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PART II: Quantum Ising and Rotor Models
Chapter 4
The Ising chain in a transverse field
This chapter describes in considerable detail the quantum critical properties
of the quantum Ising model in one spatial dimension. All of the results
in this chapter are believed to be exact, but the physically oriented reader
should not be turned off by this: technical details are kept to a minimum,
and the exact results are obtained by physical arguments which do much
to illustrate the main underlying principles. Most of the important concepts
of this book appear in the simple model under consideration: the remainder
of the book is largely a description of similar phenomena in more complicated
settings.
A physical picture of the crossovers in the non-zero temperature dynamical
properties in the vicinity of the quantum critical point is developed.
Effective classical models are shown to be adequate at low temperatures,
but distinct models appear on either side of the quantum critical point.
At higher temperatures, in the `quantum-critical' regime, no classical
description is possible and quantum and thermal fluctuations play equally
important roles: h/kB T appears as a fundamental
relaxation time in the dynamics.
Exercises
Corrections
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In Eqn. (4.30) the last term in HI is +g (hardback only)
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In Eqn. (4.32) replace -i by i; thanks to Stephen Powell.
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In Eqn. (4.36) change the sign of the right-hand-side; thanks to Stephen Powell.
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In Eqn. (4.41) replace E0+ by E0-, and make a corresponding
change in (4.44); thanks to Philip Phillips.
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