffusion equation for the orthogonal universality class, β = 1; the diffusion equation for the unitary universality class, β = 2; and universal conductance fluctuations in the good-metallic limit.

*Hans-Peter Eckle*

- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780199678839
- eISBN:
- 9780191878589
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199678839.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

This chapter introduces the Heisenberg model, a fully quantum mechanical model that describes the magnetism of localized magnetic moments. The one-dimensional version of the Heisenberg model, the ...
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This chapter introduces the Heisenberg model, a fully quantum mechanical model that describes the magnetism of localized magnetic moments. The one-dimensional version of the Heisenberg model, the Heisenberg quantum spin chain, provides a good picture of magnetic materials that belong to a class of insulating magnetic materials where the interaction of the magnetic moments in one particular direction is much larger than in the perpendicular directions, and which can be described with high accuracy as quasi- one-dimensional magnets. A detailed description of the Heisenberg quantum spin chain is followed by a discussion of its various special cases, in particular the special case of the anisotropic Heisenberg quantum spin chain, the so-called XXZ quantum spin chain. It considers the solution of eigenvalue problem of this quantum spin and leads to Bethe’s conjecture for the wave function.Less

This chapter introduces the Heisenberg model, a fully quantum mechanical model that describes the magnetism of localized magnetic moments. The one-dimensional version of the Heisenberg model, the Heisenberg quantum spin chain, provides a good picture of magnetic materials that belong to a class of insulating magnetic materials where the interaction of the magnetic moments in one particular direction is much larger than in the perpendicular directions, and which can be described with high accuracy as quasi- one-dimensional magnets. A detailed description of the Heisenberg quantum spin chain is followed by a discussion of its various special cases, in particular the special case of the anisotropic Heisenberg quantum spin chain, the so-called XXZ quantum spin chain. It considers the solution of eigenvalue problem of this quantum spin and leads to Bethe’s conjecture for the wave function.