Numerical Renormalization Group Calculations for Impurity ...

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2. INTRODUCTION TO QUANTUM PHASE
TRANSITIONS
This chapter aims to cover the basic ideas of quantum phase transitions that are
frequently used in the main body of the thesis (Chapter 4, 5, and 6).
We start to giving the definitions of the scaling limit and universality from the
viewpoint of classical phase transitions with an example of the one dimensional
Ising model and introduce universal functions that represent the physics in the
vicinity of the critical points as a function of two large (macroscopic) lengths, L τ
(system size) and ξ (correlation length). 1
We bring those concepts defined in the classical cases into quantum systems
to develop a universal critical theory for quantum phase transitions. Again the
physical properties near to the critical points are characterized by the universal
scaling function, of which the argument is the dimensionless ratio of two small
energy scales, T (temperature) and ∆ (an energy gap between the ground state
and the first excitation), instead of the classical counterparts L τ and ξ. We take
the two-point correlation,
C(x, t) ≡ 〈ˆσ z (x, t)ˆσ z (0, 0)〉, (2.1)
as an example to discuss the shape of the universal scaling function in the critical
phase (Section 1.2).
Finally, we enter the subject of the thesis, impurity quantum phase transitions,
in Section 1.3, where we mention the specific issues of impurity models, such
as the impurity contribution of the physical observables and the local response
functions at the impurity site. The universal critical theory for the impurity
model is distinguished from the one for the lattice system in a few respects. For
examples, the feature of spatial correlations, one of the important issues of the
criticality of lattice systems, is absent (or disregarded) in impurity systems and
the quantum critical behavior reveals not in the response to a uniform global field
H but rather in that to a local field h coupled solely to the impurity. All the
arguments concerning the response to the magnetic field are given for a situation
where the impurity has a single SU(2) spin Ŝ of size S and the conduction band
is considered as a spinful bath.
1 To be precise, L τ and ξ are not treated independently but form a single argument as the
dimensionless ratio L τ /ξ.